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QUICK REFRESHER GUIDE

Linear Algebra

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Calculus

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Engineering Mechanics

Statics

Dynamics

Strength of Materialss

Simple Stress And Strain

Shear Force And Bending Moment

Stresses In Beams

Deflection Of Beams

Torsion

Mohr’s Circle

Strain Energy Methods

Columns & Struts

Thermodynamics

Basic Thermodynamics

Properties of pure substances

Irreversibility & Availability

Work, Heat & Entropy

1.1

1.2

1.3.

1.4.

1.5.

1.6.

1.7

#2.

2.1

2.2

#3.

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

#4.

4.1

4.2

4.3

4.4

118 – 122

114 - 117

111 – 113

99 – 110

99 9 – 159 9

94 – 98

92 - 93

89 – 91

84 – 88

75 – 83

71 - 74

68 – 70

62 – 67

62 2 – 98

55 – 61

46 – 54

46 – 61 1

43 – 45

38 – 42

31 – 37

20 – 30

15 – 19

9 – 14

1–8

1 – 45

Page No.

Contents

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Mathemathics

#1.

Part

CONTENTS

Quick Refresher Guide

Flow through pipes Hydraulic Machines

7.6 7.7

295 – 298

290 – 294

287 – 289

274 – 286

274 4 – 298

257 – 273

250 – 256

243 – 249

237 – 242

232 – 236

225 – 231

220 – 224

220 0 – 273

199 – 219

190 - 198

185 – 189

185 5 – 219

180 – 184

176 - 179

170 – 175

160 – 169

160 0 – 184

150 - 159

145 – 149

135 – 144

123 – 134

Contents

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Heat Exchanger

Boundary Layer

7.5

8.4

Fluid Dynamics

7.4

Radiation

Fluid Kinematics

7.3

8.3

Fluid Statics

7.2

Convection

Fluid Properties

7.1

Conduction

Fluid Mechanics

#7.

8.2

Design of Machine Elements

6.3

8.1

Fatigue

6.2

Heat Transfer

Theory of Failures

6.1

#8.

Machine Design

Vibrations

5.4

#6.

Gear Trains Flywheel

5.3

Mechanisms

5.2

Theory of Machiness

4.8

5.1

I.C. Engines

4.7

#5.

Power Engineering Refrigeration

4.6

Psychrometrics

4.5

Quick Refresher Guide

404 – 405

393 – 403

Contents

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Reference Books

#

382 – 387

Operations Research

38 82 - 403

#10. Industrial Engineering

10.3

380 – 381

9.7 Computer Integrated Manufacturing (CIM)

388 – 392

362 – 379

9.6 Metrology and Inspection

Production, Planning and Control

345 – 361

9.5 Theory of Metal Cutting

Inventory Control

339 – 344

10.2

322 – 338

9.3 Forming Process

9.4 Joining Process

10.1

305 – 321

9.2 Casting

29 99 - 38 81 299 – 304

Manufacturing Engineering

9.1 Engineering Materials

#9.

Quick Refresher Guide

a − − a − − a − − a − − − − − a − − a − − − − −

a → ith row, jth column

A=

a a − a a a a a

Conventionally, single capital letter is used to denote a matrix. Thus,

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1.1.1.5 Diagonal Matrix A Square matrix in which all the elements except those in leading diagonal are zero.. −4 0 0 e.g. 0 6 0 0 0 8

Number of rows ≠ Number of columns

1.1.1.4 Rectangular Matrix

x -

Page 1

Same number of rows and columns. Order of Square matrix → no. of rows or columns Principle Diagonal (or Main diagonal or Leading diagonal): The diagonal of a square matrix (from the top left to the bottom right) is called as principal diagonal. Trace of the Matrix: The sum of the diagonal elements of a square matrix. tr (λ A) = λ tr(A) [ λ is scalar] tr ( A+B) = tr (A) + tr (B) tr (AB) = tr (BA)

1.1.1.3 Square Matrix

1.1.1.2 Row and Column Matrices x Row Matrix → [ 2, 7, 8, 9] → single row ( or row vector) 5 x Column Matrix → 10 → single column (or column vector) 13 1

x x x

Mathematics

Definition: A system of “m n” numbers arranged along m rows and n columns.

Matrix

1.1.1.1 Types of Matrices

1.1.1

1.1 Linear Algebra

Part - 1: Mathematics

Quick Refresher Guide

=A

‫ = ܂ۯ‬A

Skew Symmetric Matrix ‫ = ܂ۯ‬- A

Mathematics

Singular Matrix: If |A| = 0, then A is called a singular matrix..

Unitary Matrix: If we define, A = (A) Then the matrix is unitary if A . A = I

1.1.1.11

1.1.1.12

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= transpose of a conjugate of matrix A

Orthogonal Matrix: If A. A = I, then matrix A is said to be Orthogonal matrix..

1.1.1.10

Page 2

1.1.1.9 Triangular Matrix x A matrix is said to be “upper triangular” if all the elements below its principal diagonal are zeros. x A matrix is said to be “lower triangular” if all the elements above its principal diagonal are zeros. a 0 0 a h g g b 0 0 b f f h c 0 0 c Upper Triangular Matrix Lower Triangular Matrix

Symmetric Matrix

Note: All the diagonal elements of skew symmetric matrix must be zero. Symmetric Skew symmetric a h g 0 −h g h b f h 0 −f g f c −g f 0

1.1.1.8 Symmetric and Skew Symmetric Matrices:

Symmetric, when a = +a for all i and j. In other words

Skew symmetric, when a = - a In other words = -A

A matrix is said to be Null Matrix if all the elements are zero. 0 0 0 e.g. 0 0 0

1.1.1.7 Null Matrix (or Zero Matrix)

A Diagonal matrix in which all the leading diagonal elements are ‘1’. 1 0 0 e.g. I = 0 1 0 0 0 1

1.1.1.6 Unit Matrix (or Identity Matrix)

Quick Refresher Guide

A = −A or a = − a

Skew Hermitian matrix: It is a square matrix with complex entries which is equal to the negative of conjugate transpose.

Note: In Hermitian matrix, diagonal elements → always real

A = A or a = a

Hermitian Matrix: It is a square matrix with complex entries which is equal to its own conjugate transpose.

Mathematics

Multiplication of Matrix by a Scalar:

If A = A, then the matrix A is called idempotent matrix.

Idempotent Matrix

Determinant:

a D = det A = a

a − − − a

a − − − −

a a

− − − − −

− − − − −

=a

a a − − a

a

-a

a

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a a D = |A| = det A = − − a

Determinant of “order n”

For n = 2 ,

If A = [a ] , Element a with ith row, jth column.

An n order determinant is an expression associated with n × n square matrix.

1.1.1.18

Page 3

Two matrices can be multiplied only when number of columns of the first matrix is equal to the number of rows of the second matrix. Multiplication of (m × n) and (n × p) matrices results in matrix of (m × p)dimension [ ] × × [ ] × = [ ] × .

Multiplication of Matrices:

Every element of the matrix gets multiplied by that scalar.

1.1.1.17

1.1.1.16

Note: In Skew-Hermitian matrix , diagonal elements → either zero or Pure Imaginary

1.1.1.15

1.1.1.14

1.1.1.13

Quick Refresher Guide

The minor of an element in a determinant is the determinant obtained by deleting the row and the column which intersect that element. (where the element Cofactor is the minor with “proper sign”. The sign is given by (-1) belongs to ith row, jth column).

Minors & Co-Factors:

Mathematics

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Important Points: 1. IA = AI = A, (Here A is square matrix of the same order as that of I ) 2. 0 A = A 0 = 0, (Here 0 is null matrix) 3. If AB = 0, then it is not necessarily that A or B is null matrix. Also it doesn’t mean BA = 0. 4. If the product of two non-zero square matrices A & B is a zero matrix, then A & B are singular matrices. 5. If A is non-singular matrix and A.B=0, then B is null matrix. 6. AB ≠ BA (in general) → Commutative property does not hold 7. A(BC) = (AB)C → Associative property holds 8. A(B+C) = AB + AC → Distributive property holds

x

| |

|A| must be non-zero (i.e. A must be non-singular). Inverse of a matrix, if exists, is always unique. d −b a b If it is a 2x2 matrix , its inverse will be −c a c d

x x

=

A

x

Inverse of a Matrix

Determinant of a skew symmetric matrix (i.e. A =-A) of odd order is zero. If A is a unitary matrix or orthogonal matrix (i.e. A = A ) then |A|= ±1. If A is a square matrix of order n, then |k A| = k |A|. |I | = 1 ( I is the identity matrix of order n).

1.1.1.21

10. 11. 12. 13.

1.1.1.20 Properties of Determinants: 1. A determinant remains unaltered by changing its rows into columns and columns into rows. 2. If two parallel lines of a determinant are inter-changed, the determinant retains its numerical values but changes its sign. (In a general manner, a row or a column is referred as line). 3. Determinant vanishes if two parallel lines are identical. 4. If each element of a line be multiplied by the same factor, the whole determinant is multiplied by that factor. [Note the difference with matrix]. 5. If each element of a line consists of the m terms, then determinant can be expressed as sum of the m determinants. 6. If each element of a line be added equi-multiple of the corresponding elements of one or more parallel lines, determinant is unaffected. e.g. by the operation, R → R + pR +qR , determinant is unaffected. 7. Determinant of an upper triangular/ lower triangular/diagonal/scalar matrix is equal to the product of the leading diagonal elements of the matrix. 8. If A & B are square matrix of the same order, then |AB|=|BA|=|A||B|. 9. If A is non singular matrix, then |A |=| | (as a result of previous).

x

x

1.1.1.19

Quick Refresher Guide

Mathematics

AC = AD , doesn’t imply C = D [even when A ≠ 0]. If A, C, D be × matrix, and if rank (A)= n & AC=AD, then C=D. (A+B)T = A + B (AB)T = B . A (AB)-1 = B . A AA =A A=I (kA)T = k.A (k is scalar, A is vector) (kA)-1 = k . A (k is scalar , A is vector) (A ) = (A ) ( A ) = (A) (Conjugate of a transpose of matrix= Transpose of conjugate of matrix) If a non-singular matrix A is symmetric, then A is also symmetric. If A is a orthogonal matrix , then A and A are also orthogonal.

Rank of Matrix

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r(A.B) ≤ min { r(A), r (B)} r(A+B) ≤ r(A) + r (B) r(A-B) ≥ r(A) - r (B) The rank of a diagonal matrix is simply the number of non-zero elements in principal diagonal. 5. A system of homogeneous equations such that the number of unknown variable exceeds the number of equations, necessarily has non-zero solutions.

Note: 1. 2. 3. 4.

Rank = Number of non-zero row in upper triangular matrix using elementary transformation.

Special case: Rank of Square matrix

Other definition: The rank is also defined as maximum number of linearly independent row vectors.

Definition: A matrix is said to be of rank r when, i) It has at least one non-zero minor of order r. ii) Every minor of order higher than r vanishes.

If we select any r rows and r columns from any matrix A,deleting all other rows and columns, then the determinant formed by these r×r elements is called minor of A of order r.

1.1.1.23

Note: 9 Elementary transformations don’t change the rank of the matrix. 9 However it changes the Eigen value of the matrix.

1.1.1.22 Elementary Transformation of a Matrix: 1. Interchange of any 2 lines 2. Multiplication of a line by a constant (e.g. k R ) 3. Addition of constant multiplication of any line to the another line (e. g. R + p R )

21. If A is a square matrix of order n then (i) |adj A|=|A| (ii) |adj (adj A)|=|A|( ) (iii) adj (adj A) =|A| A

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

Quick Refresher Guide

Mathematics

Solution of linear System of Equations:

− − − a − − − a − − − − − − − − − − − − − − a − − − a

a −

⎤ ⎥ ⎥, ⎥ ⎥ ⎦

x ⎡x ⎤ ⎢−⎥ X= ⎢−⎥ , ⎢ ⎥ ⎢−⎥ ⎣x ⎦ k ⎡ k ⎢ − B = ⎢ ⎢− ⎢− ⎣k

x = b ---------------------------------------(ii)

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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a b

x +a

a

a D= b

x = b ---------------------------------------(i)

x +a

a

Let the following two equations be there

Cramer’s Rule

Inconsistent means:: → No solution

Consistent means:: → one or more solution (i.e. unique or infinite solution)

Note:

For Homogenous Equations (A X = 0) i) If r = n, the equations have only a trivial zero solution ( i.e. x = x = - - - x = 0). ii) If r < n, then (n-r) linearly independent solution (i.e. infinite non-trivial solutions).

For Non-Homogenous Equations (A X = B) i) If r ≠ r , the equations are inconsistent i.e. there is no solution. ii) If r = r = n, the equations are consistent and there is a unique solution. iii) If r = r < n, the equations are consistent and there are infinite number of solutions.

Consistency of a System of Equations:

r = rank (A), r = rank (C), n = Number of unknown variables (x , x , - - - x )

A= Coefficient Matrix, C = (A, B) = Augmented Matrix

a ⎡ a ⎢ − Where, A = ⎢ − ⎢ ⎢ − ⎣a

For the following system of equations A X = B

1.1.1.24

Page 6

If A is a non-singular matrix, then all the row/column vectors are independent. If A is a singular matrix, then vectors of A are linearly dependent. r(A)=0 iff (if and only if) A is a null matrix. If two matrices A and B have the same size and the same rank then A, B are equivalent matrices. 10. Every non-singular matrix is row matrix and it is equivalent to identity matrix.

6. 7. 8. 9.

Quick Refresher Guide

b b

a a

D =

and x =

Characteristic Equation and Eigen Values:

Mathematics

−

]X=0

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Properties of Eigen values 1. The sum of the Eigen values of a matrix is equal to the sum of its principal diagonal. 2. The product of the Eigen values of a matrix is equal to its determinant. 3. The largest Eigen values of a matrix is always greater than or equal to any of the diagonal elements of the matrix. 4. If λ is an Eigen value of orthogonal matrix, then 1/ λ is also its Eigen value. 5. If A is real, then its Eigen value is real or complex conjugate pair. 6. Matrix A and its transpose A has same characteristic root (Eigen values). 7. The Eigen values of triangular matrix are just the diagonal elements of the matrix. 8. Zero is the Eigen value of the matrix if and only if the matrix is singular. 9. Eigen values of a unitary matrix or orthogonal matrix has absolute value ‘1’. 10. Eigen values of Hermitian or symmetric matrix are purely real. 11. Eigen values of skew Hermitian or skew symmetric matrix is zero or pure imaginary. | | 12. is an Eigen value of adj A (because adj A = |A|. A ).

Note: For a given Eigen value, there can be different Eigen vectors, but for same Eigen vector, there can’t be different Eigen values.

For each Eigen value λ, solving for X gives the corresponding Eigen vector.

Eigen vectors: [

Characteristic equation: | A − λ I |= 0, The roots of this equation are called the characteristic roots /latent roots / Eigen values of the matrix A.

1.1.1.25

Eigen Values & Eigen Vectors

For Homogenous Equations D≠ 0 → trivial solutions ( x = x =………………………x = 0) D = 0 → non- trivial solution (or infinite solution)

In general, for Non-Homogenous Equations D≠ 0 → single solution (non trivial) D = 0 → infinite solution

In the above method, it is assumed that 1. No of equations = No of unknowns 2. D≠ 0

x =

Solution using Cramer’s rule:

a a

b b

D =

Quick Refresher Guide

λ is an Eigen value of the matrix then , Eigen value of A is 1/λ Eigen value of A is λ Eigen value of kA are kλ (k is scalar) Eigen value of A + k I are λ + k Eigen value of (A − k I)2 are (O−k)

Mathematics

Vector:

Its linear combination is λ x + λ x + λ x + λ x = 0 If λ , λ , λ , λ are not “all zero” → they are linearly dependent. If all "λ" are zero → they are linearly independent.

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x x

Suppose the vectors are x x x x

Page 8

Linearly Independent Vectors x If no vectors can be written as a linear combination of others, then they are linearly independent.

Linear Dependence of Vectors x If one vector can be written as linear combination of others, the vector is linearly dependent.

Any quantity having n components is called a vector of order n.

1.1.1.26

Cayley Hamilton Theorem: Every square matrix satisfies its own characteristic equation.

Properties of Eigen Vectors 1) Eigen vector X of matrix A is not unique. Let X is Eigen vector, then CX is also Eigen vector (C = scalar constant). 2) If λ , λ , λ . . . . . λ are distinct, then X , X . . . . . X are linearly independent . 3) If two or more Eigen values are equal, it may or may not be possible to get linearly independent Eigen vectors corresponding to equal roots. 4) Two Eigen vectors are called orthogonal vectors if X T∙ X = 0. (X , X are column vector) (Note: For a single vector to be orthogonal , A = A or, A. A = A. A = , ) 5) Eigen vectors of a symmetric matrix corresponding to different Eigen values are orthogonal.

13. If i) ii) iii) iv) v)

Quick Refresher Guide

Probability

Mathematics

=

. .

( ∩ ) ( )

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P(B/A)=

Page 9

Conditional Probability: If A and B are dependent events, then P denotes the probability of occurrence of B when A has already occurred. This is known as conditional probability.

If P(A∩B)= P(A) P(B) ↔ Independent events A & B

Independent Events: Two events are said to be independent, if the occurrence of one does not affect the occurrence of the other.

Addition Law of Probability: a. For every events A, B and C not mutually exclusive P(A∪B∪C)= P(A)+ P(B)+ P(C)- P(A∩B)- P(B∩C)- P(C∩A)+ P(A∩B∩C) b. For the event A, B and C which are mutually exclusive P(A∪B∪C)= P(A)+ P(B)+ P(C)

Important points: x P(A∪B)→ Probability of happening of “at least one” event of A & B x P(A∩B) )→ Probability of happening of “both” events of A & B x If the events are certain to happen, then the probability is unity. x If the events are impossible to happen, then the probability is zero.

P(A)+ P(A’)=1

Probability: P(A)=

Odds Against the Event =

n→ no. of ways not favourable to A

Where m→ no. of ways favourable to A

Odds in Favour of an Event =

Equally Likely Events: If one of the events cannot happen in preference to other, then such events are said to be equally likely.

A ∩ B =φ, P(A ∩ B) =0

Mutually Exclusive Events (Disjoint Events): Two events are called mutually exclusive, if the occurrence of one excludes the occurrence of others i.e. both can’t occur simultaneously.

Event: Outcome of an experiment is called event.

1.2.1

1.2 Probability and Distribution

Quick Refresher Guide

[ P(B)≠0]

= P(B). P(A/B),

& P are probabilities of two independent events then → probability of first event happens and second fails (i.e only first happens) P (1-P ) (1-P )(1-P ) → probability of both event fails 1-(1-P )(1-P ) → probability of at least one event occur PP → probability of both event occurs

=

(

∑ (

). ( )

). ( )

Distribution

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PDF has the following properties: x Probability density function is always positive i.e. f(x) ≥ 0 x ∫ f(x)dx = 1 (Continuous) x ∑ f(x ) = 1 (Discrete)

The set of values Xi with their probabilities P constitute a probability distribution or probability density function of the variable X. If f(x) is the PDF, then f(x ) = P(X = x ) ,

Probability Density Function (PDF) or Probability Mass Function:

1.2.2

Random variable: Real variable associated with the outcome of a random experiment is called a random variable.

This is also known as theorem of Inverse Probability.

P

If P(B ) and P(A/B ) are given then,

An event A corresponds to a number of exhaustive events B , B ,.., B .

Baye’s theorem:

If P 1. 2. 3. 4.

Important Points:

Hence P(A∩B) = P(A) P(B)

This is also known as Multiplication Theorem. For independent events A&B → P(B/A) = P(B), P(A/B )= P(A)

[ P(A)≠0]

P(A∩B)= P(A). P(B/A),

P(x ) = ∑ f(x )

(Discrete case)

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Properties of Variance 1. Var(constant) = 0 2. Var(Cx) = C Var(x) -Variance is non-linear [here C is constant]

Var (X) =E(XX )-[E(x)]

Var (X)=∫ (xx − μμ) f(x)dx (Continuous case)

Var (X)=∑(x x − μμ) f(xx )

Var (X) =E[(x − μμ ) ]

Variance (Var(X))

But E(XY) = E(X) E(Y) , if X & Y are independent

E(XY)≠ E(X) E(Y) in general

Properties of Expectation 1. E(constant) = constant 2. E(CX) = C . E(X) [C is constant] 3. E(AX+BY) = A E(X)+B E(Y) [A& B are constants] 4. E(XY)= E(X) E(Y/X)= E(Y) E(X/Y)

Expectation [E(x)]: 1. E(X) = ∑ x f(x ) (Discrete case) 2. E(X) = ∫ x f(x )dx (Continuous case)

CDF has the following properties: ( ) i) = F ′(x) =f(x)≥0 ii) 1≥ F (x) ≥0 iii) If x > x then F (x ) > F (x ) , i.e. CDF is monotone (non-decreasing function) iv) F (−∞) = 0 v) F (∞) = 1 vi) P(a≤ x ≤b) =∫ f(x)dx = ∫ f(x)dx - ∫ f(x)dx = F (b) − F (a)

If F (x) = P(X≤x) =∫ f(x)dx, then F(x) is defined as the cumulative distribution function or simply the distribution function of the continuous variable.

Continuous Cumulative Distribution function (CDF) or Distribution Function:

F (x) = F(x) = P(X≤x) = ∑

The Cumulative Distribution Function F(x) of the discrete variable x is defined by,

Mathematics

Theorem of Combined Probability: If the probability of an event A happening as a result of trial is P(A). Probability of an event B happening as a result of trial after A has happened is P(B/A) then the probability of both the events A and B happening is

Quick Refresher Guide

Discrete Cumulative Distribution Function (CDF) or Distribution Function

Mathematics

For independent events A & B → P(B/A) = P(B)

Quick Refresher Guide

Mathematics

E(xy) = E(x) . E(y)

√

e

(

)

P(x1 < x < x2) = ∫

√

e

)

e

)

,

x≥0

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Mean: x For a set of n values of a variant X=( x , x , … . . , x )

.(

dx = Area under the curve from x1 to x2

f(x) = λ e , x ≥ 0 , here λ > 0 = 0, x < 0 f(x)= , b ≥ f(x) ≥ a = 0, otherwise f(x)=

(

5. Rayleigh distribution function : f(x) =

4. Cauchy distribution :

3. Uniform distribution:

2. Exponential distribution :

x

Where μ and σ are the mean and standard deviation respectively x P(μ − σ < x < μ + σ) = 68% x P(μ − 2σ < x < μ + 2σ) = 95.5% x P(μ − 3σ < x < μ + 3σ) = 99.7% x Total area under the curve is is unity i.e. ∫ f(x)dx = 1

1. Normal Distribution (Gaussian Distribution): f(x) =

Standard Distribution Function (Continuous r.v. case):

2. Poisson Distribution: Probability of k success is P (k) = ! k→no. of success trials , n→no. of trials , P→success case probability → mean of the distribution For Poisson distribution: Mean = , variance = , and =np

Standard Distribution Function (Discrete r.v. case): 1. Binomial Distribution : P(r) = C p q Mean = np, Variance = npq, S.D. = npq

Properties of Covariance 1. Cov(x,y) = Cov(y,x) (i.e. symmetric) 2. Cov(x,x) = Var(x) 3. |Cov(x,y)| ≤ σ σ

(if covariance = 0, then the events are not necessarily independent)

If independent ⟹ covariance=0,

Cov (x,y)=E(xy)-E(x) E(y)

Covariance

3. Var(Cx±D) = C Var(x) -Variance is translational invariant [C & D are constants] 4. Var(x-k) = Var(x) [k is constant] 5. Var(ax+by) = a Var(x) + b Var(y) ± 2ab cov(x,y) (if not independent) [A & B are constants] = a Var(x) + b Var(y) (if independent)

Quick Refresher Guide

(m , σ = mean, SD of combined sample)

(∑n )σ =∑(n σ )+∑(n D )

D = m -m

(n + n )σ = n σ +n σ +n D +n D

If X , σ be mean and SD of a sample size n and m , σ be those for a sample of size n then SD of combined sample of size n +n is given by,

( , )

is called the regression coefficient of y on x and is denoted by byx.

(y − y)

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ρ

The equation of the line of Regression of x on y is (x − x) = ρ

The equation of the line of regression of y on x is y − y = ρ

(x − x)

-1 ≤ U(x, y) ≤1 U(x,y) = U(y,x) |U(x,y)| = 1 when P(x=0)=1; or P(x=ay)=1 [ for some a] If the correlation coefficient is -ve, then two events are negatively correlated. If the correlation coefficient is zero, then two events are uncorrelated. If the correlation coefficient is +ve, then two events are positively correlated. Line of Regression:

x x x x x x

Correlation coefficient = U(x,y) =

Co-efficient of variation = × 100

Important Points: 9 Mean is best measurement ( ∵ all observations taken into consideration). 9 Mode is worst measurement (∵ only maximum frequency is taken). 9 In median, 50 % observation is taken. 9 Sum of the deviation about “mean” is zero. 9 Sum of the absolute deviations about “median” is minimum. 9 Sum of the square of the deviations about “mean” is minimum.

Mode: It is defined as the value in the sampled data that occurs most frequently.

Median: When the values in a data sample are arranged in descending order or ascending order of magnitude the median is the middle term if the no. of sample is odd and is the mean of two middle terms if the number is even.

x

If X is mean for n data; X is mean for n data; then combined mean of n +n data is

x X=

For a grouped data if x , x , … . . , x are mid values of the class intervals having frequencies ∑ f , f ,….., f ,then, X= ∑

∑

Mathematics

x

The arithmetic mean, X=

Quick Refresher Guide

is called the regression coefficient of x on y and is denoted by bxy.

Mathematics

∫

f(x, y) dx dy

F(x, y)

= 1

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If fxy(x,y) = fx(x) . fy(y)

Note: X and Y are said to be independent random variable

Property: ∫

Defined as f(x, y) =

Joint Probability Density Function:

Properties of Joint Distribution Function/ Cumulative Distribution Function: 1. F (−∞, −∞) = 0 2. F (∞, ∞) = 1 3. F (−∞, ∞) = 0 { F (−∞, ∞) = P(X ≤ −∞ ; Y ≤ y) = 0 x 1 = 0 } 4. F (x, ∞) = P(X ≤ x ; Y ≤ ∞) = F (x) . 1 = F (x) 5. F (∞, y) = F (y)

Joint Probability Distribution: If X & Y are two random variables then Joint distribution is defined as, Fxy(x,y) = P(X ≤ x ; Y ≤ y)

ρ

Quick Refresher Guide

Consider an equation f(x) = 0

Solution of Algebraic and Transcendental Equation / Root Finding :

Mathematics

.

(

)

x

=x −

(

)– (

)

f(x )

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9 Regula falsi method always converges. 9 However, it converges slowly. 9 If converges, order of convergence is between 1 & 2 (closer to 1).

4. Regula Falsi Method or (Method of False Position)

9 Convergence is not guaranteed. 9 If converges, convergence super linear (more rapid than linear, almost quadratic like Newton Raphson, around 1.62).

3. Secant Method

This method is commonly used for its simplicity and greater speed. Here f(x) is assumed to have continuous derivative f’(x). This method fails if f’(x) = 0. It has second order of convergence or quadratic convergence, i.e. the subsequent error at each step is proportional to the square of the error at previous step. 9 Sensitive to starting value, i.e. The Newton’s method converges provided the initial approximation is chosen sufficiently close to the root. 9 Rate of convergence is quadratic.

9 9 9 9

2. Newton Raphson Method (or Successive Substitution Method or Tangent Method) ( ) xn+1 = xn –

Similarly x2 and x3 . . . . . are determined. 9 Simplest iterative method 9 Bisection method always converge, but often slowly. 9 This method can’t be used for finding the complex roots. 9 Rate of convergence is linear

If f(x1) = 0, then x1 is the root of f(x) = 0, otherwise root lies between a and x1 or x1 and b.

First approximation to the root is x1 =

If f(x) is continuous between a and b and f (a) and f (b) are of opposite sign then there is a root between a & b (IIntermediate Value Theorem).

This method finds the root between points “a” and “b”.

1. Bisection method

1.3.1

1.3 Numerical Methods

Quick Refresher Guide

(

)

–

(

)

, f(x ) =

(

(

)–

)

(

( ) )

………

2.

1.

(pivotal equation, b2’ is pivot point.)

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Modification of the Jacobi’s Iteration Method

(ii) Gauss-Siedel Iteration Method

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The process is repeated till the difference between two consecutive approximations is negligible. In generalized form: x(k+1) = k1 – l1 y(k) – m1z(k) y(k+1) = k2 – l2 x(k) – m2z(k) z(k+1) = k3 – l3 x(k) – m3y(k)

Note: No component of x(k) is used in computation unless y(k) and z(k) are computed.

Let us start with initial approximation x0 , y0 , z0 x1= k1 – l1y0 – m1z0 y1= k2 – l2y0 – m2z0 z1= k3 – l3y0 – m3z0

If a1, b2 , c3 are large compared to other coefficients, then solving these for x, y, z respectively x = k1 – l1y – m1z y = k2 – l2x – m2z z = k3 – l3x – m3y

(i) Jacobi Iteration Method a1x + b1y + c1z = d1 a2x + b2y + c2z = d2 a3x + b3y + c3z = d3

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+n -

U11 U12 U13 0 U22 U23 0 0 U31

Mathematics

Solve for Y from a) then solve for X from b). This method is known as Doolittle’s method. 9 Similar methods are Crout’s method and Cholesky methods.

Ax = LUX = b can be written as a)LY=b and b) UX=Y

[A]n x n = [ L ] n x n [U] n x n a11 a12 a13 1 0 0 L21 1 0 a21 b22 c23 = L31 L32 1 a31 b32 c33

L U Decomposition 9It is modification of the Gauss eliminiation method. 9Also Used for finding the inverse of the matrix.

4. Iterative Method

3.

Step 2: Eliminate y from 1st and 3rd Step 3: Eliminate z from 1st and 2nd

Quick Refresher Guide

Start with (x0, y0, z0) = (0, 0, 0) or anything [No specific condition]

Note: Number of operations: N =

Step 3: The value of x , y and z can be found by back substitution.

Step 2: Eliminate y from third equation a1x + b1y + c1z = d1’ b2’y + c2z = d2’ c3’’z = d3”

Step 1: To eliminate x from second and third equation (we do this by subtracting suitable multiple of first equation from second and third equation) a1x + b1y + c1z = d1’ (pivotal equation, a1 pivot point.) b2’y + c2’ z = d2’ b3’y + c3’ z = d3’

Here equations are converted into “upper triangular matrix” form, then solved by “back substitution” method. Consider a1x + b1x + c1z = d1 a2x + b2x + c2z = d2 a3x + b3x + c3z = d3

Gauss Elimination Method

Solution of Linear System of Equations

(i.e. opposite sign)

Mathematics

Gauss Jordon Method 9 Used to find inverse of the matrix and solving linear equations. 9 Here back substitution is avoided by additional computations that reduce the matrix to “diagonal from”, instead to triangular form in Gauss elimination method. 9 Number of operations is more than Gauss elimination as the effort of back substitution is saved at the cost of additional computation. Step 1: Eliminate x from 2nd and 3rd

1.3.2

which is an approximation to the root.

Compute x

Check if f(x0) f(x2) < 0 or f(x1) f(x2) < 0

x =x -

Select x0 and x1 such that f(x0) f(x1) < 0

Given f(x) = 0

9 It is superior to Bisection method.

Quick Refresher Guide

Numerical Integration

h {( first term + last term) + 2 (remaining terms)} 2

Mathematics

h (b − a) max |f (ξ)| ∈[ , ] 12

h {( first term + last term) + 4 (all odd terms) + 2 (all even terms)} 3

h (b − a) max |f ( ) (ξ)| ∈[ , ] 180

3h ( first term + last term) + 2 (all multiple of 3 terms) +3 (all remaining terms) 8

(b − a) max |f ( ) (ξ)| 6480 ∈[ , ]

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1.3.4 Solving Differential Equations (i) Euler method (for first order differential equation ) Given equation is y = f(x, y); y(x0) = y0

−

The error in approximating an integral using Simpson’s 3/8 rule is

f(x)dx =

Simpson’s Three Eighth Rule:

−

The error in approximating an integral using Simpson’s one third rule is

f(x)dx =

Simpson’s One Third Rule (Simpson’s Rule):

−

The error in approximating an integral using Trapezoidal rule is bounded by

Error = Exact - approximate

f(x)dx =

Trapezoidal Formula: Step size h =

1.3.3

Note: To compute any variable, use the latest available value. In generalized form: x(k+1) = k1 – l1y(k) – m1z(k) y(k+1) = k2 – l2x(k+1) – m2z(k) z(k+1) = k3 – l3x(k+1) – m3y(k+1)

In first equation, put y = y0 z = z0 which will give x1 In second equation, put x = x1 and z = z0 which will give y1 In third equation, put x = x1 and y = y1 which will give z1

Quick Refresher Guide

Mathematics

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K3 = h f(x0 + , y0 + ) K4 = h f(x0 +h, y0 + k3) K = (k1 + 2k2 + 2k3 + k4) Y(x0+h) = y0 + k

Used for finding the y at a particular x without solving the 1st order differential equation = f(x, y) K1 = h f(x0, y0) K2 = h f(x0 + , y0 + )

(ii) Runge Kutta Method

Solution is given by, Yn+1 = yn + h f(xn,yn)

Quick Refresher Guide

Limit of a Function

Mathematics

e −1 =1 x

lim

)

.........

.........

.........

.........

.........

!

!

!

!

a +. . . . . . . . . a

n(n − 1)(n − 2) x +. . . . . . . . . x 3!

.........

+

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→

→

a −1 = log a x

=

lim

→

lim(1 + x) =

→

1 x

sinx =0 x

lim 1 +

→

lim

Some Important Limits

!

+

!

+

−

−

Cosh x = 1+

!

!

!

a+x

−

+ !

+

+

+

+ !

!

!

(

.........

Sinh x = x +

Cos x = 1−

Sin x = x −

log(1 − x) = − x −

log(1 + x) = x −

!

+

e =1+x+

!

+x

x −a =x x−a

(1 + x) = 1 + nx +

Some Standard Expansions

Then lim → f(x)= ℓ i.e, “ f(x)→ℓ as x→a” implies for any ∈(>0), δ(>0) such that whenever 0< |x − a|<δ, |f(x) − ℓ|<∈

Let y = f(x)

1.4.1

1.4 Calculus

Quick Refresher Guide

a

or

→

→

→

f(x) = lim →

→

f(x)= f(a) then the function f(x) is said to be continuous.

C1

C

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C

Geometrically: There exists at least one point c between (a, b) such that tangent at c is parallel to x axis

Then there exists at least one point c between (a, b) such that ′( ) = 0

(iii) f(a) = f(b)

(ii) f’(x) exists for every value of x in open interval (a,b)

If (i) f(x) is continuous in closed interval [a,b]

Rolle’s theorem

c. |f| or |g| is continuous at a

If f and g are two continuous functions at a; then a. (f+g), (f.g), (f-g) are continuous at a is continuous at a, provided g(a) ≠ 0 b.

Properties of Continuity

3. If lim

Mathematics

form, differentiate numerator & denominator and then apply

Existence of Limits and Continuity: 1. f(x) is defined at a, i.e, f(a) exists. 2. If lim f(x) = lim f(x) = L ,then the lim f(x) exists and equal to L.

L – Hospital’s Rule x When function is of limit.

→

x −a = x−a

log(1 + x) =1 x

lim log|x| = − ∞

→

lim

→

lim

Quick Refresher Guide

( )

= f (c).

( ) ( )

(

)

( )

.

.

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iv. (f/g)′ =

Algebra of derivative:i. (f + g)′ = f′ + g ′ ii. (f − g)′ = f ′ – g ′ iii. (f. g)′ = f ′. g + f . g′

Provided the limit exists ’( ) is called the rate of change of f at x.

→

Derivative:

( ) = ( )

’( ) = lim

1.4.2

( ) ( )

then, for a < β < b

Let f1 and f2 be two functions: i) f1,f2 both are continuous in [a,b] ii) f1, f2 both are differentiable in (a,b) iii) f2’ ≠ 0 in (a,b)

f(a+h) = f(a) + h f(a+θh)

(ii) f (x) exists in the open interval (a,a+h), then there is at least one number θ (0<θ<1) such that

If (i) f(x) is continuous in the closed interval [a,a+h] and

Cauchy’s Mean Value Theorem

Geometrically, it means that at point c, tangent is parallel to the chord line.

( )

(ii) f’(x) exists in the open interval (a,b), then atleast one value c of x exist in (a,b) such that

+ 2xy

Total Derivative

x

+y

. ∆x +

+ ∆y

.

= n(n − 1)u

b) Local

x

If

If

at x = α,

< 0,y has a maximum value

> 0, y has a minimum value

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Find

Rule for finding maxima & minima: x If maximum or minimum value of f(x) is to be found, let y = f(x) x Find dy/dx and equate it to zero and from this find the values of x, say x is α, β, …(called the critical points).

a) Global

Maxima-Minima

Mathematics

is called homogenous function of

Note: If f is a monotonic function on a domain ‘D’ then f is one-one on D.

Monotonicity of a Function f(x) 1. f(x) is increasing function if for α > β , f(α) > f(β) Necessary and sufficient condition, f’ (x) > 0 2. f(x) is decreasing function if for α > β, , f(α) < f(β) Necessary and sufficient condition, f (x) < 0

∆u =

=

If u=f(x,y) ,x=φ(t), y=Ψ(t)

1.4.3

x

If u be a homogenous function of order n in x and y then, = nu x x +y

Euler’s Theorem on Homogenous Function

f(x,y) = xn ϕ

f(x,y) = a0xn + a1xn-1y + a2xn-2y2 + …………+ an yn

Any function f(x, y) which can be expressed in from xnϕ order n in x and y. (Every term is of nth degree.)

If (i) f(x) is continuous in the closed interval [a,b] and

Quick Refresher Guide

Homogenous Function

Mathematics

Lagrange’s Mean Value Theorem

Quick Refresher Guide

< 0, y has maximum value

= 0, proceed further

If

If

at x = α.

Mathematics

!

= 0,

, t=

= 0 → solve these equations. Let the solution be (a, b), (c, d)…

,s=

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2. (i) if rt− s > 0 and r < 0 → maximum at (a, b) (ii) if rt− s > 0 and r > 0 → minimum at (a, b) (iii) if rt− s < 0 at (a, b), f(a,b) is not an extreme value i.e, f(a, b) is saddle point. (iv) if rt− s > 0 at (a, b), It is doubtful, need further investigation.

1.

r=

Maxima & Minima (Two variables)

h3 f'''(0) 3!

f”(a) + . . . . . . . . .

f(x) = f(0) + x f’(0) + f′′(0)+

Maclaurian Series:

f(a + h)= f(a) + h f’(a) +

!

≠ 0Æ Neither minima nor maxima exists

Taylor Series:

=0,

ii)

=0,

iii)

i)

Point of inflexion

If at a point, the following conditions are met, then such point is called point of inflexion

Point of Inflexion

Note: Greatest / least value exists either at critical point or at the end point of interval.

> 0, y has minimum value

If

= 0, proceed further and find

≠ 0, y has neither maximum nor minimum value at x = α

If

But If

= 0, proceed further and find at x = α.

If

Quick Refresher Guide

dx =

sec

dx =

31. ∫

cos bx dx =

(a cos bx + b sin bx )

(a sin bx − b cos bx )

) where x > a

) where x
Mathematics

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sin bx dx =

37. ∫ e

∫ cos x dx = + sin 2x ∫ tan x dx = tan x − x ∫ cot x dx = − cot x − x ∫ ln x dx = x ln x − x

− sin 2x

log (

log (

tan

38. ∫ e

33. 34. 35. 36.

32. ∫ sin x dx =

dx =

30. ∫

dx =

log(x + √x − a ) 29. ∫

log(x + √x + a )

√

27. ∫ √a + x dx = √x + a +

+

28. ∫ √x − a dx = √x − a −

sin

dx = log(x + √x + a ) = sinh ( )

25. ∫ √

26. ∫ √a − x dx =

dx = log(x + √x − a ) = cosh ( )

24. ∫ √

14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

13. ∫ dx = sec x √ ∫ cosh x dx = sinh x ∫ sinh x dx = cosh x ∫ sech x dx = tanh x ∫ cosech x dx = −coth x ∫ sech x tanh x dx = −sech x ∫ cosec h x cot h x dx = −cosech x ∫ tan x dx = log sec x ∫ cot x dx = log sin x ∫ sec x dx = log( sec x + tan x) = log tan(π 4 + x 2) ∫ cosec x dx = log(cosec x − cot x) = log tan

√

√

∫ cos x dx = sin x ∫ sin x dx = −cos x ∫ sec x dx = tan x ∫ cosec x dx = −cot x ∫ sec x tan x dx = sec x ∫ cosec x cot x dx = −cosec x dx = sin ∫

12. ∫

5. 6. 7. 8. 9. 10. 11.

2. ∫ dx = log x 3. ∫ e dx = e 4. ∫ a dx = (prove it )

, n ≠ −1

Standard Integral Results

1. ∫ x dx =

1.4.4

Quick Refresher Guide

a
f(x)dx+∫

∇f. N⃗ = |∇f|cosα

The integral ∫

The integral ∫

x

)

)

is convergent if and only if p< 1

is convergent if and only if p< 1

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(

(

e dx is converges for any constant p rel="nofollow"> 0 and diverges for p≤ 0

x

dx and ∫

∫ e

x

= c where c≠0, then both integrals ∫ f(x)dx and ∫ g(x)dx converge or both

x

( ) ( )

diverge. is converges when p> 1 and diverges when p≤ 1 ∫

→

If lim

x

+ k

Gradient:

+k

Divergence:

∇f is vector function If f(x,y,z) = 0 is any surface, then ∇f is a vector normal to the surface f and has a magnitude equal to rate of change of f along this normal. Directional derivative of f(x,y,z) is maximum along ∇f and magnitude of this maximum is |∇f|.

+ĵ

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div. F = ∇. F

The divergence of a continuously differentiable vector point function F is denoted by div. F and is defined by the equation.

1.4.9



grad f = ∇f = î

The vector function ∇f is defined as the gradient of the scalar point function f(x,y,z) and written as grad f.

1.4.8

Where, l =cos α , m=cos β , n=cos γ ,

Direction cosine: l + m + n = 1

1.4.6 Convergence: x ∫ f(x)dx is said to be convergent if the value of the integral is finite. x If (i) 0 ≤ f(x) ≤ g(x) for all x and (ii) ∫ g(x)dx converges , then ∫ f(x)dx also converges x If (i) f(x) ≥ g(x) ≥ 0 for all x and (ii) ∫ g(x)dx diverges, then ∫ f(x)dx also diverges

+ ĵ

The directional derivative of f in a direction N⃗ is the resolved part of ∇f in direction N⃗.

Directional Derivative:

Vector Differential Operator or Del Operator: ∇ = î

F(P) = F(x,y,z) = f1(x,y,z) ̂ +f2(x,y,z)ĵ +f3(x,y,z)k

If corresponding to each point P of region R, there corresponds a vector defined by F(P) then F is called a vector point function for region R.

Vector Point Function:

∅(P)= ∅(x,y,z)

Those integrals for which limit is infinite or integrand is infinite in a ≤ x ≤ b in case of ∫ f(x)dx, then it is called as improper integral.

f(x)dx

Mathematics

If corresponding to each point P of region R there is a corresponding scalar then ∅(P) is said to be a scalar point function for the region R.

Where N⃗ is a unit vector in a particular direction.

f(a − x)dx → ∫ f(x)dx = 2 ∫ if f(a-x)=f(x) if f(a-x)=-f(x) if f(-x) = f(x), even function if f(x) = -f(x), odd function

/

Vector Calculus:

Scalar Point Function:

1.4.7

Quick Refresher Guide

Improper Integral

=0 4. ∫ f(x)dx =2 ∫ f(x)dx =0

3. ∫ f(x)dx =∫

/

2. ∫ f(x)dx =∫ f(a + b − x)dx → ∫ f(x)dx =∫ f(a − x)dx

1.4.5 Rules for Definite Integral 1. ∫ f(x)dx =∫ f(x)dx+∫ f(x)dx

/

Exponential

Algebraic Trigonometric

I L A T E

∫ v dx)dx

Mathematics

Note: Take that function as “u” which comes first in “ILATE”

Logarithmic

Inverse circular (e.g. tan−1 x)

Selection of U & V

39. ∫ e [f(x) + f (x)]dx = e f(x) Integration by parts: ∫ u v dx = u. ∫ v dx − ∫(

Quick Refresher Guide

+

∅

+

+ ĵ

+ k

.( f î + ∅ĵ + Ψk)

∇. f is scalar ∇. ∇= ∇ is Laplacian operator

Mathematics

f

î

φ

ĵ

Ψ

k

+

+

---------- this is Laplace equation

curl grad f = ∇ × ∇f = 0 div curl F = ∇. ∇ × F =0 curl curl F = ∇ × (∇ × f) = ∇(∇. f) - ∇ F grad div F = ∇(∇. f)= ∇ × (∇ × F) + ∇ F

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f, g are scalar functions & F, G are vector functions 1. ∇(f + g) = ∇f + ∇g 2. ∇. (F + G) = ∇. F + ∇. G 3. ∇ × (F + G) = ∇ × F + ∇ × G 4. ∇(fg) = f ∇g + g∇f 5. ∇. (fG)= ∇f. G + f. ∇G 6. ∇ × (fG) = ∇f × G + f × ∇G 7. ∇(F. G) = F × (∇ × G) + G × (∇ × F) 8. ∇. (F × G) = G.( ∇ × F) − F. (∇ × G)

1.4.14 Vector Identities:

2. 3. 4. 5.

1. div grad f = ∇ f=

1.4.13 DEL Applied Twice to Point Functions:

If ∇ × A =0, then A is said to be irrotational otherwise rotational.

1.4.12 Irrotational Vector Function

If ∇.A = 0 , then A is called as solenoidal vector function.

1.4.11 Solenoidal Vector Function

∇ × F is vector function

Curl F = ∇ × f =

The curl of a continuously differentiable vector point function F is denoted by curl F and is defined by the equation.

1.4.10 Curl:

x x

=

div.F= ∇. f = î

F = f î + ∅ĵ + Ψk

Quick Refresher Guide

∇(f/g)= (g ∇f – f ∇g)/g (F.G)’ = F’.G + F . G’ (F × G)’ = F’× G + F × G’ ∇ (fg) = g ∇ f + 2 ∇f. ∇g + f ∇ g

Mathematics

∫ F dv = î∫ ∫ ∫ fdxdydz + ĵ ∫ ∫ ∫ ∅dxdydz +k ∫ ∫ ∫ Ψdxdydz

If F(R ) = f(x,y,z)î + ∅ (x,y,z)ĵ + Ψ (x,y,z) k and δv = δxδyδz , then

Volume integral : ∫ F dv

Line integral = ∫ F(R)dR If F(R )=î f(x,y,z) + ĵ ∅(x,y,z) + k Ψ(x,y,z) dR = î dx + ĵ dy + k dz ∫ F(R )dR = ∫ ( f dx + ∅ dy + Ψ dz ) Surface integral: ∫ F⃗ . ds⃗ or ∫ F⃗ . N⃗ ds, Where N is unit outward normal to Surface.

−

dxdy

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∮ (P dx + Q dy) = ∫ ∫

If R be a closed region in the xy plane bounded by a simple closed curve c and if P and Q are continuous functions of x and y having continuous derivative in R, then according to Green’s theorem.

1.4.17 Green’s Theorem

x x

x

1.4.16 Line Integral, Surface Integral & Volume Integral

1. Dot product of A × B with C is called scalar triplet product and denoted as [ABC] Rule: For evaluating the scalar triplet product (i) Independent of position of dot and cross (ii) Dependent on the cyclic order of the vector [ABC] = A × B. C = A. B × C = B × C. A= B.C × A = C × A. B = C.A × B A × B. C = -(B × A. C) 2. (A⃗ × B⃗) × C⃗ = (extreme × adjacent) × Outer = (Outer. extreme) adjacent−(Outer. adjacent) extreme x (A⃗ × B)⃗ × C⃗ = (C⃗ . A⃗ ) B⃗ - (C⃗ . B⃗ ) A⃗ x A⃗ × (B⃗ × C⃗ ) = (A⃗ . C⃗ ) B⃗ - (A⃗ . B⃗ ) C⃗ x (A⃗ × B⃗ ) × C⃗ ≠ A⃗ × (B⃗ × C⃗ )

1.4.15 Vector product

1. 2. 3. 4.

Also note:

9. ∇ × (F × G) = F(∇ × G) − G(∇ × F)

Quick Refresher Guide

Mathematics

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∫ F .N.ds =∫ div F dv

The normal surface integral of a vector point function F which is continuously differentiable over the boundary of a closed region is equal to the

1.4.19 Gauss divergence theorem

If F be continuously differentiable vector function in R, then ∮ F. dr = ∫ ∇ × F .N ds

1.4.18 Stoke’s theorem

Quick Refresher Guide

Differential Equations of First Order First Degree:

Degree of Differential Equation: It is the degree of the highest derivative occurring in it, after expressing the equation free from radicals and fractions as far as derivatives are concerned.

Order of Differential Equation: It is the order of the highest derivative appearing in it.

Mathematics

( , ( ,

) )

Separate the variable V and x and integrate.

=V+x

To solve a homogeneous equation, substitute y = Vx

=

=

(say) ) )

= ( (

= =

=

=

(h and K are constants)

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Case II: If

y=Y+k Solve for h and k ah + bk + c = 0 a h+b k+c =0

Substitute x = X + h

≠

This is a non - homogeneous but can be converted to homogeneous equation

Case I: If

→

The differential equation:

Equations Reducible to Homogenous Equation:

x

x

2. Homogenous Equation:

∫ f(x)dx + ∫ g(y)dy = c is the solution

1. Variable separable : f(x)dx + g(y)dy = 0

Equations of first order and first degree can be expressed in the form f (x, y, y ) = 0 or y = f(x, y). Following are the different ways of solving equations of first order and first degree:

1.5.3

1.5.2

1.5.1

1.5: Differential Equations

Quick Refresher Guide

)

+a

d y dy + P(x) + Q(x)y = R(x) dx dx

+ 3x

+ y

=Q

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M (x, y) dx + N (x, y) dy = 0

→ This is a linear equation and can be solved easily

where, P & Q are functions of x only.

+ (1 − n)Pz = Q (1-n)

=z

+ Py

4. Exact Differential Equations:

Substitute, y

y

Divide by y

= 0.

+Py=Qy

3.1 Bernoulli’s Equation:

Ex:

)]

=d[

6.

M dx + ∫(terms of N not contaning x ) dy = C

−

+

= f(y) , then I.F. = e∫

( )

( )

f(D) y = X ; f(D) = 0 is called Auxiliary Equation.

+ - - - - - + k )y = X {Where, D =

+ ------- +k y=X }

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If all the roots of A.E. are real and different

Rules for Finding Complimentary Function: Case I :

−

= f(x), then I.F. = e∫

Linear Differential Equation with Constant Coefficients:

The equation can be written as (D + k D

1.5.4

Rule 4: If the equation M dx + N dy = 0 and

Rule 3: If the M dx + N dy = 0 and

Rule 2: If the equation f (x, y) y dx + f (x, y) x dy = 0 and M x – N y ≠ 0 then I.F. =

Rule 1: when M dx + N dy = 0 is homogenous in x and y and M x + N y ≠ 0 then I.F. =

log(

)]

=-d( ) = d [tan (

4. 5.

)]

1. x dy + y dx = d (x y) 2. =d( )

Rule 0: Finding by inspection

= d [log (

Note: The degree of every linear differential equation is always one but if the degree of the differential equation is one then it need not be linear.

Mathematics

Sometimes an equation which is not exact may become so on multiplication by some function known as Integrating factor (I.F.).

Integrating Factor:

4.1 Equation Reducible the Exact Equation:

Solution of exact differential equation: ∫

3.

= ∫ Q. (I. F)dx + C ⇒ y(I. F. ) = ∫ Q. (I. F)dx + C

Quick Refresher Guide

The necessary and sufficient condition for the differential equations M dx +N dy = 0 to be exact is =

ye∫

Mathematics

Integrating factor, I.F. = e∫

Commonly known as “Leibnitz’s linear equations”

Second order linear equation:

+ P(x) y = Q(x) , where P and Q are functions of x

The standard form of a linear equation of first order:

3. Linear Equations:

Solve by variable separable method.

=

(

Substitute ax +by = t, so that,

Quick Refresher Guide

y = e∝ (C cos βx + C sin βx)

m = ∝ +iβ , m = ∝ −iβ

+ -- - - - -+ C e

P.I. =

P.I. = x

P.I. =

"( )

( )

( )

⋯

)

=

ф"(

=xф(

P.I. = ф(

)

)

(

(

(

put

+ ) put

+ )

+ )

put

put D = a

=-

=-

=-

[ф’’(-

[ф’(-

[ф (-

) ≠ 0, ф’(-

) ≠ 0, ф(-

)≠0]

)

) = 0]

[ ( ) = 0, ’( ) = 0, ’’( ) ≠ 0]

[ ’( ) ≠ 0, ( ) = 0]

put D = a

.X

[ ( ) ≠ 0]

( )

put D = a

X=

When X = sin (ax + b) or cos (ax +b)

When X =

P.,. =

Rules for finding Particular Integral

y = e∝ [(C x + C ) cos βx + (C x + C ) sin βx ]

i.e. repeated imaginary root ∝ ±iβ, ∝ ±iβ

If two pairs of root are imaginary

i.e.

If one pair of roots are imaginary

y = (C + C x +C x ) e

Similarly, if m = m = m

y = (C + C x ) e

If two roots are equal i.e. m = m

+ C e

Mathematics

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Case II:

Case I:

1.5.5

Case IV:

Case III:

Case II:

So, the solution is: y = C e

(D − m ) (D − m ) - - - - - - (D − m ) y = 0

Quick Refresher Guide

P.I. =

=

=

P.I. =

x V(x) =

=

P. I. =

−

(

( )

( )

( )

( )

X=

) (D −

P.I. =

X

∫

)

x

+------ -

= D (D-1)(D-2) y

= D (D-1) y

= Dy

x=e

.

+

=X

( )

into partial fractions

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Substitute

+

Mathematics

V as in Case I, II & III

) and resolve

(

( ) +⋯]

on each terms.

) - - - - - - - (D −

V(x)

V then evaluate

( )

)

V

( )−

Cauchy-Euler Equation: (Homogenous Linear Equation)

Complete Solution: y = C.F. + P.,.

and then apply,

]

(D) [1− ( ) +

( )

= [ ( )]

(D) [1 +

( )

V where V is function of x

Factorize f(D) = (D −

1.5.6

) = 0]

When X is any other function of x

( )

= 0, ф(, m being positive integer

When X = x V(x)

Case VI:

Case V:

When X =

Case IV:

When X =

Case III:

Quick Refresher Guide

Mathematics

+ )

(

=

=

=aDy

(ax + b)

+ )

⇒

ax + b =

(

+

+ )

(

t = ln (ax + b)

+ )

D(D-1)(D-2)y

D(D-1)y

(

Legendre’s Linear Equation:

+ - - - - -- -

=X

=q,

= r,

= s,

=

+

+------ -

+

{ where D =

x) + x

(y +

x) +

(y +

x) + - - - -

,

+ - - - - - + = 0, and the roots are , ---- x) + (y + x) + - - - - - - , are distinct x) + x (y + x) + (y + x) + - - - - - - ,

) = f(x, y)

,

,

}

three equal roots.

two equal roots.

and D’ =

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(y +

1. Write A.E. + Where m = 2. CF = (y + CF = (y +

CF =

+-------

f (D, D’) = f(x,y)

Step I: Finding the C.F.

(

= f( x, y) → this is called homogenous because all

Homogenous Linear Equation with Constant Coefficients:

=p,

z = f(x, y)

Partial Differential Equation:

terms containing derivative is of same order.

1.5.9

1.5.8

After substituting these differentials, the Legendre’s equation results in a linear equation with constant coefficients.

1.5.7

After substituting these differentials, the Cauchy – Euler equation results in a linear equation with constant coefficients.

Quick Refresher Guide

( , )

f (x, y)

)

( , )

, ′ = −

f (x, y) , resolve

=− )]

( ,

)

)

into partial

Mathematics

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y + mx after integration.

fractions considering ( , ) as a function of D alone and operate each partial fraction ) on f(x, y) remembering that f(x, y) = ∫ ( , − where c, is replaced by (

4. when F(x, y) is any function of x and y. P., =

, put [ D = a, ′ = b] 1. when F( ax +by ) = 2. when F( x, y) = sin (mx +ny), put ( = − , 3. when F(x, y) = , P., = ( , ) =[ ( ,

P.,. =

Step II: Finding P.I.

Quick Refresher Guide

+

Mathematics

, Argument of

=

=

( )=

Continuity of f (z):

,

=-

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o Real and imaginary part i.e. u, v of the function is called conjugate function. o An analytic function posses derivatives of all order and these are themselves analytic.

are both “necessary and sufficient” condition for the function f(z) = u + iv to be analytic in R.

=

continuous throughout a region R , then CR equations

o A single valued function which is defined and differentiable at each point of a domain D is said to be analytic in that domain. o A point at which an analytic function ceases to possess a derivative is called Singular point. , , , are o Thus if u and v are real Single – valued functions of x and y such that

, → Cauchy-Riemann equations (CR equations)

Analytic Functions (or Regular Function) or Holomorphic Functions

=−

1.6.4

,

, are continuous functions of x and y in R.

=

,

ii)

,

,

i)

The necessary and sufficient conditions for the derivative of the function f( ) to exist for all values of in a region R.

Theorem on Differentiability:

( ) = ( ). A function = f (z) is said to be continuous at = if → Further f (z) is said to be continuous in any region R of the z-plane, if it is continuous at every point of that region. Also if w = f (z) = u(x, y) + i v(x, y) is continuous at = , then u(x, y) and v(x, y) are also continuous at x= & y = .

1.6.3

x

x x

1.6.2

w = f (z) = u(x, y) + i v(x, y)

Where z = x + iy,

1.6.1 Function of a Complex Variable: It is a rule by means of which it is possible to find one or more complex numbers ‘w’ for every value of ‘z’ in a certain domain D, then w = f (z)

+

is a complex no., where x & y are real numbers called as real and imaginary part of z.

Modulus or absolute value = | | =

=

1.6: Complex Variables

Quick Refresher Guide

Harmonic Functions:

Mathematics

,

=−

=0

,

=− → (Laplace Equation)

Methods of Constructing Analytic Functions:

( , )

( , )

dz + i ∫

( , )

dx +

dy

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1.6.7 Complex Integration o Line integral = ∫ ( ) , C need not be closed path Here, f(z) = integrand , curve C = path of integration o Contour integral = ∮ ( ) , if C is closed path

Integrate this equation to find v. f (z) = u(x, y) + i v(x, y) 3. If a real part of the analytic function f(z) is given which is harmonic function u (x, y), then , - u(0, 0) f(z) = 2u

dv = −

2. If u (x, y) is known, then to find v(x, y) we have dx + dy dv =

f (z) = ∫

dz + c

dz + c

Similarly in case v(x, y) is known, then +i f’ (z) =

( , )

Integrate with points at (z, 0) dz - i ∫ f(z) = ∫

1. If the real part of a function is given then, ’( ) = -i

1.6.6

(1) For a function to be regular, the first order partial derivations of u and v must be continuous in addition to CR equations. (2) Mean value of any harmonic function over a circle is equal to the value of the function at the centre.

Note:

+

=

Differentiating with respect to x and y respectively,

=

If f(z) = u + iv be an analytic function in some region of the z – plane then the C –R equations are satisfied.

1.6.5

Quick Refresher Guide

( )

=∫(

−

)+

∫(

+

) ( )

= f( ) − ( ), i.e.

Mathematics

Cauchy’s Integral Theorem:

( )=

.

!

!

!

∫

∫

∫

∫

(

(

(

.

)

( )

)

( )

)

( )

( )

th

th

th

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.

”( ) =

’( ) =

( )=

If f(z) is analytic within and on a closed curve and if a is any point within C, then

1.6.8.1 Cauchy’s Integral Formula:

C

D

Note: In other words, by Cauchy’s theorem if f(z) is analytic on a simple closed path C and everywhere inside C (with no exception, not even a single point) then ∮ ( ) = 0

∮஼ ݂(‫ = ݖ݀)ݖ‬0

If f(z) is analytic in a simple connected domain D, then for every simple closed path C in D,

1.6.8

Dependence on Path: In general “Complex line integration” depends not only on the end points but also on the path (however analytic function in simple connected domain is independent of path.)

integration is independent of the path

Theorem: f(z) is analytic in a simple connected domain then ∫

∴∫

If f(z) = u(x, y) + i v(x, y) and dz = dx + i dy

Quick Refresher Guide

( )

= 0 around every

Mathematics

= ∫ (

( − ) ( ) )

!

"( ) !

”( ) + - - - - - - -

(z-a) + - - - - - - -

= ∫ (

+ ( ) )

( − )+

+

( − )

=

=

=

( − )

( − ) + − − − +

( − ) + − − −

= 0, then f(z) is said to have a zero of order n at z =a.

( − )+

=------

( ) !

+

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2. Removable Singularity: If all the negative powers of (z-a) in Laurent series are zero then

1. Isolated Singularity: If z =a is a singularity of f(z) such that f(z) is analytic at each point in its neighbourhood (i.e. there exists a circle with centre a which has no other singularity 1, then z =a is called an isolated singularity).

A “singular point” of a function as the point at which the function ceases to be analytic.

1.6.13 Singularities of an Analytic Function:

if

where

=∑

f(z) =

If f(z) is analytic in the neighbourhood of a point z = a then by Taylor’s theorem.

The value of z for which f(z) = 0

1.6.12 Zeroes of Analytic Function:

( − )

= 0 and Laurent series reduces to Taylor’s series.

( − ) + − − − +

If f(z) is analytic inside the curve then

where,

f(z) =

1.6.11 Laurent’s Series: If f(z) is analytic in the ring shaped region R bounded by two concentric circles and of radii and ( > ) and with centre at a then for all z in R

f(a+h) = f(a) + h ’( ) +

Other form, put z = a + h

where

f(z) = ∑

f(z) = f(a) + f’(a) (z-a) +

1.6.10 Taylor’s Series: If f(z) is analytic inside a circle C with centre at a then for z inside C

1.6.9 Morera’s Theorem: If f(z) is continuous in a region and ∫ simple closed C then f(z) is analytic in that region.

Quick Refresher Guide

Mathematics

are missing then. The

∅( ) ø( )

)!

[( − )

( )]

ܽ

ܽ

‫ܥ‬

‫ܥ‬ ‫ܥ‬ ‫ܥ‬

C

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Note: If an analytic function has singularities at a finite number of points, then the sum of residues at these points along with infinity is zero.

Here n =order of singularity

( )=(

3. If ( ) has a pole of order n at z=a , then

Res ( ) =

ø( )

1. If f(x) has a simple pole at z=a , then Res f(a) = → [( − ) ( )] ∅( ) 2. If ( ) = where ø( ) = ( − ) ( ), ( ) ≠ 0

Calculation of Residues

1.6.14 Residue Theorem If f(z) is analytic in and on a closed curve C except at a finite number of singular point within C then ∫ f(z)dz = 2 i (sum of the residue at the singular point within C)

4. Poles: If all the negative power of (z-a) in Laurent’s series after singularity at z = a is called a pole of order n. A pole of first order is called a “simple pole”.

3. Essential singularity: If the numbers of negative power of (z-a) in Laurent’s series is infinite, then z =a is called an essential singularity. ( ) does not exist in this case →

f(z) = ∑ ( − ) singularity can be removed by defining f(z) at z = a is such a way that it becomes analytic at z =a ( ) exists finitely, then z = a is a removable singularity. → Example: f(z) = , then z = 0 is a removable singularity.

Quick Refresher Guide

Introduction

Mathematics

. ( )

. ( )

Properties of Laplace transform

: Two sided/ bilateral LT.

: One sided/ unilateral LT, where S = (σ + J ω)

. F(s)

( ) ] = s F(s) – f(0) where f(0) is initial value of f(t).

( )]=

( )

=

( )

and ℒ ∫

( )

=

( )+

∫

( )

F(s) –s f(0) - ′(0) where ′(0) is the value of [

( ) ] at t = 0

( )

and ℒ {

( )} = (−1)

(F(s))

( )

= ∫

( )

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ℒ

Integration in Frequency Domain

Differentiation in frequency domain is equal to multiplication by t in time domain.

ℒ [ t f(t) ] =

Differentiation in Frequency Domain

Integration in time domain is equivalent to division by s in frequency domain, if f(t) = 0 for t < 0.

ℒ ∫

Integration in Time domain

Similarly, ℒ [

If initial conditions are zero (i.e. f(0) = 0),differentiating in time domain is equivalent to multiplying by s in frequency domain.

ℒ[

Differentiation in Time domain

ℒ [f(t – to)] =

Time shift

ℒ [e-at f(t) ] = F(s + a) and ℒ [eat f(t) ] = F(s - a)

Frequency shift

1.7.2

ℒ[ ( )] = ( )= ∫

ℒ[ ( )] = ( ) = ∫

Definition of Laplace Transform

Laplace Transform (LT) is a method to get generalized frequency domain representation of a continuous time signal and is generalization of CTFT (Continuous Time Fourier Transform).

1.7.1

1.7: Laplace Transform

Quick Refresher Guide

( )=

→

( )

Mathematics

Final Value Theorem

( )

( ).

ℒ[ ( ) ∗

( )] =

( )

Convolution theorem ℒ[ ( ). ( )] = ( ) ∗

.

(s) where

(s) = ∫

(

+

F(s-a)

√

( − )

)

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. ( )

( ) f(t). sin at. u(t)

8. 9.

t.u(t)

6.

1

1 1

u(t) . ( )

Laplace transform of f(t), L{f(t) = F(s) 1

4. 5.

7.

( )

( ) be in the left half of

Table. Laplace Transform of Standard Functions

1

Function, f(t) ( ) u(t)

Laplace Transform of Standard Functions

)

3.

S. No 1. 2.

1.7.7

ℒ( ( )) = (

If f(t) is periodic function with period T, then

1.7.6 Laplace Transform of the Periodic Function

1.7.5

→

( )= ( ) → For applying final value theorem, it is required that all the poles of s- plane (strictly) i.e. poles on axis also not allowed.

If f(t) and its derivative ′(t) are Laplace transformable, then

1.7.4

This theorem does not apply to the rational function F(s) in which the order of numerator polynomial is equal to or greater than the order of denominator polynomial.

→

( ) are Laplace transformable, then

Initial Value Theorem

If f(t) and its derivative

1.7.3

Integration in frequency domain is equal to division by t in time domain.

Quick Refresher Guide

25

24

23

22

f (u). f (t − u)du

−

+ )

)

F(s) + f F(s)

(o ).

where f

(o ) = ∫

f(u)du

((s+∝) + ω )

√

√

((s+∝) + ω ) ∫ F(s)ds

ω

(s+∝)

F (s). F (s) where * is convolution operator

. F(s) d (−1) . ds |a|. F(as) 1 s F( ⁄a) |a|

e

(s − a ) s.F(s)-f(o ) s . F(s) − s. f(o ) –f (o ) 1 F(s) s

s

(

(

Mathematics

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t

t

. f(t)

sin ωt

f (t) ∗ f (t)=∫ e ∝ . cos ω t

21.

∝

f ta f(at)

19. 20.

e

∫ f(u)du f(t-a).u(t-a) t . F(t)

17. 18.

16.

cos hat. u(t)

12. f (t) f (t) ∫ f(u) du

sin hat. u(t)

11. 13. 14. 15.

cos at. u(t)

10.

Quick Refresher Guide

Part 2.1: Statics

Magnitude Point of application Line of action Direction

Weight of stationary train on rail when the trackis straight.

Forces on a rod resting against wall.

Forces on a ladder resting against a wall when a person stands on a rung which is not at its center of gravity.

The weight of benches in class room

A tripod carrying camera

Coplanar like parallel force

Coplanar concurrent

Coplanar non- concurrent force

Non- coplanar parallel

Non- coplanar concurrent force

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Non- coplanar Non-concurrent force Forces acting on moving bus

Example

Force system

Collinear force system: In a system, all the forces parallel to each other, if line of action of all forces lie along a single line then it is called a collinear force system.

Concurrent force system: If line of action of all the forces in a system passes through a single point it is called concurrent force system.

Coplanar force system: If all the forces in the system lie in a single plane, it is called coplanar force system.

2.1.1 Equivalent force system

Scalar and Vector: A quantity is said to be scalar if it is completely defined by its magnitude alone. e.g. length, energy, work etc. A quantity is said to be vector if it is completely defined only when its magnitude and direction is specified. e.g. force, acceleration.

x x x x

Force: A force is completely defined only when the following four characters are specified.

Statics deals with system of forces that keeps a body in equilibrium. In other words the resultants of force systems on the body are zero.

2.1 Statics

Engineering Mechanics

Part – 2: Engineering Mechanics

Quick Refresher Guide

Engineering Mechanics

dv dt

A

B P

P4

P3

P2

P1

E R A

R2

P4

P

R1

D

B

P3 C P2

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Polygon law of forces:

Triangle law of forces: If two forces acting simultaneously on a body are represented by the sides of triangle taken in order, then their resultant is represented by the closing side of the triangle taken in the opposite order.

2.1.2.1 Coplanar Concurrent Forces

2.1.2 Equilibrium and Free body diagrams

Parallelogram law of forces: If two forces acting simultaneously on a body at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram their resultant is represented in magnitude and direction by the diagonal of the parallelogram which passes through the point of intersection of the two sides representing the forces.

P

the line of action of the replaced force.

Principle of transmissibility of force: The state of rest of motion of rigid body is unaltered if a force acting on a body is replaced by another force of the same magnitude and direction but acting anywhere on the body along

Third law: For every action, there is an equal and opposite reaction.

F α m.

Second law: The rate of change of momentum of a body is directly proportional to the applied force & it takes place in the direction in which the force acts.

First Law: Every body continues in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by force acting on it.

Newton’s law of motion:

Quick Refresher Guide

Engineering Mechanics

+

+2

P1

A

ߠ

∝

E

B

ߠ

C

D

A block on a ramp

SMOOTH

600N

P

G

R2

600N W R1

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P

Sample Free body diagrams

Weight of cable not negligible

We Weight of cable negligible

Flexible cable, belt, chain or rope

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Force exerted by the cable is always a tension away from the body in the direction of the cable.

x

Following types of mechanical contacts can be found in various structures:

b

A free body diagram shows all contact and non-contact forces acting on the body.

ߚ

P1

J

Types of contacts/supports:

a

P3

P2

Free body diagram: A free body diagram is a pictorial representation used to analyze the forces acting on a free body.

P3

ߚ

c

Engineering Mechanics

In a free body diagram all the contacts/supports are replaced by reaction forces which it will exert on the structure. A mechanical system comprises of different types of contacts/supports.

∝

ߛ

∝

A cantilever beam

A ladder resting on smooth wall

Quick Refresher Guide

P P P = = sinα sinβ sinγ

P2

P1

Lami’s theorem: (only three coplanar concurrent forces) If a body is in equilibrium under the action of three forces, then each force is proportional to the sine of the angle between the other two forces.

= angle between two forces, = inclination of resultant with force P1 When forces acting on a body are collinear, their resultant is equal to the algebraic sum of the forces.

=

Resultant (R) =

tan

ߠ

P2

If a number of forces acting at a point be represented in magnitude and direction by the sides of a polygon in order, then the resultant of all these forces may be represented in magnitude and direction by the closing side of the polygon taken in opposite order

Quick Refresher Guide

SMOOTH

Smooth surfaces

Rough surfaces

Engineering Mechanics

Roller support

Freely sliding guide

Pin connection

Built in or fixed end

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x

A freely hinged pin supports a force in any direction in the plane normal to the axis; usually shown as two components Rx and Ry. A pin not free to turn also supports a couple M.

x

Collar or slider support force is normal to guide only. There is no tangential force as surfaces are considered to be smooth.

x

Roller, rocker, or ball support transmits a compressive force normal to supporting surface.

x

Rough surfaces are capable of supporting a tangential component F (frictional force as well as a normal component N of the resultant R.

x

Contact force is compressive and is normal to the surface.

x

Quick Refresher Guide

Engineering Mechanics

‫۾‬૚

R

Couple is rotated through any angle. Couple is shifted to any position. The couple is replaced by another pair of forces whose rotated effect is the same.

The algebraic sum of the components of the forces along each of the three mutually ⊥ direction is zero. The algebraic sum of the components of the moments acting on the body about each of the three mutually perpendicular axes is zero.

&

M= ¦ M=0

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Two forces can be in equilibrium only if they are equal in magnitude, opposite in direction, and collinear in action. If a system is in equilibrium under the action of three forces, those three forces must be concurrent.

These requirements are both necessary and sufficient conditions for equilibrium.

∑M = 0 ,∑M = 0 & ∑M = 0

For non-collinear force system

∑ F = 0, ∑ F = 0 & ∑ F = 0

For collinear force system

R ¦F 0

When a body is in equilibrium, the resultant of all forces acting on it is zero. Thus, the resultant force R and the resultant couple M are both zero, and we have the equilibrium equations

x

x

Condition for body in Equilibrium:

x x x

Effect of couple is unchanged if

R.d = P1.d1 + P2.d2

A

‫۾‬૛

‫܌‬૚

B ‫܌‬૛ ࢊ

Varignon’s theorem: The algebraic sum of the moments of a system of coplanar forces about a momentum center in their plane is equal to the moment of their resultant forces about the same moment center.

2.1.2.2 Coplanar Non-Concurrent Forces

A built-in or fixed end supports an axial force F, a transverse force V, and a bending moment M.

Quick Refresher Guide

Engineering Mechanics

Unstable Equilibrium

Neutral Equilibrium

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It states that if a system of forces acting on a body or a system of bodies are in equilibrium and if the system is supposed to undergo a small virtual displacement consistent with its geometrical constraints, the algebraic sum of the virtual work done by the system of forces is zero.

Principle of virtual Work

δU = F.δs

Virtual Work: The product of the force F and the virtual displacement δs in the direction of the force is called virtual work.

Virtual Displacement: It may be defined as the infinitesimally small imaginary (or hypothetical or virtual) displacement given to a body or to a system of bodies in equilibrium, consistent with the constraints. The virtual displacement may be either rectilinear or angular.

U = F cos θ × S

When the distance moved by the body is not in the direction of the force then to determine the work done, the component of the force in the direction of the distance moved may be multiplied with the distance moved. For example, if the force F is acting at an angle θ with the direction of the distance S moved, then work done is given by

U=F×S

Work = Force × distance

Work: When a force acts on a body and moves it through some distance in its own direction, then work is said to be done. Thus, work may be defined as the product of the force and the distance moved in the direction of the force. Mathematically, we can write that

2.1.3 Virtual Work

Stable Equilibrium

Neutral Equilibrium: A body is in neutral equilibrium if it stays in the displaced position after if has been displaced slightly.

Unstable Equilibrium: A body is in unstable equilibrium if it does not return to its equilibrium position and does not remain in the displaced position after it has been displaced slightly.

Stable Equilibrium: A body is in stable equilibrium if it returns to its equilibrium position after it has been displaced slightly.

There are three types of equilibrium as defined below:

Types of Equilibrium:

Quick Refresher Guide

Engineering Mechanics

Large truss in which only few forces are required Situation where method of joints fail.

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While determining the reactions at the supports, the following points should be remembered

x x

Method of section is used when

2. Graphical method

(b) Method of sections (non-concurrent force system).

(a) Method of joints (concurrent force system).

1. Analytical methods

Methods of Solution: Two methods are generally used for determining the forces in various members of a truss. These methods are

The member of a truss may be in tension or compression. A member in tension is called a tie and a member in compression a strut.

1. The members of truss are connected at the joints by friction less joints. 2. The members of truss lie in a common plane (plane truss). 3. The loads are applied only on the pins connecting the members and that the lines of action of the loads lie in the plane of the truss. 4. The weight of members is negligible as compared to the applied loads. 5. The truss is rigid and that it does not deform or change its shape upon the application of the loads.

The following assumptions are made in solving trusses:

A pin jointed frame which has just sufficient number of members to resist the loads without undergoing deformation in space is called perfect frame. If number of members in frame is less than that that required for a perfect frame then it is called deficient frame. If number of members in frame is more than that required for perfect frame then it is called redundant frame. A redundant frame is indeterminate.

where j is the number of joints, then the truss is known as a perfect truss, otherwise imperfect. The truss is called deficient or redundant, if m < (2j-3) or m > (2j – 3), respectively.

m = 2j – 3

When the number of members m in a truss satisfies the condition,

Trusses are commonly used for construction of roofs of workshop factories and bridges. The trusses are subjected to mainly three types of loads, viz, dead load, live load and wind load. The dead load is self weight of truss; live load is the load which is applied to the truss; e.g. the load acting on a bridge truss due to the passing of a train, load acting on a workshop truss due to an electric overhead, travelling crane, the wind load due to the high velocities of wind blowing in a particular region.

2.1.4 Trusses and Frames

Quick Refresher Guide

Engineering Mechanics

2

3

4

3

5

4

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If three forces act at a joint and two of them are along the some straight line then, for the equilibrium of the joint, the third force should be equal to zero.

If there are two forces acting at a joint then, for the equilibrium, forces should act along the same straight line. The two forces should be equal and opposite. If the (only) two forces acting at a joint are not along the same straight line, then for the equilibrium of the joint each force should be equal to zero.

If there is only one force acting at joint, then for the equilibrium, this force should be equal zero.

For deficient frame, m < (2j -3)

For perfect frame, m = (2j -3)

1

1

2

A frame in which all the member lies in a single plane is called plane frame. While a frame in which all the member do not lie in a single plane is called space frame.

(b) At fixed support, the reaction can take an arbitrary direction.

(a) At simply supported (i.e., pinned or roller support) support there can be only a vertical reaction.

Quick Refresher Guide

Engineering Mechanics

+2 +

+

=

∝

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Maximum height (h) =

Motion of projectile:

2.2.2 Kinematics of Curvilinear motion

Time for returning to the starting point time of flight =

Maximum height reached,

Time to reach the highest point, =

The following important results can be deduced from these equations:

When a body is projected vertically upwards, it is under the effect of the downward acceleration due to gravity, i.e., it moves with retardation. Its velocity, therefore, gradually decreases until it becomes zero; the body is then for an instant at rest and immediately begins to fall with a velocity which increases numerically but is negative. Thus, we get = − 1 = − 2 v = u − 2gs

Motion of Bodies Projected vertically upwards

Where u = initial velocity, v = final velocity, s = distance of travel, t = time and a = acceleration

=

=

=

Motion with constant acceleration:

2.2.1 Kinematics of Rectilinear Motion

In kinetics, motion of particles or rigid bodies is studied with the unbalanced force system that produces or changes this motion.

In kinematics, motion of particles or rigid bodies is studied without considering the forces that produce or change this motion.

(a) Kinematics (b) Kinetics

Dynamics can be divided into two main branches:

2.2 Dynamics

Part 2.2: Dynamics

Quick Refresher Guide

or

=

Σ =

⟶

is constant, above equation may be integrated, giving

=∫

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= angular velocity

Where r = radius of the path

=

+

′

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+

and with velocities and respectively, before coming Let the two bodies have masses into contact with each other, and velocities ′ and ′ at the end of the period of contact. Then according to the conservation of linear momentum, we have

=

Centrifugal force =

Central force motion

If the sum of the external forces acting on any system of mutually attracting and impinging bodies resolved in any direction is always zero, the total momentum of the system in that direction remains constant during the motion.

2.2.4 Kinetics of Curvilinear Motion

The resultant impulse of the external forces acting upon a body is equal to the change of momentum of the body. Both impulse and momentum are vector quantities. The units of impulse and momentum are Ns.

Ft = mv – mvo

Linear impulse of a force is defined as Ft and linear momentum is defined as mv. Thus, it may be expressed as follows:

If the forces Σ are variable, they must be given as a function of time and should be similarly integrated. Forces that cannot be expressed mathematically as a function of time may be plotted on a force-time curve, in which the area under the curve is equal to the left side of the equation.

Where ⟶ indicates vector difference of two momentum terms.

If Σ

∫ Σ

Conservation of Linear Momentum

ma = 0

= , ,

Σ

=

For a finite period of time, integrating, we get

Or

=

where ∑ indicates the sum of all forces acting on the body in the direction of motion.

F −

Where

F − ma = 0

When different forces act on a system such that it is in motion in a particular direction, the algebraic sum of all the forces acting on the system in the direction of the motion, including the inertia force taken in opposite direction to motion is zero. Thus in general

stated as follows:

is the inertia force. Example represents the D’ Alembert’s principle, which may be

Now

=0

+ −

=

or

F = ma where

According to Newton’s second law of motion,

It was pointed out first of all by D’Alembert that on the line of equation of static equilibrium, equation of dynamic equilibrium can also be established by introducing inertia force in the direction opposite to acceleration in addition to the real forces acting on the system.

Σ

From Newton’s second law of motion

Where I = mk , k being the radius of gyration.

2.2.3 Kinetics of rectilinear motion

D’ Alembert’s Principle

Engineering Mechanics

Moment of momentum (angular momentum)of the whole body = Iω

g = acceleration due to gravity

v = linear speed

Quick Refresher Guide

2.2.5 Impulse and Momentum

∝

∝

∝

Engineering Mechanics

∴ for maximum range, ∝ = 450

Range (R) =

∴ time of flight =

Time required to reach maximum height (t) =

Quick Refresher Guide

+

+

′

Z =

Z +

Z ′

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Z +

Suppose the two bodies have moments of inertia and and angular velocities Z and Z repectively, before coming, into contact, and angular velocities Z ′ and Z ′ and the end of the period of contact. Then the principle of conservation of angular momentum may be stated as

According to this principle, if a system of two rotating bodies are brought into contact for a short time period, and no external torque is applied to the system during this time, the resultant angular impulse on the system is zero.

Conservation of Angular Momentum

The value of e lies between zero and one. It is zero for perfectly inelastic bodies and one for perfectly elastic bodies.

in which the proper sign of the four velocities must be included.

=

For direct central impact Newton verified experimentally that the relative velocity after impact is in a constant ratio to the relative velocity before impact. If the bodies collide obliquely, the same fact holds for their compound velocities along the common normal at the point of contact. This ratio is known as the coefficient of restitution, and is denoted by e. Thus

Coefficient of Restitution

+

For a very short period of time after the two bodies come in contact, the mass centres continue to approach each other. This is known as the period of deformation. During this period the intensity of the force between the surfaces increases. For an instant at the end of the period of deformation, the mass centres are moving with the same velocity. If the bodies are elastic, the impulsive forces causes centres to begin separating and, after a second short interval, the surfaces of the bodies are no longer in contact. This second short period is known as the period is known as the period of restitution. Time of impact is the sum of the period of deformation and period of restitution. The time of impact is very small. For this reason, the resultant impulse of the external forces acting on the system during this time must be small and can be neglected. On the bosis of this assumption, the sum of the momentum before impact is equal to the sum of the momentum after impact, i.e, the conservation of momentum holds, thus for direct central impact, we have = F.s

, we have

=∆ represents kinetic energy. This equation represents the principle of work and energy..

U

−

)

&

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=(

Work energy principle: The work done by a force acting on a particle during its displacement is equal to the change in kinetic energy of the particle during that displacement.

Principle of work and Energy:

=

Work of the Elastic force: If a prismatic bar of area of cross section A, length and elastic constant E is stretched then the work of elastic force can be calculated by treating it as a spring of stiffness k.

One metric horse power = 735.5 watts

Power = (F cos ∝) v. v is the velocity of the point where the force F is acting. ∝ is the angle between the directions of the force and the velocity. If both are in the same direction then = . .

Where

,

Net work = change in kinetic energy

Relation between work and change of kinetic energy:

,

Denoting work by

The work done by a force is equal to the product of the force, F, and its displacement, s, provided the force is constant and the displacement, of the body is in the same direction as the force.

Work Done by a Force

Power: The rate of doing work is called power.

Kinetic Energy: This is the energy which a body possesses because of its velocity.

Potential Energy: This is the energy which a body possesses because of its position.

Energy: The capacity to do work is called energy. It is measured in N.m.

Work: If a force acts on a body and causes it to move some distance, work is said to be done by the force. Thus, work is a measure of accomplishment. Therefore, work done by a constant force is equal to the product of the force and the displacement of its point of application in the direction of the force. It is measured in Nm.

Engineering Mechanics

If two bodies suddenly collide, an impulsive force, or impact, is set up between them. When the direction of each body is along the common normal at the point where they touch, the impact is said to be direct. When the direction of motion of either or both, is not along the common normal at the point of contact, the impact is said to be oblique. If the pressure exerted on the surfaces of contact coincides with the line joining the mass centres of the bodies, the impact is central. If such is not the case, it is eccentric.

Quick Refresher Guide

2.2.6 Work and Energy

Engineering Mechanics

Collision of Elastic Bodies

Quick Refresher Guide

−

)

Engineering Mechanics

=−

=−

=−

(

(

−

)

)

is negative upwards.

is positive upwards

=−

(

−

)

U

= ∫ du = − ∫ F. dx

2

.

=

mV − 2 2 1 ( − = 2

)

be the velocities of the particle at points 1 and 2 and the corresponding distance be

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(K. E. ) + (P. E. ) = (K. E. ) + (P. E. )

U

Let v and and .

1 . .= 2

If a particle of mass m is moving with velocity , it’s kinetic energy k.E is given by.

=−

=−

′ − ′ sign indicates that Force and displacement are in opposite directions.

U

Force exerted by the spring is not a constant force but it varies linearly with the displacement from the undeformed position.

. .=

Work of the gravity force:

This principle cannot be applied where frictional force is involved.

K. E. + P. E. = K. E. + P. E.

Principle of conservation of energy: states that the sum of the potential energy and the kinetic energy of a particle (or of a system of particles) remains constant during the motion under the action of conservative forces.

W.D. by a force for displacing a body from (1) ⟶ (2) is positive (+ve) and from (1) ⟵ (2) is negative (−ve).

=(

= Net W.D. by the force for displacing a body from (1) to (2)

Quick Refresher Guide

Engineering Mechanics

d =

=

=∫

=

=∫

(

(

∫ rotating about a fixed axis.

) =

is the mass moment of inertia with respect to the axis of rotation.

)

Rigid body

ࢊ࢓

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Where

∴

=

Therefore,

O

࢘

W

The kinetic energy of the particle is, as

Consider a rigid body rotating about an axis 0 with an angular velocity w as shown in a Fig. The particle of mass dm in this body has a velocity v = rw normal to the radial line r.

Relation between Work and Kinetic Energy for Rotation

=∫

When the body rotates through the angle , the total work done is,

= Fds = Fr d

Let a couple Fr act on a body so that the body starts rotating. As the body rotates through a small angle d , the work done by the force is

Work Done by a Couple or Torque

W.D. by the springs ⟹ +ve ⟸ W.D. by the system (springs) W.D. on the system ⟹ −ve. ⟸ W.D. on the system (springs) When spring is stretched W. D. by force is "–ve" ie work is done on the spring. When the spring returns returns towards undeformed position W. D. is +ve(or)work is done by the spring.

Quick Refresher Guide

Strength of Material

A

Pparallel _ to _ area

Height

G parallel _ to _ area

L

tan T | T [T in radians]

EH

PL AE

V



E and that the cross-sectional area is constant.

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normal stress H



This expression for axial deformation assumes that the linear strain is proportional to the

§V· G LH L¨ ¸ ©E¹

The deformation, G, of an axially loaded member of original length L can be derived from Hooke’s law. Tension loading is considered to be positive, compressive loading is negative. The sign of the deformation will be the same as the sign of the loading.

V

Hooke‘s law is a simple mathematical relationship between elastic stress and strain: stress is proportional to strain. For normal stress, the constant of proportionality is the modulus of elasticity (Young’s Modulus), E.

3.1.2 Hooke’s Law: Axial and Shearing Deformations

J

H

G

Linear strain (normal strain, longitudinal strain, axial strain), H, is a change in length per unit length. Linear strain has no units. Shear strain, J is an angular deformation resulting from shear stress. Shear strain may be presented in units of radians, percent, or no units at all.

W

V

Pnormal _to _area A

Stress is the internal resistance offered by the body per unit area. Stress is represented as force per unit area. Typical units of stress are N/m2, ksi and MPa. There are two primary types of stresses: normal stress and shear stress. Normal stress,V, is calculated when the force is normal to the surface area; whereas the shear stress, W is calculated when the force is parallel to the surface area.

3.1.1 Simple Stress & Strain

Part 3.1: Simple Stress and Strain

Part – 3: Strength of Material

Quick Refresher Guide

Strength of Material

PL

¦ AE

P¦

L AE

³

PdL AE P³

dL AE

L G

GJ

0

Elastic limit Proportional limit

Yield point

Stress

Ultimate strength

VL AG

=

Rupture strength

Actual rupture strength

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=

3.1.3 Stress-Strain Diagram

Gs

The relationship between shearing deformation, Gs and applied shearing force, V is then expressed by

W

Hooke’s law may also be applied to a plane element in pure shear. For such an element, the shear stress is linearly related to the shear strain, by the shear modulus (also known as the modulus of rigidity), G.

The algebraic deformation must be observed.

Lf

The new length of the member including the deformation is given by

G

When one of the variables (e.g., A), varies continuously along the length,

G

When an axial member has distinct sections differing in cross-sectional area or composition, superposition is used to calculate the total deformation as the sum of individual deformations.

Quick Refresher Guide

Strength of Material



H lateral H axial

Vx Q E E

E

Vy

Q

Vx E

>

>

Hy



@

@

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1 V y  QV z  V x E

1 Vx  Q Vy  Vz E

Hx

Hooke’s law can be further extended for three-dimensional stress-strain relationships and written in terms of the three elastic constants, E, G, and Q. The following equations can be used to find the strains caused due to simultaneous action of triaxial tensile stresses:

Hy

Similarly, the total strain in the y direction is

Hx

Vy

Poisson’s ratio permits us to extend Hooke’s law of uniaxial stress to the case of biaxial stress. Thus if an element is subjected simultaneously to tensile stresses in x and y direction, the strain in the x direction due to tensile stress Vx is Vx/E. Simultaneously the tensile stress Vy will produce lateral contraction in the x direction of the amount QVy/E, so the resultant unit deformation or strain in the x direction will be

Theoretically, Poisson’s ratio could vary from 0 to 0.5, but typical values are 0.33 for aluminum and 0.3 for steel and maximum value of 0.5 for rubber.

Q

Poisson’s ratio, Q, is a constant that relates the lateral strain to the axial strain for axially loaded members.

3.1.4 Poisson’s Ratio: Biaxial and Triaxial Deformations

Rupture Strength: It is the stress at failure

W zx G

G

W yz

G

W xy

>



1 Vz  Q Vx  Vy E

@

Quick Refresher Guide

Strength of Material

2G1 Q

E 21 Q

= α't

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If a temperature deformation is permitted to occur freely no load or the stress will be induced in the structure. But in some cases it is not possible to permit these temperature deformations, which results in creation of internal forces that resist them. The stresses caused by these internal forces are known as thermal stresses.

ϵ=

From the above equation thermal strain can be expressed as:

Where, L is the length, α (/oC) is the coefficient of linear expansion, and 't (oC) is the temperature change.

ΔL = L.α.'t

Temperature causes bodies to expand or contract. Change in length due to increase in temperature can be expressed as

3.1.5 Thermal stresses

E 3K1  2Q

For an elastic, isotropic material, the modulus of elasticity E, bulk modulus K, and Poisson’s ratio Q are related by

The bulk modulus (K) describes volumetric elasticity, or the tendency of an object's volume to deform when under pressure; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young's modulus to three dimensions.

E

G

For an elastic isotropic material, the modulus of elasticity E, shear modulus G, and Poisson’s ratio Q are related by

J zx

J yz

Yield Point: It is the point on the stress strain curve at which there is an appreciable elongation or yielding of the material without any corresponding increase of load; indeed the load actually may decrease while the yielding occurs.

Ultimate Strength: It is the highest ordinate on the stress strain curve.

J xy

Hz

Elastic Limit: It is the point on the stress strain curve up to which material will return to its original shape when unloaded.

Proportional Limit: It is the point on the stress strain curve up to which stress is proportional to strain.

Quick Refresher Guide

Strength of Material

z

¦F

0 : V1 (2t 'x)  p(2r'x) 0

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2

0 : V 2 (2S rt )  p(2S r ) 0

Longitudinal stress = pr/2t = pd/4t

x

¦F

Hoop stress or circumferential stress = pr/t = pd/2t

Cylindrical shells

3.1.6 Thin-Walled Pressure Vessels

σ = E.α.'t

When the temperature deformation is prevented, thermal stress developed due to temperature change can be given as:

Quick Refresher Guide

x

¦F

pr 2t

Strength of Material

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V1 V 2

2

0 : V 2 (2S rt )  p(2S r ) 0

Hoop stress = longitudinal stress =

Spherical shells

Quick Refresher Guide

Strength of Material

¦F

i ª sec tion _ to º «¬ one _ end »¼

i i ªsec tion _ to º «¬ one _ end »¼

¦F d



i ªsec tion _ to º «¬ one _ end »¼

¦C

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Bending moment sign conventions

Bending moments in a beam are positive when the upper surface of the beam is in compression and the lower surface is in tension. Positive moments cause lengthening of the lower surface and shortening of the upper surface. A useful image with which to remember this convention is to imagine the beam “smiling” when the moment is positive.

M

The bending moment, M, at a section of a beam is the algebraic sum of all moments and couples located between the section and any one of its ends.

Shear force sign conventions

Shear is positive when there is a net upward force to the left of a section, and it is negative when there is a net downward force to the left of the section.

V

The direction (i.e., to the left or right of the section) in which the summation proceeds is not important. Since the values of shear will differ only in sign for summation to the left and right ends, the direction that results in the minimum no. of calculations should be selected.

The shear force, V at a section of a beam is the sum of all vertical forces acting on the beam between that section and any one of its ends. It has units of Newtons, pounds, kips, etc. Shear force is not the same as shear stress, since the area of the object is not considered.

3.2.1 Shear and Moment

Part 3.2: Shear Force and Bending Moment

Quick Refresher Guide

Strength of Material

³ wx dx

dV x dx

x1

x2

³V x dx

dM x dx

x1

x2

The shear at any section is equal to the sum of the loads and reactions from the section to the left end. The magnitude of the shear at any section is equal to the slope of the moment function at that section. Loads and reactions to the left of the section acting upward are positive The shear diagram is straight and sloping for uniformly distributed loads. The shear diagram is straight and horizontal between concentrated loads. The shear is undefined at points of concentrated loads.

The moment at any section is equal to the sum of the moments and couples from the section to the left end. The change in magnitude of the moment at any section is the integral of the shear diagram, or the area under the shear diagram. A concentrated moment will produce a jump or discontinuity in the moment diagram. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 69

x

x

The following guidelines and conventions should be observed when constructing a bending moment diagram. By convention, the moment diagram is drawn on the compression side of the beam.

x x x x

x

x

The following guidelines and conventions should be observed when constructing a shear diagram.

Both shear force and bending moment can be described mathematically for simple loadings by the preceding equations, but the formulas become discontinuous as the loadings become more complex. It is more convenient to describe complex shear and moment functions graphically. Graphs of shear and moment as functions of position along the beam are known as shear force and bending moment diagrams.

3.2.3 Shear Force and Bending Moment Diagrams

V x

M 2  M1

The change in magnitude of the moment at any point is equal to the integral of the shear function, or the area under the shear diagram up to that point.

wx

V2  V1

The change in magnitude of the shear at any point is equal to the integral of the load function, w(x), or the area under the load diagram up to that point.

3.2.2 Shear Force and Bending Moment Relationships

Quick Refresher Guide

The maximum or minimum moment occurs when the shear is either zero or changes its sign. The moment diagram is parabolic and is curved downward for downward uniformly distributed loads.

Strength of Material

Type of loading Point load U.D.L. U.V.L. or Triangular Parabolic Bending couple

Variation of S.F. Rectangle Linear Parabolic Cubic No shear variation

Variation of B.M. Inclined line for linear Square Parabola Cubic Parabola Fourth degree polynomial A vertical step at the point of application

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S.No. 1 2 3 4 5

¾ If there is any internal hinge in beam , bending moment will be zero at hinge point. Variation of S.F. and B.M. for different loadings on spans of beams:

Axial: Component (cos ∅) produces pull or push

Transverse: Components (sin ∅) produces B.M. and S.F.

I

¾ If the external load is not at right angles to the axis of the beam, the loading can be resolved axially and transversely to the beam

Note:

x

x

Quick Refresher Guide

Strength of Material



My I

Figure. Bending Stress Distribution at a Section in a Beam



Mc I

M S

Vb

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For a rectangular b x h section, the centroidal moment of inertia and section modulus are

 S

I c

This equation shows that the maximum bending stress will occur at the section where the moment is maximum. For standard structural shapes, I and c are fixed. Therefore, for design, the elastic section modulus S, is often used.

V b,max

Since the maximum stress will govern the design, y can be set equal to c to obtain the extreme fiber stress.

In the above equation, I is the centroidal area moment of inertia of the beam. The negative sign in the equation, required by the convention that compression is negative, is commonly omitted.

Vb

Bending stress varies with location (depth) within the beam. It is zero at the neutral axis, and increases linearly with distance from the neutral axis, as predicted by Equation,

For positive bending moment, the lower surface of the beam experiences tensile stress while the upper surface of the beam experiences compressive stress. The bending stress distribution passes through zero at the centroid, or neutral axis, of the cross section. The distance from the neutral axis is y; and the distance from the neutral axis to the extreme fiber (i.e., the top or bottom surface most distant from the neutral axis) is c.

3.3.1 Bending Stress

Part 3.3: Stresses in Beams

Quick Refresher Guide

bh3 12

Srec tan gular

bh 2 6

Strength of Material



y R

Or,

Vb E



y R

Radius of curvature is large compared to the dimensions of the cross section.

x x x

x

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The exact value of shear stress is dependent on the location, y, within the depth of the beam. The shear stress distribution is given by equation shown below. The shear stress is zero at the top and bottom surfaces of the beam. For a regular shaped beam, the shear stress is maximum at the neutral axis

The shear stresses in a vertical section of a beam consist of both horizontal and transverse (vertical) shear stresses.

3.3.2 Shear Stress

Section Modulus ( S = I/c):: It represents the strength of the section. Greater the value of ‘S’, stronger will be the section.

Curvature = (1/ R) = (M / EI), EI = Flexural rigidity

M/I=V/y=E/R

Equation of Pure Bending:

Neutral axis: Line of intersection of Neutral Layer with plane of cross section. It passes through C.G. of cross section. At this axis the strain changes its sign.

Neutral Layer: The layer which does not undergo any change in length (N.A.)

Pure Bending: Only B.M. but no S.F.

Points to remember:

x

The transverse sections which are plane and normal before bending remain plane and normal to the longitudinal fibres after bending (Bernoulli’s Assumption). Material is homogeneous, isotropic and obeys Hook’s Law and limits of eccentricity are not exceeded. Every layer is free to expand or contract. Modulus of elasticity has same value for tension and compression. The beam is subjected to pure bending and therefore bends in an arc of a circle.

x

The above mentioned bending stress eqn. is based on following assumptions:

Hb

Also, the strain in any fiber varies directly with its location y from the neutral axis and can be found by the equation

I

Quick Refresher Guide

QV Ib

Strength of Material

c

y1

³ ydA

y ' A'

3V 2A

3V 2bh

1.5W avg

4V 3A

4V 3Sr 2

4 Wavg 3

V Aweb

V dtweb

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W avg

For a steel beam with web thickness tweb and depth d, the web shear stress is approximated by

Wmax,circular

For a beam with a circular cross section, the maximum shear stress is

W max, rec tan gular

For a rectangular beam, the equation for Wmax, can be simplified. The maximum shear stress is 50 percent higher than the average shear stress.

Q

For rectangular beams, dA = bdy. Then, the moment of the area A’ above 1ayer y is equa1 to the product of the area and the distance from the centroidal axis to the centroid of the area.

Q

In the above equation, I is the area moment of inertia, and b is the width or thickness of the beam at the depth y within the beam where the shear stress is to be found. The first (or statical) moment of the area of the beam with respect to the neutral axis, Q, is defined by,

Figure: Dimensions for Shear stress Calculations

W xy

Quick Refresher Guide

Strength of Material

E E wea ker

nF Ac,t

V stronger

I c,t

nMcstronger

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V stronger

V wea ker

Mcwea ker I c ,t

For beams in bending, the bending stresses in the weaker and stronger materials are

F Ac,t

V wea ker

The area of the stronger material is increased by a factor of n. The transformed area is used to calculate the transformed composite area, Ac,t , or transformed moment of inertia, Ic,t. For compression and tension members, the stresses in the weaker and stronger materials are

n

The transformation method starts by determining the modulus of elasticity for each (usually two in number) of the materials in the composite beam and then calculating the modular ratio, n. Eweaker is the smaller modulus of elasticity.

Most simple composite structures can be analyzed using the method of consistent deformations, also known as the transformation method. This method assumes that the strains are the same in both materials at the interface between them. Although the strains are the same, the stresses in the two adjacent materials are not equal, since stresses are proportional to the modulus of elasticity.

A composite structure is one in which two or more different materials are used. Each material carries part of the applied load. Examples of composite structure include steel-reinforced concrete and timber beams with bolted-on steel plates.

3.3.3 Composite Beams

Figure. Dimensions of a Steel Beam

Quick Refresher Guide

Strength of Material

H max c

c

U

M EI

d2y dx 2

dT dx

------- (2)

------- (1)

y''''

y'''

y''

y'

------- (6)

------- (7)

Vx EI wx EI

d3 y dx3 d4 y dx 4

------- (5)

------- (4)

Mx EI

slope

------- (3)

d2 y dx2

dy dx

deflection

³³ Mx dx

0

Built-in Support

0

y’

0

0

V

0

0

0

M

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Hinge

Free end

0

y Simple Support

End condition

y’’

------- (8) Table. Beam Boundary Conditions

EIy

If the moment function, M(x), is known for a section of the beam, the deflection at any point on that section can be found from Eq. (8). The constants of integration are determined from the beam boundary conditions in the table shown below.

y

Using the preceding relationships, the deflection and slope of a loaded beam are related to the moment M(x), shear V(x), and load w(x) by Eqs. (3) through (7).

H max

1 U

The curvature of a beam caused by a bending moment is given by Eq. (1), where U is the radius of curvature, c is the largest distance from the neutral axis of the beam, and Hmax is the maximum longitudinal normal strain in the beam.

3.4.1 Double Integration Method

Part 3.4: Deflection of Beams

Quick Refresher Guide

Strength of Material

=

- - - - - for -ve B.M

/

= /

.

A = area of B.M.D.

= x

xL=

L

,

B

= x

xLx

=

A

,

B

=

L xLx = 2 2

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=

L

M

Step 2: Draw (BMD) / (EI) i.e., M / EI Step 3: Slope = area of (M / EI) diagram between fixed end point under consideration. Step 4: Deflection A / EI,

A

Step 1: To determine slope and deflection at any point say B.

= = distance of C.G. of B.M.D. e.g,: (Suitable for cantilevers) – from objective point of view.

Theorem 2: The intercept on a vertical line made by two tangents drawn at the two points on the deflected curve, is equal to the moment of M / EI diagram between the two points about the vertical line.

=∫

i.e., θ = area of M / EI diagram.

Theorem 1: The angle between tangents drawn at any two points on the deflected curve is equal to the area of M / EI diagram between the two points.

3.4.2 Area Moment Method

=

Curvature =

EI / =−M - - - - - for + ve B.M Slope θ = dy / dx radians EIθ = EI. dy / dx = ∫ Deflection = y, EIy = ∫ ∫ = − dM / dx = Shear force +F EI y / = dF / dx = Load + Z EI y /

x

x x x x x

Points to be remembered

When multiple loads act simultaneously on a beam, all of the loads contribute to deflection. The principle of superposition permits the deflections at a point to be calculated as the sum of the deflections from each individual load acting individually. Superposition can also be used to calculate the shear and moment at a point and to draw the shear and moment diagrams. This principle is valid as long as the normal stress and strain are related by the modulus of elasticity, E. Generally this is true when the deflections are not excessive and all stresses are kept below the yield point of the beam material.

Quick Refresher Guide

Strength of Material

=

Maximum Deflection

W C

b

B

B

C

B

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A

a

= −

= − Slope

W

=−

A

Maximum Bending Moment

SL No. 2: Cantilever subjected to point load on its span

A

SL No. 1: Cantilever subjected to point load at free end

3.4.4 Slope and deflection of beams

∴

Consider cantilever beam AB. Let ‘C’ be an intermediate point. Then the deflection at ‘C’ due to a point load ‘P’ at B say , is equal to deflection at ‘B’ due to a point load ‘P’ at C i.e.,

3.4.3 Maxwell’s Law of Reciprocal Deflections:

A = B.M.D area between fixed end and point under consideration. = distance of C.G. of M/EI from point under consideration.

Quick Refresher Guide

= −

=−

where W =

= −

= −

(total load on the cantilever)

B

(3 − )

Strength of Material

Maximum Bending Moment

a

C

-

= −

B

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A

w/unit run

SL No. 4 Cantilever subjected to uniformly distributed load up to a certain length from fixed end

Maximum Deflection

Slope

Maximum Bending Moment

A

w/unit run

SL No. 3: Cantilever subjected to uniformly distributed load.

= −

=−

Slope

Maximum Deflection

=−

Maximum Bending Moment

Quick Refresher Guide

= −

= +

=− +

where W =

( − ) =−

= −

( −

Strength of Material

a

= −

=−

=−

(3

(

B

−4

−

B

)

+

+

)

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Maximum Deflection

Slope

= −

= − ( − )

= −

( − )

Maximum Bending Moment

A

SL No. 6 Cantilever subjected to a couple at free end

Maximum Deflection

Slope

Maximum Bending Moment

A

w/unit run

SL No. 5 Cantilever subjected to uniformly distributed load up to a certain length from free end

Maximum Deflection

Slope

Quick Refresher Guide

C

Slope

Maximum Bending Moment

/

(=

)=−

= +

=−

=

B

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Maximum Deflection

A

/

W

= −

=−

Slope

Maximum Deflection

= −

Maximum Bending Moment

A

run

SL No. 8 Simply supported beam subjected to point load at centre.

/ B

Strength of Material

SL No.7 Cantilever subjected to linearly varying load up to a certain length

Quick Refresher Guide

= −

=

=−

= +

=

=

= +

=−

where W =

B

/

=+

=−

( total load on the beam )

√

B

Strength of Material

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Slope

Maximum Bending Moment

A

w/unit run

= =−

SL No. 10 simply supported beam with uniformly distributed load

Maximum Deflection

Slope

Maximum Bending Moment

A

W

SL No.9 Simply supported beam subjected to point load on its span

Quick Refresher Guide

= −

C

Maximum Bending Moment

∙

=

w/unit run

B

( at x = 0∙519 from A )

= −

= +

√

B

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A

= = −

SL No. 12 simply supported beam with linearly varying loads as shown

Maximum Deflection

Slope

Maximum Bending Moment

A

=−

Strength of Material

w/unit run

SL No.11 Simply supported beam with linearly varying load as shown

Maximum Deflection

Quick Refresher Guide

)=−

+ −

Downward

+

Deflection : upwards

Counter- clockwise

Slope: Clockwise

(=

= +

= −

−

Strength of Material

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Sign conventions used :

Maximum Deflection

Slope

Quick Refresher Guide

Strength of Material

T

= T

=

(

)

=

=

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In SI system : Power (P) is measured in watts (W)

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1. Plane normal sections of shaft remain plane after twisting.

3.5.2 Torsion of shafts

Angle of Twist,

The torsional which produces unit twist per unit length.

or

= inner diameter.

=

Torsional Rigidity: GJ, unit: kg.

= Outer diameter,

For a hollow circular shaft,

For a solid circular shaft,

=

= : ℎ As the value of Torsional modulus increases, the Torsional strength increases. For E.g.: A hollow circular shaft compared to that of a solid shaft of same area, will have more torsional strength.

=

Distribution along vertical

The stress setup at any point in a cross section is one of pure shear or simple shear. The longitudinal axis is neutral axis. The shear stress will vary linearly from zero at the centre to maximum at the outer surface (any point on periphery)

Note:

Torsional Section Modulus: x

Strength of Material

Torsion is uniform along the shaft Material of the shaft is homogeneous, and isotropic. Radii remain straight after torsion. Stress is proportional to strain i.e., all the stresses are with in elastic limit.

Power Transmitted by a Shaft:

for Hollow circular shaft

for solid circular shaft

=

x x x

2. 3. 4. 5.

Quick Refresher Guide

Assumptions:

=

J = Polar moment of inertia =

L = Length of shaft

R = Radius of shaft

= Maximum shear stress due to torsion

T = Twist of cross section

Where T = Torque applied

T

Shaft Transmitting Torque or power. L beams Portico beams Curved beams Close coiled springs.

Torsion formula:

x x x x x

e.g.:

If moment is applied in a plane perpendicular to the longitudinal axis of the beam (or) shaft, it will be subjected to Torsion.

3.5.1 Torsion

Part3.5: Torsion

Quick Refresher Guide

H.P.

= =

( )

+

=

OR

+

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=

Both the parts carry same Torque i.e., = Total angle of twist at fixed end is sum of separate angles of twist of two shafts.

Shafts in Parallel:

x x

For such shaft,

T

.

= (

+

)

2

2

2

− (

+ (

=

2 =

=

=

=

=

/4) +

/4) +

=

=

=

16

16

(

(

− 2

−

+

=

16

+

+

)

)

+

+

+

+

=(

= 1.7

ℎ

= /

ℎ

)

ℎ

.

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) /

When the areas of solid and hollow sections are equal,

For e.g. If K = 0.6,

x

Comparison of Hollow and Solid Shafts:

1 = ( 2

Equivalent Bending Moment: The bending moment to produce the maximum bending stress equal to greater principle stress ‘ ’.

=

Equivalent Torque: It is the twisting moment, which acting along produce the maximum shear stress due to combined bending and Torsion.

Principle stresses are,

=

Shear stress, =

( )

Strength of Material

Let a shaft be subjected to a bending moment of ‘M’ and twisting moment ‘T’ at a sector.

Combined Bending and Torsion:

=

If both the shafts are of same material

. .,

Now bending stress,

/

Quick Refresher Guide

If the Torque ‘T’ is applied at the junction of two shafts and resisting Torque at their remote ends, the shafts are said to be connected in parallel. For such a case, x = ; x T= +

Shafts in Series: If the driving torque is applied at one end, and the resisting torque at the other end, the shafts are said to have been connected in series.

=

Strength of Material

Composite Shafts: When two dissimilar shafts are connected together to form one shaft, the shaft is known as composite shaft.

Diameter of shaft,

Design of Shaft: To be safe against maximum permissible shear stress.

Where T = average torque in kg.m

=

1 metric ‘H.P.’ = 746 watts ≅ 0.75 kW

1 watt = 1 Joule / sec = 1 N. m/sec

where T = Average Torque in kN-m, N in rpm

. .

Metric System

=

=

Quick Refresher Guide

The ratio of the weight of a hollow shaft, and solid shaft of equally strength is 1− = (1 − ) /

x

each turn is practically a plane at right angles to the axis of helix stresses in the material are due to ‘Pure Torsion’ Bending couple is negligible axial force need not be considered at a section.

+

=

1+

+

…… (1)

)=

=

+

.

= 64

/

, deflection

= =

/ =

/64

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Stiffness of Spring: Load required to produce unit deflection.

Twist

Twist and deflection of free end:

Max. =

If m is small, then maximum shear stress can be calculated by A.M. Wahl’s formulae that takes account for initial curvature of the spring wire:

∴

16

If m is large, the effect of direct shearing force may be neglected.

Spring Index: (m) 2R/d,

R = Radius of coil, d = dia of circular wire or rod.

=

Maximum Shear Stress = (

Stresses at a section of a rod: A section of a rod is subjected to direct shear force (W) and a Torque (T = WR)

x x x x

Assumptions:

3.5.3 Close coiled helical spring subjected to Axial Pull (W)

When radius of solid shaft is equal to external radius of hollow shaft, =1−

Strength of Material

x

Quick Refresher Guide

Strength of Material

x

x x x

+τ

-τ

y

τyx σy

σy

σave

(σx , -τxy )

τxy

2y

V x V y

B

τyx τxy

and

A

2

§ V x V y · 2 ¨ ¸  W xy © 2 ¹

R

x

σ

R

2

W xy

V x V y

2

§ V x V y · 2 ¨ ¸  W xy © 2 ¹

x y-face coordinates: ( V y , W xy )

x-face coordinate: ( V x ,W xy )

(σx , τxy )

σx

R

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+τ (meaning counterclockwise around the cube) is downward - τ (meaning clockwise around the cube) is up on the axis A rotation angle of θ on the stress cube shows up as 2θ on the circle diagram and rotates in the same direction. The largest and smallest values of σ are the principle stresses, σ 1 and σ2. The largest shear stress, τmax is equal to the radius of the circle, R. The center of the circle is located at the value of the average stress, σave If σ1 =σ2 in magnitude and direction (nature) the Mohr circle will reduce into a point and no shear stress will be developed.

Notes:

σx

a Ÿ V ave

Compare the stress cube to a circle created using the circle offset

Mohr's circle gives us a graphic tool by which, we can compare the different stress transformation states of a stress cube to a circle. Each different stress combination is described by a point around the circumference of the circle.

3.6.1 Mohr’s Circle

Part 3.6: Mohr’s circle

Quick Refresher Guide

Strength of Material

σx + σy = σ1 +σ2

The summation of normal stresses on any two mutually perpendicular planes remains constant.

σ1=+ τ , σ2= -τ

If the plane contain only shear and no normal stress (pure shear), then origin and centre of the circle will coincide and maximum and minimum principal stress equal and opposite.

=

=

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Longitudinal stress =

Hoop stress or circumferential stress =

Cylindrical shells:

3.6.2 Applications: Thin-Walled Pressure Vessels

x

x

Quick Refresher Guide

=

=

σ pd = 8t 2

)

4 2

= 8

Strength of Material

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τ

Hoop stress = longitudinal stress = σ = σ =

Spherical shells:

(

=

Quick Refresher Guide

Strength of Material

³

FdL = AL

³ VdH

P2 L 2 AE

2E

V2

y dA =

(dA)2 dx/2(dA)E

P

dx

U=

N.A.

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M dx 2EI

Therefore, for the entire length of the beam we obtain:

dU =

dU = P2dx/2AE = M2y2/

y

In the beam shown in the figure consider a differential element isolated by two transverse sections at a distance dx apart. Treating this element as an axially loaded bar, where P = VdA = (My/I)dA, the energy stored in it is

3.7.2 Elastic Strain Energy in Flexural Loading

u

U AL

The strain energy per unit volume is

U

1 PG 2

For an axially loaded member below the proportionality limit, the total strain energy is given by,

Work per unit volume corresponds to the area under the stress-strain curve.

Work per volume =

T dx 2GJ

∂U ∂P

=σ

1+ 1+

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For sudden loading, free fall ‘h’ does not exist i.e., h = 0. i.e., a suddenly applied load (dynamic condition), produced a deflection which is twice as great as that obtained when the load is applied gradually.

σ

Also the stress due to gradually applied load may be applied by the impact factor to obtain the maximum stress:

This ratio is called as the impact factor.

δ 2h =1+ 1+ δ δ

The ratio of the maximum dynamic deformation G to the static deformation Gst can be given by the equation

The problem of impact is analogous to that of a falling body stopped by spring. Let us consider a free falling body of mass ‘m’ from a height h that produces a deflection G in the spring. Relationship between dynamic and static deflection can be obtained by equating the resultant work done to the zero change in kinetic energy.

3.7.5 Impact or Dynamic Loading

Interpretation: The partial derivative of the strain energy with respect to one of the external loads equals the displacement of the point of application of load in the direction of that load.

δ=

It states that the deflection caused by any external force is equal to the partial derivative of the strain energy with respect to that force.

3.7.4 Castigliano’s Theorem

U=

When the torque varies the result may be applied over a segment of length dx and integrated over the length of the bar to obtain

³

Work = force x distance = FdL

=

For a circular bar of constant cross section, the strain energy stored in the body is equal to the product of average torque and the angular deformation; that is

3.7.3 Elastic Strain Energy in Torsional Loading

Quick Refresher Guide

U = 1/2 TT = T

Strength of Material

Strain energy, also known as internal energy per unit volume stored in a deformed material. The strain energy is equivalent to the work done by the applied force. Simple work is calculated as the product of a force moving through a distance.

3.7.1 Elastic Strain Energy in Uniaxial Loading

Part 3.7: Strain Energy Methods

Quick Refresher Guide

Columns and Stanchions Struts Beam Beam column

: : : :

Vertical compression members in building compressions members in roof trusses Jib of a crane. Co-beam that is acted on by an axial compressive force in addition to transversely applied loads.

Strength of Material

I/A

X

Y For a given area, Tubular section will have maximum radius of gyration. H-Section is more efficient than I-Section.

X

Radius of gyration will be least along major axis of cross section. e.g. for a rectangular column along yy−axis Y

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Equilibrium of a column: A column is said to have buckled or failed when it reaches “Neutral Equilibrium”.

x x

x

As slenderness ratio increases, permissible stress or critical stress reduces, consequently, load carrying capacity also reduces.

Slenderness Ratio: Effective length/least radius of gyration.

Radius of gyration: r =

Long columns: Long columns will buckle in the transverse direction that has the smallest radius of gyration. Buckling failure is sudden, often without significant warning. If the material is wood or concrete, the material will usually fracture (because the yield stress is low); however, if the column is made of steel, the column will usually fail by local buckling, followed later by twisting and general yielding failure. Intermediate length columns will usually fail by a combination of crushing and buckling.

∴ Crushing load P = f . A, f = ultimate crushing stress.

Short Column: Short columns, called piers or pedestals, will fail by compression of the material. These columns fail essentially by direct crushing at ultimate load.

x x x x

Definitions:

3.8.1 Columns & Struts

3.8: Columns & Struts

Quick Refresher Guide

Strength of Material

S2 EI L2 --------- (1)

Pcr A §L· ¨ ¸ ©r¹

2

S2 E ≤ --------- (2)

Y

X X

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X

Y

The smallest force at which a buckled shape is possible. Prior to this load the column remains straight. The columns buckle in the plane of the major axis of the cross section as shown below.

Most columns have two radii of gyration, rx and ry, and therefore, have two slenderness ratios. The largest slenderness ratio will govern the design.

The quantity L/r is known as the slenderness ratio. Long columns have high slenderness ratios. The smallest slenderness ratio for which Eq. (2) is valid is the critical slenderness ratio, which can be calculated from the material’s yield strength and modulus of elasticity. Typical slenderness ratios range from 80 to 120. The critical slenderness ratio becomes smaller as the compressive yield strength increases.

L is the longest unbraced column length. If a column is braced against buckling at some point between its two ends, the column is known as a braced column, and L will be less than the full column height.

V cr

The corresponding column stress is given by the equation shown below. This stress cannot exceed the yield strength of the column material.

Pcr

Critical load: The load at which a long column fails is known as the critical load or Euler load. The Euler load is the theoretical maximum load that an initially straight column can support without transverse buckling. For column with frictionless or pinned ends, this load is given by Euler’s formula shown below.

3.8.2 Euler’s Theory of Buckling

Quick Refresher Guide

Strength of Material

Column is initially perfectly straight and is axially loaded. Section of column is uniform. The material is perfectly elastic, homogeneous, isotropic and obeys hook’s law. Length of column is very large compared to lateral dimension. Direct stress is small compared to bending stress corresponding to buckling condition. Self weight of column is ignorable. The column will fail by buckling alone.

L

L

L

L

L

L

2L

L/√2

L/2

I = actual length Eff. Length (L) L

π EI/L

π EI/4L

2 π EI/L

4 π EI/L

Critical load π EI/L

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5. One end fixed, at other end only lateral displacement and no rotation

4. One end fixed and other end free

3. One end fixed and other hinged

2. Both ends fixed

1. Both ends hinged

L = Eff. Length Boundary Condition

Effective length and critical loads for various boundary conditions compared to a column whose both ends are hinged.

Effective length of columns:

1. 2. 3. 4. 5. 6. 7.

Assumptions:

Quick Refresher Guide

L

7. One end fixed, at other end lateral displacement and partial rotation.

1.5L

2L

π EI/2.5L

π EI/4L

Strength of Material

i) The relation between slenderness ratio and corresponding critical stress is hyperbolic ii) According to Euler formulae the critical load does not depend upon strength property of material the only material property involved is the elastic modules ‘E’ which physically represents the stiffness characteristics of the material.

. ( / )

Where α = Rankine’s constant L = eff. Length σ = yield stress. Rankine’s Co-efficient: is independent of geometry and end conditions, can be modified to incorporate imperfections

P =

Basic Formula:

It is empirical formula Takes into account both direct crushing (Pc) load and Euler critical load (P ). 1 1 1 i. e. , = + P P P P .P ∴ P = P +P

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x x

3.8.3 Rankine’s formula

Note:

Euler’s formula is valid only up to proportional limit i.e., in inelastic zone, the formulae are not valid

As V and E are constant for a particular material, Euler’s formula is valid for a particular range of slenderness ratio, for e.g. for mild steel whose V = 3300 Kg/cm and E = 2.1 × 10 Kg/cm Euler formula is not valid for slenderness ratio less than 80.

Euler’s formula can also be written as σ = π E/(L/r)

Limitations of Euler’s formula:

L

6. One end pinned, at other only lateral displacement no rotation

Quick Refresher Guide

1/7500 1/9000 1/1600

Rankine’s Constant

Strength of Material

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Rankine’s formula is valid for any type of column No limitations for slenderness ratio.

3200 2500 5500

Mild steel Wrought Iron Cast Iron

x x

σ

σ π E

Material

α=

Quick Refresher Guide

Thermodynamics

Energy out

Mass out

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e.g. Gas compressed in a piston-cylinder assembly

Energy in

Energy out

Closed system: Energy transfer occurs across the boundary. No mass transfer across the boundary

Mass in

Energy in

Open system: Both energy and mass can transfer across the boundary e.g., Steam turbine, centrifugal pump.

Classification of system:

Surroundings

System

Boundary

Boundary: It separates system and surroundings

Surroundings: Everything external to the system.

Thermodynamic system is a quantity of matter or region in space considered for the analysis of a problem.

4.1.1 Thermodynamic systems

Part 4.1: Basic Thermodynamics

Part – 4: Thermodynamics

Quick Refresher Guide

Q

Gas

W

Thermal conductor

Piston

Thermodynamics

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The quasi – static process is an ‘infinite slow’ process.

Quasi – static process: The departure of the state of the system from the thermodynamic equilibrium is infinitely small.

a) Mechanical equilibrium b) Thermal equilibrium. c) Chemical equilibrium

Thermodynamic equilibrium should satisfy the following.

a) Intensive properties – independent of mass eg. Pressure, temperature, density. b) Extensive properties – depends on the mass of the system eg. Volume, energy etc.

Types of thermodynamic properties

Process: When path is completely specified, the change of state is called process.

4.1.2 Thermodynamic Processes

Path of change of state: The succession of states passed through during a change of state.

Change of state: Any operation in which one or more of the properties of the system changes.

Thermodynamic state: All the properties have definite values.

Thermodynamic property: Any characteristic of a system by which its physical condition can be described, eg. Pressure, temperature, volume, etc.

Mass Transfer X Energy Transfer X

Isolated system: Neither mass nor energy transfers across the boundary e.g. Universe

Cylinder

Quick Refresher Guide

2

1 W = ∫ pdv

2

1 V

V =

2

W=∫

T=C

V=C

Thermodynamics

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PV = constant

d) Reversible Adiabatic or Isentropic process:

P

1

c) Isothermal (constant temperature) process:

P

2

P=C

All are in thermodynamic equilibrium

V b) Constant volume process or Isochoric process :

P

V processes): Thermodynamic processes (non-flow a) Constant pressure or Isobaric process:

P

1

Quick Refresher Guide

V

P

=C

2

V

P

2

=C

V=c

V

PV = C

PV = C

T =C < 1.4)

Index of expansion (n) ∞ 0 1 1< n < 1.25 (= 1.4 )

(1 <

Thermodynamics

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Thermodynamic Process Constant volume (V = C) Constant pressure (P = C ) Isothermal (T = C) Polytrophic (PV = c ) Reversible adiabatic (PV = C)

P

P =C

Representation of thermodynamic processes on P – V diagram:

n=

P

1

PV = const n – index of expansion

e) Polytropic process (generalized process)

P

1

Quick Refresher Guide

If t = t & t = t Then t = t

C

B

As per FLTD, heat (Q) and work (W) are mutually convertible 100kJ of ‘Q’ 100 kJ of ‘W’ 100kJ of ‘W’ 100kJ of ‘Q’

U- internal Energy

W= work done

FLTD is postulated by J.P. Joule It is law of conservation of energy (energy can neither be created nor be destroyed) Energy is of 2 types 1. Energy in transit 2. Energy in storage e.g. Heat & work e.g. Internal energy For a closed system undergoing a cyclic process, FLTD states that ∮ δQ = ∮ δW For a closed system undergoing non cyclic process, FLTD: δQ = δW + ∆U For a cyclic process ∆U = 0 (i.e.: U = constant) Note: Q – heat supplied/liberated

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Heat liberated from the system (-ve)

Heat supplied to the system (+ve)

Sign convention of heat and work:

-

Thermodynamics

Definition: When a body A is in thermal equilibrium with a body B, and also separately with a body C, then B and C will be in thermal equilibrium with each other. ZLTD is the basis for temperature measurement A reference body used for quantitave measurement of temperature is called thermometer A certain physical characteristic of thermometer which changes with change in temperature is called thermometer property.

First law of thermodynamics (FLTD):

-

-

Zeroth law of thermodynamics (ZLTD):

4.1.3 Zeroth law, First law & Second law of thermodynamics

Quick Refresher Guide

Q

W

W → (−ve) W → (+ve)

Thermodynamics

V

x x

x x

2

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Steady flow – properties of the system are constant with respect to time. Flow energy or flow work: work done by the fluid on itself to cause the fluid flow. Flow work = PV kJ Flow work is a point function Enthalpy (H): H= (U+PV) kJ Where, U – internal energy(kJ) PV – flow work (kJ). Specific enthalpy, h= (u+pv) kJ/kg Where u – specific internal energy (kJ/kg

FLTD for a steady flow process

P

1

Q − W = U − U

FLTD for a non cyclic process (non-flow process)

W

PMM1

Q=0

Perpetual motion machine of first kind (PMM1)is a fictitious machine which gives continuous output without any input. It violates FLTD

W

Work done on the system (-ve)

Q Work done by the system (+ve)

Q → (+ve) Q → (−ve)

Quick Refresher Guide

,

,

,

,

(3)

(1)

Fig. SFEE

, , ,…. ṁ

=∑

, , ,…. ṁ

ṁ =

=

= [Total Energy ]

=

̇ ,

̇ ,

,

,

,

Control Surface

,

Control Volume

Thermodynamics

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[Total Energy ]

Energy balance:

ṁ =

(4)

(2)

(From Fig. SFEE)

C = velocity (m/s) v = specific volume (m /kg) = density (m /kg)

Where A = Cross-sectional area (m )

Continuity equations:

ṁ → mass flow rate at outlet

ṁ → mass flow rate at inlet

∑

In general,

ṁ + ṁ = ṁ + ṁ

ṁ = constant

Mass balance:

SFEE continued

ṁ → mass flow rate(kg/s) P → density (kg/m ). C → velocity (m/s) T → temp(℃ or 0K)

̇ ,

̇ ,

Steady Flow Energy Equation (SFEE):

pv = flow work per unit mass (kJ/kg)

Quick Refresher Guide

+

w =

q =

̇

̇

̇

̇

and

(i)

−

+

q =h −h

Boiler

HPW(

HPS(

)

)

+

+

+

+

+

(

(

+

−

−

+

(2)

(2)

)

)

- - - - - SFEE

- - - - - SFEE

datum

Thermodynamics

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q

Boiler Condenser

Applications of SFEE

Where

+

−

+

= Ẇ + ṁ

( q̇ − ẇ ) = ṁ[(ℎ − ℎ ) +

+

+

−

+

( Q̇ − Ẇ ) = ṁ[(ℎ − ℎ ) +

( Q̇ − Ẇ ) = ṁ

Q̇ + ṁ

(1)

(1)

Quick Refresher Guide

(ii)

)

)

−

)

HPF(

)

HPF(

=

)

Turbine

−

LPF(

LPW – Low Pressure Water LPS – Low Pressure Steam

)

Thermodynamics

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q =0 C ≈C Z ≈Z HPF – high pressure fluid LPF - low pressure fluid.

=

LPF(

Compressor

Turbine/Compressor

W =0 C ≈C Z ≈Z HPW – High Pressure Water HPS – High Pressure Steam

q =h −h

Condenser

LPW (

LPS(

Quick Refresher Guide

,

h >h C
(2)

(2)

h C

C =

2C (T − T )

(1)

(1)

Diffuser

(h − h ) = (C − C ) gain in enthalpy = drop in KE

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(2)

(2)

Thermodynamics

(h − h ) + (C − C ) = 0

,

,

where C = exit velocity, m/s (h − h ) = enthalpy drop, J/kg In general, C <<< C C = 2(h − h ) m/s = 2000(h − h ) where h and h are given in kJ/kg C = 44.7 (h − h ) for a gas nozzle h = C (T − T ) where C = specific heat, k

(C − C ) = (h − h ) i.e. gain in kE = drop in enthalpy exit velocity, C = C + 2(h − h )

SFEE: For Nozzle (h − h ) + (C − C ) = 0

q =0 w =0 Z ≈ Z

(1)

Nozzle /Diffuser Nozzle (1)

(i) SFEE for a throttling process: q =0 W =0 C ≈ C Z ≈Z SFEE: h −h =0 h =h Throttling process is also called is isenthalpic process.

(iii)

Quick Refresher Guide

h ,C

(2) 0

(1)

Thermodynamics

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Kelvin Planck Statement: It is impossible for a heat engine to produce net work in a complete cycle if it exchanges heat only with bodies at a single fixed temperature.

Also called as “law of degradation of energy”

Second law of thermodynamics:

(∆h) = (∆h) ̇ ( − )) ṁ C (t − t )= m (C ṁ , ̇ → mass flow rates of hot and cold fluids respectively C , C → spacific heats t , t → temperatures

, ̇

, ̇

(vi) SFEE for a heat exchanger q =0 W =0 Z ≈Z SFEE: h − h = 0 Increase in enthalpy of cold fluid = decrease in enthalpy of heat fluid

q =0 C ≈ C h ≈ h W = g(Z − Z ) i.e. work input = increase in P.E

(1)

(2)

(ii) SFEE for a water pump:

Quick Refresher Guide

(

>

)

Heat pump (HP)/refrigerator The performance of heat pump or refrigerator is represented by its COP (coefficient of performance)

Q

HP/K+1

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x

Impossible

Q ≠ i.e. (W
x x x

W

=

HE

Thermodynamics

η

Heat Engine

Impossible

Quick Refresher Guide

Thermodynamics

2′′′

1

(a)

′′ ′′′ (d) ′ (c) ′′ 2 ′

′′′

CP

V

′

′′′

3 (b)

′′

′′′

′ 4

′′ (e)

(f)

-

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At critical point, change in enthalpy, change in specific volume etc. are zero At critical point (for water) pressure, p = 220.8 bar Temperature, T = 374.14 ℃

Critical point: Water changes its phase directly to vapour with no distinction between liquid and vapour phases.

CP --- Critical Point.

3, 3’, 3”, 3’’’ ------ saturated vapour states

2, 2’, 2”, 2”’ ------- saturated liquid states

(a) - Saturated liquid curve (SLC) (b) - Saturated vapour curve (SVC) (c) - vapour dome (d) - under cooled liquid. (e) - Super heated vapour zone (f) - Gaseous Zone

T

Phase diagrams:

a) Change in temperature without phase change – sensible heat b) Change in phase at constant temperature – latent heat

Heat supplied: it causes

E.g. Water – steam mixture Atmospheric air Combustion Products of a fuel.

A substance that is homogeneous and invariable in chemical composition in all of its three phases (solid, liquid and gases) is called pure substance.

4.2.1 Properties of pure substances

Part 4.2: Properties of pure substances

Quick Refresher Guide

Thermodynamics

h

s

=

= .

Constant dryness fraction lines (quality lines)

Constant pressure lines

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a) Wet steam: (iii) = V = specific volume of dry saturated steam (directly available from steam tables ) = dryness fractio (iv) h= h + xh

Properties for pure substance (water)

a) Pressure entry b) Temperature entry

Steam Tables: Two types

Sublimation: Solid directly converts into vapour.

– specific Volume(m /kg)

s – entropy (kJ/kg K )

h – enthalpy (kJ/kg)

Mollier Diagram:

where m = mass of vapour m = mass of liquid 0≤ ≤1 x = 0 → 100% liquid x = 1 → 100% vapour

=

Dryness fraction: Wet steam characterized by dryness fraction.

For water, triple point is T = 273.16K P = 4.587mm of Hg

Triple point: the state at which all three phases solid, liquid and gas exist in equilibrium is called triple point.

Quick Refresher Guide

s=s +xs

= V −T )

)

Thermodynamics

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where = superheated steam temperature, Kelvin T T = saturated steam temperature, Kelvin C = specific heat of steam kJ/kg k Note: internal energy, u = (h –PV) kJ/kg

s = s + C ln

h = h +C (T

V=V

b) Superheated steam (

h , s ⟶ for dry saturated steam

s = s − s

h = h −h

h , s ⟶ for saturated liquid

where

(v)

Quick Refresher Guide

= lost work or lost heat = Irreversibility (I)

T

T0

S

(U E)

(A E)

.

Thermodynamics

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* Obeys all ‘Perfect gas Laws’ at low pressure and high temperatures.

Ideal gas: * Inter molecular forces are negligible

4.3.2 Ideal or perfect gases

Anergy: Minimum heat losses that are to be suffered during energy conversion process.

UE = T (∆S)

Unavailable Energy (UV) or ANERGY:

AE=Q 1−

AE=Q

Maximum work that can be obtained from the given heat source.

Available Energy (AE) or EXERGY:

I = W − W – for compressor/ refrigerator

I = W − W − for heat engines

Irreversibility: (I)

TδS

= 0 → reversible process

> 0 → irreversible process

≥ 0,

irreversible process

δS

TδS

= entropy generated in the process

Tds –

δS

δQ =

δW = PdV − TδS

δW = PdV reversible process δQ = Tds

4.3.1 Availability and irreversibility

Part 4.3: Irreversibility & Availability

Quick Refresher Guide

u

h

Thermodynamics

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Where ⃖R = universe gas constant. Its value is constant for all the gases.

PV = nR⃗T - - - - - - - - - (2)

Where R = characteristic gas constant. Its value changes from gas to gas.

PV = mRT - - - - - - - - (1)

Universal Gas Constant:

Avogadro Number, A = 6.023 × 10

h= specific enthalpy (J/kg) (e) Avogadro’s hypothesis: It states that equal volumes of different gases at same pressure and temperature contain equal no of molecules. - At NTP 22.4136 of any perfect gas has its mass equal to its molecular weight in kg. - In a gram of perfect gas, there are 6.023 × 10 molecules.

where u = specific internal energy (J/kg)

T

(d) Joule’s Law: u = f(T) only h = f(T) only

PV = mRT (c) Regnaut’s Law: C & C are constant for a given gas C – specific heat at constant pressure C – specific heat at constant volume

(a) Boyle’s Law: V ∝ at T = const (b) Charles’s Law: V∝T at P = const V ∝ ⟹ = const = R PV = RT - characteristic gas equation P – Pressure (Pa) v – Specific volume ( / ) R – characteristic gas constant (J/kg K) T – Temperature (K) For ‘m’ kg of gas

Perfect gas Laws:

Quick Refresher Guide

kJ

=

Z<1

Thermodynamics

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II.

Inter molecular forces are repulsive

Where, ̅ = molar specific volume = If Z = 1 , P ̅ = RT ⟶ a perfect gas Z ≷1 , ⟶ a real gas I. Z>1 > ̅ ̅

a) Compressibility factor, Z =

4.3.3 Real gases

γ = 1.4 for diatomic gases = 1.67 for monoatomic gases = 1.33 for triatomic gases. [γ] > [γ] > [γ]

Relation between C , C & C − C =R C =γ C J – adiabatic index C = C =

∆U = m C (T − T )

∆ H = mC (T − T )

-

Change in internal energy

and C =

Change in enthalpy

C =

Units:

kJ kg K or kg ℃ For solids – only one specific heat For liquids – only one specific heat For gases – two specific heats 1. C (specific heat at constant pressure) 2. C ( specific heat at constant volume)

C=

Specific heat (C):

R⃗ = MR – Relation between R and R⃗

In equation (2) n = no. of moles =

R⃗ = 8.314 kJ/kg mol K

Quick Refresher Guide

b → another constant. It called as co – volume. The values of a & b have been obtained using kinetic theory of gases.

Thermodynamics

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x

̅ < ̅ Inter molecular forces attractive In general 0.95≤Z ≤1.05 a) Vander waal’s equation : [V − b] = RT P+ a → is a constant. It is called forces of cohesion.

Quick Refresher Guide

V

1

V

V=C

Thermodynamics

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∆U = mC (T − T )

W = ∫ pdv = 0

P

2

2

V T = V T

W = ∫ pdv

Constant volume process or Isochoric process:

Q = ∆U +W = mC (T − T ) + p(V − V ) Q = mC (T − T ) ∵

P

1

P=C

Work done, W = ∫ pdV = p (V − ) Change in internal energy ∆u = mC (T − T ) As per FLTD,

Constant pressure or Isobaric process:

Work and Heat calculations for various thermodynamic processes are:

Change in internal energy during any process ∆U = m C (T − T )

For any closed system, non- flow work is given by W = ∫ pdV, applicable for any process.

4.4.1 Work & Heat

Part 4.4: Work, Heat & Entropy

Quick Refresher Guide

=

P P

(

)

P

1

V

P

2

2

W=∫

P T = P T

T=C

Thermodynamics

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V T = T V

For an isentropic process = 0 =0 ∴∆ = Q–W=

W = ∫ pdV =

PV = constant

=C

V =

Reversible Adiabatic or Isentropic process:

∆U = mC (T − T ) = 0 P Q = mRT ln P =W

= mRT ln

W = ∫ pdV = mRT ln

P

1

Isothermal (constant temperature) process:

= mC (T − T )

Q = ∆ U +W

Quick Refresher Guide

P

=

P P

1

V

P

2

=C

Thermodynamics

4

V

3

2

=

T

S

3

2

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4

1

Net Work Output of the cycle Total heat input to the system

x Heat supplied to the system, q = T . ∆S 2-3: Isentropic expansion

1-2: Isothermal expansion

P

1

Carnot Cycle:

=

The efficiency of a heat engine is defined as

The function of a heat engine cycle is to produce work continuously at the expense of heat input to the system. So the net work done W and heat input Q are of primary interest.

A heat engine cycle is a thermodynamic cycle in which there is net heat transfer to the system and a net work transfer from the system. The system which executes a heat engine cycle is called a heat engine.

4.4.2 Heat Engine

Q ≠ 0 in polytropic process.

T V = T V

W = ∫ PdV = ∆U = mC (T − T ) Q = ∆U + W = mC (T − T ) +

n=

PV = const n – index of expansion

Polytropic process (generalized process)

Quick Refresher Guide

Work done by the system, W = C (T − T ) Heat rejected by the system, q = T . ∆s

=

Thermodynamics

Carnot cycle consists of 4 reversible processes. Carnot cycle is an ideal cycle. No other heat engine has more efficiency than heat engine working on carnot cycle

=

=

=

=

=η

) THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 121

(B) For equal efficiency (η

(A) For equal work output (W = W ), T =

η

η

Heat engines connected in series:

x x x

=

T − sink temp, K

T − source temp, K

In general representation

T (= T ) - Sink temp, Kelvin

Where T (= T ) − source temp, Kelvin

=

=

Work done on the system, W = C (T − T )

Carnot efficiency, η

x

4-1 : Isentropic compression

x

3-4 : Isothermal expansion

x

Quick Refresher Guide

T T

-

-

-

-

Thermodynamics

s − s = C log ( )

s − s = R log ( )

s −s ≠0 s −s =0

b) P = C

c) T = C

d) Adiabatic e) Isentropic (Reversible adiabatic)

> 0,

impossible

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∮ ≤ ds ≥ 0 ∵ δq = 0 for the universe [∆s] ⋛0 i.e. > 0 → irreversible process = 0 → reversible process < →impossible process

c) ∮ = 0, reversible Principle of increase of entropy ∮ ≤ 0,

b) ∮

Clasius inequality: It states that when a closed system undergoes a cyclic process, a) ∮ < 0, for irreversible cycle

change in entropy s − s = C log ( )

s

Process a) V = C

T

1

2

ds =

= ∫ Tds For closed system

= 0 for unit mass

⇒δq = T.ds

i.e, ∮

It is outcome of SLTD The entropy of the universe always increases and it represents the degree of irreversibility associated with the process. Entropy of the system increases upon heating and decreases upon cooling. Clasius theorem: When closed system undergoes a cyclic process. =0 R → Reversible ∮

4.4.3 Entropy

Important note: Always convert temperature to “Kelvin” units.

T =

Quick Refresher Guide

Thermodynamics

S

If a mixture of air-water vapour is cooled at constant pressure, the temperature at which water vapour begins to condense is called the dew point temperature. At dew point the partial pressure of water vapour in the mixture is equal to the saturation pressure of water. The composition of air-water vapour mixture is usually specified in terms of specific humidity or relative humidity.

T

P = constant

Suppose a mixture of air-water vapour which is not saturated is cooled at constant pressure the partial pressure of water vapour remains constant till it is equal to the saturation pressure of water. With continued cooling, the water vapour begins to condense. The constant pressure cooling of a mixture is represented on a T-S diagram in Fig.

= 0.622 (

/

) = 0.622 (

/( −

)

/

=

/0.622

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Adiabatic Saturation: Consider the steady flow of an unsaturated air-water vapour mixture through an insulated device as shown in fig. called adiabatic saturator. Assume the equilibrium is attained between the water and air-water vapour mixture in the device and hence saturated air-water vapour leaves the device.

=

Relative Humidity: Relative humidity (RH) is defined as the ratio of the partial pressure of the water vapour in the mixture to the saturation pressure (p ) of water at the mixture temperature.

P = Total pressure.

Where ṁ , ṁ = Mass of water vapour and dry air, respectively p , p = Partial pressure of water vapour and air in the mixture, respectively

= ̇ / ̇

Specific Humidity: Specific humidity (SH) or humidity ratio is defined as the ratio of mass of water vapour to the mass of dry air in the mixture.

x

x

x

x

4.5.1 Dew Point Temperature:

The science which investigates the thermal properties of moist air, considers the measurement and control of the moisture content of air, and studies the affects of atmospheric moisture on material and human comfort may properly be termed ‘Psychrometrics’.

Part 4.5: Psychrometrics

Quick Refresher Guide

3

(h

− h ) + SH (h − h ) (ℎ − ℎ )

+ ṁ

h

x

x

x

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A graphical representation of the solution of the adiabatic saturation relation is called psychrometric chart. The enthalpy of air-water vapour mixture is expressed on the basis of dry air and is given by h∗ = h + SHh That is h* represents the enthalpy of 1 kg dry air and the enthalpy of the accompanying water vapour. In the psychrometric chart, the enthalpies of air and water vapour are measured with reference to 0℃. Some psychrometric charts use 0℉ as reference state for air and 32℉ as reference state for water vapour.

Psychrometric Chart:

Thermodynamics

̇

=

,

̇

1

Control volume

Adiabatic mixer ̇

,

∗

∗

− −

ℎ ℎ

∗

∗

−ℎ −ℎ

∗

∗

RH

∗

Dry bulb Temp (℃ ℃)

=

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Dehumidification: If a mixture of air-water vapour is cooled at constant pressure, the specific humidity of the mixture does not undergo any change till the dew point temperature is reached, but its relative humidity increases. Further cooling results in condensation of water vapour and the specific humidity decreases. A schematic diagram of a dehumidifier is shown in Fig.

,

̇

The adiabatic mixing process is shown in Fig.

̇

These equations can be solved to obtain

Mass balance for air : ṁ + ṁ = ṁa Mass balance for water : ṁ SH + ṁ SH = ṁ SH Energy balance : ṁ h ∗ + ṁ h ∗ = ṁ h ∗

Adiabatic Mixing of Streams: Consider the steady flow of steams 1 and 2 into the adiabatic mixer shown in Fig. The mixture leaves the device as stream 3. Considering the device as a control volume, one can write the following material and energy balance equations

The field of air conditioning uses various processes such as heating, cooling, humidification and adiabatic mixing of air-water vapour mixtures. These processes can be easily analysed with the help of a psychrometric chart.

4.5.2 Applications of Psychrometry

Liquid water

Where, ṁ = mass flow rate of water vapour / water ṁ = mass flow rate of dry air; h = specific enthalpy of dry air; h = specific enthalpy of water vapour / water Subscripts 1, 2, 3 denote the conditions at the points shown in fig. If air is treated as an ideal gas, we can write (h − h ) = C (T − T ). Assume that liquid water enters the device at the same temperature as the air leaving the device. That is T = . Then, h − h = (h − h ) = (h ) and h − h = h − h Thus, C (T − T ) + SH (h ) SH = ℎ −ℎ x The specific humidity and relative humidity of an air-water vapour mixture can be measured with an adiabatic saturator. x For all practical purposes, the adiabatic saturation temperature (T3) does not depend upon the temperature at which liquid water enters the device x The adiabatic saturation temperature (T3) does not depend upon the temperature at which liquid water enters the device x The adiabatic saturation temperature (T3) depends only on the conditions (T1, SH1) of the entering air.

SH =

These equations can be solved to obtain

Mass balance of air: ṁ = ṁ = ṁ Mass balance for water: ṁ̇ + ṁ Energy balance: ṁ h + ṁ h + ṁ h = ṁ h

Quick Refresher Guide

The adiabatic saturation relation when expressed in terms of enthalpy reduces to h∗ = h∗ . That is during adiabatic saturation h* remains constant The lines of constant h* coincide with the lines of constant wet bulb temperature, because h∗ depends on T = (=T ) only. Once T is specified p is fixed because P = p and hence SH is fixed.

x

x

saturated air – water Vapour mixture ṁ , ṁ , T , SH

Thermodynamics

Consider the device as control volume and apply material and energy balances to get

Unsaturated air – water Vapour mixture ṁ , ṁ , T , SH

1

Quick Refresher Guide

2

Heating unit

Dehumidified air 4

*h

2

h∗

h∗ kJ/kg dry air

23

4

RH

RH

Dry bulb Temp. ( C)

*h

41

1

SH

SH

x

x

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The process of humidification with cooling is extensively used in evaporative coolers or desert cooers which are used for cooling homes in hot and dry climates. The rate at which water is evaporated in the evaporative cooler is given by ṁ = ṁ (SH − SH )

Humidification with Cooling: If an unsaturated air-water vapour mixture is made to flow through porous pads soaked in water, the mixture gets saturated. Since the process occurs without any energy exchange as heat with the surroundings, it is adiabatic. The energy required for the evaporation of water comes from air-water vapour mixture resulting in a decrease in its temperature.

x

x

x

Condensed water

Cooling unit

Heating coil

Thermodynamics

3 Dehumidification of air-water vapour mixture can be achieved by cooling the mixture below its dew point temperature, allowing some water to condense, and then reheating the mixture to the desired temperature For cooling the mixture, chilled water can be sprayed into the mixture or the mixture can be made to pass over cooling coils through which a cold refrigerant is circulated. The dehumidification process is represented on a psychrometric chart in Fig. below.

Humid air 1

Refrigerant

Quick Refresher Guide

Specific Humidity(SH) kgH O/kg dry air

̇

Dry air

Porous pad

Water ̇

Dry bulb

2

.(

1

)

Cool and humid air;

A schematic diagram of an evaporative cooler is shown in fig. And the process of humidification with cooling is shown on a psychrometric chart.

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Degree of Saturation: The water vapour exists at the dry bulb temperature T of the mixture and partial pressure P . Consider now that more water vapour is added in this Control volume V at temperature T itself. The partial pressure P will go on increasing with the addition of of water vapour until it reaches a value p corresponding to state 2, after which it cannot increase further as p is the saturation pressure or maximum possible pressure of water at temperature T. The thermodynamic state of water vapour is now saturated at point 2. The air containing moisture in such a state is called saturated air.

x

Thermodynamics

where, SH and SH denote the specific humidity of air-water vapour mixture at the inlet and outlet of the cooler, respectively.

Quick Refresher Guide

S

Saturated water Vapour in saturated

Air

3

Super heated water vapour in unsaturated Air

1

Thermodynamics

=

/ /

=

=

/ I= /

=

/

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- - - - - - - - - - - - - - - - - - - - (2)

=

I = ( /0.622) / ( / )

/

- - - - - - - - - - - - - - - - - - - - (1)

I=

= 0.622 I ( / )

It can be shown that

We have

P v = P v or P V = P V

Using the perfect-gas relationship between points 1 and 2,

,

I=

/ /

Relative Humidity: Relative humidity denoted by the symbol I or RH is defined as the ratio of the mass of water vapour m in a certain volume of moist air at a given temperature mass of water vapour m in the same volume of saturated air at the same temperature. Thus if V and V are the specific volumes of water vapour in the actual moist air and saturated air respectively at temperate T and in volume V, at points 1 and 2 respectively

=

1− 1−

The ratio of the actual specific humidity ω to the specific humidity ω of saturated air at temperature T is termed as the degree of saturation denoted by the symbol μ. Thus

ωs = 0.622 [P / (p − p )]

In this state the air is holding the maximum amount of water vapour (the specific humidity being ω , corresponding to the partial pressure p ) at temperature T of the mixture. The maximum possible specific humidity, ω at temperature T is thus

T

2

Quick Refresher Guide

Thermodynamics

B

(t − t )kJ/kg

ii) h = h = (h ) 0℃ + C

(t − 0)

. .

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h = 2501 + 1.88 t kJ/kg (2500 + 1.88 ) ℎ = 1.005 +

Thus, employing the second expression and taking the latent heat of vaporization of water at 0℃ as 2501 kJ/kg, we obtain the following empirical expression for the enthalpy of the water vapour part

i) h = h = (h )

Accordingly, enthalpy of water vapour at A, at DPT of t and DBT of t, can be determined more conveniently by the following two methods:

h = 4.1868 t + (h ) + 1.88 (t − t )

Taking the specific heat of liquid water as 4.1868 kJ/kg K) and that of water vapour as 1.88 kJ/kg K in the range 0 to 60℃, we have

= specific heat of superheated vapour

= specific heat of liquid water, t = dew point temperature

t + (h ) + C

(h ) = latent heat of vaporization at DTP, C

where C

h =h =C

Again taking the reference state enthalpy as zero for saturated liquid at 0℃, the enthalpy of the water vapour part, at point A is expressed as

S

Reference state

C A

Enthalpy of Moist Air: The enthalpy of moist air h is equal to the sum of the enthalpies of dry air and associated water vapour, i.e (h = h + ω h ) per kg of dry air, where h is the enthalpy of the dry air part and h is the enthalpy of the water vapour part. h = C t = 1.005 t kJ/kg

I = /[1 − (1 − ) / ]

From eqs 1 and 2 we get

Quick Refresher Guide

) + ℎ 0℃ ℎ=( + = + ℎ 0℃ = + where = (1.005 + 1.88 ω kJ/(kg d.a.) (K) Humid specific heat is the specific heat of moist air (1 + ω) kg per kg of dry air. The term C t governs the change in enthalpy of moist air with temperature at constant specific humidity, and the term ω (h ) 0℃ governs the change in enthalpy with the change in specific humidity, i.e. due to the addition or removal of water vapour in air. Since the second term 1.88 ω is very small compared to the first term 1.005, an approximated value of C of 1.0216 kJ/kg d.a.) (K) may be taken for all practical purposes in air-conditioning calculations.

Thermodynamics

h

t

*

t*

=( *-

) per kg of Dry air

Feed Water

h*

*

t*

Outlet air

∗/

)( ∗− )

Let us compare the expressions for the wet bulb temperature t’ and the temperature of adiabatic saturation t*, i.e. ∗= − ( / ) ℎ ( ∗ − ) ∗= − (ℎ ∗ / ) ( ∗ − ) It follows that if k /f = (1/C ) Or f /k C = L = 1 = (α/D) /

∗= − (ℎ

ℎ + ( ∗ − )ℎ ∗= ℎ ∗

Therefore, since the process is strictly adiabatic, we have by energy balance

Inlet Air

Adiabatic Enclosure

For any state of unsaturated moist air, there exists a temperature t* at which the air becomes adiabatically saturated by the evaporation of water into air, at exactly the same temperature t* The leaving air is saturated at temperature t*. The specific humidity is correspondingly increased to ω*. The enthalpy is increased from a given initial value h to the value h*. The weight of water added per kg of dry air is ω* - ω which adds energy to the moist air of amount equal to ω*- ωh ∗, where h * is the specific enthalpy of the injected water at t*.

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x

x

x

Thermodynamic wet bulb temperature or temperature of adiabatic saturation:

x

x

x

Humid Specific Heat:

Quick Refresher Guide

Adiabatic Mixer

4

t

or

ω = [(ṁ ω + ṁ ω )/(ṁ

+ ṁ )] − ω

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ω =ω −ω

If may be noted that due to condensation, the specific humidity of the mixture ω , will be reduced to below ω . Correspondingly, the temperature of the air would be increased to t from t due to the release of the latent heat of the condensate. Now, if ω represents the mass of the condensate per unit mass of the mixture, we have by moisture and energy balance

It the DBT of the mixture falls below 0℃, the condensate may eventually freeze.

2

3

4

relative humidity, there is a possibility of condensation of water vapour, the mixture will then consist of saturated air and the condensate.

2

1

1

Mixing with Condensation: When large quantity of cold air mixes with a quantity of warmer air at a high

x x

x

Thermodynamics

Then t’ = t* i.e. the two temperatures are equal. The dimensionless quantity f /k C is called the Lewis number. The air and water vapour mixture at low pressures, this number is approximately equal to unity (L = 0.945). The measurable wet bulb temperature is equal to the thermodynamic wet bulb temperature. For any other kind of gas and vapour mixture these would not be the same In the case of air and water vapour mixture, the two temperatures are exactly the same.

Quick Refresher Guide

h = [(ṁ

h + ṁ

h )/ (ṁ

ṁ h + ṁ h = ṁ h

+ ṁ )] ω h

Thermodynamics

−

)=

(1.005 + 1.88 )(

(

−

(

)

−

)+

(

−

)

m denotes the mass flow rate of dry air. Generally the flow rate of dry air is measured in terms of cubic meters of air per minute (cmm). Then the mass flow rate of dry air can be calculated from m =Q ρ where Q is the volume flow rate of air. Expressing this in cmm, we have m = (cmm)ρ/60 kg d. a./s For the purpose of calculation, standard air is taken at 20℃ and 50 percent RH. The density of standard air is approximated to 1.2 kg/m d.a. The value of humid specific heat is taken as 1.0216 kJ/ (kg d.a.) K. we obtain Q = [(cmm)(1.2)(1.0216)/ 60] ∆t = 0.0204 (cmm) ∆t, kW

x

x

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When the state of air is altered along the t = constant line, such as BC moisture in the form of vapour has to be transferred to change the humidity ratio of the air. (ω − ω ) This transfer of moisture is given by =

Latent Heat Process-Humidification or Dehumidification:

x

x

where C is the humid specific heat. This heat, denoted by the subscript S, is called the sensible heat. If a building to be air conditioned r receives or loses heat due to transmission or other reasons, it is supposed to have sensible heat load.

=

=

Q = m (h − h )

t

Sensible Heat Process-Heating or Cooling:

Where h is the enthalpy of the condensate at temperature t of the mixture. The two variables to be solved are t and ω . By assuming different values of t and substituting for ω , h and h , the two equations can be solved by trial and error to obtain he final state after mixing.

or

and

Quick Refresher Guide

=

−

B

C

= −( +ℎ +ℎ ) = ℎ ( − ) = ℎ = 2500 If the building gains or loses moisture, it is supposed to have a latent-heat load. A gain of moisture will require the condensation of moisture for the dehumidification of air in the conditioning apparatus, and hence a cooling load. On the other hand, a loss of moisture will necessitate the evaporation of water for the humidification of air in the apparatus and hence a heating load. Q = [(cmm)(1.2)(2501)/ 60] ∆ω )∆ , = 50 (

The change in temperature causes a sensible heat load given by = m C (t − t ) Q = m (h − h ) The change in the humidity ratio causes a moisture transfer given by = m (ω − ω ) And a latent heat load given by Q = m (h − h ) = m h + (ω − ω ) Q = Q + Q = m (h − h ) = m [ C (t − t ) + h (ω − ω )] again, expressing the mass flow rate in cmm, we get )(1.2)/ 60] ∆ℎ = [( )∆ℎ, = 0.02 ( Which is the same as )(0.0204 ∆ + 50 ∆ ), =(

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x

Total Heat Process:

x

Thermodynamics

Because of this change in the humidity ratio, there is also a Change in enthalpy of the air given by (h − h ). In air-conditioning practice this change in enthalpy due to the change in the Humidity ratio is considered to cause a latent-heat transfer Given by Q = m (h − h )

Quick Refresher Guide

1 1 = 1 + 2451 (∆ /∆ ) 1 + tan

SHF

A

(1 – SHF)

∆t

C

∆

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∆h h

We see tan θ is the slope of the SHF line AC on the Psychrometric chart, which is purely a function of SHF

Where tan θ = ∆ω / ∆t = (1/2451) [1/SHF) – 1]

=

A zero SHF line is vertical on the psychrometric chart and implies no sensible heat transfer. An SHF of 0.75 to 0.8 is quite common in air conditioning practice in a normal dry climate. A lower value of SHF, such as 0.65, implies a high latent head load, which is quite common

The process line AC is called the sensible heat factor line or process or condition line.

Thermodynamics

Fig. Convergent – divergent nozzle.

Throat

Exit

)

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v = initial volume of 1 kg of steam at pressure p ( p = steam pressure at the throat v = volume of 1 kg of steam at pressure p (m ) A = cross – sectional area of nozzle at throat (m )

Let p = initial pressure of steam

Discharge through the Nozzle and Conditions for its Maximum Value:

The cross – section of a nozzle at first tapers to a smaller section (to allow for changes which occur due to changes in velocity, specific volume and dryness fraction as the steam expands); the smallest section being known as throat, and then it diverges to a large diameter. The nozzle which converges to throat and diverges afterwards is known as convergent – divergent nozzle. In convergent nozzle there is no divergence after the throat. In a convergent – divergent nozzle, because of the higher expansion ratio, addition of divergent portion produces steam at higher velocities as compared to a convergent nozzle. Velocity of steam at the exit of nozzle, C = 44.2 h where h = heat drop during expansion of steam.

Entry

Divergent part

(i) It transforms a portion of energy of steam (obtained from steam generating unit) into kinetic energy. (ii) In the impulse turbine it directs the steam jet of high velocity against blades, which are free to move in order to convert kinetic energy into shaft work. In reaction turbines the nozzles discharge high velocity steam. The reactive force of the steam against the nozzle produces motion and work is obtained. Convergent part

A steam nozzle may be defined as a passage of varying cross – section, through which heat energy of steam is converted to kinetic energy. Its major function is to produce a steam jet with high velocity to drive steam turbine. A turbine nozzle performs two functions:

4.6.1 STEAM NOZZLES

Part 4.6: Power Engineering

Sensible Heat Factor (SHF): The ratio of the sensible heat transfer to the total heat transfer is termed as the sensible heat factor. Thus SHF = Q / (Q + Q ) = Q /Q

SHF = [(h − h )/(h − h ) + (h − h )] = h − h /(h − h ) = [0.0204 ∆ / 0.0204∆ + 50 ∆ ] = (0.0204 ∆ /0.02∆ℎ)

Quick Refresher Guide

Thermodynamics

Quick Refresher Guide

Thermodynamics

=

=A n

)

̇

×

kW

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1. Impulse turbine. 2. Reaction turbine.

The common types of steam turbines are:

Power per wheel =

1. The steam turbine is a prime mover in which the potential energy of the steam is transformed into kinetic energy, and latter in its turn is transformed into the mechanical energy of rotation of the turbine shaft. 2. The most important classification of steam turbines is as follows: (i) Impulse turbines (ii) Reaction turbines (iii) Combination of impulse and reaction turbines. 3. The main difference between Impulse and Reaction turbines lies in the way in which steam is expanded while its moves through them. In the former type, steam expands in the nozzle and its pressure does not change as it moves over the blades while in the latter type the steam expands continuously as it passes over the blades and thus there is a gradual fall in pressure during expansion. 4. The different methods of compounding are: (i) Velocity compounding (ii) Pressure compounding (iii) Pressure velocity compounding (iv)Reaction turbine. 5. Force (tangential) on the wheel = ṁ + Nm

Definitions and Formulae

4.6.2 STEAM TURBINES

m

The value of maximum discharge is given by

=

C = velocity of steam (m/s) The steam flowing through the nozzle follows approximately the equation given below: pv = constant where, n = 1.135 for saturated steam and = 1.3 for superheated steam [For wet steam, the value of n can be calculated by Dr. Zenner’s equation, n = 1.035 + 0.1 x where x is the initial dryness fraction of steam] The discharge through the nozzle is maximum when critical pressure ratio, i.e.,

Quick Refresher Guide

β

Thermodynamics

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The steam jet issuing from the nozzle at a velocity C impinges on the blade at an angle α. The tangential component of this jet C performs work on the blade, the axial compontent C however does not work but causes the steam to flow through the turbine.

Fig. shows the velocity diagram of a single stage impulse turbine. C = liner velocity of moving blade (m/s) C = absolute velocity of steam entering moving blade (m/s) C = absolute velocity of steam leaving moving blade (m/s) C = velocity of whirl at the entrance of moving blade. = tangential component of C . C = velocity of whirl at exit of the moving blade. = tangential component of C . C = velocity of flow at entrance of moving blade. = axial component of C . C = velocity of flow at exit of moving blade. = axial component of C . C = relative velocity of steam at moving blade at entrance. C = relative velocity of steam at moving blade at exit. α = angle with the tangent of the wheel at which steam with velocity C enters. This is also called nozzle angle. β = angle which the discharging steam makes with the tangent of the wheel at the exit of moving blade. θ = entrance angle of moving blade. ϕ = exit angle of moving blade.

Fig. Velocity diagram for moving blade.

Velocity Diagram for Moving Blade

Impulse Turbines

Quick Refresher Guide

Thermodynamics

S

=

(

×

=

)

α

L

N

Q

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and the work done in the last row = of total work where n is the number of moving rotating blade rows in series. In practice more than two rows are hardly preferred. 7. The degree of reaction of reaction turbine stage is defined as the ratio of heat drop over moving blades to the total heat drop in the stage.

3. The axial thrust on the wheel due to difference between the velocities of flow at entrance and outlet : Axial force on the wheel = ṁ C − C 4. Energy converted to heat by blade friction = loss of kinetic energy during flow over blades − = ṁ 5. Optimum value of ratio of blade speed to steam is, ρ = 6. The blade efficiency for two stage turbine will be maximum when, ρ = In general optimum blade speed ratio for maximum blade efficiency or maximum work done is given by ρ =

2. Stage efficiency, η

M

Fig. 6.3

1. Blade or diagram efficiency, η

Important Formulae:

P

As the blades move with tangential velocity of C , the entering steam jet has a relative velocity C , (with respect to blade) which makes an angle θ with the wheel tangent. The steam then glides over the blade without any shock and discharges at a relative velocity of C is the at an angle ϕ with the tangent of the blades. The relative velocity at the inlet C same as the relative velocity at the C if there is no frictional loss at the blade. The absolute velocity (C ) of leaving steam makes an angle β to the tangent at the wheel. To have convenience in solving the problems on turbines it is a common practice to combine the two vector velocity diagrams on a common base which represents the blade velocity (C ) as shown in Fig. This diagram has been obtained by superimposing the inlet velocity diagram on the outlet diagram in order that the blade velocity lines C coincide.

Quick Refresher Guide

becomes maximum when, ρ = cos α

P P

(

)

V

(1)

(2)

LPL

(3)

ST

(1)

LPV

T (3)

(4)

S

(2)

(1)

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(3)

(4)

(2)

B

HPL

(4)

HPV

B- boiler ST – Steam Turbine SC – Steam Condenser P – Pump HPL – High Pressure Liquid LPL – Low Pressure Liquid HPV- High Pressure Vapour LPV – Low Pressure Vapour

Basis for steam turbine power plant Working substance is steam.

4.6.3 Rankine Cycle -

Thermodynamics

and hence (η ) = 9. The state point may be defined as that point on h – s diagram which represents the condition of steam at that instant. 10. Theoretical efficiency of reheat cycle is given by ( ) ( ) η =( , neglecting pump work. )

η

8. The blade efficiency of the reaction turbine is given by η =2−

Quick Refresher Guide

Effects of Reheating 1. Net work output increases 2. Efficiency of the plant increases 3. Life of steam turbine blades increases

(∵ W <<< W )

-

-

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Purpose of regeneration: To increase the efficiency of the plant by increasing the mean temperature of heat addition. Regeneration: some amount of steam is extracted from the turbine and is used to increase the temperature of feed water. This process is called ‘bleeding’.

Rankine Cycle with Regeneration:

-

SSC =

0.95< W < 0.98 Specific steam consumption (KJ/KWh): The amount of steam consumed by the steam power plant per unit power output ̇ SSC = Where ṁ = mass flow rate of steam, kg/h P = net power output, kW SSC = =

-

-

Work ratio, W = 1 −

In general W <<< W = ∴η

-

-

Thermodynamics

P − boiler pressure P - Condenser pressure Rankine cycle contains 2 – isentropic processes(expansion & pumping ) 2 – isobaric processes (boiling & condensation) 1-2 isentropic expansion in steam turbine work developed per kg of steam, W = h − h 2 − 3 isobaric condensation. Heat rejected by the steam, q = h − h 3 – 4 isentropic pumping Work supplied per kg of steam, W = h − h = V = V (P − P ) 4 − 1: Isobaric heat adition Heat supplied per kg of steam in the boiler q =h − h Net work done = q − q = W − W ( ) ( ) η = = × 100

Quick Refresher Guide

(7) m

M (2)

(4) (M – m) (3)

(6) (5)

(1)

M kg/s

Regenerator

Efficiency increases Work output decreases

HP T

Condenser

(M –m) kg/s

Reheated steam

ST

LPT

Condenser

Thermodynamics

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Rankine cycle with Reheating:

-

Effects of Regeneration:

= mass flow rate of bled steam

S

M-m

Pump

m kg/s

M = total mass flow rate of steam

T

M kg/s

Boiler

M kg/s

Quick Refresher Guide

-

Thermodynamics

(5)

(6)

1-2 : expansion in HPT 2-3 : Reheating Process 3-4 : Expansion in LPT

T

d

Cascade Heat exchanger

Pump

2

Turbine

Pump

(4)

(3)

c

ST

a

b Condenser

Steam Turbine

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4

3

ST

2 working fluids are being used In general, Hg − working fluid for primary cycle Steam - working fluid for secondary cycle 1

s

(2)

(1)

Purpose of reheating: To avoid blade erosion by increasing the dryness fraction of the steam at the end of expansion process in steam turbine. To avoid blade erosion , dryness fraction should be greater than or equal to 0.88 x ≥ 0.88 Reheating involves partial expansion of steam in HPT and then extract this steam for reheating in to the boiler, feed the reheated steam to LPT for further expansion (refer Fig. )

Binary power cycles:

-

-

-

Quick Refresher Guide

V

4

c

T

S

Steam

3

1

2

S

b

a

2

Compression ratio (r) =

= (r )

Basic cycle for gas turbine plant and jet propulsion systems. Cycle contains – 2 isentropic and 2 constant pressure processes. Pressure ratio (r ) =

1

3

d

4

3

4

Thermodynamics

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=

q = C (T − T ) KJ/Kg

2-3 constant pressure heat addition

W → work input to the compressor

= r

W = C (T − T ) KJ/Kg

1-2 : isentropic compression through the required pressure ratio(r )

-

P

2

4.6.4 Brayton (or) Joule Cycle:

T

1

Quick Refresher Guide

η

= 1−

= W −W =q −q

ν

=1−

(r )

ν

1

.

(r )

W

=

.

(r )

T

(r )

W

(

(

KJ/Kg

----------

(r )

.

----------

( )

(r )

− T

)

.

r →

( )

) on Brayton cycle efficiency

.

r →

( ) ( )

Pressure ratio for W

Pressure ratio for η

η

−

Thermodynamics

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= C

=

=

(r )

Effect of

For steam turbine plants, 0.95 ≤ W ≤ 0.98

For gas turbine plants, 0.45 ≤ W ≤ 0.55

Work ratio (W ) : W =

W

q = C (T − T )KJ/Kg

4-1 Isobaric (constant pressure ) heat rejection.

W = C (T − T )KJ/Kg

3-4 Isentropic expansion process work developed by the turbine,

Quick Refresher Guide

COP

HP/K+1

= COP

+1

Sink

Source

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Fig. T-s diagram of refrigeration cycle

Simple Vapor Compression Cycle:

Relative COP =

Carnot COP =

Refrigerator

=

HP/K+1

Relative COP = COP

COP

W

Actual COP=

=

Sink

W = work input

T = sink temperature

T = source temperature

Q = heat extracted from sink

Source Q = heat supplied to source

Thermodynamics

Actual COP=

Carnot COP =

Heat Pump

W

4.7.1 Refrigerator

Part 4.7: Refrigeration

Quick Refresher Guide

Thermodynamics

Heat extracted at low temperature Work supplied

+

ℎ −ℎ ℎ −ℎ

=

(

. ∗

)

kg/min – ton

(m / min – ton)

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Piston displacement =

Let the specific volume of the vapor at B i.e at suction of the compressor be, v and let the volumetric efficiency of the compressor be η , then piston displacement required per min.

(b) Piston Displacement:

Or, mass of refrigerant in circulation,

Refrigeration effect = (ℎ − ℎ ) KJ/Kg of refrigerant

(a) Mass of Refrigerant in Circulation:

= (ℎ − ℎ ) = (ℎ − ℎ )

= (ℎ − ℎ ) + (ℎ − ℎ )

=

So, COP =

= (ℎ − ℎ ) (adiabatic compression).

Now, heat rejected to the condenser, =

Work of compression =

= (ℎ − ℎ )

Heat extracted at low temperature = Heat transfer during the process A-B = refrigerating effect.

COP =

COP of Vapor Compression Cycle:

i) Process B-C (B’-C’ or B”-C”): Isentropic compression of the vapor from state B to C. If vapor state is saturated (B), or superheated (B”), the compression is called dry compression. If initial state is wet (B’), the compression is called wet compression as represented by B’-C’. ii) Process C-D (C’-D or C”-D): Heat rejection in condenser at constant pressure. iii) Process D-A: An irreversible adiabatic expansion of vapor through the expansion value. The pressure and temperature of the liquid are reduced. The process is accompanied by partial evaporation of some liquid. The process is shown by dotted line. iv) Process A-B (A-B’ or A-B”) : Heat absorption in evaporator at constant pressure. The final state depends on the quantity of heat absorbed and same may be wet (B’) dry (B) or superheated (B”).

Figure shows a simple vapor compression refrigeration cycle on T-s diagram for different compression processes. The cycle works between temperatures T and T representing the condenser and evaporator temperatures respectively. The various processes of the cycle A-B-CD (A-B’-C’-D and A-B”-C”-D) are as given below:

Quick Refresher Guide

(

)

(KW / ton)

∗

=m

( −

∗

(

−

∗

)

)

(ℎ − ℎ ) (KJ /min- ton)

Thermodynamics

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Fig. Reversed Carnot cycle

Process d-a: Isentropic expansion of the working fluid. The temperature of the working fluid falls from T to T .

Process c-d: Isothermal compression of the working fluid during which heat is rejected at constant high temperature T .

The temperature of the fluid rises from T to T .

Process b-c: Isentropic compression of the working fluid with the aid of external work.

Process a-b: Absorption of heat by the working fluid from refrigerator at constant low temperature T during isothermal expansion.

Reversed Carnot cycle is shown in below Figure. It consists of the following processes.

4.7.2 Reversed Carnot Cycle

Total heat rejected =

/

(KW/ton)

)( −

) − (ℎ − ℎ ) (KJ⁄min − ton)

(

Heat rejected in condenser = (ℎ − ℎ ) (KJ / Kg)

(e) Heat Rejected in Condenser:

Q

(d) Heat Rejected to Cylinder Jacket:

Or Power required =

Work of compression =

(ii) If the compression is polytropic (Pv = C).

Hence, Power required =

Work of compression = (ℎ − ℎ ) KJ/Kg

(c) Power Required by Compressor: (i) If the compression is isentropic, then,

Quick Refresher Guide

(

−

(

− ) )− ( −

=

)

=

(

−

)

−

Thermodynamics

Reversed Brayton Cycle

(b) Air refrigeration system

(a) Air refrigeration system

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4.7.3

Practically, the reversed Carnot cycle cannot be used for refrigeration purpose as the isentropic process requires very high speed operation, whereas the isothermal process requires very low speed operation.

=

=

COP of Refrigerator:

Quick Refresher Guide

Thermodynamics

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Process 5-6 represents the isentropic expansion of air in the expander. Process 6-2 represents the absorption of heat from the evaporator at constant pressure.

Process 1-2 represents the suction of air into the compressor. Process 2-3 represents the isentropic compression of air by the compressor. Process 3-5 represents the discharge of high pressure air from the compressor into the heat exchanger. The reduction in volume of air from v to v is due to the cooling of air in the heat exchanger.

The working of air-refrigeration cycle is represented on p-v and T-s diagrams in Fig. (b) and (c).

(c) Air refrigeration system

Quick Refresher Guide

Cast iron

Cast iron, aluminum alloy

Cast iron, aluminum alloy

Silicon cast iron

Steel

Alloy steel

White metal

Nickel alloy steel

Cylinder head

Piston

Piston rings

Connecting rods

Crank shaft

Bearing

Cylinder liner

x

x

x

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Clearance Volume (Vc): The nominal volume of the space on the combustion side of the piston at top dead centre Cylinder Volume (V) : The sum of piston swept volume and clearance volume

Vs = A × L

Piston Swept Volume (Vs): The nominal volume generated by the piston when travelling from one dead centre to the next one.

Engine’s Terminology:

Material used

Connecting rod

Cylinder volume ‘ V ’

Cylinder

Crank

Crank Pin

Cylinder

Stroke volume Vs

Clearance volume, Vc

Name of the part

Crank Shaft

Crank case

Bottom dead center BDC

Piston Gudgeon of wrist pin

Top dead center T.D.C

Cylinder head Exhaust Valve Exhaust manifold

Exhaust stroke

Expansion stroke

Compression stroke

Suction stroke.

Intake Closes

( )

BDC

Exhaust Opens

Intake Closes

Overlap

Intake opens

Suction valve closed.

Exhaust valve open

Both valves closed

Both valves closed

Exhaust valve closed

Suction valve open

Valve position

BDC ( )

Exhaust Opens

Exhaust Opens

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Intake Closes

Exhaust Opens

For four-stroke S.I. engine

Valve timing diagrams: x

r = V/Vc On the basis of the number of stroke engine can be four-stroke engine or can be two – stroke engine. On the basis of the working cycle it can be spark ignition (otto cycle) engine or it can be compression ignition engine (diesel cycle).

Four – stroke Engine: Stroke

x

x

Exhaust valv ve open

Suction Valve Intake of suction manifold

I.C. Engine classification:

Compressio n

Engine Components: The I.C. Engine Figure below showing its various components.

Intake valv ve open

4.8.1 Basics of I.C. Engine

V = Vc + Vs

Thermodynamics

Compression Ratio (r): It is ratio of cylinder volume to clearance volume.

Power

x

ve open Exhaust valv

Part 4.8: I.C. Engines

Quick Refresher Guide

Compressio n

Thermodynamics

Intake valv ve open

Quick Refresher Guide

Power

x

It contains only ports not valves

Poor part load efficiency

C. I. Engines

Based on diesel cycle

Fuel has low self

x It contains valves

x Better part load efficiency

S.I. Engines

x Based on otto cycle

x Fuel has high self

Heavier

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Higher max. efficiency

x Lighter

is between 14 to 22

Compression ratio

x Lower max. efficiency

is between 6 to 10.5

x Compression ratio

ignition temperature

Lower thermal efficiency

x Higher thermal efficiency

ignition temperature

Less volumetric efficiency

flywheel is needed.

x More volumetric efficiency

flywheel is needed

moment hence lighter

shaft

Shaft

More uniform turning

each revolution of crank

in two revolutions of crank

uniform hence heavier

It has one power stroke in

x It has only one power stroke

x Turning moment is not

two strokes of the piston.

in four strokes of the piston.

The cycle is completed in

Inlet port closes

x The cycle is completed

port open

Thermodynamics

Exhaust port closes

Two-stroke cycle

Exhaust

Inlet port open

B.D.C

T.D.C

Rotation

Four-stroke cycle

Inlet port closes

Exhaust port closes

Ignition occurs

For two-stroke engine

Compression

Quick Refresher Guide

Expansion x

x

x

x

x

x

×

×

. .

): Brake thermal efficiency is the ratio of

×

. . . .

energy in the brake

): It is a ratio of brake power to the indicated horse power.

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Specific fuel consumption (sfc): It is expressed in grams per horsepower-hour or per kWh.

=

Relative efficiency or efficiency ratio: It is defined as ratio of thermal efficiency of the actual cycle to that of the ideal cycle.

=

f.p. is usually assumed constant. At part loads b.p. is changed, thus from b.p. & f.p., ip. can be calculated. Volumentric efficiency ( ): It is defined as the ratio of the air actually induced at ambient conditions to the swept volume of engine.

f. p. = friction power

f. p. = i. p. – b. p.

=

=

=

Mechanical efficiency (

b. p. = break power.

=

Brake thermal efficiency ( power to the fuel energy.

Thermodynamics

): It is a ratio of energy in the indicated horse power to the

QLHV = Lower Heat Calorific Value

mf = Mass of fuel

I. P. = Indicated power

=

Indicated thermal efficiency ( fuel energy.

Performance parameter:

For a muticylinder engine a smaller flywheel is required

Quick Refresher Guide

=

for 4s

. . /

=

×

kg/kWh

kg/kWh

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The working medium is a perfect gas throughout, i.e., it follows the law pV= mRT. The working medium has constant specific heats. The working medium does not undergo any chemical change throughout the cycle.

Assumptions in ideal or air standard cycle

x x x

Thermodynamics

In line engines : all cylinders are arranged linearly and transmit power to a single crankshaft Radial engines: air cooled aircraft engines, odd cylinders are employed for balancing, pistons of all cylinders are coupled to same crankshaft.

> 1 rich mixture

< 1 lean mixture

= 1 chemically correct

=

Equivalent ratio:

k = No. of cylinders

n = N for 2s

n=

=

Mean effective pressure,

‘N’ – r.p.m.

‘L’ – length of cylinder

Mean Piston speed = 2LN

4.8.2 Air Standard Cycles

x x

. .

. .

Fuel-Air Ratio: It is relative proportion of the fuel and air in the engine.

NOTE:

x

x

x

isfc =

bsfc =

Quick Refresher Guide

5

V

(

) )

S

= 1.30

= 1.4

= 1.67

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r

Exchaust gas

Air

Monoatamic gas

Thus the efficiency of otto cycle depends only on compression ratio (r), and the efficiency increases with increasing compression ratio and . The efficiency at compression ratio 5 is 47.5% and at compression ratio 10 is 60.2%

=1−

=1−(

= Constant

Q

4

= Constant 3

Thermodynamics

Remark Adiabatic and reversible compression Combustion Adiabatic and reversible expansion Exhaust stroke

1

2

Q

Work done = heat added – heat rejected = cv(T3 – T2) – cv(T4 – T1)

6

1

4

W.D. in expansion 3-4 = area 3-4-6-5-3

Process 1–2 2–3 3–4 4–1

2

3

Thermal efficiency, =

P T

The kinetic and potential energies of the working fluid are neglected. The operation of the engine is frictionless. All the process are reversible.

The constant volume or Otto cycle:

x x x

Quick Refresher Guide

)(

(

)

P

(

5

(

)

(

(

.

=

)

)

)

r

=

Reversible adiabatic =

−

T

1

2

5

4

6

3

Thermodynamics

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=

,r=

=

= cut off ratio

=1−

=

=

1

0

Isothermal

Constant pressure or Diesel cycle:

=

)

(∝

mep =

Mean effective pressure (mep)

Quick Refresher Guide

S

Thermodynamics

( (

)(

) ( )

)

x

x

x

2

(

,

=

(

4

(∝

) ( ) ∝ )

) (

(

(

)

)

V

)

1

5

T

1

P = constant (Diesel)

f

2

v = constant (otto )

g

5

s

v = constant

4

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(

If ρ = 1 in above equation it becomes otto cycle and when α = 1, it becomes diesel cycle.

=

=1−

=

=

P

3

The name dual combustion is derived from the fact that it incorporates the features of both otto and diesel cycles. High speed diesel engine is based on this.

The dual combustion or mixed or limited pressure cycle

The efficiency decreases as cut off ratio increases. If cut off ratio is greater than 10% of stroke, smoking occurs in an actual engine because there is no sufficient time for the combustion process to be completed before the exhaust valve opens.

=

mep =

Mean Effective Pressure (mep)

The efficiency of the diesel cycle is different from that of the Otto cycle only by the bracketed term, which is always greater that unity.

Quick Refresher Guide

(

) (

(

)(

)

)

(

)

Thermodynamics

x

5

4

6 "

"

"

′

′

′

S

For constant maximum pressure and same heat input.

1

2

"

3

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x

T

ηotto > ηdual > ηdiesel

For same compression ratio and same heat input: The heat rejected in the Otto cycle is less than that in the diesel cycle and dual combustion cycle thus the efficiency of the Otto cycle is more than the diesel and the dual combustion cycle for same compression ratio and same heat input.

Comparison of Otto, Diesel, and dual Combustion (Limited - pressure) Cycles:

=

mep =

Quick Refresher Guide

Constant volume

′

′

4

3

T

1

′ 2

5

Constant volume

Constant pressure

6

4

3

′ 6 S For same maximum pressure and temperature > >

1

2

′

Constant pressure

S

Thermodynamics

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x

T

Quick Refresher Guide

Theory of Machines

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Kinematic Pair: Two links or elements connected with a joint that allows the relative motion between the links.

Quatenary Link: It connects with four other links.

Ternary link: It connects with three other links in a system

Binary link: It connects with two other links

Classification of links: The links or the kinematic elements are the basic building block. They are classified as follows:

Link / Kinematic Element: A rigid body or a resistant body that forms the part of a mechanism.

DOF of a rigid body in plane motion are 3 consisting of 2 translatory freedoms and one rotational freedom.

Degrees of Freedom: Number of independent co-ordinates that are required to specify the system completely. DOF of a rigid body in spatial motion are 6, consisting of 3 translatory freedom and 3 rotational freedoms

Planar Mechanism: A mechanism that is constrained to move in a single plane or in parallel planes is referred as a planar mechanism or plane mechanism.

Mechanism: A system that consists of links and joints and converts one form of motion to another form (or) A system of links and joint that converts the available form of motion to the desired form.

5.1.1 Introduction

Part 5.1: Mechanisms

Part – 5: Theory of Machines

Quick Refresher Guide

Theory of Machines

Form Closure: The two links are held together by the shape of the links and they cannot be detached easily. Force Closure: The contact is maintained by an external force either the gravity force or spring force and the two bodies can be separated easily.

−

−

−

−

−

−

−

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=

For a planar kinematic chain.

=

Degrees of Freedom of a Kinematic chain: A kinematic chain is formed by connecting number of links with number of pairs. Let ‘n’ be the no. of links and J is the number of pairs of class n. Then as for grubler’s criterion the DOF of a spatial kinematic chain is given by

A mechanism is obtained by fixing any one link in a kinematic chain.

A kinematic chain is formed by connecting number of links with kinematic pairs so that there exists definite relation between the motion of various links. They can be of two types closed kinematic chains and open kinematic chains.

Kinematic chain:

x

x

Based on type of closure: Closure means the way the two bodies are held together to have continuous contact. Two types of joint closures exists they are form closure and force closure.

Based on Nature of contact: Based on the nature of contact between the two links the kinematic pairs are classified as lower pairs and higher pairs. When the two bodies have surface to surface contact they are referred as lower pairs. When the contact between the bodies is a point or line contact they are referred as higher pairs.

Based on Nature of relative motion: Based on the relative motion that exists between the two links the pairs can be classified as Rotary/ Revolute pairs, Sliding/ Prismatic pairs, Cylindrical pairs, special pairs and so on.

Class-n kinematic pair allows n degrees of freedom for the pair and constrains 6-n (3-n) degrees of freedom. A pair that constrains all the degrees of freedom of the second link relative to the first link is not considered as a kinematic pair. It is a rigid joint.

Based on Degrees of Freedom: A kinematic pair allows few degrees of freedom and constraints some of them. When two bodies are joined together one of the body (base) has all the DOF, whereas the other body looses some DOF and has few DOF relative to the base body. Depending on the allowed degrees of freedom and constrained degrees of freedom they are classified into Class I, Class II ….. Class V incase of spatial kinematic pairs and class I and class II in planar kinematic pairs.

Classification of kinematic pairs:

Quick Refresher Guide

Theory of Machines

−

−

−

−

−

.

Grashoffs or Class –I Non grashoffs or Class-II Special Grashoffs or Class-III

C

D

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Nomenclature of four bar mechanisms:

A

B

Inversion: By fixing one link in a kinematic chain a mechanism is obtained. By fixing different links, different mechanisms are obtained. Inversion is the process of obtaining different mechanism by fixing different links in a kinematic chain.

If l + s < p + q If l + s > p + q If l + s = p + q

Grashoff’s Condition: Grashoff’s condition checks the link proportions and classifies the chains mechanism

When l is the length of the longest link, s is that of the shortest link and p, q are the lengths of the remaining two links. Though a chain is formed by satisfying the above condition it may not result in useful mechanism, if one barely satisfy the condition.

Four bar chain/quadric cycle chain: It is the basic chain that consists of four links and four turning pairs. It is the basic chain from which many one DOF mechanism can be derived. The necessary condition to form a four bar chain based on their length is l < s + p + q.

Positive degrees of Freedom: Mechanism

Negative degrees of freedom: Super structure/ Preloaded structure

Zero degrees of freedom: Structure

Classification based on degrees of freedom:

Note: A mechanism has six (three) degrees of freedom less compared to that of the kinematic chain from which it is obtained.

= ( − )−

For a planar mechanism

= ( − )−

Degrees of Freedom of a Mechanism: As one link is fixed in a kinematic chain to get a mechanism. Grublers equation for the DOF of a mechanism is as follows for spatial mechanism

Quick Refresher Guide

Theory of Machines

Shortest link fixed-Double crank mechanism Link adjacent to shortest link is fixed crank-Rocker mechanism Link opposite to the Shortest link is fixed: Rocker-Rocker mechanism

Mechanical advantage =

=

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∴

Mechanical Advantage: The ratio of load to effort is known as mechanical advantage.

III inversion: Oldham’s Coupling. Useful for connecting two parallel shafts with little offset.

II inversion: Elliptical trammel. Useful for tracing the elliptical curves

Inversions of Double Slider Crank chain: I inversion: Scotch Yoke-mechanism. Useful for generating trignometric functions

IV inversion: Hand pump

III inversion: Crank and slotted level type quick return motion mechanism and oscillating cylinder engine

II inversion: wit-worth Quick return motion mechanism and rotary engine.

Inversions of a single slider crank chain: I inversion: An I.C. engine mechanism/compressor mechanism

Equivalent linkage: By replacing any pair in a kinematic chain with its equivalent (from the same class). An equivalent chain can be obtained, by replacing any turning pair in a four bar chain with a sliding pair a slider crank chain can be obtained. By replacing any two turning pairs with sliding pairs a double slider crank chain can be obtained.

In parallelogram connection both long links and short links are opposite to each other. In deltoid connection both long links and short links are side by side.

Inversion of Special Grashoff’s Chain: Parallelogram or anti-parallelogram connection will result in double crank or drag link mechanism. Deltoid connection will result in crank-Rocker mechanism.

By fixing any link it results in Rocker-Rocker mechanism.

Inversions of Non-Grashoffs 4-bar chain: l + s > p + q

x x x

Inversions of Grashoff’s 4-bar chain (l + s < p + q): The mechanisms obtained from the Grashoff’s kinematic chain are based on the positions of the shortest link

A four bar mechanism is as shown in figure; AD, the link 1 is known as the fixed link. AB, the link 2 acts as input link. The link 3 BC is the coupler and the link 4 CD is the output link. The input and output links can be interchanged. If the input/output link can have complete rotation about its centre it is known as crank. If it has only a partial revolution it is known as a rocker or an oscillatory link. Based on this the mechanisms can be classified as C-C, C-R, R-C, R- R mechanisms.

Quick Refresher Guide

Fig. 5.1.2.1

IDC

A

0

ODC

Theory of Machines

√

√

(

) /

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α = angular acceleration of the connecting rod

α = −ω sin θ

ω = angular velocity of connecting rod.

ω = ω.

Angular velocity & Angular Acceleration of connecting rod:

At θ = 180° , when direction of motion is reversed a = rω 1 −

θ = 180° i.e. at ODC, a = rω −1 +

a = rω cos θ + If n is very very large a = rω cos θ as in case of SHM When θ = 0° i.e. at IDC, a = rω 1 +

Acceleration of Piston:

is neglected if ‘n’ is quite large If V = rω sin θ

If n is large compared to sin θ, V = rω sin θ +

V = rω sin θ +

Velocity of Piston:

Let x = displacement of piston from IDC (Inner Dead Centre).

Figure shows a slider crank mechanism in which the crank OA rotates in the clockwise direction. ℓ & r are the lengths of the connecting rod & the crank respectively.

B

5.1.2 Dynamic Analysis of Slider-crank Mechanism

Quick Refresher Guide

Theory of Machines

B

Fig. 5.1.2.2

+

( + )

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2. Thrust on the sides of cylinder: It is the normal reaction on the cylinder walls. F = F sin β = F tan β

1. Force (thrust) along the connecting rod: Let F = Force in the connecting rod shown in Fig (1.2.2) Then equating the horizontal components of forces. F × cos β = F or F =

Force on the piston due to gas pressure, F =p A −p A Inertia force, F = m × a. = mrω cos θ + Net (Effective) force on the piston, F=F −F In case friction resistance F is also taken into account, Force on the piston, F = F − F − F In case of vertical engines, the weight of the piston or reciprocating parts also acts as force and thus, Force on the piston, F = F + mg − F − F

Let A = area of the cover end A = area of the piston rod p = pressure on the cover end p = pressure on the rod end m = mass of the reciprocating parts.

F

A

An engine is acted upon by various forces such as weight of reciprocating masses and connecting rod, gas forces, forces due to friction & inertia forces due to acceleration & retardation of engine elements, the least being dynamic in nature. The effect of the weight & the inertia effect of the connecting rod is neglected.

Engine Force Analysis:

The –ve sign indicates that the sense of angular acceleration of the rod is such that it tends to reduce the angle β. Thus, in the given case, the angular acceleration of the connecting rod is clockwise.

Quick Refresher Guide

Theory of Machines

C

Fig. 5.1.3.1

D

⊥ DC c

b

⊥ BC

a, d

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In the velocity diagram the vector bc indicates the velocity of C relative to B and ω l = bc gives ω . Similarly dc = ω l from which ω can be obtained.

Construction procedure for velocity polygon: A and D are fixed points having zero velocity mark, a, d at a convenient location and they act as reference for the velocity polygon. Velocity of B relative to A will be l ω perpendicular to AB in the direction of ω so draw ab ⊥ to AB⃗ with a length l ω . Velocity of C relative to B will be ⊥ to BC but sense is not known hence draw a line ⊥ to BC passing through b. Velocity of C relative to D will be ⊥’lr to DC sense is not known. So draw a line ⊥lr to DC through d. These two lines will intersect at C that completes the velocity polygon.

A

B

⊥ AB

Consider the mechanism shown in Figure 5.1.3.1. The pre requisite for the velocity analysis is the knowledge of position of all the links which is available from the position analysis or from configuration diagram. If the link 2 rotates with ω rad/sec in counter clock wise direction. The velocity of other links is obtained as follows.

5.1.3 Velocity Analysis in 4-bar Mechanism:

√

Turning moment on crank shaft T = F sin θ +

4. Thrust on the Bearing The component of F along the crank (in the radial direction) produces a thrust on the crank shaft bearings. F = F cos(θ + β) = cos(θ + β)

3. Crank Effort: Force is exerted on the crankpin as a result of the force on the piston. Crank effort is the net effort (force) applied at the crank pin perpendicular to the crank which gives the required turning moment on the crank shaft. Let F = crank effort As Ft × r = F r sin(θ + β) F = F sin(θ + β) = sin(θ + β)

Quick Refresher Guide

Theory of Machines

nn  1 2

I12

2

Fig. 5.1.3.2

I23

3

I13

1

nn  1 44  1 6 = 2 2

I34

4

I14

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I24

For Four bar mechanism, n = 4, N =

There are three types of instantaneous centers namely fixed, permanent and neither fixed nor permanent.

N=

The number of instantaneous centers in a mechanism depends upon number of links. If N is the number of instantaneous centers and n is the number of links.

Instantaneous Centre: The instantaneous centre, for a plane body moving in a two dimensional plane is a point in its plane around which all other points on the body are rotating at the instant. This point itself is the only point that is not moving at that instant.

3. When BC and CD are parallel i.e. they are in same line, ω = 0 and ω l = ω l

2. When AB and DC are parallel i.e. they are in the same line, ω = 0 and ω l = ω l .

1. When the link AB and BC are parallel to each other Velocity of polygon will be a straight line Velocity of B is equal to the velocity of C. and ω = 0 ∴l ω =l ω If both AB and DC are on the same side of AB both have the angular velocity in the same sense. If they are on opposite side i.e. BC crosses AD; AB and DC will have velocities in the opposite sense

Cases:

Capital letters deals with the configuration diagram is AB, CD are respective link positions, lower case letters indicate the points on the velocity diagram

Quick Refresher Guide

Theory of Machines

O

dT

3

Fig. 5.1.3.3

Q

B1

Z2

A1

dT

P1 B2

= VB/A Z2(dt)2

= A1B1 dT

= OQ dT - AO dT

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? Arc B1B2

= Arc QB2 – Arc AA1

We have Arc B1B2 = Arc QB2 – Arc QB1

3. B1 to B2 due to acceleration Ar to link 2 this component in the coriolis component of acceleration.

2. A1 to B1 due to outward velocity of slider VB/A.

1. B to A1 due to rotation of link 2.

Assume link 2 having constant angular velocity Z2, in its motions from OP to OP1 in a small interval of time Gt. During this time slider 3 moves outwards from position B to B2. Assume this motion also to have constant velocity VB/A. Consider the motion of slider from B to B2 in 3 stages.

A on link 2

B on link 3

2

P

Coriolis Acceleration: To illustrate this let us take an example of crank and slotted lever mechanisms.

Neither fixed nor permanent instantaneous center I13, I24

Permanent instantaneous center I23, I34

Fixed instantaneous center I12, I14

Quick Refresher Guide

Theory of Machines

fcr

Z2

Fig. 5.1.3.4

(d) Rotation CCW slider moving down

Z2

fcr

(b) Rotation CW slider moving down

Z2

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(c) Rotation CCW slider moving up

fcr

Z2

(a) Rotation CW slider moving up

Z2

fcr

Figure 5.1.3.4 shows the direction of coriolis acceleration in different situation.

The direction of coriolis component is the direction of relative velocity vector for the two coincident points rotated at 90o in the direction of angular velocity of rotation of the link.

fcrB/A = 2Z2 VB/A Coriolis acceleration

? ½ f (dt)2 = VB/A Z2 (dt)2

= ½ f (dt)2

? Displacement B1B2 = ½ at2

The tangential component of velocity is Ar to the link and is given by Vt = Zr. In this case Z has been assumed constant and the slider is moving on the link with constant velocity. Therefore, tangential velocity of any point B on the slider 3 will result in uniform increase in tangential velocity. The equation Vt = Zr remain same but r increases uniformly i.e. there is a constant acceleration Ar to rod.

Quick Refresher Guide

Gears for neither parallel nor intersecting shafts: 1. Worm Gears: Worm gears are used when large gear reductions are needed. It is common for worm gears to have reductions of 20:1, and even up to 300:1 or greater.

1. Bevel Gears: Bevel gears are useful when the direction of a shaft's rotation needs to be changed. The teeth on bevel gears can be straight, spiral or hypoid.

Gears for connecting intersecting shafts:

4. Rack and Pinion: Racks are straight gears that are used to convert rotational motion to translational motion by means of a gear mesh.

3. Double helical gears (Herringbone gears): Herringbone gear is a special type of gear which is a side to side (not face to face) combination of two helical gears of opposite hands. Their advantage over the simple helical gear is that the side-thrust of one half is counter-balanced by that of the other half.

2. Helical gears: The teeths on helical gears are cut at an angle to the face of the gear. Because of the angle of the teeth on helical gears, they create a thrust load on the gear when they mesh.

Gears for connecting parallel shafts: 1. Spur gears: They are common types of gears with straight teeth.

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Addendum: The radial distance between the Pitch Circle and the top of the teeth.

Gear Terminology:

x

x

x

Gear Classification: Gears may be classified according to the relative position of the axes of revolution.

5.2.1 Gears

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Module: Millimeter of Pitch Diameter to Teeth. m =

Line of Action: That line along which the point of contact between gear teeth travels, between the first point of contact and the last.

Land: The top surface of the tooth.

Gear: The larger of two meshed gears. If both gears are of the same size, they are both called "gears".

Flank: The working surface of a gear tooth, located between the pitch diameter and the bottom of the teeth

Face Width: The width of the tooth measured parallel to the gear axis.

Face: The working surface of a gear tooth, located between the pitch diameter and the top of the tooth.

Diametral Pitch: Teeth per mm of diameter. DP = T/D

Dedendum: The radial distance between the bottom of the tooth to pitch circle.

Contact Ratio: The ratio of the length of the Arc of Action to the Circular Pitch.

Clearance: The distance between the top of a tooth and the bottom of the space into which it fits on the meshing gear.

Circular Thickness: The thickness of the tooth measured along an arc following the Pitch Circle

Circular Pitch: Millimeter of Pitch Circle circumference per tooth. Pc =

Chordal Thickness: The thickness of the tooth measured along a chord passing through the points where the Pitch Circle crosses the tooth profile.

Chordal Addendum: The distance between a chord, passing through the points where the Pitch Circle crosses the tooth profile, and the tooth top.

Center Distance: The distance between centers of two gears.

Base Circle: The base circle of an involute gear is the circle from which involute teeth profiles are derived.

Backlash: Play between mating teeth.

Arc of Recession: That arc of the Pitch Circle between the Pitch Point and the last point of contact of the gear teeth.

Arc of Approach: Is the arc of the Pitch Circle between the first point of contact of the gear teeth and the Pitch Point.

Theory of Machines

Gears are machine elements that transmit motion by means of successively engaging teeth. The gear teeth act like small levers. Gears are highly efficient (nearly 95%) due to primarily rolling contact between the teeth.

Quick Refresher Guide

Arc of Action: Is the arc of the Pitch Circle between the beginning and the end of the engagement of a given pair of teeth.

Theory of Machines

Part 5.2: Gear Trains

Quick Refresher Guide

ra 2  r 2 cos2 I  r sin I

R A 2  R 2 cos2 I  ra 2  r 2 cos2 I  R  r sin I

Path of recess:

Length of path of contact

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Contatratio

Page 172

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Length of arc of approach

Lenght of path of approach cos I Lenght of path of recess Length of arc of recess cos I Length of path of contact Length of arc contact cosI Contact Ratio (or Number of Pairs of Teeth in Contact): The contact ratio or the number of pairs of teeth in contact is defined as the ratio of the length of the arc of contact to the circular pitch.

Arc of contact: Arc of contact is the path traced by a point on the pitch circle from the beginning to the end of engagement of a given pair of teeth.

I = Pressure angle.

R = Radius of pitch circle of wheel.

r = Radius of pitch circle of pinion,

R A = Radius of addendum circle of wheel

ra = Radius of addendum circle of pinion,

R A 2  R 2 cos2 I  R sin I

Path of approach:

Path of contact: The length of path of contact is the length of common normal cut-off by the addendum circles of the wheel and the pinion.

Working Depth: The depth to which a tooth extends into the space between teeth on the mating gear.

Root Diameter: The diameter of the Root Circle.

Root Circle: The circle that passes through the bottom of the tooth spaces.

Ratio: Ratio of the numbers of teeth on mating gears.

Profile Shift: An increase in the Outer Diameter and Root Diameter of a gear, introduced to lower the practical tooth number or acheive a non-standard Center Distance.

Pressure Angle: Angle between the Line of Action and a line perpendicular to the Line of Centers.

and

PC Circular pitch S u m m = Module.

Quick Refresher Guide

Theory of Machines

ZB

2

ZC

2

Z A DA ZB DB ZC DC Z A m t A ZB m t B ZC m tC Z A t A ZB t B ZC tC

2

=

=

GEAR 'B' (Idler gear)

ZB

GEAR 'C'

N 2 T2

2S N1 T1 60

2S N 2 T2 60 T2 Ÿ T1

N1 N2

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N1 T1

P

In an ideal gear box, the input and output powers are the same so;

Train Value: It is reciprocal of velocity ratio.

Velocity Ratio =

If A is the driving wheel and C is driven wheel, then

or in terms of rev / min N A t A N B t B N C tC

v ZA

GEAR 'A'

ZA

v

ZC

The velocity v of any point on the circle must be the same for all the gears, otherwise they would be DC DA DB slipping.

t = number of teeth on the gear, D = Pitch circle dia meter, N = speedin rpm D m = module = t and module must be the same for all gears , otherwise they would not mesh. D D D m= A = B = C tB tC tA D A = m t A; DB = m t B and DC = m t C Z = angula r velocity. D v = linear velocity on the circle. v = Z = Z r 2

Simple Gear Trains: The typical spur gears are shown in diagram. The direction of rotation is reversed from one gear to another. It has no affect on the gear ratio. The teeth on the gears must all be of the same size so if gear A advances one tooth, so does B and C.

The combination of gear wheels by means of which motion is transmitted from one shaft to another shaft is called a gear train. The gear trains are of the following types:

5.2.2 Gear Trains

Where:

Pitch Circle: The circle, the radius of which is equal to the distance from the center of the gear to the pitch point.

Pitch Point: The point of tangency of the pitch circles of two meshing gears, where the Line of Centers crosses the pitch circles.

Mathematically,

Theory of Machines

Pinion: The smaller of two meshed gears.

Quick Refresher Guide

v

Theory of Machines

Power out Power In

2S u N 2 T2 u 60 2S u N1 T1 u 60

N 2 T2 N1 T1

ND

t A u tC

tA + tB = tC + tD

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RA + RB = RC + RD and

If R and T=Pitch circle radius & number of teeth of the gear

Reverted Gear train: The driver and driven axes lies on the same line. These are used in speed reducers, clocks N A tB u tD and machine tools. GR

The gear ratio may be written as : N In t B t D GR u N Out t A t C

Z B ZC ZA tB tD GR u ZD t A tC Since Z 2 u S u N

6LQFHJHDU%DQG&DUHRQWKHVDPHVKDIW

Compound Gear Trains: Compound gears are simply a chain of simple gear trains with the input of the second „‡‹‰–Š‡‘—–’—–‘ˆ–Š‡ˆ‹”•–Ǥ

If we use a convention that anti-clockwise is positive and clockwise is negative we can determine the holding torque. The direction of rotation of the output shaft depends on the design of the gear box.

T1 + T2 + T3 = 0

The total torque must add up to zero.

Because the torque in and out is different, a gear box has to be clamped in order to stop the case or body rotating. A holding torque T3 must be applied to the body through the clamps.

K

It follows that if the speed is reduced, the torque is increased and vice versa. In a real gear box, power is lost through friction and the power output is smaller than the power input. The efficiency is defined as:

Quick Refresher Guide

Theory of Machines

Add

Revolve C by –1 revolution, keeping the arm fixed

Revolve all once

Action

1

0

1

A

1



tC tB

tC tB

1

B

0

-1

1

C

t ·

tC . tB

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Note: that if C revolves -1, then the direction of B is opposite so 

The number of revolutions made by B is ¨¨1  C ¸¸ © tB ¹

§

Step 3 is simply add them up and we find the total revs of C is zero and for the arm is 1.

Step 2 identify that C should be fixed and rotate it backwards one revolution keeping the arm fixed as it should only do one revolution in total. Work out the revolutions of B.

Step 1 is to revolve everything once about the center.

3

2

1

Step

Suppose gear C is fixed and the arm A makes one revolution. Determine how many revolutions the planet gear B makes.

The diagram shows a gear B on the end of an arm. Gear B meshes with gear C and revolves around it when the arm is rotated. B is called the planet gear and C the sun.

Epicyclic Gear Train: Epicyclic means one gear revolving upon and around another. The design involves planet and sun gears as one orbits the other like a planet around the sun.

Quick Refresher Guide

Theory of Machines

f

h

p

FIG. 5.3.1.1

crank angle,

j

k

e

f

+ −

where

is the

SUCTION

p

COMPRESSION

Fig.5. 3.1.2

EXPANSION

+

EXHAUST

f

e

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Fig. 3.1.3

3rd

d

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2nd

c

As observed in the foregoing paragraphs, the turning-moment diagram for a single cylinder engine varies considerably & a greater variation of the same is observed in case of four stroke, single-cylinder engine. For engines with more than one cylinder, the total crank shaft torque at

0

1st

b

Mean Torque

Turning Moment Diagram for Multi-cylinder Engine:

In case of a four-stroke IC engine, the diagram repeats itself after every two revolutions instead of one revolution as for a steam engine. It can be seen from the diagram (Fig. 5.3.1.2) that for the majority of the suction stroke, turning moment is –ve but becomes +ve after point ‘p’. During the compression stroke, it is totally –ve. It is +ve throughout the expansion stroke & again –ve for most of the exhaust stroke.

0

Turning moment

The area ℎ, ℎ , & represent fluctuations of energy of the fly wheel. When the crank is at b, the flywheel has absorbed energy while the crank has moved from a to b and thereby, the speed of the engine is maximum. At c, the flywheel has given out energy while the crank has moved from b to c and thus, the engine has a minimum speed. Similarly, the engine speed is

Similarly, during the period of crank travel from to , excess work is again developed and is stored in the flywheel and the speed of the engine increases. During the crank travel from to , the loss of work is made up by flywheel and the speed again decreases.

During the crank travel from or the work needed for the external resistance is proportional to ℎ , whereas the work produced by the engine is represented by the area under ℎ . Thus, during this period, more work has been taken from the engine that is produced. The loss is made up by the flywheel which gives up some of its energy & the speed decreases during this period.

When the crank turns from angle to (Fig. 5.3.1.1), the work done by the engine is represented by area ℎ . But the work done against the resisting torque is represented by the area ℎ . Thus, the engine has done more work than what has been taken from it. The excess work is represented by the area ℎ. This excess work increases the speed of the engine and is stored in the flywheel.

The mean torque against which the engine works is given by = mean torque and is the mean height of the turning-moment diagram.

Note that the area of the turning-moment diagram is proportional to the work done per revolution as the work is the product of turning-moment & the angle turned.

It can be observed from Fig. 5.3.1.1 that during the outstroke ( ) the turning moment is maximum when the crank angle is little less than 90° (π/2) & zero when the crank angle is zero & 180° (π). Similar turning moment diagram is obtained during the instroke ( ).

e

g

Turning Moment Diagram for Single-cylinder Double acting Steam Engine:

Torque

−

Turning Moment Diagram for Single-cylinder Four stroke Engine:

A turning moment diagram also known as a crank effort diagram is the graphical representation of the turning moments for different positions of the crank.

5.3.1 Turning Moment Diagram

again maximum at d & minimum at a. Thus, there are two maximum & two minimum speeds for the turning-moment diagram.

Quick Refresher Guide

The difference between the greatest & the least speeds of the engine over one revolution is known as the fluctuation of speed

Theory of Machines

A flywheel is a device which serves as a reservoir to store energy when the supply of energy is more than the requirement and releases energy when the requirement is more than the supply. Thereby, it controls the fluctuation of speed of the prime mover during each cycle of operation.

Part 5.3: Flywheel

Quick Refresher Guide

Torque

Theory of Machines

+

−

− −

−

+ +

−

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There are two types of flywheels: disc type & arm type. In the arm type of flywheel, the weight of the flywheel is mainly located in the rim & the arms & boss do not contribute much in storing the energy.

5.3.2 Size of Flywheel

The difference between the greatest speed & the least speed is known as the maximum fluctuation of speed & the ratio of the maximum fluctuation of speed to the mean speed is the coefficient of fluctuation of speed.

Whereas the ratio of this maximum fluctuation of energy to the work done per cycle is defined as the co-efficient of fluctuation of energy.

The difference between the maximum & minimum kinetic energies of the fly wheel is known as the maximum fluctuation of energy.

The least of these energies is the least kinetic Energy of the fly wheel & for the corresponding crank position, the speed is minimum.

The greatest of these energies is the maximum kinetic energy of the flywheel & for the corresponding crank position, the speed is maximum.

=0

E+ E+

d e

−

E+

Flywheel energy E E+

c

Crank position a b

The energies of the flywheel corresponding to positions of the crank are as follows.

Let , be the areas in work units of the portions above the mean torque ae of the turning moment diagram (Fig. 5.3.1.3) these areas represent quantities of energies added to the below ae represents quantities of energies taken from the flywheel. Parallely areas , flywheel.

Fluctuation of Energy:

Fig. 5.3.1.3 shows the turning moment diagrams for a multi-cylinder engine. The mean torque line ab intersects the turning moment curve at a, b, c, d & e. The area under the wavy curve is equal to the area . As discussed earlier, the speed of the engine will be maximum when the crank positions correspond to b, d & minimum corresponding to a, c, e.

any instant is given by the sum of the torques developed by each cylinder at the instant. For example, if an engine has two cylinders with cranks at 90°, the resultant turning moment diagram has a less variation than that for a single cylinder. In a three-cylinder engine having its cranks at 120°, the variation is still less.

Quick Refresher Guide

Theory of Machines

=

, is the average speed, ks =

is the coefficient of fluctuation of speed

is the density of the rim &

is its peripheral speed.

=

)::

.

.

.

.

&

=

,

4 for four stroke IC engine.

= 4 for steam engine &

= mean speed =

= mean torque =

=

where,

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where,

=

=

Excess energy developed by the engine between two cranks positions.

Co-efficient of Fluctuation of energy (

= .

If b & d be the respective width and diameter of the flywheel & t its thickness, then

Where,

Hoop stress,

The Hoop stress in the flywheel can be determined by assuming it is as a ring.

where,

=

= mean speed = = Kinetic energy of the Flywheel at mean speed = Maximum fluctuation Energy = Co-efficient of fluctuation of speed = ( / ). (radius of gyration)

= minimum speed =

I = moment of Inertia of the Flywheel = maximum speed =

The whole weight of the flywheel is assumed to be concentrated in the rim of the flywheel, therefore it is usual practice to neglect the weight of the arms and the boss in the design of the flywheel.

Quick Refresher Guide

Theory of Machines

= kx.

x(t)

. . .5.4.1.1

+

̇( )

/

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x(t) = x(0) cos

Thus for an initial displacement but with no initial velocity the motion is sinusoidal with an amplitude x(0) and frequency Zn,

=

where x(0) is the initial displacement from the equilibrium position; ̇ (0) is the initial velocity. The frequency Zn is called the undamped natural frequency and is given by

x(t) = x(0) cos

Equation (5.4.1.1) is called the equation of motion. The equation is unchanged if gravity effects are included. The solution of the equation of motion gives,

Equilibrium position

m ̈ + kx(t) = 0

Thus from Newton’s second law of motion using a free – body diagram,

Then there will be a force on the mass from the spring of –kx(t).

If at some time t the mass is displaced an amount x(t) in the positive direction as shown.

If the spring has a linear stiffness k, then

A free vibration is one that dies away with time due to energy dissipation. Usually there is some initial disturbance. Following this initial disturbance the system vibrates without any further input. This is called the transient vibration or free vibration. Consider the motion of the spring/mass system when it is initially disturbed and then allowed to vibrate freely. The displacement of the mass with time, x(t), is measured from the static equilibrium position, i.e. the rest position.

5.4.1 Free Vibration

Vibration refers to oscillations about an equilibrium point. A system vibrates when, it is possible for energy to be converted from one form to another and back again. There are three types of vibrations viz. Free Vibrations, Damped Vibrations & Forced Vibrations

Part 5.4: Vibrations

Quick Refresher Guide

Theory of Machines

x(t)

̇ ( ) + kx(t) = 0

̈( ) + 2

( )+

( )=0

ξ=

√

(0)

ℎ

[ (0) + [ ̇ (0) +

1−

− 1+

[ ̇( )

[ ̇( )

(0)] ]

+

( )]

( )]

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Logarithmic decrement: For damping ratio ξ < 1.0 then vibration will occur and the motion is defined by

x(t) =

>1

x(t) =

=1

x(t) =

(0)

If the initial conditions are x(0) and ̇ (0) then for ξ<1

. . . 5.4.2.2 is the undamped natural frequency as before and the viscous damping ratio is defined

. . . 5.4.2.1

The solution of equation (5.4.2.2) has different forms depending on the value of ξ.

Where as

It is useful to divide equation (5.4.2.1) by m so that rearranging we obtain,

Equilibrium position

m ̈+

Thus the damping coefficient c, of the damper, results in an additional force − ̇ ( ) on the mass. Thus from Newton’s second law of motion using a free body diagram, the equation of motion is,

Real vibration systems have a source of energy dissipation and it is convenient to represent this by a massless viscous damper as shown. This produces a drag force opposing the motion which depends on the velocity of the mass.

5.4.2 Damped Vibrations

The undamped natural frequency does not depend on the initial conditions or the amplitude of motion. It only depends on the mass and stiffness.

Quick Refresher Guide

t

(0)

x(t )

t

1−

x(t

,

)

[ ̇( )

( )]

Theory of Machines

)/ (

)] = 2

/ 1−

=

=

√

F sin ωt

c

̇( ) +

( )=

. . . 5.4.3.1

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̈( ) +

The equation of motion when the force input f(t) is F sin ωt is

x(t)

k

It is a sinusoidally varying force that has a particular frequency which is popularly known as harmonic excitation.

A forced vibration is usually defined as being one that is kept going by an external excitation. We now come to look at the vibration of a one degree of freedom system when there is an externally applied force. The system will respond to the force. The response will depend on the particular forcing function. We will look at a forcing function that illustrate most of the main effects resulting from external forces.

5.4.3 Forced vibration

For small values of δ,

This equation can be rearranged to give,

δ = ln[x(

It can be shown that, if the amplitudes on any two successive peaks are measured, the ratio of these amplitudes is constant. For any value of m, the log decrement will be

and looks like

x(t) =

Quick Refresher Guide

Theory of Machines

) (

) ] /

and tan ϕ = ( )

. . .5. 4.3.2

=

/

ξ=

√

−

−

0.5

1.0

o∞

=

2.5

/

3.0

. . . 5.4.3.4

…5. 4.3.3

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M.F o 0 as

M.F o 1 as

2.0

o0

= 0.1

1.5

is known as magnification factor (MF)

−

−

0

0

1

2

3

4

5

/

and

and tan ϕ =

The equation may be presented in graphical form,

Where

=

It is common to non-dimensionalize these equations so that

X is the displacement amplitude and ϕ is the phase angle between displacement and force.

Where x = [(

x(t) = X sin (ωt + ϕ)

The steady state solution for x(t) can be shown to be

The mathematical solution of the equation of motion may be achieved in various ways. It will be found that after an initial transient (depending on initial conditions and start up effects from applying the sinusoidally varying force) the motion becomes a steady sinusoidal displacement. This situation is known as the steady state.

Quick Refresher Guide

=

1−2

Theory of Machines

0.5

1.0

Resonant frequency

0

(1 − 2ξ )

2.0

2.5

= 0.1

3.0

is a maximum, when (X/X ) /dt is zero. This can be shown to be

1.5

Resonance

(kX) + (cωX) = kX 1 + . . . 5.4.3.5

=

. . . 5.4.3.6

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TR =

The ratio of transmitted force to that of disturbing force is known as Transmissibility ratio (TR). Mathematically, it is

F =

The force to be isolated is transmitted through spring and damper. Its equation is

Vibration Isolation: Vibration forces generated by machines and other causes are often unavoidable; however, their effects on a dynamical system can be minimized by proper isolator design. An isolation system reduces the excessive vibration transmission to the delicate objects from its supporting structure.

For small values of ξ, X/X is equal to 1/(2 ξ).

However note that there is no real solution for ω when ξ > 1/√2, i.e the response continuously falls with frequency. The final point of interest is the response amplitude at resonance, X/X

ω = ω

Resonance occurs, i.e. X/ when

0

1

2

3

4

5

Resonance:

At resonance the response peak equals 1/2ξ for small ξ. The value of ξ may be determined from the response curve for small ξ. The phase ϕ varies from 0 to – 180 degrees, i.e. the displacement lags the force.

There is thus no resonance (i.e. no peak in the response) when ξ >1/√2

The response curve has a resonance. The resonance is at a frequency

Notes:

Quick Refresher Guide

Machine Design

Maximum principle (or normal) stress theory (also known as Rankine’s theory). Maximum shear stress theory (also known as Guest’s or Tresca’s theory). Maximum principle (or normal) strain theory (also known as Saint Venant theory). Maximum strain energy theory (also known as Haigh’s theory). Maximum distortion energy theory (also known as Hencky and Von Mises theory). Octahedral Shearing Stress theory.

= Yield stress in tension as determined from simple tension test

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FOS = Factor of Safety

= Ultimate stress

Where,

=

=

Since the limiting strength for ductile materials is yield point stress and for brittle materials is ultimate stress, the maximum principle or normal stress ( ) is given by

The elastic failure or yielding occurs at a point in a member when the maximum principle or normal stress reaches the limiting strength of the material in a simple tension test irrespective of the value of other two principle stresses, i.e., when =

Maximum principle or Normal Stress Theory (Rankine’s Theory) for Brittle materials

Brittle materials, do not exhibit identifiable yield strength, and are typically classified by ultimate tensile and compressive strengths, Sut and Suc, respectively (where Suc is given as a positive quantity)

Ductile materials have identifiable yield strength that is often same in compression as in tension (Syt = Syc = Sy ).

1. 2. 3. 4. 5. 6.

But the problem of predicting the failure stresses for members subjected to bi-axial or tri-axial stresses is much more complicated, that a large number of different theories have been formulated. The principle theories of failure for a member subjected to tri-axial stress are as follows:

The strength of machine members is based upon the mechanical properties of the materials used. Since these properties are usually determined from simple tension or compression tests, predicting failure in members subjected to uniaxial stress is both simple and straight-forward.

6.1.1 Theories of Failure under Static Load

Part 6.1: Theory of Failures

Part – 6: Machine Design

Quick Refresher Guide

Machine Design

−

−

)=

)=

or

[

− (

+

)] =

+

+

=

=−

3(1 − 2 )

)] =

)] =

2

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This theory may be used for ductile materials.

But in fact with this type of loading (i.e.,) when there is uniform pressure all round (hydrostatic pressure), no failure occurs.

=

And in that case failure is predicted when

=

This theory breaks down for a case when,

1 [ + + −2 ( + 2 [ + + −2 ( +

According to this theory, the maximum energy which a body can store without deforming plastically is constant for that material irrespective of the manner of loading.

The failure or yielding occurs when the strain energy per unit volume in a strained material reaches the limiting strain energy (i.e. strain energy at the yield point ) per unit volume as determined from simple tension test.

Maximum Strain Energy Theory (Beltrami’s or Haigh’s Theory) for ductile materials

This theory over-estimates the elastic strength of ductile materials.

1

According to the above theory, the elastic failure will occur, when

The elastic failure occurs when the greatest principle (or normal) strain reaches the elastic limit point (i.e. strain at yield point) as determined from a simple tensile test.

Maximum Principle Strain Theory (Saint Venant’s Theory)

(

(

The elastic failure occurs when the greatest shear stress reaches a value equal to the shear stress at elastic limit in a simple tension test.

Maximum Shear Stress Theory (Guest’s or Tresca’s Theory) for ductile materials.

However, for brittle materials which are relatively strong in shear but weak in tension or compression, this theory is generally used.

Since this theory ignores the possibility of failure due to shearing stress, it is not used for ductile materials.

Quick Refresher Guide

Machine Design

−

) +( −

) +( −

) = 2

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Fig. 6.1.2 Octahedral surfaces

According to this theory, the critical quantity is the shearing stress on the octahedral plane. The plane which is equally inclined to all the three principle axes is called the octahedral plane.

Octahedral Shearing Stress Theory

Note: The maximum distortion energy is the difference between the total strain energy and the strain energy due to uniform stress.

This theory is mostly used for ductile materials in place of maximum strain energy theory.

Fig. 6.1.1 The distortion-energy (DE) theory for plane stress states

(

The failure or yielding occurs at a point in a member when the distortion strain energy (also called shear strain energy) per unit volume in the stressed material reaches the limiting distortion energy (i.e. distortion energy at yield point) per unit volume as determined from a simple tension test. Mathematically, the maximum distortion energy theory for yielding is expressed as

Shear Strain energy or Maximum Distortion Energy Theory (Hencky and Von Mises Theory)

Quick Refresher Guide

−

=

−

) +(

−

) ]

Machine Design

)

Nature and Properties of the material Type of loading Shape of member Temperature of member, etc.

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x x x x

Mode of failure of a ductile material differs from that of brittle material. It depends on a large number of factors like

6.1.3 Significance of theories of failure

5. Maximum distortion energy theory + − =

=

≥0>

4. Maximum strain energy theory + −2 =

(b) For unlike stresses − = ℎ >0> − =

3. Maximum shear stress theory (a) For like tensile stresses > >0 −0 = =

2. Maximum principle strain theory )= ( −

(

as zero, the above equations reduce to

1. Maximum principle stress theory =

Taking

6.1.2 Theories of failure for two dimensional stresses:

This theory is supported quite well by experimental evidences and is identical to Von Mises theory.

Failure is said to occur when

√

) +(

= Octahedral shearing stress

Where,

= [(

Quick Refresher Guide

Machine Design

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The maximum shear stress theory gives design values on the safe side and is widely used in design with ductile materials.

Results of many laboratory tests on ductile material shows shear stress from torsion tests varies between 0.55 and 0.6 of the yield strength determined from tension tests. This result agrees with shear strain energy theory and octahedral shear stress theory. The maximum shear stress theory predicts that the shear yield value is 0.5 times the tensile yield value, which is about 15% less than the value predicted by the other two theories.

In practice, in complex loading conditions, the factor associated with failure has to be identified and precautions taken to ensure that this factor does not exceed maximum allowable value determined on the basis of suitable tests (uniform tension or torsion) on the material in the laboratory.

If the loading conditions are suitably altered, a brittle material may be made to yield before failure. Hence, design of a member requires the determination of the mode of failure (yielding or fracture), and the factor (such as stress, strain and energy) associated with it. Full scale tests simulating all conditions would be ideal but not practicable.

Quick Refresher Guide

Machine Design

Maximum stress Nominal stress

1+

2

= 1+

2

2

Stress Concentration due to Holes and Notches:

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1+

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=

The stress concentration in the notched tension member, is influenced by the depth a of the notch and radius r at the bottom of the notch. The maximum stress, which applies to members having notches that are small in comparison with the width of the plate, may be obtained by the following equation,

Fig. 6.2.3 Stress Concentration due to notches

When a/b is large (a), the ellipse approaches a crack transverse to the load and the value of Kt becomes very large. When a/b is small (b), the ellipse approaches a longitudinal slit and the increase in stress is small. When the hole is circular (c), then a/b = 1 and the maximum stress is three times the nominal value.

=

and the theoretical stress concentration factor,

=

Consider a plate with transverse elliptical hole and subjected to a tensile load as shown in the figure.

In static loading, stress concentration in ductile materials is not so serious as in brittle materials, because in ductile materials local deformation or yielding takes place which reduces the concentration. In brittle materials, cracks may appear at these local concentrations of stress which will increase the stress over the rest of the section. In cyclic loading, stress concentration in ductile materials is always serious because the ductility of the material is not effective in relieving the concentration of stress caused by cracks, flaws, surface roughness, or any sharp discontinuity in the geometrical form of the member. If the stress at any point in a member is above the endurance limit of the material, a crack may develop under the action of repeated load and the crack will lead to failure of the member.

Machine Design

From the stress-distribution, the stress at the point away from the hole is practically uniform and the maximum stress is induced at the edge of the hole. The maximum stress is given by

Fig. 6.2.2 Stress Concentration due to holes

Quick Refresher Guide

Methods of Reducing Stress Concentration: x Maintain or improve the spacing of the stress flow lines that tend to bunch up and cut very close to the sharp re-entrant corner, by providing 9 Fillets and 9 Notches ( when not possible to use large radius fillets as in case of ball and roller bearing mountings)

x

x

The value of Kt depends upon the material and geometry of the part.

K =

The theoretical or form stress concentration factor is defined as the ratio of the maximum stress to the nominal stress at the same section based upon net area.

Theoretical or Form Stress Concentration Factor:

In the above member with different cross-section under a tensile load, the nominal stress in the right and left hand sides will be uniform but in the region where the cross-section is changing, a re-distribution of the force within the member must take place. The maximum stress occurs at some point on the fillet and is directed parallel to the boundary at that point.

Fig. 6.2.1

It occurs for all kinds of stresses in the presence of fillets, notches, holes, keyways, splines, surface roughness or scratches etc.

Whenever a machine component changes the shape of its cross-section, the simple stress distribution no longer holds good and the neighbourhood of the discontinuity is different. This irregularity in the stress distribution caused by abrupt changes of form is called stress concentration.

6.2.1 Stress concentration

Part 6.2: Fatigue

Quick Refresher Guide

Machine Design

The stress concentration effects of a press fit may be reduced by making more gradual transition from the rigid to the more flexible shaft

Fig. 6.2.5

Fig. 6.2.6

The stresses which vary from a minimum value to a maximum value of the opposite nature (i.e. from a certain minimum compressive to a certain maximum tensile or from a minimum tensile to a maximum compressive) are called alternating stresses.

stresses.

The stresses which vary from zero to a certain maximum value are called repeated

Fig. 6.2.4

The stresses which vary from a minimum value to a maximum value of the same nature, (i.e. tensile or compressive) are called fluctuating stresses.

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The variable stress, in general, may be considered as a combination of steady (or mean or average) stress and a completely reversed stress component.

x

x

x

6.2.2 Dynamic loading

x

Quick Refresher Guide

+

−

=

For completely reversed stresses, R = – 1 For repeated stresses, R = 0. R cannot be greater than unity.

=

= Stress ratio.

= Endurance limit for completely reversed stresses, and

= Endurance limit for any stress range represented by R.

−

= 0) in each cycle.

Machine Design

Design of rapid changes in cross section, keyways, holes, etc. where stress concentrations occur THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 193

x

Discontinuities can arise because of:

Fatigue failure is due to crack formation and propagation. A fatigue crack will typically initiate at a discontinuity in the material where the cyclic stress is a maximum.

The fatigue of material is effected by the size of the component, relative magnitude of static and fluctuating loads and the number of load reversals.

When a material is subjected to repeated stresses, it fails at stresses below the yield point stresses. Such type of failure of a material is known as fatigue.

6.2.3 Fatigue

Where,

=

4. Relation between endurance limit and stress ratio

x x x

=

3. Stress ratio,

∴

Note: For repeated loading, the stress varies from maximum to zero (i.e.

=

2. Reversed stress component or alternating or variable stress,

=

1. Mean or average stress,

Quick Refresher Guide

residual tensile stresses elevated temperatures temperature cycling corrosive environment high-frequency cycling

Stress-life method Strain-life method Linear-elastic fracture mechanics method

- high-cycle fatigue

N > 103

Based on stress levels only Least accurate approach, especially for low-cycle applications. Easiest to implement for a wide range of design applications, Has ample supporting data, and Represents high-cycle applications adequately.

Machine Design

For steel and iron, the S-N diagram becomes horizontal at some point. The strength at this point is called the endurance limit [maximum value of the completely reversed bending stress which a polished standard specimen can withstand without failure for infinite number of cycles (usually THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 195

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Fig. 6.2.9 S-N bands for representative aluminum alloys, excluding wrought alloys with Sut < 38 kpsi

Fig. 6.2.8 S-N diagram for steel, normalized; Sut = 116 kpsi; maximum Sut = 125 kpsi.

Fig. 6.2.7 Test-specimen geometry for the R.R. Moore rotating beam machine.

beam machine) of standard laboratory controlled specimens. The loading often is that of sinusoidally reversing pure bending. The laboratory-controlled specimens are polished without geometric stress concentration at the region of minimum area.

Quick Refresher Guide

The strength-life (S-N) diagram provides the fatigue strength Sf versus cycle life N of a material. The results are generated from tests using a simple loading (R. R.Moore high-speed rotating-

Fatigue Strength and the Endurance Limit:

9 Assumes a crack is already present and detected. It is then employed to predict crack growth with respect to stress intensity. 9 Most practical when applied to large structures in conjunction with computer codes and a periodic inspection program.

Fracture mechanics method:

9 More detailed analysis of the plastic deformation at localized regions where the stresses and strains are considered for life estimate 9 Good Method for low-cycle fatigue applications 9 In applying this method, several idealizations must be compounded, and so some uncertainties will exist in the results.

Strain-life method:

9 9 9 9 9

Stress-life method:

- low-cycle fatigue

N ≤ 103

These methods attempt to predict the life in number of cycles to failure, N, for a specific level of loading.

x x x

Fatigue-Life Methods

x x x x x

Machine Design

Elements that roll and/or slide against each other (bearings, gears, cams, etc.) under high contact pressure, developing concentrated subsurface contact stresses that can cause surface pitting or spalling after many cycles of the load. Carelessness in locations of stamp marks, tool marks, scratches, and burrs; poor joint design; improper assembly; and other fabrication faults. Composition of the material itself as processed by rolling, forging, casting, extrusion, drawing, heat treatment, etc. Microscopic and submicroscopic surface and subsurface discontinuities arise, such as inclusions of foreign material, alloy segregation, voids, hard precipitated particles, and crystal discontinuities.

Conditions that accelerate crack initiation:

x

x

x

Quick Refresher Guide

Machine Design

.

=

The variation in the size of the component should be as gradual as possible. The holes, notches and other stress raisers should be avoided. The proper stress de-concentrators such as fillets and notches should be provided wherever necessary. The parts should be protected from corrosive atmosphere. A smooth finish of outer surface of the component increases the fatigue life. The material with high fatigue strength should be selected. The residual compressive stresses over the parts surface increases its fatigue strength.

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When a machine member is subjected to cyclic or fatigue loading, the value of fatigue stress concentration factor shall be applied instead of theoretical stress concentration factor. Since the determination of fatigue stress concentration factor is not an easy task, therefore from experimental tests it is defined as

Fatigue Stress Concentration Factor:

x x x x

x x x

Factors to be Considered while Designing Machine Parts to Avoid Fatigue Failure:

= Yield point stress.

= Endurance limit stress for completely reversed stress cycle, and

Where,

= .

For Steel,

Factor of safety (FOS) =

When a component is subjected to fatigue loading, the endurance limit is the criterion for failure. Therefore, the factor of safety should be based on endurance limit. Mathematically,

Factor of Safety for Fatigue Loading:

Modifying factors are defined and used to account for differences between the specimen and the actual machine part with regard to surface conditions, size, loading, temperature, reliability, and miscellaneous factors.

Endurance Limit Modifying Factors:

The term endurance limit is used for reversed bending only while for other types of loading, the term endurance strength may be used when referring the fatigue strength of the material. It may be defined as the safe maximum stress which can be applied to the machine part working under actual conditions.

107 cycles)]. S′e and occurs somewhere between 106 and 107 cycles. For non-ferrous materials that do not exhibit an endurance limit, a fatigue strength at a specific number of cycles, S′f , may be given. The strength data are based on many controlled conditions that will not be the same as that for an actual machine part. What follows are practices used to account for the differences between the loading and physical conditions of the specimen and the actual machine part.

Quick Refresher Guide

Quick Refresher Guide

Machine Design

≤1

−1 −1

is plotted on the horizontal axis and the alternating stress σ

is plotted

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To establish the factor of safety relative to the Soderberg Criteria

If the point of the combined stress is below the Soderberg line then the component will not fail. This is a very conservative criteria based on the material yield point S

Soderberg Line

The Mean Stress σ on the vertical axis.

Mean Stress

Modified Goodman Line

Goodman Line

Gerber Line

Soderberg Line

A number of interaction criteria are used to quantify the combined stress and the relevant design factors of safety. These are plotted together below

The mean stress can have a significant effect on the failure due to fatigue and must be considered in combination with the alternating stress.. (Under normal fatigue loading conditions the mean stress is small compared to the alternating stress.)

Combination of Mean Stress and Fluctuating stress:

0≤

=

The notch sensitivity of a material is a measure of how sensitive a material is to notches or geometric discontinuities. Mathematically, it is expressed as

Notch Sensitivity Factor:

=

Fatigue stress concentration factor,

Stress Amplitude

=

1

Machine Design

=

S S S N

= The Modified fatigue strength = The ultimate tensile strength = The yield tensile strength = The Factor of Safety applicable for Fatigue

=1

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x x x x

+

To establish the factor of safety relative to the Gerber Criteria

If the point of the combined stress is below the Gerber line then the component will not fail. This is a less conservative criteria based on the material Ultimate strength S

Gerber Line

+

1

To establish the factor of safety relative to the Goodman Criteria

If the point of the combined stress is below the relevant Goodman line then the component will not fail. This is a less conservative criteria based on the material Ultimate strength yield point S

Goodman Line/Modified Goodman Line

+

Quick Refresher Guide

6.3.1 Design of riveted joints

Lap

80-88

76-84

Triple riveted

55-60

Double riveted

72-80

Single riveted

60-72

Triple riveted

50-60

Efficiencies (in %) Double riveted

Single riveted

Machine Design

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Figure 6.3.1.7 Important design parameters of riveted joint

These parameters are shown in figure 6.3.1.7. g

a. Pitch: This is the distance between two centers of the consecutive rivets in a single row. (usual symbol p) b. Back Pitch: This is the shortest distance between two successive rows in a multiple riveted joint. (usual symbol p or p ) c. Diagonal pitch: This is the distance between the centers of rivets in adjacent rows of zigzag riveted joint. (usual symbol d ) d. Margin or marginal pitch: This is the distance between the centre of the rivet hole to the nearest edge of the plate. (usual symbol m)

Few parameters, which are required to specify arrangement of rivets in a riveted joint are as follows:

Joints

Table 6.3.1.Efficiencies of riveted joints (in %)

Butt (double strap)

1. Lap joints 2. Butt joints

Riveted joints are mainly of two types

Types of riveted joints and joint efficiency:

Part 6.3: Design of Machine Elements

Quick Refresher Guide

Machine Design

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Figure 6.3.1.3 Failure of a rivet by shearing

d for double strap butt joint = 2s where s = allowable shear stress of the rivet material.

b) Shearing of the rivet: The rivet may shear as shown in Figure 6.3.1.3 . The maximum force withstood by the joint to prevent this failure is d for lap joint, single strap butt joint P =s

Figure 6.3.1.2 Failure of plate in tension (tearing)

t = thickness of the plate

d = diameter of the rivet hole

p = pitch

where s = allowable tensile stress of the plate material

P = s (p − d)t

a) Tearing of the plate: If the force is too large, the plate may fail in tension along the row (see figure 6.3.1.2). The maximum force allowed in this case is

Strength of a riveted joint is evaluated taking all possible failure paths in the joint into account. Since rivets are arranged in a periodic manner, the strength of joint is usually calculated considering one pitch length of the plate. There are four possible ways a single rivet joint may fail.

Strength of riveted joint:

Quick Refresher Guide

Machine Design

(

,

,

)

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efficiency (η)=

s pt. Thus

Efficiency of the single riveted joint can be obtained as ratio between the maximum of P , P and P and the load carried by a solid plate which is

Efficiency:

Figure 6.3.1.5 Tearing of the plate at the edge

d) Tearing of the plate at edge: If the margin is too small, the plate may fail as shown in figure 6.3.1.5. To prevent the failure a minimum margin of m = 1.5d is usually provided.

Figure 6.3.1.4 Failure of rivets by

where s = allowable bearing stress between the rivet and plate material.

c) Crushing of rivet: If the bearing stress on the rivet is too large the contact surface between the rivet and the plate may get damaged. (see Figure 6.3.1.4). With a simple assumption of uniform contact stress the maximum force allowed is P = s dt

Quick Refresher Guide

Machine Design

d s (valid for t < 8

)

d

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1. Design of a butt joint: The main failure mechanism of welded butt joint is tensile failure. Therefore the strength of a butt joint is

6.3.2 Design of welded joints

(p : usually 0.33p + 0.67d mm)

In order to design boiler joints, a designer must also comply with Indian Boiler Regulations (I.B.R.).

Margin (m): m = 1.5d

But p ≥ 2d in order to accommodate heads of the rivets.

s (p − d)t = s × 2

Pitch (p): Pitch is designed by equating the tearing strength of the plate to the shear strength of the rivets. In a double riveted lap joint, this takes the following form.

However, d should not be less than t, in any case. The standard size of d is tabulated in code IS: 1928-1961.

s t=

Otherwise d is obtained by equating crushing strength to the shear strength of the joint. In a double riveted zigzag joint, this implies

Diameter of the hole (d): When thickness of the plate (t) is more than 8 mm, Unwin’s formula is used, = 6√t mm.

The design parameters in a riveted joints are, d, p, and m

Design of rivet joints:

The efficiency should be calculated taking all possible failure mechanism into consideration.

(i) shearing of rivets in both rows (ii) crushing of rivets in both rows (iii) shearing of rivet in one row and crushing in the other row.

Further, the joint may fail by

P = s (p − d)t + min{P , P }

In a double or triple riveted joint the failure mechanisms may be more than those discussed above. The failure of plate along the outer row may occur in the same way as above. However, in addition the inner rows may fail. For example, in a double riveted joint, the plate may fail along the second row. But in order to do that the rivets in the first row must fail either by shear or by crushing. Thus the maximum allowable load such that the plate does not tear in the second row is

Quick Refresher Guide

Machine Design

√

= throat area.

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Figure 6.3.2.2 Design of a single transverse fillet

For a double transverse fillet joint the allowable load is twice that of the single fillet joint.

A

where s = allowable shear stress

P=s A

consideration the permissible load carried by a transverse fillet weld is

2. Design of transverse fillet joint: Consider a single transverse joint as shown in figure 6.3.2.2. The general stress distribution in the weld metal is very complicated. In design, a simple procedure is used assuming that entire load P acts as shear force on the throat area, which is the smallest area of the cross section in a fillet weld. If the fillet weld has equal base and . With the above height, (h, say), then the cross section of the throat is easily seen to be

Figure 6.3.2.1 Design of a butt joint

For a square butt joint t is equal to the thickness of the plates. In general, this need not be so (see figure 6.3.2.1).

l = length of the weld.

t = thickness of the weld

where s = allowable tensile strength of the weld material.

=

Quick Refresher Guide

Machine Design

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4. Design of circular fillet weld subjected to torsion: Consider a circular shaft connected to a plate by means of a fillet joint as shown in figure-6.3.2.5. If the shaft is subjected to a torque,

Figure 6.3.2.4 Design of combined transverse and parallel fillet joint

A ′= throat area along the transverse direction.

where A = throat area along the longitudinal direction.

P = 2s A + s A ′

In designing a weld joint the design variables are h and l. They can be selected based on the above design criteria. When a combination of transverse and parallel fillet joint is required (see figure-6.3.2.4) the allowable load is

Figure 6.3.2.3 Design of a parallel fillet joint

P = 2s A .

√

3. Design of parallel fillet joint: Consider a parallel fillet weld as shown in figure 6.3.2.3. Each weld carries a load P 2. It is easy to see from the strength of material approach that the maximum shear occurs along the throat area (try to prove it). The allowable load carried by each of the joint is s A where the throat area A = . The total allowable load is

Quick Refresher Guide

Machine Design

=

)

= throat thickness

≪ d, τ

=

[(d + 2t =

) −d ]

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Shaft is a common and important machine element. It is a rotating member, in general, has a circular cross-section and is used to transmit power. The shaft may be hollow or solid. The shaft is supported on bearings and it rotates a set of gears or pulleys for the purpose of power transmission. The shaft is generally acted upon by bending moment, torsion and axial force. Design of shaft primarily involves in determining stresses at critical point in the shaft that is arising due to mentioned loading.

6.3.3 Design of Shafts

Fig. 6.3.2.5 Design of a fillet weld for torsion

=s

The throat dimension and hence weld dimension can be selected from the equation

When, t

=

I = polar moment of area of the throat section.

t

(

d = outer diameter of the shaft

where T = torque applied.

τ

shear stress develops in the weld in a similar way as in parallel fillet joint. Assuming that the weld thickness is very small compared to the diameter of the shaft, the maximum shear stress occurs in the throat area. Thus, for a given torque the maximum shear stress in the weld is

Quick Refresher Guide

Machine Design

(

)

d : Outer diameter of the shaft

M: Bending moment at the point of interest

σ =

(

)

α: Column-action factor(= 1.0 for tensile load)

F: Axial force (tensile or compressive)

σ =

( / )

for L/K > 115

for L/K < 115

K = least radius of gyration, L = shaft length

n = 1.6 for ends partly restrained, as in bearing

n = 2.25 for fixed end

n = 1.0 for hinged end

.

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Where,

α=

α=

Here, α is defined as,

The term α has been introduced in the equation. This is known as column action factor. What is a column action factor? This arises due the phenomenon of buckling of long slender members which are acted upon by axial compressive loads.

Where,

Axial Stress

k: Ratio of inner to outer diameters of the shaft ( k = 0 for a solid shaft because inner diameter is zero )

Where,

Bending stress

Basic stress equations:

The stress at any point on the shaft depends on the nature of load acting on it. The stresses which may be present are as follows.

Design based on Strength

Quick Refresher Guide

)

T: Torque on the shaft

=

Machine Design

(

)

+ (

)

= τ

=

=

(

)

C

M+

(

)

+ (C T)

in the above equation, the final form is,

+τ

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The bearings are classified broadly into two categories based on the type of contact they have between the rotating and the stationary member

They permit relative motion of two members in one or two directions with minimum friction, and also prevent the motion in the direction of the applied load.

Bearings are machine elements which are used to support a rotating member viz., a shaft. They transmit the load from a rotating member to a stationary member known as frame or housing.

6.3.4 Design of Bearings

Therefore, the shaft diameter can be calculated in terms of external loads and material properties. However, the above equation is further standarized for steel shafting in terms of allowable design stress and load factors in ASME design code for shaft.

τ

Substituting the values of σ and τ

τ

Design of the shaft mostly uses maximum shear stress theory. It states that a machine member fails when the maximum shear stress at a point exceeds the maximum allowable shear stress for the shaft material. Therefore,

Maximum shear stress theory

The net normal stress can be either positive or negative. Normally, shear stress due to torsion is only considered in a shaft and shear stress due to load on the shaft is neglected.

σ =

Both bending and axial stresses are normal stresses, hence the net normal stress is given by,

Combined Bending and Axial stress

τ : Shear stress due to torsion

Where,

τ (

= yield stress in compression

Stress due to torsion

σ

Quick Refresher Guide

Machine Design

(c) Fitted

.

is called the bearing characteristic or the bearing

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is known as the clearance ratio.

modulus. This a very important dimensionless parameter.

In the above relation the parameter

r, c = radius and radial clearance respectively.

p = pressure and

N = rev/min or rev/sec,

= viscosity,

where f = coefficient of friction,

f = 2π .

The relation between bearing friction and viscosity of the lubricant in a circular journal bearing which is running truly concentric is given by,

Petroff’s Equation:

(b) Partial

Figure 6.3.4.1 Various types of journal bearings.

(a) Full

The radial bearings are also called journal or sleeve bearings. The portion of the shaft inside the bearing is called the journal and this portion needs better finish and specific property. Depending on the extent to which the bearing envelops the journal, these bearings are classified as full, partial and fitted bearings. As shown in Fig. 6.3.4.1

Among the sliding contact bearings radial bearings find wide applications in industries and hence these bearings are dealt in more detail here.

Journal / sleeve bearings:

The sliding contact bearings have surface contact and come under lower kinematic pair.

a. Sliding contact b. Rolling contact

Quick Refresher Guide

Machine Design

B

Figure 6.3.4.2

μN/P

C

O

e

Line of centres

O′

Journal

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c is the radial clearance and is the difference in the radii of the bearing and the journal.

In a journal bearing, the following nomenclature is used.

As a result of the lubricant pressure, a minimum film thickness h occurs, not at the bottom of the journal, but displaced in the direction of rotation, as shown in the Figure 6.3.4.3. This is because the lubricant pressure in converging gap reaches maximum at a point to the left of the bearing centre.

Figure 6.3.4.3

h β c = radial clearance

Bearing

When a journal starts rotating in a bearing as shown in Figure 6.3.4.3, below the lubricant is forced into a wedge shaped (strictly a curved wedge) space by a pumping type of action and the pressure built up in the wedge supports the load on the journal.

However, if we are operating in the region BC, the lubrication is stable and is known as thick film or hydrodynamic lubrication.

f

A

left of point B, the lubrication is not stable and is known as boundary lubrication.

It has been shown experimentally that the coefficient of friction varies as shown in Figure 6.3.4.2. In which a curve of f versus is plotted. It is found that the operation of a bearing to the

If a lubricant film is enclosed in a wedge or tapered gap between the stationary and moving members, the oil film is drawn into the wedge shape generating a pressure that can support a load.

Hydrodynamic lubrication:

Quick Refresher Guide

Machine Design

also = 1 −

.

=

/

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Basic mechanism of braking: The illustration below explains the working of mechanical brakes. An element dA of the stationary member is shown with the braked body moving past at velocity v. When the brake is actuated contact is established between the stationary and moving member and a normal pressure is developed in the contact region. The elemental normal force dN is equal to the product of contact pressure p and area of contact dA. As one member is stationary

In mechanical brakes artificial resistances are created using frictional contact between the moving member and a stationary member, to retard or stop the motion of the moving member.

Mechanical Brakes: Mechanical brakes are invariably used based on the frictional resistance principles.

Types of Brakes: Different types of brakes are used in different applications. Based on the working principle used, brakes can be classified as mechanical brakes, hydraulic brakes, electrical (eddy current) magnetic and electro-magnetic types.

A brake is a device by means of which artificial resistance is applied on to a moving machine member in order to retard or stop the motion of the member or machine

6.3.5 Brakes

When a bearing is installed there is no way of knowing whether it is one of the 90 per cent that are good or one of the 10 per cent that will not attain the rating life. In other words, one can have but 90 per cent confidence that the bearing will achieve or exceed its rating life, usually designated L .

where L is the millions of revolutions that 90% of a group of bearings (which are apparently identical) will complete before any of them develops evidence of fatigue.

L

L = (b) Roller bearings

(a) Ball bearings

Rated life of a ball or roller bearings:

A bearing in which the radii of the bearing and the journal are equal, is known as a fitted bearing.

β shown in Figure is the angular length of a partial bearing, if it is not a full circle.

∈ is the eccentricity ratio =

h is the minimum film thickness and it occurs on the line of centres. (The film thickness at any other point is normally designated as h)

e is the eccentricity and is the distance between the centers O and O′ of the journal and bearing respectively.

Quick Refresher Guide

Machine Design

The braking torque The actuating force needed The energy loss and temperature rise

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Figure 6.3.5.1

Brakes Classification: Various geometric configuration of drum brakes are illustrated below:

Drum brakes basically consists of a rotating body called drum whose motion is braked together with a shoe mounted on a lever which can swing freely about a fixed hinge H. A lining is attached to the shoe and contacts the braked body. The actuation force P applied to the shoe gives rise to a normal contact pressure distributed over the contact area between the lining and the braked body. A corresponding friction force is developed between the stationary shoe and the rotating body which manifest as retarding torque about the axis of the braked body.

There are two major classes of brakes, namely drum brakes and disc brakes. Design and analysis of drum brakes will be considered in detail in following sections, the discussion that follow on disc or plate clutches will form the basis for design of disc type of brakes.

x x x

Design and Analysis: To design, select or analyze the performance of these devices knowledge on the following are required.

The moment of the frictional force relative to the point of motion contributes to the retardation of motion and braking. The basic mechanism of braking is illustrated above.

and the other is in relative motion, a frictional force dF is developed between the members. The magnitude of the frictional force is equal to the co-efficient of friction times the normal force dN

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Machine Design

Figure 6.3.5.2

(b) Free-body diagram

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For a short shoe we assume that the pressure is uniformly distributed over the contact area. Consequently the equivalent normal force F = p. A, where = p is the contact pressure and A is the surface area of the shoe. Consequently the friction force Ff = f.Fn where f is the co-efficient of friction between the shoe lining material and the drum material.

¾ Short Shoe Analysis:

(a) Brake assembly

Preliminary Analysis: The figure shows a brake shoe mounted on a lever, hinged at O, having an actuating force F , applied at the end of the lever. On the application of an actuating force, a normal force F is created when the shoe contacts the rotating drum. And a frictional force F of magnitude f. F , f being the coefficient of friction, develops between the shoe and the drum. Moment of this frictional force about the drum center constitutes the braking torque.

When the area of contact becomes larger, the contact may no longer be with a uniform pressure, in which case the shoe is termed as long shoe. The shoes are either rigid or pivoted, pivoted shoes are also some times known as hinged shoes. The shoe is termed rigid because the shoes with attached linings are rigidly connected to the pivoted posts. In a hinged shoe brake – the shoes are not rigidly fixed by hinged or pivoted to the posts. The hinged shoe is connected to the actuating post by the hinge, G, which introduces another degree of freedom.

Drum Brakes are classified based on the shoe geometry. Shoes are classified as being either short or long. A short shoe is one whose lining dimension in the direction of motion is so small that contact pressure variation is negligible, i.e. the pressure is uniform everywhere.

Quick Refresher Guide

Machine Design

F = 0, R = p A − F

F = 0, R = fp A

For a given direction of rotation the shoe in which self energization is present is known as the leading shoe When the direction of rotation is changed, the moment of frictional force now will be opposing the actuation force and hence greater magnitude of force is needed to create the same contact pressure. The shoe on which this is prevailing is known as a trailing shoe

¾

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Short and Long Shoe Analysis: Foregoing analysis is based on a constant contact pressure p. In reality, constant or uniform constant pressure may not prevail at all points of contact on the shoe. x In such case the following general procedure of analysis can be adopted x x

Self Locking: At certain critical value of f.c. the term (b-fc) becomes zero. i.e no actuation force need to be applied for braking. This is the condition for self-locking. Self-locking will not occur unless it is specifically desired.

x

x

Leading and trailing shoe:

With the shown direction of the drum rotation (CCW), the moment of the frictional force f. F c adds to the moment of the actuating force F . As a consequence, the required actuation force needed to create a known contact pressure p is much smaller than that if this effect is not present. This phenomenon of frictional force aiding the brake actuation is referred to as self – energization.

The principle of self energizing and leading and trailing shoes

Self – energizing:

i.e.

The reaction forces on the hinged pin (pivot) are found from a summation of forces,

F = F (b − fc)/a

Substituting for F and solving for the actuating force, we get,

∑M = F a − F b + f F c = 0

Applying the equilibrium condition by taking moment about the pivot ‘O’ we can write

A quasi static analysis is used to determine the other parameters of braking.

T = f Fn. r = f.p.A.r

The torque on the brake drum is then,

Quick Refresher Guide

Machine Design

Estimate or determine the distribution of pressure on the frictional surfaces Find the relation between the maximum pressure and the pressure at any point For the given geometry, apply the condition of static equilibrium to find the actuating force, torque and reactions on support pins etc.

The torque transmitted The actuating force. The energy loss The temperature rise

Machine Design

Figure 6.3.6.3 A single-Surface Axial Disk Clutch

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2 2πpfr dr = πpf r − r 3 Integrating the normal force between the same limits we get the actuating force that need to be applied to transmit this torque.

T=

The total torque that could be transmitted is obtained by integrating this equation between the limits of inner radius r to the outer radius r

= f. p. 2. π. r. dr . r

i.e. T = dF. r = f. dN. r = f. p. A. r

Now the torque that can be transmitted by this elemental area is equal to the frictional force times the moment arm about the axis that is the radius ‘r’

dF = f.2π.r.dr.p

The frictional force dF on this area is therefore

dN = 2π.r.dr.p

The normal force on this elemental area is

dA = 2Π.r dr

Uniform pressure condition: Assuming uniform pressure and considering an elemental area dA

the contact pressure appropriately and uniform pressure condition may no longer prevail. Hence the analysis here is based on uniform wear condition

Quick Refresher Guide

The applied force can keep the members together with a uniform pressure all over its contact area and the consequent analysis is based on uniform pressure condition. However as the time progresses some wear takes place between the contacting members and this may alter or vary

Figure 6.3.6.2

Frictional Contact axial or Disc Clutches: An axial clutch is one in which the mating frictional members are moved in a direction parallel to the shaft. A typical clutch is illustrated in the figure below. It consist of a driving disc connected to the drive shaft and a driven disc connected to the driven shaft. A friction plate is attached to one of the members. Actuating spring keeps both the members in contact and power/motion is transmitted from one member to the other. When the power of motion is to be interrupted the driven disc is moved axially creating a gap between the members as shown in the figure.

Friction Clutches: As in brakes a wide range of clutches are in use wherein they vary in their are in use their working principle as well the method of actuation and application of normal forces. The discussion here will be limited to mechanical type friction clutches or more specifically to the plate or disc clutches also known as axial clutches.

1. 2. 3. 4.

To design analyze the performance of these devices, a knowledge on the following are required.

A Clutch is a machine member used to connect the driving shaft to a driven shaft, so that the driven shaft may be started or stopped at will, without stopping the driving shaft. A clutch thus provides an interruptible connection between two rotating shafts

6.3.6 Clutch

x x x

General Procedure of Analysis:

Quick Refresher Guide

2πprdr

2 r −r 3 (r − r )

Machine Design

is assumed to be proportional to the product of pressure p and

r

must then occur at the smallest radius r ,

2πprdr =

2π p

r rdr = 2πp r

for p.

r (r − r )

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Similarly the Torque

F =

and integrating equation dN = 2πprdr

i.e The axial force F is found by substituting p = p

In the previous equations substituting this value for the pressure term p and integrating between the limits as done earlier we get the equation for the torque transmitted and the actuating force to be applied.

p=p

Hence pressure at any point in the contact region

K=p

The largest pressure p

pr = constant = K

Combining these equation, assuming a constant angular velocity ω

And the velocity at any point on the face of the clutch is V = r.ω

R = pV = constant

The constant-wear rate R velocity V.

According to some established theories the wear in a mechanical system is proportional to the ‘PV’ factor where P refers to the contact pressure and V to the sliding velocity. Based on this for the case of a plate clutch we can state

Uniform Wear Condition:

T = fF .

Equation 1 and 2 can be combined together to give equation for the torque

F = π r − r .p

F =

Quick Refresher Guide

f2πp

r rdr = fπp

¾

r r −r

Machine Design

T = N. f. F . R

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The geometry of a flywheel may be as simple as a cylindrical disc of solid material, or may be of spoked construction like conventional wheels with a hub and rim connected by spokes or arms

Geometry of Flywheel

A flywheel is an inertial energy-storage device. It absorbs mechanical energy and serves as a reservoir, storing energy during the period when the supply of energy is more than the requirement and releases it during the period when the requirement of energy is less than the supply.

6.3.7 Flywheel

n2 = number of driven discs

Where n1 = number of driving discs

Note that N = n1 + n2 – 1

R is the mean or equivalent radius (m).

F is the actuating force (N).

f is the coefficient of friction

N is the number of frictional discs in contact.

T is the torque (Nm).

By rearranging the terms the equations can be modified and a less general form of the equation can be written as

As both side surfaces of the clutch plate is used for transmitting the torque, a term ‘N’ is added to include the number of surfaces used for transmitting the torque

Operation of Clutch: When the driver releases the clutch pedal, power can flow through the clutch. Springs in the clutch force the pressure plate against the friction disc. This action clamps the friction disc tightly between the flywheel and the pressure plate. Now, the pressure plate and friction disc rotate with the flywheel.

Basically, the clutch needs three parts. These are the engine flywheel, a friction disc called the clutch plate and a pressure plate. When the engine is running and the flywheel is rotating, the pressure plate also rotates as the pressure plate is attached to the flywheel. The friction disc is located between the two. When the driver has pushed down the clutch pedal the clutch is released. This action forces the pressure plate to move away from the friction disc. There are now air gaps between the flywheel and the friction disc, and between the friction disc and the pressure plate. No power can be transmitted through the clutch.

T = fF .

(r + r ) 2 Single Clutch and Multiple Disk Clutch

The equation can be given as

Substituting the values of actuating force F

T=

Quick Refresher Guide

Machine Design

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The equation is better solved by geometric proportions i.e by assuming inside to outside radius ratio and radius to thickness ratio.

Where γ is material’s weight density

πγ r −r t 2g

γ W =π r −r t g g

I =

Combing the two equations we can write

m=

For a solid disc geometry with inside radius r and outside radius r , the mass moment of inertia I is m I = mk = r +r 2 The mass of a hollow circular disc of constant thickness t is

The latter arrangement is a more efficient of material especially for large flywheels, as it concentrates the bulk of its mass in the rim which is at the largest radius. Mass at largest radius contributes much more since the mass moment of inertia is proportional to mr

Figure 6.3.7.2 Arm Type Flywheel

Figure 6.3.7.1

Small fly wheels are solid discs of hollow circular cross section. As the energy requirements and size of the flywheel increases the geometry changes to disc of central hub and peripheral rim connected by webs and to hollow wheels with multiple arms.

Quick Refresher Guide

Machine Design

3+v 8

γ ω g r +r −

r +r − r r −r r

1 + 3v r 3+v

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The point of most interest is the inside radius where the stress is a maximum. What causes failure in a flywheel is typically the tangential stress at that point from where fracture originated and upon fracture fragments can explode resulting extremely dangerous consequences, Since the forces causing the stresses are a function of the rotational speed also, instead of checking for stresses, the maximum speed at which the stresses reach the critical value can be determined and safe operating speed can be calculated or specified based on a safety factor. Consequently ω F. O. S (N) = N = ω

Figure 6.3.7.3

γ = material weight density, ω = angular velocity in rad/sec. v = Poisson’s ratio, is the radius to a point of interest, r and r are inside and outside radii of the solid disc flywheel.

σ =

3+v 8

γ σ = ω g

Flywheel being a rotating disc, centrifugal stresses acts upon its distributed mass and attempts to pull it apart. Its effect is similar to those caused by an internally pressurized cylinder

Stresses in Flywheel

Quick Refresher Guide

=

(

(

)

)

= 1/ (reciprocal of density)

;

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Units: No units (ratio)

For gases, standard fluid is hydrogen or air.

For liquids, standard fluid is water at 40C

Specific gravity, G =

Specific gravity (G):

Units: m3/kg

Specific Volume ( ):: Volume occupied by a unit mass of fluid,,

Units: N/m3, Dimensions: ML2T2 or FL3

where g = acceleration due to gravity

Mass density and specific weight of a fluid are related as:

Also represents force exerted by gravity on a unit volume fluid.

Specific Weight ( ):: Weight of the substance per unit volume.

As pressure increases mass density increases. (As large number of molecules are forced into a given volume)

Absolute quantity i.e., does not change with location L2T

CGS:

or stokes

=

2. Pump impellers 4. Ship propellers.

(

)

(

)

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It can also be expressed as a line surface: Force per unit length.

A liquid forms an interface with a second liquid or gas. This liquid – air interface behaves like a membrane under tension. The surface energy per unit area of interface is called Surface Tension.

Adhesion: force of attraction between the molecules of different liquids (or) between the liquid molecules and solid boundary containing the liquid.

Cohesion: Force of attraction between the molecules of the same liquid.

Surface tension:

=

(negative sign indicates a decrease in volume with increase in pressure) Coefficient of compressibility

=

Compressibility: Change in volume (or density) due to change in pressure. Compressibility is inversely proportional to Bulk Modulus K.

1. Turbine runners 3. Hydraulic structures like spillways and sluice gates

Cavitation: Occurs in a flow system, dissolved gases (vapour bubbles) carried into a region of high pressure and their subsequent collapse gives rise to high pressure, which leads to noise, vibrations and erosion. Cavitation occurs in

Kinematic viscosity depends on both pressure and Temperature

Dimensions:

Units: S I:

cm2/sec

Mass Density ( ):: It is defined as mass per unit volume. Unit: kg / m3, Dimension: M / L3 m2/sec

Kinematic Viscosity ( ) =

7.1.2 Properties of fluids:

or FL T

It is independent of pressure. For Liquids dynamic viscosity decreases with temperature because molecular momentum increases and cohesion is negligible in gases.

Dimensions: ML T

Conversion: 1 poise = 0.1 Pa.sec.

CGS: Poise = dyne.sec/cm2

Units: SI: Pa.sec or N.sec/m2 MKS: kg/m.sec

Viscosity: A measure of its resistance to shear or angular deformation. A property by virtue of which it offers resistance to the movement of one layer of fluid over the adjacent layer. It is due to Intermolecular cohesion and Transfer of molecular momentum between layers.

Both liquids and Gases come under the fluids.

i) Liquid: has definite volume but no shape for all practical purposes incompressible ii) Gas: has no shape and volume highly compressible iii) Vapour: A gas whose temperature and pressure are such that it is very near to the liquid phase e.g.: Steam

Fluid Mechanics

Since the density of fluid varies with temperature, specific gravity must be determined and specified at a particular temperature.

Specific gravity of water = 1.0, Mercury = 13.6

Quick Refresher Guide

Dynamic Viscosity ( )::

Fluid Mechanics

It is defined as a substance which deforms continuously even with a small amount of shear force exerts on it, whereas a solid offers resistance to the force because very strong intermolecular attraction exists in it.

7.1.1 Fluids

Part 7.1: Fluid Properties

Part – 7: Fluid Mechanics

Quick Refresher Guide

=

, where d = dia of bubble.

, where d = dia of drop let

Water

mercury

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σ = surface tension

Figure. 7.1.3

h

. . . (7.1.1)

For e.g., Mercury depressive with convex upwards is capillary rise or fall

Capillary rise: If the adhesion > cohesion

Units: cm or mm of liquids

Figure. 7.1.2

h

Capillarity: The phenomenon of rise or fall of a liquid surface relative to the adjacent general level of liquid in small diameter tubes. The rise of liquid surface is designated as capillary rise and lowering is called capillary depression. It happens due to both cohesion and adhesion.

Soap bubble: △

C)

=

Figure. 7.1.4

U

dθ θ Stationary plate

or τ = μ (u / y) . . . (7.1.2)

F

Non viscous (frictionless) and incompressible Used in the mathematical analysis of flow problems Does not exist in reality Does not offer shear resistance when fluid is in motion. x x

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Possess the properties such as viscosity, surface tension and compressibility. Resistance is offered when they are set in motion.

Real Fluid:

x x x x

Ideal Fluid or Perfect Fluid:

7.1.4 Types of fluids:

According to Newton’s law of viscosity, for a given shear stress acting on fluid ( ), the rate at which fluid deforms (u / y) is inversely proportional to viscosity ( ).

where τ = Shear stress; (du / dy) = Velocity gradient; μ = Dynamic viscosity

Differential form: =

∝

Where F is the Force required to move the surface Area ‘A’

∝

Shear stress ∝ time rate of deformation (angular deformation)

Gap filled with fluid

moving plate

7.1.3 Newton’s Law of Viscosity:

For tube dia. > 12mm capillary effects are negligible. Hence the dia. of glass tubes used for measuring pressure (manometers, piezometer etc.) should be large enough.

(i) Liquid jet (ii) droplet (iii) soap bubble. A) Liquid jet: Increase in Pressure inside and outside of liquid jet △ = , where d = dia of jet

Liquid drop: △

θ = 1300 → Mercury and gases

Due to cohesion, surface tension pressure changes occur across a curved surface of

B)

d = dia. of tube θ = 00 → Water and glass

Fluid Mechanics

θ = angle of contact between liquid and boundary

Quick Refresher Guide

Surface tension is due to cohesion between liquid molecules. As temperature increases ⟶ surface tension decreases (because cohesion decreases)

Fluid Mechanics

Units: N/m Dimensions: FL-1 or MT-2.

Quick Refresher Guide

Fluid Mechanics

Figure. 7.1.5

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Where A and B are constants depend upon the type of fluid and conditions imposed on flow. Based on power index ‘n’ and constant B Non – Newtonian fluids are i) B = 0 and n > 1 (represented by OE in Figure 7.1.5) Dilatant Fluids, e.g.: Butter, quick sand ii) B = 0 and n < 1 (represented by OC in Figure 7.1.5) Pseudoplastic e.g.: Blood, Paper Pulp, Polymeric solutions such as rubber, suspension paints. iii) = = 1(represented by PD in the Figure 7.1.5) Bingham plastic Eg: Sewage sludge, drilling mud require minimum shear stress τ known as yield stress before they start flowing. iv) Thixotropic Fluids: Printers ink, lipstick

Time dependent fluid i.e., viscosity depends upon both shear stress and duration of application.

Viscosity increases or decreases with time. e.g.: Paints and enamels, when subjected to high shear by the brush during application of paints, the apparent viscosity is reduced the paint covers the surface smoothly and brush marks disappears subsequently.

+

Relationship between shear stress and

x

velocity gradient is =

Do not follow the Newton’s law of viscosity

du / dy

Ideal fluid

C Pseudoplastic

Dilatant

G

x

E

Bingham plastic D

Newtonian

Non – Newtonian Fluids:

B

Q Thixotropic

P

τ

Which obeys Newton’s Law of Viscosity Newtonian fluids have constant viscosity (Viscosity is independent of shear stress) There will be linear relationship between shear stress and resulting rate of deformation. e.g.: air, water, light oils and gasoline

Newtonian Fluids:

x x x

Quick Refresher Guide

1 MPa = 10 bar

1 bar = 105 N/m2 = 100kPa

Fluid Mechanics

piston

Figure 7.2.1

P

F

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= / A = area of plunger Wt. ‘W’ lifted W = (F/a) × A Where A = Area of piston.

Assumption: Pressure variation due to height neglected and friction force is neglected.

iv) Application of Pascal’s Law-Hydraulic Press:

P

W

iii) If the fluid is in motion and fluids is ideal (frictionless) then no shear stresses, hence the pressure at any point is same in all the directions.

ii) If the fluid is in motion, shear stresses occur and normal stresses are no longer same in all directions at a point of a real fluid.

i) Viscosity of fluid has no effect on fluids at rest, therefore ideal and real fluids behave in a similar manner.

Pascal’s Law: Intensity of pressure at any point in a fluid at rest is same in all the directions.

Absolute Pressure = Atmospheric Pressure – Vacuum Pressure.

Absolute pressure = Atmospheric Pressure + Positive Gauge Pressure.

Dimensional Formula: ML-TT-2 (or) FL-2

Units: N/m2 (Pascal)

7.2.1 Fluid Pressure The normal force exerted by a fluid per unit area of the surface

Part 7.2: Fluid Statics

Quick Refresher Guide

= vertical distance measured from a datum (positive upward)

= specific weight of the fluid

=−

Fluid Mechanics

=

ℎ.

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d) Aneroid Barometer: used to measure local atmospheric pressure. (Absolute pressure)

c) Mechanical Gauges: Generally used for measuring high pressures where high precision is not required. Eg. Bourdon pressure gauge measures gauge pressures.

- This is used only to measure the difference of low pressures between two points where better accuracy is required. It generally consists of an air cock at top. iii) Differential Manometer: A U – Tube manometric liquid heavier than the liquid for which the pressure difference is to be measured, and is not immiscible with it (generally mercury). iv) Micro Manometer: Modified form of a simple manometer whose one limb is made of large cross sectional area. - Measurement of very small pr. differences with very high precision is made possible.

ii) Inverted U – Tube Manometer: Consists of an inverted U – Tube containing a light liquid.

which the pressure is to be measured.

as negative pressures. Contains liquid of specific gravity greater than that of the fluid of

the gauge point and other is open to the atmosphere. Can measure both positive as well

i) U – Tube Manometer: Consists of a U – shaped bend unit whose one end is attached to

liquid (whose pressure is to be found) by the same or another column of liquid.

b) Manometer: Pressure measuring device based on the principle of balancing the column of a

suction pressures and pressures of gases.

Suitable for measuring moderate gauge pressures of liquids. Not suitable for high pressures,

called pressure head and the pressure

inserted in the wall of a pipe or a vessel. The height upto which the liquid rises in the tube is

a) Piezometer: It consists of a glass tube, open at one end to the atmosphere and another end

Measurement of Fluid Pressure:

The linear variation of pressure with depth below the free surface is known as hydrostatic pressure distribution.

Where

∗

Intensity of Pressure (Variation of pressure in a static fluid):

Quick Refresher Guide

Figure 7.2.2

F = γ.A.

b) Vertical Plane θ

F = γ.A.

c) Inclined Plane

F

Fluid Mechanics

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NOTE: i) CP always lies below CG. ii) As the depth of immersion is more, CP comes closer to CG. iii) From eq. it is evident that the position of CP is independent of θ, i.e., θ may be varied by rotating the surface provided h remains unchanged.

Figure 7.2.3

θ

Centre of Pressure (CP) on Inclined Plane:

Note: i) Force F always acts normal to the plane surface. ii) The value of F is independent of the angle of inclination of the plane as long as the depth of centroid (ℎ ) is unchanged. iii) Total Force F = Area × Pressure at the centroid = . ̅ iv) Pressure Prism Concept: i) Total force F = volume of pressure prism = area of pressure diagram X width of plane ii) Force F acts at the C.G. of the pressure prism. v) Centre of pressure (C.P): The point of application of resultant force (F). ℎ= ̅+ ̅ = M.I. of the section about an axis parallel o X passing through C.G. of the area. v) C.P is always below the C.G as the depth of immersion is increased, the C.P. approaches the C.G

M N F = γ.A. MN → Plane surface to paper

F

a) Horizontal Plane

Forces on Plane Surface:

7.2.2 Forces on Submerged Bodies:

Quick Refresher Guide

Fluid Mechanics

F

Figure 7.2.5

A

E

B

C

D

= wt. of liquid prism represented by ABCDE

= wt. of liquid prism vertically above it.

= vertical component

Force on vertical Projection of the given area

= Horizontal component

Total Force = F =

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CB

(i) The Buoyancy force is equal to the weight of the fluid displaced by the body. (ii) The Buoyancy force acts through the CG of the displaced volume called ‘Centre of Buoyancy (C )’.

The resultant force exerted on a submerged or floating body in a static liquid is called ‘Buoyancy Force (F )’.

7.2.3 Buoyancy

L

H

h

Forces on Curved Surfaces:

i) Dam: Pressure prism concept is easier for plane rectangular surfaces. = × ℎ × ℎ/2 (1m length of dam) F acts at h = 2/3 h from top ii) Lock Gates: (*) Lock Gates are used in docks and at the tail end of a river for navigational purposes. F = Hydrostatic Force = F − F R = Resultant Reaction

Forces on plane surface (Applications):

Quick Refresher Guide

Fluid Mechanics

G B

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Neutral Equilibrium: (i) When CG coincides with CB.

BM > BG

M

Unstable Equilibrium: (i) When CG lies above CB. (ii) The over turning couple produced due to a slight disturbance will cause the body to move away from its original position.

to bring the body to its original position.

A slight rotational displacement generates forces which oppose the change of position and tend

Stable Equilibrium:

(i) Buoyancy Force F = W Here W = Weight of body (ii) CB lies above CG

Submerged body is in stable equilibrium when

B = center of buoyancy

G = center of gravity

Stability of Submerged Bodies:

(iii) A floating body displaces a volume of fluid whose weight is equal to the weight of the body.

Quick Refresher Guide

Fluid Mechanics

(v)

d

W

0

B’

M

Figure 7.2.6

G

T

c

b

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Where I = second moment of area of water plane (m4) about an axis passing through centre of area and perpendicular to the axis of tilted longitudinal axis = ∫ V = volume of liquid displaced by the body (m3).

Figure 7.2.7

Y

dA

X

Meta Centric Height ( ) (a) The distance between the centre of gravity ‘G’ and the meta centre ‘M’ of the floating body i.e., GM as θ → 0 is known as Meta Centric Height. (b) The Meta Centric Height is independent of magnitude of angular rotation (θ) (as long as it is small) and is given by GM = (I / V) – BG, if ‘M’ is above G (stable) GM = BG – (I / V), if ‘M’ is below G (unstable) Y

B = Centre of Buoyancy (CB) ‘B’ is the shift in CB due a tilt through small angle ‘θ’. The new CB (B ) cuts the vertical axis of the body (line cd) vertically at a point ‘M’ (Meta centre).

a

Buoyancy Force (F ) is equal to the weight of the liquid displaced by the body and acts through the CG of the displaced liquid. (iii) The body is in stable equilibrium if the meta centre lies above its CG. *i.e., BM > BG) (iv) Meta Centre

(ii)

7.2.4. Floating Bodies: Stability

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Fluid Mechanics

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(vi) Period of Rolling: Time Period = /( . ) k = radius of gyration ‘m’. = meta centric height in ‘m’. If GM is large the Time Period of roll will reduce i.e., more stable vessel will have less period of roll. Period of Rolling (T) is inversely proportional to stability and directly proportional to radius of gyration (k). The meta centric height of ocean-going vessel is usually 30 cm to 1.2 m. for war ships it ranges from 1 m to 1.5 m. As meta centric height increases less comfort to passenger (because less period of roll) but more stability.

= / is called Meta Centric radius. (c) The distance It depends upon the geometry of vessel and draft (depth of submergence). For e.g: For a vessel M.I. does not change but ‘V’ (displaced vol.) becomes less when load is removed.

Quick Refresher Guide

Fluid Mechanics

= 0,

=0

Unsteady

V = f (x, y, z, t)

V = f (x, y, t)

V = f (x, t)

Eg:

Flood flows

flow between plates

shooting flows

V = f (x)

V = f (x, y)

V = f (x, y, z)

Steady

th

Differential Equation of Stream Line dx / u = dy / v = dz / w dx / u = dy / v (or) u dy – V dx = 0

th

th

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Type of Flow 3 – dimensional 2 – dimensional

Flow Pattern: (a) Stream Line: An imaginary curve drawn through a flowing fluid in such away that the tangent to it at any point gives the direction of the velocity of flow at that point. Also called as flow line, since flow is along the stream lines.

1 – dimensional

2 – dimensional

3 – dimensional

Type of Flow

Table: 1, 2 and 3 – dimensional flows:

(ii) Non Uniform Flow: If the velocity of flow of fluid changes from point to point in the flowing fluid at any instant of time.

(b) (i) Uniform Flow: When the velocity of flow of fluid does not change both in magnitude and direction from point to point in the flowing fluid, at any given instant of time. i.e., ( / ) = 0 e.g.: Flow of liquids under pressure through long pipe lines of constant diameter.

(ii) Unsteady Flow: Flow parameters at any point change with time. i.e., ≠ , ≠ 0, ≠ 0, ( ) ≠ 0 e.g.: Flow in which the quantity of liquid per second is not constant.

e.g.: Flow of fluid through a pipe at constant rate of discharge.

= 0,

Classification of Flow: (a) (i) Steady Flow: At any point of the flowing fluid, various characteristics such as velocity, pressure density temperature etc., do not change with time. Mathematically, = 0, . . , = =0

7.3.1 Velocity of a Fluid Flow: = lim → / V = f(x, y, z, t) In x-direction, = →

Part 7.3: Fluid Kinematics

Quick Refresher Guide

Figure 7.3.1

V V

X

Fluid Mechanics

+

.

.

+

+

. .

+

+

. .

.

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(d) Tangential and Normal Acceleration:

Note: In steady flow the local acceleration is zero, but the convective acceleration is not necessarily zero and hence total or substantial acceleration is not necessarily zero. However if flow is uniform also, convective acceleration is zero and therefore total acceleration is zero.

(c) Convective Acceleration: Terms that represent the rate of increase of velocity due to particle’s change of position. In uniform flow convective acceleration is zero.

(b) Local Acceleration or Temporal Acceleration: Expressions that represent rate of increase of velocity with respect to time. / . e.g: , / ,

+

a = a =

Acceleration of a fluid particle (a) (a) Cartesian Co-ordinates: v=i.u+j.v+k.w + . + .( / ) + a =

(d) Streak Line: A line traced by a fluid particle passing through a fixed point in a flow field. e.g.: The trail of a colour dye injected at a point. In steady flow a streak line, stream line and a path line are all identical.

(c) Path Line: The line traced by a single fluid particles as it moves over a period of time.

(b) Stream Tube: An imaginary tube formed by a group of stream lines passing through a small closed curve. There can be no flow across a stream tube. Only in a steady flow, a stream tube is fixed in space.

Stream lines indicate tracing of motion of a group of particles. There can be no fluid flow across a stream line. For steady flow, stream line pattern remains same at different times. For unsteady flow it varies from time to time and hence instantaneous.

0

Y

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Fluid Mechanics

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(b) In one dimensional analysis (Flow through a stream tube): For compressible fluids, p A V = p A V .

Assumptions: (1) Flow is steady. (2) Flow is incompressible. (3) Velocity is uniform over a cross section.

velocity vector ’.V=0

Basis: Principle of conservation of mass – mass can neither be created nor destroyed (a) Differential form (in Cartesian co – ordinates): (i) For compressible fluids, ( / ) + ( ( )/ ) + ( ( )/ ) = 0, (ii) For incompressible fluids, ( u/ ) + + = 0 vertically the divergence of

7.3.2 Continuity Equation:

Note: If the stream lines are (i) Equidistant, Tangential convective acceleration is zero. (ii) Straight (not curves), Normal convective acceleration is zero. If the stream lines are straight and parallel to each other, there is no acceleration. If the stream lines are curved and equidistant there will be only normal convective acceleration. If the stream lines are curved and converging, then both normal and tangential convective accelerations. If stream lines are diverging, instead of acceleration will be corresponding retardation.

= . ( ) = convective normal acceleration )/ For steady flow, a = (V . a = V /r

Normal Acceleration, a = + ( / ) Where r = radius of curvature of stream line V = tangential component of velocity V = normal component of velocity / = local tangential acceleration / = local normal acceleration . ( ) = convective tangential acceleration

Tangential Acceleration is developed when the magnitude of velocity changes with respect to space and time. Normal Acceleration is developed when a fluid particle moves in a curved path i.e., simply due to change in direction of velocity of fluid particle, regardless of whether the magnitude of the velocity is changing (or) not. For steady flow L.A. = zero ‘O’ ∴ a = V . ( ) Tangential Acceleration, a = + .( )

Unlike velocity vector, the acceleration vector has no specific orientation with respect to stream line i.e., it need not always be tangential to stream line. Therefore, at any point it may have acceleration components both tangential and normal.

Quick Refresher Guide

Fluid Mechanics

.

= ∫

.

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Stream Function ( ): It is a scalar function of space and time such that its partial derivative with respect to any direction gives the velocity component at right angles (in counter clockwise direction) to this direction. i.e., ∂ψ/∂x = v and ∂ψ/wy = − Laplace Equation for ‘ψ’: For an irrotational flow, (∂ ψ/ ∂x ) + (∂ ψ/ ∂y ) = 0

(b) Vorticity (ξ): The limiting value of circulation divided by the area of closed contour, as the area tends to zero. Vorticity = circulation / area = [(∂v/ ∂x) − (∂u/ ∂y)] Note: Vorticity = 2 × rotation component W . i.e., ξ = 2 ω . (zeta)

Circulation and Vorticity: (a) Circulation (Γ): Flow along a closed curve (i.e., the flow in eddies in vertices) It is the line integral of the tangential component of the velocity taken round a closed contour. Γ (Capital Gamma Greek) = [(∂v/ ∂x) − (∂u/ ∂y)] δ . δ

For fluids or flows of large viscosity, flow is invariably rotational. For fluids such as air and water having small viscosity, the flow in the region away from boundary may be treated as irrotational. In the case of rapidly converging or accelerating flows, flow may be treated as irrotational.

(b) Irrotational Flow: If the liquid particles while moving in the direction of flow do not rotate about their centers. Mathematically, in irrotational flow curl of velocity vector ∇ × V = 0. True irrotational flow exists only in ideal fluids. If at every point in the flowing fluid, the rotation components ω , ω and ω are equal to zero, then the flow is known as irrotational. For ω = 0, (1/2) [(∂w/dy)) ( ∂v/∂z)]=0 ∴ (∂w/∂y)=( ∂y/∂z) For ω = 0, (1/2) [(∂u/dx) ( ∂w/∂x)]=0 ∴ (∂u/∂z)=( ∂w/∂x) For 2-dimension irrotational flow − = 0∴ = w = 0,

Rotational and Irrotational Motions: (a) Rotational Flow: If the fluid particles while moving in the direction of flow rotate about their mass centers. e.g: Liquid in rotating tanks where the velocity of each particle varies directly as the distance from the centre of rotation.

Q=∫

Assumptions for equation = . (i) Flow is steady (ii) Flow is incompressible (iii) Flow is one dimensional over the cross section (iv) Velocity is uniform at the across section (v) No branching of steam tube. When there is variation of velocity across the cross section of a tube, for an incompressible fluid discharge,

For incompressible fluids, A V = A V .

Quick Refresher Guide

∂y =

∂x

(This will be true if ‘ψ’ is a continuous function and its second

Fluid Mechanics

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Flow Net can be drawn for (i) Steady Flow (ii) Irrotational flow and (iii) When the flow is not governed by gravity force.

Stream Lines, Equipotential Lines and Flow Net: Stream line is given by ψ = constant. Equipotential line is given by ϕ = constant. Stream lines and equipotential lines intersect each other orthogonally at all points of intersection. (Reason: The product of slopes of tangents of these two lines is – 1.) A grid obtained by drawing a series of stream lines and equipotential lines is known as “Flow Net”

Laplace Equation for ‘ ’: (∂ ϕ/ ∂x ) + (∂ ϕ/ ∂y ) + (∂ ϕ/ ∂z ) = 0 Any function ‘ϕ’ that satisfies the Laplace equation is a possible irrotational flow. Velocity potential exists only for irrotational flows. Lines of constant ‘ϕ’ are called equipotential lines and these lines will form orthogonal grids with constant ψ lines.

Potential Function ( ): A scalar function of space and time such that its negative derivative with respect to any direction gives the fluid velocity in that direction. =− ,v = − ,w = − .

derivative exists) ∴ Any function ‘ψ’ which is continuous is a possible case of fluid flow. It is constant along a stream line. The difference of stream functions for two stream lines is equal to the flow rate between them.

Also

Quick Refresher Guide

Fluid Mechanics

.( .( .(

/ / /

) + ) + ) +

.( .( .(

/ ) / ) / )

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Integration of Euler’s equation of motion: Integration of Euler’s equation yields the Energy equation under the following assumptions.

Note: In derivation of above equations, no assumption has been made that the mass density ′ ′ is constant. Hence these equations are applicable to compressible or incompressible, non-viscous fluids in steady or unsteady state of flow.

X, Y, Z are components of body force per unit mass at a point.

– ( / ) .( / ) = ( / ) + .( / ) + – ( / ) .( / ) = ( / ) + .( / ) + – ( / ). ( / ) = ( / ) + . ( / ) +

Only pressure and gravity forces (self weight which is a body force) are considered. The equations are,

7.4.2 Euler’s equations of motion

a) According to Newton’s Second Law of motion, F = Ma ∴ Ma = F + F + F + F + F + F b) Reynolds equations of motion: For laminar or viscous flows, turbulence forces are also neglected. ∴ Ma = F + F + F + F c) Navier – Stoke’s equations: For laminar or viscous flows, turbulence forces are also neglected. ∴ Ma = F + F + F d) Euler’s equations of motion: If the fluid is ideal, viscous forces are also insignificant. ∴ Ma = F + F

Forces acting on a fluid in motion: The various forces that may influence are due to gravity, pressure, viscosity, turbulence, surface tension and compressibility.

a) Body or Volume forces: These are proportional to the volume of body. Ex: weight, centrifugal force etc. b) Surface forces: These are proportional to the surface area. Ex: Pressure force, shear or tangential force, turbulence force, force of compressibility etc. c) Linear Forces: Proportional to length. Ex: surface tension

A fluid in motion is broadly subjected to various forces as detailed below.

7.4.1 Equations of Motion

Dynamics of fluid flow is defined as that branch of science which deals with the study of fluids in motion considering the forces which cause the flow.

Part 7.4: Fluid Dynamics

Quick Refresher Guide

Fluid Mechanics

Flow is steady Flow is irrotational Flow is incompressible Flow is non-viscous i.e., density is constant Flow is continuous Velocity is uniform over a cross section Fluid is ideal.

+

+

=

+

+ℎ

Here h = energy loss

+

− (energy equation)

.

Here U = Velocity at point.

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∴ ∝ = ( / )∫

Kinetic energy correction factor (∝): In one dimensional method of analysis, the average velocity ‘V’ is used to represent the velocity at a cross section. The actual velocity distribution in the cross section may be non-uniform. Hence, the kinetic energy calculated by using ‘V’ must be multiplied by a correction factor, to obtain kinetic energy at the cross section due to nonuniform velocity distribution.

a) Datum head: A liquid particle ‘Z’ metres above a reference datum is same said to possess a potential head or datum head ‘Z’. b) Pressure head: P/W in metres. (It is due to energy possessed by a body). c) Velocity head: V /2g (It is due to kinetic energy). d) Piezometric head: (P/W) + Z.

Energies in fluid motion:

c) When the flow is steady but may not be irrotational i.e., rotational flow: In this case, Bernoulli’s equation is applicable only to particular stream line that is the value of constant is different for different stream lines. d) Basis for Bernoulli’s equation is ‘Law of conservation of Energy’. Therefore, it is also called ‘Energy equation’.

In the above equation each term represents “Energy per unit weight”.

∴

b) For real fluids there will be some loss of energy between two points.

1. 2. 3. 4. 5. 6. 7.

Assumption made are:

Note: It is applicable to all points in the flow field i.e., for all the stream lines, the value of constant is same.

a) Integration of Euler’s equation for steady, incompressible and frictionless flow yields the Bernoulli’s equation. ( / ) + ( /2 ) + = constant → This is for ideal fluids. i.e., total energy of a particle remains constant.

7.4.3 Bernoulli’s Equation:

a) There exists a force potential (:) which is defined as that whose negative derivative with respect to any direction gives the component of body force per unit mass in that direction. ∴ X = (∂:/∂x), Y = (∂:/∂y), Z = (∂:/∂z), b) The flow is irrotational, i.e., the velocity potential exists (or) the flow may be rotational, but is steady.

Quick Refresher Guide

= 1.01 to 1.20

For laminar flow through a pipe For turbulent flow through a pipe

Fluid Mechanics

A

(h )

Figure 7.4.1

h Stagnation head

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b) Venturimeter: It consists essentially of the following parts:

2. A pitot tube measures both static pressure and stagnation pressures.

1. A pitot measures stagnation pressure head (or the total head) at dipped end.

Actual velocity V = c 2 ℎ, Where c = pitot tube constant With U – tube manometer reading ‘x’, h = x [(S /s) − 1] Note:

We have, h = V /2g. ∴ V = 2 ℎ

Stagnation pressure head = Static pressure head ‘h ’ + Dynamic pressure head ‘h’ = h + h (h = difference between stagnation pressure and static pressure)

Dynamic head (h)

In the Figure 7.4.1 A is the stagnation point. The pressure at stagnation point is called ‘Stagnation pressure’. Static pr. head = /

a) Pitot Tube: Pitot tube consists of a glass tube bent through 90 . The lower end of the tube faces the direction of flow. The liquid rises up in the tube due to pressure exerted by the liquid flow. It is used to measure velocity of flow at any section of a pipe or channel. The basic principle: If the velocity of flow at a particular point is reduced to zero, known as stagnation point, the pressure there is increased due to conversion of kinetic energy into pressure energy and level of water rises.

1. Flow measurement:

The application of Bernoulli’s Equation includes:

7.4.4 Practical applications of Bernoulli’s theorem:

= 1.0 = 2.0

For uniform velocity distribution

Quick Refresher Guide

Taper angle20°°

Figure 7.4.2

Throat

=

a a

2gh

(a − a )

Taper Angle 5°

Figure 7.4.3

area ‘a0’

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area ‘a’

c) Orifice meter: It consists of a plate having a sharp edged circular hole known as orifice which is fixed inside the pipe whose discharge is required.

Where Q = theoretical discharge, a = area of cross section at inlet, a = area of cross section at throat, h = venturi head. (P /W) − (P /W), i.e., h = pressure head at inlet pressure head of throat. Actual discharge = coefficient of discharge C × Q . ‘C ’ for venturi metre = 0.98 (generally) “C ” is a function of Reynold’s number, diameter ratio, roughness of surface, velocity distributions at inlet and outlet. h = X (Sm/S – 1), if U – tube manometer used. h = X (1 – Sm/S), if inverted U – tube manometer is used. X = difference in levels of manometric liquid. S = specific gravity of manometric liquid. S = specific gravity of liquid in pipe. h = head in terms of flowing fluid.

Q

Fluid Mechanics

Diverging cone

Use: For determining discharge in pipes i.e., rate of flow of liquid.

i) An inlet section (U – tube) ii) Converging cone iii) Throat iv) Divergent cone Pressure taps are provided at the inlet section and throat.

Inlet

Converging cone

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Fluid Mechanics

Stream lined Convergent nozzle

(for maximum range θ = 45 )

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b) Vortex Motion: A rotating of fluid is known as ‘Vortex’ and the motion of rotating mass of fluid is known as ‘Vortex motion’. Different types of Vortex motion are described below :

b) Range (L) =

a) Maximum vertical elevation of jet profile, H = (V sin θ/2g)

Since the pressure head is zero at every point, the sum of velocity and datum head is constant.

Bernoulli’s equation is used for analysis.

L Figure 7.4.5

H

a) Free Liquid Jet: It is a steady curvilinear flow of liquid with a free surface in which at all points the pressure is atmospheric. Under the action of gravity the liquid jet traverses a parabolic path known as trajectory.

2. Analysis of fluid flow

Basic equations are same as those for venturimeter. Coefficient of discharge is almost same as that for a venturimeter. e) Other flow measurement devices are: i) Rota meter ii) Elbo meter (or) pipe bend meter

Figure

d) Nozzle meter: Used for measuring discharge. When compared to venturimeter diverged part is omitted, and therefore a greater dissipation of energy.

Use: For measuring discharge through pipes. a a 2gh (a − a )

Discharge Q =

Quick Refresher Guide

Fluid Mechanics

)

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iii) Spiral Vortex motion: A combination of cylindrical vortex motion and radial flow.

a) On any horizontal plane the fluid pressure increases with the square of the radial distance from center of vortex motion. b) Surfaces of constant pressure are paraboloids of revolution. c) The free surface is a special surface of constant pressure and is also a paraboloid of revolution. d) Forced vortex is basically a rotational motion.

Important points:

Equation of pressure variation: ∂p/ ∂r = rw

Relation: V =r w, w = angular velocity i.e., velocity of flow is directly proportional to its radial distance from the axis of rotation.

ii) Forced Vortex motion: Forced vortex motion occurs when a constant torque is applied to a fluid mass. Ex: Cylinder rotated about its vertical axis at a constant angular velocity

Ex: Motion of air mass in tornado.

The equation v.r = constant, is applicable only to the region farther away from the central region of free vortex motion which occurs frequently is known as ‘Rankine Vortex motion’.

Flow field of a vortex motion is everywhere irrotational except at the axis and therefore it is also called irrotational vortex motion or potential vortex motion.

As r → 0, v →f , the point where the velocity ‘V’ becomes infinity is called singular point.

i.e., ∫ = 0, i.e., v.r = constant (i.e., the velocity of flow in a free vortex motion varies inversely with the radial distance from center of vortex motion).

(

No external torque is required to be exerted on the fluid mass. Therefore, the rate of change of angular momentum of the flow must be zero.

4) Flow of water in a turbine casing before it enters the guide vanes.

3) Flow of liquid in a centrifugal pump casing after it has left the impeller.

2) A wash basin (or) bath tub draining water through an outlet at a bottom.

Ex: 1) A whirl pool in a river.

i) Free Vortex motion: Is that in which the fluid mass rotates without any external force being impressed and the expenditure of energy from any external source takes place.

Quick Refresher Guide

Fluid Mechanics

/

.

for sphere

.=

/

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δ = 11.6 v/V.,

Laminar Sub-layer : If the plate is very smooth, even in the region of turbulent boundary layer, there is a very thin layer adjacent to the boundary, in which flow is laminar. This layer is known as “Laminar Sub-layer”.

a) It increases as the distance from leading edge increases. b) It decreases with the increase in the velocity in the direction of flow of the approaching stream of fluid. c) Greater is the kinematic viscosity of fluid, greater is the Boundary Layer thickness. d) Boundary layer thickness is affected by pressure gradient in the direction of flow. If the pressure gradient is negative (converging flow), it accelerates the retarded fluid in the Boundary Layer and hence its growth is retarded. If pressure gradient is positive (divergent flow), the fluid in the Boundary Layer is further decelerated and hence assists in thickening of boundary layer. In the later case back flow and separation may be caused.

Factors affecting boundary layer thickness along a smooth plate:

for flat plates

Reynolds number is more than 5 ×

… for flow over sphere.

… for flow over flat plate.

Reynolds number is more than 2 ×

Turbulent Boundary Layer:

Where V = free stream velocity x = distance from leading edge. Q = kinematic viscosity of fluid.

For flow over flat plates Re = Vx / Q

Reynolds number is less than 2 ×

Reynolds number is less than 5 ×

Laminar Boundary Layer: The boundary layer is called laminar boundary layer if the flow in the boundary layer exhibits all the characteristics of a laminar flow, irrespective of whether the incoming flow is laminar or turbulent.

Note: Outside the boundary layer velocity is constant and therefore velocity gradient and shear stress are zero.

= .

In this region velocity gradient ‘du/dy’ exists and hence the fluid exerts a shear stress on the wall in the direction of motion given by

When a solid body is immersed in a fluid, there is a narrow region of the fluid in the neighborhood of the solid body, where the velocity of fluid varies from zero to free stream velocity. This narrow region of fluid is called “Boundary Layer”.

7.5.1 Boundary Layer

Part 7.5: Boundary Layer

Quick Refresher Guide

=∫ ( − / )

= ∫ ( / )( − / )

Nominal & displacement thickness

−

/

Here τ = shear stress at surface.

θ Relative magnitudes of δ, , θ

0.99 u

U

δ

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This is applied to (i) Laminar Boundary (ii) Transition boundary layer and (iii) Turbulent boundary layer flows. Assumptions: 1) steady flow 2) Two dimensional flow 3) Incompressible

=

Von Karman momentum Integral equation of Boundary Layer:

y

u

U U=0.99 y

δ

=∫ ( / )

Note: Shape factor of Boundary Layer = δ∗ /θ

∴

Energy Thickness ( ): The distance from the actual boundary such that the energy flux corresponding to the main stream velocity ‘V’ through this distance δ is equal to the deficiency or loss in energy due to the boundary layer formation.

∴

Momentum Thickness (θ): The distance from the actual boundary such that the momentum flux corresponding to the main stream velocity ‘V’ through this distance ‘θ’ is equal to the deficiency or loss

∗

(or)

δ =K xv/V /

Transition from Laminar Boundary Layer between R Critical R = 5 × 10

(

)

.

∴

.

.

=

(

= .

(

(

.

. ) ) .

)

− if 10 < R < 10

− − if 5 × 10 < Re < 10

.

= 1.3 × 10 to 4 × 10

= +ve … condition for attached flow.

= 0 … condition for separation.

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(ii)

Thus, (i)

Separation of Boundary Layer: Boundary Layer separation takes place due to adverse or positive pressure gradient. i.e., when dp/dx > 0 and / ≤ 0. In fact separation starts when / = 0

As per schlitching, C

Average drag coefficient C

(ii) Local drag coefficient C =

(i) =

R = 5 × 10 to 10 . Velocity distribution is logarithmic or

Usually thicker than laminar ones.

Turbulent Boundary Layer:

If R lies between 1.3 × 10 to 4 × 10

.

=

The velocity Distribution is infact parabolic for laminar boundary layer. Being smaller in thickness it can be assumed as linear distribution. Relation between θ, δ and δ∗ for Laminar Boundary Layer = , δ∗ = , δ∗ = 3θ ⟶ for linear distribution velocity

=

F = ∫ τ B dx = C (LB) Where B = width of plate L = length of plate Average drag coefficient (or) Total frictional drag coefficient C Where R

Fluid Mechanics

/ ∴ , C = local or shear stress coefficient.

Total horizontal force (skin friction drag) on one side of plate

As per Blasius K = 5, ∴ = 5 Further, = = . /

=

Transition from laminar Boundary Layer

(iii)

(ii)

(i)

‘K’ is a constant, x = distance from leading edge

Displacement Thickness( ∗ ): The distance by which the boundary surface would have to be displaced outwards so that the total actual discharge would be same as that of an ideal fluid past the displaced boundary

Laminar Boundary Layer thickness:

Quick Refresher Guide

Thickness of Boundary Layer:

Fluid Mechanics

Nominal Thickness ( ): The distance ‘y’ from the boundary surface at which velocity of fluid is approximately equal to 0.99 times free stream velocity.

Quick Refresher Guide

= −ve … condition for detached flow.

C D

P min Effective pressure gradient on boundary layer separation

B

Fluid Mechanics

Fluid Mechanics

− terms would entirely depend upon the

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Drawbacks: 1. It does not provide any information regarding the number of dimensionless groups to be obtained as a result of dimensional analysis. 2. The method becomes rather cumbersome when a large number of parameters are involved.

Rayleigh’s Method: Gives a special form of relationship among the dimensionless groups.

Methods of Dimensional analysis:

i) For either dynamic or kinematic similarity to exist geometric similarity is a prerequisite. ii) If two flow fields are dynamically similar, they are necessarily, kinamatically and geometrically similar

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Various force ratios:

Note:

c) Dynamic similarity: Corresponding forces have the same ratio throughout the flow field i.e., similarity of forces.

Total number of variables = number of Independent variables + one dependent variable.

The empirical equations involving numerical coefficients are dimensionally non homogeneous

Fundamental dimensions: Mass (M), Length (L), time (T) and temperature (Q). In place of mass, the force is also considered as a fundamental quantity.

Dimensional Homogeneity: Dimensional analysis is based on Fourier’s “Principle of Dimensional Homogeneity”. An equation is said to be dimensionally homogeneous if the form of the equation does not depend upon the units of measurement. To satisfy this condition, the dimensions of each side of equation must be same.

Types of similarity: For complete similarity to exist between the model and its prototype. It is necessary that it must be geometrically, kinematically and dynamically similar.

Model and prototype: Prototype is the full size machine. Model is the tool for studying the behavior of a prototype. It may be larger, smaller or even of the same size as the prototype.

As per Buckingham theorem ‘m’ is the number of repeating or Fundamental variables, but the number of fundamental variables depend on the system used. If for a phenomenon ‘F’ and ‘L’ are chosen fundamental dimensions m = 2. But F = MLT . Hence if M-L-T system is chosen, m = 3. The dimensional matrix method avoids the confusion over value of ‘m’ depending on system.

The number of dimensionless groups = n r, where ‘r’ Rank of dimensional matrix. It is the order of the determinant whose value is not equal to zero.

Matrix approach Number of dimensionless groups:

Note: The nature or form of the dimensionless selection of repeating variables.

1. A variable describing geometry of flow such as diameter, length, 2. A fluid property ex: Viscosity, density, surface tension, elasticity and vapour pressure. 3. A variable characterizing the fluid motion. Eg. Velocity, acceleration, discharge.

Selection of repeating variables:

Number of π terms = (n-m) where n = total number of variables, m = number of repeating variables

The method expresses the equation in terms of dimension less groups (π terms),

Buckingham – Theory (or method of repeating variables):: It offers an advantage over Rayleigh’s method in letting us know, in advance of the analysis as to how many dimensionless groups are to be expected.

Quick Refresher Guide

a) Geometric similarity: Similarity of shape, to satisfy this condition the ratios of the corresponding lengths in the model and its prototype must be same. = = This ratio is also known as scale factor b) Kinematic similarity: Similarity of motion. Streamline pattern in model must be same as that in its prototype. The ratios of kinematic quantities flow characterizing such as time, velocity, acceleration and discharge must be same at all corresponding points. i.e., t = , = = , = , =

Dimensional analysis is a method to describe a physical phenomenon by a dimensionally correct equation among certain variables which affect the phenomenon. It reduces the number of variables and arranges them into dimensionless groups useful for phenomenon which defy analytical solution and must be solved experimentally.

7.5.2 Dimensional Analysis

A

At the separation point shear stress is zero and velocity gradient, ∂v / ∂y = 0

(iii)

Quick Refresher Guide

Fluid Mechanics

∆

2.

∆

. It is derived from ratio of Inertia and pressure forces.

th

th

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th

A fluid moving relative to a rigid boundary exerts force on the boundary. Shear stresses acting on surface and pressure acting normal to the boundary give rise to forces which would add up vectorially to give a resultant force on the body. The component of the resultant force in the direction of relative velocity ‘V’ of the flow past the body is called ‘Drag’. The component of the resultant normal to the relative velocity is known as ‘Lift’.

7.5.3 Lift & Drag

the significance of various forces.

f) Weber Number (W): Inertia force/surface tension force = Significance: In formation of water droplet or bubbles, flows of shallow depth over spillways, dams etc., for dynamic similarity to exist between a model and a prototype, (R) = (R)m, (F) = (F)m, (E)p = (E)m, (M)p = (M)m and (W) = (W)m, depending on

Significance: For compressible flows, high speed flows, motion of objects like aeroplane and projectiles through air at super sonic speeds.

e) Mach number (M): Stream velocity / Acoustic velocity in the fluid medium =

d) Cauchy number: Inertia force/Elastic force =

Reciprocal of Euler number is called Newton number. Significance: In cavitation studies, the pressure force is important in addition to the viscous force, and the dynamic similarity will be obtained when the Reynolds and Euler numbers are kept the same for model and prototype.

=

c) Euler number (R): V /

b) Froude Number ( ): Inertia force/gravity force = V/ g Significance: For open channel flow where free surface is present, wave action as in breakwaters and ships, hydraulic structures such as spillways, stilling basins, weirs and notches, forces on Bridge piers and off shore structures.

significance: i) completely submerged flow (air planes, torpedo etc) ii) completely submerged flow (air planes, for flow through pipes and plates) iii) viscous flow (settling of particles fluids) flow in flow meters in pipes (venturimeter, orifice meter etc).

a) Reynolds number (Re): Inertia force/viscous force =

Quick Refresher Guide

Fluid Mechanics

P

Lift

Drag

Resultant

A = characteristic area of the body

A = Characteristic area

where C = lift coefficient

(

/ )

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where R

C or (C ) = ( , R )

For incompressible flow,

C =

For an airfoil or hydro foil vane, characteristic area is plain area

F = C . A.

Lift: Lift force occurs normal to the direction of relative motion ‘V ’.

A vertical disc normal to the flow is mainly subjected to pressure drag since the wake size is very large. ii. Streamlined body is predominantly subjected to shear drag and pressure drag is very less due to very small wake. iii. The total drag in sphere is approximately 1/3 of that of vertical disc. iv. The total drag on streamlined body is approximately about 1/40th of that of vertical disc.

i.

Comparison of drag force on vertical disc, sphere and streamlined body:

Note: A new cricket ball is roughened to convert approaching flow laminar boundary layer to turbulent one. Due to this C reduces and the drag resistance and hence the ball can be bowled at greater speeds.

A = Frontal or projected area for blunt shaped objects such as spheres, cylinders, cars, projectiles, Missiles etc., = Plan area used for thin flat surfaces (frictional forces are predominant e.g. vanes and hydrofoils). = Wetted area. E.g. for boats, barges and ships

= Fn (geometry, Re) for incompressible and non free surface flows

C = Fn (geometry, Re, F , M) In general

Where C = total drag coefficient

Total drag F =

Drag: Total drag is made of two parts. Frictional drag (Due to-shear stresses) + Pressure drag (due to pressure difference).

(b) Bluff body: If the surface of body does not coincide with stream lines.

(a) Streamlined body: A body whose surface ‘coincides with streamlines.

Vo

Lift force occurs only when the axis of body is inclined to the direction of flow. If the axis of body is parallel to the direction of flow, lift force is zero, only drag force acts. If fluid is assumed ideal and body is symmetrical such as sphere or cylinder, both the drag and lift are zero.

Quick Refresher Guide

Fluid Mechanics

=

or R =

=

=

Proportional to the velocity of flow independent of the pressure, proportional to the area of surface in contact, independent of the nature of the surface in contact, greatly affected by the variation of the temperature of the flowing fluid.

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vi) dependent on the nature of the surface in contact.

v) proportional to area of surface in contact,

iv) slightly affected by the variation of the temperature of the flowing fluid,

iii) proportional to the density of the flowing fluid,

ii) independent of the pressure,

i) proportional to (velocity) . Where the index n varies from 1.72 to 2.0,

The frictional resistance in the case of turbulent flow is:

The reason for the frictional resistance in the case of laminar flow being independent of the nature of the surface in contact, is that when a fluid flows past a surface with velocity less than critical velocity, a film of almost stationary fluid is formed over the surface, which prevents the flowing fluid to come in contact with the boundary surface. Similarly in the case of laminar flow the resistance is due to viscosity only and the viscosity of a fluid depends on its temperature.

i) ii) iii) iv) v)

The frictional resistance in the laminar flow is:

a) Major Losses due to friction. b) Minor Losses due to various fittings, transitions, changes in velocity to change in crosssections.

As fluid flows through a pipe certain resistance is offered to the flowing fluid, resulting in a loss of energy. Broadly these are of two types.

R =

In case of pipes ‘L’ is replaced by diameter ‘D’.

R =

Inertia force = Mass × Acceleration = Viscous force = × = ×

The flow in pipes will generally be under a pressure greater than atmospheric, whether a flow is Laminar or Turbulent generally depends upon the Reynolds number, that is ratio of inertia force to viscous force.

7.6.1 Losses in pipes

Part 7.6: Flow through pipes

Quick Refresher Guide

Fluid Mechanics

V = mean velocity in pipe f = friction factor, which is a function of R and relative roughness.

where L = Length of pipe of diameter ‘D’

=

=

=

)

0. 5

.

)

(

)

)

)

(

≈ 0.5

(

(

(

Head Loss =

V = Velocity in pipe

vena contracta

V = Velocity at vena contracta

V = Velocity in contracted section

Expansion from section ‘1’ to ‘2’

Explanation

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Entrance to a pipe from a reservoir

Sudden Contraction

Sudden expansion

Minor Losses in Pipes: Situation

a) The frictional resistance varies approximately with the square of velocity b) The frictional resistance varies with the nature of the surface.

The above equation is derived based on experiment of Froude, which revealed that

The ratio “ = S ” represents the energy slope which is equal to the hydraulic gradient in uniform flow. In long pipe lines, ‘h ’ forms a major part of the total loss.

h =

Major Loss of Head ( ): The basic equation used is Darcy Weis Bach Equation.

Quick Refresher Guide

.

[

(

)

]

)

‘C ’ lies between 0.6 to 0.666

h =

(

V 2 A

a

‘K’ is a constant

‘K’ is a constant

V = Velocity in pipe

Fluid Mechanics

+

.

=

.

=

.

Q

Figure 7.6.1

Q

+

+

+ …

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Pipes in Parallel: When a main pipe line divides into two or more parallel pipes which again join together downstream side and continue as a main line, the pipes are said to be parallel.

=

Equivalent Pipe: Often a compound pipe consisting of several pipes of varying diameters and lengths is to be replaced by a pipe of uniform diameter, known as equivalent pipe. The uniform diameter of equivalent pipe is known as equivalent diameter.

Q=

Q

+ ⋯ + ⋯ (neglecting minor losses)

ii) Discharge through each pipe will be same,

i.e., H =

i) The difference in liquid surface levels in the two reservoirs is equal to sum of the head losses in all the sections.

Pipes In Series Or Compound Pipe: If a pipe line connecting two reservoirs is made up of several pipes of different diameters D , D and D etc., and lengths L , L and L , etc… all connected end to end, then the system is called pipes in series, in such a case.

An obstruction to flow

Bends, pipe fittings

Conical expansion

At exit of a pipe

Quick Refresher Guide

Fluid Mechanics

=

=⋯

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It is a function of flow conditions and geometry of flow CASE LOWER 1 2000 Pipe flow 2 1000 Flow through parallel plates 3 500 Open Channel flow 4 1 Flow around a sphere

Critical Reynolds number: The Reynolds value below which the flow can be certainly considered to be laminar is known as critical Reynolds number

The laminar flow of a fluid is one in which the fluid moves in layers (laminae), each layer sliding over the adjacent one. The viscosity plays a dominant role on the fluid motion.

7.6.2 Viscous flow

Note: The efficiency corresponding to maximum transmission of power is not maximum.

Corresponding to the maximum power transmitted, the efficiency of power transmission is η = 2/3 or 66.7 %

i.e., loss of head should be 1/3 of total head to have maximum power transmitted. Efficiency of power transmission η =

= 0, ∴ H = 3 h

The condition for maximum power transmitted through pipe:

( − ℎ ) H = total head at the entrance to pipe = ℎ = head loss in pipe Q = discharge through pipe

Hydraulic transmission of power: The pipes carrying water under pressure from one point to other may be used to transmit hydraulic power.

′d = / d = diameter of each parallel pipe D = diameter of single main pipe line n = No. of parallel pipes

Note: If there are ‘n’ pipes of same diameter laid in parallel in place of a single pipe then diameter of each parallel pipe.

i.e., h =

a) Rate of discharge in main line is equal to sum of discharge in each of parallel pipes. Q=Q + Q + … b) Head loss is constant

Note: The pipes are connected in parallel in order to increase the discharge passing through the main.

Quick Refresher Guide

R

D

Linear

(

−

)=

[1 − ( / ) ]

)

=

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i.e., τ =

(

Shear stress at boundary τ =

i.e., τ = τ

Linear variation zero at centre and maximum at boundary:

Shear stress distribution is linear variation:

=

General equation for velocity distribution is:

.e., V =

Mean velocity (V) occurs at a distance of 0.707 R from centre of pipe and is equal to half of maximum velocity which occurs at centre.

Figure 7.6.2

Shear Stress

Velocity distribution is parabolic variation

Velocity

0.707R

Laminar flow through circular pipes:

i.e., pressure gradient in the direction of flow is equal to the shear stress gradient in the normal direction.

∂P / ∂X = ∂τ / ∂y

For a two dimensional, steady uniform flow relation between shear and pressure gradients is:

Fluid Mechanics

=

because V =

D

= Q (P − P )

V=

[Bγ − γ ]

Figure 7.6.3

Shear Distribution

Shear

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−

Head loss hf = 12 μVL/ γB

Shear stress at boundary = 6 μV/B

Average velocity V = (2/3) V

Velocity distribution is parabolic variation

Velocity Distribution

B

Flow between two stationary parallel plates:

i.e., friction factor in laminar flow is a function of Reynold’s number only.

Friction factor for laminar flow, f = 64 / R

P and P are pressure between two points.

P = γ Qh =

Power (P): In laminar flow the power required to overcome frictional resistance i.e., (h ) in a pipe of length ‘L’ and diameter ‘D’ carrying a discharge ‘A’ of a fluid of specific weight ‘r’ and viscosity ‘ ’ is

i.e., ℎ =

Head loss ( ): The variation of head loss ‘h ’ due to uniform laminar flow in a length ‘L’ of a pipe of diameter ‘D’ is given by Hagen – Poiseuille equation

(it has unit of velocity)

=

The basic equations which govern the motion of incompressible viscous fluid in laminar flow are “Navier stoke’s equation” Shear velocity = V =

Quick Refresher Guide

Fluid Mechanics

Quick Refresher Guide

Fluid Mechanics

=

( − )

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where γ and γ are the specific weights of the solid and the fluid respectively. The Eq. expresses the Stokes’ law. It’s validity lies within the Reynolds number ≤ 0.1.

(γ − γ) = 3πμVd and the terminal velocity,

Thus, the submerged weight (weight of particle minus the buoyant force) = Resisting force

A small solid particle falling through a fluid under its own weight will accelerate until the net downward force on it is zero. In other words, when the submerged weight of the particle is equal to force given by Eq. it will have reached the steady state of its motion. No further acceleration is then possible and the particle is said to have reached its terminal velocity.

It has been experimentally found that the Stokes’ law expressed by Eq. is valid only if the Reynolds number is less than 0.1. It is used to determine the fall of relatively small bodies through fluids of relatively high viscosity such as the fall of dust particles, mist droplets in the atmosphere and the settlement of silt in reservoirs.

μ = dynamic viscosity of fluid, d = diameter of sphere.

Where V = velocity of the sphere relative to the undisturbed fluid,

F = 3πμ Vd

The force resisting the motion of sphere (i.e., the drag on the sphere) was then obtained by Stokes as

Laminar flow around a sphere – Stokes Law:

Quick Refresher Guide

Fluid Mechanics

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factor ‘K’ = blade velocity coefficient. K =

If vane surface is very smooth, V = V If friction of blade surface is considered, V = KV . The effect of blade friction is expressed by a

The direction of fluid jet changes as if flows over the smooth surface of the vane, and at the outlet tip it emerges with a relative velocity ‘V ’ inclined at angle ‘β ’ as shown in Fig. 7.1. β = Blade (or) vane angle at outlet. (∠A B C )

U = Velocity of curved vane V = Absolute velocity of jet at inlet (A C ) u = Vane velocity at inlet (A B ) V = Relative velocity at inlet (B C ) V = Velocity of whirl at inlet (A D ) Component of ‘V ’ in the direction of blade motion i.e., along ‘u’ V = Velocity of flow at inlet (D C ) Normal component of ‘V ’ to the blade motion. α = Angle made by jet with direction of vane or blade motion ∠C A D β = Angle made by relative velocity vector at inlet in the direction of motion of vane at inlet. ∠C B D

In order to obtain a higher efficiency it is always desirable that the relative velocity vector ‘V ’ strike the vane tangentially.

Jet Striking a Moving Vane Tangentially at One Tip

Figure 7.7.1

7.7.1 Dynamic Force on a Curve Blade Moving In Translation

Part 7.7: Hydraulic Machines

Quick Refresher Guide

Fluid Mechanics

(

±

.

).

=

(

±

).

∆(

)

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Consider the flow through the runner of a turbo machine. Let the flow enter the runner blade with an absolute velocity ‘V ’ at an angle ‘α ’ and leave it with absolute velocity ‘V ’ at an angle α . It is preferred to draw velocity triangle on the blade tips. As shown in Fig. 7.7.2

7.7.2 Theory of turbo machines When a fluid flows through the runner of a turbo machine, its radius usually varies along its path. Hence it is desirable to compute torque rather than the force.

Curved Vanes Mounted on a Wheel: For rotation of the wheel at constant angular speed ‘ω’ the blade tip velocity will be u = w ,u = w ‘r ’ and ‘r ’ being the radii at the inlet and outlet respectively.

=1−

=

∴ Vane Efficiency =

Note:- If blade friction is negligible, then the workdone by the jet = Change in kinetic energy.

=

=

Vane Efficiency:

Note: If is the change in the whirl velocity ‘V ’ between inlet and outlet sections that produces the force on the vane and is responsible for doing work. Force on vane: ‘V ’ denotes the fate of flow passing through control volume. Forces on Vanes: Case (a): If α < 90° Force on the vane = ρQ [V − (−V )] = ρQ [V + V ] = γQ/g [V + V ] = W/g [V + V ] ‘W’ = Wt of fluid striking the vane/sec = γQ γ = sp. weight of fluid Case (b): If α > 90°, Force on the vane = ρQ [V − V ] = W/g [V − V ] Workdone per second = Power (on the vane) Workdone / second = Power = F.u = W/g (V + V ). u if α < 90° = W/g (V − V ). u if α > 90° W = γQ

Quick Refresher Guide

outlet

Vr2

U2

V2

Vw2

INLET VELOCITY TRIANGLE

α1

Vw1

Figure 7.7.2 Outlet Velocity Triangle

r1

Vr1

β1

U1

Vf2

β1 Vr1

Vf1

Fluid Mechanics

P = γQH

− − − − − − (iii)

=( Efficiency of pump =

)

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=

Efficiency of turbine =

Total operating head H = H + ∑h → for a turbine The heat produced, H = H − ∑h → for a centrifugal pump.

Where H = Head utilized by the turbine comparing (ii) and (iii) =

Further

→ Euler’s equation for turbo machines. (This is the torque exerted by the fluid upon the runner of the turbo machine). Power delivered to the runner, P=Tω = ρQ (V r − V r ). ω = ρQ [V u − V u ] − − − − − − − (ii)

The rate of flow passing through the control volume depends upon only the radial component of the absolute velocity V (or) V . i.e., on velocity of flows V and V . Q=A.V = (2 πr . b ) V = (2 πr . b ) V r , r are radii at inlet and outlet. b and b are width of flow passages with in the runner at inlet and outlet. The torque exerted on the axis of rotation by the fluid depends upon only the tangential components of the absolute velocity V and V i.e., on velocity of whirl. From moment of momentum (or) angular momentum equations. [V r − V r ] = −−−−−( )

Inlet

α1 V1

Vw

Quick Refresher Guide

Fluid Mechanics

Figure 7.7.4

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Outlet Velocity Triangles: At the outlet tip any of the three velocity triangle are possible depending upon the magnitude of ‘U’, corresponding to which it is a slow, medium or fast runner.

Inlet Velocity Triangle: It is a straight line. Where V = V − U = V − U, V = V

Notations: V = Velocity of jet at inlet = C 2 (C = Co. eff. of velocity = 0.97 to 0.99 takes care of loss in nozzle) V = Absolute velocity of jet leaving the bucket. U = Absolute velocity of bucket considered along the direction tangential to the pitch circle. V = (V − U ) = Velocity of incoming jet relative to the bucket. V = Velocity of jet leaving the bucket relative to the bucket. V = K. V , K = blade friction coeff. V = Velocity of whirl at the inlet tip of bucket. V = Velocity of whirl at the outlet tip of bucket. = (V Cos β2 − u ) β = β = Angle of blade at outlet tip α = α = Angle made by absolute velocity with peripheral velocity at outlet U = U = U πDN/60, D = dia of wheel, N = speed of wheel in r.p.m. Since the inlet and outlet tips of the bucket are at the same radial distance from the center of shaft.

A

Velocity Triangle: g ((i)) The followingg figure g shows the velocity triangles g at the tips of buckets.

It is a tangential flow impulse turbine used for high heads. Whole of the pressure energy in this turbine is converted to kinetic energy. The wheel revolved in open air at atmospheric pressure when water strikes on the series of buckets, i.e., there is no difference of pressure at the inlet and outlet of runner.

7.7.3 Pelton Wheel

Quick Refresher Guide

V2

Vw2

α2

Vf2 α2

Figure 7.7.5

U2 = U

β2

β2

Vr2

is positive)

V2 = Vf2

U2 = U

Vf2 Vw2

V2

α2

Fluid Mechanics

. .

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Kinetic Energy/unit mass =

iv) Hydraulic Efficiency w. r. t unit mass of water/sec =

C. Work done/Sec (or) Power developed by buckets (runner): (For α < 90°):= ρQ [V - u] [1 + K cosβ] u i) Power developed P = ρQ (V + V )u ii) Power developed per unit mass/sec = (V − u) (1 + K Cosβ)u ) ( )( iii) Power developed/unit weight of water per sec =

B. Force exerted by the fluid on the buckets:= [V − V ] If ∝ < 90° , = [V − (−V )] = [V − V Cosβ − u] [V − u) + V Cosβ] = = [V − K. V Cosβ] = V [1 + KCosβ] [V − ][1 + ] = Similarly if ∝ = 90°, V = 0. ∴ F = ρQ [V − u] if ∝ > 90°, F = ρQ [V − V ]

1. Work done and efficiencies of Pelton Wheel: A. Velocity of Whirl (V ):- The components of absolute velocities V = V Cos ∝ and V = V Cos ∝ are velocity of whirls. These are responsible for force exerted and Work done.

∴ Power = F.u. = ρQ (V − u) (1+KCosβ) u The transfer of energy from the water to the buckets takes place according to the momentum principle. The dynamic force exerted by water on the buckets in a direction tangential to the pitch circle is the force which produced a torque and causes the rotation of the runner. ∴ The change of momentum in a direction tangential to the periphery of the runner shall be computed.

α =α;β =β (α < 90°, V negative) (α = 90°, V = 0) (α > 90°, V Torque = F.r., Power = F.r.w. = (Tw) = F.u.

U2 = U

β2

Vr1

Quick Refresher Guide

(

)(

)

Fluid Mechanics

(

(

)(

) )

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metres. = 0.542 — G) Jet Ratio (m):- The ratio of pitch diameter ‘D’ of the Pelton Wheel to the diameter of the jet (d). i.e. m = (D/d). It is an important parameter in the design of Pelton Wheel. For maximum efficiency jet ratio varies from 11 to 14 and is 12 for most cases. H) Number of buckets for a Pelton Wheel Runner = 15 + = 15 + 0.5 , m = jet ratio. I) Number of Jets:- It is obtained by dividing the total rate of flow through the turbine by the rate of flow of water through a single jet. J) Important dimensions of a Pelton Wheel in terms of jet diameter ‘d’: Axial Width B = 3d to 5d, radial length L = 2d to 3d M = 1.1 to 1.25d, β = 10° to 20°,

2. Working Proportions of Pelton Wheel:A) Spouting Velocity or Ideal Velocity of Jet = 2gH Where H = Net head at base of Nozzle. B) Actual Velocity of Jet (V) at inlet = V = Cv 2gH Where C = (Coeff. Of velocity = 0.98 (or) 0.99 for nozzle. C) The velocity of wheel for maximum efficiency U = 0.5 V (Theoretical) However in actual practice, U = ∅ 2 Where ∅ = Speed Ratio = 0.43 to 0.47 D) Angle through which the jet of water gets deflected in buckets = 165° (unless otherwise specified). E) The mean diameter or the pitch diameter ‘D’ of the Pelton Wheel is given by, ( ), = = F) Least dia of jet given by

vii) Mechanical Efficiency:- (η ) . . ( ) . . η = . . The mechanical losses are i) Friction in bearings ii) Windage loss i.e. the friction between the wheel and atmosphere in which it rotates.. viii) Overall Efficiency:Shaft horse power η = energy svailable at the entrance to the turbine For Pelton Wheel it lies between 85 to 90%

vi) Hydraulic Efficiency w. r. t unit wt. =

∴ Maximum Hydraulic Efficiency η =

(Hydraulic Efficiency represents the effectiveness of wheel in converting the Kinetic Energy of the jet into mechanical energy of rotation). v) Conditions for Maximum Efficiency:- (β = outlet blade angle) = 1, =

∴ Hydraulic Efficiency =

Quick Refresher Guide

Fluid Mechanics

Spiral casing Stay Ring Guide Mechanism Runner

U

Vf = V

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work done = ρaV (Vw U − Vw U ) ρaV = mass of water striking per second = Maximum output under specified conditions is obtained by Vw equal to zero. Now work done = ρaV (Vw U ) work per unit weight of water = (Vw U ) / g Hydraulic efficiency of Francis Turbine

Figure 7.7.6

Vr

Work done and efficiencies of Francis turbine:

Francis Turbine: It is a mixed flow type of reaction turbine. Water enters the runner radially at its outer periphery and leaves axially at its centre.

‘A’ and ‘B’ above are most efficient, ‘C’ and ‘D’ have an advantage that they require lesser excavation for their installation

(A) Straight divergent tube (B) Moody spreading tube (C) Simple Elbow tube (D) Elbow Tube having circular cross section at inlet and rectangular at outlet.

Types of draft tube:

i) ii) iii) iv)

The hydraulic elements of a reaction turbine are as follows:

Reaction turbine means that the water at the inlet of the turbine possesses kinetic energy as well as pressure energy. As the water flows through the runner, a part of pressure energy goes on changing into kinetic energy. Thus runner is completely enclosed in an air tight casing and the runner is always full of water.

7.7.4 Reaction Turbines

Depth D = 0.8d to 1.2d. Multiple Jet Pelton Wheel:- A Pelton Wheel having more than one jet spaced around its runner.

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Fluid Mechanics

).

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Between the guide vanes and the runner the water in a Kaplan turbine runs through a right angle into the axial direction and then passes through the runner. The runner consists of four or

Kaplan Turbine: Kaplan is also a reaction type of turbine and hence it operates in an entirely closed conduct from the head race to the tail race. It is a type of propeller turbine. It is an axial flow turbine, suitable for low heads and hence required a large quantity of water to develop large amount of power.

viii) The number of runner vanes should be either one more or one less than the number of guide vanes, in order to avoid setting up of periodic impulse.

v) The runner dia at outlet (D ) varies from D/3 to 2D/3. usually taken as D/2. Tangential velocity at outlet. U V = πD N / 60 vi) Normally width of wheel at outlet B = 2B . vii) Generally runner is designed to have the velocity of whirl Vw at outlet equal to zero. i.e., Vw = 0. and β = 90°. Then runner vane angle ‘β’ at outlet is given by tan β =

iv) Velocity of whirl Vw at inlet can be determined from relation ηh = (

A) Design of a Francis Turbine Runner: i) Determine required discharge from the relation P = ηo W.Q.H / 75 P = power in kW W = Sp. Wt. of water in kN/m ii) If B = Width of wheel at inlet, D = dia of runner, k = vane thickness coefficient. discharge Q = K πD B × V iii) Tangential velocity of runner at inlet U =

i) The ratio of width ‘B’ of the wheel to the diameter ‘D’ of the runner is represented by ‘n’, that is n = B/D. It varies from 0.1 to 0.45. ii) Flow Ratio : The ratio of Velocity of flow V at inlet tip of the vane to the spouting velocity 2 is known as flow ratio ‘Ψ’. ∴ Flow ratio Ψ = V / 2gH. It varies form 0.15 to 0.30. iii) Speed ratio ‘I’ = U 2gH, varies from 0.60 to 0.90.

Working Proportions of Francis Turbine:

ρaV = W / g W = Wt. of water striking per second. = × = S.H.P / WQH ⟶ in SI system (S.H.P in W) Over all efficiency S.H.P / (WQH/75) ⟶ S.H.P in H.P ∴ power ‘P’ developed by runner Note: 1 HP ≈ 746 W P = ηo. W.Q.H W = Sp. wt. of water The overall efficiency of Francis Turbine ranges from 80 to 90%

Note: If Vw is not equal to zero, ηh = (Vw U − Vw U ) / gH The value of ‘ηh’ varies from 85 to 95%. Mechanical Efficiency : If ‘P’ is the horse power developed by runner shaft, mechanical efficiency. = (

ηh = Vw U / Gh

Quick Refresher Guide

Fluid Mechanics

Type of Turbine Pelton Wheel with single jet. Pelton Wheel with two or more jets Francis Turbine Kaplan (Or) Propeller Turbine.

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Specific Speed 10 to 35 35 to 60 60 to 300 300 to 1000

Also performance of a turbine can be predicated.

(1) For selecting type of turbine.

Significance:

1. Significance of specific speed (Ns) : Defined as the speed of a turbine which is identical in shape, geometrical dimensions, blade angles, gate opening etc., with the actual turbine but of such a size that it will develop unit horse power. When working under unit head. It is useful in comparing the different types of turbines as every type of turbine has different specific speed. / Mathematically, Ns= / Note: (1) If ‘P’ is in metric horse power, the specific speed is obtained in M.K.S units. (2) If ‘P’ is taken in Kilowatts, the specific speed is obtained in S.I. units.

= √ / / Where N = original speed of rotation of turbine runner P = shaft horse power H = net head Note: (i) In SI system, if P = power in KW, (‘N’ speed of rotation in r.p.m, ‘H’ in m, ‘Ns’ has the dimensional formula of M1/2 L-1/4 T-5/2 (ii) Homologous Turbine : A turbine which is geometrically and kinematically similar i.e., having similar stream lines (or) similar velocity triangles.

Specific Speed: The speed of a homologous turbine developing unit power under unit head.

7.7.5 Specific Speed and Performance of Turbines

= ( − ) 2 = /4 Where flow ratio Ψ = 0.70 for Kaplan Turbine. iii) Velocity flow at inlet and outlet are equal i.e. Vf = Vf . iv) Area of flow at inlet = Area of flow at outlet = π (D − d )/4

i) Ratio n = d / D where ‘d’ = Hub (or) Boss diameter, D = runner outside dia. It varies from 0.35 to 0.60. ii) Discharge ‘Q’ through the runner is given by

Working Proportions of a Kaplan turbine:

six blades. Servometer is used for adjustment of runner blades. Eddy losses, which are inevitable in Francis and Propeller turbines are almost eliminated in a Kaplan Turbine.

Quick Refresher Guide

Fluid Mechanics

/

(

√

) /

(for in S.I. System)

=

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Cavitation is found to occur in turbines as well as in various hydraulic structures such as penstocks, gates, valves, spillway etc due to low pressures.

7.7.6 Cavitation in Turbines

=

=

4. Similitude in Turbines: Scale models are often used in designing and other studies relating to turbines. Geometric similarity is assured by having geometrically similar velocity vector diagram. It is usual to neglect viscous effects in the model studies. The model and prototype characteristic relationships are usually expressed in terms of the following relationship between the variables: =

=

(C) Unit power (Pu) : Pu = P/H3/2 Where P = power developed by turbine under a head ‘H’ Use of Unit Quantities : If a turbine is working under different heads the behavior of the turbine can be easily known from the values of Unit Quantities. = , = / / ∴ =

(B) Unit Discharge (Qu): Where Q = discharge passing through a given turbine under a head ‘H’

3. Unit Quantities : In order to predict behaviors of a turbine working under varying conditions of head, speed, output and grate opening, the results are expressed in terms of quantities which may be obtained when the head on the turbine is reduced to unity. The conditions of turbine under unit head are such that the efficiency of the turbine remains unaffected. The important unit quantities are given below. (A) Unit Speed : It is defined as the speed of a turbine working under a unit head. (ie., under a head of 1 m). Denoted by ‘N ’. We have N = N√ Where N = speed of turbine under a head ‘H’ H = head under which a turbine is working U = tangential velocity.

=

2. Shape Number : A dimensionless form of specific speed is known as Shape Number Shape Number = Dimensionless specific speed.

Note: Some of the authors report Ns as 35 to 50. The higher the specific speed, the smaller the runner diameter as well as overall size of runner, due to which the weight and the cost of runner are reduced.

Quick Refresher Guide

Fluid Mechanics

−

−

)/ . (H − H ) is also called as barometric head.

.

If ‘σ’ for a turbine is greater than’σ ’, no cavitation.

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1. Classification of Centrifugal Pumps: The basis for classification of Centrifugal Pumps depends upon the following characteristics features: i) Working Head ii) Type of Casing iii) Relative direction of flow through impeller iv) Number of impellers per shaft v) Number of entrances to the impeller vi) Type of Liquid Handled vii) Disposition of Shaft

A rotodynamic pump is a reaction turbine “in reverse”. Reaction Turbines placed below the tail water while coupled to electric motor and rotated in reverse direction will work as pumps. A centrifugal Pump is a machine which increase the pressure energy of a fluid.

Centrifugal Pumps are rotodynamic type of pumps in which the dynamic pressure developed enables the lifting of liquids from a lower level to a higher level.

7.7.7 Centrifugal Pumps

--------------- (ii)

-------------- (i)

obtained by equation (ii) should be increased by 10 percent.

= 0.28 +

For Kaplan turbines, values of

For Propeller Turbines

For Francis Turbines σ = 0.625 (N / 444)

Complete similarity in respect of cavitation can be ensured if the value of σ is same in both the model and the prototype. Moreover it has been found that σ depends on Ns of the turbine, and for a turbine of particular Ns the factor σ can be reduced upto a certain value upto which its efficiency dose not remains constant. A further decrease in the value of σ results in a sharp fall in no. The value of ‘σ’, at this turning point is called critical cavitation factor ‘σc’, The value of ‘σc’ for different turbines may be determined with the help of the following empirical relationship.

Where H is atmospheric pressure head, H is vapour pressure head, H is suction pressure head (or height of runner outlet above tail race) and H is working head of turbine.

=(

D. Thoma of Germany has developed a dimensionless parameter called Thoma’s cavitation factor, which is expressed as

In order to determine whether cavitation will occur in any portion of the turbine,

Due to cavitation the metal of the runner vanes and the draft tube is gradually eaten away in these zones, which results in lowering the efficiency of the turbine. As such the turbine components should be so designed that as far as possible cavitation is eliminated.

In reaction turbines the cavitation may occur at the runner exit or the inlet to the draft tube, where the pressure is considerably reduced.

Quick Refresher Guide

Fluid Mechanics

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= / Where N = Actual speed of pump in r.p.m.

Specific Speed: “The speed of a geometrically similar pump when delivering one m /sec against a head of one metre”. (Unit discharge)

Classification based on specific speed:

ii) Multi Stage Centrifugal Pump: - Two or more impellers mounted to a single shaft in single casing. - Pr. is built up in stages. - Used for high working heads. - No. of stages depends upon the head. -

- Low lift pumps.

i) Single Stage Centrifugal Pump: - One impeller mounted to the shaft.

Classification based upon No. of impellers:

i) Radial Flow Pump: Almost all centrifugal pumps are manufacture with radial flow impellers. ii) Mixed Flow Pump: Used for irrigation purposes. - Large Q at low ‘H’. - Liquid flows in the impeller with a combination of radial and axial flow. - These look like a screw and also called as screw impeller pumps. iii) Axial Flow Pump: The flow through the impeller is in the axial direction only. - Very large Q at low heads - There is no centrifugal action.

Classification based upon the direction of flow:

a) Working Head: Based upon the head at which water is delivered the pumps are Classified as i) Low Lift Centrifgal Pumps: - Heads upto 15m, - Impeller surrounded by volute. - No guide vane, - Shaft generally horizontal. - Entrance of water to the impeller from one side or both sides depending upon the quantity of discharge. ii) Medium lifts Centrifugal Pumps: - Works against the head as high as 40 m -They are provided with guide vanes. iii) High lift Centrifugal Pumps: -They deliver liquids at the heads of above 40 m

Classification based upon working head:

viii) Specific Speed N = / ix) Dimensionless Parameter K

Quick Refresher Guide

2400-5000 3400-15000

80-160 110-500

d) Mixed flow (screw runner) e) Axial flow or propeller runner

= 90 ,

=0

- For shock free entry of liquid.

Vr1 β1 u1

α1 = 900

Figure 7.7.7

α1

Vr2

U ) V

α = 90° (W = γ Q) =0 V = U − V cotα β

U )

triangle

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H = Total or manometric head.

Speed Ratio : ∅ =

Work done / unit wt. =

∴ Work Done = W/g (V

U −V

Inlet velocity triangle V1 = Vf1

Vf2

Work-done / sec by the impeller on the liquid = W/g (V

inlet of vane

Tangent to impeller of

Outlet of vane

Tangent to impeller at

V2

(In case of diffusion pump, the direction of absolute velocity of liquid leaving the impeller coincides with the tangent to the inlet tip). U2 Vw2 Outlet velocity α2 β2

The relative velocities V and V are parallel to the vane at the inlet and outlet tips.

∴

Liquid enters the impeller radially i.e., Abs. Velocity of liquid at inlet is in radial direction.

Work done by the impeller:

* Ns = 30 Ns

900-1500 1500-2400

50-80

300-900 rpm

*Ns With Q = Litres/sec H = 1m

30-50

Ns /sec, H = 1m 10-30

c) High Speed radial flow

Q=

Fluid Mechanics

b) Normal Speed radial flow

a) Slow speed radial flow runner

Types of Impeller

Q = Discharge m /sec . = Delivery head (Total or manometric) in ‘m’.

Quick Refresher Guide

Flow Ratio : Ψ =

High N impeller 1.25 0.25

∴ U and V can be worked out and substituded in the

Fluid Mechanics

+

+

(

)

=

.

(

∆

Xη Xη

=

)

(

.

.

is neglected).

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ABD = Inlet velocity triangle.

The efficiency of a pump decreases due to variation in design discharge and speed because of loss of head due to shock at the inlet.

5. Head loss due to shock:

iv) Overall Efficiency η = η

iii) Mechanical Efficiency η

=

ii) Volumetric Efficiency :- η =

=

(or) η

i) Manometric Efficiency η = = ⁄

4. Efficiencies:

Also H = h + h + h + h + V

b) Manometric head (or) Total Head (or) Gross Head (or) Effective head “Total Head required to be produced by the pump to satisfy external requirements”. H = Energy given to the liquid by the impeller - Losses of head in the pump. . = - Losses of head in the pump

Static head is the net total vertical height through which the liquid is lifted by the pump.

a) Static Head: H =h +h h = Static suction lift h = Static delivery lift

3. Head of pump:

= Change in K.E. + Centrifugal head + Static pr. Head (energy produced by impeller)

=

i) Fundamental Equation of Centrifugal Pump:- Work done per Kg is also given by

expression of work done. Low N ∅ 0.95 Ψ 0.10

Quick Refresher Guide

Fluid Mechanics

=

∅

2

/

(

(

.

)

/

H = Total head

N = speed in r.p.m

=

(√

)

)

… . (1)

.

/

/

) / THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 271

=(

Non – Dimensional specific speed (or) Shape number of the pump:

“The speed in r.p.m. of a geometrically similar pump of such a size that under corresponding conditions it would absorb 1 KW power when working against a head of 1 metre.

√

(Most commonly used) (Dimension formula)

V = 1.5 to 3.5 m/sec

* Also on power criteria. The uncommon expression for specific speed. Ns =

Specific Speed : Ns =

d =

V = velocity of flow in suction pipe. = 1.5 to 3 m/s (usually) Dia. of delivery pipe:

=

=

D = 0.5 D This equation is similar to (1) for ∅ =2/√3

Dia. of suction pipes:

D =

.

= 2gH

=

Usually ∅ = 0.95 – 1.25 ∴ The above eq. can be used to check an existing pump to determine the head developed if D and N are known. b) Inlet dia. of impeller = = 0.5 D usually c) Least dia. of impeller from min. starting speed consideration

∴

U = Nu ∅ 2gH

a) Outside dia of impeller U =πD

6. Working proportions of impeller and pipes:

∴ Loss of head due to shock at entrance =

Also EC is parallel to AB. ∴ Tangential change in velocity AE will take place suddenly and results in shock causing head loss.

AB is parallel to the vane inlet tip. If the flow is reduced or increased from BD to CD while the speed of rotation remains the same, the vel. triangle is represented by ACD and AC is the new relative at the inlet remains unchanged.

Quick Refresher Guide

Fluid Mechanics

=

+

−

[

+ℎ +ℎ

]

=

/

h = head loss in strainer

ℎ = ht. of inlet above the sump.

V = vel. In suction pipe.

=

−

−ℎ −ℎ

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∴

It is not possible to create at the pump inlet, an absolute pressure lower than the vapour pressure. If P = vapour pressure of liquid in Abs, units then P + P = P in limiting case.

+

When pumps are installed above the level of sump. Pressure at inlet < P . Applying Bernoulli’s eq. at pump inlet and the liquid level in sump, the absolute pressure head.

10. Limitation of Suction Lift:

Pressure developed is ∝ specific weight of the liquid in contact. If air is in contact, pressure developed is for air and cannot pump the liquid. ∴ Priming (i.e. filling the liquid in suction pipe, impeller casing and in delivery pipe upto delivery valve with the liquid to be pumped is essential. - Gain in pressure head between the outlet of the impeller and the outlet of the pump is given by K(V /2g), K = 0.4 for volute casing K = 0.7 for turbine pump or diffusion casing.

9. Priming of Pump:

When a large quantity of liquid is required to be pumped against a relatively small head, two or more pumps are connected in parallel. Pumps in parallel are so arranged that early pump works separately lifting liquid from a common sump and delivering it to a common collecting pipe. + + + ⋯ … …. = = . If = = H = constant.

8. Pumps in Parallel:

Head produced by a C.P. depends on the rim speed of the impeller, which can be increased by increasing rotative speed or the dia. of the impeller or both, requiring large impellers. A multi stage pump consists of two or more identical impellers mounted on a common shaft and enclosed in the same casing. All the impellers are connected in series. Total head. H = n (H ) H = head gained in the impeller = n (H + H ) H = head gained in the guides ∴H=H +H ,Q=Q =Q

7. Multi Stage Pumps in series:

* For pumps discharging large quantities at low heads. Such as axial flow pumps the N will have large value.

Quick Refresher Guide

−

−ℎ

Fluid Mechanics

=K −[

−

−h −h −

+h +h ] −

+

+

=

(

) (

)

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Suction specific speed S=N Range of ‘S’ for cavitation free operation of C.P. and propeller pumps. S = 4700 to 6700

“When hot liquids are to be pumped the pumps have to be installed at liquid surface or even below the liquid surface” In first case h = 0, in second case h ≤ 0 indicating that there is +ve pr. at pump inlet.

Critical τ = 0.103

= atm. Pr. head, H = vapour pr. head, h = suction head. H h = head loss in suction, H = manometric head, H = total suction head. N = sp. speed.

=

Thomas’s cavitation factor =

If the pressure at the suction side of the pump drops below the vapour pressure of the liquid then cavitation may occur. Cavitation in a centrifugal pump results in sudden drop of head and efficiency.

12. Cavitation in Centrifugal Pumps:

−ℎ −ℎ = But R.H.S is the total suction head −ℎ −ℎ H = ∴ NPSH = H i.e. Total suction head ∴ NPSH is the head required to make the liquid flow through the suction pipe into the impeller.

∴ NPSH =

=

NPSH = Abs. Pr. Head at pump inlet – vapour pr. Of liquid to be pumped – vel. Head in the suction pipe.

11. Net Positive Suction Head (NPSH):

Suction lift in no case shall be greater than the above value otherwise vaporizations of liquid due to reduction in pressure takes place leading to cavitation. * Usually h = 6 to 8 m for water at 10° − 20° C h =0m for water at 65℃ Positive pressure shall be provided at pump inlet if water is at 65℃. i.e., the pump to be installed below sump water level.

ℎ =

Quick Refresher Guide

Heat Transfer

(

)

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Thermal conductance =

KA 1 = Rt dx ∴ Unit thermal resistance = dx/K

Electric current (I), Heat flow rate (Q),Voltage (V), Temp. difference (dt), Resistance – dx/KA – called thermal resistance (Rt)

∴ Heat flow rate Q =

.

In heat transfer the driving force is the temperature difference.

Thermal resistance:

¾ Valid for all matter – solid, liquid or gas ¾ Is a vector expression indicating that heat flow rate is normal to an isotherm and is in the direction of decreasing temperature. ¾ It cannot be derived from first principle ¾ Helps to define the transport property “K”

Features of Fourier equation:

¾ Steady state conduction ¾ One directional ¾ Bounding surfaces are isothermal in character i.e, constant and uniform temps are maintained at the two faces. ¾ Isotropic and homogeneous material and K is constant ¾ Constant temp gradient and linear temp profile ¾ No internal heat generation.

Following are the assumptions of Fourier equation

¾ dt/dx is the temp gradient ¾ -ve sign indicates that the heat flow is in the direction of negative temperature gradient and that serves to make the heat flow positive. ¾ Thermal conductivity “K” is one of its transport properties. Others are the viscosity associated with the transport of momentum, diffusion coefficient associated with the transport of mass. ¾ K provides an indication of the rate at which heat energy is transferred through a medium by conduction process.

For 1-D steady state heat conduction through homogeneous material without heat generation is given by Q = – K A (dt/dx) ⇒ q = Q/A = - K (dt/dx)

8.1.1 Fourier equation

Part 8.1: Conduction

Part – 8: Heat Transfer

Quick Refresher Guide

Heat Transfer

+

+

=

=

Q

A

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K

δ

A = Constant C.S. area

K = Constant thermal conductivity

Let δ = isotropic wall thickness

Conduction through a plane wall:

8.1.3 Steady state conduction

Liquids have low “K” and high “ρc” and so low “α” Solids have high “K” and low “ρc” and so high “α” α indicates rate at which heat is distributed in a material The relative magnitude of thermal diffusivity is a measure of rapidity with which the material responds to sudden temp changes. ¾ In insulators the heat conduction takes place due to vibration of atoms about their mean positions ¾ In metals besides atomic vibrations, there are large no of free electrons which also participate in the process of heat conduction.

¾ ¾ ¾ ¾

=0

For one dimensional, steady state with q = 0

α = K/ρc called thermal diffusivity and is a physical property of the material. It is more useful in unsteady conduction situations.

+

8.1.2 General heat conduction equation

¾ For most materials K = K0 (1 + βt). K0 – Thermal conductivity at 0℃ β – Constant (+ve or –ve) +ve for non metals and insulation materials -ve for metallic conductors exception in Al and Uranium

Freon12 – 0.0083 (min), silver – 410 w/m°K (max), Al – 225, Brass – 107, Copper = 385, Steel – 20 to 45, Concrete – 1.2, Brick – 0.65, earth – 0.3, glass – 0.75, plaster – 0.8, wood – 0.052, saw dust – 0.07, cork – 0.03, air – 0.024, ash – 0.12, Ice – 2.25, water – 0.55 – 0.7

K for different materials are

Unit thermal conductance = K/dx

Quick Refresher Guide

Heat Transfer

(

)

+

(

)

/

=

where R = Thermal resistance = δ / KA

(

)

…

=

(

)

Rt = δ /K A … … …

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i. There is perfect contact between adjacent layers ii. The temp is continuous at the interface iii. There is no fall of temp at the interface However in real systems, the contact surfaces touch only at a discrete locations due to surface roughness, void spaces etc which are usually filled with air. So there will not be a single plane of contact. This implies that the area for heat flow at the interface will be small compared to the geometric area of the face. Due to this apparent decrease in the heat flow area and also due to the presence of voids, there occurs a large resistance to heat flow. This resistance is referred as thermal contact resistance and it causes temp drop between two materials at the interface.

Heat flow through a multi – layer composite wall can be calculated based on the assumption that

Thermal Contact Resistance

Rt = δ /K A, Rt = δ /K A,

=

Therefore

Under steady state conditions heat flow does not vary across the wall i.e., it is same for every layer.

Conduction through a composite wall:

=

From temp distribution equation it is seen that it is linear across the wall and it is independent of the material because it does not involve thermal conductivity. From H.T. equation it can be written as

=

Taking Fourier equation and substituting dt/dx, we get

dt/dx = (T − T )/δ

Differentiating the above equation w.r.t.’x’

=

on integration and substituting the B.C’s, the temp distribution equation becomes

=0

The general heat conduction equation of one – dimensional, steady state without heat generation is

Assume one dimensional heat conduction through homogeneous material and wall is insulated on its lateral faces but different constant temperatures T1 and T2 are maintained at boundary surfaces.

Quick Refresher Guide

/

) ( /

T

) ( /

Rt

)

T

Rt

T h

Hot fluid

T

hall

T

T h

= 0 ⇒ (1/ )

(

/

)=0

( / ) ( / )

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=0

Steady state, one dimensional with no heat generation equation in spherical co-ordinates is

Conduction through sphere:

Where A is the logarithmic mean area = (A2 – A1) / log (A2 / A1)

Q = KA (T1 – T2) / δ = KA (T1 – T2) / (r2 – r1)

Writing the above “Q” equation in the form of plane wall

Q = (T − T )/R where R − [log(r /r1)]/2πKL

=

on integration and substituting the B.C’s we get the temp distribution equation as

+

Heat Transfer

Cold fluid

For steady state unidirectional with no internal heat generation the equation becomes

Conduction through a cylindrical wall:

=(

Rt

Heating and cooling of fluids:

Quick Refresher Guide

=

(

(

)

)

=

Heat Transfer

+ =

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8.1.4 Heat transfer from extended surfaces (fins)

Similarly for sphere the critical radius works out to be r = 2K/h

When r > r the effect of wall thickness dominates the overall thermal resistance

the above relation for “S” is valid when a, b, c > (1/5) dx

×A

Critical thickness = r − r

a, b, c are inside dimensions

= 0.79 A

Critical radius = r = 2K/h for sphere

dx = wall thickness

if not S

Critical radius = r = K/h for cylinder

Stotal = 2/dx (ab + bc + ca) + 4 X 0.54 (a + b + c) + 8 x 0.15 x dx

X

=−

Critical thickness of insulation:

T

Q = -Km A (T1 – T2) where Km = Ko (1 + β T ), T = (T1 + T2)/2

On separating the variables and solving it

β = constant

K0 = thermal conductivity at 00C

Q = -K0(1 + β t)A dt/dx

Fourier law of heat conduction through a plane wall can be expressed as

Effect of variable thermal coductivity:

The insulation radius at which resistance to heat flow is minimum is called critical radius and corresponding thickness of insulation is critical thickness.

= 0.15 X dx

= 0.54 X length of edge

Heat Transfer

Note: For same inside capacity and same amount of fabrication material, same temp and same material the heat loss is lowest in cylindrical furnace because of lowest shape factor. The ratio of R.T or S.F are cubical : spherical : cylindrical is 1 : 0.71 : 0.439

Quick Refresher Guide

A complete rectangular furnace has 6 walls, 12 edges and 8 corners. The shape factor for complete furnace is

S

S

Shape factor for an edge is

The unit of shape factor is length units. From the above it is seen that for a prescribed temp difference (T − T ), bodies with the same shape factor will allow heat transfer proportional to material “K”

S = 4Π r1 r2/(r2 – r1) for sphere

S = 2ΠL/ log (r2/r1) for cylinder

Q = K ∆TS, where S = A/δ for plane wall

Shape factor : In general all the factors relating to geometry of the section are grouped together into a single constant called the shape factor. Shape factors for different sections are

Where rm = √r r

Where Am = 4π r r = 4π r

Q = (KA ∆ T / δ) = {(K Am ∆ T) / (r − r )}

If we write the above H.T. equation in the form of plane wall

Where Rt = {(r2 – r1) / 4π K r r }

Heat Transfer rate =

From the above it is seen that it is hyperbolic

=

on integration and substituting the B.C’s we get the temp distribution equation as

Quick Refresher Guide

Heat Transfer

…

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On substitution

T = Ta at x = ∝ and θ = 0

T = To at x = 0 and θ = θo = To – Ta

The boundary conditions are

(i) Infinitely long fin (L → α)

This is the general heat dissipation equation from the fin surface

+

(

)

=

hpKA .

. tan h (ml)

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The boundary conditions are

(iii) Finite Fin without tip insulation

Q

The heat transfer equation is

=

applying the above BC’s, the temp distribution equation is

dt / dx = 0 at x = 1

Θ = θo at x = 0

(ii) Short Fin Insulated at the Tip

The general solution for the above equation is

=

Heat Transfer

Generally tapered fin is preferred because it has more lateral area near the base where the difference in temperature is high.

The boundary conditions are

Thickness is small when compared to L and b and H.T is one dimensional Homogeneous and isotropic fin material and K is constant Uniform h.t.c which also includes radiation effects No heat generation within the fin Joint is perfect Steady state heat dissipation

Take a small element of fin which has thickness dx and is located at a distance “x” from wall. Heat conducted into the element at plane “x” = Heat Conducted out from fin at (x + dx) + heat convected from the fin between plane x and (x + dx)

¾ ¾ ¾ ¾ ¾ ¾

Assumptions

m

Perimeter = πd

m m

x

m <m <m

Ac = πd /4

For circular Fin

Perimeter = 2 (b + t)

Ac = b.t − −

Q = - K. Ac (- m θ) = K. Ac (m θ) = hpK A . θ

Steady flow of heat along a Rod:

For rectangular Fin

On simplification

Q = - K Ac (dt/dx)x = 0

For estimating heat transfer

(T – Ta) = (To – Ta)e

Quick Refresher Guide

So the only way is to increase the surface area of H.T. This can be done by providing extensions to the surfaces which are called extended surfaces or fins.

Generally H.T from a surface is calculated based on Q = hA (T – Ta), from this it is seen that H.T. can be increased by increasing h or A or (T – Ta) or combination, but h depends on geometry, fluid properties, flow rate so we cannot increase much amount of “h” and also (T – Ta) is difficult to increase

Quick Refresher Guide

Heat Transfer

) {( / )( {( / ) (

=

h p K A .θ

( /

)}

)}

) ( / ) ) ( )

(

(

=

=

. .

(

)

hPKA . θ

.

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Note: For calculating error in measurement of temp of gas with thermometric well

¾ Fins are generally made of Al (even though its thermal conductivity is less than that of copper) because of its lower cost and weight ¾ ρ/Ac increases, the ε also increases and to accommodate this we use thin fins at less pitch ¾ Fins will always be used where the h.t.c is less. Suppose if there is fluid at one side and gas at the other side, it is preferred to use fins in gas side. This is the reason for not using fins in steam condenser tubes: ε = η × Pl/A = η × A /A for fin insulated at tip.

But from practice the fins on surfaces is justified only if the ratio PK / hAc > 5

¾ For improving H.T, the ε should be greater than 1 pK/hA > 1

From the effectiveness equation the following observations were made

For long fin =

H.T. with long fin = hA (t0 − ta) =

H.T. without fin = hA (t − t ) = h . A . θ

Effectiveness of fin: It is the ratio of the H.T. with fin to H.T. without fin.

=

(K A /h p). 1/ = (1/m )

For fin insulated at tip, η

For long fin, η

η

Efficiency of Fin: It is the ratio of performance of an actual fin to that of an ideal or fully effective fin

Fin performance:

Q

(

The heat transfer equation is

=

(

Applying the BC’s, the temp distribution equation is

1. θ = θo at x = 0 2. heat conducted to the fin at x = L equals to the heat convected from the end of the fin to surroundings.

Quick Refresher Guide ( ) ( )

hp/KA = h/Kt,

t = thickness of the wall

Heat Transfer

⟶O

never equal to 380 and K never tends to infinity

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But in reality R

∴ Heat transfer is mainly controlled by convection only

∴K⟶α⟶R

for ex:- quenching of small metal billet casting in bath. In this case convection is more predominant than conduction. So conduction resistance is neglected.

but in general, convection resistance is more than conduction resistance

- surface convection resistance =

- internal conduction resistance =

H.T in cooling or heating of a body depends upon

THC in solids with Infinite ‘K’

¾ Change in temperature during unsteady state may follow a periodic or non-periodic variation ¾ In non-periodic the variation in temp is neither according to any definite pattern nor is in repeated cycles. Ex:- heating of ingot in a furnace ¾ In periodic variation, temp changes in repeated cycles and the conditions get repeated after some fixed time interval. Ex : temperature variations in the cylinder of I.C engine

For steady state the main assumption is temp at both sides is constant over a period of time. But in reality it is not possible to maintain constant temp and so is called unsteady state heat conduction, it is of time dependent

8.1.5 Transient Heat Conduction

m=

T = Temperature of the pipe wall

T = Gas temperature

T = Temperature at the bottom of the well

Where

=

Quick Refresher Guide

Heat Transfer

[−ℎ

/

]

⇒

=

/ℎ

= (1/ℎ ) (

)

(−

.

)

.

− 1]

=−

= 0.368

has time units and called time constant

and

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The equation of THC

∴ The lumped parameter solution for T.H.C is

If temp distribution is given, Bi is found from graph and then we can calculate Fo, from Fo ⟶ τ can be calculated and vice versa

T−T = exp(−Bi Fo) T −T

This needs graphs correlating (T – Ta) / (To – Ta) and l/Bi for different values of x/L or r/R etc

THC in solids with finite

∴ In practical conditions temp of thermocouple is to be recorded after 4τ*

∴ The lower the value of τ*, the better the response of thermocouple

τ* = 1/Bi.Fo if Bi Fo increases ‘τ’ decreases.

Ti – Ta = 0.368 (Ti – Ta)

=e

τ/τ* = 1

∗

The above can be reached by decrease wire dia, density and specific heat or by increasing the value of h, therefore a thin wire should be used in thermocouples

∴ For rapid response the index [hAτ / ρVC] should be as large as possible to make the exponential term reach zero faster.

T.H.C in infinite thick solids (Bi ⟶ α) (infinite in all directions)

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Heat Transfer

The important application of lumped heat parameter is thermocouple and thermometer. In general we expect that the temp of measuring instrument should reach the temp of source as early as possible. That is the response time should be as min as possible.

Response Time of a Temperature Measuring System:

Q = ρVC (T − T )[e

And on simplification

For total heat flow rate, integrate Q with time interval τ = 0 to τ = τ

Instantaneous heat flow rate Q = −hA (T − T )Exp (−B . F )

=

Quick Refresher Guide

It signifies the degree of penetration of heating or cooling effect through solid. For example if α/l is small, a large time τ is required to obtain a significant change of temp.

Fourier Number (FO): F0 =

That is for plates or cylinders or spheres if Bi < 0.1, the body temperatures differs only by 5% at any time.

Biot Number (Bi): It gives an indication of internal conduction resistance to the external convection resistance. A small value of Bi indicates the system has a small conduction resistance, so when the conduction resistance is small, almost uniform temp exists within the system or solid body. The convection resistance predominates and convection heat exchange controls the transient phenomenon. So the requirement for lumped heat parameter is Bi < 0.1

Also ρVCτ/hA = (hl/K) x (∝τ/l ) = Biot number x Fourier number

= thermal resistance x thermal capacitance

=− /

−ρVc/hA has a dimensions of time so is called thermal time constant. It indicates the rate of response of a system to sudden change in temperature

∴ Rise or fall of temp takes place exponentially

=

on substitution and simplification, the temp distribution equation becomes

∴ Rate of change of I.E = Convection H.T

Ti = initial temperature, Tα = temp of surroundings

K = Thermal Conductivity, C = Specific heat

A = surface area, V = volume, ρ = density.

Let us consider a body of

In Newtonian heating or cooling the temperature throughout solid is considered to be uniform at a given time, such an analysis is called lumped heat capacity analysis.

¾ Heat treatment of metals by quenching, time response of thermocouples, thermometers etc can be analyzed by this method. ¾ The process in which R << R is called Newtonian heating or cooling

Note:- The problems which comes under this type requires solids of large K, with areas that are large in proportion to their volumes like thin metallic wires and plates

Quick Refresher Guide

( √ ∝

)

= −1.13 KA (T − T ) τ/∝

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erf – error function – taken from table

= erf[∝ τ/R ]

Under similar conditions the temp at the center of cylinder or sphere of radius “R” is

/2√∝ τ ≥ 0.5

For using infinite solution to body of finite thickness subjected to one dimensional is

= erf[ x/2 ∝ τ]

The temperature distribution at any time τ on a plane wall at distance x from surface is

Q

Total H.T. rate at the surface will be obtained by integrating τ = 0 to τ = τ

=

Heat Transfer

=( −

) ℎ

= ℎ. (

−

), ℎ = 1/

Surface condition (roughness and cleanliness) Geometry and orientation of the surface Thermo-physical properties of fluid Nature of fluid flow Boundary layer configuration Prevailing thermal conditions

ℎ

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Reynolds Number (Re): It is the ratio of Inertia force to the viscous force. It is indicative of the relative importance of inertial and viscous effects in a fluid motion. At low Reynolds number, the viscous effects dominate and so the fluid motion is laminar. At high Reynolds number, the inertial

T T are surface and fluid temp before touching surface

T T are inlet and exit temp of a heat exchanger

Mean film temp = Tf = (Ts + Tα) / 2

∴ Bulk temp = Tb = (Ti + To) / 2

The physical properties (μ,K,ρ,Cp) of a fluid are temp dependent. But there is no definite rule to take properties at a particular temp. the accuracy will dependent upon the temp at which properties are taken.

Bulk and mean film temp:

¾ ¾ ¾ ¾ ¾ ¾

Also “h” depends on

Where h = avg. h. t. c. = (1/L) ∫ h dx

=

But for total heat transfer

q = h (T − T )

Let us take an arbitrary shape of area A at temp Ts, over which a fluid flows at a velocity V and at temp T the local heat flux according to Newton’s law of cooling is

8.2.2 Heat transfer coefficient

¾ The energy transfer in convection is mainly due to bulk motion of fluid particles. ¾ If this motion is mainly due to density variations then it is called free or natural convection. ¾ If this motion is produced by some super imposed velocity field then it is called forced convection.

Convection is the mode of heat transfer between a surface and a fluid moving over it.

Instantaneous heat flow At the surface (x = 0) is

Part 8.2: Convection

Quick Refresher Guide

8.2.1 Convection

Heat Transfer

On solving and substituting boundary conditions

dt/dx = (1/∝) dt/dτ

Quick Refresher Guide

Heat Transfer

. .

x Re ℎ / )(

/ )

=

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n = positive index

δ = thickness of thermal boundary layer

where δ = thickness of velocity boundary layer

δ/δ = (Pr )

‘Pr’ is a connecting link between velocity and thermal boundary layers respectively.

The kinematic viscosity indicates the momentum transport by molecular friction and thermal diffusivity represents the heat energy transport through conduction. So ‘Pr’ provides a measure of relative effectiveness of momentum and energy transport by diffusion.

= kinematic viscosity / thermal diffusivity

Pr = μ Cp/K = ργCp/K = γ/(K/ρCp) = γ/α

Prandtl number (Pr): It is indicative of the relative ability of the fluid to diffuse momentum and internal energy by molecular mechanisms.

Grashof number has a role in free convection similar to that played by Reynolds number in forced convection.

μ = fluid viscosity

ρ = density

×

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Presence of n – shields reduces the radiant heat transfer by a factor of (n + 1) i.e if configuration factor is equal to “zero”

¾ There will be two surface resistances for each radiation shield and one for each radiating plane, if all “ε” are equal, all surface resistances are equal. If n shields are used then no. of surface resistances is equal to (2n + 2), and the no. of space resistances are equal to (n + 1)

If n radiation shields are inserted between two planes then

(

∆t = temp difference (t − t )

=

Then the ratio of H.T with one shield to without shield is ½ (half)

=

Stanton Number (St): It is the ratio of heat transfer coefficient to the flow of heat per unit temperature rise due to the velocity of the fluid.

Nu = hl/K = conduction resistance / convection resistance = (l/K) / (l / hl)

g = acceleration due to gravity

)

If “n” shields are placed between two bodies

= ( . . . .∆ )(

Heat Transfer

Nusselt number (Nu): Establishes the relation between convective film coefficient h, thermal conductivity K, and characteristic length l of the physical system

δ >>>>> δ . . . . . . liquid metals

δ ≈ δ . . . . . . . . gases

δ <<<<< δ . . . . . . . . . oils

Quick Refresher Guide

β = fluid coefficient of thermal expansion

l = characteristic length

=

)

= ( . . . .∆ )

= Byoyant force x (

=

. . . .∆

Grashof number (Gr): Indicates the relative strength of the buoyant to viscous force.

=

effects lead to turbulent flow and the associated turbulence level dominates the momentum and energy flux.

Quick Refresher Guide

Heat Transfer

/

α, ρ, τ are always positive and lies between 0 and 1 for non reflectivity surfaces ρ = 0, α + τ = 1 and perfect reflecting surfaces ρ = 1, α + τ = 0 for black body α = 1, ρ + τ = 0 for opaque bodies τ = 0, ρ + α = 1 for transparent or diathermanous bodies τ = 1, ρ + α = 0

+ =1

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In diffused reflection, the incident beam is reflected in all the directions that is directional independence of reflected beam.

A body that reflects all the incident thermal radiations is called specular body. Specular reflection implies that angle between the reflected beam and normal equals to the angle made by incident radiation with normal ex:- highly polished surfaces

1. 2. 3. 4. 5.

τ = transmissivity

ρ = reflectivity

α = absorptivity

where

+

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(i) Monochromatic emissivity: It is the ratio of the monochromatic emissive power of a surface to the monochromatic emissive power of a black surface at the same wavelength and temp. For a gray body the monochromatic emissivity is independent of the wavelength of the emitted radiation. (ii) Total emissivity: Ratio of the total emissive power of a surface to the total emissive power of black surface at the same temp (iii) Normal total emissivity: Ratio of normal component of the total emissive power of a surface to the normal component of the total emissive power of a black body at the same temp.

Emissivities of real bodies: Emissivity of a surface indicates its ability to emit radiation energy in comparision with a black surface of the same temp level.

For many materials the emissivity is different for the various wavelengths of the emitted energy. The radiating bodies exhibiting this behaviors are called the selective emitters.

Gray Body: When the emissivity of non-black surface is constant at all temps and throughout the entire range of wavelength, the surface is called a gray body.

Kirchoff’s Law: It can be stated as “the emissivity ε and absorptivity α of a real surface are equal for radiation with identical temps and wave lengths. Therefore it states that a perfect absorber is also a perfect radiator.

… … for gray bodies, F = shape factor

⇒

−

=

=

+

… … for black body with shape factor F

−

… … for gray bodies with emissivity ε

=

−

−

∴ Qa + Qr + Qt = Qo

=

let a surface 1 at T1 is completely enclosed in black body at T2 then

= 5.67 × 10

Where σ = radiation coefficient or Stephen Boltzmann constant

E =σ T

According to Stephen – Boltzmann law the amount of radiant energy emitted per unit time from unit area of black surface is proportional to the fourth power of its absolute temp.

Stefan-Boltzmann’s Law: The total emissive power “E” of a surface is defined as the total radiant energy emitted by the surface in all directions over the entire wavelength range per unit surface area per unit time

8.3.2 Total Emissive Power

=

+

Heat Transfer

Note: A small hole leading into a cavity thus acts very nearly as a black body because all the radiant energy entering through it gets absorbed.

Quick Refresher Guide

The total radiant energy (Qo) comprising upon a body would be partially or totally absorbed by it (Qa), reflected from its surface (Qr), or emitted through it (Qt) according to the characteristics of the body. But by the conservation of energy principle the total sum must be equal to the incident radiation.

¾ Electromagnetic waves are emitted as a result of vibrational movements of the molecular, atomic or sub – atomic particles comprising the matter. The emission occurs when the body is excited by an oscillating electrical signal, electronic or neutronic bombardment, chemical reactions etc. ¾ Two radiations are distinguished by C (speed of light) = λ (wavelength) x f (frequency) ∴ if λ is less, f is high and vice versa Energy emitted by a photon = e = mc = h.f ⇒ m = hf/c ∴ momentum = mc = h.f/c J/s where h = planck’s constant = 6.625 x 10 ¾ Thermal radiation has properties similar to the light ¾ At high temperature bodies emit more radiation energy than at low temp.

The salient features of Radiation are

8.3.1 Salient Features of Radiation

Part 8.3: Radiation

Quick Refresher Guide

Heat Transfer

=

=

/

( /

)

=

/

(

( /

)( /

) )

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Net heat exchange between two bodies is =

(B) Infinite long concentric cylinders: Consider two large concentric cylinders of area A1 and A2 emissivities ε1 and ε2 and surfaces are maintained at temp T1 and T2

Net heat exchange between two bodies is =

(A) Infinitely long parallel plate:.

8.3.4 Heat exchange between non-black bodies

(i) The value of shape factor depends only on the geometry and orientation of surfaces with respect to each other. (ii) Net heat exchange between surfaces A1, A2 is Q12 = A1 F12 σ1 T − A2 F21 σ2 T If both are black surfaces and are maintained at same temp i.e. σ1 = σ2 = σb and T1 = T2 = T Q12 = 0 = A1 F12 σ1 T − A2 F21 σ2 T So A1 F12 σ1 T = A2 F21 σ2 T A1 F12 = A2 F21 . . . . . . . . . called Reciprocity Theorem (iii) The shape factor of convex surface 1 with respect to enclosure surface 2 i.e. F12 is unity. Therefore A1 F12 = A2 F21 ∴ Shape factor of convex surface with respect to itself is “0” i.e. F11 = 0 (iv) The shape factor for concave surface with respect to itself is not zero ∴ F11 ≠ 0 (v) If the transmitting surface is subdivided the shape factor for that surface with respect to the receiving surface is not equal to the sum of the individual shape factors. A1 F12 = A3 F32 + A4 F42 A1 ≠ A3 ≠ A4 and F12 ≠ F32 + F42 But if it is reverse A2 F21 = A2 F23 + A2 F24 that if the receiving surface is F21 = F23 + F24 divided, then the sum is equal (vi) Shape factor is the fraction of total radiation leaving the radiating surface and falling upon a particular receiving surface. ∴ F11 +F12 + F13 + F14 + . . . . . . . . . . . . . + F1n = 1 F21 + F22 + F23 + F24 + . . . . . . . . . . . . . . + F2n = 1 ............................................ Fn1 + Fn2 + Fn3 + Fn4 + . . . . . . . . . . . . . . . + Fnn = 1

The properties of shape factor are

F

It is defined as “the fraction of radiative energy that is diffused from one surface element and strikes the other surface directly with no intervening reflections. It is also called geometrical factor, configuration factor or view factor, it is represented by F means the shape factor form a surface A to another surface A . Therefore F is

8.3.3 Shape Factor

Quick Refresher Guide

Heat Transfer

(

)

=

(

−

)

Eb

=

(

)

=

J

J

A ,< < < < A ,

≈0

Eb

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(E) Heat exchange between two black surfaces enclosed by an insulated (adiabatic) surface Eb − Eb Q =

F

(C) Radiation heat exchange between large concentric cylinders or sphere is A ≠A (D) The radiation exchange between a small body in a large enclosure

Q

Eb

(B) Radiant energy exchange between two infinite parallel gray planes 1/ 1 −∈ /∈ 1 −∈ /∈

=

Eb

(A) Radiant energy exchange between two black surfaces

1/

The radiation problems can be easily obtained by reducing the actual system to an equivalent electrical network and then solving that network. Some of the terms in this technique are

8.3.5 Electrical network approach for radiation exchange

T −T =f A F where F12 = 1 2 called equivalent emissivity or interchange factor (D) Small body in a large enclosure: = ( − ) = f A σ (T − T ) where f =

(C) Small gray bodies: The size of the bodies is very small when compared to distance between them. That is the energy emitted by body 1 would be partly absorbed by body 2 and remaining is lost in space and no reflections. Therefore the energy exchange between two bodies is = − ( − ) =

Quick Refresher Guide

1

/

/

2

Heat Transfer

th

th

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th

Radiation shields: It is a thin shield placed between two planes that would neither remove nor add any heat to the system but used to reduce heat transfer rate.

(G) Radiation H – E between 3 gray surfaces. The figure is same as above but Eb3 ≠ T3 and Q12 equation is same as above

(F) Heat exchange between two gray surfaces enclosed by an insulated surface When third body is black =0 Eb = J ,

/

3

Quick Refresher Guide

Heat Transfer

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C = Heat capacity = mc

∆t = temp drop or rise of a fluid across the H.E

t = fluid temp

Cp = Sp. Heat

Let m = mass flow rate (kg/s)

8.4.2 Performance Analysis

D) Physical state of heat exchanging (condensation and evaporation) ¾ Condenser ¾ Evaporator

B) Relative direction of motion of fluids : According to flow of fluids, the H.E are classified into 3 categories ¾ Parallel flow (co-current) ¾ Counter flow (counter current) ¾ Cross flow C) Mechanical design of H.E. surface ¾ Concentric tubes ¾ Shell and tube ¾ Multiple shell and tube passes

A) Nature of heat exchange process: ¾ Direct contact or open heat exchanger: this is done by complete physical mixing (simultaneous heat and mass transfer takes place) ex: water cooling towers and jet condensers in steam power plants. ¾ Regenerator: here hot and cold fluids flows alternately when hot fluid passes, the heat is transferred to the solid matrix and then the flow of hot fluid, is stopped next cold fluid is passed on to the matrix which takes heat from solid matrix ex:- open hearth and blast furnaces. ¾ Recuperator: the fluids flow simultaneously on either side of a separating wall. Examples are super heaters, condensers, economizers and air pre-heaters in steam power plants, automobile radiators

Heat exchangers are classified according to

8.4.1 Heat Exchanger Classification

The heat ‘transferred in the heat exchanger may be in the form of latent heat (ex: boilers and condensers) or sensible heat (ex: heaters and coolers)

(i) Boilers (evaporators), super heaters and condensers of power plant (ii) Automobile Radiators (iii) Evaporator of an ice plant (iv) Water and air heaters and coolers

Heat exchanger is a process equipment designed for the effective transfer of heat energy between two fluids (hot and cold fluid). Examples are

Part 8.4: Heat Exchanger

Quick Refresher Guide

/

/

⇒

= 1/ ∑

=

( /

/

/

)

this is without fouling factor

−

1/∑

−

=

= (ℎ ℎ )/(ℎ + ℎ )

The U is constant throughout the H.E. C and m are constant The H.E is perfectly insulated The C.S of stream is uniform No conduction of heat along the tubes of H.E.

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¾ ¾ ¾ ¾ ¾

Assumptions made in LMTD are

8.4.3 LMTD (Log Mean Temp Difference)

∴

When thickness is small r = r and ‘δ/KA’ is neglected

For cylindrical type we can calculate U based on outer and inner area

Where h = scale h.t.c

Fouling factor = 1/ℎ =

(i) The “U” depends upon flow rate and properties of fluid, material thickness, surface condition of tubes and the geometrical configuration of the H.E. (ii) When fluids with low values of K and h will give low value of U when flowing in one side (iii) When highly conductive fluids passes on both sides will give high value of U (iv) For efficient and effective design there should be no thermal resistance in the heat flow path

Note:-

During operation the tube surfaces get covered by deposits of ash, soot, dirt and scale etc causes rust formation and deposition of fluid impurities is called fouling

/

∑

− ∆ /∑

As we know that

θ = mean temp difference

A = surface area

called logarithmic mean temp difference

> θ

and for same HT rate (As) counter < (As) parallel

Heat Transfer

)/2 − (

+

)/2

The counter flow H.E. preferred than parallel flow because

When overall h.t.c is varying linearly as U = a + b θ, HT is given by

=

( (

/

) )

LMTD will generally be used when θ1/θ2 > 1.7 H.T. rate is same in bath parallel and counter flow types when one of the fluid temp remains constant

+

=

,

=

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And F should be less than 1.0

Where

For multi tube and multi-pass H.E. we should use a correction factor “F” which can be found out through graphs which are plotted between P and Z

¾ The exchange of heat may rise the temp of the cold fluid to more nearly the temp of the hot fluid ¾ LMTD is higher and accordingly more heat can be transferred ¾ The unknown exit temp of the cooling water may be found from an energy balance on two fluids

x

x x

Note:

AMTD = (θ1 + θ2)/2 = (

If the variation in temp of the fluids are relatively small then the temp variation curves are approximately straight lines and the accurate results can be obtained through arithmetic mean temp difference (AMTD)

In general θ

Applying L – Hospital’s rule and solving, it becomes θ1 = θ2 = θ

Then θ = 0/log 1 = 0/0 indeterminate

Note: If heat capacities of both the liquids are same then θ1 = θ2

)

Where θ =

∴ Heat transferred from hot to cold fluid is = U.A. θ

(

Q=UAθ

where U = overall heat transfer coefficient

So by using LMTD heat transfer is given by

¾ The K.E and P.E changes are negligible ¾ The flow conditions are steady

Quick Refresher Guide

For the HE to be in equilibrium condition always heat lost by hot fluid is equal to heat gained by cold fluid, so Q = Q = Q

(∆ )

Heat Transfer

Heat gain by cold fluid = Q = C (∆t )

Heat lost by hot fluid = Q =

Subscripts c, h indicates cold and hot fluids

Quick Refresher Guide

Heat Transfer

)

(

)

=

( (

)

(

))

=

.

(

(

(

(

)) ))

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1. For boiling and condensation for one fluid change of temp is zero that means capacity rate Cmax → ∝ Therefore C = Cmin / Cmax = 0 = 1 − exp (− NTU) for both parallel and counter flow 2. In gas turbine, exhaust gases after expansion in the turbine are used to heat the compressed air so both the fluids have same capacity rate C = Cmin / Cmax = 1 For parallel flow HE ε = [1 – exp (−2NTU)]/2 so max value of ‘ε’ is 50%. For counter flow HE, C⇒ 1, (1 – C) tends to zero On substitution = /1+ max value of ε is 100%

Limiting values of capacity Ratio

Effectiveness =

Effectiveness for the counter flow HE:

Effectiveness =

Effectiveness for the parallel flow HE:

¾ NTU: It is a measure of the size of H.E. It provides some indication of the size of the H.E NTU = UA / Cmin

(

¾ Capacity ratio is the ratio of minimum to maximum capacity rate (m C or m C ) Capacity ratio = C = m C /m C when m C < m C = m C /m C when m C > m C So a fluid with smaller value of capacity rate experiences the greater change in temp. In counter flow when m C < m C Tc2 tends to Th1 ¾ Effectiveness of H.E (ε): is the ratio of actual heat transfer to the maximum possible heat transfer ( ) ( ) = = =

The concept of LMTD will not work when outlet temp of both the fluids are not known. Normally in many mechanical applications we do not know exit temps of both the fluids. At this situations we use NTU approach which is based on the concept of capacity ratio, effectiveness and number of transfer units.

8.4.4 Effectiveness and NTU

Quick Refresher Guide

Manufacturing Engg

(c)Close-packed hexagonal (cph) Fig. 9.1.1 Some common unit cells.

(b)Face-centered cubic (fcc)

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Point Defect: If an imperfection is restricted to the neighbourhood of a lattice point, the imperfection is referred to as a point defect. Point defects are mostly of 3 types, Vacancy, Interstitial impurity and Substitutional impurity as depicted in below figure.

These imperfections govern most of the mechanical properties of the crystalline solids. The imperfections in a crystal lattice structure are classified as follows:

Crystal Imperfections: Crystals are rarely perfect, i.e, the lattices are not without imperfections.

When a liquid metal solidifies by cooling, the atoms arrange themselves in regular space lattices, form a crystal. The crystallization starts simultaneously at various places within the liquid mass. Most metals have only one crystal structure. A few metals, however, can have more than one type of crystal structure. Such elements are called allotropic metal. Iron is an example of allotropic metal.

(a)Body-centered cubic (bcc)

The three commonly observed crystal structures in metals are Body centered cubic (BCC), Face centered cubic (FCC) and Closed pack hexagonal (CPH). The FCC and CPH have the most dense packing.

The properties of a material are significantly dependent on the arrangement of atoms. In all metals and in many non-metallic solids, the atoms are arranged in a well –ordered pattern. Such solids are commonly called crystalline solids. In a crystal, we can identify the unit cell, the repetition of which forms the whole crystal. The structure of a crystal is identified and described by this unit cell.

9.1.1 Crystal Structure

The properties of a material are intimately connected to its basic molecular structure. Some knowledge of this structure is therefore essential for understanding the various macroscopic properties exhibited by material.

Part 9.1: Engineering Materials

Part – 9: Manufacturing Engineering

Quick Refresher Guide

(b)Interstitial impurity (c)Substitutional impurity Fig. 9.1.2 Types of point

Manufacturing Engg

Y

Fig 9.1.3 Edge and screw dislocations

(b) Screw dislocation

Y

X

Displaced d atoms

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Fig. 9.1.4 Twinning mechanism

Twin planes

Surface Defect: When an imperfection extends over a surface, the imperfection is known as surface defect. The most common type of surface defects is twins.

(a) Edge dislocation

X

X’

Line Defect: If an imperfection extending along a line has a length much larger than the lattice spacing, the imperfections is called as line defect or a dislocation. Dislocations are of two types, edge dislocations and screw dislocations. Y’

(a)Vacancy

Quick Refresher Guide

Manufacturing Engg

0

b a

Strain, ∈

d e

c

f

g

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Ductile Materials: Fig 9.1.5 shows the stress-strain diagram for a ductile material such as mild steel. The curve starts from the origin O showing thereby that there is no initial stress or strain in the test specimen. Upto point ‘a’ Hooke’s law is obeyed and stress is proportional to strain. Therefore, oa is a straight line and point a is called the limit of proportionality and the stress at point a is called the proportional limit stress, σ . The portion of the diagram between ab is not a straight line but upto point b, the material remains elastic, i.e. on removal of the load, no permanent set is formed and the path is retraced.

Fig. 9.1.5 Typical stress-strain diagram for a ductile material

Stress 0

σ σ σ σ

σ

Tensile Test/ Stress-Strain Diagrams: The uni-axial tension test is done to obtain different strength related parameters for the engineering materials. The results and discussion of the test are discussed as under:

Continuity: If the material grains in the body are properly bonded to adjacent grains and fill the total volume, we consider it continuous.

Homogeneity: If a body has similar properties throughout its volume it is known as homogeneous. Thus homogeneity ensures the uniformity of properties from point to point.

Isotropy: It is the faculty of having similar properties in all directions. If the properties are different in different directions, the material is termed as anisotropic, e.g. wood.

A common means of distinguishing one material from another is by evaluating their physical properties. These include characteristics such as density, melting point, optical properties etc. More often however, the properties that describe how a material responds to applied loads assume dominant position in material selection. These mechanical properties are usually determined by subjecting prepared specimen to standard laboratory tests.

9.1.2 Physical and Mechanical Properties of Materials

Quick Refresher Guide

Manufacturing Engg

Loading

b

∈

(b)

b

∈

Unloading

Residual Strain (b)

0

Loading

a

Fig. 9.1.6 Loading and Unloading paths

(a)

Unloading

a

σ

σ

0

σ

Fig. 9.1.7

0.2% ∈

∈

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Sometimes it is not possible to locate the yield point quite accurately in order to determine the yield strength of the material. For such materials the yield point stress is defined at some particular value of the permanent set. It has been observed that if load is removed in the plastic range then the unloading path line is parallel to the straight portion of the stress-strain diagram as shown in Fig. 9.1.6 (b). The commonly used value of permanent set for determining the value of yield strength for mild steel is 0.2 per cent of the maximum strain as shown in Fig. 9.1.7.

0

σ

Ultimately the specimen breaks at point g, known as the breaking point, and the corresponding stress is called the nominal breaking stress based upon the original area of cross section. Whereas the true stress at fracture is the ratio of the breaking load to the reduced area of crosssection at the neck. The initial portions of the diagram are shown in Fig. 9.1.6 on exaggerated scale.

The point b is called the elastic limit point and the stress corresponding to that is called the elastic limit stress, σ . In actual practice, the point a and b are so close to each other that it becomes difficult to differentiate between them. Beyond the point b, the material goes to the plastic stage until the upper yield point ‘c’ is reached. At this point there is a cross to a lower value to point d, called the lower yield point. Corresponding to point c, the stress is known as upper yield point stress, σ and corresponding to point d, the stress is known as lower yield point stress, σ . At point d the specimen elongates by a considerable amount without any increase in stress and upto point e. The portion de is called the yielding of the material at constant stress. From point e onwards, the strain hardening phenomena becomes predominant and the strength of the material increases thereby requiring more stress for deformation, until point f is reached. Point f is called the ultimate point and the stress corresponding to this point is called the ultimate stress, σ . It is the maximum stress to which the material can be subjected in a simple tensile test. At point f the necking of the material begins and the cross-sectional area starts decreasing at a rapid rate. Due to this local necking, the stress in the material goes on decreasing inspite of the fact that the actual stress intensity goes on increasing.

Quick Refresher Guide

Manufacturing Engg

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Annealing: Annealing is a heat treatment process in which the iron based alloys are heated above the transformation range, held there for proper time and then cooled slowly (at the rate of 300C 1500C per hour) in furnace. Annealing can be of various types:

The brief description of different types of heat treatment process is as under:

7. To produce hard surface on a ductile interior

6. To increase resistance to wear, heat and corrosion

5. To improve magnetic and electrical properties

4. To improve mechanical properties

3. Stress relieving

2. To change and refine grain size

1. To improve machinability

Heat treatment are generally employed for the following purposes:

Heat treatment is the controlled heating and cooling of metals for the purpose of altering their properties and can perform without a concurrent change in product shape. Heat treatment can alter mechanical properties of steels by changing the size and shape of the grains.

9.1.3 Heat Treatment

Hardness: Hardness is a very important but hard to define property of materials. It can be defined as resistance to permanent deformations under static and dynamic loading.

Toughness: Toughness or modulus of toughness is defined as the work per unit volume required to fracture a material. The tensile test can provide one means of measuring toughness, since the total area under stress-strain curve represent the desired value.

Ductility: The amount of plasticity that a material can exhibit is a significant feature when evaluating its suitability for certain manufacturing process. For metal deformation process, the more plastic a material is, the more it can be deformed without rupture. The ability of a material to change shape without fracture is known as ductility.

Anelasticity: This is often called delayed elasticity. Here the recovery is complete but not instantaneous and takes some time.

3. The load deformation curve has same shape while loading or unloading

2. The recovery is complete and immediate, i.e. no permanent set is left after unloading.

1. While loading or unloading, the deformation and recovery are instantaneous.

Elasticity: A material can be called perfectly elastic, if and only if:

Quick Refresher Guide

Process Annealing: In this process, recrystallization of ferrite is done at sub-critical temperature. During this process, hardness of steel is decreased as the ductility is increased. The steel is heated to about 6500C and cooled freely in air. This process is generally used on rolled products such as flats, sheets etc. Patenting: This process is similar to sub annealing with only difference is that it is applied to steel wires. Spheroidizing: This is an annealing process in which high carbon steel, tool steels containing a large amount of free cementite, which makes them brittle, are heated 200C to 400C below lower critical temperature, held there for considerable time and then cooled very slowly at room temperature at furnace itself. Isothermal annealing: This annealing process is used for plain and high carbon steels. In this a ferrous alloy is austenitised, and then cooled and held at transformation temperature at which austenite transforms to a relative soft ferrite carbide aggregate. Normalizing: Normalizing is defined as the process in which Iron base alloys are heated 400C to 500C above the transformation range and held there for a specified period and followed by cooling in still air at room temperature.

Manufacturing Engg

Carburisation: In this process, the carbonaceous medium is a solid carburiser. The chief carburiser for pack carburising is activated charcoal with grain size varying from 3.5 to 10 mm in diameter. Nitriding: It is a process of saturation of surface of steel with Nitrogen by holding it for a prolonged period (up to 100 hrs) at a temperature ranging from 4800C to 6500C in an atmosphere of Ammonia. Cyaniding: It is a process in which both carbon and nitrogen in the form of cyaniding salt are added to the surface of low and medium carbon steel to increase its hardness and wear resistance. Induction Hardening: In process of induction hardening a high frequency current of about 1000 to 10000 cycles per second is passed through a copper inductor block which acts as a primary coil of the transformer. In this the heated part is cooled rapidly with sprays of water delivered through the numerous small holes in the inductor block. This helps in obtaining hard and wear resistance surface while having soft core. Flame Hardening: This process is also same as Induction hardening with difference that heating of the specimen is carried out by flame instead of induction effect. The heating is generally accomplished by oxy-acetylene flame.

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x

x

x

x

x

Case Hardening: Case hardening is type of heat treatment process which is supported by a tough and shock resisting core. Below are some case hardening techniques:

Tempering: Tempering is defined as the reheat process done at sub-critical temperatures. Such reheating permits the trapped martensite to transform into troostite or sorbite depending upon the tempering temperature and relieves the internal stress.

Hardening: Hardening is the heat treatment process in which steel is heated at 20 0C above the transformation range, soaking at this temperature for a considerable period to ensure thorough penetration of the temperature inside the component, followed by continuous cooling to room temperature by quenching in water, oil or brine solution.

x

x

x

x

x

Quick Refresher Guide

Manufacturing Engg

Melting and pouring – Furnaces & Solidification Casting – Shakeout, removal of risers and gates Heat treatment Cleaning and finishing – Fettling

o o o o

Flask: This holds the sand mould intact. According to position:– Drag – lower moulding flask; Cope – Upper moulding flask; Cheek – intermediate moulding flask used in the three piece moulding. Made of metal for long term use Pattern: It is a replica of the final object to be made with some modifications. Parting line: It is the dividing line between the two moulding flasks that makes up the sand mould. Bottom board: Used when at the start of the mould making process, first the pattern is kept on the bottom board and sand particles sprinkled on it and then the ramming is done in the drag. Pouring basin: Funnel shaped cavity at the top of the mould into which the molten metal is poured. Reduces the eroding force of the liquid metal stream coming directly from the furnace Strainer: A ceramic strainer in the sprue removes dross. Splash core: A ceramic splash core placed at the end of the sprue also reduces the eroding force of the liquid metal stream Skim bob: It is trap placed in a horizontal gate to prevent heavier and lighter impurities from entering the mould. Sprue: A Passage through which the molten metal from the pouring basin reaches the mould cavity. Runner system: It has channels that carry the molten metal from the sprue to the mold cavity. Gate: It is an actual entry point through which molten metal enters mould cavity THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750, [email protected] © Copyright reserved. Web: www.thegateacademy.com Page 305

x

x

x

x

x x

x

x

x x

x

Casting Terms

Inspection – Defects, Pressure tightness, Dimensions

Mold and core making

o

o

Pattern making

o

¾ The sequence of operations involved in casting are –

¾ Temp is also applied to the part that is made by this process

¾ Casting is a process in which molten metal flows by gravity or other force into a mold where it solidifies in the shape of the mold cavity.

9.2.1 Casting

Part 9.2: Casting

Quick Refresher Guide

Manufacturing Engg

Core: It is used for making hollow cavities or otherwise defines the interior surface of the castings. Chaplet: It is used to support cores inside the mould cavity to take care of its own weight and overcome the metallostatic forces. Chill: This is a metallic object which is placed in the mould to increase the cooling rate of castings to provide uniform or desired cooling rate. Riser: It is a reservoir of molten metal provided in the casting so that hot metal can flow back into the mould cavity when there is a reduction in volume of metal due to solidification.

Single piece pattern Split pattern or two piece pattern Gated Pattern Cope and drag pattern Match plate pattern Loose piece pattern Follow board pattern Sweep pattern Skeleton pattern

Dimensional accuracy required

Molding process

Size and shape of the casting

x

x

x

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Wood, Metals, Plastics and Polyurethane foam are generally used materials

Production Quantity

x

Pattern materials: The selection of pattern material is based on the following factors.

x x x x x x x x x

the number of castings required and the moulding procedure adopted

Classifications of Patterns: Various types of patterns depending upon the complexity of the job

Shrinkage allowance Machining allowance or finishing allowance Draft or Taper allowances Distortion allowance Shake or Rapping allowance

System sand Rebounded sand Facing sand Parting sand core sand

x x x x x

To strengthen the weak patterns

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Hot Strength: At high temperature, the strength of sand that is required to hold the shape of the mould cavity.

Dry Strength: It should retain the mould cavity and at the same time withstand the metallostatic forces when the sand is at dry condition.

Green Strength: The green sand should have enough strength so that the constructed mould retains its shape. (Where Green refers moisture contained in the sand)

Refractoriness: It is the ability of the moulding material to withstand the high temperatures of the molten metal so that it does not cause fusion.

Required processing properties of moulding material:

Moulding sand

x

Variety of moulding materials:

9.2.3 Moulding Materials

Diagonal black stripes with clear varnish

Pattern allowances:

Seats of and for loose pieces & loose core prints

Core prints for machined openings

Core prints for unmachined openings

Black Yellow stripes on black

Surfaces to be machined

Unfinished surfaces and cast surfaces

Code

Yellow

Red or Orange

Colour

Green

x x x x x

Manufacturing Engg

Pattern color Code: The patterns are normally painted with contrasting colours such that the mould maker would be able to understand the functions clearly.

Quick Refresher Guide

Pattern is a replica of the object to be made by the casting process, with some modifications. The size of pattern is slightly greater than the casting by an amount called allowances.

9.2.2 Pattern

x

x

x

x

Quick Refresher Guide

Manufacturing Engg

x x x Testing sand properties: Moisture content:

Based on initial condition: Green sand, Dry sand, Loam sand, Facing sand, and Parting sand.

Moulding sand composition: Main ingredients of moulding sand: Silica grains, clay, water and

Silica sand: (Silica grains)

x

Used as binding agents and this is mixed with the moulding sands to provide the strength, because of their low cost and wider utility

Kaolinite or fire clay (Al2O32SiO22H2O) Bentonite (Al2O34SiO2H2OnH2O) Southern bentonite Calcium as adsorbed ion Low dry strength but higher green strength

Develops plasticity and strength Amount of water used should be properly controlled. Usual percentages of water is from 2 % to 8 %

x

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Used for providing better surface finish to the castings

Coal dust:

Additives:

x x x

Water:

x x x x x

Types of Clay:

x

Clay:

x

Silica grains forms the major portion of the moulding sand (96%) and rest being the other oxides such as alumina, sodium (Na2O + K2O) and magnesium oxide (MgO + CaO) Impurities (oxides) should be minimum because it affects the fusion point of the silica sands.

Measured amount of calcium carbide in a container along with a separate cap consisting of measured quantity of moulding sand is kept in the moisture teller. The apparatus is shaken vigorously such that the following reaction takes place.

x x

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CaC2 + 2H2O Æ C2H2 + Ca (OH)2

It utilizes calcium carbide to measure the moisture content.

Method III: (Moisture Teller)

Sand is dried by suspending the sample on a fine metallic screen and allowing hot air to flow through the sample. Time taken for removal of moisture in a matter of minutes compared to earlier method.

Method II: (Moisture Teller)

To test the moisture of moulding sand a carefully weighed test sample of 50 g is dried at a temperature of 105oC to 110oC for 2 hours at the time all the moisture in the sand would have been evaporated. Now weigh the sample % of moisture – (weight difference in grams) x 2

Method I:

x

x

x

x x

x

Iron oxide:

Based on its origin: Natural sand, Synthetic sand & Special sands:

additives

Increase resistance to deformation, skin hardness and expansion defects such as scab.

x

Types of molding sands:

Improves surface finish, decreases metal penetration, reduces burn-on, increases the chilling effect of the mould It decreases green strength and permeability while improving the hot strength. It reduces collapsibility and makes the shake-out of the mould difficult.

Organic binding materials used with mould and core sands.

x

Mold hardness: It is the hardness of mold which is measured similar to Brinell hardness test.

It reduces the expansion defects while improving the flowability of the moulding sand

It widens the range of water that can be added to get proper green strength.

Manufacturing Engg

Starch and dextrin:

x

x

Saw dust or Wood flour:

Quick Refresher Guide

Cohesiveness: Ability of sand particles to stick to each other. Increases with increase in clay and decreases with increase in grain size.

Adhesiveness: Ability of molding sand to adhere to the surface of molding boxes.

Permeability: The ability of sand to allow the gas to pass through the mould. It depends on size and shape of grains, moisture content and degree of compaction.

Quick Refresher Guide

It is determined by dissolving or washing it off the sand. To determine the clay %, a 50 g sample is dried at 105 to 110oC and the dried sample is taken in a one litre glass flask and added with 475 ml of distilled water and 25 ml of a one percent solution of caustic soda( NaOH 25 g per litre) Sample is thoroughly stirred and washed for many times. And the sand is removed from the flask and dried by heating. Clay % calculated as = (Difference in weight of the dried sand and sample) x 2

x x x

V uH p u AuT

Permeability Number P =

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Test is carried out on the universal sand strength testing machine. It refers to the stress required to rupture the sand specimen under compressive loading. The standard specimen put on the strength testing machine and the force required to cause the compression failure is determined.

501.28 pu T

Where V = Volume of air = 2000 cm3, H = Height of the sand specimen = 5.08 cm, p = air pressure, (g/cm2), A = cross – sectional area of sand specimen = 20.268 cm2, T = time in minutes for the complete air to pass through.

Permeability Number P =

Permeability is expressed in terms of permeability number. The standard permeability test is to measure time taken by a 2000 cm3 of air at a pressure typically of 980 Pa, to pass through a standard sand specimen confined in a specimen tube. The standard specimen size is 50.8 mm in diameter and a length of 50.8 mm.

Green Compression Strength:

x

x x

fi – amount of sand retained on the ith sieve;

Where Mi – multiplying factor for the ith sieve;

The dried clay – free sand grains are placed on the top sieve of a sieve shaker which contains a series of sieves one upon the other with gradually decreasing mesh sizes. The sieves are shaken continuously for a period of 15 min. Grain Fineness Number is a quantitative indication of the grain distribution Grain fineness Number – amount retained on each sieve is multiplied by the respective weightage factor, summed up, and then divided by the total mass of the sample. ∑M F GFN = f

Permeability:

x x x

x

Sand grain size:

x

x

x x

Manufacturing Engg

The amount (pressure) of acetylene generated is proportional to the moisture present in the sample.

Clay Content:

x

Quick Refresher Guide

The standard specimen dried between 105 and 110

When it sinks completely – reading is 0 which means very soft mould.

x

Finer grains would have lower permeability and they provide better surface finish to the casting produced. Distribution of the grain size – widely distributed sand would have higher permeability.

x x

Round grains would have lower permeability compared to angular grains

x

Purity of sand grains improves the refractoriness. Finer grains & impurities in the sand tend to lower the refractoriness.

x x

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Higher the grain size, higher would be the refractoriness

x

Grain size – refractoriness:

Angular sand grains require higher amounts of binder.

x

Grain shape- permeability, binder amount:

Coarse grains would have more void space between the grains which increase the permeability.

x

Grain Size – Permeability

Sand grain shape and size

Various parameters influencing the sand properties:

The preparation of sand is thorough mixing of its various ingredients. In the mixing process the clay is uniformly enveloped around the sand grains and moisture is uniformly distributed which is done by MULLER machine.

Sand Preparation:

The depth of indentation can be directly measured on the scale which shows units 0 to 100. When no penetration occurs – reading is 100 which means hardest mould

x x

A spring loaded steel ball with a mass of 0.9 kg is indented into the standard specimen prepared.

x

for 2 hours which is used for testing.

Stress required to shear the specimen along the axis is then represented as the green shear strength.

Mould Hardness:

x

Manufacturing Engg

Test is carried out on the universal sand strength testing machine with a different adapter fitted in the universal machine so that the loading now be made for the shearing of the sand sample.

Dry Strength:

x

x

Green shear strength:

Quick Refresher Guide

Manufacturing Engg

The green compression strength increases with a decrease in the grain size because the fine grain size provides a large surface area for the binder to act.

Additional amount of water increases the plasticity and dry strength but reduces the green strength.

x

Increases Reaches a maximum and then decreases

Decreases

Reaches a maximum and then decreases

Decreases

Clay content

Moisture content

Degree of ramming

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The defects in castings may arise due to the defects in one or more of the following x Design of casting and pattern x Molding sand and design of mold & core x Metal Composition x Melting and pouring

9.2.4 Defects in castings

Three methods used for ramming the sand into the moulding flasks x Jolting x Squeezing x Sand slinging

Moulding machines: Used for production work involving large batches of the same type of casting is to be produced. Eliminate arduous labor, offer high quality casting by improving the application and distribution of forces, and manipulate the mold in a carefully controlled fashion.

Increases

Increases

Decreases

Increase of grain fineness number

Strength

Permeability

The degree of ramming increases the bulk density or the mould hardness. Increased ramming increases the strength Permeability of green sand decreases with the degree of ramming.

Factors

x x x

Method of preparing sand mould

Initially the green compression strength increases with increase in water content and then after reaching the optimum value it starts decreasing.

x

Clay & water – green compression strength:

Clay type and amount of Moisture content

x

Grain size – strength:

Quick Refresher Guide

Gating and risering

Manufacturing Engg

x

x

x

poor gating design such as straight sprues in unpressurized gating, abrupt bends and other turbulence low permeability of the mould.

– –

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Ö Reason – the high pouring temperature which increases the gas pick-up.

Ö Caused by release of gases (Hydrogen) during pouring, consists of many small gas cavities (small diameter & long pin hole) formed at or slightly below the surface of the casting.

Pin hole porosity:

Ö Remedy – Choose the appropriate pouring temperature and improve gating practices by reducing the turbulence

Higher pouring temperatures

–

Ö Reasons

Ö The atmospheric and other gases absorbed by the molten metal in the furnace, in the ladle, and during the flow in the mould, when not allowed to escape, would be trapped inside the casting and weaken it.

Air inclusions:

Ö Remedy – proper venting and adequate permeability

Ö Causes – moisture in the mould, lower venting and lower permeability of the mould.

Ö Because of moisture left in mould is converted into steam by heat in the molten metal, part of which entrapped in the casting ends up as blow hole or ends up as open blow when it reaches the surface.

Ö Spherical, flattened or elongated cavities present inside the casting (B Blow holes) or Open blows) on the surface (O

Blow holes and open blows:

Gas defects: These defects are caused to a great extent by the lower gas passing tendency of the mould which may be due to lower venting, lower permeability of the mould or improper design of the casting, lower permeability caused by finer grain size of the sand, higher clay, higher moisture, or by excessive ramming of the moulds.

The following are the major defects which are likely to occur in sand castings:

x

Quick Refresher Guide

Manufacturing Engg

x

x

x

x

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Ö Reasons – caused either due to faulty mould making or because of the faulty moulding flask.

Ö When the molten metal leaks out of the mould

Run out

Ö Remedy – Choice of correct type and amount of bentonite

Ö Reasons – clay in the moulding sand have low refractoriness or high pouring temperature.

Ö Fusion of sand grains with the molten metal, giving a brittle, glassy appearance on the casting surface.

Fusion

Ö When two metal streams while meeting in the mould cavity, do not fuse together properly – discontinuity or weak spot in the casting.

Cold shuts:

Ö When the metal is unable to fill the cavity completely and thus leave unfilled cavities.

Mis-runs

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Ö by the lower fluidity of the molten metal or when the section thickness of the casting is too small.

™ Causes of above two defects:

x

x

Pouring Metal defects

Ö Appear as rough, irregular projections on the surface containing embedded sand.

Ö Remedy – choose appropriate grain size, together with a proper mould wash.

Scabs

Ö Reason – improper ramming of the cope flask.

Ö Dropping of loose moulding sand or lumps normally from the cope surface into the mould cavity

Drop

Ö Remedy – proper ramming

Ö Reason – the faulty mould making procedure adopted

Ö Under the influence of the metallostatic forces, the mould wall may move back causing a swell in the dimensions of the casting.

Swell

Ö Remedy – proper choice of facing sand ingredients and the pouring temperature

Ö Reasons – moulding sand has got poor expansion properties and hot strength or the heat in the pouring metal is too high facing sand applied does not provide the necessary cushioning effect.

Ö Under the influence of the heat, the sand expands, thereby moving the mould wall backwards and in the process when the wall gives away, the casting surface may have this marked as a small line.

Rat tails and buckles

Manufacturing Engg

Ö Sort of projection on the casting which occur when a portion of the mould face or core lifts and the metal flows beneath in a thin layer.

x

x

x

x

Quick Refresher Guide

Ö Reasons – either the grain size of the sand is too coarse or no mould wash has been applied to the mould cavity, and by higher pouring temperatures.

Ö When the molten metal enters the gap between the sand grains – rough casting.

Metal penetration

Ö Remedies – by the proper choice of moulding sand and using appropriate moulding method for preventing the erosion of mould and by altering the gating design to reduce the turbulence in the metal, by increasing the size of gates or by using multiple ingates.

Ö Reasons – caused by the erosion of moulding sand by the flowing molten metal at high velocity.

Ö Appear as rough spots and areas of excess metal in casting

Cuts and Washes:

Moulding material defects: These defects occur because the moulding materials are not of requisite properties or due to improper ramming.

Ö Remedy – proper riser design.

Ö Occurs near the top of the casting (pipe)

Ö Caused by solidification shrinkage that restricts the amount of molten metal available in the last region to freeze.

Ö It is a depression in the surface or an internal void in the casting.

Shrinkage cavities (Pipe)

Quick Refresher Guide

Ö Remedy – adding flux and proper slag trapping methods.

Ö Slag (undesirable oxides and impurities) entering the mould cavity will be weakening the casting and also spoil the surface of the casting.

Slag inclusions:

Ö Improving the mould design.

Ö To increase the fluidity of the metal by changing the composition or raising the pouring temperature.

Remedy

Ö Remedy – by ensuring proper alignment.

Ö Causes – misalignment of flasks

Ö Results in mismatch of the sections of a casting usually at a parting line.

Shift

Ö Remedy – proper metallurgical control and chilling practices.

Ö Caused by the chilling of the casting

Hot spots

Ö Reason – when the casting is poorly designed and abrupt sectional changes take place; no proper fillets and corner radii are provided, and chills are inappropriately placed.

Ö Metal has low strength at higher temperatures; any unwanted cooling stress may cause the rupture of the casting.

Ö Internal or external ragged discontinuities or cracks on the casting surface, caused by hindered contraction occurring immediately after the metal has solidified.

Hot tears:

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Ö the heat to raise the temperature to the melting point

The heat energy required is the sum of

9.2.5 Heating and Pouring

x

x

x

Manufacturing Engg

the difference between the temperature at pouring and the temperature at which freezing begins (the melting point for a pure metal or the liquidus temperature for an alloy)((Super Heat)

If the rate is too high – turbulence will occur in the flow

x

Aggravates mold erosion, the gradual wearing away of the mold surfaces due to impact of the flowing molten metal.

x

P V2  w 2g

constant

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But as the metal moves through the gating system, a loss of energy occurs because of the friction between the molten metal and the mould walls.

g – Gravitational constant, m/s2

w – Specific weight of liquid, N/m3

V – Liquid Velocity, m/s

P – Pressure, Pa

Where h – potential head, m

h

The liquid metal that runs through the various channels in the mould obeys the Bernoulli’s theorem which states that the total energy head remains constant at any section.

9.2.6 Gating design

Tends to accelerate the formation of metal oxides that can be become entrapped during solidification

x

Effects of Turbulence:

Turbulence in the fluid flow is characterized by erratic variations in the magnitude and direction of the velocity throughout the fluid.

If the rate is too slow – metal will chill and freeze before filling the cavity

x

Pouring rate refers to the volumetric rate at which the molten metal is poured into the mold.

x

Pouring Temperature is the temperature of the molten metal as it is introduced into the mold.

Factors affecting the pouring operation include: pouring temperature, pouring rate and turbulence

Pouring – Introduction of molten metal into the mold, including its flow through the gating system and into the cavity, is a critical step in the casting process.

H – total heat required to raise the temperature of the metal to the pouring temperature, J; ρ – density, g/cm3; Cs = weight specific heat for the solid metal, J/g-C; Tm = melting temperature of the metal, C; To = starting temperature- usually ambient, 0C ; Hf = Heat of fusion, J/g; Cl=weight specific heat of the liquid metal, J/g-C; Tp = pouring temperature, 0C and V= Volume of metal being heated, cm3

Ö the heat to raise the molten metal to the desired temperature for pouring ( − )+ + − =ρ

Ö the heat of fusion to convert it from solid to liquid

Ö When the heat removal capacity is increased such as in the case of green sand moulds.

Metallurgical defects

x

x

Quick Refresher Guide

Manufacturing Engg

Quick Refresher Guide

Manufacturing Engg

Factors affecting fluidity – Pouring temperature relative to melting point, metal composition, viscosity of the liquid metal, and heat transfer to the surroundings.

Higher pouring temperature relative to the freezing point - increases the fluidity of the molten metal – which also causes some casting problems – oxide formation, gas porosity, and penetration of liquid metal into the interstitial spaces between the grains of sand forming the mold.

Metal Composition – w.r.t. the metal’s solidification mechanism – best fluidity is obtained by metals that freeze at a constant temperature (pure metals and eutectic alloys).

The freezing mechanism, metal composition also determines heat of fusion – the amount of heat required to solidify the metal from the liquid state.

Higher heat of fusion tends to increase the measured fluidity in casting.

x

x

x

x

x

Permeability

Heat content

With increase in:

Freezing range

Surface tension

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Pure Metals: - solidifies at a constant temperature equal to its freezing point, which is the same as its melting point.

9.2.7 Solidification and Cooling

Fluidity

Increase of

Viscosity

With decrease in:

Factors

Spiral mold test – standard testing method to measure the fluidity of the molten metal.

x

Property

The molten metal flow characteristics are often described by the term fluidity, a measure of the capability of a metal to flow into and fill the mold before freezing.

x

Fluidity:

V – Velocity of metal flows, m/s

A – Area of cross section, m2

Where Q – rate of flow, m3/s

Q A1V1 A2V2

Law of continuity says that the volume of metal flowing at any section in the mould is constant.

Quick Refresher Guide

Manufacturing Engg

Local solidification time: - actual freezing takes time – during which the metal’s latent heat of fusion is released into the surrounding mold. Total solidification time – time taken between pouring and complete solidification.

§V · Cm ¨ ¸ © A¹

n

x

To perform its function of feeding molten metal to the main cavity, the metal in the riser must remain in the liquid phase longer than the casting, then only effects of shrinkage are minimized.

Rule is used for designing the riser in a mold.

Rule indicates that a casting with a higher volume-to-surface area ratio will cool and solidify more slowly than one with a lower ratio.

Cm – Mold constant (min/cm2)

n-exponent taken to have a value = 2

A- Surface area of the casting cm2

V- volume of the casting, cm3

T- Total solidification time, min

T

Dependent on the size and shape of the casting by an empirical relationship Chvorinov’s rule) (C

x

x

It is the time required for the casting to solidify after pouring.

Exception in the above category, Cast iron containing high carbon content, whose solidification is complicated by a period of graphitization during the final stages of

x

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Solidification shrinkage occurs in nearly all metals because the solid phase has a higher density than the liquid phase.

x

(3) thermal contraction of the solidified casting during cooling to room temperature

(2) contraction during the phase change from liquid to solid – Solidification shrinkage

(1) liquid contraction during cooling prior to solidification

Shrinkage occurs in three steps:

Shrinkage:

-

-

-

-

Solidification time

Quick Refresher Guide

Manufacturing Engg

( )

To function, the riser must remain molten until after the casting solidifies.

The riser represents waste metal that will be separated from the cast part and remelted to make subsequent castings. It is desirable for the volume of metal in the riser to be minimum. Since the geometry of the riser is normally selected to maximize the V/A ratio, this tends to reduce the riser volume as much as possible.

x

x

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Top riser – connected to the top surface of the casting.

Side riser – attached to the side of the casting by means of a small channel

Different forms of riser:

Riser is used in a sand-casting mold to feed liquid metal to the casting during freezing in order to compensate for solidification shrinkage.

x

Riser Design:

Figure (a) External chill to encourage rapid freezing of the molten metal in a thin section of the casting and (b) the likely result if the external chill were not used

( )

External Chills are metal inserts in the walls of the mold cavity that can remove heat from the molten metal more rapidly than the surrounding sand in order to promote solidification.

Internal Chills are small metal parts placed inside the cavity before pouring so that the molten metal will solidify first around these parts.

Another way to encourage directional solidification is to use Chills.

Directional Solidification: In order to minimize the damaging effects of shrinkage, it is desirable for the regions of the casting most distant from the liquid metal supply to freeze first and for solidification to progress from these remote regions toward the risers. It is attained by locating sections of the casting with lower V/A ratios away from the riser, freezing will occur first in these regions and the supply of liquid metal for the rest of the casting will remain open until these bulkier sections solidify.

freezing, which causes expansion that tends to counteract the volumetric decrease associated with the phase change.

Quick Refresher Guide

Manufacturing Engg

(Ts) riser ≥ (Ts) casting

Shrinkage volume consideration method

x

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Navel research method

Modulus method

x x

Caine’s Method

x

Riser can be designed by using four methods:

o

Design of riser is such that the time taken for solidification of liquid metal in the riser should be more than the time taken for solidification of liquid metal in the cavity.

Blind riser – entirely enclosed within the mold

Open riser – exposed to the outside at the top surface of the cope.-dis adv: allowing more heat to escape, promoting faster solidification.

Quick Refresher Guide

Manufacturing Engg

(b) Stresses on

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Fig. 9.3.1 Forces and stresses during rolling

(a) Details of rolling

+

+

Determination of Rolling Pressure: Figure 9.3.1(a) shows a typical rolling operation for a strip with an initial thickness t which is being rolled down to a final thickness t . Both the rolls are of equal radius R and rotate with the same circumferential velocity V. The origin of the coordinate system xy is taken at the midpoint of the line joining the centres O and O . (The operation is two-dimensional, and so the position of O along the axis mutually perpendicular to Ox and Oy is of no significance. In our analysis, we shall assume that the width of the strip is unity.) The entry and exit velocities of the strip are V and V , respectively. In actual practice, V > V > V . Therefore, at a particular point in the working zone, the velocity of the strip will be equal to V, and this point will hereafter be referred to as the neutral point.

(i) The rolls are straight and rigid cylinders. (ii) The width of the strip is much larger than its thickness and no significant widening takes place, i.e., the problem is of plane strain type. (iii) The coefficient of friction is low and constant over the entire roll job interface. (iv) The yield stress of the material remains constant for the entire operation, its value being the average of the values at the start and at the end of rolling.

The basic objectives of the analysis we give here are to determine (i) the roll separating forces, (ii) the torque and power required to drive the rolls, and (iii) the power loss in bearings. An analysis considering all the factors in a real situation is beyond the scope of this text, and therefore the following simplifying assumptions will be made:

9.3.1 Rolling

The forming process can be grouped under two broad categories, namely, cold forming and hot forming. If the working temperature is higher than the re-crystallization temperature of the material, then the process is called hot forming. Otherwise the process is called as cold forming. During hot working, a large amount of plastic deformation can be imparted without significant strain hardening. This is important as large amount of strain hardening renders the material brittle.

Forming can be defined as process in which the desired size and shape are obtained through plastic deformation of material. The stresses induced during the process are greater than yield strength but less than fracture strength of the material.

Part 9.3: Forming Process

Quick Refresher Guide

tan

1− θ

1−

(

e ,

e

)

,

)

in the regions before and after the

Manufacturing Engg

ln +λ

σ =σ =σ

σ =σ =0

Entry

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Assuming that the width of the strip is unity, the total force F trying to separate the rolls can be obtained by integrating the vertical component of the force acting at the roll-strip interface.

= σ" > σ

σ =σ

θ θ Fig. 9.3.2 Pressure distribution in rolling

Neutral points

Determination of Roll Separating Force

O

Exit

Typical distributions of pressure p are shown in Fig. 9.3.2. The peak pressure at the neutral point is normally called the friction hill. This peak pressure increases with increasing coefficient of friction.

For typical values of the parameters in a rolling operation, we find that the roll pressure p increases continuously from the point of entry till the neutral point is reached and decreases continuously thereafter.

λ =

The pressure at the neutral point can be determined from either of the above equations for pressure. So, the value of λ corresponding to the neutral point (λ ) is obtained by equating the right-hand sides of these equations. Thus,

λ=2

=

=

(

Quick Refresher Guide

The expressions for the non-dimensional roll pressure neutral point we obtain are

p/(2k)

Manufacturing Engg

p

R dθ + ∫ p

R dθ

+ 2 ∫ pR θ dθ

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where Z is the angular speed of the roll.

P = Tω,

Once the driving torque T is determined, the driving power per roll P is obtained as

σ t − σ t + ∫ pR θ dθ

σ t −σ t

σ t − σ t + 2 ∫ pR sin θ dθ

T ≈F R =

Accordingly,

≈

F =

Figure 9.3.3 shows the deformation zone along with the forces acting on it, including an equivalent horizontal force F which represents the net frictional interaction between each roll and the strip (the reaction - F of F has to be overcome by the roll driving torque T). F can be determined by considering the horizontal equilibrium of the system. Thus,

Fig. 9.3.3 Equilibrium of deformation zone

The driving torque is required to overcome the torque exerted on the roll by the interfacial friction force.

Driving Torque and Power

The integrations in above equation are normally computed numerically.

i.e., F = ∫

≈∫ pR dθ (since θ is small),

F = ∫ pR cos θ dθ

Since the angle θ is normally very small, the contribution of the roll-strip interface friction force is negligible in the vertical direction. Thus,

Quick Refresher Guide

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Moving plate

Fixed plate

(b) Stresses on element

+

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Substituting dσ , we get

σ + p = 2K or dσ = −dp , K is the allowable shear stress

where τ is the frictional stress. To make the analysis simpler, −p and σ are considered as the principle stresses. The problem being of a plane strain type, Thus,

hdσ + 2τ dx = 0,

unity (in the z-direction)]. Figure 9.3.4 b shows the same element with all the stresses acting on it. Considering the equilibrium of the element in the x-direction, we get

Fig. 9.3.4 Forces and stresses during forging

(a) Details of forging operation

Workpiece

At the instant shown in Fig. 9.3.4(a), the thickness of the work piece is h and the width is 2l. Let us consider an element of width dx at a distance x from the origin. [In our analysis, we take the length of the work piece as

The forging force F attains its maximum value at the end of the operation. The coefficient of friction between the work piece and the dies (platens) is constant. (iii) The thickness of the work piece is small as compared with its other dimensions, and the variation of the stress field along the y-direction is negligible. (iv) The length of the strip is much more than the width and the problem is one of plane strain type. (v) The entire work piece is in the plastic state during the process.

(i) (ii)

Figure 9.3.4(a) shows a typical open die forging of a flat strip. To simplify our analysis, we shall make the following assumptions:

Forging of strip:

In this section, our analysis is mainly devoted to determining the maximum force required for forging a strip and a disc between two parallel dies. Obviously, it is a case of open die forging.

9.3.2 Forging

Quick Refresher Guide

dx

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ln p =

=

+C

∫ dx + C (0 ≤ x ≤ x ).

p = 2Ke

/

(0 ≤ x ≤ x )

p=

+C

∫ dx + C (x ≤ x ≤ l)

p = 2K exp(2μx /h) + (x − x )

or

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A x = x , τ = p = K Using this along with the expression for p , we get

p = 2K exp(2μx /h)

(x − x ).

Also,

p–p =

If = p = p at x = x , then C = p − 2Kx /h. Thus,

or

∫ dp =

For the sticking zone, we have

or

C = ln 2K

Now, at x = 0, σ = 0, i.e., p = 2K (from the yield criterion). So,

or

∫

For the sliding (non-sticking) zone, we have

However, it should be noted that this assumption is incorrect as the shear stresses τ act on the planes on which – p is acting (Fig. 9.3.4b).

τ = K.

and, for x ≤ x ≤ l

τ= p

Near the free ends, i.e., when x is small (and also at x ≈ 2l; the problem being symmetric about the midplane, we are considering only one-half in our analysis, i.e., 0 ≤ x ≤ l), a sliding between the workpiece and the dies must take place to allow for the required expansion of the workpiece. However, beyond a certain value of x (in the region 0 ≤ x ≤ l), say, x , there is no sliding between the workpiece and the dies. This is due to the increasing frictional stress which reaches the maximum value, equal to the shear yield stress, at x = x and remains so in the rest of the zone, x ≤ x ≤ l. Hence, for 0 ≤ x ≤ x ,

dp =

Quick Refresher Guide

x =

ln

= ln

1 − ln +

,

x ≤x≤l

Manufacturing Engg

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Fig. 9.3.5 Drawing of cylindrical rod (i) a bell-shaped entrance zone for proper guidance of the workpiece, (ii) a concial working zone, (iii) a straight and short cylindrical zone for adding stability to the operation, and (iv) a bell-shaped exist zone. The final size of the product is determined by the diameter of the stabilizing zone (d ), the other important die dimension being the half-cone angle (α). Sometimes, a back tension F is provided to keep the input workpiece straight. The work load, i.e., the drawing force F, is applied on the exit side, as shown in Fig. 9.3.5, A die can handle jobs having a different initial diameter (d ) which, in turn, determine the length of the job –die

Job

Die

In a drawing operation, in addition to the work load and power required, the maximum possible reduction without any tearing failure of the workpiece is an important parameter. In the analysis that we give here, we shall determine these quantities. Since the drawing operation is mostly performed with rods and wires, we shall assume the workpiece to be cylindrical, as shown in Fig. 9.3.5 A typical drawing die consists of four regions, viz.,

9.3.3 Drawing

where p and p are the pressures obtained earlier.

F = 2 ∫ p dx + ∫ p dx ,

The total forging force per unit length of the workpiece is given as

p = 2K

Substituting this value of x in equation, we obtain

or

or

2K exp(2μx /h) = K

Quick Refresher Guide

Manufacturing Engg

=

,

=1−

r

t

Fig. 9.3.6 Deep drawing

r

r

r

c

Punch F

Die

−t

Job

Blank holder

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The radii of the punch, the job, and the die are r , r and r , respectively. Obviously, without taking the thickening and thinning into account, the clearance between the die and punch (r − r ) is equal to the job thickness t. The corners of the punch and the die are provided with radii r and r respectively. A clearance (c) is maintained between the punch and the blank holder.

r

F

F

The major objectives of our analysis are (i) to correlate the initial and final dimensions of the job and (ii) to estimate the drawing force F. Figure 9.3.6 shows the drawing operation with the important dimensions.

From the point of view of analysis, the process of deep drawing is very complex. In this process, various types of forces operate simultaneously. The annular portion of the sheet metal workpiece (See Fig. 9.3.6) between the blank holder and the die is subjected to a pure radial drawing, whereas the portions of the workpiece around the corners of the punch and the die are subjected to a bending operation. Further, the portion of the job between the punch and the die walls undergoes a longitudinal drawing. Though in this operation varying amount of thickening and thinning of the workpiece is unavoidable, we shall not take this into consideration in our analysis.

9.3.4 Deep Drawing

A and A being the initial and the final cross-sectional areas of the workpiece.

when the true strain is = ln = ln

D=

interface. The degree of a drawing operation (D) is normally expressed in terms of the reduction factor in the cross-sectional area. Thus,

Quick Refresher Guide

+

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(b) Radial stress due to blank holding

=0

+ ln r

+ ln

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=

Using this in the expression for σ , we have

C=

Now, at r = r , σ = μF /(πr t), as already mentioned. Hence,

= C − ln r

Integrating, we obtain

+

(We have not used on Mises’ criterion to avoid mathematical complexity.)

(σ − σ ) = 2K

As σ and σ are the principle stresses, the equation we obtain by using Tresca’s yield criterion is

rdσ + σ dr − σ dr = 0

Now, considering the radial equilibrium of the element shown in Fig. 9.3.7(a), we get

To start with, let us consider the portion of the job between the blank holder and the die. Figure 9.3.7(a) shows the stress acting on an element in this region. It should be noted that the maximum thickening (due to the decreasing circumference of the job causing a compressive hoop stress) takes place at the outer periphery, generating a line contact between the holder and the job. As a result, the entire blank holder force F is assumed to act along the circumference Fig. 9.3.7(b). Thus, the radial stress due to friction can also be represented by an equivalent radial stress 2 F /(2πr t) at the outer periphery.

Fig. 9.3.7 Analysis of deep drawing operation.

Z (a) Stresses acting on element during drawing

dθ

-JJob axis

Quick Refresher Guide

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=

O

Z

rd

J

Friction

σ |r = r

Fig. 9.3.8 Effect of friction at corners.

Job axis

+ ln

= e

/

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In a bending operation, apart from the determination of work load, an estimate of the amount of elastic recovery (spring back) is essential. When the final shape is prescribed, a suitable amount of overbending is required to take care of this spring back. In this section, we shall work out

9.3.5 Bending

F ≈ σ 2πr t

where β is between 0.02 and 0.08. An estimate of the drawing force F (neglecting the friction between the job and the die wall) can easily be obtained,

F = βπr K,

Normally, the blank holder force is given as

There is a further increase in the stress level around the punch corner due to bending. As a result, the drawn cup normally tears around this region. However, to avoid this, an estimate of the maximum permissible value of (r /r ) can be obtained with σ equal to the maximum allowable stress of material. Since r is the final outside diameter of the product, it is easy to arrive at such an estimate. This estimate is based on the consideration of fracture of the material. However, to avoid buckling (due to the compressive hoop stress in the flange region), (r − r ) should not, for most materials, exceed 4t.

where is the coefficient of friction between the workpiece and the die.

Thus,

As the job slides along the die corner, the radial stress, increases to σ due to the frictional forces, as shown in Fig. 9.3.8 This increment can be roughly estimated by using a belt-pulley analogy.

r=r

So, the radial stress at the beginning of the die corner (i.e., at r = r = r + t) is given by

Quick Refresher Guide

Manufacturing Engg

Fig. 9.3.9 Details in bending

+

+

Punch Job

t

Die

= ln 1 +

(

( .

. )

)

= ln 1 +

.

(

/)

.

has ), we

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Determination of Work Load: Since the job undergoes plastic bending, the stress distribution at the cross section along the centre line (XX) is as shown in Fig. 9.3.10(a). This distribution is obtained by neglecting all other effects of curvature except the shift of neutral line. It is obvious that in the zone on either side of the neutral plane the strain level is within the elastic range. When the strain (both in the tensile and the compressive zones) reaches the yield limit, plastic deformation starts. Assuming the yield stress to be σ (same in both tension and compression) and linear strain

assuming a 5% shift of the neutral plane. Depending on the ductility of the job material, ϵ a limiting value beyond which a fracture takes place. The limiting value of ϵ (= ϵ can determine the smallest punch radius for a given job thickness.

ϵ

A radius r is provided at the nose of the punch and, accordingly, the die centre has a radius (r + t), where t is the job thickness. The portions of the die, in contact with the job during the operation, are also provided with some radius, say r . The angle between the two faces of the punch and the die is α. At the instant shown, the angle between the two bend surfaces of the job is (π – 2θ). As we shall subsequently show, the bending force F is maximum at some intermediate stage, depending on the frictional characteristics. The degree of a bending operation is normally specified in terms of the strain in the outer fibre. The width of the job w (in the direction perpendicular to the plane of the paper) is much larger as compared with t, and hence a plane strain condition can be assumed. It is obvious that the stock length should be calculated on the basis of the length of the neutral plane of the job. Since the radius of curvature involved in a bending operation is normally small, the neutral plane shifts towards the centre of curvature. Usually, a shift of 5-10% of the thickness is assumed for the calculation of strain and stock length. Thus, the strain in the outer fibre of the bend is given as

+

F

these quantities and also illustrate how the stock size for a given job is computed. Figure 9.3.9 shows a bending operation with characteristic dimensions.

Quick Refresher Guide

M

Plastic tension zone

Ellastic zone

Plastic compression zone

-

+ 0.45t

Neural line

-

(a) Bending stress distribution

Fig. 9.3.10 Mechanics of bending

(c) Forces and moments in bending

(b) Simplified bending stress distribution

X

-

-

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P

+1

+

Neutral line

+

Quick Refresher Guide

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=σ

=σ

= σ

+ n ln 1 +

+ n ln 1 +

+ nϵ

.

(

.

/)

.

0.1σ + 0.55σ

+ − 0.45σ

+ (0.45t)

+

,

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F=

Substituting N in terms of M, we obtain

= N cos θ + μN sin θ or F = 2N(cos θ + μ sin θ)

with the other forces. The normal and frictional forces exerted by the die and the punch at their contact lines (since r is small as compared with the other dimensions, the finite contact between the job and the punch can be idealized as a line) are N and N, respectively. As t is small, the moment due to N is negligible. Hence M = Nl/cos θ. One-half of the bending force per unit width is given as

Fig. 9.3.11 Free body diagram of half the job

Now, let us consider the right half of the job (of unit width) and the forces and moments acting on it (see Fig. 9.3.11). Since P arises from the shift of the neutral plane which is very small, it can be neglected in comparison

P=

M = (0.55t)

The loading due to this stress distribution can be represented by a bending moment M and a force P (per unit width of the job), shown in Fig. 9.3.10(c), Expressed as

σ

σ

hardening, the stress distribution will be as shown in the figure. The magnitude of σ and σ is different due to the shift of the neutral plane. For the sake of simplicity, the stress distribution for large plastic bending is idealized as shown in Fig. 9.3.10(b). When the strain hardening rate is n, then

Quick Refresher Guide

(− sin 2θ + μ cos 2θ)

Manufacturing Engg

= tan

μ

=

[1 + cos(tan

μ) + μ sin(tan

μ)]

d, = 2r,

x

l

l

B

B

dx

45

45

x

C

C

Dead zone

O

Product

Fig. 9.3.12 Analysis of extrusion process

(a)Details of extrusion process

A

A

σ + dσ

45

P

P

σ

(b)Stresses on element

μP

μP

2r

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Ram

Die

d(r, = 2r)

Determination of work Load from Stress Analysis With a flat-face die and high friction between the material and the container wall, a dead zone, shown in Fig. 9.3.12(a), develops where no flow of material takes place. We assume that the dead zone can be approximated by a half-cone angle of 45 . The material undergoing deformation can be divided into two regions, namely, (i) section AA to BB, where the flow of material is considered as a rigid body motion and (ii) section BB to CC, where the flow is analogous to that in a drawing operation (of course, with a compressive load). Figure 3.15(b) shows an element in the region BB-CC along with the stresses acting on it. Comparing Fig. 9.3.12(b) with

The basic nature of the deformation in extrusion is, to some extent, similar to that in drawing. Here instead of applying a tensile load at the exit end, a compressive load is applied at the other end. However, a number of complexities arise as the die is commonly flat-face (i.e., the equivalent half-cone angle is very large unlike in the drawing die). Consequently, with the same assumptions as in drawing, the results become highly inaccurate. In our analysis here, we shall determine the work load and the frictional power loss for a simple forward extrusion with a flatface die. For doing this, we shall use two approaches; of these, one is in line with that used for drawing, whereas the other is based on the energy consideration. Since both involve rather drastic assumptions, we shall compare the results obtained from the two approaches.

9.3.6 Extrusion

F

Thus, the maximum work load per unit width is given as

θ=θ

Since M is independent of θ. It is obvious that F reaches a maximum when

=

Now, differentiating F with respect to θ, we get

Quick Refresher Guide

2(r, = d )

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(

)

+ 1−

(

)

,

(

)

−1 ,

=

/√

BB

+ +σ

(considering von Mises’ yield

|

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Finally, we get the work load F as

σ AA d = σ BB d + πd lK

which can be solved to obtain the value of . Considering the axial equilibrium of the region (Fig. 9.3.13), we get

Fig. 9.3.13 Frictional load during extrusion

|

It should be noted that equation was derived assuming that the half-cone angle α is small, whereas here we have taken α = 45 . This introduces some inaccuracy in the analysis but the solution becomes unwieldy without this assumption.

=

criterion). Hence [taking K = σ /√3 ]

K. At the section BB, the value of p is given by p = −σ

Let us now consider the stresses acting on the boundaries of the region between the sections AA and BB. The frictional stress at the container wall is assumed as the shear yield stress K, i.e., p=

where ϕ = 1 + (because tan α = tan 45 = 1)

=

where σ = 0 and −F /A is nothing but the compressive stress at the section BB. Thus, considering the compressive nature here, we have

=

The figure for drawing the similarity between extrusion and drawing is easily discernible. The only change here is that σ is compressive. Therefore, following the same analysis as in drawing equation can be rewritten as

Quick Refresher Guide

Manufacturing Engg

√

σ dV l

−1

1+

Die

Sintering

9.3.7 Punching and Blanking

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Compacting x

Mixing or Blending

x x

Powder manufacture

x

The powder metallurgy has four basic steps:

Powder metallurgy is the name given to the process by which fine powdered materials are blended, pressed into desired shape (compacted) and then heated (sintered) in a control atmosphere to bond the contacting surfaces of the particles and establish desired properties. This method is used for mass production of small, intricate parts of high precision, often eliminating need of additional machining.

9.3.8 Powder Metallurgy

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σ d V 1 + d ln

Clearance c

Fig. 9.3.14 Details of punching process.

d

Job holder

Figure 9.3.14 shows a simple punching operation. As in deep drawing, so here too the job is held by job holders to prevent any distortion and to provide a support. It should be noted that the punch and die corners are not provided with any radius (unlike in the deep drawing operation) as the objective in this process is to cause a rupture of the material. A clearance c is provided between the punch and the die. Hence, the die diameter d = d + 2c, where d is the diameter of the punch.

Job

Punch

As we have already noted, the punching and blanking processes cannot, strictly speaking, be grouped under the forming operations. In these processes, a finite volume from a sheet metal is

√

√

σ d V ln

dr

− 1 dr

Manufacturing Engg

removed in block by using a die and a punch. The shape and size of the portion removed are determined by the geometry of the die and the punch. If the final product happens to be the removed portion, then the operation is termed as blanking. On the other hand, if the pierced sheet metal is the final product, then the operation is called punching. Since the basic mechanics of material removal is the same in both the operations, we shall discuss these under a single heading, namely punching.

Quick Refresher Guide

Powder Manufacture: The properties of powder metallurgy are highly dependent on the characteristics of metal or material powders that are used. Some important properties or characteristics include chemistry and purity, particle size, size distribution, particle shape and the surface texture of the particle. The commercial powder is produced by some form of melt atomization where a liquid is fragmented into molten droplets which then solidify into particles.

=

P =P +

So, the total power loss is

P = P +P

−

− 1 − ln

So, the total power loss in friction is

Finally, P = πσ V d

= 2π σ V d ∫

P = ∫ 2π V d μ

Making use of the foregoing relations in the expression for P , we get

σ =σ

The yield criterion can be written as p = σ + σ , where σ is expressed, with ϕ=1+ , as

where V = V

P = ∫ √2πdμp √2 V dr

To determine the loss in the first region, we can use either of the two approaches followed in computing the work load. According to the first of these approaches, the frictional power loss in the conical region is given as

P = πd lKV =

The total frictional power loss can be found by summing up the frictional power losses in the conical and the cylindrical regions of the work material. The contribution from the second region can be expressed as

Frictional Power Loss

F = d σ BB + d lσ √

Quick Refresher Guide

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When product is composed of more than one material, the sintering temperature may even be above the melting temperature of one or more components.

Sintering: In the sintering operation, the pressed powder compacts are heated in a controlled atmosphere to a temperature below the melting point but high enough to permit sold state diffusion, and held for sufficient time to permit bonding of particles. Most metals are sintered at temperatures of 70%-80% of melting temperature.

Compacting: It is one of the most important process of powder metallurgy. Loose powder is compressed and densified into a shape known as a green compact, usually at room temperature. High product density and the uniformity of that density throughout the compact are generally desired characteristics. In addition the compact should posses sufficient green strength for inprocess handling and transport to the sintering furnace.

Blending or mixing operations can be done either dry or wet, where water or other solvent is used to improve mixing, reduce dusting and lessen explosion hazards.

The final product chemistry is often obtained by combining pure metal or non metal powders rather than using prealloyed material. Sufficient diffusion must occur during the sintering operation to produce a uniform chemisty and structure in the final product.

Powder Mixing and Blending: It is rare that a single powder will posses all of the characteristics desired in a given process and product. Most likely, the starting material will be a mixture of various grades and sizes of powder, with additions of lubricants or binders.

Quick Refresher Guide

Manufacturing Engg

/

/

,

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The greatest hurdle in solid phase welding is posed by the surface oxide layers and oil films. The liquid films can be removed by heating in hot welding, and by means of scratch brushing in cold welding. The oxide films can also be reduced to a certain extent by scratch brushing. Moreover, these oxide layers (being hard and brittle) fracture when the pressure is applied. A lateral movement is very useful (as in ultrasonic welding) since this tends to roll together the fragmented oxide layer into a relatively thick agglomerate. This results in a more metal-to-metal

The strength of a welded junction increases with increasing bulk deformation. Moreover, no weldment takes place below a certain critical deformation. The amount of deformation necessary for obtaining a specific strength decreases with increasing temperature. A strong weld may be made with only 10% deformation if the working temperature is quite close to the melting point of the material. The ratio of the oxide hardness and the parent metal hardness also effectively governs the amount of necessary deformation.

respectively).

× 100% (for a circular specimen of original and final diameters d and d ,

× 100% (for a sheet of original and final thickness t and t , respectively),

The surface deformation that takes place during welding is difficult to measure. As such, in pressure welding, the bulk deformation is used as an index of the surface deformation and is expressed as

In the solid phase welding processes, the four important factors are (i) surface deformation (ii) surface films, (iii) recrystallization, and (iv) diffusion.

where l (≫d) is the length of the crack.

σ =

where E is the modulus of elasticity of the material and d is the lattice spacing in the cleavage plane. However, in a brittle solid, the failure takes place by the extension of the cracks already present, and the bulk strength is much reduced from that given by above equation. In this case, the bulk strength (σ ) is expressed as

σ =

The solid state welding processes may be carried out both at the room temperature and at an elevated temperature without, of course, melting any part of the joining surfaces. For a better understanding of the quality of a solid phase joint, it is worthwhile to recapitulate the strength and cohesion of metals. A defect-free crystal fails by a cleavage along a crystallographic plane where the interatomic force is the weakest. As a result, two new surfaces are produced, and the surface energy γ is defined as the work done in order to create these surfaces. The strength of a single crystal (σ ) is found to be

9.4.1 Principles of Solid phase welding

Unlike the manufacturing process employed to produce a single component, the joining process are used to assemble different members to yield the desired complex configuration. Such a complex geometry is either too difficult to obtain or impossible to obtain by using only the manufacturing processes.

Part 9.4: Joining Process

Quick Refresher Guide

Manufacturing Engg

Work

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+

Arc

Fig. 9.4.1 Arc welding scheme

Electric source

−

Electrode

Emission and Ionization of Electric Arc: First of all, let us see how an electric arc is created and maintained between two electrodes of opposite polarity. Figure 9.4.1 schematically shows an electric circuit used for arc welding where the work is the positive electrode (called the anode) and the electrode rod is the negative electrode (called the cathode). Initially, a good contact is made between the electrode and the work. Thereafter, the electrode is withdrawn. As a result, the metallic bridges start breaking, thus increasing the current density per bridge.

Heat Source: A heat source, suitable for welding, should release the heat in a sharply defined, isolated zone. Moreover, the heat should be produced at a high temperature and at a high rate. The most common sources of heat include (i) the electric arc (as in various arc weldings), (ii) the chemical flame (as in gas welding), (iii) an exothermic chemical reaction (as in thermit welding), and (iv) an electric resistance heating (as in electroslag and other resistance welding processes). The general characteristics of these heat sources are now discussed.

(i) the characteristics of the heat source, (ii) the nature of deposition of the filler material in the fusion zone, known as the weld pool, (iii) the heat flow characteristics in the joint, (iv)the gas metal or slag metal reactions in the fusion zone, and (v) the cooling of the fusion zone with the associated contraction, residual stresses, and metallurgical changes.

In a fusion welding process, the material around the joint is melted in both the parts to be joined. If necessary, a molten filler material is also added. Thus, a fusion welding process may be either autogeneous or homogeneous. Metallurgically, there are three distinct zones in a welded part, namely, (i) the fusion zone, (ii) the heat affected unmelted zone around the fusion zone, and (iii) the unaffected original part. The most important factors governing a fusion welding process are

9.4.2 Principles of Fusion welding

A solid phase welding done at the room temperature does not allow recrystallization and gain growth at the interface. This reduces the ductility of the joint to some extent. An increase in working temperature not only increases the ductility but also eliminates some other defects. The phenomenon of diffusion, though it has not been studied extensively, has an important bearing on the performance of a solid phase weld. The shape and the size of the voids at the interface are modified considerably depending on the amount of diffusion.

contact area. An excessive oxide contamination is always harmful, resulting in a poor joint efficiency.

Quick Refresher Guide

Manufacturing Engg

7.61

Nickel

5.0

2.2

2.3

4.5

4.4

4.4

4.1

Φ(eV)

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(i) Cathode spot: This is a relatively very small area on the cathode surface, emitting the electrons. (ii) Cathode Space: It is a gaseous region adjacent to the cathode and has a thickness of the order of 10 cm. This region has the positive space charge, so a voltage drop is necessary as the electrons are to be pulled across this region. (iii) Arc Column: This is the visible portion of the arc consisting of plasma (hot ionized gas) where the voltage drop is not sharp. (iv)Anode Space: This, again, is a gaseous region (thickness ≈10 cm) and is adjacent to the anode surface where a sharp drop in the voltage takes place. This is because the electrons have to penetrate the anode surface after overcoming the repulsion of the therminicallyemitted electrons from the anode surface. (v) Anode Spot: This is the area on the anode surface where the electrons are absorbed. This area is larger than the cathode spot.

Arc structure, characteristics, and power: Structurally, we can distinguish five different zones in an electric arc. These are as follows.

4.3

Potassium

8.1

Tungsten

5.1

7.83

Iron

Sodium

7.9

6.0

Ionization potential (V)

Copper

Aluminium

Metal

Table 9.4.1 Ionization potential and thermionic work function of some common metals

Where e = charge of an electron, k = Boltzmann’s constant, and ϕ (when measured in electron volts) as the thermionic work function. ϕ, in fact represents the kinetic energy necessary to ‘boil’ out an electron. The values of ϕ for some common metals are shown in Table 9.4.1

β = ϕe/(kθ)

where I is in amp/cm , θ, is the absolute temperature, C is a constant, and β is given by

I = Cθ exp(−β/θ),

The rate at which the electrons are emitted from a hot surface is given by

Finally, the current density rises to such a high value that the bridges start boiling. Under such conditions, the electrons come out of both the surfaces by a process known as thermionic emission. Obviously, the electrons (having the negative charge) coming out of the anode are pulled back, whereas those coming out of the cathode are also attracted towards the anode.

Quick Refresher Guide

Manufacturing Engg

Cathode space

Cathode

Fig. 9.4.2 Potential across arc space

Potential

Arc column

Anode space

100

Current (amp)

50

500

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V = A + Bl,

As a first approximation, we can assume the conductivity of the arc column to be independent of the arc length l. The electrode drops are also independent of the arc length. Hence, we can write the voltage drop across the entire arc as

Fig. 9.4.3 Current--voltage characteristic of arc

0

Voltage

It has been experimentally found that, for given spacing (and, of course, electrode materials), the voltage reduces up to a current value of 50 amp (against the ohmic law of constant resistance) and increases thereafter, as shown in Fig. 9.4.3. This can be explained as follows. Up to 50 amp of current, the shape of the arc is almost cylindrical and the surface to volume ratio of a cylinder decreases with increasing radius. Thus, a thick, high current arc loses less heat and essentially burns hotter. This result in a higher conductivity (and consequently lower resistance) as compared with a thin, low current, arc. However, beyond 50 amp of current, the arc bulges out and the current path becomes more than the arc gap which again increases the resistance of the arc. Due to these two opposite effects, i.e., higher temperature and longer current path, the voltage drop remains constant over a wide range of the current values.

Cathode

Anode

The potential drop across an arc is schematically shown in Fig. 9.4.2. The voltage drop shown in this figure is for given spacing, current, and electrode materials. A change in the materials alters all the values. However, a change in the spacing and the current essentially changes only then drop in the arc column.

Quick Refresher Guide

Manufacturing Engg

+

,

(ρ = density, c = specific heat).

+

,

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Q = CVI,

In arc welding with short circuit metal transfer, the heat input rate is easily seen to be

where h = plate thickness.

Q = 8kθ h

For a two-dimensional heat source, the corresponding equation is given by

=

α = thermal diffusivity of the work material (m /sec)

v = speed of welding (m/sec),

θ = melting point of the work material above the initial temperature (℃),

k = thermal conductivity of the work material (W/m-℃),

w = width of the weld (m),

where Q = rate of heat input (W),

Q = πwkθ

Heat flow characteristics: A study of heat flow characteristics can provide an estimate of the minimum heat input rate required to form a weld of a given width. Moreover, a recognition of the major variables controlling the thermal cycle (i.e., the heating and the cooling rate of the heat affected zone) is essential for a successful fusion welding. In the fusion welding processes, the heat source is moving, except in spot welding where the source is stationary. Once the steady state is reached, even with a moving heat source, the temperature distribution relative to the source becomes stationary. The most convenient way of analyzing such a problem with a moving source is to assume the source as stationary and the workpiece to move with the same velocity in the opposite direction. This speed is called the welding speed. Two different types of heat sources can be considered. In most cases, the heat is liberated in a small zone which is idealized as a point source, and the heat flow from the source is three-dimensional. In a few cases, e.g., in butt welding of relatively thin plates, the heat is liberated along a line and heat source is idealized as a line source. In such situations, the heat flow is two-dimensional The available results include those of infinite, semi-infinite, and finite medium, each with point and line sources. Of these results, the most useful is the one which gives the minimum heat input rate necessary for maintaining a given width of the weld. For a three dimensional heat source, this is given as

The voltage-current relationship of an arc (Fig. 9.4.3) determines the required characteristics of the power source.

where A is the electrode drop and Bl represents the column drop.

Quick Refresher Guide

C = fraction of total time during which the arc is on.

I = arc current (A),

V = arc voltage (V),

Manufacturing Engg

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The soldering and brazing processes are carried out by allowing a molten filler material to flow in the gap between the parent bodies. Obviously, the filler material has to have a melting point much lower than that of the parent bodies. When the filler material is a copper alloy (e.g., copper-zinc and copper-silver), the process is called brazing. A similar process with a lead-tin alloy as the filler material is called soldering. The most common heat source for these processes is electrical resistance heating.

Soldering and Brazing

Three different processes, namely, brazing, soldering and adhesive bonding are grouped under solid/liquid state welding. The physical phenomena associated with each of these processes are essentially the same, and differ mainly in the metallurgical aspects. In these processes, the bulk material is not melted. Also, a molten filler material is used to provide the joint.

9.4.3 Principles of Solid/Liquid State Joining

(iii) Metallurgical Changes: These changes are due to the heating and subsequent cooling of the weld and the heat affected zones of the parent materials. Such changes significantly affect the quality of the weld. The wide variety of changes that may take place depend on various factors, e.g., a) the nature of material, i.e., single-phase, two-phase, b) the nature of the prior heat treatment, if any, and c) the nature of the prior cold working.

(i) Contraction: During the freezing of the weld pool, a decrease in the volume takes place. Moreover, the direction of freezing, and thus the effect of contraction, depends on the type of joint. (ii) Residual Stress: During the fusion welding of plates, as the weld pool contracts on cooling, this contraction is resisted by the rest of the plates (which have not melted). As such, a tensile stress is generated in the weld, and this is balanced by the compressive stress in the parent metal. This residual stress may result in the cracking of a brittle material and is not important as far as a ductile material is concerned.

The three important effects intimately connected with the cooling of a fusion weld are (i) contraction, (ii) residual stress, and (iii) metallurgical phase transformation. All these effects significantly control the quality of a weld.

Cooling of fusion weld

where

Quick Refresher Guide

Manufacturing Engg

Variety of work materials Variety of part shapes and geometric features Dimensional accuracy Good surface finishes.

Wasteful of material Time consuming

Tipped type

o

Milling cutter Grinding wheel etc.

o o

Single point cutting tools are fed axially at a uniform feed per revolution. But multipoint cutting tools are fed perpendicularly to the cutter axis of rotation.

x

x x

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Orthogonal cutting or 2D cutting Oblique cutting or 3D cutting

Method of machining

The life of multipoint cutting tool is more than single point cutting tool.

MRR obtained from multipoint cutting tool is more than that from single point cutting tool

Drill bit

o

Multipoint cutting tools

Ground type

o

Single point cutting tools

x

x

x

x

The process of removing metal can be done by using two types of cutting tools.

x x

Limitations

x x x x

Advantages of Machining:

Machining is a manufacturing process in which a sharp cutting tool is used to cut away material to leave the desired part shape. The predominant cutting action in machining involves shear deformation of the work material to form a chip; as chip is removed, a new surface is exposed.

9.5.1 Machining

The material removal processes are family of shaping operations in which excess material is removed from a starting work part so that what remains is the desired final geometry.

Part 9.5: Theory of Metal Cutting

Quick Refresher Guide

vchip

Work

v

Tool

Rake angle

Flank surface

Clearance angle

Rake surface

∝

Manufacturing Engg

Primary shear zone: the uncut layer deforms into a chip after it goes through a severe plastic deformation in the primary shear zone. Secondary shear zone: after the chip formation, the chip flows over the rake surface of the tool and the strong adhesion between the tool and the newly-formed chip surface results in some sticking. Due to that, it undergoes a further plastic deformation at the interface between tool and chip.

Nose radius

Temperature of cutting region

x

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Side cutting edge angle r

Coefficient of friction between tool and chip

x

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End cutting edge angle φs

End relief angle

Side rake angle θe

αs

φe

Type of cutting fluid

x

Back rake angle

αb

Surface finish of tool

Effective rake angle of the tool

x

Brittle work material Small rake angle Large uncut chip thickness Very low or very high cutting speed Lack of an effective cutting fluid Low stiffness of the machine tool

Manufacturing Engg

According to ASA system, the single point cutting tool can be designated as Notation Name of the angles Order of Reading

x x x x x x

x

Depth of cut

x

Ductile work material Low cutting speeds Low rake angle and high feed Heavy depth of cut Absence of cutting fluid Stronger adhesion between chips and tool face

Favorable factors to get Discontinuous chip:

x x x x x x

Side relief angle

Feed rate

x

Ductile work material Small uncut chip thickness, fine feed High cutting speed Large rake angle Suitable cutting fluid Using sharp cutting edge

Favorable factors to get Continuous chip with built-up edge:

x x x x x x

θs

Cutting speed

x

Continuous chip Continuous with built-up edge Discontinuous chip Non-homogeneous chip

Favorable factors to get continuous chip

x x x x

Types of Chips:

Quick Refresher Guide

Minor factors

Work material properties.

x

Major Factors

Chip formation influenced by the following factors:

x

x

When the zone under the cutting action is carefully examined, primary shear zone and secondary shear zone are formed.

Mechanism of chip formation:

Chip thickness Uncut thickness

9.5.2 Mechanics of basic Machining Operation

Quick Refresher Guide

Downward direction

ϕ

π

θs Y

π

ϕ

π

∝

π

θ

π

π

∝

Y

π

Z

Manufacturing Engg

To give protection to the tool point

x

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To control the direction of chip flow

x

Purpose:

™ Back rake angle (αb):

Importance of various angles:

Nose radius (r): Radius curvature of the tool tip. It provides strengthening of the tool nose and better surface finish.

Side cutting edge angle (φs)) or Lead angle: It is the angle between the side cutting edge and the side of the tool shank.

End cutting edge angle (φe)): It is the angle between the end cutting edge and a line normal to the tool shank.

Side relief angle (θs)): It is the angle between the portion of the side flank immediately below the side cutting edge and a line perpendicular to the base of the tool, and measured at right angle to the side flank.

End relief angle (θe)): It is the angle between the portion of the end flank immediately below the end cutting edge and a line perpendicular to the base of the tool, and measured at right angle to the end flank.

Side rake angle (αs)): It is the angle between the tool face and a line parallel to the base of the tool and measured in a plane perpendicular to the base and the side cutting edge.

Back rake angle (αb)): It is the angle between the face of the tool and a line parallel to the base of the tool and measured in a plane (perpendicular) through the side cutting edge.

X

X

π

Z ,V

Quick Refresher Guide

Rake angle influences the cutting forces, power and surface finish:

x

back rake

shear

cutting force -

back rake

compression

Heat Dissipation

Heat Dissipation

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l1>l2

For same depth of cut

Cutting force: With decrease in rake angle the shear plane angle (φ) decreases and for the same depth of cut, the extent of the shear plane increase and thereby the cutting force increases (μ at the tool-chip interface assumed to be constant) +ve

rakee

rakee

Heat conduction: Tool with large rake angle do not conduct heat readily whereas tools with negative rake angles have better heat conductivity.

+

Strength of tool: Tool with large rake angle is weak and the tool point may break off readily and on the contrary tools with negative rake angle are stronger.

Effect on performance:

Controls the directions of the resultant force on the tool and chip flows.

Manufacturing Engg

x

Quick Refresher Guide

Manufacturing Engg

Positive Back Rake angle Promote chip formation giving the tool more wedge shape but weakens the tool section causing heat concentration near the chip and reduces tool life. Less rough surface Less curling and more open chips More continuous chips When machining low strength ferrous and non-ferrous materials and workhardening materials Using low power machines The setup lacks strength and rigidity and cutting at low speeds When machining long shafts of small diameters

ve rake angle

For rigid set ups and when cutting at high speeds When there are heavy impacts loads such as interrupted machining

More rough surface More curling of chips The chips break in short intervals Machining high strength alloys

Negative Back Rake angle Chip is highly distorted that forces increased causes tool vibration, chatter, poor finish, machining accuracy and machine power.

subjected to impact

The back rake angle controls the point of initial contact between the tool and the workpiece and there by exercises a control over strength of tool point.

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ƒ

Power consumption: With large rake angle the cutting forces are less thereby the power required for making the cut is also reduced.

Quick Refresher Guide

Manufacturing Engg

angle )

( +ve side rake

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( +ve side rak ke )

¾ As the side rake angle decreases, the heat generation also increases (because power consumption increases ) but the temperature at the tool point decreases because the area of heat dissipation increases with decrease in side rake angle.

¾ A fairly large part of the power required to part and bend the chip is transformed into heat.

Heat Generation:

Work Piece

¾ As the side rake angle increases, the amount of chip that bends decreases, and hence the power required to part and bend the chip decrease as the side rake angle increases

Power consumption:

Effect on performance:

Purpose: By providing a shearing action for chip removal, this angle enables the tool to cut more freely.

™ Side rake angles (αs)

Quick Refresher Guide

and

More rough surface

Discontinuous chips

x

x x

x x

Less rough surface(ie) there is an improvement in surface finish, a reduction in cutting forces and also an increase in tool life.

broken Continuous chips

Negative side Rake angle

More Positive angle

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More rough surface and reduction in tool strength.

Less roughness, however it should be Increase in the height of feed maintained such that it does not rub marks. with the workpiece

Less Positive angle

To avoid rubbing between the edge of the tool and the workpiece. If a large part of the end cutting edge is in contact with work which increases the radial force and this may lead to chatter and vibration. Main function of end cutting edge angle is to prevent chatter and vibration Too greater an angle leaves the tool pointy, which results in a tool which is not able to conduct away the heat fast enough. No relation with the power consumption but it affects the tool life. On finishing tool, a small flat is ground on the front portion of the edge next to the nose radius. Tools like, cut off tools and necking tools often have no end cutting edge angle.

Purpose:

more

Positive side Rake angle

¾ Side rake angle has a greater effect on the chips breakage than does the back rake angle

¾ The side rake angle increases, the size of built up edge that adheres to the top face near to the cutting edge becomes smaller and therefore, the resulting surface finish is smooth.

Surface finish:

¾ For a high feed/rev, the side rake angle should be decreased in order to increase the area of heat dissipation.

The feed/rev increase, the chip thickness increases and consequently the power expended to part the chip also increases resulting in higher amount of heat generated.

™ End cutting edge angle(φe)

x x

Manufacturing Engg

¾ The thickness of the chip is largely determined by the amount of feed/rev.

Feed/rev:

Quick Refresher Guide

Edge Leaving Gradually

Depth of cut Edge Entering Gradually

The angle distributes the cutting stresses favourably at the start and at the end of a cut. With γs the tool first contacts the workpiece some distance away from the tip and hence

Manufacturing Engg

x

x

x

Feed pressure

Direection of feed.

Direection of feed.

L

Back pressure

Feed Pressure

Side Pressure

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Depth of cut (d) L

action of the tool

L → ength of the cutting

Chip produced will be thinner and wider which will distribute the cutting and heat produced over more of the cutting edge. A gentle back pressure is created against the tool by the introduction of side cutting edge angle and thereby the chatter and vibration are reduced in the machine tool. The back pressure keeps the cross slide tight against the lead screw which prevents any backlash.

the starting load is better withstood and also γs allows the tool to enter and leave the workpiece gradually, so there is no extra load comes into play and therefore tool life increases.

x x

™ Side cutting edge angle(φs)

Quick Refresher Guide

Manufacturing Engg

Less rough surface

More rough surface

Tool will rub against the work which promotes the formation of wear land and finish on the work will be spoiled.

o

x

x x

harder

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To prevent the end of the tool from rubbing on the work. Interrupted cutting requires additional tool strength due to impact loading; therefore lower flank angles are used. It has marginal effect on the cutting forces, power and surface finish

the

Used to materials

Used to machine the softer materials

machine

For brittle tool materials

Lower values

Used for tougher tool materials

Higher values

If it is too large the cutting edge will be very weak and obviously the tool life will be lowered.

Tool becomes dull quickly.

o

™ End relief angle (θe)

x

Side cutting edge cannot dig into the work properly.

If the side relief is too small then,

x

o

Allows the tool cutting edge to penetrate into the metal and promotes free cutting by preventing the side flank f the tool from rubbing against the work.

x

Slight increase in tool life. Could cause chatter in a non-rigid setup.

Thinner continuous chips

More Positive angle

Thicker discontinuous chips

Less Positive angle

It has a significant influence on the life of the tool under normal conditions. Because by producing a wider chip it distributes the cutting force and heat produced over larger cutting edge. No influence on the power consumed or on the total force necessary to cut the metal for a given cut of feed. Size of this angle influences chatter (ie) greater angles cause chatter Helps to avoid the formation of a built up edge on the tool. No side cutting edge angle is desirable when machining castings and forgings with hard and scaly skins, because the least amount of tool edge should be exposed to the destructive action of the skin.

™ Side relief angles (θs)

x x x

x

x

Quick Refresher Guide

If tool tip is below centre line, the tool tip will dull rapidly

-

iii) Always add θ to back rake angle algebraically

ii) Always subtract θ from relief angle algebraically

i) off set is taken as positive if the tip is above centre line and negative if the tip is below centre line

The considerations are

Let Of = Off set distance; r = radius of work; θ = Angular change; Sin θ = Of/r

To reduce the above two effects the tool is filled at slightly inclined position, due to this the back rake and end relief angles will get changed. The effective rake and relief angles can be calculated as follows:

If tool tip is above centre line, the tool will destroyed due to friction

-

Chatter will result if nose radius is too large; because a large nose radius thins the chip and increases radial pressure.

x

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Slight reduction in cutting forces is usually obtained.

x

The increase in nose radius avoids high heat concentration at a sharp point, improvement in tool life and an improvement in surface finish is obtained.

™ Nose radius (r) x

Manufacturing Engg

During turning cylindrical jobs the tool tip must coincide with centre line of job, if not the effects are:

Effect of rake and relief angle at different settings:

Quick Refresher Guide

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Friction coefficient: The friction coefficient should be low to enable the easy flow of chip during the machining.

Chemical stability or inertness: It should have less affinity towards the workpiece material, so that any adverse reactions contributing to tool wear are avoided.

Toughness: The tool should not fracture or fail due to the sudden impact load on tool which is created in interrupted cutting operations.

Wear resistance: The tool should give an acceptable tool life which is obtained before the tool is resharpened or replaced.

Hot Hardness: The tool should be able to retain its hardness and strength even at elevated temperatures encountered in cutting operations.

A cutting tool must have certain characteristics in order of produce good-quality and economical parts.

9.5.3 Cutting tool materials

Nose radius equal to feed

If nose radius is increased the feed marks tend to be wiped out.

x

Depth of cut

Feed marks are closer together when feed rate is reduced; a feed rate equal to the nose radius gives a poor finish.

Depth of cut

w/p

Chip thickness decreases at point of tool.

Manufacturing Engg

x

w/p

Chip has uniform thickness

Quick Refresher Guide

Manufacturing Engg

Reduce friction and wear, thus improving tool life and surface finish. Reduce forces and energy consumption Cool the cutting zone, thus reducing workpiece temperature and distortion. Wash away the chips. Protect the newly machined surfaces from environmental corrosion.

x x

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Water Oils

Types of cutting fluid:

x x x x x

Purposes of cutting fluid:

Cutting fluids: Cutting fluids are sometimes called cutting oil or coolant or lubricant.

Columbium – 50 % ; Titanium – 30 %; Tungsten – 20%

UCON: It is developed by union carbide in USA. It is consists of

Cubic boron nitride (CBN)

Diamond: Single crystal diamond & Poly-crystalline diamond (compacts)

Cermets:

Ceramics and sintered oxides: There are two principle families of ceramic cutting tool materials – Alumina base ceramic and silicon-nitride (SiN) base ceramic

Cemented Carbides: It is produced by powder metallurgy technique with sintering at 1500oC. The two basic groups of carbides used for machining operations are tungsten carbide and titanium carbide.

Cast nonferrous tools (Stellite):

1 % - Vanadium – used to maintain keenness of cutting edge.

4 % - Chromium – used to increase strength

18% - Tungsten – used to increase hot hardness and stability

Composition of General HSS:

High speed steel (HSS)

Composition C = 0.8 to 1.3%; Si = 0.1 to 0.4%; Mn = 0.1 to 0.4%

Carbon steels (High carbon steels):

Tool materials:

Cost and method of manufacture: Cost should be low and method of manufacture should be easy

Quick Refresher Guide

Plastic deformation

Gradual Wear (Tool wear)

Flank wear

Crater Wear

x

x

x

x

x

Change in the magnitude of the cutting force resulting in changes in machine and workpiece dimensions to change

Change in the cutting temperature.

o

o

C

Where V = Cutting speed, m/min

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C = a constant equal to the intercept of the curve and the ordinate or the cutting speed. Actually it is the cutting speed for a tool life of one minute.

T = Tool life, minutes (it is time that it takes to develop a certain flank wear land)

VT n

The selection of the correct cutting speed has an important bearing on the economics of all metal-cutting operations. To measure the tool life the Taylor tool life relationship is used

Change of the quality of the machine surfaces

Other criteria are sometimes used to evaluate tool life:

x

o

It can be defined as the time a newly sharpened tool will cut satisfactorily before it becomes necessary to remove it and regrind or replace.

x

Tool life:

Fracture

x

Tool failure: The useful life of a cutting tool may end in a variety of ways:

9.5.4 Tool life

x

Manufacturing Engg

Soluble Oil: It contains 80% water, soap and mineral oil. Soap helps as an emulsifying agent. o Straight Oil: It contains mineral oils, Kerosene and low viscosity petroleum products. o Mixed Oil: It is a combination of straight, mineral and fatty oils. emulsions, semi-synthetics, and synthetics

o

Quick Refresher Guide

logV1  logV2 logT2  logT1

Manufacturing Engg

y

C

For minimum production cost For Maximum production rate For maximum profit or maximum efficiency

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R4 – tool changing cost/piece

R3 – Machining cost/piece

R2 – set up and idle time cost /piece

R1 – material cost/piece

R – Total cost/piece

R = R1 + R2 + R3 + R4 + R5

For minimum production cost: The total cost of a part can be written in the form

The maximum production rate consideration will lead to higher tool cost, where as the minimum cost consideration will lead to reduced production rate.

x x x

Economics of machining: Economics of machining is not but selection of cutting speed for optimum conditions which can be done based on the following criteria:

™ Cutting fluid

™ Workpiece material

™ Tool geometry

™ Tool material

™ Process variables

Variables affecting tool life:

Where d is the depth of cut, f is the feed rate (mm/rev)

VT n d x f

Cutting speed is the most significant process variable in tool life; however, depth of cut and feed rate are also important. Thus tool life relation can be modified as follows.

Values of n and C differ depending on work and tool materials

n = slope of the curve n = tanθ =

Quick Refresher Guide

Manufacturing Engg

1

Tm SLD § V · n = y1 ¨ ¸ Tct T 1000fV © C ¹

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R5= y2 x No. of times tool failed

Tool regrinding cost/piece (R5):

Tool changing cost/piece = y1 x Tct x

Tm T

πLD 1000fV

Number of times the tool has to be changed =

Tool changing cost (R4):

R3 = y1Tm = y1

Machining cost (R3): The machining cost/piece is given by the product of the machining time/piece and the cost/unit time of labour and overheads.

R2 = y1Ts

Set-up and idle time cost (R2): It is given by the product of the set-up and idle time and the cost/unit time of labour and overheads. This is also independent of the cutting conditions f and v

Material cost R1: The material cost does not depend on the cutting conditions and remains as constant (R1)

Tct – tool changing time, min

Tm- machining time/piece, min

Ts – Setup time and idle time/piece, min

f – Feed, mm/rev

T – Tool life, min

V – Cutting speed (m/min)

D – Work diameter (mm)

L – Length of work (mm)

y2 – total cost per grind including depreciation

y1 – Cost/min of labour and overheads,

R5 – tool regrinding cost/piece.

Quick Refresher Guide 1

SLD § V · n ¨ ¸ 1000fV © C ¹ 1

dR dV 0

1

SLD SLD § V · n SLD § V · n + y1 ¨ ¸ ¨ ¸ Tct + y 2 1000fV 1000fV © C ¹ 1000fV © C ¹

Manufacturing Engg

ª º ny1 C« » ¬ 1 - n y1Tct  y2 ¼

n

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y · § 1 ·§ Corresponding Tool-life T = ¨  1¸¨¨ Tct  2 ¸¸ y1 ¹ © n ¹©

Vopt

Differentiating the above equation w.r.t. V, (In above equation the first two terms are constant)

To obtain minimum total cost per piece,

Total Cost per piece R = R1 + y1Ts + y1

= y2

Quick Refresher Guide

Manufacturing Engg

Tolerance

Basic Size

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Fig. 9.6.1Unilateral Tolerance

Tolerance

Unilateral System: In this system, the dimension of a part is allowed to vary only on one side of basic size i.e., tolerance lies wholly on one side of the basic size either above or below it.

(i) Unilateral system (ii) Bilateral system

There are two systems of writing tolerances:

Systems of writing tolerances

Tolerance can also be defined as the amount by which the job is allowed to go away from accuracy and perfectness without causing any function trouble, when assembled with its mating part.

Tolerance: The permissible variation in size or dimension is called tolerance. Thus, the word tolerance indicates that a worker is not expected to produce the part to the exact size, but a definite small size error is permitted. The difference between the upper limit and the lower limit of a dimension represents the margin for variation in workmanship and is called a “Tolerance zone”.

Limits: The limits of size of a dimension of a part are two extreme permissible sizes, between which the actual size of a dimension may lie. They are fixed with reference to the basic size of that dimension. The high limit (upper limit) for that dimension is the largest size permitted and the low limit is the smallest size permitted for that dimension.

Every production process involves a combination of three elements viz, men, machine and materials. Each of these elements has some inherent variations as well as some unnatural variations. The natural variations are due to chance causes, which are difficult to trace and control. The unnatural variations are due to assignable cause which can be easily traced, controlled and reduced to economic minimum. These variables result in variation of size of components.

9.6.1 Limits, Fits and Tolerances

Part 9.6: Metrology and Inspection

Quick Refresher Guide

.

.

.

Manufacturing Engg

Fig. 9.6.2 Bi-lateral tolerance

Basic Size

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(i)

Fig. 9.6.3 Relation between Cost and Tolerance

Work Tolerance

Relationship between Tolerance and Cost: The relationship between tolerance and cost of production is shown in Fig. 9.6.3 If the tolerance are made closer and closer, the cost of production goes on increasing, because to manufacture the component with closer tolerances, we need:

e.g., 25± . , 25 . In this system it is not possible to retain the same fit when tolerance is varied and the basic size of one or both of the mating parts is to be varied. This system is used in mass production where machine setting is done for the basic size.

.

Tolerance

Bilateral System: In this system, the dimension of the part is allowed to vary on both the sides of the basic size i.e., the limits of tolerance lie on either side of the basic size; but may not be necessarily equally disposed about it, Tolerance

25 . , 25 . , 25 . , 25 . etc. Unilateral system is preferred in interchangeable manufacture, especially when precision fits are required, because: (i) It is easy and simpler to determine deviations. (ii) Another advantage of this system is that Go gauge ends can be standardized as the holes of different tolerance grades have the same lower limit and all the shafts have same upper limit. (iii) This form of tolerance greatly assists the operator, when machining of mating parts. The operator machines to the upper limit of shaft (lower limit for hole) knowing fully well that he still has some margin left for machining before the parts are rejected.

.

Examples of unilateral tolerances are:

Quick Refresher Guide

Cost

Manufacturing Engg

th

th

th

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For shaft

The relationship of deviation with tolerance (IT) is given by,

It is denoted by ‘EI’ for hole and ‘ei’ for shaft.

Lower Deviation: It is the algebraic difference between the lower limit of size and the corresponding basic size. It is a positive quantity when the lower limit of size is greater than the basic size and negative quantity when the lower limit of size is less than the basic size.

Upper Deviation: It is the algebraic difference between the upper (maximum) limit of size and the corresponding basic size. It is a positive quantity when the maximum limit of size is greater than the basic size and a negative quantity when the upper limit of size is less than the basic size as shown in Fig. 9.6.4 It is denoted by ‘ES’ for hole and ‘es’ for a shaft.

Deviation: Deviation is the algebraic difference between the size (actual, maximum etc.) and the corresponding basic size.

Zero line: It is a straight line drawn horizontally to represent the basic size. In the graphical representation of limits and fits, all the deviations are shown with respect to the zero line (datum line). The positive deviations are shown above the zero line and negative deviations below as shown in Fig. 9.6.4

Actual Size: Actual size is the dimension as measured on a manufactured part.

Basic or Nominal Size: It is the standard size of a part with reference to which the limits of variation of a size are determined. It is referred to as a matter of convenience. The basic size is the same for the hole and its shaft. It is the designed size obtained by calculations for strength.

When an assembly is made of two parts, one is known as male-surface and the other mating part as a female (enveloping) surface. The male surface is called as ‘Shaft’ and the female surface as ‘Hole’.

Hole: The term hole refers not only to the diameter of circular hole but also to any internal dimension of a component.

Shaft: The term shaft refers not only to the diameter of a circular shaft but also to any external dimension of a component.

Terminology for Limits and Fits:

Fits: Fit may be defined as the degree of tightness or looseness between two mating parts to perform a definite function when they are assembled together.

(ii) Trained and highly skilled operators: (iii) Tight inspection and more precise testing and inspection devices. (iv) It needs more concentration of the operators, frequent checking and more time which slows down the rate of production. (v) Close supervision and control is essential.

Quick Refresher Guide

ES = EI + IT.

IT = ES – EI

es = ei + IT

Manufacturing Engg

Low limit size

High limit size

Basic size

High limit size

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Fig. 9.6.5 Upper deviation as fundamental deviation

Low limit size

The fundamental deviation for the hole is denoted by capital letters A, B, C, …. 2 C and the same for shaft is denoted by small letters a, b, c, …… zc etc. as explained later. Zero line Fundamental Deviation Tolerance Zone (Upper Deviation) Tolerance

Zero line

Tolerance

Fig. 9.6.4 Lower deviation as fundamental deviation

Fundamental Deviation (Lower Deviation)

Tolerance Zone

Fundamental Deviation. Fundamental deviation is that one of the two deviation (either the upper or the lower) which is the nearest to the zero line for either hole or a shaft. It fixes the position of the ‘Tolerance Zone’ in relation to the zero line as shown in Fig. 9.6.5

For hole

i.e.,

IT = es – ei (upper deviation – lower deviation)

Quick Refresher Guide

Manufacturing Engg

Slide Fit Easy slide Fit Running Fit Slack running Fit Loose running Fit

(a) Push fit (b) Wringing Fit

Transition Fit (a) Force Fit (b) Tight Fit (c) Shrink fit

Interference Fit

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(a) (b) (c) (d) (e)

Clearance Fit

Fits

These are further classified in the following manner:

(1) Clearance Fit (2) Transition Fit and, (3) Interference Fit.

On the basis of positive, zero and negative values of Clearance, there are three basic types of fits:

Classification of Fits

Accordingly, a fit may result either in a movable joint or a fixed joint. For example, a shaft running in a bearing can move in relation to it and thus forms a movable joint, whereas, a pulley mounted on the shaft forms a fixed joint.

It is the relationship between the two mating parts with respect to the amount of loose or tightness which is present when they are assembled together.

Fit: Fit may be defined as a degree of tightness or looseness between two mating parts to perform a definite function when they are assembled together.

Standard Tolerance Unit: A unit which is a function of basic size and which is common to the formula defining the different grades of tolerances. It is denoted by the letter ‘i’ and expressed in terms of microns. It serves as a basis for determining the standard tolerance (IT) of the system.

Tolerance grade: The tolerance grade is an indication of the degree of accuracy of manufacture and it is designated by the letters ‘IT’ followed by a number, where ‘IT’ stands for “International Tolerance grade”. Tolerance grades are IT0, IT0, IT1, up to IT16, the larger the number larger will be the tolerance.

Tolerance Zone: It is the zone bounded by two limits of size of a part in the graphical representation of tolerance. It is defined by its magnitude and by its position in relation to the zero line as shown in Fig. 9.6.5

Basic Hole: Basic hole is the hole whose lower deviation is zero i.e., its low limit is the same as the basic size. It is denoted by a letter ‘H’.

Basic shaft: Basic shaft is the shaft whose upper deviation is zero. Thus the upper limit of the basic shaft is the same as the basic size. It is denoted by letter ‘h’.

From Fig. 9.6.5 it is clear that when the tolerance zone is above the zero line, lower deviation is the fundamental deviation. While, when the tolerance zone is below the zero line, upper deviation is the fundamental deviation.

Quick Refresher Guide

Quick Refresher Guide

Manufacturing Engg

Shaft Tolerance

Zero line

Hole Tolerance

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Interference Fit: In this type of fit the minimum permissible diameter of the shaft is larger than the maximum allowable diameter of the hole. Thus the shaft and the hole members are intended to be attached permanently and used as a solid component.

1. Slide Fit: This type of fit has a very small clearance, the minimum clearance being zero. Sliding fits are employed when the mating parts are required to move slowly in relation to each other e.g., tailstock spindle of lathe, feed movement of the spindle drill in a drilling machine, sliding change gears in quick change gear box of a centre lathe etc. 2. Easy Slide Fit: This type of fit provides for a small guaranteed clearance. It serves to ensure alignment between the shaft and hole. It is applicable for slow and non-regular motion, for example, spindle of lathe and dividing heads, piston and side valves, spigots etc. 3. Running Fit: Running fit is obtained when there is an appreciable clearance between the mating parts. The clearance provides a sufficient space for a lubrication film between mating friction surfaces. It is employed for rotation at moderate speed, e.g., grear box bearings, shaft pulleys, crank shafts in their main bearings etc. 4. Slack running Fit: It is obtained when there is a considerable clearance between the mating parts. This type of fit may be required as compensation for mounting errors e.g., arm shaft of I.C. engine, shaft of certifigual pump etc. 5. Loose running Fit: Loose running fit is employed for rotation at very high speed, e.g., idle pulley on their shaft such as that used in quick return mechanism of a planar.

Minimum Clearance: It is the difference between the maximum size of shaft and minimum size of hole.

Maximum Clearance: It is the difference between the minimum size of shaft and maximum size of hole.

Clearance fit exists when the shaft and the hole are at their maximum metal conditions, The tolerance zone of the hole is above that of the shaft as shown in Fig. 9.6.6

Fig. 9.6.6 Clearance Fit

minimum clearance

Clearance Fit: In this type of fit shaft is always smaller then the hole i.e., the largest permissible shaft diameter is smaller than the diameter of the smallest hole. So that the shaft can rotate or slide through with different degrees of freedom according to the purpose of mating part.

Max. Clearance

Manufacturing Engg

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Fig. 9.6.8 Transition fit

Hole

Shaft

Transition Fit: Transition fit lies mid way between clearance and interference fit. In this type the size limits of mating parts (shaft and hole) are so selected that either clearance or indifference may occur depending upon the actual sizes of the parts. Push fit and wringing fit are the examples of this type of fit.

1. Force Fit: Force fits are employed when the mating parts are not required to be disassembled during their total service life. In this case the interference is quite appreciable and, therefore, assembly is obtained only when high pressure is applied. This fit, thus, offers a permanent type of assembly, e.g., gears on the shaft of a concrete mixture, forging machine etc. 2. Tight Fit: It provides less interference than force fit. Tight fits are employed for mating parts that may be replaced while overhauling of the machine, for example, stepped pulleys on the drive shaft of a conveyor, cylindrical grinding machine etc. 3. Heavy force and Shrink Fit: It refers to maximum negative allowance. Hence considerable force is necessary for the assembly. The fitting of the frame on the rim can also be obtained first by heating the frame and then rapidly cooling it in its position.

Elastic strains developed on the mating surfaces during the process of assembly prevent relative movement of the mating parts. For example, steel tyres on railway car wheels, gears on intermediate shafts of trucks, bearing in the gear of a lathe head stock, drill bush in jig plate, cylinder liner in block, steel rings on a wooden bullock cart wheels etc.

Fig. 9.6.9

Allowance

Allowance (+) (clearance)

Allowance may be positive (clearance) or negative (interference).

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4. It has absolute value without sign.

3. The tolerance is provided on a Allowance is to be provided on the dimension of a part as it is not possible dimension of mating parts to obtain desired to make part to exact specified type of fit. dimension.

2. It is the difference between higher and It is the intentional difference between the lower limits of a dimension of a part. lower limit of hole and higher limit of shaft.

1. It is the permissible variation in It is the prescribed difference between the dimension of a part (either a hole or a dimensions of two mating parts (hole and shaft). shaft).

Tolerance

Difference between Tolerance and Allowance

Allowance (−ve) (Interference)

The allowance may be positive or negative. The positive allowance is called clearance and the negative allowance is called interference.

Fig. 9.6.7 Interference fit

Allowance is the prescribed difference between the dimensions of two mating parts for any type of fit.

Allowance

1. Wringing Fit: A wringing fit provides either zero interference or a clearance. These are used where parts can be replaced without difficulty during minor repairs. 2. Push Fit: The fit provides small clearance. It is employed for parts that must be disassembled during operation of a machine for example, change gears, slip bushing etc.

In this type of fit the tolerance zones of the hole and shaft overlap completely or in part.

Quick Refresher Guide

It is the intentional difference between the lower limit of hole and higher limit of the shaft.

Manufacturing Engg

Holee

Shaft

Interference

Quick Refresher Guide

Manufacturing Engg

(b) Transition Fit

Zero line

(c) Interference Fit

Hole basis system is preferred in mass production, because it is convenient and less costly to make a hole of correct size due to availability of standard drills and reamers. It is much more easy to vary the shaft sizes according to the fit required. It requires less amount of capital and storage space for tools needed to produce shafts of different sizes. Gauging of shafts can be easily and conveniently done with adjustable gap gauges.

3.

4.

5.

6.

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Gauges differ from measuring instruments in the following respects :

Gauges are easy to employ and can be used in many cases by unskilled operators. For checking the component with a gauge it is not necessary to make any calculations or to determine the actual dimension of the part, the time involved for checking/inspection is thus considerably reduced. For these reasons they find wide application in engineering particularly for mass production.

Gauges are scaleless inspection tools at rigid design, which are used to check the dimensions of manufactured parts. They also check the form and relative positions of the surfaces of parts. They do not determine (measure) the actual size or dimensions of part. They are only used to determine whether the inspected part has been made within the specified limits. These gauges consist of two sizes corresponding to their maximum and minimum limits. For gauging hole limits plug gauges and for gauging shafts snap gauges are used.

Being internal measurement, gauging of holes cannot be easily and conveniently done.

It needs large amounts of capital and storage space for large number of tools required to produce holes of different sizes.

It is rather difficult to vary the hole size according to the fit required.

This system is not suitable for mass production because it is inconvenient, time consuming and costly to make a shaft of correct size.

Limits on the shaft are kept constant and those on the hole are varied to have necessary fit.

Size of shaft whose upper deviation is zero (hshaft) is assumed as assumed as basic size.

Shaft Basis System

(i) No adjustment is required in their use (ii) They are not general purpose instruments but are specially made for some particular component, which is to be produced in sufficiently large quantities.

Zero line

Limits on the hole are kept constant and those of shaft are varied to obtain desired type of fit.

2.

9.6.2 Gauges and Gauge Design

Size of hole whose lower deviation is zero (Hhole) is assumed as the basic size.

1.

Hole Basis System

The hole basis system is most commonly used because it is more convenient to make correct holes of fixed sizes, since the standard drills, taps, reamors and broaches etc. are available for

(a) Clearance Fit (b) Transition Fit (c) Interference Fit Fig. 9.6.11 Shaft Basis System (Upper deviation of shaft is zero)

Shaft tolerance

Hole Tolerance

In this system the upper deviation (fundamental deviation) of shaft is zero i.e., the high limit of shaft is the same as basic size and the various fits are obtained by varying the low limit of shaft and both the limits of hole.

Shaft basis System: In the shaft basis system the shaft is kept constant and the sizes of the hole are varied to give various types of fits.

Fig. 9.6.10 Hole Basis System (fundamental deviation of hole is zero)

(a) Clearance Fit

Shaft tolerance

Hole tolerance

In this system lower deviation of the hole is zero i.e., the low limit of hole is the same as basic size. The high limit of hole and the two limits of size for the shaft are then varied to give the desired type of fit, as shown in Fig. 9.6.10

Hole basis System. In the hole basis system the hole is kept constant and the shaft sizes are varied to give the various types of fits.

(1) Hole basis system. (2) Shaft basis system.

Difference between ‘Hole Basis’ and ‘Shaft Basis’ Systems

There are two systems of fit for obtaining clearance, interference or transition fit. These are:

producing holes and their sizes are not adjustable. On the other hand size of shaft produced by turning, grinding etc. can be very easily varied.

Quick Refresher Guide

Shaft basis system is used when the ground bars or drawn bars are readily available. These bars do not require further machining and fits are obtained by varying the sizes of hole.

Manufacturing Engg

A limit and fit system is the system of series of standard allowances to suit specific range of basic size. If these standard allowances are selected properly and assigned to mating parts ensures specific classes of fit.

Systems of Obtaining Different Types of Fits

Quick Refresher Guide

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While checking, each of these two gauges is offered in turn to the work. A part is considered to be good, if the GO gauge passes through or over the work and NO-GO gauge fails to pass under the action of its own weight. This indicates that the actual dimension of the part is within the specified tolerance. If both the gauges fail to pass, it indicates that hole is under size or shaft is over size. If both the gauges pass, it means that the hole is over size or the shaft is under size.

The difference between the sizes of these two gauges is equal to the tolerance on the workpiece. GO gauges check the Maximum Metal Limit (MML) and NO-GO gauge checks the Minimum Metal Limit (LML). In the case of a hole, maximum metal limit is when the hole is as small as possible, that is, it is the low limit of size. In case of hole, therefore, GO gauge corresponds to the low limit of size, while NO-GO gauge corresponds to high limit of size. For a shaft, the maximum metal limit is when the shaft is on the high limit of size. Thus, in case of a shaft GO gauge corresponds to the high limit of size and NO-GO gauge corresponds to the low limit size.

Limit Gauges: Limit gauges are very widely used in industries. As there are two permissible limits of the dimension of a part, high and low, two gauges are needed to check each dimension of the part, one corresponding the low limit of size and other to the high limit of size of that dimension. These are known as GO and NO-GO gauges.

Standard Gauges: If a gauge is made as an exact copy of the mating part of the component to be checked, it is called as standard gauge. For example, if a bushing is to be made which is to mate with a shaft. Shaft is the mating part. Then, the bushing is checked by a gauge which is a copy of the mating part in form of its surface and size. A standard gauge cannot be used to check an interference fit. It has limited applications.

(1) According to their type: (a) Standard gauges (b) Limit gauges (2) According to purposes: (a) Workshop gauge (b) Inspection gauge (c) Reference or master gauge. (3) According to the form of the tested surface: (a) Plug gauge (b) Snap, Gap or Ring gauge. (4) According to their design: (a) Single limit and double limit gauges (b) Single-ended and double-ended gauges (c) Fixed and adjustable gauges.

Plain Gauges: Plain gauges are used for checking plain, that is, unthreaded holes and shafts. They are classified in the following ways.

(iii) They give quick results about the conformance or non-conformance of the part with the specified tolerances.

Manufacturing Engg

Tolerance

HL

HOLE

GO

Fig. 9.6.12 Limit Gauges

GO √

NOT √

Quick Refresher Guide

HL

NO GO

(b) For Shaft

Tolerance

Manufacturing Engg

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(1) GO gauges should be designed to check the maximum material limit, while the NO-GO gauges should be designed to check the minimum material limit.

It states that

Taylor’s Principle of Gauge Design

Gauges can also be made from steel, special wear resisting material, like hard chrome plated surfaces and tungsten carbide, Invar etc. Glass gauges were used during World War. Chromium plating makes the gauge corrosion and wear resistant. Also, the size of worn gauging surface can be increased by this method. The gauge surfaces can also be plated to provide hardness, toughness and stainless properties.

A good quality high carbon steel is usually used for gauge manufacture. Suitable heat treatment can produce a high degree of hardness coupled with stability. High carbon steel is relatively inexpensive, it can be readily machined and brought to a high degree of accuracy and surface finish.

(a) Hardness to resist wear. (b) Stability to ensure that its size and shape will not change over a period of time. (c) Corrosion resistance, (d) Machinability to enable it to be machined easily into the required shaft and to the required degree of accuracy. (e) Low coefficient of linear expansion to avoid effect of temperature. (f) The parts of the gauge which are to be held in the hand should have low thermal conductivity.

The material used for manufacturing the gauges must fulfil the following requirements, either by virtue of its own properties, or by a heat treatment process.

Material for Gauges

(a) For Hole

LL

Quick Refresher Guide

LL

Manufacturing Engg

NO GO

Max. Limit

Fig. 9.6.13

GO Mini. Limit

Tolerance Max. Zone Limit NO GO

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For example, suppose the bush to be inspected has a curved axis and a short ‘GO’ plug gauge is used to check it. The short plug gauge will pass through all the curves of the bent busing. This will lead to a wrong result that the workpiece (hole) is within the prescribed limits. Actually, such a busing with a curved hole will not mate properly with its mating part and thus defective. A GO plug gauge with adequate length will not pass through a curved busing and the error will be detected. A long plug gauge will thus check the cylindrical surface not in one direction, but in a number of sections simultaneously. The length of the ‘GO’ plug gauge should not be less than 1.5 times the diameter of the hole to be checked.

According to this rule, GO plug gauge should have a full circular section and be of full length of the hole it has to check. This ensures that any lack of straightness, or roundness of the hole will prevent the entry of full length GO plug gauge. If this condition is not full filled, the inspection of the part with the gauge may give wrong results.

(2) ‘GO’ gauges should check all the related dimensions (roundness, size, location etc). Simultaneously whereas ‘NO-GO’ gauge should check only one element of the dimension at a time.

The difference in size between the GO and NO GO plug gauges, as well as the difference in size between GO and NO-GO Snap gauges is approximately equal to the tolerance of the tested hole or shaft in case of standard gauges.

Similarly, the ‘GO Snap gauge’ on the other hand corresponds to the high limit of shaft, while ‘NO-GO Snap’ gauge corresponds to the low limit of shaft.

GO

Tolerance Zone

Now, the plug gauges are used to check the hole, therefore the size of the GO plug gauge should correspond to the low limit of hole, while that of NO-GO plug gauge should corresponds to the high limit of hole.

Quick Refresher Guide

Manufacturing Engg

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Linear measurement applies to measurement of lengths, diameters, heights and thicknesses including external and internal measurements. The line measuring instruments have series of accurately spaced lines marked on them, e.g. scale. The dimension to be measured is aligned

9.6.3 Linear Measurement

The measuring surfaces of GO gauges rub constantly against the surfaces of the work pieces during checking. This results in wearing of the measuring surfaces of gauges. The GO gauge thus looses its initial size. The size of the GO plug gauge is reduced due to wear and that of ring or snap gauge is increased. Hence a wear allowance is provided to the gauges in the direction opposite to that of the wear. In case of GO plug gauge wear allowance is added, while in ring or snap gauge it is subtracted. For ‘NO-GO’ gauges wear allowance is not provided as they are not subjected to much wear as GO gauges. Wear allowance is usually taken as 50% of work tolerance. Wear allowance is applied to a normal GO gauge diameter before gauge tolerance is applied.

Wear Allowance

Now suppose the hole to be checked has an oval shape. While checking it with the cylindrical ‘NO-GO’ gauge the hole under inspection will overlap (hatched portion) the plug and thus will not enter the hole. This will again lead to wrong conclusion that the part is within the prescribed limits. It will be therefore more appropriate to make the ‘NO-GO’ gauge in the form of a pin as shown in Fig. 9.6.15

Fig. 9.6.15 Checking an oval shape

NO-GO Plug Gauge

Hole

GO Plug Gauge Fig. 9.6.14 Checking a bush with curved axis

Bush

Quick Refresher Guide

Manufacturing Engg

Hold the calliper gently and near the joint Hold it square to the work Apply only light gauging pressure Handle it gently to avoid disturbing the setting for accurate measurement.

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(i) (ii) (iii) (iv)

The accurate use of callipers depends upon the sense of feel that can only be acquired by practice. While using callipers the following rules should be followed:

To measure the diameter of a circular part it is essential that the measurement is made along the largest distance or true diameter. The steel rule alone is not a convenient method of measuring directly the size of the circular part. A calliper is used to transfer the distance between the faces of a component to a scale or micrometer. It thus converts an end measurement situation to the line system of the rule.

Callipers

Steel rule is most commonly used in workshop for measuring components of limited accuracy. The marks on a good class rule vary from 0.12 mm to 0.18 mm wide, so that we cannot expect to obtain a degree of accuracy much closer than within 0.012 mm. The quickness and ease with which it can be used and its low cost, makes it a popular and widely used measuring device.

Steel rule is a line measuring instrument. It is a part replica of the international prototype meter. It compares an unknown length to be measured with the previously calibrated length. It is made of hardened steel or stainless steel having series of equally spaced line engraved on it.

Steel Rule

(i) Non-precision instruments such as steel rule, calliper etc. (ii) Precision measuring instruments, such as vernier instruments, micrometers, dial gauges etc.

The graduated instruments include rules, vernier callipers, vernier height gauges, vernier depth gauges, micrometers, dial indicator etc. The non-graduated instruments include callipers, trammels, telescopic gauges, surface gauges, straight edges, wire gauges, screw pitch gauges, radius gauges, thickness gauges, slip gauges etc. They can also be classified as :

(i) Graduated (ii) Non-graduated.

The direct measuring instruments are of two types:

(i) Direct measuring instruments (ii) Indirect measuring instruments

The instruments used for linear measurements can be classified as:

with the graduations of the scale. Linear measuring instruments are designed either for line measurements or end measurements. In end measuring instruments, the measurement is taken between two end surfaces as in micrometers, slip gauges etc.

Quick Refresher Guide

Firm joint (Fixed joint) callipers Spring type callipers.

Manufacturing Engg

A reference or datum surface for testing flatness of surfaces. Reference surfaces for all other measuring instruments having flat bases e.g., for mounting V-blocks, angle plates, sine bars, height gauges, dial gauges, comparators etc.

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Micrometer The accuracy of the vernier calliper is 0.02 mm ; most engineering precision work has to be measured to a much greater accuracy than this, especially to achieve interchangeability of

Least count (L.C.) is the difference between the value of main scale division and vernier scale division. Thus least count of a vernier instrument = Value of the smallest division on the main scale – The value of the smallest division on the vernier scale.

Least Count: Vernier instruments have two scales, main scale and the vernier scale. The main scale is fixed and the vernier scale slides over the main scale. When zero on the main scale coincides with the zero on the vernier scale, the vernier scale has one more division than that of the main scale with which it coincides. So, the value of a division on vernier scale is slightly smaller than the value of a division on the main scale. This difference is the least count.

The vernier callipers consists of two scale: one is fixed and the other is movable. The fixed scale, called main scale is calibrated on L-shaped frame and carries a fixed jaw. The movable scale, called vernier scale slides over the main scale and carries a movable jaw. The movable jaw as well as the fixed jaw carries measuring tip. When the two jaws are closed the zero of the venier scale coincides with the zero of the main scale. For precise setting of the movable jaw an adjustment screw is provided. Also, an arrangement is provided to lock the sliding scale on the fixed main scale.

Vernier Callipers

The mass production which is a characteristic of modern engineering manufacture makes it necessary to manufacture component part with close dimensional tolerance to make them inter changeable. Interchangeability can be achieved only by precision dimensional control of the parts being manufactured. Thus, to measure the dimensions of the part with close accuracy precision instruments play an important role.

Precision Linear Measurements

Surface plates are massive and highly rigid in design. They have truly flat level planes. They are generally made up of C.I. free from blow holes, inclusions and other surface defects and are heat treated to relieve internal stresses.

(i) (ii)

Surface plate forms the basis of measurement. They are extensively used in workshops and metrological laboratories where inspection is carried out. They are used as:

Surface plate

(i) (ii)

Callipers can be classified as:

Quick Refresher Guide

Manufacturing Engg

. By reducing the pitch of the screw thread or by

Pitch of the spindle screw Number of divisions on the spindle

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There are many instruments which can be used for measuring the angles. The selection of an instrument to be used for angular measurement depends upon the component and the accuracy of measurement required. For example, the ordinary bevel protractor with Vernier scale can read to 2 minutes accuracy and optical protractor is accurate to 2 minutes. These are usually not adequate for metrological work and for high precision work to within a few seconds. To

Instruments for Angular Measurements

Angular measurements are frequently necessary for the manufacture of interchangeable parts. The ships and aeroplanes can navigate confidently without the help of the sight of the land, only because of precise angular measurement. Precise angular measuring devices can be used in astronomy to determine the relation of the stars and their approximate distances.

9.6.4 Angular Measurement

L. C. of micrometer =

Least count is the minimum distance which can be measured accurately by the instrument. The micrometer has a screw of 0.5 mm pitch, with a spindle graduated in 50 divisions to provide a . direct reading of = = 0.01 mm. Least count of a micrometer is thus, the value of one division on a spindle, which is connected to the screw.

Least Count of Micrometer

increasing the number of divisions on the circumference of screw, the length value of one circumferential division can be reduced and accuracy of measurement can be increased considerably.

measured by such arrangement will be

Micrometers work on the principle of screw and nut. We know that when a screw is turned through nut through one revolution, it advances by one pitch distance i.e., one rotation of screw corresponds to a linear movement of a distance equal to pitch of the thread. If the circumference of the screw is divided into number of equal parts say ‘n’, its rotation through one division will length. Thus, the minimum length that can be cause the screw to advance through

Principle of Micrometer

(a) Outside Micrometer (b) Inside Micrometer (c) Screw Thread Micrometer and, (d) Depth gauge Micrometer.

component parts. To achieve this greater precision, measuring equipment of a greater accuracy and sensitivity must be used. Micrometer is one of the most common and most popular forms of measuring instrument for precise measurement with 0.01 mm accuracy. Micrometers with 0.001 mm accuracy are also available. Micrometers can be classified as :

Quick Refresher Guide

Manufacturing Engg

Vernier bevel protractor Optical bevel protractor Universal bevel protractor Sine bar Angle gauges Clinometer Angle dekkor Auto collimator, etc.

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– – – – – – – –

The following instruments are generally are generally used for angular measurements

obtain these fine accuracies for high precision work, use is made of sine bar, angle gauges, and optical instruments. The spirit level and the dividing head are also employed.

Quick Refresher Guide

Manufacturing Engg

Remarkable flexibility for manufacturing diverse components in the same setup by easy and quick manipulation of software. High rates of production with consistent high quality Uninterrupted production with negligible supervision. Economical production even where product demand is only moderate in volume Drastic reduction of lead times where drastic changes in design are called for. Integrating and fine tuning of all factory functions

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CAD/CAM (Computer Aided Design/Computer Aided Manufacturing) technology was initiated in aerospace engineering but is now applicable in most of the fields. It can be defined as the use of computers to translate a product’s specific requirements into final physical product. With this system, a product is designed, produced and inspected in one automatic process. CAD/CAM plays a key role in areas such as design, analysis, production, planning, detailing, documentation,

9.7.2 CAD/CAM Technology

x x x x x

x

Advantages of Computer Integrated Manufacturing (CIM)

Computer aided inspection and reporting provides a feedback loop.

Computer Aided Inspection and Reporting

Computer aided manufacturing (CAM) helps indentify manufacturing problems and opportunities. Distributed intelligence in the form of microprocessors could be used to control machines and material handling and collect the data on current shop conditions.

Manufacturing

Computer aided process planning helps to establish optimum manufacture routines and processing steps, sequences and schedules so that the process is optimum.

Manufacturing Planning

Product design for which interactive computer aided design (CAD) system allows the drawing and analysis tasks to be performed. These computer graphics systems are very useful to get data out of designer’s mind into a presentable form and enable analysis in fraction of time required otherwise and with greater accuracy. Design process is speeded up considerably.

Product Design

Manufacturing by CIM technique could be separated into four blocks:

Computer Integrated Manufacturing comprises a combination of software and hardware for product design, production planning, production control, and production equipment and production process. It is an attempt to integrate the many diverse elements of discrete parts manufacturing into one continuous process like stream.

9.7.1 Computer Integrated Manufacturing (CIM)

Part 9.7: Computer Integrated Manufacturing (CIM)

Quick Refresher Guide

Manufacturing Engg

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Manufacture: With CAD/CAM system the complete NC part programming process can be carried out interactively. Source programs in languages such as APT can be produced.

Design and Design Analysis: CAD system would be best suited for office drawings where frequent modifications are required on drawing and several parts to be repeated. Once a drawing is entered in CAD system, later modifications can be done quickly, and detailed drawings can be prepared quickly from a general arrangement, NC tapes can be produced. Also, it is very convenient to calculate properties like weight, volume, centre of gravity, moment of inertia etc because 3D models can be easily produced.

Application Areas of CAD/CAM

NC part programming, tool fabrication, assembly, jig and fixture design, quality control and testing.

Quick Refresher Guide

Part 10.1: Production, Planning and Control

Plant location

Plant layout or expansion

New product planning

o

o

o

Production planning

Capital and cash planning

Inventory planning

o

o

o

Overtime decision

Job scheduling

Machine maintenance

Inventory planning

o

o

o

o

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1. Extrapolation

Quantitative Methods of Forecasting:

Purchasing

o

Short range forecast is generally for less than 1 year:

¾ Short range forecast

Sales planning

o

Medium range forecast is generally from 1 to 5 years. The medium range forecasting is useful in following areas:

¾ Medium range forecast

Capital planning

o

Long range forecast consists of time period of more than 5 years. The long range forecasting is useful in following areas:

¾ Long range forecast

Types of Forecasting:

The main purpose of forecasting is to estimate the occurrence, timing, or magnitude of future events. Once, the reliable forecast for the demand is available, a good planning of activities is needed to meet the future demand. Forecasting thus provides the input to the planning and scheduling process.

10.1.1 Forecasting

Industrial Engineering

Part – 10: Industrial Engineering

Quick Refresher Guide

Industrial Engineering

=

1 n D

=

WD

W =1

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Ft = α Dt-1 + α(1- α) Dt-2+ α(1- α)2 Dt-3+ α(1- α)3 Dt-3………..

In exponential smoothing method of forecasting, the weightage of data diminishes exponentially as the data become older. In this method all past data is considered. The weightage of every previous data decreases by (1-α), where α is called as exponential smoothing constant.

4. Exponential smoothing

It may be noted that when more weight is given to the recent values, the forecast is nearer to likely trend. Weighted moving average is advantageous as compared to simple moving average as it is able to give more importance to recent data.

and

Wi= Relative weight of data for ith period

Where:

F

In this method, more weightage is given to the relatively newer data. The forecast is the weighted average of data.

3. Weighted moving average

n= Number of periods included in each average

Di = Actual demand for ith period

where:

F

Forecast for (t+1)th period is given by:

In this method, mean of only a specified number of consecutive data which are most recent values in series.

2. Simple moving average

Extrapolation is one of the easiest ways to forecast. In this method, based on the past few values of production capacity, next value may be extrapolated on a graph paper.

Quick Refresher Guide

period

α= Smoothing constant (0≤α ≤1)

Dt= Actual demand for

Ith

Di= One period ahead forecast made at time t

Industrial Engineering

t= time period

dt= Forecasted demand for time period t

Ft= Forecast for time period t

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Where a, b and c are parameters

Ft=a +bt+ct2

4. Quadratic Forecaster

Where a, b, u and v are parameters and N is periodicity

Ft= a+ bt+ uCos (2∏/N)t + vSin (2∏/N)t

3. Cyclic Forecaster with Growth

Where a, u and v are parameters and N is periodicity

Ft= a+ uCos (2∏/N)t + vSin (2∏/N)t

2. Cyclic Forecaster

Where a and b are parameters

Ft= a+bt

1. Linear Forecaster

Let:

Statistical forecasting is based on the past data. We evaluate the expected error for the statistical technique of forecasting. Some common regression functions are as follows:

Statistical Forecasting:

x Higher value of α gives more weightage to past data as compared to smaller value.

x Higher value of α gives more robust forecast and response more quickly to changes.

x Smaller is the value of α, more is the smoothing effect in forecast.

Comments regarding Smoothing constant α:

Where:

Quick Refresher Guide

Industrial Engineering

∑

|D − F | n

∑

(D − F ) n

∑

(D − F ) n

Work station design

Process design

Product development and design

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o

o

o

¾ Preplanning

Phases of PPC:

Production planning and control is one of the most important areas of industrial management. This aims at achieving the efficient utilization of resources in any organization through planning, coordination and control of production activities.

10.1.2 Production Planning and Control

BIAS =

BIAS is measure of the average of all errors in the forecast. Its intern. A positive bias indicates underestimation while a negative BIAS indicates over-estimation

3. BIAS

MSE =

The average of square of all errors in the forecast is termed as MSE. Its interpretation is same as MAD.

2. Mean Sum of Square Error (MSE)

n= number of periods considered for calculating the error

Dt= Forecasted demand for period t

Ft= Actual demand for period t

Where:

MAD =

This is calculated as the average of absolute value of difference between actual and forecasted value.

1. Mean Absolute Deviation (MAD)

Many factors affect the trend in data therefore it is impossible to obtain an exact right forecast. Below are the tools that are used to determine the error in the forecasted value.

Accuracy of Forecast:

Quick Refresher Guide

Factory layout and location

Method

Machine

Men

o

o

o

Conduct the analysis to determine the material needed for the product.

Determine the required manufacturing operations and their sequence.

Determine the lot size.

Determine the scrap

Estimate product cost.

Prepare different forms of production control.

o

o

o

o

o

o

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Scheduling involves fixing the priorities for different jobs and deciding the starting and finishing time of each job. Main purpose of scheduling is to prepare a time-table indicating the time and rate of production, as indicated by starting and finishing time of each activity. Scheduling will be discussed in detail in next section.

2. Scheduling

Conduct an analysis of the product to determine the part/ component/ subassemblies required to be produced.

o

Routing is the process of deciding sequence of operations (route) to be performed during production process. The main objective of routing is the selection of best and cheapest way to perform a job. Procedure for routing is as follows:

Distribute machine loading and schedule charts

Issues authorization to start work at the pre-determined date and time.

o

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Gantt chart is a graphical tool for representing a production schedule. Normally, Gantt chart consists of two axis. On X-axis, time is represented and on Y-axis various activities or tasks, machine centres and facilities are represented.

¾ Gantt Chart

Scheduling Methods:

Scheduling is used to allocate resources over time to accomplish specific tasks. It should take account of technical requirement of task, available capacity and forecasted demand. The output plan should be translated into operations, timing and schedule on the shop floor. Detailed scheduling encompasses the formation of starting and finishing time of all jobs at each operational facility.

10.1.3 Scheduling

Report any production related problems.

Co-ordinating with other departments.

o o

Expediting corrective measures.

o

Physical control of work-in-progress through checking

Identification of delays, disruptions or discrepancies.

Watching progress of production process.

This is the final stage of production planning and control. It is used for ensuring that the work is carried out as per plans and due dates are met. The main objective is to arrest deviations from the plan. Another objective is to integrate different production activities to meet the production target. The following activities are done ine expediting phase:

o

Dispatching

o

Industrial Engineering

Ensuring right material, tools, parts, jigs and fixtures are available.

Collecting and issuing work centre

4. Expediting

o

o

o

o

1. Routing

Evaluation

o

Quick Refresher Guide

Dispatching is the selection and sequencing of available jobs to be run at the individual workstations and assignments of those jobs to workers. Functions of dispatching are as under.

3. Dispatching

o

Expedition

o

Industrial Engineering

Production Planning and Control Steps

Inspection

o

¾ Control

Material

o

¾ Planning Different Resources

o

Quick Refresher Guide

Industrial Engineering

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In the sections to follow, we will discuss some of the commonly used inventory models. For these models, following notations are used : Q = Quantity ordered each time Q∗ = Optimum quantity of inventory ordered for minimum total cost D = Annual demand of parts (in unit) C = Cost of inventory per unit item C = Carrying cost per unit of individual item, expressed as a percentage of unit cost

10.2.2 Variables in Inventory Models:

Shortage cost is the cost incurred due to a stock out situation. There may be lost in sales due to customer dropping/ postponing the idea of purchase or customer may go to some other producer.

¾ Shortage cost / Stock out cost

Holding costs are incurred due to maintaining an inventory level in the organization. It is due to the interest on the held up capital in inventory, insurance cost, rent, obsolescence, deterioration etc.

¾ Holding cost / Carrying cost

Ordering cost is the cost associated with the placement of an order for the acquisition of inventories. The expenses incurred in the purchase department are its main constituents. Setup cost is used when inventory is made within the organization.

¾ Ordering cost/ Setup cost

Unit cost of inventory is the price which is paid to the supplier for procuring one unit of the inventory. For parts manufactured in house, cost of inventory is the direct manufacturing cost.

¾ Unit cost of inventory

In inventory models various cost elements are considered. Generally, these costs are dependent upon the timing and quantity to order. The different types of costs relevant to inventory models are explained under:

Inventory Costs:

3. Inventory ensures smooth flow of production process.

2. Inventory guards against stock out situations.

1. Inventory is required to meet anticipated demand.

Inventory may be defined as any resource that has certain value and which can be used at a later time, when the demand of item will arise. Below are some functions of Inventory:

10.2.1 Inventory

Part 10.2: Inventory Control

Quick Refresher Guide

Industrial Engineering

Demand for the inventory is deterministic, i.e., it is known with certainty. Demand rate is constant and known beforehand. All orders are placed in single lot. No stock-out shortages or back orders are allowed. No quantity discount is allowed. Thus, purchase cost per unit is fixed. Lead time is constant and it is independent of demand. Inventory is controlled from one point of the system, i.e., in a stockroom or in a warehouse.

)

= C D

( / )

+

=O

…..(iii)

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For total cost minimization; + =0 or, DC −

(

What level of inventory should be ordered (i.e., Q∗ ), so that total cost will be minimum? To answer this, equation (ii) is differentiated with respect to Q and equated to zero. Second differential should be positive for cost minimization:

This is because the inventory level is uniformly decreasing from Q to zero (Fig. 10.2.1). Hence, from (i): + C …..(ii) TC = C

Number of orders to be placed in a year =

D Demand in a year = Q Quantity ordered each time Q Q + 0 = Average inventory carried during the year = 2 2

Let us further assume that lead time is zero; which means that the inventory is delivered instantaneously after the order is placed. Total cost during the year is the sum of the inventory carrying cost during year and total ordering cost. Thus, TC = (Ordering Cost) (Number or Orders placed in a year) + (Carrying cost per unit) (Average inventory level during year) ….. (i)

1. 2. 3. 4. 5. 6. 7.

Assumptions of the Model I

This is one of the oldest developments in material management. Ford Harris developed it in 1915 and later R.H. Wilson in 1943, popularized it among researchers and practitioners.

Model 1: Uniform Demand Rate, Infinite Production

10.2.3 Inventory Models:

C = Ordering, set – up or procurement cost per order R = Reorder point TC = Total annual costs T = Lead time C = Cost of shortages due to non-availability of inventory

Quick Refresher Guide

Q=

or,

=

(a positive quantity)

Industrial Engineering

Q

0

Cost

Lowest Total Cost

t

t

(EOQ)

Quantity (Q)

Carrying Cost ( )

t

Ordering Cost ( )

Fig. 10.2.2 Economic Order Quantity

∗

Total Cost (TC)

Fig. 10.2.1 EOQ model with uniform demand

t

Average Inventory Level(Q/2)

Inventory Ordered (Q)

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Q/2

Quantity (Q)

Q∗ =

For total inventory cost minimization, we have defined Q as Q∗ and we will call it as economic order quantity (EOQ).

(TC) = C D −

=

or,

Quick Refresher Guide

)∗

∗

1

1.25

1.0 ( /

∗

1.5 )

2.0

Fig.10.2.3 Sensitivity of EOQ model

0.5

Industrial Engineering

+

+

+

=

=

2C C D

. . . (iv)

=

=

=

) )∗

∗

+

∗

+ + Q

=

+

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Now, let us summarize the concept of EOQ. Economic Order Quantity (EOQ) is that size of order which is able to minimize the total cost of carrying inventory and cost of ordering for a given period under the assumption of known and certain demand.

If we increase or decrease the EOQ by twice, the increase in total cost is only 25% (Fig.10.2.3). Thus, total cost is not very sensitive in the vicinity of EOQ. The physical significance of this observation is quite important. If there is a slight error in deciding the EOQ, the total inventory cost is insignificantly affected.

( (

Sensitivity of EOQ model It is important to note that the total cost curve is quite flat near the EOQ zone (Fig. 2.2). Slight change in the value of Q is near the EOQ point (i.e., when Q ~ Q∗ ), the change in total cost is insignificant. Mathematically, dividing (ii) by (iv):

Operating Policy of Inventory Control Delivery is instantaneous. Therefore, reordering should be done just at the time inventory stock is zero. This policy has a pre-assumption that lead time is zero. If in case, lead time is known and constant, the order should be placed exactly ahead of lead time so that the instantaneous supply of Q∗ arrives when the stock depletes to zero.

=

=

(TC)∗ = C D

Minimum Total Cost, (TC) will be obtained by putting Q∗ as Q in equation (ii). Thus,

(

2 1.5

Quick Refresher Guide

∗

=

=

Industrial Engineering

∗

=

×

×

=

∗

=D

=

days

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d∗ = 365

Optimum Number of Days Supply ( ∗ ) The optimum number of days for which an order is to be made is sometimes required. Since annual demand (D) is for 365 days, therefore for each optimal order, the supply is for ∗ days. Therefore,

or, N ∗ =

Optimum Number of Orders ( ∗ ) The optimum number of orders per year is obtained by dividing annual demand by economic order quantity.

=

or, t ∗ =

Optimum Ordering Interval ( ∗) Since, EOQ = demand rate × Optimum ordering interval or, Q∗ = Dt ∗

Therefore, for high cost items (i.e., high value of C ), the average inventory level should be low.

or, Average Inventory is proportional to

or, Average Inventory,

As, Q∗ =

High Cost Item Inventory

Other Observations of Basic EOQ Model

Quick Refresher Guide

Industrial Engineering

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Optimize (i.e., minimize or maximize) objective function:

A LPP may be expressed as:

Problem:

¾ Formulation of Linear Programming Problem

The optimal solution of linear programming problem is that set of feasible solution, which satifies the objective of the problem (Maximizing or minimizing).

¾ Optimal Solution

A set of values of decision variables, which satisfies the constraint sets. There may be many feasible solutions to the linear programming problem.

¾ Feasible Solution

The restrictions that are expressed in form of an equation or inequality are termed as constraints.

¾ Constraint

The function that is to be maximized or minimized is called as objective function.

¾ Objective Function

The decision variable means the variable whose quantitative values are required to be found to maximize or minimize the objective function.

¾ Decision Variable

Below mentioned definitions are important for the linear programming problem.

Linear programming is a technique based on mathematical theory for specifying ways to use the limited resources or constraint of a system to obtain a particular objective such as highest profit, least cost and least time. The linear programming problem can be solved using Graphical method (max 2 variables) and Simplex method (any number of variables)

10.3.1 Linear Programming:

The operations research has many fields. Some of the important field will be covered in the coming sections.

Operations research, is a branch of industrial engineering that uses advanced analytical methods such as mathematical modeling, statistical analysis, and mathematical optimization to arrive at optimal or near-optimal solutions to complex decision-making problems. It is often concerned with determining the maximum (of profit, performance, or yield) or minimum (of loss, risk, or cost) of some real-world objective.

Part 10.3: Operations Research

Quick Refresher Guide

Z=∑

Cx

Industrial Engineering

−C x

Cx

−C x

Cx

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Its equivalent is:

5x + 3x = 24

For Example:

The equal to (=) sign in a constraint may be handled by adopting two constraint sets with ≥ and ≤ signs.

¾ How to deal with equal to sign

Minimize Z = ∑

is equivalent to

Maximize Z = ∑

Thus,

Similarly, a minimization problem may be transformed into a maximization problem by changing the sign of the decision coefficients.

Minimize Z = ∑

is equivalent to

Maximize Z = ∑

Thus,

The maximization objective function is equivalent to a minimization objective function except with changed sign.

¾ Conversion of a maximization problem into minimization problem

= is called as equal to

≥ is called as greater than or equal to

≤ is called as less than or equal to

(‘≤’ or ‘=’ or ‘≥’) means that either of the three notations is required.

j = decision variable number; j = 1, 2, . . . n.

i = Constraint number; i = 1, 2, . . . m.

b = Requirement or availability of ith constraints.

a = Exchange coefficient of jth variable in the ith constraint set.

C = Unit contribution to the jth decision variable in the objective function.

x = jth decision variable about which the decision maker is interested.

Here,

a x (‘≤’ or ‘=’ or ‘≥’) b : for all i; i = 1, 2, . . . . m

x ≥ 0; j = 1, 2, . . . n (non negativity constraints)

∑

Subject to the constraints:

Quick Refresher Guide

Industrial Engineering

∑

C X

. . . . (1)

X = 1; for all j; j = 1, 2, . . . n

. . . . (2)

X = 1; for all i; i = 1, 2, . . . n

. . . . (4)

. . . . (3)

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X ≥0

The non – negativity constraint is:

∑

Similarly for each operator, there may be only one assignment of job. For this, the constraint set is:

∑

Since one job (i) can be assigned to any one of the operators, we have following constraint set:

Minimize Z = ∑

1 If job i is assigned to operator j 0 Otherwise The objective is to minimize the total cost of assignment. If job 1 is assigned to operator 1, the cost is (C X ). Similarly, for job 1, operator 2 the cost is (C X ). The objective function is: X =

C = Unit cost for assigning job ‘i’ to operator ‘j’

j = Index for operators, j = 1, 2, . . . n

i = Index for job, i = 1, 2, . . . n

Let there be n jobs which are to be assigned to n operators so that one job is assigned to only one operator.

¾ Formulation of Assignment Problem

Assignment problem pertains to problem of assigning n jobs on n machines. This problem can be effectively used for any other problem in which n items (or persons) are to be assigned to other n items, so that each one of the first group is assigned to one distinct item from the second group.

Assignment Problem

The graphical method to solve LPP is useful when there are only two decision variables. This is because more than two coordinates are difficult to be represented on a graph paper. Simplex technique can handle any number of variables.

1. Graphical Method 2. Simplex Method

The linear programming problem may be solved by two methods:

and, 5x + 3x ≤ 24

5x + 3x ≥ 24

Quick Refresher Guide

Yes x Assignment of row with column of matrix

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The most important issue in the waiting line problem is to decide the best level of service that an organization should provide.

Waiting lines are one of the important phenomenon in daily life. It affects people who need service at different places. Some examples of waiting lines are railway reservation counters, telephone booth, doctor’s clinic, Aircraft landing at airports etc.

10.3.2 Queuing Models:

Fig. 10.3.33 Solution method of Assignment Problem

Revise the solution

No

Is the solution optimal?

Generate Opportunity Cost Matrix

Formulate the Assignment Matrix

Start

The assignment problem is solved in the following manner:

Find

Fig. 10.3.32 Mathematical Formulation of Assignment Problem

¾ Solution of Assignment Problem

Arrival Pattern at the system: The number of arrivals per unit time (rate of arrival) is estimated by Poisson’s distribution. The probability distribution of inter-arrival times, which is the time between two consecutive arrivals, may also be governed by a probability distribution.

Size of input source: The size of the input source can be considered as limited or infinite.

For x = 0, 1, 2, 3, . . . . . .

1/λ : Mean time between arrivals or inter-arrival time.

λ : Average arrival time.

x : Number of arrivals per unit time.

P(x) : Probability of x arrival.

!

Queue discipline: Queue discipline is the rule by which customers waiting in queue would receive service. Some examples are FIFO (First-In-First-Out), LIFO (Last-InFirst-Out) or SIRO (Service-In-Random-Order)

o

o

Single Channel Single Phase

Service Facility

Size: The queue may be considered to be limited when its length cannot exceed a certain number. It may be limited or infinite

Number of waiting lines o

o

Queue

P(t) = λe

It can be shown mathematically that the probability distribution of inter-arrival time is governed by the exponential distribution when the probability distribution of number of arrivals is Poisson distribution. The corresponding exponential distribution for interarrival time is given by:

where:

P(x) =

For a given arrival rate(λ), a discrete Poisson distribution is given by:

o

o

Arrival Characteristics

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x

x

x

= 1; for all ; = 1, 2, . . . n

∑

=∑

¾ Characteristics of queuing model

Minimize

= 1; for all ; = 1, 2, . . . n

Industrial Engineering

Subject to ∑

Quick Refresher Guide

The important aspect of the queuing problem is the arrival pattern. The arrival pattern is generally governed by probabilistic distribution. The organizations have to come to a compromise between a good service and less cost in running the service points.

Industrial Engineering

∑

Quick Refresher Guide

Multi Channel Single Phase

o

Industrial Engineering

Queue Configuration: The queue length is unrestricted and there is a single queue. Queue Discipline: Customers are patient. Service Discipline: First-Come-First-Serve (FCFS) Service Facility: There is one server, whose service times are distributed as per exponential distribution. Service is continuously provided without any prejudice or breakdown, and all service parameters are state independent.

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2. 3. 4. 5.

1. Arrival Process: The arrival is through infinite population with no control or restriction. Arrivals are random, independent and follow Poisson distribution. The arrival process is stationary and in single unit (rather than batches).

Let us understand the M/M/1 model first. Following set of assumptions is needed:

Arrival process Queue configuration Queue discipline Service discipline, and Service facility.

Queuing models may be formulated on the basis of some fundamental assumptions related to following five features:

(

)

=

(

)

(

)

+ =

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=

3. Average line length or expected number of units in the system, L = L + units being served

System Related Operating Characteristics

. . . (10.3.2)

. . . (10.3.1) 2. Average waiting time or expected time in queue, W = =

L =

1. Average line length or expected number of units in queue,

Queue Related Operating Characteristics

Following are the lists of operating characteristics, which may be derived for steady state situation and for ρ < 1:

w = Random variate for time spent in the system.

n = Number of units in the system

ρ = = Traffic intensity

μ = Mean service rate in units per period

λ = Mean arrival rate in units per period

Following are the lists of parameters:

e: Queue discipline

There are two major parameters in waiting line: arrival rate (λ) and service rate (μ). They follow Poisson and exponential probability distribution, respectively. When arrival rate (λ) is less than service rate (μ), i.e., traffic density ρ = is less than one, we may have a real waiting line

It is the measure of performance of a waiting line application. How well the model performs, may be known by evaluating the operating characteristics of the queue. We analyze the steady state of the queue, when the queue has stabilized after initial transient stage. Similarly, we do not consider the last or shutting down stage of the service.

Operating Characteristics

situation, because otherwise there would be an infinitely long queue and steady state would never be achieved.

¾ Single-Line-Single-Server Model

¾ ¾ ¾ ¾ ¾

Industrial Engineering

Despite being simple, this model provides the basis for many other complicated situations. It provides insight and helps in planning process. Waiting line for ticket window for a movie, line near the tool crib for checking out tools, railway reservation window, etc., are some direct applications of this model.

Relevance of This Model

Quick Refresher Guide

d: Maximum allowed customers in the system

c: Number of servers in the queuing model

b: Probability distribution of the service time

a: Probability distribution of the inter-arrival time

In a queuing model notation, M is traditionally indicative of exponential distribution. Therefore, (M/M/1): (∞/ FIFO) indicates a queuing model when the inter-arrival time and service time are distributed exponential with distribution (equivalent to this: M stands for Poisson arrivals and departures). There is 1 server, the permissible number of customers in the system are infinite and the service discipline is first-in-first-out (FIFO).

Where:

(a/b/c) : (d/e)

Kendall (1951) proposed a set of notations for queuing models. This is widely used in literature. The common pattern of notations of a queuing model is given by:

¾ Kendall Notations

Single Channel Multi Phase

o

Quick Refresher Guide

. . . (10.3.6)

. . . (10.3.7)

6. Expected number of units in queue for busy system, L =

7. Expected time in queue for busy system, W =

=(

)

All time estimates are assumed to be deterministic.

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o

¾ Assumptions of CPM

A project is a collection of some linked activities that are performed in an organized manner with well known start and finish points, to achieve some specific results that fulfil the needs of an organization. Project management is the domain that deals with planning, organizing, staffing, controlling and directing a project for its effective execution. Critical Path Method (CPM) and Project Evaluation and Review Technique (PERT) are two tools for the project management.

10.3.3 CPM and PERT

V

13. Variance of time in the system, . . . (10.3.13)

. . . (10.3.12)

12. Variance of number of units in the system, V =

)

. . . (10.3.11)

11. Probability density function for time spent in the system, f(w) = (μ − λ)e ( ) w≥0

(

. . . (10.3.10)

. . . (10.3.9)

9. Probability of system being occupied or busy, P(n > 0) = 1 − P = ρ

10. Probability of n units in the system, (Geometric distribution) P =P ρ

. . . (10.3.8)

8. Probability of no unit in the system (i.e., system is idle), P =1−ρ

Probabilities Related Operating Characteristics

. . . (10.3.5)

Industrial Engineering

5. Utilization of service facility, U= =ρ

4. Average waiting time or expected time in the system . . . (10.3.4) W = =

Quick Refresher Guide

The longest path of the network is the indicator of project duration

o

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1. Activity: Distinct part of a project, involving some work, whose completion requires some amount of time. Examples of activity are: drilling a hole, starting a bus, issuing the work order, floating a tender, etc. 2. Activity Duration: It is the physical time required to complete an activity. In CPM, it is the best estimate of the time to complete an activity. In PERT, it is the expected time or average time to complete an activity. 3. Critical Activity: This activity has no room for schedule deviation. In case of deviation or slips, the entire project completion will slip. An activity with zero slack is also same. 4. Critical Path: The sequence or chain of critical activities for the project constitutes critical path. It is the longest duration path through the network. 5. CPM: Project management technique that is used when activity times are deterministic (Critical Path Method). 6. Crashing: The process of reducing an activity time by adding fresh resources and hence usually increasing cost. Crashing is needed for finishing the task before estimated time.

¾ Terminology used in CPM/PERT

We have seen in step 5 that the critical path determines the project completion time. It the project time needs to be compressed, we have to focus on activities on critical path. Similarly, if any activity of the critical path gets delayed by t time, then the total project will be delayed by t time. Same is not true for activities, not lying on critical path. This is due to slack (or float) associated with them. This offers flexibility in scheduling the resources. From the to time, some resources from non-critical activities may be diverted to the critical activity.

Step 7: Use regular monitoring, evaluation and control of the progress of the project by replanning, rescheduling and relocation of resources (Such as money, manpower, etc.).

Step 6: Determine slack (or float) for each activity, not contained on the critical path.

Step 5: Identify longest path (time-wise) on the network. It is the critical path of the network. The project completion takes time equal to critical path time.

Step 4: Assign time-estimates and/or cost-estimates to all the activities of the network.

Step 3: Prepare the network of activities and/or events.

Step 2: Establish the interdependence and sequence of specific activities (also called as precedence relationship).

Step 1: Break the project in terms of specific activities and/or events. Find the time of each activity. In CPM, it is a deterministic estimate, while in PERT it is a probabilistic, three time estimate.

Critical path is the path on the network of project activities which takes longest time form start to finish. The way by which we construct and analyze CPM or PERT network, the analysis is called as critical path analysis (CPA). A general methodology for CPA is as follows:

¾ Methodology of CPM

CPM can be represented as directed graph in which time (or cost) estimates are deterministic.

The precedence relationship in between activities is known.

Industrial Engineering

o

o

Quick Refresher Guide

Industrial Engineering

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¾ Project Evaluation and Review Technique (PERT)

7. Crash Cost: Cost associated with an activity when it is completed in the possible time (Crash time), which is lesser than the expected or normal time. 8. Dummy Activity: An activity that consumes no time but shows precedence among activities. It is useful for proper representation in the network. 9. Earliest Finish (EF) Time: The earliest time that an activity can finish, from the beginning of the project. 10. Earliest Start (ES) Time: The earliest time that an activity can start, from the beginning of the project. 11. Event: It is the beginning, completion point, or milestone accomplishment within the project. An activity begins and ends with events. An event triggers an activity of the project. 12. Expected Activity Time: The average activity time that is used in the project scheduling. 13. Free Slack (float): The length of time upto which an activity can be delayed for channeling resources or re-adjustments, without affecting the starts of the succeeding activities. 14. Immediate Predecessor: An activity, which should immediately precede the activity under consideration. 15. Latest Finish (LF) Time: It is the latest time that an activity can finish, from the beginning of the project, without causing a delay in the completion of the project. 16. Latest Start (LS) Time: It is the latest time that an activity can start, from the beginning of the project, without causing a delay in the completion of the project. 17. Most Likely Time ( ): It is the time for completing an activity that is the best estimate; under the given conditions (used in PERT). 18. Normal Cost: Cost associated with an activity when it is completed in normal time. 19. Optimistic Time ( ): It is the time for completing an activity if everything in the project goes well (used in PERT). 20. Pessimistic Time ( ): It is the time for completing an activity if everything in the project goes wrong (used in PERT). 21. Predecessor Activity: An activity that must occur before another activity in the project which is decided on precedence relationship. 22. Project: Set of activities which are interrelated with each other and are to be organized for a common goal or objective during a specified time-frame. 23. Project Network: A visual representation of the interdependence between different activities of a project which are normally associated with a time-wise sequencing. 24. PERT: It is the project management technique used when activity times are probabilistic. (Program Evaluation and Review Technique). 25. Resource Allocation Methods: Allocation of resources to the activities so that project completion time is as small as possible and resources are well utilized. 26. Slack: It is the amount of time that an activity or a group of activities can delay in getting completed without causing a delay in the completion of the project. An activity having slack cannot be critical activity. 27. Successor Activity: It is the activity that must occur after another activity (which is predecessor). 28. Total Slack (Float): The time upto which an activity can be delayed without affecting the start of the succeeding activities. 29. Updating: It involves some revision of the project schedule after partial completion with revised information. 30. Variance: It is the measure of the deviation of the time distribution for an activity.

Quick Refresher Guide

Industrial Engineering

Optimistic time (t ) Pessimistic time (t ) Most likely time (t )

Event oriented Focussed on time More suitable for new projects

2. 3. 4.

More suitable projects

for

repetitive

Focussed on time-cost trade-off

Activity oriented

Time estimates are deterministic

CPM

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Time estimates are probabilistic

PERT 1.

S.N.

Table: Differences between PERT and CPM

σ =

v =

The variance (V ) and standard deviation (σ ) for the activity are:

Once, expected time (t ) is known from the three time estimates, the algorithm for network calculations is similar to CPM approach.

t =

In PERT, it is assumed that the three time estimates are random variables, distributed as Beta-distribution. The probability of most likely time is four times that of either of the remaining two. Mathematically, the expected time (t ) for an activity is related with the three time estimates as follows.

Optimistic time for an activity is that estimate for the completion of the activity which happens when every best thing happens to facilitate the execution. Thus, when everything goes well, the estimate is optimistic time. On the other extreme, when every thing goes worst, the duration of time-estimate is the pessimistic time. Most likely time is in between the optimistic and pessimistic times. Under normal circumstances, this is the probable time in which an activity is completed.

x x x

PERT allows uncertainty in the estimates for time of each activity. there are three time estimates in PERT. These are.

¾ Time estimate in PERT

PERT incorporates probabilistic time estimates for each activity. It employs a betadistribution for the time estimates. The procedure for making the network and determining the critical path is same as CPM. However, there is a specific calculation approach for finding the most expected time for every activity and for finding the measure of certainity in meeting this estimate.

Quick Refresher Guide

Advance Engineering Mathematics – Erwin Kreyszig

Advance Engineering Mathematics – Dr. HK Dass

x

x

Engineering Mechanics – I. H. Shames

Engineering Mechanics – Beer & Johnson

Engineering Mechanics – CBS Publishers

x

x

x

Strength of Materials – Pytel & Singer

Strength of Materials – Timoshenko and Young

x

x

Theory of Machine – S. S. Rattan

Theory of machine – R. S. Khurmi

x

x

Machine Design – Sharma & Agarwal

x

Power Plant Engineering – Arora & Domkundwar

Referigeration & Airconditioning – Arora & Domkundwar

I.C. Engines – V. Ganesan

x

x

x

Fluid Mechanics – Schaum’s series

Fluid Mechanics – A. K. Jain

Fluid Mechanics – R.K. Bansal

x

x

x

Reference Books

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Fluid Mechanics – K. Z. Schaffer

x

Fluid Mechanics & Hydraulic Machines:

Engineering Thermodynamics – P. K. Nag

x

Thermodynamics & its Applications:

Mechanical Design – Shigley

x

Machine Design:

Theory of Machine – Norton

x

Theory of Machine:

An Introduction to the Mechanics of Solids – Stephen H Crandall

x

Strength of Material:

Engineering Mechanics Vol. I & Vol. II – Meriam & Kraige

x

Engineering Mechanics:

Higher Engineering Mathematics – Dr. BS Grewal

x

Mathematics:

Reference Books

Quick Refresher Guide

Industrial Engineering – Banga & Sharma

Reference Books

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x

Industrial Engineering:

Tolerance system – Mahajan

Manufacturing Science – A.K. Ghosh

x x

Production Engineering – R. K. Jain

x

Manufacturing Engineering:

Heat Transfer – Arora & Domkundwar

Heat transfer – D.S. Kumar

x x

Heat Transfer – J. P. Holman

x

Heat Transfer:

Quick Refresher Guide