Heat And Mass Formula Sheet

Cooling of Electronic Systems MCTR1004 Spring 2016

Formula Sheets Chapter -1 Mechanisms of Heat Transfer & C p ∆T Under steady conditions, conservation of energy: Q& = m

where m& = ρAV

dT Fourier’s law of heat conduction: Q& cond = −kA dx ∆ T For a plane wall: Q& cond = kA L & Newton’s law of cooling: Qconvection = hAs (Ts − T∞ ) Radiation: Q& emit. max = σAs Ts4 where σ = 5.67 × 10 −8 W / m 2 .K 4 Q& = εσA (T 4 − T 4 ) , Q& = αQ& , ε = α, ε = α=1 for black body rad

s

s

surr

absorbed

incident

Chapter-3 Steady Heat Conduction One-dimensional heat transfer through a simple or composite body exposed to convection from both sides to mediums at temperatures T∞1 and T∞2 can be expressed as: T −T L 1 1 Q& = ∞1 ∞ 2 , Rtotal = Rconv .1 + R wall + Rconv .2 = + + Rtotal h1 A kA h2 A Elementary thermal resistance relations: L Conduction (plane wall): Rwall = kA ln(r2 / r1 ) Conduction (Cylinder): Rcyl = 2πLk r −r Conduction (Sphere): Rsph = 2 1 4π r1 r2 k 1 Convection resistance: Rconv = hA R 1 = c Interface resistance: Rint erface = hc A A 1 2 Radiation resistance: R rad = , hrad = εσ (Ts2 + Tsurr )(Ts + Tsurr ) hrad A The temperature drop across any layer can be determined from: ∆ T = Q& R 1 Constriction/Spreading resistance: RConstriction = π dk k ins 2k , rcr , sphere = ins Critical radius of insulation: rcr , cylinder = h h

Fins: The temperature distribution along the fin for very long fins and for fins with negligible heat transfer at the fin are given by: T ( x ) − T∞ − x hp / kAc =e Very long fin: Tb − T∞ MCTR1004

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Cooling of Electronic Systems MCTR1004 Spring 2016 T ( x) − T∞ cosh a( L − x) a = hp / kAc , p is the perimeter, and = , Tb − T∞ cosh aL Ac is the cross-sectional area of the fin. The rates of heat transfer for both cases are given to be dT Very long fin: Q& long fin = − kAc hpkAc (Tb − T∞ ) x =0 = dx dT Adiabatic fin tip: Q& insulated tip = −kAc hpkAc (Tb − T∞ ) tanh aL x =0 = dx Corrected length: L c = L + Ac / p Fin efficiency: Actual heat transfer rate from the fin η fin = Ideal heat transfer rate from the fin if the entire fin were at the base temperatur e tanh aL 1 η fin −long fin = & η fin −adiabatic = aL aL −T T 1 Q& fin = η finQ& fin.max = η fin hAfin (Tb − T∞ ) = b ∞ & R fin = R fin η fin hAfin Fin effectiveness: Q& fin Q& fin Heat transfer rate from the fin of base area Ab ε fin = = = Heat transfer rate from the surface of area Ab Q& no fin hAb (Tb − T∞ )

Adiabatic fin tip:

Overall effectiveness: Q& total , fin h( Aunfin + η fin A fin )(Tb − T∞ ) = ε fin , overall = & hAno fin (Tb − T∞ ) Q

&

ε fin =

total , no fin

A fin Ab

η fin

Chapter 4- Transient Heat Conduction 1- Heating or cooling (no heat generation): The temperature of a lumped body of arbitrary shape of mass m, volume V, surface area As, density ρ, and specific heat Cp initially at a uniform temperature Ti that is exposed to convection at time t = 0 in a medium at temperature T∞ with a heat transfer coefficient h is expressed as

T (t ) − T∞ = e −bt Ti − T∞

hAs h = ρ C pV ρ C p Lc 2- Heating with heat generation: ‫ܧ‬ሶ ൌ ݄‫ܣ‬௦ ሺܶ௦௦ െ ܶஶ ሻ T (t ) − T∞ hAs h = 1 − e −bt where b = = Tss − T∞ ρ C p V ρ C p Lc where b =

hLc V , and Lc = is the characteristic length. k As Convection heat transfer between the body and its environment at time t: Q& (t ) = hAs [T (t ) − T∞ ] Total amount of heat transfer: Q = mc p [T (t ) − Ti ]

Biot number: Bi =

Maximum heat transfer: Qmax = mc p (T∞ − Ti )

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Cooling of Electronic Systems MCTR1004 Spring 2016

Chapter 9 - Natural Convection In natural convection, any fluid motion occurs by natural means such as buoyancy. The volume expansion coefficient of a substance represents the variation of the density of that substance with temperature at constant pressure, and for an ideal gas, it is expressed as β = 1/T, where T is the absolute temperature in K. The flow regime in natural convection is governed by a dimensionless number called the Grashof number, which represents the ratio of the buoyancy force to the viscous force acting on the fluid and is expressed as

where Lc is the characteristic length, which is the height L for a vertical plate and the diameter D for a horizontal cylinder. The correlations for the Nusselt number Nu = hLc /K in natural convection are expressed in terms of the Rayleigh number defined as

Nusselt number relations for various surfaces are given in Table. All fluid properties are evaluated at the film temperature of Tf = ½ (Ts + T∞). The outer surface of a vertical cylinder can be treated as a vertical plate when the curvature effects are negligible. The characteristic length for a horizontal surface is Lc = As/p, where As is the surface area and p is the perimeter. The average Rayleigh number and Nusselt number for vertical isothermal parallel plates of spacing S and height L is given as

The optimum fin spacing for a vertical heat sink and the Nusselt number for optimally spaced fins is

All fluid properties are to be evaluated at the average temperature Tave = (Ts + T∞)/2. Arrays of printed circuit boards used in electronic systems can often be modeled as parallel plates subjected to uniform heat flux. The modified Rayleigh number for uniform heat flux on both plates and the Nusselt number at the upper edge of the plate where maximum temperature occurs are determined from:

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The optimum fin spacing for the case of uniform heat flux on both plates is given as

The critical surface temperature TL occurs at the upper edge of the plates All fluid properties are to be evaluated at the average temperature Tave = (TL + T∞)/2. In a horizontal rectangular enclosure with the hotter plate at the top, heat transfer is by pure conduction and Nu = 1. When the hotter plate is at the bottom, the following simple correlations are used for air:

These relations can also be used for other gases with 0.5 < Pr < 2. For water, silicone oil, and mercury: For vertical horizontal enclosures, the Nusselt number can be determined from:

For aspect ratios greater than 10, the following Eqs. should be used.

Again all fluid properties are to be evaluated at the average temperature = (T1+T2)/2. The airflow in the analysis of electronic equipment can be assumed to be laminar. The natural convection heat transfer coefficient for laminar flow of air at atmospheric pressure is given by a simplified relation of the form:

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Chapter 7 (Forced Convection-External)

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Sphere:

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Square

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Chapter 8 (Forced Convection-Internal)

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