Flowsheet Optimization

Flow sheet Optimization MIDLANDS STATE UNIVERSITY COLLEGE OF APPLIED SCIENCES

FACULTY OF ENGINEERING DEPARTMENT OF CHEMICAL AND PROCESSING ENGINEERING Compiled by PENDUKA VALENTINE AND OBEY ZVAVANHU 1

INTRODUCTION



Process Engineering is nothing else but it is the practical and creative application of the chemical engineering knowledge to a certain process so that existing process can be improved and some new design for it can be developed which will be more cost effective and more profitable one



Process flowsheet is the use of computer aids to perform a steady state and mass balancing, sizing and cost calculation for a chemical process.It is an essential and core component of process design. Process is split into three component i.e. synthesis analysis and optimization. 2

PRESENTATION OUTLINE •

Overview.

•

Introduction

•

Optimization basics.

•

Essential features of optimization problems.

•

Development of process(mathematical models) constrains in optimization problems. 

Degrees of freedom



Formulation of optimization problems



Objective and decision variables

•

Types of optimization models.

•

Classification of optimization problems.

•

Steps used to solve optimization problems.

•

Optimization example.

•

Introduction to LINGO.

•

Merits and demerits of optimization.

•

Safety and environment.

•

Summary .

•

References.

3

OVERVIEW 

The chemical industry has undergone significant changes during the past 20 years due to the increased cost of energy and raw materials, more stringent environmental regulations, and intense worldwide competition. Modifications of both plant-design procedures and plant operating conditions have been implemented in order to reduce costs and meet constraints. One of the most important engineering tools that can be employed in such activities is optimization. As plant computers have become more powerful, the size and complexity of problems that can be solved by optimization techniques have correspondingly expanded. A wide variety of problems in the operation and analysis of chemical plants (as well as many other industrial processes) can be solved by optimization. Real-time optimization means that the process-operating conditions (set points) are evaluated on a regular basis and optimized 4

OVERVIEW  

Optimization is concerned with selecting the best among the entire set by efficient quantitative methods Engineers work to improve the initial design of equipment and strive for enhancements in the operation of the equipment once it is installed in order to realize the most production, the greatest profit, the maximum cost, the least energy usage, and so on.



optimization of operating conditions is carried out monthly, weekly, daily, hourly, or even every minute and plant operations determines the set points for each unit at the temperatures, pressures, and flow rates that are the best in some sense.



For example, the selection of the percentage of excess air in a process heater is quite critical and involves a balance on the fuel-air ratio to assure complete combustion and at the same time make the maximum use of the heating potential of the fuel. 5

INTRODUCTION Objectives

On completion of this course unit, you are expected to be able to:  Understand the fundamentals of optimization  Formulate and solve a linear program (LP)  Formulate a nonlinear program (NLP) to minimize or maximize an objective function by adjusting continuous decision variables in the model of a process.  Be able to understand the nature of algorithms that optimize the process while simultaneously converging the recycle loops and design specifications associated with the process simulation.  Begin to understand the advantages and disadvantages of converting design specifications associated with a simulation model to equality constraints in the NLP.  Be able to formulate and solve a variety of optimization problems in LINGO 6

Optimization Basics WHAT IS OPTIMIZATION

The

purpose of optimization is to maximize (or minimize) the value of a function (called objective function) subject to a number of restrictions (called constraints). Optimization is applied to improve all product and process design at various design stages. EXAMPLES

Maximizereactor conversion Subject to reactor modeling equations kinetic equations limitations on T, P and x 2. Minimize cost of plant Subject to mass & energy balance equations equipment modeling equations environmental, technical and logical

1.

7

constraints

Optimization basics  Some

common objective in optimization of an industrial process are:

1.

Achieve lower capital cost design.

2.

Increase production.

3.

Reduce unit operation cost.

4.

Reduce environmental impact.

5.

Reduce energy consumption. 8

ESSENTIAL FEATURES OF OPTIMIZATION PROBLEMS The solution of optimization problems involves the use of various tools of mathematics hence, the formulation of an optimization problem requires the use of mathematical expressions. Every optimization problem contains three essential categories: 1.

An objective function to be optimized (revenue function, cost function, etc.)

2.

Equality constraints (equations)

3.

Inequality constraints (inequalities)

No single method or algorithm of optimization exists that can be applied efficiently to all problems. The method chosen for any particular case will depend primarily on 4.

the character of the objective function.

5.

the nature of the constraints.

6.

the number of independent and dependent variables. (Edgar and Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York, 1988). 9

DEVELOPMENT OF PROCESS (MATHEMATICAL) MODELS CONSTRAINTS IN OPTIMIZATION PROBLEMS

Arise from physical bounds on the variables, empirical relations, physical laws, and so on. Two general categories of models exist: 1. Those based on physical theory 2. Those based on strictly empirical descriptions Models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed before the system is constructed. On the other hand, an empirical model can be devised that simply correlates input output data without any physiochemical analysis of the process. 10

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Degrees of freedom Degrees of freedom are the number of input variables you need to specify A degree of freedom analysis is incorporated in the development of each subroutine that simulate a process unit.  Over Specified Problem: many variables are determining in > 1 way values must be reconciled(data reconciliation and rectification)  Fitting Data  Nvariables>>Nequations  Equally Specified Problem: proceed solving the problem  Units in Flow sheet  Nvariables=Nequations  Under Specified Problem: some values are underdetermined-can be manipulated to optimize the process.  Optimization  Nvariables<


Number of Decision Variables  ND=Nvariables-Nequations



Objective Function is optimized with respect to ND Variables Minimize Cost  Maximize Investor Rate of Return Subject To Constraints  Equality Constraints  Mole fractions add to 1  Inequality Constraints  Reflux ratio is larger than Rmin 





Upper and Lower Bounds  Mole fraction is larger than zero and smaller than 1 13

Formulation of Optimization Problems 

Step 1  Determine the quantity to be optimized and express it as a mathematical function (this is your objective function)  Doing so also serves to define variables to be optimized (input variables or optimization variables)



Step 2  Identify all stipulated requirements, restrictions, and limitations, and express them mathematically. These requirements constitute the constraints



Step 3  Express any hidden conditions. Such conditions are not stipulated explicitly in the problem, but are apparent from the physical situation, e.g. 14 non-negativity constraints

From a mathematical point of view, chemical engineers encounter 3 situations when solving equations •

Any optimization problem can be represented as, min or max f(x), s.t. c(x)=0 , g(x) •

f(x) – is the objective function

•

c(x) – is the set of m equations in n variables x. the equality constraints.

•

g(x) – is the set of r inequality constrains. These are bound to the feasible region of operation 15

OBJECTIVE AND DECISION VARIABLES  Objective 1.

measures

Profitability measures

Return

on investment (ROI)



Venture profit (VP)



Payback period (PBP)

Annualized

cost (Ca)

Other rigorous measure are those that involve time value Net

present value (NPV)

Investor's

rate return (IRR) 16

Other involve costs, safety control and pollution effects

Types of optimization models Feasible Region Unconstrained Optimization refers to the case where no inequality constraints are present and all equality constraints can be eliminated by solving for selected dependent variables followed by substitution for them in the objective function  No constraints  Uni-modal  Multi-modal Constrained Optimization When constraints exist and cannot be eliminated in an optimization problem, more general methods must be employed  Constraints  Slack  Binding 17

Classification of Optimization Problems 

Linear Programs (LP’s) A mathematical program is linear if f(x1,x2,……,xN) and gi(x1,x2,……,xN)≤0 are linear in each of their arguments: f(x1,x2,……,xN) = c1x1 + c2x2 + …. cNxN gi(x1,x2,……,xN) = ai1x1 + ai2x2 + …. aiNxN where ci and aij are known constants. Linear Programs (LP’s) can be solved to yield a global optimum. Solver routines can guarantee a truly optimal solution. 18

The accepted procedure, linear programming (LP), has become quite popular, solving a wide range of industrial problems. It is increasingly being used for online optimization. For processing plants, there are several different kinds of linear constraints that may arise, making the LP method of great utility. 1. Production limitation due to equipment throughput restrictions, storage limits, or market constraints. 2. Raw material (feedstock) limitation. 3 . Safety restrictions on allowable operating temperatures and pressures. 4. Physical property specifications placed on the composition of the final product. 5. Material and energy balances of the steady-state model. The optimum in linear programming lies at the constraint intersections, which was generalized to any number of variables and constraints by George Dantzig

19

Classification of Optimization Problems continued Non-Linear Programs (NLP’s) A mathematical program is non-linear if any of the arguments are non-linear. For example: min 3x + 6y2 s.t. 5x + xy ≥ 0 general case for optimization that occurs when both the objective function and constraints are nonlinear Other established types of constrained optimization methods include the following types of algorithms: 1. Penalty functions with augmented Lagrangian method (an enhancement of the classical Lagrange multiplier method) 2. Successive quadratic programming  Integer Programming Optimization programs in which ALL the variables must assume integer values. The most commonly used integer variables are the zero/one binary integer variables. 

20

Integer

variables are often used as decision variables, e.g. to choose between

Classification of Optimization Problems continued 

Mixed Integer Linear Programs (MILP’s) Linear programs in which SOME of the variables are real and other variables are integers Can

be solved as individual LP’s by fixing the integer variables, thus a global optimum can be identified. 

Mixed Integer Non-Linear Programs (MINLP’s) Non-linear programs in which SOME of the variables are real and other variables are integers Can

be solved as individual NLP’s by fixing the integer variables, but depending on the nature of the NLP’s it may not be possible to find a global optimum. 21

Solving for a Recycle Loop  Newton-Raphson Solving

for a root

F(xi)=0  Optimization Minimize/Maximize

w.r.t. ND variables (d) s.t.

constraints F(xi)

= 0, G(xi) < 0, H(xi) > 0

Six Steps Used to Solve Optimization Problems Analyze the process itself so that the process variables and specific characteristics of interest are defined (i.e., make a list of all of the variables). 1.

2. Determine the criterion for optimization and specify the objective function in terms of the above variables together with coefficients. This step provides the performance model (sometimes called the economic model when appropriate). 3. Develop via mathematical expressions a valid process or equipment model that relates the input-output variables of the process and associated coefficients. Include both equality and inequality constraints. Use well-known physical principles (mass balances, energy balances), empirical relations, implicit concepts, and external restrictions. Identify the independent and dependent variables (number of degrees of freedom). 4. If the problem formulation is too large in scope, (a) break it up into manageable parts and/or (b) simplify the objective function and model. 5. Apply a suitable optimization technique to the mathematical statement of the problem. 6. Check the answers and examine the sensitivity of the result to changes in the coefficients in the problem and the assumptions. 23

Optimization Example •

Coal Conversion Plant –

What are the optimal production rates of gaseous and liquid fuels that maximize the net 2x1profit kg coal/s for power Gaseous Fuel of the plant?

generation of gasification x1 kg gas. fuel/s plant (value of power breaks Net profit $3/kg Coal even with the of gaseous fuel gasification air cost of coal (maximum used in power capacity generation) 4 kg x1 ≤ 4 x1 kg coal/s coal/s) Byproducts Coal pre(negligible value) treatment coal in (maximum Liquid Fuel 3x1 + 2x2 Coal capacity x2 kg liquid fuel/s kg coal/s liquefaction 2x2 kg coal/s 18 kg coal/s) Net profit $5/kg (maximum of liquid fuel capacity 12 kg 2x2 ≤ 12 coal/s) Byproducts 3x1 + 2x2 ≤ 18 (negligible value)

Optimization Example •

Coal Conversion Plant (Cont’d) Objective function Constraints

– –

• • • • x2

Pretreatment capacity 3x1 + 2x2 ≤ 18 Gasification capacity x1 ≤ 4 Liquefaction capacity 2x2 ≤ 12 Non-negativity x1 ≥ 0 x2 ≥ 0

10

x1 = 4

3x1 + 2x2 = 18

8

max z = 3x1 + 5x2

2x2 = 12

6 4 2

x1

0 0

2

4

6

8

Optimization Example •

Coal Conversion Plant (Cont’d) LINGO Input

Model: max = 3*x1 + 5*x2; 3*x1 + 2*x2 <= 18; x1 <= 4; 2*x2 <= 12; x1 > 0; x2 > 0; end

LINGO Output Rows= 6 Vars= 2 No. integer vars= 0 ( all are linear) Nonzeros= 11 Constraint nonz= 6( 3 are +- 1) Density=0.611 Smallest and largest elements in absolute value= 1.00000 18.0000 No. < : 3 No. =: 0 No. > : 2, Obj=MAX, GUBs <= 2 Single cols= 0 Optimal solution found at step: Objective value:

1 36.00000

Variable X1 X2 Row 1 2 3 4 5 6

Value 2.000000 6.000000

Reduced Cost 0.0000000E+00 0.0000000E+00

Slack or Surplus 36.00000 2.000000 0.0000000E+00 0.0000000E+00 2.000000 6.000000

Value of objective function: 36 Value of variable x1:

2

Value of variable x2:

6

Dual Price 1.000000 0.0000000E+00 1.500000 1.000000 0.0000000E+00 0.0000000E+00

Optimization Example • x2

Coal Conversion Plant (Cont’d) –

Graphical solution

10 8

10

x1 = 4

3x1 + 2x2 = 18

x2

8

Z = 36 = 3x1 + 5x2

2x2 = 12

6

6

Z = 20 = 3x1 + 5x2 4

4

Z = 10 = 3x1 + 5x2

2

2

x1

0 0

2

4

6

8

x1

0 0

2

4

6

Maximum profit Z = 36 for x1 = 2 and x2 = 6

8

INTRODUCTION TO LINGO WHAT IS LINGO ???????



LINGO is a simple tool for utilizing the power of linear and nonlinear optimization to formulate large problems concisely, solve them, and analyze the solution.



Optimization helps you find the answer that yields the best result; attains the highest profit, output, or happiness; or achieves the lowest cost, waste, or discomfort. Often these problems involve making the most efficient use of your resources including money, time, machinery, staff, inventory, and more.



Optimization problems are often classified as linear or nonlinear, depending on whether the relationships in the problem are linear with respect to the variables. 28

INTRODUCTION TO LINGO Starting LINGO 

This section illustrates how to input and solve a small model in Windows. The text of the model’s equations is platform independent and will be identical on all platforms.

29

EXAMPLE 2 

For our sample model, we will create a small product-mix example. Let’s imagine that the Mutare Bottling Company produces two models of soft drinks Coke and Sprite. Delta can sell every Coke drink producing a profit contribution of $1 000 000, and each Sprite drink for a contribution of $1 500 000. At the Sprite factory, the sprite drink production line can produce, at most, 10 000 bottles per day. At the same time, the coke drink production line can turn out 12 000 bottles per day. Furthermore, Delta has a limited supply of daily labor. In particular, there is a total of 160 hours of labor available each day. Coke drinks require 6 hour of labor, while Sprite drinks are relatively more labor intense requiring 8 hours of labor. The problem for Delta is to determine the mix of Coke and Sprite drinks to produce each day to maximize total profit without exceeding line and labor capacity limits. 30

In general, an optimization model will consist of the following three items: Objective Function- The objective function is a formula that expresses exactly what it is you want to optimize. In business oriented models, this will usually be a profit function you wish to maximize, or a cost function you want to minimize. Models may have, at most, one objective function. In the case of our Delta Beverages example, the objective function will compute the company’s profit as a function of the output of Coke and Sprite. Variables - Variables are the quantities you have under your control. You must decide what the best values of the variables are. For this reason, variables are sometimes also called decision variables. The goal of optimization is to find the values of a model’s variables that generate the best value for the objective function, subject to any limiting conditions placed on the variables. We will have two variables in our example−one corresponding to the number of Coke to produce and the other corresponding to the number of Sprite to produce. Constraints - Almost without exception, there will be some limit on the values he variables in a model can assume—at least one resource will be limited (e.g., time, raw materials, your department’s budget, etc.). These limits are expressed in terms of 31 formulas that are a function of the model’s variables. These formulas are referred to as constraints because they constrain the values the variables can take. In our Delta

Entering the Model 

We will now construct the objective function for our example. We will let the variables COKE and SPRITE denote the number of Coke and Sprite drinks to produce, respectively. Delta’s objective is to maximize total profit. Total profit is calculated as the sum of the profit contribution of the Coke ($1 000 000) multiplied by the total Coke bottles produced (COKE) and the profit contribution of the Sprite ($1500 000) multiplied by the total Sprite bottles produced (SPRITE). Finally, we tell LINGO we want to maximize an objective function by preceding it with “MAX =”. Therefore, our objective function is written on the first line of our model window as: MODEL: MAX = 10 000 * COKE + 15 000 * SPRITE;

32

Entering the Model 

Next, we must input our constraints on line capacity and labor supply. The number of Coke and Sprite bottles produced must be constrained to the production line limits of 10 000 and 12 000 bottles respectively. Do this by entering the following two constraints just below the objective function: COKE <= 10 000; SPRITE <= 12 000 ;



In words, the first constraint says the number of Coke bottles produced daily (COKE) must be less-than-or-equal-to (<=) the production line capacity of 10 000. Likewise, the second constraint says the number of Sprite bottles produced daily (SPRITE) must be less-than-or-equal-to (<=) its line capacity of 12 000 .



The final constraint on the amount of labor used can be expressed as: COKE + 6 * SPRITE+ 8 <= 160;

Specifically, the total number of labor hours used (COKE + 6 * SPRITE) must be less-than-or-equal-to (<=) the amount of labor hours available of 160.

33

SHOW CASE ON WINDOW MODEL:

OBJECTIVE FUNCTION

MAX = 10 000 * COKE + 15 000 * SPRITE; COKE <= 10 000; SPRITE <= 12 000 ; COKE + 6 * SPRITE + 8 <= 160; END: VARIABLES

CONSTRAINTS 34

RESULTS INTERPRETATION

35

RESULTS INTERPRETATION  From

the above LINGO Solution Page:

 CompuQuick

should build 100 Standards and 30 Turbos each day for a total daily profit of $14,500.

But in our case we incorporated an CHEMICAL ENGINEERING industrial example so as to suite our aims and requirements as Engineers

36

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ADVANTAGES OF OPTIMIZATION 1.

improved plant performance.

2.

improved yields of valuable products (or reduced yields of contaminants).

3.

reduced energy consumption.

4.

higher processing rates.

5.

longer times between shutdowns.

6.

lead to reduced maintenance costs

7.

less equipment wear

8.

better staff utilization.

9.

helpful to systematically identify the objective, constraints, and degrees of freedom in a process plant.

10.

improved quality of designs, faster and more reliable troubleshooting, 38 and faster decision making is achieved

DISADVANTAGES OF OPTIMIZATION 1. Application 2. Involve

to systems is cumbersome

complex mathematical models

3. Optimization

at large scale is dependent on high quality expensive

machines 4. Involve

high computational input of data onto computer packages such as ASPEN, MATLAB and LINGO

5. Requires

more time to simulate changes.

6. Produce

a compromising product conversion that sometimes produce high toxins

39

SAFETY, HEALTH AND ENVIRONMENT  Optimization

transforms existing engineering disciplines and practices to those that lead to sustainability. It incorporates development and implementation of products, processes, and systems that meet technical and cost objectives while protecting human health and welfare and elevates the protection of the biosphere as a criterion in engineering solutions.

 Optimization

of flow sheets of chemical manufacturing process is described as inherently safer if it reduces or eliminates hazards associated with materials used and operations, and this reduction or elimination is a permanent and inseparable part of the process technology. (Kletz, 1991; Hendershot, 1997a, b) 40

Comparison between IS and (GE) Strategy/Tenet (Based on IS)

Example Concepts

Inherent Safety (IS)

Green Engineering (GE)

Substitution

Reaction chemistry, Feedstocks, Catalysts, Solvents, Fuel selection

√√√√

√√√√

Minimization

Process Intensification, Recycle, Inventory reduction, Energy efficiency, Plant location

√√√

√√√√

Number of unit operations, DCS configuration, Raw material quality, Equipment design

√√√√

√√√√

Moderation (1) [Basic Process]

Conversion conditions, Storage conditions, Dilution, Equipment overdesign

√√√√

√√√

Moderation (2) [Overall Plant]

Offsite reuse, Advanced waste treatment, Plant location, Beneficial co-disposal

√√√

√√√

Simplification

√√√√ = Primary tenet/concepts √√ = Some aspects addressed

√√√ = Strongly related tenet/concepts √ = Little relationship

41

PROBLEM

 A first

pass optimization using mill data indicates that the boiler is generating more steam than the heating value of the fuel will provide ( i.e. efficiency greater than 100%).which one is an appropriate response to this situation:

1.

Ignore the problem as insignificant.

2.

Replace the boiler simulation model with one that will give you realistic results.

3.

Recommend a certificate of appreciation for outstanding performance be presented to the boiler operating crew.

4.

Double check the accuracy of the measurements and arrange for test to be performed on the boiler fuel. 42

Summary On completion of this part, you should: 1. Understand the different types of optimization problems and their formulation 2. To design optimized models realizing safety and environmental concepts as explain in Green Engineering (GE) and Inherent Safety (IS) 3. Be able to formulate and solve a variety of optimization problems in LINGO

REFERENCES 1. [Latour,

Hydro Proc.,58(6), 73, 1979, and Hydro. Proc.,58(7), 219, 1979]. 2.(Edgar and Himmelblau, “Optimization of Chemical Processes”, McGraw-Hill, New York, 1988). 3.Towler . G and Sinnott, “Principles, Practice and Economics of Plant and Process Design” Copyright, Elsevier Inc,(2008) 4.Seider, W.D., J.D. Seader, D.R. Lewin, S. Widagdo “Product and Process Design Principles”, 3rd edition Wiley (2008). 5.Eden, M. R. "ASPEN Lab Notes", Department of Chemical Engineering Auburn University. 6. EPA Contract 3W-0500-NATA – OPPT, Green Engineering Program 7. NSF/Lucent Technologies Industrial Ecology Research Fellowship (BES-9814504) 8. National Center for Clean Industrial and Treatment Technologies (CenCITT) 44