Chapter 1 USES OF OPTIMIZATION FORMULATION OF OPTIMIZATION

Chapter 1 • USES OF OPTIMIZATION • FORMULATION OF OPTIMIZATION PROBLEMS • OVERVIEW OF COURSE 1

Chapter 1 OPTIMIZATION OF CHEMICAL PROCESSES T. F. EDGAR, D. M. HIMMELBLAU, and L. S. LASDON UNIVERSITY OF TEXAS MCGRAW-HILL – 2001 (2 nd ed. ) PART I – PROBLEM FORMULATION II – OPTIMIZATION THEORY AND METHODS III – APPLICATIONS OF OPTIMIZATION APPENDICES (MATRIX OPERATIONS) 2

PHILOSOPHY OF BOOK Chapter 1 • Most undergraduates learn by seeing how a method is applied • Practicing professionals need to be able to recognize when optimization should be applied (Problem formulation) • Optimization algorithms for reasonably-sized problems are now fairly mature • Focus on a few good techniques rather than encyclopedic coverage of algorithms 3

Chapter 1 The Nature and Organization of Optimization Problems 4

WHY OPTIMIZE? 1. Improved yields, reduced pollutants Chapter 1 2. Reduced energy consumption 3. Higher processing rates 4. Reduced maintenance, fewer shutdowns 5. Better understanding of process (simulation) But there always positive and negative factors to be weighed 5

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Chapter 1 OPTIMIZATION • Interdisciplinary Field Max Profit Min Cost Max Efficiency • Requires 1. Critical analysis of process 2. Definition of performance objective 3. Prior experience (engr. judgment) 8

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Chapter 1 Min reflux to achieve separation Figure E 1. 4 -3 Flooding constraint Optimal Reflux for Different Fuel Costs 12

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Chapter 1 Material Balance Reconciliation 17

Chapter 1 Least squares solution: opt. m. A is the “average” value any constraints on m. A? 18

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Chapter 1 THREE INGREDIENTS IN OPTIMIZATION PROBLEM 20

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TABLE 1 Chapter 1 THE SIX STEPS USED TO SOLVE OPTIMIZATION PROBLEMS 1. 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. 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). 22

Chapter 1 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). 23

Chapter 1 4. If the problem formulation is too large in scope: (A)Break it up into manageable parts and/or (B)Simplify the objective function 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. 24

EXAMPLES – SIX STEPS OF OPTIMIZATION Chapter 1 specialty chemical 100, 000 bbl/yr. how many bbl produced per run? Step 1 define variables Q = total # bbl produced/yr (100, 000) D = # bbl produced per run n = # runs/yr 25

Step 2 develop objective function Chapter 1 inventory, storage cost = k 1 D production cost per run = k 2 (set up cost) + k 3 D operating cost per unit (could be nonlinear) 26

Step 3 Chapter 1 evaluate constraints D>0 Step 4 simplification – none necessary 27

Step 5 computation of the optimum Chapter 1 analytical vs. numerical solution 28

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Chapter 1 Step 6 Sensitivity of the optimum subst Dopt into C 30

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Chapter 1 RELATIVE SENSITIVITY (Percentage change) 32

Chapter 1 PIPELINE PROBLEM 33

Chapter 1 Equality Constraints 34

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min (Coper + Cinv. ) Chapter 1 subject to equality constraints need analytical formula for f substituting for ∆p, 36

Chapter 1 (constraint eliminated by substitution) 37

optimum velocity Chapter 1 non-viscous liquids gases (effect of ρ) 3 to 6 ft/sec. 30 to 60 ft/sec. at higher pressure, need to use different constraint (isothermal) for large L, ln ( ) can be neglected exceptions: elevation changes, slurries (settling), extremely viscous oils (laminar flow, 38 f different)

Chapter 1 Heat Exchanger Variables (given flow rate of one 1. heat transfer area fluid, inlet 2. heat duty temperatures, one 3. flow rates (shell, tube) outlet temp. , phys. 4. no. passes (shell, tube) props. ) 5. baffle spacing 6. length 7. diam. of shell, tubes 8. approach temperature 9. fluid A (shell or tube, co-current or countercurrent) 10. tube pitch, no. tubes 11. velocity (shell, tube) 12. ∆p (shell, tube) 13. heat transfer coeffs (shell, tube) 14. exchanger type (fins? ) 15. material of construction 39