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Chapter 1

USES OF OPTIMIZATION FORMULATION OF OPTIMIZATION PROBLEMS OVERVIEW OF COURSE. 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

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Chapter 1

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

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

  3. PHILOSOPHY OF BOOK • 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 Chapter 1

  4. Chapter 1 The Nature and Organization of Optimization Problems Chapter 1

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

  6. Chapter 1

  7. Chapter 1

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

  9. Chapter 1

  10. Chapter 1

  11. Chapter 1

  12. Chapter 1 Min reflux to achieve separation Figure E1.4-3 Optimal Reflux for Different Fuel Costs Flooding constraint

  13. Chapter 1

  14. Chapter 1

  15. Chapter 1

  16. Chapter 1

  17. Material Balance Reconciliation Chapter 1

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

  19. Chapter 1

  20. THREE INGREDIENTS IN OPTIMIZATION PROBLEM Chapter 1

  21. Chapter 1

  22. TABLE 1 THE 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. • 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). Chapter 1

  23. 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). Chapter 1

  24. 4. If the problem formulation is too large in scope: • Break it up into manageable parts and/or • 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. Chapter 1

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

  26. Step 2 develop objective function inventory, storage cost = k1D production cost = k2 + k3 D per run (set up operating cost) cost per unit (could be nonlinear) Chapter 1

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

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

  29. Chapter 1

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

  31. Chapter 1

  32. RELATIVE SENSITIVITY (Percentage change) Chapter 1

  33. PIPELINE PROBLEM Chapter 1

  34. Equality Constraints Chapter 1

  35. Chapter 1

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

  37. (constraint eliminated by substitution) Chapter 1

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

  39. Heat Exchanger Variables (given flow rate of one fluid, inlet temperatures, one outlet temp., phys. props.) • heat transfer area • heat duty • flow rates (shell, tube) • no. passes (shell, tube) • baffle spacing • length • diam. of shell, tubes • approach temperature • fluid A (shell or tube, co-current or countercurrent) • tube pitch, no. tubes • velocity (shell, tube) • ∆p (shell, tube) • heat transfer coeffs (shell, tube) • exchanger type (fins?) • material of construction Chapter 1

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