Optimization technique chapter

H.J. Kelley. Method of gradients. In G. Leitman, editor, Optimization Techniques, Chapter 6. Academic Press, New York, 1962. 

A non-linear regression analysis is employed using die Solver in Microsoft Excel spreadsheet to determine die values of and in die following examples. Example 1-5 (Chapter 1) involves the enzymatic reaction in the conversion of urea to ammonia and carbon dioxide and Example 11-1 deals with the interconversion of D-glyceraldehyde 3-Phosphate and dihydroxyacetone phosphate. The Solver (EXAMPLEll-l.xls and EXAMPLEll-3.xls) uses the Michaehs-Menten (MM) formula to compute v i- The residual sums of squares between Vg(,j, and v j is then calculated. Using guessed values of and the Solver uses a search optimization technique to determine MM parameters. The values of and in Example 11-1 are ... 

As has been discussed in Chapter One, mathematical programming (or optimization) is a powerful tool for process integration. For an overview of c mization and its application in pollution prevention, the reader is referred to El-Halwagi (1995). In this chapter, it will be shown how optimization techniques enable the designer to ... 

Chapter 3, Geometry Optimizations, describes how to locate equilibrium structures of molecules, or, more technically, stationary points on the potential energy surface. It includes an overview of the various commonly used optimization techniques and a consideration of optimizing transition strucmres as well as minimizations. 

This chapter also introduces several optimization techniques. The emphasis is on robustness and ease of use rather than computational efficiency. 

The rest of this chapter is a series of examples and problems built around semirealistic scenarios of reaction characteristics, reactor costs, and recovery costs. The object is not to reach general conclusions, but to demonstrate a method of approaching such problems and to provide an introduction to optimization techniques. 

The sum of squares as defined by Equation 7.8 is the general form for the objective function in nonlinear regression. Measurements are made. Models are postulated. Optimization techniques are used to adjust the model parameters so that the sum-of-squares is minimized. There is no requirement that the model represent a simple reactor such as a CSTR or isothermal PER. If necessary, the model could represent a nonisothermal PFR with variable physical properties. It could be one of the distributed parameter models in Chapters 8 or 9. The model... 

If we have very little information about the parameters, direct search methods, like the LJ optimization technique presented in Chapter 5, present an excellent way to generate very good initial estimates for the Gauss-Newton method. Actually, for algebraic equation models, direct search methods can be used to determine the optimum parameter estimates quite efficiently. However, if estimates of the uncertainty in the parameters are required, use of the Gauss-Newton method is strongly recommended, even if it is only for a couple of iterations. 

In the last few years, optimization techniques have become more widely used in the pharmaceutical industry. Some of these have appeared in the literature, but a far greater number remain as in-house information, using the same techniques indicated in this chapter, but with modifications and computer programs specific to the particular company. An excellent review of the application of optimization techniques in the pharmaceutical sciences was published in 1981 [20]. This covers not only formulation and processing, but also analysis, clinical chemistry, and medicinal chemistry. 

The optimization can be carried out using nonlinear optimization techniques such as SQP (see Chapter 3). The nonlinear optimization has the problems of local optima if techniques such as SQP are used for the optimization. Constraints need to be added to the optimization in order that a mass balance can be maintained and the product specifications achieved. The optimization of the side-rectifier and side-stripper in a capital-energy trade-off determines the distribution of plates, the reflux ratios in the main and sidestream columns and condition of the feed. If a partitioned side-rectifier (Figure ll.lOd) or partitioned side-stripper (Figure 11.lid) is to be used, then the ratio of the vapor flowrates on each side of the partition can be used to fix the location of the partition across the column. The partition is located such that the ratio of areas on each side of the partition is the same as the optimized ratio of vapor flowrates on each side of the partition. However, the vapor split for the side-rectifier will only follow this ratio if the pressure drop on each side of the partition is the... 

This section of the book is devoted to representative applications of the optimization techniques presented in Chapters 4 through 10. Chapters 11 through 16 include the following major application areas ... 

Each chapter presents several detailed studies illustrating the application of various optimization techniques. The following matrix shows the classification of the examples with respect to specific techniques. Truly optimal design of process plants cannot be performed by considering each unit operation separately. Hence, in Chapter 15 we discuss the optimization of large-scale plants, including those represented by flowsheet simulators. 

This chapter contains examples of optimization techniques applied to the design and operation of two of the most common staged and continuous processes, namely, distillation and extraction. We also illustrate the use of parameter estimation for fitting a function to thermodynamic data. 

All of the various optimization techniques described in previous chapters can be applied to one or more types of reactor models. The reactor model forms a set of constraints so that most optimization problems involving reactors must accommodate steady-state algebraic equations or dynamic differential equations as well as inequality constraints. 

Presentation of each optimization technique is followed by examples to illustrate an application. We also have included many practically oriented homework problems. In university courses, this book could be used at the upper-division or the first-year graduate levels, either in a course focused on optimization or on process design. The book contains more than enough material for a 15-week course on optimization. Because of its emphasis on applications and short case studies in Chapters 11-16, it may also serve as one of the supplementary texts in a senior unit operations or design course. 

I hope that this chapter provides a fitting introduction to the complex task of active pharmaceutical ingredient product design through ciystallization, and most importantly that it will stimulate work and encourage further growth in the application of thermodynamic models and optimization techniques in this area. 

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