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Modeling Simplification Techniques in Optimization: A Comprehensive Guide

Explore various simplifications in mathematical modeling for optimization, understand empirical and probabilistic models, and learn about modeling precautions to ensure accuracy. Discover factorial design and least squares fitting.

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Modeling Simplification Techniques in Optimization: A Comprehensive Guide

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  1. Chapter 2 Chapter 2 Developing Models for Optimization

  2. Chapter 2

  3. Chapter 2

  4. Chapter 2

  5. Chapter 2

  6. Everything should be made as simple as possible, but no simpler Chapter 2

  7. Chapter 2

  8. Chapter 2

  9. Chapter 2

  10. Chapter 2

  11. TERMINOLOGY OF MATHEMATICAL MODELS There are many additional ways to classify mathematical models besides those used in Chapter 2. For our purposes it is most satisfactory to first consider grouping the models into opposite pairs: deterministic vs. probabilistic linear vs. nonlinear steady state vs. nonsteady state lumped parameter vs. distributed parameter black box vs. fundamental (physical) Chapter 2

  12. Common Sense in Modeling What simplifications can be made? How are they justified? Types of Simplifications Chapter 2 • Omitting Interactions • Aggregating Variables • Eliminating Variables • Replace Random Variables with Expected Values • Reduce Detail of Mathematical Description

  13. Chapter 2

  14. Chapter 2

  15. Precautions in Model Building • Limits on availability of data and accuracy of data • Examples: Kinetic coefficients • Mass transfer coefficients • (2) Unknown factors present or not present in scale up • Examples: Impurities in plant streams • Wall effects • (3) Poor measures of deviation between ideal and actual • models • Examples: Stage efficiency Chapter 2

  16. (4) Models used for one purpose used improperly for another purpose Example: Invalidity of kinetic models (5) Extrapolation – using the model outside of the regions Where it has been validated Chapter 2

  17. Chapter 2

  18. Chapter 2

  19. 2. Empirical Models Chapter 2 3. Probabilistic concepts applied to small physical subdivisions of the process Not often used

  20. Chapter 2

  21. Chapter 2

  22. Chapter 2

  23. Chapter 2

  24. Chapter 2

  25. Chapter 2 FIGURE E2.3b Variation of overall heat transfer coefficient with tube-side flow rate Wt for ws = 4000. FIGURE E2.3a Variation of overall heat transfer coefficient with shell-side flow rate ws = 80,000.

  26. Semi-empirical Model Fitting Heat exchanger data, p. 54 curve D in Eqn (3), Figure 2.6 Chapter 2

  27. Quadratic Curve Fitting Least squares analysis leads to 3 linear equations in 3 unknowns (n data points) Chapter 2 What about (coefficients must appear linearly)

  28. Chapter 2

  29. Factorial Design and Least Squares Fitting Chapter 2 for data matrix on p. 65 (see Fig. E2.6)

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