1. Developing Models for Optimization
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. Chapter 2 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. Chapter 2 Common Sense in Modeling Types of Simplifications
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. Chapter 2 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. Chapter 2 (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. Chapter 2 2. Empirical Models
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 FIGURE E2.3a 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. Chapter 2 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. Chapter 2 Factorial Design and Least Squares Fitting
for data matrix on p. 65 (see Fig. E2.6)