Developing models for optimization

1. Developing models for optimization Lec. 3

2. Why we need models? We develop mathematical models for two main purposes: 1. Enhance understanding - Does the pressure affect the reaction rate - Does the traffic flow affect income levels - Does CO2 affect the earth’s temperature? 2. Enhance prediction accuracy - What is the reactor volume for a reaction, -rA = k1CA/(1+k2CA) - How much investment is required to produce 300,000 tons/yr of ethylene?

3. CLASSIFICATION OF MODELS Two general categories of models exist: • Those based on physical theory. • Mathematical models based on physical and chemical laws (e.g., mass and energy balances, thermodynamics, chemical reaction kinetics) are frequently employed in optimization applications. These models are conceptually attractive because a general model for any system size can be developed even before the system is constructed. 2. Those based on strictly empirical descriptions (so-called black box models). - Empirical models are attractive when a physical model cannot be developed due to limited time or resources. Input-output data are necessary in order to fit unknown coefficients in either type of the model.

4. CLASSIFICATION OF MODELS • In addition to classifying models as theoretically based versus empirical, we can generally group models according to the following types: • Linear versus nonlinear. • Steady state versus unsteady state. • Lumped parameter versus distributed parameter. • Continuous versus discrete variables.

5. Linear versus nonlinear • Equations (and hence models) are linear if the dependent variables or their derivatives appear only to the first power: otherwise they are nonlinear. • In practice the ability to use linear models is of great significance because they are an order of magnitude easier to manipulate and solve than nonlinear ones.

6. Linear versus nonlinear (example: • A coal combustion pilot plant is used to obtain efficiency data on the collection of particulate matter by an electrostatics precipitator (ESP). The ESP performance is varied by changing the surface area of the collecting plates. Two models of different complexity have been proposed to fit the performance data; Examine models 1 and 2 for the electrostatic precipitator. Is model 1 linear in A? Model 2? • Model 1 is linear in the coefficients, and model 2 is nonlinear in the coefficients.

7. Steady state versus unsteady state • steady state are time-invariant, static, or stationary. These terms refer to a process in which the values of the dependent variables remain constant with respect to time. • Unsteady state processes are also called non steady state, transient, or dynamic and represent the situation when the process-dependent variables change with time. • A typical example of an unsteady state process is the operation • of a batch distillation column, which would exhibit a time-varying product composition. • A transient model reduces to a steady state model when = 0.

8. Distributed versus lumped parameters • Briefly, a lumped parameter representation means that spatial variations are ignored and that the various properties and the state of the system can be considered homogeneous throughout the entire volume. • A distributed parameter representation, on the other hand, takes into account detailed variations in behavior from point to point throughout the system.

9. Distributed versus lumped parameters • In next Figure, compare these definitions for a well-stirred reactor and a tubular reactor with axial flow. • In the first case, we assume that mixing is complete so no concentration or temperature gradient occurs in the reactor, hence a lumped parameter mathematical model would be appropriate. • In contrast, the tubular reactor has concentration or temperature variations along the axial direction and perhaps in the radial direction, hence a distributed parameter model would be required.

10. Continuous versus discrete variables • Continuous variables can assume any value within an interval; • Discrete variables can take only distinct (cetrain values).

11. HOW TO BUILD A MODEL • Model building can be divided into four phases: (1) problem definition and formulation, (2) preliminary and detailed analysis, (3) evaluation, and (4) interpretation application. • Keep in mind that model building is an iterative procedure.

12. How to Determine the Form of a Model • Models can be written in a variety of mathematical forms. • Optimization methods can help in the selection of the model structure as well as in the estimation of the unknown coefficients. • The best model presumably exhibits the least error between actual data and the predicted response in some sense.

13. Graphical presentation of data assists in determining the form of the function of a single variable (or two variables). • The response y versus the independent variable x can be plotted and the resulting form of the model evaluated visually.

14. Fitting Models by Least Squares • least squares estimation, is used to calculate the values of the coefficients in a model from experimental data. • In estimating the values of coefficients for a model, keep in mind that the number of data sets must be equal to or greater than the number of coefficients in the model.

15. The oldest (and still the most frequent) use of Ordinary Least Square was linear regression, which corresponds to the problem of finding a line (or curve) that best fits a set of data points. • In the standard formulation, a set of N pairs of observations {Yi ,Xi } is used to find a function relating the value of the dependent variable (Y ) to the values of an independent variable (X ). Which we explained as the empirical model

18. solution