Developing models for optimization Lec 5 week 6

Example (1) Classify the following model

Solution The model is linear if the ratios are independent of temperature. If they are not, then the model is nonlinear. The model is Unsteady state. The model is Distributed.

Example (2) Classify the following equation (y = dependent variable; x, z = independent variables)

Solution Linear if v x is independent of v y ; otherwise, nonlinear Steady state Distributed

Degree of freedom The degrees of freedom in a model is the number of variables that can be specified independently and is defined as follows: where N F = degrees of freedom N v = total number of variables involved in the problem. N E = number of independent equations. It is important to eliminate redundant information and equations before any calculations are performed using the developed model.

Degree of freedom A degrees-of-freedom analysis separates modeling problems into three categories: 1.N F = 0: The problem is exactly determined. If N F = 0, then the number of independent equations is equal to the number of process variables and the set of equations may have a unique solution, in which case the problem is not an optimization problem. For a set of linear independent equations, a unique solution exists. If the equations are nonlinear, there may be no real solution or there may be multiple solutions.

Degree of freedom 2. N F > 0: The problem is underdetermined. If N F > 0, then more process variables exist in the problem than independent equations. The process model is said to be underdetermined.

Degree of freedom 3. N F < 0: The problem is overdetermined. If N F < 0, fewer process variables exist in the problem than independent equations, and consequently the set of equations has no solutions. The process model is said to be over-determined.

Example: the process flow chart for a series of two distillation columns, with mass flows and splits defined by x 1, x 2,..., x 5. Write the material balances, and show how many independent variable and degree of freedom that the process model comprises.

There are 3 equations one of them can be obtained by adding the other two, then the over all that we have two equations (NE) and three un known (NV) Then NF = NV-NE = 1 For Column (1) NF = 1

For Column (2) we have three unknown (NV) and only one equation (NE) then the degree of freedom (NF) is two. And for the two columns there are three degree of freedom

Example (2) Suppose you want to design a hydrocarbon piping system in a plant between two points with no change in elevation and want to select the optimum pipe diameter that minimizes the combination of pipe capital costs and pump operating costs. Prepare a model that can be used to carry out the optimization. Identify the independent and dependent variables that affect the optimum operating conditions. Assume the fluid properties (p, p) are known and constant, and the value of the pipe length (L) and mass flow rate (m) are specified. In your analysis use the following process variables: pipe diameter (D), fluid velocity (v), pressure drop (Ap), friction factor (f).

The model equations are as follows:

The above model have: NV = 4 NE =3 NF = 1

Example Define how many degree of freedom are there in the next figure

solution With 12 variables and 9 independent linear equality constraints, 3 degrees of freedom exist that can be used to maximize profits. These degree of freedom can be over comes by assuming some constrians