Mer439 - Design of Thermal Fluid Systems Optimization Techniques Professor Anderson

1. Mer439 - Design of Thermal Fluid Systems Optimization Techniques Professor Anderson Spring Term 2012

2. Modeling/Simulation Need Identified Problem Definition Workable Design Concept Generation Optimization/ Optimal Design Concept Selection Optimization in Design

3. Optimization Set of all “workable” or “functional designs” (Allowed by physics, orange border) x2 * Optimal Design U(x1,x2) = Umax x1

4. Lingo • Objective Function: represents the quantity (U) which is to be optimized (the “objective”) as a function of one or more independent variables (x1, x2, x3…) • Design Variables: The independent variables (x1, x2, x3…) that the objective function depends on. • Constraints: Relations which limit the possible (physical limitations) or the permissible (external constraints) solutions to the objective function.

5. Mathematical Formulation • Objective Function of n independent design variables: For U( x1, x2, x3…xn) Find Uopt • Equality Constraints: Gi( x1, x2, x3…)=0 i=1,2,…,m • Inequality Constraints: Hj(x1, x2, x3…) < or > Cj j=1,2,…l If n>m → An Optimization problem results If n=m → A unique solution exists…just solve all equations simultaneously If n<m → The problem is “over-constrained” no solution which satisfies all of the constraints is possible

6. Acceptable Designs Set of all “workable” or “functional designs” (Allowed by physics, orange border) x2 H1 : X1> c1 H2 : X2 < c2 * Set of all “acceptable” designs. (allowed by constraints, yellow border) Optimal Design U(x1,x2) = Umax x1

7. Example • Set up a mathematical statement to optimize a water chilling system. The requirement of the system is that it cool 20 kg/s of water from 13 to 8 oC, rejecting the heat back to the atmosphere though a cooling tower. We seek a system with a minimum first cost to perform this duty.

8. Classification of Optimization Techniques • Calculus based Techniques • Lagrange Multipliers • “Programming” methods • Linear Programming • Geometric Programming • Search Methods • Elimination Methods • Exhaustive • Fibonacci • golden section search • “Hill Climbing” techniques • Lattice Search • Steepest ascent

9. The aptly named Exhaustive Search x2 H1 : X1> c1 H2 : X2 < c2 * Note: None of the search points exactly hits the optimum. The space between search points is known as the “interval of uncertainty” Optimal Design U(x1,x2) = Umax x1

10. Search Methods • Types of Approaches • Elimination Methods • Hill Climbing Techniques • Constrained Optimization

11. Unconstrained Search with Multiple Variables.

12. Lattice Search 1 3 * * 2 4 * *

13. Lattice Search

14. Univariate Search

15. Example (Univariate Search) • Find the minimum value for y, where: • Use only integer values of x1 and x2 and start w/ x2 = 3

16. The Project • What exactly are you trying to optimize? → “What is your Objective Function?” • What is the Absolute maximum that one would be willing to pay? → “Is there a cost inequality constraint that we can use to help limit our design domain?” • What are your “design variables” ? • What is the nature of your functions? (continuous / discrete) (linear/non-linear) etc.

17. Presentation Must Include • A clear representation of your Objective Function • Clear Representations of your constraints • A description of your optimization methods • Evidence of a Sanity Check on your proposed solution