1. Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery Lecture 4

2. Optimization of thermal processes2007/2008 Overview of the lecture Method of Lagrange Multipliers −Lagrange function −Necessary condition for a general problem −Sufficient condition −Example: cascade of heat exchangers Multivariable optimization with inequality constraints −Kuhn-Tucker conditions

3. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers (general problem) Minimize subject to Lagrange function n decision variablesm additional variables (Lagrange multipliers)

4. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers (necessary condtions for general problem) Necessary condtions We solve these n+m equations for: Possible relative extreme point Lagrange multipliers vector n+m variables

5. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers (sufficient condtion for general problem – Hancock formulation) where: If each root of the determinantal equation is positive (negative), then X * is minimum (maximum). If some of the roots are positive while the others are negative, the point X * is NOT an extreme point. Polynomial in z

6. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers EXAMPLE Find the dimensions of a cylindrical tin to maximize its volume such that the total surface area is equal to A 0 =24 . Maximize subject to So the constraint is

7. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers EXAMPLE contd Necessary condition Possible extreme point

8. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers EXAMPLE contd So, for A 0 =24  we have Is it an extreme point? Let’s apply sufficiency condition:

9. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers EXAMPLE contd This equation has one root: Which is negative, so the point corresponds to the maximum of. To calculate the determinant apply Sarrus method Polynomial in z (1 st order)

10. Optimization of thermal processes2007/2008 Method of Lagrange Multipliers HOMEWORK: Minimize subject to: by: a)direct substitution (see the previous lecture) b)Lagrange multipliers method (make sure you have found a minimum!) Compare the results.

11. Optimization of thermal processes2007/2008 A design problem: cascade of heat exchangers Choose the heat transfer surfaces (A 1,...,A n ) in the cascade of four heat exchangers to maximize the outlet temperature T 4. The total heat transfer surface is constrained: Assume the same mass flow rates and heat capacity at all stages. Use Lagrange Multipliers method.

12. Optimization of thermal processes2007/2008 A design problem: cascade of heat exchangers Stage n, heat transfer surface A n n The heat (energy) balance: The kinetics of heat transfer between the flows: k – heat transfer coefficient q – mass flow rate c p – heat capacity After elimination of :

13. Optimization of thermal processes2007/2008 A design problem: cascade of heat exchangers So, our optimization problem is: maximize subject to constraints: Lagrange function:

14. Optimization of thermal processes2007/2008 A design problem: cascade of heat exchangers When we employ necessary condition we arrive at: Introducing: Constraints

15. Optimization of thermal processes2007/2008 A design problem: cascade of heat exchangers variables iteration Maximum outlet temperature

16. Optimization of thermal processes2007/2008 Multivariable optimization with inequality constraints Minimize subject to Extreme point The region where the constraints are satisfied is called feasible region

17. Optimization of thermal processes2007/2008 Multivariable optimization with inequality constraints We can transform the inequality constraint to equality constraint with the use of so called slack variables: Minimize subject to where Vector of slack variables Now, we can use method of Lagrange Multipliers...

18. Optimization of thermal processes2007/2008 Multivariable optimization with inequality constraints We are looking for the stationary points in the way presented earlier: If then

19. Optimization of thermal processes2007/2008 Multivariable optimization with inequality constraints But means that i.e. the stationary point lies on the boundary of the feasible region. Feasible region Stationary point In this case the constraint is said to be active. Otherwise, the constraint is inactive. Stationary point Inactive constraint

20. Optimization of thermal processes2007/2008 Multivariable optimization with inequality constraints (Kuhn-Tucker conditions) The slack variables are not needed These conditions are only necessary. They are sufficient for convex programming For minimization

21. Optimization of thermal processes2007/2008 Thank you for your attention