1. Optimization of thermal processes2007/2008 Optimization of thermal processes Maciej Marek Czestochowa University of Technology Institute of Thermal Machinery Lecture 2

2. Optimization of thermal processes2007/2008 Overview of the lecture Extreme points (maximum or minimum) Unconstrained optimization and differential calculus methods −Necessary and sufficient conditions Applications to engineering −Optimum design of two-stage compressor −Optimum pipe diameter

3. Optimization of thermal processes2007/2008 Extreme points x f(x) a b A1A1 A2A2 A3A3 B1B1 B2B2 A 1, A 2, A 3 – relative maxima B 1, B 2 – relative minima A 2 – global maximum B 1 – global minimum x f(x) ab Relative minimum is also global minimum

4. Optimization of thermal processes2007/2008 Unconstrained optimization (differential calculus methods) Find x * which minimizes f(x). Necessary condition The necessary condition is satisfied in all stationary points. Be careful! There are stationary points that are not extreme: f(x) x Stationary point f’(x)=0

5. Optimization of thermal processes2007/2008 Unconstrained optimization (differential calculus methods) Sufficient condition Let But Then: If n is even f(x * ) is minimum value if f (n) (x * ) > 0 f(x * ) is maximum value if f (n) (x * ) < 0 If n is odd, x * is not an extreme point Exercise: Determine the maximum and minimum values.

6. Optimization of thermal processes2007/2008 Compressors Heat exchangers Optimum design of two-stage compressor Work input Objective: find pressure p 2 to minimize work input. c p – specific heat of the gas (constant pressure) T – temperature p 1 – initial pressure p 3 – final pressure

7. Optimization of thermal processes2007/2008 Optimum design of two-stage compressor Necessary condition Is it really optimum?

8. Optimization of thermal processes2007/2008 Optimum design of two-stage compressor Sufficient condition Hence: relative minimum at p 2

9. Optimization of thermal processes2007/2008 Optimum pipe diameter S L p1p1 p2p2 D G [kg/s], U [m/s] Given : G [kg/s],  [kg/m 3 ],  [kg/ms] Amount of mass transfered in unit time DensityViscosity Objective: find the most economical pipe diameter, i.e. minimize the total cost Investment costOperation cost

10. Optimization of thermal processes2007/2008 Optimum pipe diameter Investment and maintenance cost A,n – given constants Operation cost B – given constant, t – time, N – power of the engine Total cost as a function of D for fixed L and t. This is the function we are going to minimize (objective function) What about constraints?

11. Optimization of thermal processes2007/2008 Optimum pipe diameter Work output Pressure drop Power output Velocity Efficiency Now, all we need is pressure drop  p

12. Optimization of thermal processes2007/2008 Optimum pipe diameter Pressure drop depends on the flow conditions −laminar flow (Hagen-Poiseuille formula) −turbulent flow (Darcy-Weisbach formula) −pipe surface (rough or smooth) Let’s assume that the flow is laminar and the pipe is smooth. Then: H-P formula So the total cost can be finally expressed as:

13. Optimization of thermal processes2007/2008 Optimum pipe diameter Necessary condition Sufficient condition for the relative minimum Homework : do the calculations!

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