Optimization of thermal processes
Lecture 2
Maciej Marek
Czestochowa University of Technology
Institute of Thermal Machinery
Optimization of thermal processes
2007/2008
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
Optimization of thermal processes
2007/2008
Extreme points
A1, A2, A3 – relative maxima
B1, B2 – relative minima
A2
f(x)
A3
A1
A2 – global maximum
B1 – global minimum
B2
f(x)
B1
a
b
x
a
Relative minimum is also
global minimum
Optimization of thermal processes
2007/2008
b x
Unconstrained optimization
(differential calculus methods)
Find x* which minimizes f(x).
Necessary condition
f ( x* )  0
The necessary condition is satisfied in all stationary points.
Be careful! There are stationary points that are not extreme:
f(x)
Stationary point
f’(x)=0
x
Optimization of thermal processes
2007/2008
Unconstrained optimization
(differential calculus methods)
Sufficient condition
Let
f ( x* )  f ( x* )  ...  f ( n1) ( x* )  0
But
f ( n ) ( x* )  0
Then:
• If n is even
• f(x*) is minimum value if f(n)(x*) > 0
Exercise:
f ( x)  12 x5  45 x 4  40 x3  5
• f(x*) is maximum value if f(n)(x*) < 0
• If n is odd, x* is not an extreme point
Optimization of thermal processes
Determine the maximum
and minimum values.
2007/2008
Optimum design of two-stage compressor
Work input
 p  ( 1) /   p ( 1) / 

W  c pT  2 
 3 
 2
 p1 

 p2 

Objective: find pressure p2 to minimize work input.
cp – specific heat of the gas
(constant pressure)
Compressors
T – temperature
p1 – initial pressure
p3 – final pressure
Heat exchangers
Optimization of thermal processes
2007/2008
cp
cV
Optimum design of two-stage compressor
dW
0
dp2
Necessary condition
( 1) / 

dW
 1
 1
1/ 
 
 c pT
 p2 
dp2
  1  p1 



( 1) /    1
 ( p3 )
( p2 )(1 2 ) /    0


p2 
Optimization of thermal processes
p1 p3
Is it really optimum?
2007/2008
Optimum design of two-stage compressor
d 2W
d 2W
 0 or
0
2
2
dp2
dp2
  1
dW
 c pT    
2
dp2
  p1 
2
( p3 )
 d 2W 
 2 
 dp2  p2 
( 1) / 

p1 p3
Sufficient condition
( 1) / 
1

1  2

2c p T
( p2 ) (1 ) / 

( p2 )(13 ) /  

 1

p1(3 1) / 2 p3( 1) / 2
 0 as
 >1
Hence: relative minimum at p2
Optimization of thermal processes
2007/2008
Optimum pipe diameter
p1
D
G [kg/s], U [m/s]
S
p2
L
Density
Viscosity
Given: G [kg/s],  [kg/m3],  [kg/ms]
Amount of mass transfered in unit time
Objective: find the most economical pipe diameter, i.e.
minimize the total cost
 K  K IM  KOP
Investment cost
Optimization of thermal processes
Operation cost
2007/2008
Optimum pipe diameter
K IM  ALD n
KOP  BtN ( D)
Investment and maintenance cost
A,n – given constants
Operation cost
B – given constant, t – time, N – power of the engine
 K  K IM  K OP  ALD n  BtN ( D )
 K  C1 D n  C2 N ( D )
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?
Optimization of thermal processes
2007/2008
Optimum pipe diameter
Wu  F  L  ( p1  p2 )  S  L  p  SL
Pressure drop
Work output
Wu
L
Nu 
 p  S
 p  S  U
t
t
Velocity
Power output
G   SU  SU 
G
Nu  N  N 

p  G
N

Optimization of thermal processes
Nu

Efficiency
Now, all we need is
pressure drop p
2007/2008
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:
128G  L
p 
 D 4
N ( D)
H-P formula
1
D4
So the total cost can be finally
expressed as:
Optimization of thermal processes
1
 K  C1 D  C3 4
D
n
2007/2008
Optimum pipe diameter
d ( K )
0
d ( D)
d 2 ( K )
0
d ( D)
Necessary condition
Sufficient condition for the
relative minimum
Homework: do the calculations!
Optimization of thermal processes
2007/2008
Thank you for your attention
Optimization of thermal processes
2007/2008