Optimal Location of Multiple Bleed Points in Rankine Cycle
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Sincere Efforts for Best Returns…..
A MATHEMATICAL MODEL
A
Turbine
B
S
G
Yj-11,hbj-1
yj, hbj
Yj-2,hbj-2
C
OFWH
1 ,hf (j)
1- yj
hf (j-1)
OFWH
OFWH
1- yj – yj-1
hf (j-2)
C
1- yj – yj-1- yj-2
hf (j-3)
n number of OFWHs require n+1 no of Pumps…..
The presence of pumps is subtle…
ANALYSIS OF ‘ith’ FEED WATER HEATER
• Mass entering the
turbine is
m SG
STEAM
IN
STEAM TURBINE
m SG
Mass
of steam
leaving the turbine is
m cond
 n 
 m SG 1   yi 
 i 1 
STEAM
OUT
yi,
y(i-1)
hbi
hb(i-1)
mie ,
hfi
mi,i,
hf(i-1)
y1,
hb1
Contributions of Bleed Steam
m SG
 The power developed by
the bleed steam of ith
heater before it is being
extracted is given by
hs
TURBINE
yi
hbj
 SG yi (hs  hbi )
Wbi  m
OPTIMIZATION METHODOLOGY (Contd..)
The work done by the bleeds of all feed water heaters is
given by:
n
n
i 1
i 1

W
 bi  m SG  yi (hs  hbi )
ANALYSIS OF ‘ith’ FEED WATER HEATER
Mass balance of the heater at inlet and exit is given by:
n
m i ,i  m SG (1   y j )
yi , hbi
j i
n
m i ,e  m SG (1   y j )
j i 1
ith heater
hfi
m i ,e
• Energy balance of the feed heater gives:
m i ,i h fi1  m SG yi hbi  m i ,e h fi
hf
i-1
m i ,i



h f j  h f j 1

yi   1 
hb j  h f j 1
j i 1

n
  1  h
 
  h
n
j i
f j
 hf
j 1
bj
 hf
j 1

 j
 n 
 j

yi   1 
  1 

j i 1
 j 
 j i 
 j 

n
m cond
m cond
 n 
 m SG 1   yi 
 i 1 
 n  n   j  n   j  
 m SG 1     1    1   
 i 1  j i 1  j  j i   j  


 
 




A
D
T
 cond hB  hC 
Q out  m
 SG hA  hD 
Q in  m
i
i-1
B
C
0
S
T-S DIAGRAM FOR REGENERATION CYCLE
Therefore the thermal efficiency of the cycle is
 cond hB  hC 
Q out
m
  1   1
 SG hA  hD 
Qin
m
m cond
 n  n   j  n   j  
 m SG 1     1    1   
 i 1  j i 1  j  j i   j  


 
 

Modified Heywood’s Model
Maximize:
m SG
1

n 
n  n 
m cond 










j
j
1  

1


1



 
 

 





i 1 
j

i

1
j

i
j
j 



 


Or Maximize:
 n   j  n   j 
  1    1  





i 1  j i 1
j

i
j
j 





n
Maximum Bleed Steam Power Model
• Fundamentally, the steam is generated to produced
Mechanical Power.
• However, after expanding for a while, the scope for
internal utilization of some steam for feed water heating
looks lucrative.
• To have a balance between above two statements.
• Any optimal cycle should lead to:
• Maximization of the combined power generated by all
the bleed streams.
Therefore the work bone by bled steams can be
written as
n
n
i 1
i 1

W
 bi  m SG  yi (hA  hbi )


 n 
h f j  h f j 1


Wbi  m SG    1 

hb j  h f j 1
i 1
i 1  j i 1
 
n
n
  1  h
 
  h
n
j i
f j
 hf
bj
 hf
j 1
j 1
(h


A
 hbi )
OPTIMIZATION MODEL
Optimization problem can now be expressed as :
Maximising the function


 n 
h f j  h f j 1


Wbi  m SG    1 

hb j  h f j 1
i 1
i 1  j i 1
 
n
n
  1  h
 
  h
n
j i
Where hfi = f(p(i)) , hf(i+1) = f(p(i+1)) and hbi = f(pi, s)
And subjected to following constraints:
 h
n
i 1
fi

 h f (i 1)  (hD  hC )
hfi , hf(i+1) , hbi
>0
f j
 hf
bj
 hf
j 1
j 1
(h


A
 hbi )
Artificial Intelligence Technique Applied to
Optimization of OFWHs
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Best Blue Print for Carnotization of Rankine cycle…
OPTIMIZATION PROCEDURE

Suitable method to find the value of the variable that
maximize the objective function.

The design variables and the constraints show that the
system optimization is a non-linear programming
problem.

For such problems, a Monte Carlo
simulation
technique has been found to be quite efficient.
Monte Carlo Method
• A Random Walk Method.
• Solve a problem using statistical sampling
• Name comes from Monaco’s gambling resort city
Example of Monte Carlo Method
Area of a square : 400 square units.
Area of Circle: 314.15 square units.
D = 20 units
D = 20 units
Ratio of circle to square 

4
 0.7853981
Example of Monte Carlo Method
Generate 20 random number in the range 1 to 400.
Locate them inside circle or outside circle based on their value.
Count the points lying inside the circle.
Area = D2
16 
  0.8    3.2
20 4
Increasing Sample Size Reduces Error
n
Estimate
Error
1/(2n1/2)
10
2.40000
0.23606
0.15811
100
3.36000
0.06952
0.05000
1,000
3.14400
0.00077
0.01581
10,000
3.13920
0.00076
0.00500
100,000
3.14132
0.00009
0.00158
1,000,000
3.14006
0.00049
0.00050
10,000,000
3.14136
0.00007
0.00016
100,000,000
3.14154
0.00002
0.00005
1,000,000,000
3.14155
0.00001
0.00002
START
Flow Chart for optimisation
Calculate Yi ,
fraction of bled
steam extracted at
each pressure Pi
INPUT n, Tmax ,
Pmax, Pmin, max
no. of generation
For generation
i = 1 to Maximum
no of generation
Generate ‘n’
pressures in between
Pmax & Pmin
randomly
NO
If i = Max
Calculate the bleed
work & Efficiency of
the cycle
YES
η OPT = η (i)
POPT = P(i)
Calculate hbi, hfi
at each pressure Pi
and hboi, ht1, hc1, hc2.
Go to 1
If eff =
high &
∑ ∆h <
hd– hc
NO
η OPT = η ( i-1)
Popt= P( i-1)
YES
set Popt
OUTPUT:
efficiency,Popt
End
Number of generation and Efficiency
0.39
0.385
CYCLE EFFICIENCY
0.38
0.375
0.37
0.365
0.36
0.355
0.35
0
500
1000
1500
2000
2500
3000
3500
NUMBER OF GENERATION
4000
4500
5000
RESULTS
S.NO
1
2
3
4
5
GENERATIONS = 5000 /// NO OF FEED HEATERS = 6
Pmax Tmax Tmin Pmin
Thermal Efficiency
Mpa
oC oC
Mpa
ΔH = constant ΔT = constant Simulated
12.75
23.5
12.74
15
16.5
535
540
565
550
535
25.7
26
23.97
40
40
0.0033
0.0034
0.00298
0.0074
0.01
43.4
30.8
44.39
46.73
38.8
47.6
36.97
46.72
50.285
43.08
50.64
53.54
51.2
53.99
49.022
RESULTS
S.NO
1
2
3
4
5
Pmax
12.75
23.5
12.74
15
16.5
Tmax Tmin
535
540
565
550
535
25.7
26
23.97
40
40
Pmin
0.0033
0.0034
0.00298
0.0074
0.01
W bledsteam Popt
234.2
302.09
242.16
270.3
249.589
7.28 , 4.88 , 3.42 , 3.09 , 1.326 , 0.276
5.3 , 2.23 , 1.096 , 0.779 , 0.651 , 0.0674
8.28, 4.88, 3.7792, 2.0907, 0.48, 0.2597
7.28 , 4.88 , 3.42 , 3.09 , 1.326 , 0.276
5.293, 2.33, 1.096, 0.779, 0.651, 0.0674
RESULTS
S.NO
1
2
3
4
H2
Pmax
1465
1500
1550
1600
16.75
16.75
16.75
16.75
NO OF FEED HEATERS = 6
Tmax
Tmin
Pmin
535
535
535
535
25.7
25.7
25.7
25.7
0.0033
0.0033
0.0033
0.0033
THERMAL EFFICIENCY
Simulated
49.19
49.022
49.022
49.022
40
1100
39
1050
38
1000
37
950
36
900
35
Work output(KJ/kg)
Efficiency(%)
Thermal Efficiency
850
Work output
34
800
0
1
Work
2 output
3
4
5
6
No of Feed water Heaters
Effect of no of feed water heaters on thermal efficiency and work output
of a regeneration cycle
Closed Feed Water Heaters (Throttled
Condensate)