Optimal Location of Multiple Bleed Points in Rankine Cycle
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Transcript of Optimal Location of Multiple Bleed Points in Rankine Cycle
Optimal Location of Multiple Bleed Points in Rankine Cycle
P M V SubbaraoProfessor
Mechanical Engineering DepartmentI I T Delhi
Sincere Efforts for Best Returns…..
C
OFWH
OFWH
OFWH
Turbine
yj, hbjYj-11,hbj-1 Yj-2,hbj-2
1- yj – yj-1- yj-2
hf (j-3)
1- yj – yj-1
hf (j-2)
1- yj
hf (j-1)
1 ,hf (j)
A MATHEMATICAL MODEL
SG
A
B
C
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 STEAM TURBINE
n
iiSGcond ymm
1
1
y1,
hb1
yi,
hbi
y(i-1)
hb(i-1)
mie ,
hfi
mi,i,
hf(i-1)
STEAM IN
STEAM OUT
SGm
SGm
Mass of steam leaving the turbine is
Contributions of Bleed Steam
The power developed by the bleed steam of ith heater before it is being extracted is given by
hs
yi
hbj
TURBINE
)( bisiSGbi hhymW
SGm
OPTIMIZATION METHODOLOGY (Contd..)
The work done by the bleeds of all feed water heaters is given by:
n
ibisiSG
n
ibi hhymW
11
)(
ANALYSIS OF ‘ith’ FEED WATER HEATER
)1(1
,
n
ijjSGei ymm
)1(,
n
ijjSGii ymm
yi , hbi
hfi h f i-1
ith heater
eim , iim ,
Mass balance of the heater at inlet and exit is given by:
• Energy balance of the feed heater gives:
fieibiiSGfiii hmhymhm ,1,
n
ij jfjb
jfjfn
ij jfjb
jfjf
i hh
hh
hh
hhy
1
1
1 1
1 11
n
ij j
jn
ij j
jiy
111
n
iiSGcond ymm
1
1
n
i
n
ij j
jn
ij j
jSGcond mm
1 1
111
T-S DIAGRAM FOR REGENERATION CYCLE
S
A
B
0
Di
i-1
C
T
CBcondout hhmQ
DASGin hhmQ
Therefore the thermal efficiency of the cycle is
DASG
CBcond
in
out
hhm
hhm
Q
Q
11
n
i
n
ij j
jn
ij j
jSGcond mm
1 1
111
Modified Heywood’s Model
Maximize:
n
i
n
ij j
jn
ij j
jcond
SG
m
m
1 1
111
1
n
i
n
ij j
jn
ij j
j
1 1
11
Or Maximize:
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
ibiAiSG
n
ibi hhymW
11
)(
n
ibiA
n
ij jfjb
jfjfn
ij jfjb
jfjf
SG
n
ibi hh
hh
hh
hh
hhmW
1 1
1
1 1
1
1
)(11
OPTIMIZATION MODELOptimization problem can now be expressed as :
Maximising the function
Where hfi = f(p(i)) , hf(i+1) = f(p(i+1)) and hbi = f(pi, s)
And subjected to following constraints:
)(1
)1( CD
n
iifif hhhh
hfi , hf(i+1) , hbi >0
n
ibiA
n
ij jfjb
jfjfn
ij jfjb
jfjf
SG
n
ibi hh
hh
hh
hh
hhmW
1 1
1
1 1
1
1
)(11
Artificial Intelligence Technique Applied to Optimization of OFWHs
P M V SubbaraoProfessor
Mechanical Engineering DepartmentI I T Delhi
Best Blue Print for Carnotization of Rankine cycle…
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.
OPTIMIZATION PROCEDURE
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
D = 20 units
D = 20 units
Area of a square : 400 square units.
Area of Circle: 314.15 square units.
7853981.04
square tocircle of Ratio
Example of Monte Carlo Method
Area = D2
2.38.0420
16
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.
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
If eff = high &
∑ ∆h < hd– hc
η OPT = η (i)POPT = P(i)
η OPT = η ( i-1)Popt= P( i-1)
YES NO
START
For generation i = 1 to Maximum no of generation
Generate ‘n’ pressures in between
Pmax & Pmin
randomly
Calculate Yi , fraction of bled
steam extracted at each pressure Pi
Calculate the bleed work & Efficiency of
the cycle
End
set Popt
If i = Max
INPUT n, Tmax ,
Pmax, Pmin, max no. of generation
OUTPUT: efficiency,Popt
NO
YES
Go to 1
Flow Chart for optimisation
Calculate hbi, hfi at each pressure Pi and hboi, ht1, hc1, hc2.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.35
0.355
0.36
0.365
0.37
0.375
0.38
0.385
0.39
NUMBER OF GENERATION
CY
CLE
EF
FIC
IEN
CY
Number of generation and Efficiency
RESULTS
S.NO Pmax Tmax Tmin PminMpa oC oC Mpa ΔH = constant ΔT = constant Simulated
1 12.75 535 25.7 0.0033 43.4 47.6 50.642 23.5 540 26 0.0034 30.8 36.97 53.543 12.74 565 23.97 0.00298 44.39 46.72 51.24 15 550 40 0.0074 46.73 50.285 53.995 16.5 535 40 0.01 38.8 43.08 49.022
GENERATIONS = 5000 /// NO OF FEED HEATERS = 6 Thermal Efficiency
RESULTS
S.NO Pmax Tmax Tmin Pmin W bledsteam Popt
1 12.75 535 25.7 0.0033 234.2 7.28 , 4.88 , 3.42 , 3.09 , 1.326 , 0.2762 23.5 540 26 0.0034 302.093 12.74 565 23.97 0.00298 242.164 15 550 40 0.0074 270.35 16.5 535 40 0.01 249.589 5.293, 2.33, 1.096, 0.779, 0.651, 0.0674
7.28 , 4.88 , 3.42 , 3.09 , 1.326 , 0.276
5.3 , 2.23 , 1.096 , 0.779 , 0.651 , 0.06748.28, 4.88, 3.7792, 2.0907, 0.48, 0.2597
RESULTS
S.NO H2 Pmax Tmax Tmin Pmin THERMAL EFFICIENCYSimulated
1 1465 16.75 535 25.7 0.0033 49.192 1500 16.75 535 25.7 0.0033 49.0223 1550 16.75 535 25.7 0.0033 49.0224 1600 16.75 535 25.7 0.0033 49.022
NO OF FEED HEATERS = 6
34
35
36
37
38
39
40
0 1 2 3 4 5 6
No of Feed water Heaters
Eff
icie
nc
y(%
)
800
850
900
950
1000
1050
1100
Wo
rk o
utp
ut(
KJ
/kg
)
Effect of no of feed water heaters on thermal efficiency and work output of a regeneration cycle
Thermal Efficiency
Work output
Work output
Closed Feed Water Heaters (Throttled Condensate)