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)