32

CHAPTER 3

DEVELOPMENT OF COMPUTER PROGRAMFOR STEAM PROPERTIES

The thermodynamic properties of steam at different

temperatures/pressures upto the critical point of the steam

(374.140C/221.2 bar) are readily available in the literature. In order to

carry out the thermodynamic analysis of steam power plant based on

supercritical/ultra supercritical/advanced ultra supercritical cycle

without reheat/with single reheat/with double reheat, it is required to

find the thermodynamic properties of steam in these ranges. Though,

complicated thermodynamic relations are available for each of several

regions of steam (ISI Steam Tables in SI units [10]) to calculate different

thermodynamic properties at different pressures and temperatures, it is

a tedious job to find them. To avoid this tedious job, the present

investigation has successfully developed a software program for finding

the values of these properties at a given temperature and pressure in any

of these regions.

3.1 SUBREGIONS OF PRESSURE-TEMPERATURE ANDTEMPERATUE-ENTROPY DIAGRAMS

ISI steam tables [10] presented formulations which describes the

thermodynamic properties of ordinary water substance throughout the

whole of region that extends in pressure from the ideal gas limit (zero

33

pressure) to a pressure of 1000 bar and that extends in temperature

from

Figure 3.1: Illustration of sub regions on the pressure- temperature diagram

Figure 3.2: Illustration of sub regions on the temperature- entropy diagram

34

273.16K (0.010 C) to 1073.15K (8000C). This whole region is divided into

six sub regions, numbered 1 to 6 and shown on the pressure-

temperature plane in Figure 3.1 and on the temperature- entropy plane

in Figure 3.2.

3.2 FORMULATION AND EQUATION

In order to develop a computer program which is capable of

predicting different thermodynamic properties water/steam at different

temperatures and pressures, the formulations and equations presented

in ISI steam tables[10] have been reproduced here under. For better

understanding of the formulation and equations the reduced

dimensionless quantities used are presented below.

3.2.1 Reduced Dimensionless QuantitiesP/Pc1 = β, the reduced pressure

T/Tc1 = θ, the reduced temperature

V/Vc1 = χ, the reduced volume

h/(Pc1 Vc1) =ε, the reduced enthalpy

s/( Pc1 Vc1/Tc1) = σ, the reduced entropy

g/(Pc1 Vc1) = ε - θ σ = ξ, the reduced free enthalpy(Gibbs Function)

f/( Pc1 Vc1) = ξ-βχ =ψ, the reduced free energy(Helmholtz function)

R1Tc1/( Pc1 Vc1) = I1, the reduced ideal gas constant

Ps/Pc1 = βk(θ), the reduced saturated pressure, where Ps=Ps(T)

Ps/Tc1 = θk(β), the reduced saturation temperature, where Ts=Ts(P)

Tt/Tc1= θk, the reduced saturation temperature

35

Pt/Pc1 = βt(θt), the reduced triple point pressure

3.2.2 Thermodynamic RelationsThe known thermodynamic relations

s= - (∂g/∂T)p=-(∂f/∂T)v

v=+(∂g/∂p)T

p=-(∂f/∂v)T

h= g+Ts= f+pv+Ts

When written in terms of the reduced dimensionless quantities become

σ = -(∂ξ/∂θ)β =-(∂ψ/∂θ)x

χ = (∂ξ/∂θ)θ

β= -(∂ψ/∂χ) θ

ε=ξ+θσ/ασ=ψ + βχ + θσ

The reduced specific heat capacities are given by

= -θ (∂2ξ/∂θ2)β

= -θ ((∂2ψ/(∂θ2)x + θ ((∂2ψ/(∂x ∂θ))2/((∂2ψ/(∂x2))θ

= -θ (∂2ψ/∂θ2)x

= -θ ((∂2ξ /(∂θ2) β + θ ((∂2ξ /(∂θ ∂β))2/((∂2ξ /(∂β 2))θ

3.3 SPECIFICATION OF THE SUB-REGIONS3.3.1 The Subregions are specified in the following table andillustrated in Figure 3.1 and 3.2

Temperature range Pressure range Subregion

36

The functions βK(θ) and βL(θ) are equations in boundaries between sub

regions, the K-function being the equation for the saturation line and the

L-function the equation for the boundary between sub regions 2 and 3.

3.3.2 Equations for Boundaries between Sub-regions

3.3.2.1 The K-Function (Saturation line):

This function is given in 3.5

3.3.2.2 The L-function: Reduced pressure along the boundary between

sub regions 2 and 3

βL= βL(θ)

= 2 1 1 2 2 1

2 1

( ) ( ) ( )( )( )

L

θ t ≤ θ ≤ θ 1

O ≤ β < βK(θ)

β = βK(θ)

βK(θ) < β ≤ β2

2

6

1

θ 1 < θ < 1

O ≤ β ≤ β2(θ)

β2(θ) < β < βK(θ)

β = βK(θ)

βK(θ)< β ≤ β2

2

3

5

4

1 ≤ θ < θ 2 O ≤ β < β2(θ)

βL(θ) < β ≤ β2

2

3

θ 2 ≤ θ ≤ θ 3 O ≤ β ≤ β2 2

37

Where 2dd

= ( )I IL L

= 2 1 ( 2 1

2 1

2 )L

Derived forms for βL and IL convenient for computer use, are in 3.8.2.

3.3.3 Constants relating to boundaries between subregions

3.3.3.1 Primary constants: The constant L and the constants relating

to the K-function are given in A.1.1.5 Saturation Line (Appendix A.1).

3.3.3.2 Expressions for values of derived constants:Expressions for the values of the derived constants θt, θ1, θ2, θ3, β1, β2 are

given in A.1.1.5 Saturation Line (Appendix A.1).

3.3.3.3 Numerical values of derived constants:

The numerical values of θ1, θ2, θ3, β1 and β2 are given in 3.8.1, and the

numerical values of the constants relating to the derived forms for βL and

β1L are given in 3.8.2

3.4 SUB – FORMULATIONS

3.4.0 For each subregion these are set out below

a) The canonical function and

b) The derived functions and the relations between the canonical

and derived functions.

The functions of each sub formulation are identified by the same number

as that identifying the subregion.

3.4.1 Subregion 1

38

1 0 1( , ) ( , )A

Where 0 ,( / ) ( / )

t tA A Ad d d

1 ,( / )

t tA

1 1( , ) ( / )x x

1 1 1( , )

1 1( , ) ( / )

3.4.2 Subregion 2

2 1( , ) ( , )B o

2 2( , ) ( / )x x

2 2( , ) ( / )

2 2 2( , )

3.4.3 Subregion 3

3 1 1( , ) ( )C ox x

3( , ) ( / )x x

3 3( , ) ( / )xx

3 3 3 3( , )x x

3 3( , )A x x

3.4.4 Subregion 4

39

4 1( , ) ( , ) ( , )C o Dx x x

4 ( , ) ( / )A xx

4 4( , ) ( / )xx

4 4 4 4( , )x x

4 4( , )A x x

3.4.5 Subregions 5 and 6

( )K

q = Dryness fraction = f f f

g f g f g f

x xx x

xf = x4(θ,βK(θ))

xg = x3(θ,βK(θ))

σf = σ4(θ,βK(θ))

σg = σ3(θ,βK(θ))

εf = ε4(θ,βK(θ))

εg = ε3(θ,βK(θ))

3.4.6 Subregion 5

xf = x1(θ,βK(θ))

xg = x2(θ,βK(θ))

σf = σ1(θ,βK(θ))

σg = σ2(θ,βK(θ))

εf = ε1(θ,βK(θ))

εg = ε2(θ,βK(θ))

40

3.4.7 Subregion 6

xf = x1(θ,βK(θ))

xg = x2(θ,βK(θ))

σf = σ1(θ,βK(θ))

σg = σ2(θ,βK(θ))

εf = ε1(θ,βK(θ))

εg = ε2(θ,βK(θ))

3.5 THE K-FUNCTION (SATURATION LINE)

This equation given the saturation line which is also a boundary between

sub-regions. The equation for the reduced saturation pressure. βK as a

function of the reduced temperature θ is

βK(θ) =exp

52

12 2

(1 )(1 )1/

[1 6 (1 ) 7(1 ) ] 8(1 ) 9VKV

k k k k

The constants of the K-Function are given in A.1.1.5 Saturation Line

(Appendix A.1)

3.6 CANONICAL FUNCTIONS

3.6.1 The A-Function

The reduced free enthalpy (Gibbs function)

10 121 1711

1

17 17( , ) (1 ln )29 12

vA o

vA Av A Z Y Z

41

+ 2 10 19 112 13 14 15 16( 6 ) ( 7 )A A A A a A a

- 18 1 2 317 18 18( 8 ) *( )a A A A

- 18 2 320 10 11( 9 ) ( )A a a a

+ 3 20 421 12 22( )A a A

Where

Z = Y + (a3y3-2a4 θ+2a5 β)1/2

Y = 1-a1 θ2-a2 θ-6

3.6.2 The B-FunctionReduced free enthalpy (Gibbs Function)

51 13 3

1 11 121

( , ) ln (1 ln ) ( )vB o ov

vI B B B x x

- 18 2 2 18 10 312 22 23 31 32( ) ( )B x B x B x B x B x

- 41 25 42 14 4 32 28 24 551 52 53( ) ( )B x B x B x B x B x

-12 11 4 24 18 5

61 62 71 7214 4 19 5

61 71

( ) ( )1 1B x B x B x B x

b x b x

-24 14 6 6

81 82954 27 6

081 82

( )1 ( )

vv

VL

B x B x B xb x b x

Where x=exp(b(1-θ)) and βL= βL(θ), the expression for which is given in

3.3.2.2

42

The number of terms n(μ) and l(μ) and the exponents z(μ,v) and x(μ, )

are listed below:

μ n (μ) z(μ,v) l(μ) x(μ, ) μ

v =1 v =2 v =3 =1 =2

1 2 13 3 - - - - 12 3 18 2 1 - - - 23 2 18 10 - - - - 34 2 25 14 1 - - - 45 3 32 28 24 - - - 56 2 12 11 - 1 14 - 67 2 24 18 - 1 29 - 78 2 24 14 - 2 54 27 8

3.6.3 The C-FunctionReduced free energy (Helm hotz function)

111

1 0122

( , ) lnvC oo o ov

Vx C C x C x C x

+6

111 1 17

2ln 1v

vV

C x C x C x

+7

1 221 2 28

2ln ( 1)v

vv

C x C x C x

+9

1 331 3 30

2ln ( 1)v

vv

C x C x C x

+ 5 2340 41 50( ) ( 1) lnC C x C

+4 8

6 2 16 7

2 0( 1)v v

v vv vx C C

43

3.6.4 The D-FunctionReduced free energy (Helm holtz function)

4 4 232

1 53 0 0

( ) v vD v v

v vx D Y x Y D x

Where Y=1-θ /(1- θ1)

3.6.5 Constants Relating to Canonical FunctionsThe values of the constants presented in 3.6.1, 3.6.2, 3.6.3 and 3.6.4 are

given in Appendix A.1.

3.7 VALUES OF THE CONSTANTSThe numerical values of the primary constants of all sub-regions,

saturation line constants, boundaries between sub regions 2 and 3 and

expressions for values of derived constants are given in Appendix A.1.

3.8 DERIVED CONSTANTS

3.8.1 Numerical Values of Derived Constants

β2 = 4.52079566 x 100

θt =4.21999073 x 10-1

θ1 = 9.626911787 x 10-1

θ2 = 1.333462073 x 100

θ3 = 1.657886606 x 100

I1 = = 4.260321148 x 100

β1 = 7.475191707 x 10-1

βt = 2.763311032 x 10-1

3.8.2 Derived form of the L-Function and Values relating toConstants

When the L-function is rearranged to give;

44

BL = BL θ = L0 = L1 θ + L2 θ2

and consequently

1 2( ) 2I ILL L

d L Ld

Then the derived constants L0 , L1 and L2 have numerical values.

L0 =1.574373327 x 101

L1 = -3.417061 x 101

L2 = 1.931380707 x 101

3.9 DERIVED FUNCTIONS

3.9.1 Subregion 1

Reduced volume,

V/Vc1 = x1= A

x1= 5 2 10 19 11711 5 12 13 14 15 6 16 7( ) ( )A a z A A A A a A a

- 1 28 11 17 18 19( ) ( 2 3 )a A A A

- 18 2 2020 9 10 11( ) 3( )A a a a

- 18 2 420 9 10 11( ){ 3( ) }A a a a

+ 2 20 321 12 223 ( ) 4A a A

Where

Z = Y + (a3y3-2a4 θ+2a5 β)1/2

45

Y = 1-a1 θ2-a2 θ-6

Reduced Entropy,

S/(Pc1 Vc1 /Tc1) = σ1 = 1A

σ1 =10

21

2(ln ( 1) )Vo V

VA V A

+51 17

3 411[{5 /12 ( 1) } ]A z a Y Y a Z

+9 19 2 18

13 14 15 6 16 72 10 ( ) 19 ( )A A A a A a

-11 2 10 2 3

8 17 18 1911( ) ( )a A A A

+ 17 2 320 9 10 11(18 20 ) ( )A a a a

+ 3 21 421 2220A A

Where Y1 = -2a1θ + 6a2 θ-7

Reduced Enthalpy,

h/(Pc1 Vc1) = ε1 = 1 1A o

ε1 =10

1

1( 2) V

o o VV

A V A

+ 51 1 1711 4 317( /12) 50 ( 1)29 12

Z YA z y a a yy z

+ 2 19 19 212 14 15 6 6 16 7 7(9 )( ) (20 )( )A A A a a A a a

- 11 11 2 2 38 8 17 18 19(12 )( ) ( )a a A A A

+ 18 2 3 3 20 420 9 10 11 21 12 22(17 19 ) ( ) 21A a a a A a A

46

u = h-P1V:

The reduced enthalpy (Gibbs Function)

1 1 11 1( )gPcVc

1 1 11 1( )f

PcVc The reduced free energy Helmholtz function.

3.9.2 Subregion 2

Reduced Volume,

V/Vc1 = x2 = ( )B

( )

( )

( )

(1 ) ( , )5 8

1 ( , ) 12 1 2

11 1 6 2 ( , )

1

( 2)

( )

nz V

vnz V V

vV x

B xx I B x

B x

+6

109

011( / ) V

L VVB x

Where x=exp(b(1-θ))

βL = βL(θ) the impression for which is given 3.3.2.2 and the numbers of

terms n(µ) and (μ) and the exponents z(μ,v) and x(μ, ) are listed in

3.6.2.

Reduced Entropy,

S/(Pc1 Vc1/Tc1 ) σ2 = - 1( / )B

( )5 52 ( , )

2 1 11 1 1

ln ln ( 1) ( , )n

v z vo ov v

V VI B v B b z v B x

47

-b

( )

( )

( )

( )

( , )

( , ) 1

1 2 ( , )8

1

6 2 ( , )

1

( , )( , )

(

(

lx

vnz v

v lv x

v

lx

v

x b xB x z v

l b x

b x

+16

109

0

10( / ) vLL v

V L

Vb B x

Reduced Enthalpy,

h/(Pc1,Vc1) = 2 1 2B o

( )5 51 ( , )

21 1 1

( 2) (1 ( , ) )n

v z vo o ov v

V VB B v B Z v b x

-

( )

( )

( )

( )

( , )

( , ) 1

1 2 ( , )8

1

6 2 ( , )

1

( , )1 ( , )

lx v

nz v

v lV x

lx

b x v b xB x Z v b

b x

b x

+ β(β/ βL)1016

90

101 vLv

V L

Vb B x

u = h-P1V

The reduced enthalpy (Gibbs function)

g/(Pc1 Vc1) = 1 2o The reduced enthalpy (Gibbs function)

48

f/(Pc1 Vc1) = 2 2 2x Reduced Helmholtz function

3.9.3 Subregion 3

Reduced Pressure,

P/Pc1 = β3

θ3(θ,x) = - 3 / x

= -11

11 12

2(1 ) v

ovV

Co v C x Co x

-6

111 1 17

2(1 ) 1v

vV

C v C x C x

-7

1 221 2 28

2(1 ) ( 1)v

vV

C v C x C x

-9

1 331 3 310

2(1 ) ( 1)v

vV

C v C x C x

+4

6 23 5 241 6

05 ( 1) 6 v

vV

C x x C

Reduced Entropy,

S/(Pc1 Vc1/Tc1) = 3 = 1( )cx

61

3 1 11 1 17 502

lnvv

VC x C x C x C

-7

121 2 28

2ln 1v

vV

C x C x C x

-9

1 231 3 310

2ln ( 1)v

vV

C x C x C x

49

+ 5 23 2440 41 50( )(22 23 ) lnC x C x C

+ 4 8

6 36 7

0 0( 2) ( 1) ( 1)v v

v vV Vx V C V C

Reduced Enthalpy,

h/(Pc1 Vc1) = ε3 = 0 1 3 3C x

ε3 =11

10 00 012 50 11

2( ) v

ovV

C C C C x VC x

- 6

11 012 17 17 50 11 21

2( ) ln ( 2 )v

vVC x C C x C C C C x

+6 7

1 11 2 28

2 2( 1) 2 2 ln ( 1)v v

v vV V

V C x C x C x

+7

128 21 31 2

2(2 3 ) ( 2) v

vV

C C C x V C x

-3 9

1 23 28 310

2( 3 ) ln ( 1)v

vVC x C C x

+9

1 3310 31 3 310

23 ( 3) 2 ln ( 1)v

vV

C C x V C x C x

+ 5 22 5 2340 41 40 41(23 28 ) (24 29 )C C x C C x

+ 4 8

6 26 7

0 0( 3) (1 )( 1)v v

v vV Vx V C C V

The reduced gibbs function (or)

g/(Pc1,Vc) = 3 3 3( , )x x

u = h-pv

50

3.9.4 Subregion 4

Reduced Pressure,

P/Pc1 = β4 =4 4 2

1 32 13 5

3 0 0

v Vv

V VVD vY x Y D x

Reduced Entropy,

S/(Pc1 Vc1/Tc1) = σ4 = 3 ( )D x

σ4 =

4 4 21 31

53 0 0

31

32

1

v vv

V VD vY x Y D x

Reduced enthalpy,

h/(Pc1 Vc1) = ε4 = 4 4

13 1

3 0(1 ) / (1 ) v

VD v v Y Y x

- 2

315 1

0(31 ) 32 / (1 ) V

VV

Y D V Y x

Reduced Gibbs function g/(Pc1 Vc1)= ξ4 = ε4 - θσ4

Reduced Helmholtz function g/(Pc1 Vc1)= ψ4 = ξ4 - βx4

Internal energy u=h – pv

Sub Region 5

σf = σ4 = (θ,βK(θ))

σg = σ3 = (θ,βK(θ))

εf = ε4 = (θ,βK(θ))

εg = ε3 = (θ,βK(θ))

Sub Region 6

xf =x1(θ,βK(θ))

51

xg =x2(θ,βK(θ))

σf = σ1 = (θ,βK(θ))

σg = σ2 = (θ,βK(θ))

εf = ε1 = (θ,βK(θ))

εg = ε2 = (θ,βK(θ))

3.10 ALGORITHM TO FIND THE STEAM PROPERTIES

The present investigation, after thoroughly understanding the sections

3.1 to 3.9, in order to write a computer program to find the properties

such as enthalpy, entropy and specific volume etc., a computer program

to find the properties of the water/steam at different pressures and

temperatures an algorithm has been developed and presented below. The

program first identifies the thermodynamic state and identifies the region

which it belongs to and makes use of formulations of that region in order

to find the different property value.

1) Read the properties P1, T1 and data2) Compute = P1/Pc, = T1/Tc1, and L using the K-function and L

function equations.3) If (t < = < = 1) // check the following conditions.

{

If (O< = <= k) then Set Fumble = 2;

else If (=) then Set Fumble = 6;

else If () < <= ) then Set Fumble = 1;

}

else

4) If (1 < = < = ) // check the following conditions.{

52

If (O < = < L ) then Set Fumble = 2;

else If (L< <then Set Fumble = 3;

else If (=) then Set Fumble = 5;

else If (< <=then Set Fumble = 4;

}

else

5) If ( < = < ) // check the following conditions.{

If (O< = <=L then Set Fumble = 2;

else If (L<< = ) then Set Fumble = 3;

}

else

6) If (2< = < ) // check the following conditions.{

If (O< = <=2then Set Fumble = 2;

}

7) Case open (Fumble)Case 1

{

Compute h,s,v,u,f,g using the equations given for the derived functionin the subregion 1

}

Case 2

{

Compute h,s,u,v,f,g using the derived function given in the subregion2

}

Case 3

{

Compute h,s,u,v,f,g using the derived function given in the subregion3

}

Case 4

{

53

Compute h,s,u,v,f,g using the derived function given in the subregion4

}

Case 5

{

Compute h,s,u,v,f,g using the derived function given in the subregion5

}

Case 6

{

Compute h,s,u,v,f,g using the derived function given in the subregion6

}

8) Write h,s,u,v,f,g9) END.

By making use of the algorithm, a computer program in C++ language

has been developed by the present investigation for computing the values

of steam properties at different operating parameters of pressure and

temperature and is presented in Appendix A.2.

The program so developed by the present investigation has been

validated with the help of the data available in the standard text books

for subcritical conditions and with the data, “IAPWS Industrial

Formulation 1997 for the Thermodynamic Properties of Water and

Steam”[128] and also compared with known data[112,114] available for

supercritical condition is presented in detail in Appendix A.3.