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Thermal Efficiency of the combined (binary) cycle with T-s and h-s charts

 

> 

restart; with(ThermophysicalData); with(Units[Standard]); with(plots)

Introduction
 

The figure shows the scheme of a simplest ideal combined (binary) cycle, i.e. steam turbine (6-7) cycle with use of superheated water steam. At this plant the boiler (6-9) burner is replaced with gas turbine unit (Brayton cycle: 1-4) equipped with air compressor (1-2), combustion chamber (2-3), gas turbine (3-4) and one more electric generator.

Calculations
 

Steam turbine cycle
 

Input data:

> 

t__6 := (480+273.15)*Unit('K'); p__6 := 9*Unit('MPa'); p__7 := 4.76*Unit('kPa'); wf__6789 := H2O

753.15*Units:-Unit('K')

  

9*Units:-Unit('MPa')

  

4.76*Units:-Unit('kPa')

  

H2O

 (2.1.1)

 

Specific entropy of live steam (the turbine inlet):

> 

s__6 := Property(entropy, pressure = p__6, temperature = t__6, wf__6789)

6.593467590*Units:-Unit(kJ/(kg*K))

 (2.1.2)

Specific enthalpy of live steam (the turbine inlet):

> 

h__6 := Property(enthalpy, pressure = p__6, temperature = t__6, wf__6789)

3336.426849*Units:-Unit(kJ/kg)

 (2.1.3)

Outlet steam specific entropy from the turbine (an ideal process of the steam extension)

> 

s__7 := s__6

Dryness of steam in outlet of the turbine:

> 

x__7 := Property(Q, P = p__7, entropy = s__7, wf__6789)

.7712559666

 (2.1.4)

Outlet wet steam temperature from the turbine

> 

t__7 := Property(temperature, pressure = p__7, Q = 1, wf__6789); t__7-273.15*Unit('K')

32.0015827*Units:-Unit(`°C`)

 (2.1.5)

Outlet wet steam specific enthalpy from of the  turbine:

> 

h__7 := Property(enthalpy, T = t__7, Q = x__7, wf__6789)

2004.437563*Units:-Unit(kJ/kg)

 (2.1.6)

Specific work of steam in the turbine:

> 

w__st := h__6-h__7

1331.989286*Units:-Unit(kJ/kg)

 (2.1.7)

Specific enthalpy of water at saturated line at temperature in the condenser

> 

hw__7 := Property(enthalpy, T = t__7, Q = 0, wf__6789)

134.1008689*Units:-Unit(kJ/kg)

 (2.1.8)

Specific entropy of water at saturated line at temperature in the condenser

> 

sw__7 := Property(entropy, T = t__7, Q = 0, wf__6789)

.4642623028*Units:-Unit(kJ/(kg*K))

 (2.1.9)

 

Specific enthalpy of condensate

> 

h__8 := hw__7

Pressure of feed water

> 

p__9 := p__6

Specific entropy of feed water (an ideal process in the pump):

> 

s__9 := sw__7

Temperature of feed water:

> 

t__9 := Property(temperature, pressure = p__9, entropy = s__9, wf__6789); t__9-273.15*Unit('K')

32.2161406*Units:-Unit(`°C`)

 (2.1.10)

Specific enthalpy of feed water:

> 

h__9 := Property(enthalpy, pressure = p__9, temperature = t__9, wf__6789)

143.1237785*Units:-Unit(kJ/kg)

 (2.1.11)

Specific useful work of the feed pump:

> 

w__p := h__9-hw__7

9.022909600*Units:-Unit(kJ/kg)

 (2.1.12)

Specific heat supplied to the boiler:

> 

q__b := h__6-h__9

3193.303070*Units:-Unit(kJ/kg)

 (2.1.13)

Hence the thermal efficiency of the steam turbine cycle:

> 

(w__st-w__p)/q__b

.4142940231

 (2.1.14)

Gas turbine cycle
 

Input data:

> 

t__1 := (15+273.15)*Unit('K'); p__1 := .1*Unit('MPa'); p__2 := Unit('MPa'); t__3 := (1100+273.15)*Unit('K'); t__5 := (130+273.15)*Unit('K'); wf__12345 := air

288.15*Units:-Unit('K')

  

.1*Units:-Unit('MPa')

  

Units:-Unit('MPa')

  

1373.15*Units:-Unit('K')

  

403.15*Units:-Unit('K')

  

air

 (2.2.1)

Specific enthalpy of fresh air

> 

h__1 := Property(enthalpy, pressure = p__1, temperature = t__1, wf__12345)

414.3778321*Units:-Unit(kJ/kg)

 (2.2.2)

Specific entropy of fresh air

> 

s__1 := Property(entropy, pressure = p__1, temperature = t__1, wf__12345)

3.849952078*Units:-Unit(kJ/(kg*K))

 (2.2.3)

Outlet air specific entropy, temperature and specific enthalpy from the compressor

> 

s__2 := s__1
> 

t__2 := Property(temperature, pressure = p__2, entropy = s__2, air); t__2-273.15*Unit('K')

279.4631181*Units:-Unit(`°C`)

 (2.2.4)
> 

h__2 := Property(enthalpy, pressure = p__2, temperature = t__2, wf__12345)

683.6031240*Units:-Unit(kJ/kg)

 (2.2.5)

Inlet gas pressure, specific entropy and specific enthalpy to  the gas turbine

> 

p__3 := p__2
> 

s__3 := Property(entropy, pressure = p__3, temperature = t__3, wf__12345)

4.867024310*Units:-Unit(kJ/(kg*K))

 (2.2.6)
> 

h__3 := Property(enthalpy, pressure = p__3, temperature = t__3, wf__12345)

1610.338970*Units:-Unit(kJ/kg)

 (2.2.7)

Outlet gas pressure, specific entropy, temperature and specific enthalpy from the gas turbine

> 

p__4 := p__1
> 

s__4 := s__3
> 

t__4 := Property(temperature, pressure = p__4, entropy = s__4, wf__12345); t__4-273.15*Unit('K')

498.0105159*Units:-Unit(`°C`)

 (2.2.8)
> 

h__4 := Property(enthalpy, pressure = p__4, temperature = t__4, wf__12345)

916.8176736*Units:-Unit(kJ/kg)

 (2.2.9)

Specific heat supplied to the combustion chamber:

> 

q__1 := h__3-h__2

926.7358460*Units:-Unit(kJ/kg)

 (2.2.10)

Specific work of the gas turbine

> 

w__gt := h__3-h__4

693.5212964*Units:-Unit(kJ/kg)

 (2.2.11)

Specific work of the air compressor

> 

w__c := h__2-h__1

269.2252919*Units:-Unit(kJ/kg)

 (2.2.12)

Hence the thermal efficiency of the gas turbine cycle:

> 

(w__gt-w__c)/q__1

.4578392067

 (2.2.13)

Combined (binary) cycle
 

Outlet gas pressure and specific enthalpy from the steam boiler

> 

p__5 := p__4
> 

h__5 := Property(enthalpy, pressure = p__5, temperature = t__5, wf__12345)

530.4505082*Units:-Unit(kJ/kg)

 (2.3.1)

 

Ratio of gas and steam mass flow

> 

m := (h__6-h__9)/(h__4-h__5)

8.264944217

 (2.3.2)

Specific heat supplied to the combustion chamber:

> 

q__1 := m*(h__3-h__2)

7659.420071*Units:-Unit(kJ/kg)

 (2.3.3)

Specific work of the gas turbine cycle

> 

w__gtc := h__3-h__4-h__2+h__1

424.2960045*Units:-Unit(kJ/kg)

 (2.3.4)

Specific work of the steam turbine cycle

> 

w__stc := h__6-h__7-h__9+h__8

1322.966376*Units:-Unit(kJ/kg)

 (2.3.5)

 

Hence the thermal efficiency of the combined (binary) cycle is higher than separate steam or gas turbine cycles:

> 

(m*w__gtc+w__stc)/q__1

.6305632987

 (2.3.6)

``

T-s chart
 

A "dead" T-s chart:

> 

NULL

> 

t__triple := 273.16*Unit('K'); t__critical := 647.096*Unit('K')

Specific entropy of liquid at the saturated line as a function of temperature

> 

wspSSWT := proc (T) options operator, arrow; Property(entropy, temperature = T, Q = 0, wf__6789) end proc
> 

p1 := plot([wspSSWT(t), t, t = t__triple/Unit('K') .. t__critical/Unit('K')], color = "Blue", thickness = 1, linestyle = "dash", legend = "Water on saturated line")

Specific entropy of vapor at the saturated line as a function of temperature

> 

wspSSST := proc (T) options operator, arrow; Property(entropy, temperature = T, Q = 1, wf__6789) end proc
> 

p2 := plot([wspSSST(t), t, t = t__triple/Unit('K') .. t__critical/Unit('K')], color = "Red", thickness = 1, linestyle = "dash", legend = "Steam on saturated line")
> 

s__9 := Property(entropy, pressure = p__9, temperature = t__7, wf__6789)

Temperature as a function of pressure and specific entropy

> 

wspTPS := proc (p, s) options operator, arrow; Property(temperature, pressure = p, entropy = s, wf__6789) end proc
> 

p3 := plot([s, wspTPS(p__6/Unit('Pa'), s), s = s__9*Unit('kg')*Unit('K')/Unit('J') .. s__6*Unit('kg')*Unit('K')/Unit('J')], thickness = 1, color = "Black", legend = "Boiler")
> 

p4 := pointplot([[s__6*Unit('kg')*Unit('K')/Unit('J'), t__6/Unit('K')], [s__6*Unit('kg')*Unit('K')/Unit('J'), t__7/Unit('K')]], connect = true, thickness = 1, color = ["Red", "Blue"], legend = "Steam expansion in turbine")
> 

p5 := pointplot([[s__6*Unit('kg')*Unit('K')/Unit('J'), t__7/Unit('K')], [s__9*Unit('kg')*Unit('K')/Unit('J'), t__7/Unit('K')]], connect = true, thickness = 1, color = ["Red", "Blue"], legend = "Condenser")
> 

p6 := pointplot([[s__1*Unit('kg')*Unit('K')/Unit('J'), t__1/Unit('K')], [s__1*Unit('kg')*Unit('K')/Unit('J'), t__2/Unit('K')]], connect = true, thickness = 3, color = "Brown", legend = "Compressor")
> 

p7 := pointplot([[s__4*Unit('kg')*Unit('K')/Unit('J'), t__4/Unit('K')], [s__4*Unit('kg')*Unit('K')/Unit('J'), t__3/Unit('K')]], connect = true, thickness = 3, color = "Green", legend = "Gas turbine")
> 

wspgTPS := proc (p, s) options operator, arrow; Property(temperature, pressure = p, entropy = s, wf__12345) end proc
> 

p8 := plot([s, wspgTPS(p__2/Unit('Pa'), s), s = s__2*Unit('kg')*Unit('K')/Unit('J') .. s__3*Unit('kg')*Unit('K')/Unit('J')], thickness = 3, color = "Orange", legend = "Combustion chamber")
> 

p9 := plot([s, wspgTPS(p__1/Unit('Pa'), s), s = s__2*Unit('kg')*Unit('K')/Unit('J') .. s__3*Unit('kg')*Unit('K')/Unit('J')], thickness = 3, color = "Gold", legend = "Boiler and Atmosphere")
> 

s__5 := Property(entropy, pressure = p__5, temperature = t__5, wf__12345)
> 

p10 := pointplot([s__5*Unit('kg')*Unit('K')/Unit('J'), t__5/Unit('K')], thickness = 5, color = "Red", legend = "Exit of the flue gas")

A live T-s chart:

> 

plots:-display(p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, labels = [s*[J/(kg*K)], T*[K]], size = [1200, 700], gridlines)

 

h-s chart
 

``

Specific enthalpy of liquid at the saturated line as function of temperature

> 

wspHSWT := proc (T) options operator, arrow; Property(enthalpy, temperature = T, Q = 0, wf__6789) end proc
> 

p11 := plot([wspSSWT(t), wspHSWT(t), t = t__triple/Unit('K') .. t__critical/Unit('K')], color = "Blue", thickness = 1, linestyle = "dash", legend = "Water on saturated line")

Specific entropy of vapor at the saturated line as function of temperature

> 

wspHSST := proc (T) options operator, arrow; Property(enthalpy, temperature = T, Q = 1, wf__6789) end proc
> 

p12 := plot([wspSSST(t), wspHSST(t), t = t__triple/Unit('K') .. t__critical/Unit('K')], color = "Red", thickness = 1, linestyle = "dash", legend = "Steam on saturated line")
> 

wspHPS := proc (p, s) options operator, arrow; Property(enthalpy, pressure = p, entropy = s, wf__6789) end proc
> 

p13 := plot([s, wspHPS(p__6/Unit('Pa'), s), s = s__9*Unit('kg')*Unit('K')/Unit('J') .. s__6*Unit('kg')*Unit('K')/Unit('J')], thickness = 1, color = "Black", legend = "Boiler")
> 

p14 := pointplot([[s__6*Unit('kg')*Unit('K')/Unit('J'), h__6*Unit('kg')/Unit('J')], [s__6*Unit('kg')*Unit('K')/Unit('J'), h__7*Unit('kg')/Unit('J')]], connect = true, thickness = 1, color = ["Red", "Blue"], legend = "Steam expansion in turbine")
> 

p15 := plot([s, wspHPS(p__7/Unit('Pa'), s), s = s__9*Unit('kg')*Unit('K')/Unit('J') .. s__6*Unit('kg')*Unit('K')/Unit('J')], thickness = 1, color = "Blue", legend = "Condenser")
> 

p16 := pointplot([[s__1*Unit('kg')*Unit('K')/Unit('J'), h__1*Unit('kg')/Unit('J')], [s__1*Unit('kg')*Unit('K')/Unit('J'), h__2*Unit('kg')/Unit('J')]], connect = true, thickness = 3, color = "Brown", legend = "Compressor")
> 

p17 := pointplot([[s__4*Unit('kg')*Unit('K')/Unit('J'), h__4*Unit('kg')/Unit('J')], [s__4*Unit('kg')*Unit('K')/Unit('J'), h__3*Unit('kg')/Unit('J')]], connect = true, thickness = 3, color = "Green", legend = "Gas turbine")

Specific enthalpy as a function of pressure and specific entropy

> 

wspgHPS := proc (p, s) options operator, arrow; Property(enthalpy, pressure = p, entropy = s, wf__12345) end proc
> 

p18 := plot([s, wspgHPS(p__2/Unit('Pa'), s), s = s__2*Unit('kg')*Unit('K')/Unit('J') .. s__3*Unit('kg')*Unit('K')/Unit('J')], thickness = 3, color = "Orange", legend = "Combustion chamber")
> 

p19 := plot([s, wspgHPS(p__1/Unit('Pa'), s), s = s__2*Unit('kg')*Unit('K')/Unit('J') .. s__3*Unit('kg')*Unit('K')/Unit('J')], thickness = 3, color = "Gold", legend = "Boiler and Atmosphere")
> 

p20 := pointplot([s__5*Unit('kg')*Unit('K')/Unit('J'), h__5*Unit('kg')/Unit('J')], thickness = 5, color = "Red", legend = "Exit of the flue gas")

A live chart:

> 

plots:-display(p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, labels = [s*[J/(kg*K)], h*[J/kg]], size = [1200, 700], gridlines)

 

Created by Valery Ochkov, 09/08/2016

See please more https://www.ptcusercommunity.com/thread/128064

Literature: Thermal Engineering Studies with Excel, Mathcad and Internet
Authors: Ochkov, Valery, Orlov, Konstantin, Voloshchuk, Volodymyr
Editor: Rogalev, Nikolay (Ed.)

Springer, 2016

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