07 td rankin klauzijus

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8.5 Idealisations in the Clausius Rankine Cycle or How to Assess a Real-LifeSteam Power Plant?

Recall the steps of the idealised Claudius-Rankine cycle depicted in Figure 8.5.1 and

8.5.2.

• Step 21→ : Ideally a feedwater pump adiabatically (or isentropically) compresses

boiling water that is added to a boiler. The operation of the feed water pump re-

quires work which is given by:

012fwp >−= hhw . (8.5.1)

• The enthalpy value 1h follows from the steam tables and is given by the specific

enthalpy of saturated liquid water at a pressure 1p which ideally is equal to the

condenser pressure:

( )11 phh f= . (8.5.2)

The enthalpy value 2h can approximately be determined from the following equa-

tion:

( ) ( )12112 v ppphh f −+= , (8.5.3)

where ( )1v pf is the specific volume of liquid water at the pressure 1p which can

either be obtained from the steam tables or be approximated as kg1l .

• Step 32 → : The boiler evaporates the water at a constant pressure (isobarically)

so that steam results which is eventually superheated and then fed into a turbine.

This requires heat input which is given by (the specific enthalpies can typically be

obtained from the Mollier diagram):

023 >−=+ hhq . (8.5.4)

• Ideally the turbine operates adiabatically (isentropically), generates power and the

resulting specific work is given by:

034tur <−= hhw . (8.5.5)

• The steam from the turbine outlet streams through a condenser (at a constant

pressure, i.e. isobarically) and turns into liquid boiling water which is added to the

feed water pump. This releases an amount of specific heat which is given by:

041 <−=− hhq . (8.5.6)

This closes the loop of the idealised Clausius-Rankine cycle.

2

2

boiler2

q+

FWP

condenser

turbine

3

4

1

1q

-

wtur

wfwp (η fwp)

(η tur)

3

4

Figure 8.5.1: Schematic of the components and steps during a Clausius-Rankine cycle.

h

c

1

2

3

4

s s

T

4

3

1

2

c

Figure 8.5.2: The Clausius-Rankine cycle in ( )sh, and ( )sT , representation.

A more realistic representation of the processes in a steam power plant requires us

to take various departures from the ideal Claudius-Rankine cycle into account. These

result from irreversibilities in the various components such as:

3

• friction and

• heat losses.

More specifically, there will always be fluid friction in the various pipes which connect

the feed water pump, the boiler, the turbine, as well as the condenser. And there will

also be fluid friction in the boiler and in the condenser (which, naively speaking, is

nothing but a complicated system of pipes). Such fluid friction will lead to pressuredrops as indicated qualitatively in the ( )sT , -representation shown in Figure 8.5.3. As

a result, the steam leaves the boiler at a somewhat lower pressure, and the pressure

at the turbine inlet is again lower than that at the boiler outlet (see path 32 ′→′ for

both of these issues), and there is an additional (but usually very slight) pressure drop

in the condenser 14 →′ .

1

2

3

4

s

T

irreversibilityof the feed water pump

pressure dropin the boiler

irreversibilityin the turbine

4'

3'

pressure drop in the condenser

2'

4''

c

Figure 8.5.3: Taking pressure drops, heat losses and irreversibilities of the Clausius-Rankine cycle

into account.

This is a sad truth of life we have to somehow compensate. Therefore, the water

needs to be pumped to a sufficiently high level to begin with which requires the use of

a stronger pump which, in turn, requires a higher work input.

Naturally, there will not only be pressure drops we have to cope with but also heat

losses from the steam to the surroundings as it flows through the various compo-

nents. Consequently, in order to maintain the same level of net work output, more

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heat has to be added to the boiler which increases the running costs and decreases

the efficiency.

Finally, neither the pump nor the turbine operate perfectly. It was mentioned already

that more work is required to operate the pump at higher pressure levels. However,

the pure operation of a pump, being a non-reversible machine, naturally requires

more work input than theoretically necessary to reach this desired pressure level. In

other words, the pump does not operate adiabatically / isentropically. Rather entropy

increases during its operation as indicated by the slightly curved line 21 ′→ in Figure

8.5.3. Similarly, the turbine does not operate adiabatically / isentropically either. As

a result of irreversibility the entropy level will also increase as shown in path 43 ′→′

of Figure 8.5.3.

It is customary to assess the irreversible losses in the pump as well as in the turbine

through so-called adiabatic efficiencies which are given by the ratio of theoretical and

real work input or the ratio of real work output to theoretical output, respectively. In

terms of specific enthalpy data which correspond to the points of Figure 8.5.3 they

are defined as follows (the point 4 ′′ lies along the vertical line through 3′ at the same

pressure level as point 4′ ):

112

12realfwp

adiafwp

fwp <−′−==η

hhhh

w

w , ( )434

34

34

adiatur

realtur

tur ,,1 ppsshhhh

hh

w

w ′=′==′′<′−′′′−′

==η . (8.5.7)

But this is not the end of a very intricate situation. Many other things have to be con-

sidered when dealing with a real-life power plant. For example, the liquid in the con-

denser is frequently subcooled. This is done to prevent cavitation, rapid vaporisa-

tion and condensation of the liquid water near the blades at the low-pressure side of

the pump, all of which can lead to potential damage. Similarly, the turbine blades can

be destroyed when they are hit by high speed water droplets as it will happen if a

state in the wet-vapour region is reached resulting in partial condensation. Also:

Water / steam (which preferably is of relatively high purity) can get lost on its way

through the various plant components and loss due to such leaks has to be replaced.

Finally, additional power necessary to operate the plant (for example for the operation

of the fans which transport air to the boiler furnaces) should be considered when the

net performance of the plant is assessed.

8.6 Numerical Treatment of a Somewhat Realistic Power Cycle

Consider the idealised power cycle as indicated in Figure 8.6.1. The objective is tooperate the plant between bar150 turbine inlet pressure at C600 ° and bar1.0 con-

denser pressure.

5

boiler + superheater

FWP

turbine

condenserI

II

II III

I

III

IV

I

II

III

IV

T

s

150 bar

0.1 bar

600 °C

c

Figure 8.6.1: Numerical example of an idealised Clausius-Rankine cycle.

We find the following enthalpy data for the points indicated in Figure 8.6.1:

( )kgkJ

192bar1.0fI == hh (from the steam tables), (8.6.1)

( ) ( )kgkJ

207mkJ

101.0150kgm

10kgkJ

192v3

23

3IIIIIII =⋅−+=−+= −pphh , (8.6.2)

( )kgkJ

3580C600,bar150III ≈°h , ( )kgkJ

2115,bar1.0 IIIIV ≈= ssh , (8.6.3)

(the latter two values were obtained from the Mollier diagram). Consequently, we

obtain

• for the work necessary to operate the feed water pump:

( ) ( )kgkJ

15mkJ

101.0150kgm

10v3

23

3IIIIIIIfwp ≈⋅−=−=−= −pphhw , (8.6.4)

• for the heat added in the boiler and in the superheater:

kgkJ

3373kgkJ

207kgkJ

3580IIIII =−=−=+ hhq , (8.6.5)

• for the work delivered by the turbine:

kgkJ

1465kgkJ

3580kgkJ

2115IIIIVtur −=−=−= hhw (8.6.6)

• for the heat which is released in the condenser:

kgkJ

1923kgkJ

2115kgkJ

192IVI −=−=−=− hhq . (8.5.7)

This results in an efficiency of %43 :

6

( )43.0

3373151465

effortwork net""profit

IIIII

IIIIIIIVfwptur =−=−

−−−=

−===

+ hh

hhhh

q

wwe , (8.5.8)

if we do not neglect the work of the feed water pump.

Now consider the data of the more realistic cycle shown in Figure 8.6.2 where pres-

sure drops, heat losses, and irreversibilities of the pump as well as of the turbine

have been taken into account.

FWP

boiler +superheater

turbine

condenser

pipes

1

2

3 4

5

6

ηt=0.87

ηfwp=0.87

0.09 bar38 C

159 bar35 C

152 bar625 C

150 bar600 C

1

2

36 = 7

6''

T

s

pipes

160 bar

0.1 bar

pipes

160159

150 bar600 °C5

0.1

0.09 bar

152

c

4 625 °C

Figure 8.6.2: Numerical example of a less idealised Clausius-Rankine cycle.

We record the following enthalpy data:

( ) ( )kgkJ

146C35C35,bar159 f3 =°≈° hh † (from the steam tables), (8.6.9)

( )kgkJ

3645C625,bar1524 ≈°h , ( )kgkJ

3580C600,bar1505 ≈°h (8.6.10)

( )kgkJ

2115,bar1.0 56 ≈=′ ssh , (8.6.11)

(the last three values were obtained from the Mollier diagram). For the pump work

input we find:

( ) ( )kgkJ

19mkJ

1009.0160kgm

1085.01v

32

33

fwp

121

fwp

adiafwpreal

fwp ≈⋅−=η

−=

η= −ppw

w . (8.6.12)

This is a consequence of the assumption that the pump operates with almost no dead

volume and that liquid water is almost incompressible:

† This approximation is justified because the isobaric lines in the liquid region of the (T,s)-diagram are denselypacked. Note that they are not drawn to scale in Figure 8.6.2. Rather they were separated considerably tomake them easier to discern.

7

( ) ( )121

2

1

1

2

1

adiafwp vdvdv0dvvddv pppppppw −==+=+−=−= ∫∫ ∫∫∫

′′

. (8.6.12)

Next we take a look at the turbine work output:

( ) ( )kgkJ

1245kgkJ

3580211587.056turadiaturtur

realtur −=−⋅=−′η=η= hhww . (8.6.13)

For the boiler heat input which is required to guarantee the necessary plant perform-

ance we find:

kgkJ

3499kgkJ

146kgkJ

364534 =−=−=+ hhq , (8.6.14)

and the efficiency becomes:

35.03499

191245effort

work net""profitrealfwp

realtur =−=

−===

+q

wwe , (8.6.15)

which is considerably lower than the efficiency of the idealised Clausius-Rankine

process.

8.7 Efficiency Improvements Revisited

In Section 8.2 it was already briefly discussed how the efficiency of the Clausius-

Rankine Cycle can be improved. In general, the idea behind all these improvements

is

• to increase or

• to decrease

the average temperature at which heat is transferred to or rejected from the working

fluid, i.e. the steam / the water, respectively. Technically speaking, this enlarges thearea enclosed in the ( )sT , -diagram representation of the cycle which, as outlined in

Section 5.6, represents the work gained. Typically three measures are taken to im-

prove on the efficiency that way: Lowering of the condenser pressure, superheating of

the steam, and increasing the boiler pressure. A few technical details will be outlined

in the following subsections.

8.7.1 Lowering the Condenser Pressure

Recall that the steam is in a state of wet vapour when it enters the condenser. In other

words, it exists as a saturated mix of liquid and water vapour at a saturation tem-

perature which corresponds one-to-one to the pressure inside of the condenser. How

the pressure and the temperature are correlated to each other can be read off from

the steam tables (say). Consequently, lowering the condenser pressure will straight-

forwardly result in a temperature decrease at which heat is rejected. The effect of

8

lowering the pressure from a level 4p to a level 4p′ is illustrated in the ideal cycle

shown in Figure 8.7.1. The advantage, i.e. the increase in obtainable work is given

by the shaded area. Note that there is also a disadvantage. It takes a somewhat

higher heat input to run the plant under the new conditions: 22 →′ . However, since

the isobaric lines are very densely spaced in the liquid water region this additional

heat input is usually very small as compared to the gain that results from a lower con-

denser pressure.

1'

12'

2

3

4

4'p' 4<

p 4

s

T

Increase in wnet

Figure 8.7.1: Lowering the condenser pressure in an idealised Clausius-Rankine cycle.

In practice the condensers of steam power plants operate well below atmosphericpressure, i.e. bar1 , in order to take as much advantage of the pressure effect as

possible. However, a lower limit is imposed by the fact that the condenser pressure

cannot be lower than the saturation pressure of water corresponding to the tempera-

ture of the cooling medium used, which in many cases is water from a near-by river at(say) C15 ° . Clearly, this temperature may vary considerably during the seasons and

the power plant owners need to cover a certain range of temperatures to guarantee

reliable operation. Also, there should be a certain temperature difference between

the incoming water vapour and the cooling medium in order to enable a sufficientlyeffective heat transfer. If we account for all this by a temperature difference of C10 °(say) we would require a condenser pressure greater than bar032.0 (see steam ta-

bles).

Clearly, lowering the condenser pressure will raise the potential for air leakage into

the condenser. Another side-effect of lowering the condenser pressure is the in-

crease of the moisture content of the steam as it will necessarily result in the final

stages of the turbine, shortly before entry into the pipes which lead to the condenser

(technically speaking, a decrease in pressure will lead to lower dryness or x -values,

i.e. to higher moisture, as can quantitatively be seen in the Mollier chart). In fact, tur-

9

bine manufacturers dislike the presence of large quantities of moisture, i.e. high-

speed / high temperature water droplets, in their machinery which will lead to erosion

and, consequently, puts extremely high demands on the materials used for manufac-

ture. A way of correcting this problem, at least partially, is superheating and the use

of reheat stages which will be discussed later.

8.7.2 Superheating the Boiler Outlet Steam

If the temperature of the steam provided by the boiler to the inlet of the turbine is in-creased from a level 3T to a level 3T ′ this will have a beneficial impact on the net

amount of work gained as illustrated by the shaded area of Figure 8.7.2.

Increase in wnet

1

2

3

4 4'

3'

T

s

Figure 8.7.2: Raising the average temperature of the turbine inlet steam by superheating in the boiler.

In fact, superheating allows us to catch two birds with one stone since it will lead to a

decrease of the moisture content at the turbine outlet as can also be seen from Fig-

ure 8.7.2 (the state 4′ corresponds to a higher x -value (dryness), i.e. a lower mois-

ture content, than 4 ).

However, there are technical limits imposed to superheating which have to do with

metallurgical and materials constraints. Presently, the highest steam temperaturepractically allowed at the turbine inlet are about C620 ° . Higher values would result in

damage of the expensive metal blades unless extremely expensive high temperature

resistant alloys are used. In the future fracture resistant ceramic blade materials

could become a serious alternative to reach even higher levels of efficiency by super-

heating.

8.7.3 Increasing the boiler pressure

Figure 8.7.3 illustrates the impact of increasing the boiler pressure from a level 3p to

a level 3p′ on the efficiency of an idealised Clausius-Rankine cycle for a fixed turbine

inlet temperature. There is obviously a competition going on between an increase

(top shaded region) and a decrease (shaded region on the right) in the net amount of

10

work gained. A close inspection of typically used isobaric levels indicates (see also

the following example) that there is still a beneficial effect, i.e. effectively some work

gained.

Increase in wnet

Decreasein wnet

1

2

3

4

3'

4'

2'

T

s

Tmax

Figure 8.7.3: Increasing the boiler pressure and its effect on efficiency.

Also note that increasing the pressure will lead to an increase of the moisture (lower

dryness, i.e. x -higher values, at point 4′ as compared to point 4 ). As it was men-

tioned before, this is undesirable but it can be corrected by additional superheating

and by the use of reheat stages.

Over the years typical boiler pressures have increased from bar27 in the twenties to

more than bar300 today. The latter leads to the use of supercritical ClausiusRankine

cycles, i.e. cycles where the boiling process does not lead through the wet vapour

region: Figure 8.7.4.

T

Criticalpoint

1

2

3

4

s

Figure 8.7.4: A supercritical Clausius-Rankine cycle at supercritical boiler pressures.

11

8.7.4 A Numerical Example

Figure 8.7.5 shows a typical idealised Clausius-Rankine cycle together with numeri-

cal data which, initially, will use to describe the performance of a non-optimised

steam power plant. Then, in a second step, we shall explore by how much the effi-

ciency of that plant can be increased if

(i) the condenser pressure is lowered plus

(ii) the steam in the boiler is considerably superheated plus

(iii) the boiler pressure is raised to a considerably higher level.

All relevant data can be read off from Figure 8.7.6.

T, C

350

s1=s2 s3=s4

30 ba

r0.7

5 bar

3

41

2

s

Figure 8.7.5: An idealised, non-optimised Clausius-Rankine cycle.

1 11

22 2

3

33

444

s ss

150 bar

30 bar30 bar

0.1 bar 0.1 bar0.1 bar

TT TT3=350 C

T3=600 C T3=600 C

(a) (b) (c)

Figure 8.7.6: The idealised, Clausius-Rankine cycle of Figure 8.7.5, (i) with a lower condenser pres-

sure, (ii) plus superheat, (iii) plus high boiler pressure.

For the non-optimised cycle we note the following enthalpy data:

( ) ( )kgkJ

384bar75.0bar75.0 f1 =≈ hh (from the steam tables), (8.7.1)

12

( ) ( )kgkJ

387mkJ

1075.030kgm

10kgkJ

384v3

23

3I2112 =⋅−+=−+= −pphh , (8.7.2)

( )kgkJ

3115C350,bar303 ≈°h , ( )kgkJ

2400,bar75.0 34 ≈= ssh , 89.04 ≈x (8.7.3)

(from the Mollier chart). Consequently, we obtain


( )kgkJ

3kgkJ

38438712fwp =−=−= hhw , (8.7.4)

• for the heat added to the boiler:

kgkJ

2728kgkJ

387kgkJ

311523 =−=−=+ hhq , (8.7.5)


kgkJ

715kgkJ

3115kgkJ

240034tur −=−=−= hhw (8.7.6)


kgkJ

2016kgkJ

2400kgkJ

38441 −=−=−=− hhq . (8.7.7)


26.02728

3715effort

work net""profit fwptur =−=−

===+q

wwe (8.7.8)

if we take the work of the feed water pump into account. We now turn to the first sce-nario of Figure 8.7.6, in other words, we lower the condenser pressure to bar1.0 and

find that:

( ) ( )kgkJ


( ) ( )kgkJ

195mkJ

101.030kgm

10kgkJ

192v3

23

3I2112 =⋅−+=−+= −pphh , (8.7.10)

( )kgkJ

3115C350,bar303 ≈°h , ( )kgkJ

2130,bar1.0 34 ≈= ssh , 81.04 ≈x (8.7.11)



( )kgkJ

3kgkJ

19219512fwp =−=−= hhw , (8.7.12)

13


kgkJ

2920kgkJ

195kgkJ

311523 =−=−=+ hhq , (8.7.13)


kgkJ

985kgkJ

3115kgkJ

213034tur −=−=−= hhw , (8.7.14)


kgkJ

1938kgkJ

2130kgkJ

19241 −=−=−=− hhq . (8.7.15)

This results in an efficiency of roughly %34 :

336.02920

3985effort


===+q

wwe (8.7.16)

which, obviously, is already quite an improvement when compared to the previous

situation. Next we look at how this can further be improved by superheating (see the

second situation in Figure 8.7.6). We update our enthalpy data as follows:

( )kgkJ

3680C600,bar303 ≈°h , ( )kgkJ

2380,bar1.0 34 ≈= ssh , 91.04 ≈x (8.7.17)

and obtain (only changes will be reported):


kgkJ

3485kgkJ

195kgkJ

368023 =−=−=+ hhq , (8.7.18)


kgkJ

1300kgkJ

3680kgkJ

238034tur −=−=−= hhw , (8.7.19)


kgkJ

2188kgkJ

2380kgkJ

19241 −=−=−=− hhq . (8.7.20)

This results in an even better efficiency of roughly %37 :

372.03485

31300effort


===+q

wwe . (8.7.21)

Finally we turn to the third situation of Figure 8.7.6 where the boiler pressure is raisedbut the inlet temperature of the boiler remains at C600 ° :

14

( ) ( )kgkJ

207mkJ

101.0150kgm

10kgkJ

192v3

23

3I2112 =⋅−+=−+= −pphh , (8.7.22)

( )kgkJ

3580C600,bar1503 ≈°h , ( )kgkJ

2115,bar1.0 34 ≈= ssh , 80.04 ≈x (8.7.23)



( )kgkJ

15kgkJ

19220712fwp =−=−= hhw , (8.7.24)


kgkJ

3373kgkJ

207kgkJ

358023 =−=−=+ hhq , (8.7.25)


kgkJ

1465kgkJ

3580kgkJ

211534tur −=−=−= hhw , (8.7.26)


kgkJ

1923kgkJ

2115kgkJ

19241 −=−=−=− hhq . (8.7.27)


430.03373

151465effort


===+q

wwe , (8.7.28)

which is clearly superior to all the previous results.

8.7.5 Reheating Revisited

The principle of reheating and the efficiency benefits gained from it were already

mentioned in Section 8.2. From a more mature point of view it can be viewed as an

answer to the following question: How can we take advantage of the increased effi-

ciencies at higher boiler pressures without facing the problem of high moisture levels

which might lead to damage of the expensive turbine blades by high speed water

droplets? The fact that reheating will indeed lead to higher values of dryness, x , is

shown schematically in Figure 8.7.7.

15

3

4

1

2

s

5

6

high-pressureturbine reheating

low-pressure turbine

T

FWP

condenserboiler

super-heater

high-pressureturbine low-pressure

turbine

reheater

12

3

45

6

Figure 8.7.7: On the principle of reheating illustrated for the idealised Clausius-Rankine cycle.

Surely, point 6 gets closer to the saturated vapour line (where 1=x ) because the

reheat stage 54 → has been added. Otherwise we would end in point 6′ . Also note

that reheat is definitely a necessary measure to be taken in order to achieve higher

efficiencies while reducing the moisture. The other alternative (see Section 8.7.2),

namely to superheat, has only a limited potential since extremely high boiler tem-

peratures would lead to the destruction of the inlet stage of the turbine.

Note that due to the technical realisation of reheating (use of only one superheater)

the inlet temperature of the second low-pressure turbine stage is typically the same

as the inlet temperature of the first high-pressure turbine stage. Furthermore note that

from an efficiency point of view the optimum reheat pressure is about one-fourth of

the maximum cycle pressure.

s

T

Tav,reheat

Figure 8.7.8: Approximation of an isothermal average temperature by multi-reheat stages.

As indicated in Figure 8.7.8 the average temperature during reheating can be in-

creased by adding additional reheat stages. As the number of stages increases the

ideal isothermal process at the maximum temperature is approximated more and

16

more and this would lead to a further increase of the efficiency while reducing the final

moisture level at the same time. However, it is impractical to use more than two re-

heat stages mainly because of the additional costs and added complexity so that the

gain in efficiency cannot be justified.

8.7.6 Numerical Example of a Reheat Clausius-Rankine Cycle

Consider the situation shown in Figure 8.7.9. A reheat stage has been added to the

Clausius-Rankine cycle from the last problem of Section 8.7.4. Note that the tem-peratures at points 3 and 4 are the same, namely C600 ° . We also require that the

moisture content of the steam at the outlet of the low-pressure turbine does not ex-ceed 10.4% , i.e. 0.8960.104-16 ==x .

3

4

1

2

s

5

6

ReheatingT, C

150 bar

0.1 bar

600

Figure 8.7.9: A reheat cycle.

Consequently, the enthalpy data has to be expanded as follows:

( ) ( )kgkJ


( ) ( )kgkJ

207mkJ

101.0150kgm

10kgkJ

192v3

23

3I2112 =⋅−+=−+= −pphh , (8.7.30)

( )kgkJ

3580C600,bar1503 ≈°h , (from the Mollier chart). (8.7.31)

For reasons of accuracy the enthalpy of point 6 should be obtained by calculation

and not directly from the Mollier chart:

( ) ( ) ( )kgkJ

2336kgkJ

2392896.0192bar1.0bar1.0 fg6f6 =⋅+=+= hxhh . (8.7.32)

17

Now the missing enthalpy in points 4 and 5 can be read off from the chart:

( )kgkJ

3675 ,C600 65 ≈=° ssh , ( )kgkJ

3155 , 354 ≈== sspph . (8.7.33)

Consequently, we obtain

• for the heat added to the boiler plus reheater:

( )kgkJ

3893kgkJ

3155367520735804523 =−+−=−+−=+ hhhhq , (8.7.34)

• for the work delivered by the two turbines:

( )kgkJ

1764kgkJ

36752336358031554634tur −=−+−=−+−= hhhhw (8.7.35)


kgkJ

2144kgkJ

2336kgkJ

19261 −=−=−=− hhq . (8.7.36)


449.03893

151764effort


===+q

wwe . (8.7.37)

07 td rankin klauzijus

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