International Journal of Energy Research Volume 33 Issue 11 2009 [Doi 10.1002_er.1528] Yong Sung...

INTERNATIONAL JOURNAL OF ENERGY RESEARCHInt. J. Energy Res. 2009; 33:989–998Published online 6 March 2009 in Wiley InterScience(www.interscience.wiley.com). DOI: 10.1002/er.1528

Distribution of size in steam turbine power plants

Yong Sung Kim1, Sylvie Lorente2 and Adrian Bejan1,�,y

1Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC 27708-0300, U.S.A.2 Department of Civil Engineering, University of Toulouse, INSA, 135 Avenue de Rangueil, 31077 Toulouse, France

SUMMARY

This paper shows that the mass inventory for steam turbines can be distributed between high-pressure (HP) and low-pressure (LP) turbines such that the global performance of the power plant is maximal. This is demonstrated for twodesign classes. For an HP turbine in series with an LP turbine, the optimal intermediate pressure (IP) is a geometricaverage of HP and LP. The total mass is distributed in a balanced way based on the total mass of turbines. For a trainconsisting of many turbines expanding the steam at nearly constant temperature, the pressure ratio between consecutiveIP should be constant, and more mass should be distributed at HPs. This approach to discovering the configuration ofthe power plant should be used in conjunction with classical approaches that account for vibration, centrifugal forceand blade length. Copyright r 2009 John Wiley & Sons, Ltd.

KEY WORDS: constructal; distributed energy systems; size effect; turbine trains

1. INTRODUCTION

Power generation is an extremely active field offundamental and applied research (e.g. References[1–4]). Like all technologies, power plants areevolving. They are becoming more efficient andlarger (Figure 1). There is a very clear relationshipbetween thermodynamic performance and ‘size’,which in Figure 1 is represented by the net outputof the plant; larger plants operate closer to theCarnot limit than smaller plants, Figure 2. Thissize effect is present in the performance of otherenergy conversion systems [6,7], for example, inheat exchanger design [8], automotive design [9]

and refrigeration and liquefaction plants [10,11]. Itis explained by the relationship between theresistance encountered by a flow (fluid, heat) andthe size of the cross-section (duct, surface) piercedby the flow. Larger cross sections offer lessresistance. This holds for the cross sections ofpipes with fluid flow, and for the heat transferareas of heat exchangers. The thermodynamicimperfection of a flow system is intimately tied tothe size of its hardware [12].

Power plants are also evolving internally.Their structure has been changing in time.The emergence of new organs (superheater,regenerator, feedwater heater) is aligned very

*Correspondence to: Adrian Bejan, Department of Mechanical Engineering and Materials Science, Duke University, Durham,NC 27708-0300, U.S.A.yE-mail: [email protected]

Contract/grant sponsor: Doosan Heavy Industries & Construction Co., Ltd., Changwon, South Korea

Received 7 January 2009

Accepted 9 January 2009Copyright r 2009 John Wiley & Sons, Ltd.

clearly with time and the stepwise increase inthermodynamic performance. This aspect of theevolution of power plant technology is one of theexamples of how constructal theory unitesengineering with the other flow designs of nature [9].

We focus on another aspect of power plantevolution, which has not been addressed in apredictive sense before. In late 1800s, the design ofsteam power plants was based on a single turbine.Contemporary power plants have turbine ‘groups’

Figure 1. The evolution of power plant design in the 20th century [5].

Figure 2. The relationship between plant efficiency and power output (from Figure 1).

Y. S. KIM, S. LORENTE AND A. BEJAN990

Copyright r 2009 John Wiley & Sons, Ltd. Int. J. Energy Res. 2009; 33:989–998

DOI: 10.1002/er

or ‘trains’, consisting of a high-pressure (HP)turbine followed by one or more intermediate-pressure (IP) and low-pressure (LP) turbines. Whyare these changes happening? Why is not thesingle-turbine design surviving? What is the bestway to divide one turbine into a train of smallerturbine?

The answer comes from the relationshipbetween the imperfection of an organ and thesize of the organ. We see that this holds for theentire power plant (Figure 2) and for a singleturbine (Figure 3). However, if larger turbines aremore efficient, why not use a single large turbine asopposed to a group of smaller turbines?

In this paper, we answer these questions byadopting the constructal design proposal to viewthe whole installation in a distributed energysystem [14]. The installation has a total size (e.g.total mass for all the turbines), and must meshwith the rest of the power plant, between clearlydefined pressures (PH, PL), and at temperatures nogreater than a specified level (TH), cf. Figure 4.Usually, the inlet steam temperature (T2) of the LPturbine can be higher than the inlet steamtemperature (TH) of the HP turbine becausethe reheater is installed in a lower gastemperature region in the boiler and theallowable temperature of the tube material is

Figure 3. The relationship between turbine isentropic efficiency and size [13].

DISTRIBUTION OF SIZE IN STEAM TURBINE POWER PLANTS 991


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fixed. However, we assume that T2 is equal to TH

to simplify the analysis. What the installation doesnot have is configuration.

We give ourselves the freedom to image that wecan distribute the available mass over the pressureinterval occupied by the installation. Our objectiveis to distribute the mass inventory in such amanner that the total power produced by theinstallation is greater and greater. In this directionof design evolution, the installation obtains itsconfiguration—more mass in turbines at somepressures, as opposed to less mass in theremaining turbines.

2. TWO TURBINES IN SERIES

Consider the train of two turbines shown inFigure 4. The overall pressure ratio is fixed,PH/PL. The steam enters the two turbines at thehighest allowable temperature level, T1 5T2 5TH.The power outputs from the two turbines are (cf.Reference [10])

W1 ¼ _mcPðT1 � T10 Þ

¼ _mcPTHZ1 1�Pi

PH

� �R=cP" #

ð1Þ

W2 ¼ _mcPðT2 � T20 Þ

¼ _mcPTHZ2 1�PL

Pi

� �R=cP" #

ð2Þ

The heat transfer rate to the reheater is

QR ¼ _mcPðT2 � T10 Þ ¼W1 ð3Þ

The overall efficiency of the power plant is

Z ¼W1 þW2

QB þQR¼

1

QB

W1 þW2

1þQR=QBð4Þ

where QB is the heat transfer administrated tothe _m stream in the boiler, before state 1. Forsimplicity, we recognize that in modern powerplants the ratio QR/QB is of order 1/10, and thisallows us to approximate Equation (4) as

ZQB ffi ðW1 þW2Þ 1�W1

QB

� �ð5Þ

where QB is fixed because state 1 is fixed (PH, TH).Equation (5) can be written further as

ZQB

_mcPTH¼ Z1 1�

Pi

PH

� �R=cP" #(

þZ2 1�PL

Pi

� �R=cP" #)

� 1�_mcPTH

QBZ1 1�

Pi

PH

� �R=cP" #( )

ð6Þ

The expression shown above can be maximizedwith respect to Pi, by solving @Z/@Pi 5 0, and in thelimit _mcPTH=QBoo1 (which is consistent withQRooQB), the optimal IP pressure is

Pi ffi ðPHPLÞ1=2 Z2

Z1

� �cP=2R

�

(1þ

_mcPTH

QB

"1

21þ

Z2Z1

� �

�Z2Z1

� �1=2PL

PH

� �R=2cP#)

ð7Þ

Figure 4. Train of two turbines, high pressure and low pressure.



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Simpler forms in the limit represented byEquation (7) are available for Z2=Z1 ffi 1 (note:Z1,2o1),

Pi ffi ðPHPLÞ1=2 1þ

_mcPTH

QB1�

PL

PH

� �R=2cP" #( )

ð8Þ

which in the limit _mcPTH=QB ¼ 0 reduces to

Pi ¼ ðPHPLÞ1=2 ð9Þ

The analysis that follows is based on the simplerversion, Equation (9), in order to make thedemonstration entirely analytically. In reality,the limit _mcPTH=QB ¼ 0 is not reached. Forexample, the boiler of a 500MW power plantreceives _m ¼ 473 kg s�1 and QB 5 106 kW, whileTH 5 813K [13]. In this case, the dimensionlessgroup _mcPTH=QB is equal to 1.26. However, itseffect on Equation (8) is weak because the quantityshown between fg is essentially constant and oforder 1.

The isentropic efficiencies increase mono-tonically with the sizes of two turbines (M1, M2),cf. Figure 3. The Z(M) curves must be concavebecause they both approach Z5 1 in the limitM-N. It is reasonable to assume that near Zp1the Z(M) data for turbine designs are curvefitted by

Z1 ¼ 1� a1e�b1M1 ð10Þ

Z2 ¼ 1� a2e�b2M2 ð11Þ

where (a, b)1,2 are four empirical constants thatdepend on the pressure level of the turbine. Thetotal power produced by the two turbines isW ¼ Z1W1;rev þ Z2W2;rev, where the functionsZ1ðM1Þ and Z2ðM2Þ are known. The total mass ofthe ensemble is fixed,

M ¼M1 þM2 ð12Þ

There is one degree-of-freedom in the makingof Figure 4, namely, the dividing of M into M1

and M2. The optimal way to divide M isdetermined using the method of Lagrangemultipliers. We form the aggregate function

F ¼Wþ lM ð13Þ

for which W and M are the expressions (1, 2)and (12). We solve the system @F=@M1 ¼ 0 and

@F=@M2 ¼ 0, eliminate the multiplier l, andobtain

b1M1 � b2M2 ¼ lna1b1W1;rev

a2b2W2;rev

� �ð14Þ

Equations (14) and (12) pinpoint the massallocation fractions M1/M and M2/M. The firstand most important conclusion is that there mustbe a balance between M1 and M2.

For example, if we use the HP and IP dataof Figure 3 for Z1 of Equation (10), we obtainapproximately a1 ffi 0:2 and b1 ffi 4:6� 10�5 kg�1.If we use the LP data of Figure 3 in conjunctionwith Equation (11), we estimate that a2 ffi 0:13and b2 ffi 2:1� 10�5 kg�1. If, in addition,Equation (9) holds, then W1,rev 5W2,rev and thismeans that on the right hand side of Equation (14)we have a1b1W1,rev/(a2b2W2,rev)ffi34, which isconsiderably greater than 1. In conclusion,Equation (14) states that b1M1 is greater thanb2M2, namely, b1M1ffib2M213.5.

Combining Equations (12) and (14) yields theoptimal mass distribution equations for HP andLP turbines

M1 ¼ 0:044Mþ 7:3� 104 kg ð15Þ

M2 ¼ 0:956M� 7:3� 104 kg ð16Þ

M1 equals M2 when M5 1.6� 105 kg. Inconclusion, there is an optimal way to distributemass along the train of turbines. According toEquations (15) and (16), the total mass is allocatedin a balanced way; with more mass at the HP endwhen total mass M is small, and with more mass atthe LP end when the total mass is large. What weshowed here for a group of two turbines also holdsfor groups of three or more.

The penalty associated with not using a train ofturbines can be calculated with reference toFigure 4. We form the ratio W(M2 5 0)/Wmax,where Wmax is the W maximum that correspondsto the constructal design, Equation (9). The ratio is

WðM2 ¼ 0ÞWmax

¼1

21þ

PL

PH

� �R=2cP" #

o1 ð17Þ

This ratio is always smaller than 1. In conclusion,the penalty is more significant when the overallpressure ratio is greater.



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3. MULTIPLE TURBINES

Consider the limit where the number of turbines isextremely large (i5 1,2,y,N) and their individualsizes are small, Figure 5. Each turbine andsubsequent reheater can be modeled as anisothermal expander receiving the heat transferrate Qi and the steam _mðTH;Pi þ DPiÞ, anddelivering the power Wi and the steam _mðTH;PiÞ.With reference to the elemental system (i) shown inFigure 5, the first law requires

Wi ¼ Qi þ _mðhiþ1 � hiÞ ð18Þ

Because the steam is modeled as an ideal gas,the enthalpy of the inflow is the same asthe enthalpy of the outflow, and the first lawreduces to

Wi ¼ Qi ð19Þ

If the isothermal expander (i) operatesreversibly, then

Qi;rev ¼ _mTHðsout � sinÞi ¼ _mTHRDPi

Pið20Þ

and Wi;rev ¼ Qi;rev. The actual system operatesirreversibly with the efficiency

Zi ¼Wi

Wi;revo1 ð21Þ

and its power output is

Wi ¼ _mTHRZiDPi

Pið22Þ

where DPi ¼ Piþ1 � Pi. Of interest is the totalpower delivery

WT ¼ _mTHRXNi¼1

ZiPiþ1 � Pi

Pið23Þ

which in combination with the boiler heatinput (QB, fixed) and the total heat input tothe isothermal train (Q5W), yields the efficiencyratio

Z ¼W

QB þWð24Þ

In order to maximize Z we must maximize totalpower WT given by Equation (23). Attractive arelarges value of (DP/P)i and Zi, in each stage ofisothermal expansion. The data of Figure 3suggest that, in general, Zi depends on both Mi

and Pi, such that

@Zi@Mi

40 and@Zi@Pi

40 ð25Þ

The masses of the N turbines are constrainedby the total mass of the turbine train, which

Figure 5. Train of N turbines.



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is fixed

MT ¼XNi¼1

Mi ð26Þ

Another constraint is the DPi’s must all up to theoverall pressure difference, DPT 5PH–PL:

DPT ¼XNi¼1

ðPiþ1 � PiÞ ð27Þ

In summary, we must maximize the sum (23)subject to the constraints (26) and (27). Accordingto Lagrange’s method of undeterminedcoefficients, this problem is equivalent to seekingthe extremum of the function

F ¼XNi¼1

ZiPiþ1

Pi� 1

� �þ lMi þ mðPiþ1 � PiÞ

� �

ð28Þ

where Zi 5 Zi(Mi, Pi), and l and m are twoLagrange multipliers. The function F is a linearcombination of the sums (23), (26) and (27). Itdepends on 2N12 variables, namely, l, m,M1,y,MN and P1,y,PN. Its extremum is foundby solving the system of 2N equations

@F

@Mi¼@Zi@Mi

Piþ1

Pi� 1

� �þ l ¼ 0

i ¼ 1; 2; . . . ;N ð29Þ

@F

@Pi¼@Zi@Pi

Piþ1

Pi� 1

� �� Zi

Piþ1

P2i

þ m ¼ 0

i ¼ 1; 2; . . . ;N ð30Þ

This system establishes the N masses (Mi) and theN IPs (Pi) as functions of the undeterminedcoefficients l and m. In principle, thesecoefficients can be determined by substituting thesolutions for Mi(l, m) and Pi(l, m) into theconstraints (26) and (27).

Here, we make analytical progress on a simplerpath by linearizing the function Zi(Mi) in thevicinity of the design range in which all theturbines are expected to operate (namely,near Zt�1 in Figure 3). We write that for allN turbines the efficiency–mass relationship isunique,

Zi ffi aþ bMi þ cPi ð31Þ

where (a, b, c) are constants. The first result of thelinearization is that the N Equations (29) reduce to

Piþ1

Pi¼ constant ð32Þ

In view of Equation (27), we conclude that thetotal pressure interval DPT is divided intoN pressure intervals such that

Piþ1

Pi¼

PH

PL

� �1=N

ð33Þ

or

DPi

Pi¼

PH

PL

� �1=N

�1 ð34Þ

This is in agreement with what we found earlier inEquation (9).

The distribution of MT among the N turbinesfollows from Equation (30), in which we substituteEquations (31) and (32). The result is

ZiPi¼

PL

PH

� �1=N

mþ cPH

PL

� �1=N

�1

" #( )

constant ð35Þ

This shows that in the (Mi, Pi) range whereEquation (31) is valid, Equation (35) becomes

aþ bMi þ cPi ¼ Pi � constant ð36Þ

If the effect of Pi on Zi is negligible, as in thecase of the HP and IP data of Figure 3, thenEquation (36) reduces to

aþ bMi ffi Pi � constant ð37Þ

The mass of the individual turbine should increaselinearly with the pressure level of that turbine.

In particular, if the linear approximation (31)reveals that a/booMi, as in the calculations shownunder Equation (14), then Equation (37) statesthat the Mi’s must be distributed in proportionwith the Pi’s, and, in view of Equation (34), inproportion with the DPi’s:

Mi � Pi � DPi ð38Þ

Equation (38) indicates that more mass should beplaced in the expanders at higher pressures.



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4. RESULTS AND DISCUSSION

These is a considerable volume of work on turbinedesign, where the size is selected from otherconsiderations such as the distribution of maximalstresses due to centrifugal forces, and the flaringout of the flow passages to accommodate theexpansion of the steam along the turbine train [15].The work reported in the present paper suggeststhat the design of future concepts of turbine trainconfiguration must combine the traditional con-siderations [16,17] with size allocation principleillustrated in this paper.

This paper also highlights the need for moreextensive and more exact information on how thesize of each turbine affects its thermodynamicperformance. The data that we used (Figure 3) arefew and provide a narrow view of the size effectthat is needed for future design. These data alsorequire an understanding of how the multipleturbines are arranged in the power plant.

As the power generation capacity of the plantincreases, the boiler and the turbine also increasein size. It is easer to increase the size of the boilerthan the size of the turbine. Today, in a 1000MWpower plant there is still a single boiler, while thenumber of turbines is five or six. Each pressurestage employs one or several (two or three)turbines in parallel. For example, in power plantswith more than 500MW capacity, the LP stageconsistently employs two or more turbines.

Several physical limitations require the use ofmultiple turbines in parallel at a single pressurestage. A major limitation posed by centrifugalstresses is the length of the last blade. The currentmaximum length is approximately 1.3m. The dataof Figure 3 come from a current design for a1000MW power plant with an LP stage consistingof four turbines in parallel, each turbine with lastblades 1.14m long. The ordinate of Figure 3indicates the efficiency of the single LP turbine,while the abscissa represents the total massemployed for each pressure stage. If the mass ofthe LP turbines is divided by 4, then the LP data ofFigure 3 move closer to the HP and IP data.

Note also that the efficiency of one turbine isaffected by the operation of the turbine upstreamof it. The irreversibility of a turbine is due to six

losses: deviation from the ideal velocity ratio,rotational loss, diaphragm-packing leakage loss,nozzle end loss, moisture and supersaturating lossand exhaust loss [15]. The last two losses arepresent only in the LP turbines, not upstream,therefore the efficiency of the single LP turbineshould be lower than the efficiency of HP and IPturbines, cf. Figure 3. Furthermore, because theHP turbine is installed at the head of the train, itsefficiency should be lower than the IP turbineefficiency because of entrance losses due to theconfiguration of the steam passage. This too isconfirmed by the data of Figure 3.

The optimal IP Equations (7)–(9) may not beattainable in a design with two turbines modelbecause of the properties of water. For example,the IP in a 500MW power plant is approximately4MPa, while the IP based on Equation (9) is0.35MPa. The discrepancy between the IPs is dueto the ideal gas model used for steam. If we modelas isentropic the expansion through the HPturbine, because of the saturated steam curve ofwater the attainable IP pressure is 3.15MPa. Theexpansion cannot proceed beyond 3.15MPabecause of engineering limitations such as waterdroplet impingement on the blades. This limitationimpacts on the mass distribution of turbines fortwo turbines model. When the number of pressurestages increase to three or more, the expansionlimitation due to steam properties diminishes inimportance.

5. CONCLUSION

The main conclusion is that the total turbine massmust be divided in a certain way when two or moreturbines are used in series. The allocating of massis synonymous with the discovery of the config-uration of the turbine train. Important is that inthis article the distribution of ‘size’ along theturbine train came from the pursuit of globalthermodynamic performance. The allocation ofmass is driven by the size effect on turbineefficiency: larger turbines operate closer to thereversible limit (e.g. Figure 3).

Future extensions of this work should take intoaccount the physical limitations that make



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necessary the use of several turbines in parallel at apressure stage. Such work could address thedistribution of turbine mass per pressure stage,instead of per turbine. This work would be assistedfurther by the availability of more extensive dataof the type sampled in Figure 3. To that end, itwould be useful to construct models that accountfor the trend exhibited by the data. In other words,the analytical form of Z(M, P), which wasapproximated here in Equations (10) and (31),should be derived from a model that accounts forirreversibility and finite size.

NOMENCLATURE

a1,2 5 constantsb 5 constantb1,2 5 constantsc 5 constantcp 5 specific heat at constant pressure

(J kg�1K�1)F 5 aggregate function, Equation (28)h 5 enthalpy (J kg�1)HP 5 high pressureIP 5 intermediate pressureLP 5 low pressure_m 5mass flow rate (kg s�1)M 5mass (kg)N 5 number of turbinesP 5 pressure (Pa)Q 5 heat transfer rate (W)QB 5 heat transfer rate to the

boiler (W)QR 5 heat transfer rate to the

reheater (W)R 5 ideal gas constant (J kg�1K�1)s 5 entropy (J kg�1K�1)TH 5 highest allowable temperature

(K)W 5 power (W)

Greek letters

DP 5 pressure drop (Pa)Z 5 turbine isentropic efficiencyl, m 5Lagrange multipliersF 5 auxiliary function, Equation (13)

Subscripts

H 5 highi 5 ith turbinei 5 intermediatein 5 inletL 5 lowmax 5maximumout 5 outletrev 5 reversibleT 5 total1 5 inlet of the first turbine10 5 outlet of the first turbine2 5 inlet of the second turbine20 5 outlet of the second turbine

ACKNOWLEDGEMENTS

This research was supported by Doosan HeavyIndustries & Construction Co., Ltd., Changwon, SouthKorea.

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