Finite-time optimizations of a heat engine

EnergyVol. 15, No. 11, pp. 979-985, 1990 Primedia Great Britain.Au rightsreserved

FINITE-TIME

0360-5442/W $3.00+ 0.00 Copyright0 1990PergamonPressplc

OPTIMIZATIONS

OF A HEAT ENGINE

WON Y. LEE and SANGS. KIM Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology (KAIST), Seoul, Korea SEUNG H.

WON?

Fluid Machinery Laboratory, Korea Institute of Energy and Resources, P.O. Box 5, Daejeon, Korea (Received 8 December 1989) Ah&act-We

study the power and efficiency of a finite-time heat engine. We consider an endoreversible Camot heat engine coupled to heating and cooling fluids and determine the maximum power output and efficiency to obtain a bound on the power-conversion systems. While the maximum power and the operating temperatures of the heat engine are functions of the heat-reservoir temperatures, heat-reservoir capacity rates and heattransfer rates, the efficiency at maximum power depends primarily on the initial temperatures of the heating and cooling fluids. The efficiency at maximum power provides a measure of the power available in a practical heat engine.

INTRODUCTION

We are interested in the analysis of the efficiency and power output of an energy-conversion process. It is important to formulate the appropriate criteria for energy efficiency in planning and the implementation of methods for the conservation of energy. It is well known that no engine could be better than the classical Camot heat engine. The necessary conditions for the classical Camot efficiency [rlcornot = (I” - TL)/T’] are that all processes are reversible and are operated in an infinitely slow cycle for the working fluid to come into thermal equilibrium with the heat reservoirs. Under these conditions, the power output is zero since it takes an infinite time to do a finite amount of work. We know that a practical heat engine is not as efficient as the classical Camot engine because of irreversibility. However, even if the Camot engine is operated reversibly, the Camot efficiency can not be realized if the heat which derives the practical engineering engine is supplied by a source and a sink with finite heat capacity rates. It is the objective of this study to investigate such a system. Curzon and Ahlbom’ were the first to calculate the real-&mot engine efficiency at maximum power output, which differs from the classical Camot engine efficiency. Rubine et a12v3solved the problem of maximum efficiency and defined an endoreversible engine, i.e., an internally reversible engine. Salmon et al and Andresen435 developed thermodynamics based on finite time duration. Finite-time constraints are essential for real process. A major objective of finite-time thermodynamics is to understand irreversible finite-time processes and to establish bounds on efficiency and maximum work for such processes. Another primary goal of finite-time thermodynamics is to establish general operating principles for systems which serve as models for real processes6 Ondrechen et al’ analyzed the maximum work from a finite reservoir by using sequential Camot engines. Morurkewich and Berry’ studied optimal paths for a heat engine based on a dissipative system applying optimal control. Watowich et al9 also

studied an intrinsically irreversible, light-driven engine. WU’~~” studied a finite-time heat engine, with both the heat source and sink having finite-time heat-capacity rates and obtained numerical solutions for the efficiency at maximum power. Wu did not obtain analytical optimal solutions. In this paper, we analytically present the optimum power and efficiency of a finite-time Camot heat engine operating between two reservoirs with finite heat-capacity rates. tTo whom all correspondence should be addressed. 979

WON Y. LEE et al

980 THEORETICAL

MODEL

AND

OPTIMIZATION

A heat engine alternatelyconnected to two reservoirs Our first system is a Camot heat engine which is alternately connected to a heating fluid and a cooling fluid with finite heat-capacity rates as in a Stirling engine. This particular cycle is shown in Fig. 1 for the temperature-entropy plane. The heat engine operates in a cyclic mode with fixed time &,. allotted for eacy cycle. Thus, after time rcy hat elapsed, the working fluid returns to the initial state. The heat-engine cycle consists of two isothermal (4-1, 2-3) and two isentropic processes (l-2, 3-4). We assume that the inlet temperatures of a heat source and a heat sink (Tni, TL1), the heat &A,) of the heat exchangers and the heat capacity rates conductance (&A,, (tinCpH, tiLCPL) of a heat source and a heat sink are fixed but are otherwise arbitrary. Therefore, the temperature distributions of the heating and cooling fluids are not constant through the heat exchangers. The rate of heat flow from the heat source to the heat engine is proportional to the logarithm of the mean temperature difference (LMTDn) and is equivalent to the decreasing rate of heat input from the heating fluid. If the heat input lasts tH set per cycle, the rate of heat input is Q, = Qnltn = G-A-~LMTDH = &&.r(Tni

- THZ),

(I)

where LM’D+ = [(G

- Tw) - (%

- Tw)IIln[(TH1 - Tw)l(Gn - Tw)l;

(2)

tH is the duration of heat transfer, Un the overall heat-transfer coefficient, AH the surface area of the heat exchanger between the heat source and the heat engine; TH, and Tm are, respectively, the inlet and outlet temperatures of the heating fluid at the heat source and Tw is the temperature of the working substance; ti n is the mass flow rate of the heating fluid and CPH the specific heat of the heating fluid. Similarly, for condensation, the rate of heat flow from the heat engine to the heat sink is

& = Q&c

= ULALLMTD~. = &G&z

- TL,),

(3)

- LVG - WI;

(4

where LMTlA-, = [(Tc - &_.A- (Tc -

L)lhdG

tc is the duration of heat transfer, UL the overall heat-transfer coefficient, AL the surface area of the heat exchanger between the heat engine and heat sink; T,, and Tu are, respectively, the inlet and outlet temperatures of the cooling fluid at the heat exchanger; TC is the temperature of the working substance; &, is the mass flow rate of the cooling fluid and C,, the specific heat of the cooling fluid. The time taken to complete the whole cycle is tcy = tH + tC + t12 + tH = (t,., + t&l

+ r).

Entropy Fig. 1. Finite-time Carnot cycle operating behveen heat source and sink with finite heat-capacity rates.

(5)

optimizationsof a heat engine

Finite-time

981

Since the times fi2 and tu required for the two isentropic processes of the cycle are negligibly small relative to fH and fo,i’*‘i the time ratio r is almost zero. Equation (5) then becomes fey = ti.#+ to.

(6)

Using Eqs. (1) and (3), the total time f,.+is given by the expression

QL

QH

t, =

~HcPH(THI

-

&2)

+ &&‘I_(~2

-

&al)

(7)

*

Using Eq. (l), the outlet temperature of the heating fluid is given as a function of Tw and THI, viz. THZ = TW + (THI - TV) exp(-~HAH/&&ib (8) Similarly, using Eq. (3) TLzbecomes Tu = To - (Tc - TL1) exp( - LILAJljti_CPL). Using Eqs. (8) and (9) to eliminate

(9)

T, and T,, the total time becomes

Ql_

QH

(10)

- T,) + LE( Tc - TLI) ’

f’Y=HE(Tni where HE=IjlHCpn[l

-exp(-UHAH/AHCrH)J,

LE = &_CPL[l - exp(-&_AJrit,C,,)]. The net output of work W is W=QH-QL. Since the Carnot cycle is endoreversible net entropy change is zero and

(II) Tw and T,, the

and operates between the temperatures

QJTc-

QHITw=

(12)

From Eqs. (11) and (12), QH

= WTwI(Tw- Tc),

Q,

= WTcI(Tw

-

(13)

Tc).

Using Eq. (13), Eq. (10) becomes 1cy

=

[

WTHI

-

WTw WT, Tw)Vw- Tc)+ WT, - L)Uw - Tc)

I

(14)

and the average power is given by

CY

[

HE(

THI -

Tw)(Tw

-1

Tc

TW

p+

-

Tc)

+ WT,

-

TLIWW

-

Tc)

1

For fixed THI and TLI, P is thus a function of Tw and Tc only. Using the optimization method of Curzon and Ahlbom,’ the maximum becomes HELExy(THi - TLI --x -y) P= LE THly + HE TL~x+ xy(HE - LE) ’

(15)

’

power

output

where x = THI - Tw, y = Tc - TLI. Maximizing P with respect to x and y yields

api ax= 0,

aplay = 0.

It can be shown that

aWa2~0,

aWay2<0.

(17)

982

WONY.LEE~~~

From Eqs. (16) and (17), x and y are determined

by

X -=

THl

-1

-=Y

TLl

'

+1

Tw and Tc are given by

From Eq. (18), the optimum~values

Tw=

~t(T~l)~'~,

(19)

T,=

GU~J)".~,

(20)

where C _ (HE THl)O.’+ (LE TL,)O.’ 1HEo.5 + LEO.5 * Substituting Eqs. (19) and (20) into Eq. (15), the optimum power for the irreversible engine becomes p = HE LE(pfi; - pi:)2 Inax (HE’.’ + LE0.5)2 ’

heat

(21)

The efficiency at maximum power is em = (Tw - T,)/Tw = I-

(TLJTn,)‘.‘.

(22)

It may be seen from Eq. (22) that the efficiency at the maximum power output point depends only on the initial temperatures of the heating and cooling fluids. For the case of a heat engine operating between a heat source and sink with infinite heat-capacity rates, i.e., ri?uCpn+m and tiLCPL-* 00, the temperature distributions of the heating and cooling fluids are constants through the heat exchangers, viz. THI=

THY=

THY

(23)

TL1= TL2= TL. As GtnCrn and tiLCPL approach infinity, Eqs. (19)-(22)

become, respectively,

Tw = C,( TH)‘.‘,

(24)

Tc = C,( TL)‘.‘,

(25)

UHULAHAL(Tofi’- Tof5)2 P max= [(&AH)“.’ + ( ULAL)0.5]2. VM =

l-

(26)

(TL/TH)O.',

(27)

where c = 2

(UHAHTH)'.'+

(ULALG_)~.~

(UHAH)‘.’ + (ULAL)‘.’

*

These equations are identical with those of the Curzon and Ahlborn.’ A heat engine connected to two heat reservoirs Thus far, we have studied the optimal conversion of heat from a finite source and a finite sink, assuming that the heat engine is alternately connected to a heating fluid and to a cooling fluid. We now treat a heat engine that is continuously connected to a heating and a cooling fluid as in a Rankine cycle engine. Thus, the time taken for the heat transfer Qn is the same as the time required for the transfer of an amount QL of heat, i.e., tH = tc.

983

Finite-time optimizations of a heat engine

for one cycle in the heat-transfer

It follows from Eq. (14) that the time required are, respectively,

cH =HE( THl-

W&v

(28)

Tw)( Tw - Tc) ’

WT, TL1)(TwTc)'

tc= LE(T,-

processes

(29)

Combining Eqs. (28) and (29) yields

Tc=

LE TWTLI

(30)

LETw+HETw-HETH,’

Substituting Eq. (30) into Eq. (28), the power output becomes p =

P is then a function of

HWHI

- Tw)(LE Tw + HE Tw - HE THI- LE LE Tw+HE Tw-HE THI

TL1)

(31)

Tw only. Maximizing P with respect to Tw yields i3P/aT,=O.

(32)

It can also be shown that From Eqs. (31) and (32),

Tw is given by Tw =

(33)

~~(THI)‘.‘>

where C _ HE(

TH1)".5 + LE( TL,)".5

3-

Substituting Eq. (33) into Eq. (30),

HE+LE

’

Tcis found to be T,= C,(TL,)o.5.

(34)

Using Eqs. (31), (33) and (34), the maximum power becomes HELE(

Pmax=

T$- T;:)2

HE+LE

(35)

’

The efficiency at the point of maximum power is given by tl,,, = (Tw - Tc)/Tw = I-

(TLJTH~)“.~.

(36)

It is important that Eq. (36) is the same as Eq. (27), even though these equations are based on somewhat different heat-engine models. Hence, Eq. (36) can be applied to the operations of heat-engine in general. If we consider a heat engine operating between a heat source and sink with infinite heat capacity rates, i.e., tiZ&p~-* 01, tjz~Cp~+ W, the maximum power and the operating temperatures are given by

Tw= C4(TH)o.5,

(37)

T,= C,(TL)o.5,

(38)

p =V,LI,AHA,(~~~~TO;')' max UHAH+ULA, ' = 1 - (TL/TH)o.5, t7rn

(39) (W

where

C = [Q-~H(TH)~.~ 4

(Q-A,

+

+ GA,)

The efficiency at maximum power is also equivalent ICIL5:ll.n

Q_A.(T~)0.51 *

to the efficiency of Curzon and Ahlbom.’

WON Y. LEE Ct d

984

Table 1. Observed efficiencies of real heat engines.

Larderello geothermal steam plant (Italy)l

353

523

16

32.5

17.8

298

840

36

64.4

40

298

698

28

57.3

34.8

West Thurrock conventional coal-fired plant (U. K.)’

steam

Central steam power stations in the U.K.13

The efficiency at maximum power formulated in this study is compared with observed effiiciencies of actual power plants in Table 1. Our efficiency affords a much more accurate estimation of real heat engines than does the classical Camot efficiency. Therefore, the formulae derived in this paper serve as a better guide to the observable performance than the Camot relation.

CONCLUSIONS

The heat which drives real engines is supplied by heat reservoirs with finite heat-capacity rates. Therefore, the prediction of the maximum available power and efficiency from such a system is of practical importance. Using a model for a Camot heat engine operating between two reservoirs with finite heat-capacity rates, we have calculated the optimum finite-time power and efficiency at the point of maximum power. We find that the optimum values for the operating temperatures and power output depend on the initial temperatures of the heating and cooling fluids as well as on the heat-transfer coefficient and the heat-capacity rates. The efficiency at maximum power depends only on the initial temperatures of the heating and cooling fluids, which is equivalent of the efficiency of Cunon and Ahlbom’ for the case of infinite heat-capacity rates. The formulae derived in this paper provide a useful guide to the achievable performance for real power-conversion systems.

REFERENCES 1. F. L. Curzon and B. Ahlbom, Am. J. Phys. 43,22 (1979). 2. M. H. Rubin, Phys. Rev. A19, 1272 (1979). 3. M. H. Rubin and B. Andresen, J. Appl. Phys. 53, 1 (1982).

Finite-time optimizations of a heat engine 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

P. Salmon and A. Nitzan, J. Chem. J’hys. 74,3% (1981). B. Andresen, R. S. Berry, A. Nitzan, and P. Salmon, Phys. Rev. AlS, 2086 (1977). M. J. Ondrechen, M. H. Rubin, and Y. B. Band, J. Chem. Phys. 78,427l (1983). M. J. Ondrechen, B. Andresen, M. Mozurkewich, and R. S. Berry, Am. .I. Phys. 49,681 (1981). M. Mozurkewich and R. S. Berry, J. Appl. Phys. 54, 3651 (1983). S. J. Watowich, K. H. Hoffmann, and R. S. Berry, J. Appl. Phys. S&2893 (1985). C. Wu, Energy 13, 681 (1988). C. Wu, Int. J. Heut Fluid Flow 10, 134 (1989). L. Y. Bronicki, ORC-HP-Technology Working Fluid Problems, VDI, Duesseldorf (1984). A. Bejan, Int. J. Heat Muss Transfer 31, 1211 (1988). A. Bejan, Entropy Generation through Heat and Fluid Flow, Wiley, New York, NY (1982). NOMENCLATURE

area of the heat exchanger operating between the heat source and the heat engine = Surface area of a heat exchanA_ ger operating between the heat sink and the heat engine C,, Cz, C,, C.,= Constants defined in Eqs. (19), (24) (33) and (37) respectively = Specific heat of the heat source CPH = Specific heat of the heat sink C = Constants defined in Eq. (10) HE, LE = Log mean temperature differLMTDn ence between the heat source and the heat engine at the heat exchanger = Log mean temperature differLMTDL ence between the heat sink and the heat engine at the heat exchanger = Mass flow rate of the heat mH source at the heat exchanger = Mass flow rate of the heat sink m_ at the heat exchanger = Power generated from the heat P engine = Optimum power generated by P“PX the finite-time heat engine = Heat transferred from the heat Q” source to the heat engine = Heat transferred from the heat QL engine to the heat sink = Rate of input heat transfer = Rate of output heat transfer 2 = (112+ 134)/(1” + cc) = Entropy i = Temperature T =Temperature of the working TC substance in the condenser

AH

= Surface

TH

= Temperature of the heat source with infinite heat capacity TH, = Inlet temperature of the heating fluid at the heat exchanger with finite heat capacity rates THz = Outlet temperature of the heating fluid at the heat exchanger with finite heat capacity rates = Temperature of the heat sink with TL infinite heat capacity TI.1 = Inlet temperature of the cooling fluid at the heat exchanger with finite heat capacity rates = Outlet temperature of the cooling fluid TU at the heat exchanger with finite heat capacity rates of the working substance KV =Temperature in the evaporator = Total time required for the whole heat t, engine cycle = Time required to transfer QH tH tc = Time required to transfer Q,_ = Time required for the isentropic exh2 pansion process 134 = Time required for the isentropic pumping process = Overall heat transfer coefficient at the UH heat exchanger between the heat source and the heat engine = Overall heat transfer coefficient at the heat exchanger between the heat sink and the heat engine W = Output work generated by the heat engine X = T”, - T, = Tc - TL, Y mm= Classical Carnot efficiency b, =Thermodynamic cycle efficiency at tlnl maximum power

u.

985