Power optimization of a finite-time Carnot heat engine

Energy Vol. 13, No. 9, pp. 681-687, Printed in Great Britain

1988

036%5442/M $3.(w) + 0.00 Pergamnn Press plc

POWER OPTIMIZATION OF A FINITE-TIME CARNOT HEAT ENGINE CHIH WV Department

of Mechanical

Engineering,

U.S. Naval Academy,

(Received

1 February

Annapolis,

MD 21402. U.S.A

1988)

Abstract-The power output of a simple, finite-time Carnot heat engine is studied. The model adopted is a reversible Carnot cycle coupled to a heat source and a heat sink by heat transfer. Both the heat source and the heat sink have finite heat-capacity rates. A mathematical expression is derived for the power output of the irreversible heat engine. The maximum power output is found. The maximum bound provides the basis for designing a real heat engine and for a performance comparison with existing power plants.

INTRODUCTION

Among the important topics in thermodynamics has been the formulation of criteria for comparing the performance of real and ideal processes. Carnot showed that any heat engine absorbing heat from a higher temperature reservoir to produce work must transfer some heat to a sink reservoir of lower temperature. He also showed that no heat engine could be better than the Carnot heat engine. The early tradition was carried on by Clausius, Kelvin and others in using thermodynamics as a tool to find limits on work, heat transfer, efficiency, coefficient of performance, energy effectiveness, and energy figure of merit of energy conversion devices. The basic laws of thermodynamics were all conceived about irreversible processes. However, the subsequent development of thermodynamics has turned away from the process variables of heat and work toward state variables since Gibbs. The Carnot-Clausius-Kelvin view emphasizes the interaction of a thermodynamic system with its surroundings, while the Gibbs view makes the properties of the system dominant and focuses on equilibrium states. Contemporary classical thermodynamics gives a fairly complete description of equilibrium states and reversible processes. The only facts that it tells about real processes are that these irreversible processes always produce less work and more entropy than the corresponding reversible processes. Reversible processes are defined only in the limit of infinitely slow execution. In the real engineering world, actual changes in enthalpy and free energy in an irreversible process rarely approach the corresponding ideal enthalpy and free energy changes. No practicing engineer wants to design a heat engine that runs infinitely slowly without producing power. The need to develop power in real energy-conversion devices is one reason why high percentages of ideal, reversible performance are seldom approached. Classical equilibrium thermodynamics can be extended to quasi-static processes. Conventional irreversible thermodynamics has become increasingly powerful, but its microscopic view does not lend itself to the macroscopic view preferred by practicing power engineers. This is a significant extension, since quasi-static processes happen in finite time, produce entropy and provide a better approximation to real processes than provided by the equilibrium thermodynamics. System parameters in equilibrium thermodynamics are masses, volumes, temperatures, pressures, and heat capacities, which can be easily measured. To model irreversible, time-dependent, real processes rigorously, the set of parameters must further include transport coefficients, relaxation times, etc. In general, irreversible thermodynamic problems are too difficult to solve exactly. The literature of finite-time thermodynamics started from Curzon and Ahlborn.’ They treated a real Carnot engine power output being limited by the rates of heat transfer to and from the working substance. They showed theoretically that the heat-engine efficiency at maximum power output is given by a different expression than the well known Carnot 681

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CHIH

WV

efficiency, and they cited cases for which the efficiency of existing engines is well described by their result. Rubin defined an endoreversible engine. In his simple model of the irreversible heat engine, all of the losses are associated with the transfer of heat to and from the engine and there are no internal losses within the engine itself. Because of the finite conductivity of the heat-transfer material, the engine is operated not between the temperatures of the available high and low temperature heat reservoirs, TH and TL, but between the temperatures of the working fluid on the warm and cold sides of the heat engine cycle, Tw and Tc. The temperatures Tw and Tc depend on the rate of heat flow and also on the power output of the machine. The efficiency of the engine also depends on its power output. Andresen et aLs5 Salamon et al,‘j and Callen’ developed thermodynamics in finite time to find the extremes for imperfect heat engines. A step Carnot cycle was defined and potentials for finite-time processes were constructed to determine the optimal performance of a real heat engine. Morurkewich and Berry8 studied optimization of a real heat engine based on a dissipative system. Band et al9 determined the optimal motion of a piston fitted to a cylinder containing a gas pumped with a given heating rate and coupled to a heat bath during finite times. Rubin” explored standards of performance for real energy-conversion processes and reviewed the argument against the use of infinitely slow reversible process standards. Rubin and Andresen” also found the optimal configuration for a class of heat engines with finite cycling times and suggested that figures of merit based on these optimal configurations may be more useful than those based on reversible processes. Rubin12 then treated thermodynamic variables of the working fluid as dynamic variables and used mathematical techniques from optimal-control theory to re-analyze the same class of irreversible heat engines as Curzon and Ah1born.r Wu13 has applied the finite-time thermodynamic cycle to an ocean thermal energy conversion system (OTEC). Wur4 also extended the cycle to a cascade cycle.

FINITE-TIME

THERMODYNAMICS IRREVERSIBLE

AND OPTIMIZATION HEAT ENGINE

OF AN

A practical heat engine is not as efficient as the classical Carnot heat engine. To achieve the theoretical Carnot cycle efficiency, the isothermal heating and cooling processes of the cycle must be carried out infinitely slowly to ensure that the working substance is in thermal equilibrium with its heat reservoirs. The power output of the cycle approaches zero since it requires an infinite time to get a finite amount of work. To obtain finite power, the cycle must be speeded up. In the other extreme, if the heat engine speed were infinitely fast, the heat would flow directly from source to sink and no mechanical work would be performed by the heat engine. Hence, the power output would again be zero and the heat engine efficiency would also be zero. Somewhere between these two extremes, the heat engine has a maximum power output. The efficiency of the heat engine under the condition of maximum output is evaluated in an analysis. Let the heat engine cycle be made of two isothermal and two isentropic processes, as indicated in Fig. 1. The cycle is a modified Carnot cycle with an irreversible isothermal expansion process from state 3 to state 4, an irreversible compression cooling process from state 4 to state 1, and an isentropic compression process from state 1 to state 2. Process 2-3 is irreversible because heat flows from the high temperature heat-source reservoir to the working fluid at temperature Tw across a temperature difference, as illustrated schematically in Fig. 1. Similarly, in the irreversible heat rejection process 4-1, heat flows across a temperature difference from the working fluid at a temperature Tc to the low temperature heat-sink reservoir. We note that both the heat source and heat sink have finite heat-capacity rates. Therefore, the temperature distributions of the heating fluid (heat source) and the cooling fluid (heat sink) are not constants throughout the heat exchangers, as shown in Fig. 1. The rate of heat flow from the high temperature reservoir to the system is proportional to the log mean temperature difference LMTDH. If tH is the time required to transfer an amount QH

Carnot heat engine power optimization

I

Cool

workinq

683

temperature

Fig. 1.

of heat, then Q, = QdtH = W&(LMTDH),

(1)

where LMTDu = [(T5 - Tw) - (G - Tw)]/ln[(T, - T,)/(T, - T,)]; U, is the overall heat transfer coefficient including conduction, convection, and radiation modes; A, is the surface area of the heat exchanger between the heat source and the system; T5 and T6 are the inlet and outlet temperatures of the heating fluid of the heat source, respectively. A similar expression holds for the rate of heat flow QJt,_ from the system to the low temperature reservoir, where LMTDr = [(Tc - T,) - (Tc - TJ]Iln[(To - T,)/(T, - &)I; t, is the time required to transfer the heat; V, is the overall heat transfer coefficient; AL is the surface of the heat exchanger between the system and the heat sink; T, and TX are the inlet and outlet temperatures of the cooling fluid of the heat sink respectively. The usual way to create an isentropic process is to pass the working fluid through the isentropic device so quickly that the system exchanges little heat with the surroundings. Therefore, the time required for the two isentropic processes, tT4 and t12, of the cycle are negligibly small relative to tH and t,_. The total time t required for the whole cycle is t = t, + tL + t3, + t*2 = tH + t1,,

(3)

where tj4 << tH, t12 << t,, t34 << tL, t12 << tL. Since QH, QL, and the output work W are related by the Carnot heat engine operating between the temperatures T, and Tc, Eq. (3) becomes t = Q,[ &A,(LMTDH)]-’ 1 W =GLMFD~(T~-

TW

+ Qc[ ULA,(LMTDL)]-’ 1 W T,.)+U,A,LMTD,(T,-

TC

(4)

G)

or 1

&A,

1 TW LMTDu ( Tw - T,) + tl,A,

1 i%f&

Tc ( Tw - T,)

1 -l

’

(5)

CHIH Wu

684

where P is the power output of the reversible heat engine. We consider the inlet and outlet temperatures of the heat source and heat sink (T5, T6,T,,T8)and the heat conductances (&A,, &_A,)of the heat exchangers are to be fixed. P is then a function of Tw and Tc only. Maximizing P with respect to the two as yet undetermined working fluid temperatures T, and Tc yields

aPlaT,=O,

(6)

aP/aT,=o.

(7)

Solving Eqs. (6) and (7) numerically, the optimum intermediate temperatures can be found. Substituting the optimum intermediate temperatures into Eq. (5) yields the optimum power delivered by the irreversible heat engine Pm,,. It can also be shown that the second derivatives of the power output of the system with respect to Tw and Tc are less than zero, i.e.,

a2PlaT2,<0,

(8)

a2PiaT2,<0.

(9)

Equations (8) and (9) verify that the power output of the irreversible heat engine is indeed the maximum, where P is the power output of the reversible heat engine. For the case of a heat source and heat sink with infinite heat capacity, the temperature distributions of the heating and cooling fluids are constants throughout the heat exchangers, as is shown in Fig, 2. The case then simplifies to that of Curzon.’ Equations (l), (2) and (5) become, respectively,

Qdh

= &MT,

- Tw),

(10)

QJk = 4_A,(Tc - TL), 1

1

Tw

(11) 1

T-C

&A,(&-Tw)(Tw-Tc)+U,A,(Tc-T,)(Tw-T,) where TH = 7;= T6and TL= T,= T8. The optimum intermediate temperatures

1

'

(12)

are found’ to be

Tw = C(T,)".',

(13)

T,= C(T,)O.',

(14)

where

C = [(U,A,T,)".'+ (ULA,TL)"."][(UHA,)".5 + (ULA,)".5]-1.

Warm

working

fluld

*l------

Entropy s Fig. 2.

temperafure

(15)

Carnot heat engine power optimization

Substituting Eqs. (13) and (14) into Eq. (12) yields the optimum irreversible heat engine and the efficiency at optimum power, Pmax= (UHA,ULA,)[(T;S q =

685

power delivered

- T’t.‘)]*[( CIHA,)“.5 + (ULAL)0.5]-2,

P,,,,,l(Q,JL)= l- (TLIT#‘.‘.

by the (16) (17)

Curzon and Alhborn’ claimed that large power plants are operated closer to this efficiency than to the Carnot efficiency and illustrated their claim by comparisons with a coal-fired steam-power plant, a nuclear reactor, and a geothermal steam-power plant. An efficiency and power output analysis on the finite-time Carnot heat engine are performed in the following numerical example.

NUMERICAL

EXAMPLE

We take the inlet and outlet temperatures of the heating and cooling fluids to be 7; = 1500 K, T6= 1200 K, G = 293 K, and T8= 303 K, let &AH = 1 MW/K and &A, = 1 MW/K. Then P = (x - y)abuv/(xzJ + uyu) = z,

(18)

u = (T.‘.- T,)/[ln(ir; -x)/(T6

-x)1 = LMTD”,

(19)

u = (T, - T,)l[ln(y

- &)I = LMTR,

(20)

where

- T,)l(y

a = YiddUA_,

(21)

b = &A,,

W-9

x = T,, y = Tc.

Fig. 3

CHIHwu

686

P is a function of x and y. A plot of the P(x, y) surface is shown in Fig. 3. Numerical solution for optimum power gives: x = 988.3 K = Tw = T,, LMTDH = 339.9 K, LMTDL = 160.9 K, Qn = 339.9 MW, Q, = 160.9 MW, P,,x = 91.90 MW, r,~= 0.5356. The traditional Carnot cycle efficiency operating between the heat source and heat sink has a value of 0.7475. qCarnot = 1 - (T,/T,) = 1 - (303/1200) = 0.7475. Although the Carnot heat engine has the maximum efficiency of 0.7475, which is larger than the efficiency 0.5356 for the finite-time Carnot heat engine, it does not produce power at all.

CONCLUSION

An irreversible heat engine may be modeled by using an irreversibility factor and a time factor to simulate the primary heat-transfer processes for the rate of energy exchange between the heat engine and its surroundings. This approach gives a much more realistic prediction of heat-engine efficiency than does the ideal Carnot cycle. The power optimization process also provides a power bound for designing a real heat engine and for performance comparisons between existing heat engines.

REFERENCES

1. 2. 3. 4. 5.

6. 7. 8. 9.

10. 11. 12. 13. 14.

F. L. Curzon and B. Ahlborn, Am. J. Phys. 43, 22 (1975). M. H. Rubin, Phys. Rev. A19, 1272 (1979). B. Andresen, P. Salamon, and R. S. Berry, J. Chem. Phys. 66, 1571 (1977). B. Andresen, P. Salamon, and R. S. Berry, Phys. Today 37, 62 (1984). B. Andresen, P. Salamon, A. Nitzan, and R. S. Berry, Phys. Rev. AU, 2086 (1977). P. Salamon, B. Andresen, and R. S. Berry, Phys. Rev. AC, 2094(1977). H. B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd ed., Wiley, New York, NY (1985). M. Mozurkewich and R. S. Berry, J. Appl. Phys. 54, 3651 (1983). Y. B. Band, 0. Kafri, and P. Salamon, J. A&. Phys. 53, 8 (1982). M. H. Rubin, Am. J. Phys. 46, 537 (1978). M. H. Rubin and B. Andresen, J. Appl. Phys. 53, 1 (1982). M. H. Rubin, Phys. Rev. A19, 1277 (1979). C. Wu, J. Ocean Engng 14, 349 (1987). C. Wu, Paper No. 87-WA/DSC-6, ASME Winter Annual Meeting, Boston, MA (1987).

NOMENCLATURE AH = Surface

area of the heat exchanger between the source and the heat engine AL = Surface area of the heat exchanger between the heat sink and the heat engine a = Ratio of l&A, to &A, defined in Eq. (21) b = Defined in Eq. (22) C = Constant defined in Eq. (15) LMTDH = Log mean temperature difference in the heat exchanger between the heat source and the heat engine LMTD, = Log mean temperature difference in the heat exchanger between the heat sink and the heat engine P = Power generated by the heat engine P,,,,, = Maximum power generated by the heat engine Qr, = Heat transferred from the heat source to the heat engine

Q,_ = Heat transferred from the heat engine to the heat sink QH = Rate of heat transfer from the heat source to the heat engine Q,_ = Rate of heat transfer from the heat engine to the heat sink s = Entropy t = Total time required for the whole heat-engine cycle tH = Time required to transfer QH t,_= Time required to transfer QL tlz = Time required for the isentropic pumping process t34= Time required for the isentropic expansion process T = Temperature Tc = Temperature of the working fluid in the condenser TH = Temperature of the heat source TL = Temperature of the heat sink Tw = Temperature of the working fluid in

Carnot heat engine power optimization

the boiler & = Inlet temperature of the heating fluid T6 = Outlet temperature of the heat fluid T, = Inlet temperature of the cooling fluid TX= Outlet temperature of the cooling fluid u = LMTD, defined in Eq. (19) lJ, = Overall heat-transfer coefficient in the heat exchanger between the heat source and the heat engine U,_= Overall heat transfer coefficient in

the heat exchanger between the heat sink and the heat engine u = LMTDL defined in Eq. (20) W = Output work generated by the heat engine x = Tw Y = Tc 2 = P, defined in Eq. (18) q = Thermodynamic cycle efficiency of the finite-time heat engine rlcarnot= Carnot cycle efficiency

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