# Stirling Engine with Heat Transfer Considerations

Посмотреть архив целикомПросмотреть файл в отдельном окне: 37cbec8c447009e26371606ceac57ee7.pdf

Energies 2012, 5, 3573-3585; doi:10.3390/en5093573

OPEN ACCESS

energies

ISSN 1996-1073

www.mdpi.com/journal/energies

Article

Performance Analysis and Optimization of a Solar Powered

Stirling Engine with Heat Transfer Considerations

Chieh-Li Chen 1, Chia-En Ho 1 and Her-Terng Yau 2,*

1

2

Department of Aeronautics and Astronautics, National Cheng Kung University, No.1,

University Road, Tainan City 70101, Taiwan; E-Mails: chiehli@mail.ncku.edu.tw (C.-L.C.);

darrenisneverstop@gmail.com (C.-E.H.)

Department of Electrical Engineering, National Chin-Yi University of Technology, No. 57, Sec. 2,

Zhongshan Rd., Taiping Dist., Taichung City 41170, Taiwan

* Author to whom correspondence should be addressed; E-Mail: pan1012@ms52.hinet.net or

htyau@ncut.edu.tw; Tel.: +886-4-2392-4505 (ext. 7229); Fax: +886-4-2392-4419.

Received: 23 July 2012; in revised form 13: August 2012 / Accepted: 4 September 2012 /

Published: 17 September 2012

Abstract: This paper investigates the optimization of the performance of a solar powered

Stirling engine based on finite-time thermodynamics. Heat transference in the heat

exchangers between a concentrating solar collector and the Stirling engine is studied.

The irreversibility of a Stirling engine is considered with the heat transfer following

Newton's law. The power generated by a Stirling engine is used as an objective function for

maximum power output design with the concentrating solar collector temperature and the

engine thermal efficiency as the optimization parameters. The maximum output power of

engine and its corresponding system parameters are determined using a genetic algorithm.

Keywords: heat transfer; irreversibility; optimization; maximum power output; genetic

algorithms; Stirling engine

1. Introduction

Classical thermodynamics, a field that concerns thermal equilibrium problems, involves the

performance indices of a time-invariant system. These include, for example, efficiency, delivered

power, and similar factors, for a quasi-static thermodynamic process. As an extension and

generalization of classical thermodynamics, the finite-time thermodynamics, which takes time into

Energies 2012, 5

3574

account, can be used to describe the dynamics of energy and entropy flow in a non-equilibrium system.

Taking into consideration time-invariant quantities, such as power, refrigeration rate, power density,

entropy production rate, and others, finite-time thermodynamics is effectively adopted to optimize the

performance of practical systems. It has been successfully utilized in a wide range of disciplines, an

important of which is the optimization of practical performance of thermodynamic cycles, such as the

Carnot cycle, the Otto cycle, the Diesel cycle, the Brayton cycle, the Stirling cycle, and others.

In 1957, Novikov [1] was the first pioneer in the field of finite-time thermodynamics, and Curzon

and Ahlborn [2] proposed a more adequate thermal efficiency analysis for practical thermal processes,

where the upper bound on the thermal efficiency was modified accordingly by considering the loss due

to thermal resistance. Ondrechen et al. [3] performed a thermal analysis of a heat source of finite

extent. In 1985, DeVos [4] extended the scope of finite-time thermodynamics using various heat

transfer laws.

In 1991, Angulo-Brown [5] introduced an ecological optimization criterion as an objective function

to optimize the performance of a heat engine, taking into account the maximum output power and the

rate of production of entropy. [5] showed that the production rate of entropy is greatly reduced by

costing part of output power, when the optimum thermal efficiency is approximated as the average of

the respective efficiency suggested by Carnot and that suggested by Curzon and Ahlborn [2].

In 1991, Ibrahim et al. [6] considered irreversible parameters - the isentropic ratio of the isothermal

processes and thus optimized the Carnot cycle by a more practical manner. Based on the assumption

that the temperature of a gas varies linearly with the temperature of the surface of the wall of the

cylinder that contains it, Klein [7] proposed the net output power and an optimized compression ratio

for designing an engine with the greatest possible power output. Wu and Kiang [8] performed a finite

heat transfer analysis to investigate the effects of a nonisentropic compression process, expansion

process, turbine efficiency and compressor efficiency and a heat exchanger on the output power

optimization. In 1993, Chen and Yan [9] evaluated the maximum output power with the consideration

of the irreversible factors, which are heat leakage, finite heat transfer between the heat reservoir and

the heat engine in the compression and expansion processes. In 1995, Ait-Ali [10] considered the range

of operating temperatures to optimize the output power of an endoreversible Carnot engine.

Angulo-Brown et al. [11] used the Clausius inequality to modify the parameters as well as a linear

time-temperature relationship, and took into account the power loss, to obtain analytic solutions for

both the output power and the thermal efficiency. In 1998, Chen et al. [12] noted that in an

Atkinson cycle, the thermal efficiency at the point of maximum power density is always better than

that at the point of maximum power. Chen et al. [13] also investigated the effect of heat transfer on an

Otto cycle. In 1999, [14] proposed a generalized Otto cycle and quantified the degree of irreversibility

in a study of performance optimization in various heat transfer modes. Lin et al. [15] also represented

an effective method for improving heat engine performance in practical operation for a dual

combustion cycle using the finite-time thermodynamics approach. In 2004, Hou [16] examined the

effect of heat transfer on a dual combustion cycle. Zhou et al. [17] studied the effect of a generalized

heat transfer law on the optimization of power output of a generalized Carnot engine with internal and

external irreversibility. In 2007, Hou [18] proved that the expansion ratio in an Atkinson cycle exceeds

the compressive ratio in an Otto cycle. In 2009, Lu et al. [19] applied the energy equilibrium equation

for a collecting plate to optimize a solar power generating system, in which the efficiency of the light

Energies 2012, 5

3575

collection unit of the solar concentrator was optimized at such operating temperature corresponding to

the working fluid temperature.

The finite-time thermodynamics, as its name indicates, applies in many range of fields, whenever

heat transfer occurs in a device or a system of finite extent or within a limited period. Under some

suitable assumptions, this work proposes a constrained model to optimize the performance of a

practical heat engines with irreversible thermodynamic process. In finite-time thermodynamics, the

optimization of performance indices involves the optimization of an overall system on the assumption

of irreversibility. Comparing with the classical thermodynamics, it is more adequate when applied to

practical applications, and is useful in studies of the best use of energy.

This paper investigates the maximum power output design problem for a solar powered Stirling

engine, where the thermal efficiency and solar collector temperature are considered as the design

parameters. The heat transfer between the solar collector to the engine and the surroundings is studied

such that the result can provide an adequate prediction for overall system thermal efficiency

in practice.

2. Maximum Power Analysis of Stirling Engine with Solar Collector

The thermal model for a Stirling engine with solar collector which mainly consists of a spherical

reflector and an absorber that acts as a heat source is shown in Figure 1 [20]. A collector, which is

connected to an expansion chamber of the heat engine, is directly heated, and the heat is released to the

ambient by radiation and natural convection as the surface temperature of the collector is increased.

Figure 1. Heat flows involved in a Stirling engine with solar collector.

Considering the heat loss on the collector surface and from the law of conservation of energy,

it yields:

?I sun Ac ? hn ?Tw ? Ts ?Ac ? ?? ?Tw4 ? Ts4 ?Ac ? qin

(1)

Energies 2012, 5

3576

The heat absorption efficiency on the collector surface is defined as:

? collector ?

q in

I sun ? Ac

(2)

In this model, heat is transferred to the expansion chamber, where a high conductive coefficient and

extremely thin surface of the collector is applied such that the temperature on the interior surface of the

expansion chamber is almost equal to that in the collector. The working fluid is considered as the ideal

gas and the heat transfer process follows Newton’s linear heat transfer law. In each cycle, Qin is the

heat absorbed by the working fluid, and Qout is the heat released to the ambient. The temperature of the

corrector is denoted as Tw and the ambient temperature is denoted as Ts . The heat from the collector to

the Stirling engine satisfies:

Qin ? ? 2 ?Tw ? T3 ?t 34 ? h f Ac ?Tw ? T3 ?t 34

(3)

and the heat released by the Stirling engine to the ambient is given by:

Qout ? ? 1 ?T1 ? Ts ?t12

(4)

where t12 and t34 are the times required for engine heat rejection and engine heat accumulation,

respectively. T1 and T3 are the temperature of the working fluid in the isothermal heat rejection and

addition process, respectively.

In the isothermal and endothermic process, the entropy terms in the thermodynamic relation yield:

? dP ?

TdS ? T ?

? dV

? dT ? v

(5)

PV ? nRT

(6)

The ideal gas equation is:

Substituting Equation (6) into Equation (5) yields:

TdS ?

nRT

dV

V

(7)

Integrating Equation (7) over states 3 and 4 yields:

4

4

nRT

dV

V

3

? TdS ? ?

3

(8)

Entropy is defined as:

? ?Q ?

S ??

?

? T ? int,rev

(9)

Substituting Equation (9) into Equation (8) yields:

Q34 ? Qin ? nRT3 ln

V4

V3

In each cycle, the incremental entropy in the working fluid inside a Stirling engine is given by:

(10)

Energies 2012, 5

3577

?S ? ?

?Q

T

? S gen

(11)

Since entropy behaves analogously to heat, it is independent of the integration path itself.

Therefore, the net change in entropy between the initial and final states of a complete cycle is

identically zero:

?S ? ?

?Q

T

? S gen ?

Qin Qout

?

? S gen ? 0

T3

T1

(12)

From the Second Law of thermodynamics:

S gen ? 0

(13)

The Clausius inequality yields:

?Q

?T

?0

(14)

From Equations (12) to (14):

Qin Qout

?

?0

T3

T1

(15)

Now, let:

Qin Qout

?

T3

?T1

? ?1

(16)

where ? is an irreversible property of a Stirling engine, such as thermal resistance, friction, heat loss.

Consistent with the second law of thermodynamics, such irreversible factors, which cannot be ignored,

bring about an increase in entropy in each cycle. Given an identical amount of heat transfer Qin , for an

endoreversible heat engine ? ? 1 , Qout is expressed as:

rev

Qin Qout

?

?0

T3

T1

(17)

rev

Qout ? ? ? Qout

(18)

Relating Equation (16) to (17) yields:

In a Stirling engine, the heat that is released to low-temperature thermal storage (cold chamber) in a

reversible isothermal process is:

rev

Qout

? nRT1 ln

V1

V2

(19)

Back-substituting Equation (19) into Equation (18) yields:

Qout ? ?nRT1 ln

Let compressive ratio rv be defined as:

V1

V2

(20)

Energies 2012, 5

3578

rv ?

V4 V1

?

V3 V2

(21)

Equating Equation (3) with Equation (10) and substituting Equation (21) into Equation (3) yields:

h f Ac ?Tw ? T3 ?t 34 ? nRT3 ln rv

(22)

The endothermic time of the heat engine is given by:

t 34 ?

nRT3 ln rv

h f Ac ?Tw ? T3 ?

(23)

Equating Equation (4) with Equation (20) and substituting Equation (21) into Equation (4) yields:

? ?T1 ? Ts ?t12 ? ?nRT1 ln rv

(24)

The time for which the heat engine is exothermic is:

t12 ?

?nRT1 ln rv

? 1 ?T1 ? Ts ?

(25)

Suppose that in the heat regenerating process, the temperature of the working fluid, Tfluid, varies

linearly with time:

dT fluid

dt

? ? K1

(26)

where K1 > 0, which is the average rate of change of temperature, and is independent of time but

dependent on the material of the heat regenerator, is called the heat regenerative time coefficient: a

positive or negative sign shows that the temperature increases or decreases with time, respectively.

The duration of the heat regenerative process from state 2 to state 3 is:

t23 ?

?T3 ? T2 ?

K1

(27)

Similarly, that of the heat regenerative process from state 4 to state 1 is:

t 41 ?

?T4 ? T1 ?

K1

(28)

Accordingly, time for a complete Stirling cycle is represented as:

tc ? t12 ? t 23 ? t34 ? t 41

(29)

Substituting Equations (23), (25), (27) and (28) into Equation (29) yields:

tc ?

nRT3 ln rv

?nRT1 ln rv ?T3 ? T2 ?

?T ? T ?

?

?

? 4 1

?1 ?T1 ? Ts ?

K1

h f Ac ?Tw ? T3 ?

K1

(30)

In each cycle, the effective energy in a collector is:

Qin ? qin ? tc

(31)

Energies 2012, 5

3579

Substituting Equations (10), (21) and (30) into Equation (31) yields the total amount of heat applied

to a Stirling engine in each cycle, which is:

qin ?

1

?

hf

h f ?T1

Ac ?Tw ? T3 ? ? 1T3 ?T1 ? Ts ?

?

2h f ?T3 ? T1 ?

(32)

nRT3 ln rv K1

The thermal efficiency of a heat engine is defined as:

P

Q ? Qout

T

? Strling ? Stirling ? in

? 1? 1 ?

Qin

Qin

T3

(33)

hf

2h f

into Equation (32) yields:

Substituting ? ? ? and K ?

nRK1

1

qin ?

hf

? 2 ?1 ? ? Stirling ?

K

?

?

Ac ?Tw ? T3 ? ? ?1 ? ? Stirling ?T3

? ln rv

? Ts ?

?

?

?

?

1

? ?1 ? ? Stirling ??

?

?1 ?

?

?

?

.

(34)

The heat that is released to the ambient in a Stirling cycle is given by:

qout ?

Qout

tc

(35)

Substituting Equations (20), (30) and (33) into Equation (35) yield:

qin ?

h f ?1 ? ? Stirling ?

? ?1 ? ? Stirling ?

K ? ?1 ? ? Stirling ??

?

?1 ?

?

?

? ?1 ? ? Stirling ?T3

? ln rv ?

?

? Ts ?

?

?

?

?

2

1

Ac ?Tw ? T3 ?

?

(36)

The output power that is provided by a Stirling engine in a Stirling cycle is:

PStirling ? ? Stirling ? qin

(37)

From the collector temperature and the thermal efficiency, the amount of heat that is applied to the

collector is determined using Equation (11); then, Equation (34) is solved for the endothermic

temperature of the heat engine, which is expressed in a quadratic form as:

T3 ?

? b ? b 2 ? 4ac

? f ?Tw ,? Stirling ?

2a

where:

a?

b?

?1 ? ?

Stirling

?

?

(38)

sAc ?1 ? ? Stirling ?

?

? ? 2 Ac ?1 ? ? Stirling ? ?

sAc Tw ?1 ? ? Stirling ?

?

? sAc Ts

Energies 2012, 5

3580

c ? ?Ts ? Ac? 2 ?1 ? ? Stirling ?Tw ? sAcTwTs

In the above three parameters, the parameter s is defined as:

s?

hf

qin

?

K

ln rv

? ?1 ? ? Stirling ??

?1 ?

?

?

?

?

From Equation (33), the rejection temperature of the heat engine is:

T1 ?

?1 ? ?

Stirling

?T

3

(39)

?

Given the collector temperature, the heat transferred from the collector to the heat engine is

determined. Partial differentiation of Equation (34) with respect to endothermic temperature yields the

optimized endothermic temperature:

?q in

?0

?T3

T3.opt ?

(40)

? ?Tw ?1 ? ? Stirling ? ? ?Ts

?1 ? ?

Stirling

??1 ?

??

?

(41)

Back-substituting Equation (41) into Equation (34) yields the amount of heat applied at the optimal

endothermic temperature:

qin ,opt ?

hf

? ?1 ? ? Stirling ?

K

1

?

?

Ac ?Tw ? T3,opt ? ? ?1 ? ? Stirling ?T3,opt

? ln rv

? Ts ?

?

?

?

?

2

? ?1 ? ? Stirling ??

?1 ?

?

?

?

?

.

(42)

3. Results and Discussion

For a given solar intensity, the relation between the thermal efficiency of a Stirling engine and the

collector temperature are studied in this section. Thermal efficiency of an engine can be improved by

either increasingg T3 of the isothermal heat addition process, or reducing T1 of the isothermal heat

rejection process. In general, the heat rejection temperature will not fall below the ambient temperature,

so the intended thermal efficiency can only be improved by elevating T3. Since the collector receives

limited energy for a given solar intensity and it loses heat to the surroundings by both radiation and

convection, a temperature upper bound will exist for the collector as well as the T3 of the thermal

process. Also, the temperature difference between them will determine the heat been accumulated by

the engine from the collector. In this study, a solar powered Stirling engine with the solar

intensity = 4000 W/m2 is considered with system parameters listed in Table 1.

Equation (1) is used to determine the heat loss from the collector to the surroundings which can be

used to determine a reasonable heat available for the engine. Figure 2 reveals the heat loss by the

collector with respect to collector temperature. When the collector temperature exceeds 698 K, the heat

loss from the collector exceeds the amount of solar power can possibly been accumulated.

Energies 2012, 5

3581

Table 1. System parameters for optimization study.

System Parameters

Solar intensity

The upper bound of thermal efficiency

The temperature of collector

The coefficient of convection in the expansion chamber

The coefficient of natural convection

The product of heat transfer coefficient and heat transfer area

between cold-end chamber to the surroundings

Surface radiation emission rate

Boltzmann constant

Collector wall absorption rate

Degree of irreversible factor in heat engine

Ambient temperature

Collector area

Values

Isun = 4000 W/m2

0.57

450–698 K

hf = 90 W/(m2·K)

hn = 5 W/(m2·K)

?1 = 50 W/K

? = 0.12

? = 5.67 ? 10?8 W/(m2·K4)

? = 0.9

? ?1

Ts = 293 K

Ac = 1 m2

Figure 2. Heat loss from collector to the surroundings.

Figure 3 plots the output power as a function of the collector temperature at thermal efficiencies of

0.1, 0.3 and 0.5. Figure 4 illustrates available heat for the engine for different isothermal heat addition

process temperature T3. The maximum power available according to the optimal hot-end temperature

determined by Equation (41) is also indicated in Figure 4. Accordingly, the output power generated by

the engine is shown in Figure 5. Figures 3–5 reveal that a higher T3 will result higher thermal

efficiency; however, it will also reduce the heat power been accumulated from the collector.

Therefore, the maximum power output of the engine requires a trade-off between thermal efficiency

and isothermal heat addition process temperature of the thermal process. Figures 6 and 7 illustrate the

maximum output power of the engine with respect to thermal efficiency and collector temperature,

respectively. They show that the solar powered Stirling engine will generate maximum power at a

thermal efficiency of between 0.3 and 0.4 and a collector temperature of between 550 K and 600 K.

Energies 2012, 5

3582

Figure 3. Engine output power.

Figure 4. Available power from the collector.

Figure 5. Engine output power.

Energies 2012, 5

3583

Max. engine

output power

(W)

Figure 6. Maximum engine output power with respect to thermal efficiency.

Figure 7. Maximum engine output power with respect to collector temperature.

Table 2. System parameters for maximum output power.

System Parameters

Solar intensity (Psolar)

Collector wall absorption rate

Power accumulated by collector

Collector temperature

Radiation heat loss

Convection heat loss

Collector efficiency

The input power of heat engine

Thermal efficiency

The temperature of the working fluid in the isothermal heat addition process (T3)

The temperature of the working fluid in the isothermal heat rejection process (T1)

The output heat of heat engine

The power generated by the heat engine (PStirling)

The total power loss of the system

Overall system thermal efficiency (PStirling/Psolar)

Values

4000 W/m2

0.9

3600 W/m2

560.4 K

621 W/m2

1337 W/m2

0.41

1642 W

0.363

517.4 K

329.7 K

1047 W

596 W

3404 W/m2

0.149

Energies 2012, 5

3584

Finally, the optimum design parameters are determined using a genetic algorithm. The maximum

output power of a Stirling engine and the corresponding parameters are determined using the Matlab

Toolbox Genetic Algorithm Toolbox, with the output power as an objective function, and the thermal

efficiency and the collector temperature as optimized parameters. As shown in Table 2, the simulation

yields a maximum output power of 596 W at a thermal efficiency of 0.363 and a collector temperature

of 560.4 K, which also confirms the expecting result from Figure 6 and 7.

4. Conclusions

This work adopts finite-time thermodynamics to optimize the performance of a solar powered

Stirling engine. The genetic algorithm is used to reveal the maximum output power of a Stirling engine

by determining the thermal efficiency, the collector temperature, and their corresponding values of the

optimized parameters. The results of the simulation herein described can be directly applied to

optimize the design of a practical solar powered Stirling engine. The proposed work investigates the

maximum power output of a Stirling engine under the solar concentrated heat of 4000 W/m2.

According to the optimal design, it reveals that maximum power can be generated if the engine is

designed with efficiency 0.363 and temperature of collector is kept to be about 560.4 K, which can be

achieved using a temperature regulating control loop.

Acknowledgments

Part of this research is financial supported by the National Science Council, Taiwan, under Grant

No. NSC 100-2221-E-006-104-MY2.

References

1.

2.

3.

4.

5.

6.

7.

8.

9.

Novikov, I.I. The efficiency of atomic power stations. J. Nucl. Energy II 1958, 7, 125–128.

Curzon, F.L.; Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys.

1975, 43, 22–24.

Ondrechen, M.J.; Rubin, M.H.; Band, Y.B. The generalized Carnot cycle-A working fluid

operating in finite time between finite heat source and sinks. J. Chem. Phys. 1983, 78, 4721–4727.

De Vos, A. Efficiency of some heat engines at maximum power conditions. Am. J. Phys. 1985,

53, 570–573.

Angulo, B.F. An ecological optimization criterion for finite time heat engines. J. Appl. Phys. 1991,

69, 7465–7469.

Ibrahim, O.M.; Klein, S.A.; Mitchell, J.W. Optimum heat power cycles for specified boundary

conditions. J. Eng. Gas Turb. Power 1991, 113, 514–521.

Klein, S.A. An explanation for observed compression ratios in internal combustion engines.

J. Eng. Gas Turb. Power 1991, 113, 511–513.

Wu, C.; Kiang, R.L. Power performance of a nonisentropic Brayton cycle. J. Eng. Gas Turb.

Power 1991, 113, 501–504.

Chen, J.C.; Yan, Z.J. Optimal performance of endoreversible cycles for another linear heat

transfer law. J. Phys. D Appl. Phys. 1993, 26, 1581–1586.

Energies 2012, 5

3585

10. Ait, A.M. Maximum power and thermal efficiency of an irreversible power cycle. J. Appl. Phys.

1995, 78, 4313–4318.

11. Angulo, B.F.; Rocha Martinez, J.A.; Navarrete Gonzalez, T.D. A non-endoreversible Otto cycle

model: Improving power output and efficiency. J. Phys. D Appl. Phys. 1996, 29, 80–83.

12. Chen, L.G.; Lin, J.X.; Sun, F.R.; Wu, C.I. Efficiency of an Atkinson engine at maximum power

density. Energy Convers. Manag. 1998, 39, 337–341.

13. Chen, L.G.; Wu, C.; Sun, F.R.; Cao, S. Heat transfer effects on the net work output and efficiency

characteristics for an air-standard Otto cycle. Energy Convers. Manag. 1998, 39, 643–648.

14. Chen, L.G.; Sun, F.R.; Wu, C. Effect of heat transfer law on the performance of a generalized

irreversible Carnot engine. J. Phys. D Appl. Phys. 1999, 32, 99–105.

15. Lin, J.X.; Chen, L.G.; Wu, C.; Sun, F.R. Finite time thermodynamic performance of a dual cycle.

Int. J. Energy Res. 1999, 23, 765–772.

16. Hou, S.S. Heat transfer effects on the performance of an air standard dual cycle.

Energy Convers. Manag. 2004, 45, 3003–3015.

17. Zhou, S.; Chen, L.; Sun, F.; Wu, C. Optimal performance of a generalized irreversible Carnot

engine. Appl. Energy 2005, 81, 376–387.

18. Hou, S.S. Comparison of performances of air standard Atkinson and Otto cycles with heat transfer

considerations. Energy Convers. Manag. 2007, 48, 1683–1690.

19. Lu, J.; Ding, J.; Yang, J. Optimal performance for solar thermal power system. Energy Power

Eng. 2009, 1,110–115.

20. Bejan, A. Advanced Engineering Thermodynamics; Wiley-Interscience: New York, NY, USA,

1997.

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article

distributed under the terms and conditions of the Creative Commons Attribution license

(http://creativecommons.org/licenses/by/3.0/).

OPEN ACCESS

energies

ISSN 1996-1073

www.mdpi.com/journal/energies

Article

Performance Analysis and Optimization of a Solar Powered

Stirling Engine with Heat Transfer Considerations

Chieh-Li Chen 1, Chia-En Ho 1 and Her-Terng Yau 2,*

1

2

Department of Aeronautics and Astronautics, National Cheng Kung University, No.1,

University Road, Tainan City 70101, Taiwan; E-Mails: chiehli@mail.ncku.edu.tw (C.-L.C.);

darrenisneverstop@gmail.com (C.-E.H.)

Department of Electrical Engineering, National Chin-Yi University of Technology, No. 57, Sec. 2,

Zhongshan Rd., Taiping Dist., Taichung City 41170, Taiwan

* Author to whom correspondence should be addressed; E-Mail: pan1012@ms52.hinet.net or

htyau@ncut.edu.tw; Tel.: +886-4-2392-4505 (ext. 7229); Fax: +886-4-2392-4419.

Received: 23 July 2012; in revised form 13: August 2012 / Accepted: 4 September 2012 /

Published: 17 September 2012

Abstract: This paper investigates the optimization of the performance of a solar powered

Stirling engine based on finite-time thermodynamics. Heat transference in the heat

exchangers between a concentrating solar collector and the Stirling engine is studied.

The irreversibility of a Stirling engine is considered with the heat transfer following

Newton's law. The power generated by a Stirling engine is used as an objective function for

maximum power output design with the concentrating solar collector temperature and the

engine thermal efficiency as the optimization parameters. The maximum output power of

engine and its corresponding system parameters are determined using a genetic algorithm.

Keywords: heat transfer; irreversibility; optimization; maximum power output; genetic

algorithms; Stirling engine

1. Introduction

Classical thermodynamics, a field that concerns thermal equilibrium problems, involves the

performance indices of a time-invariant system. These include, for example, efficiency, delivered

power, and similar factors, for a quasi-static thermodynamic process. As an extension and

generalization of classical thermodynamics, the finite-time thermodynamics, which takes time into

Energies 2012, 5

3574

account, can be used to describe the dynamics of energy and entropy flow in a non-equilibrium system.

Taking into consideration time-invariant quantities, such as power, refrigeration rate, power density,

entropy production rate, and others, finite-time thermodynamics is effectively adopted to optimize the

performance of practical systems. It has been successfully utilized in a wide range of disciplines, an

important of which is the optimization of practical performance of thermodynamic cycles, such as the

Carnot cycle, the Otto cycle, the Diesel cycle, the Brayton cycle, the Stirling cycle, and others.

In 1957, Novikov [1] was the first pioneer in the field of finite-time thermodynamics, and Curzon

and Ahlborn [2] proposed a more adequate thermal efficiency analysis for practical thermal processes,

where the upper bound on the thermal efficiency was modified accordingly by considering the loss due

to thermal resistance. Ondrechen et al. [3] performed a thermal analysis of a heat source of finite

extent. In 1985, DeVos [4] extended the scope of finite-time thermodynamics using various heat

transfer laws.

In 1991, Angulo-Brown [5] introduced an ecological optimization criterion as an objective function

to optimize the performance of a heat engine, taking into account the maximum output power and the

rate of production of entropy. [5] showed that the production rate of entropy is greatly reduced by

costing part of output power, when the optimum thermal efficiency is approximated as the average of

the respective efficiency suggested by Carnot and that suggested by Curzon and Ahlborn [2].

In 1991, Ibrahim et al. [6] considered irreversible parameters - the isentropic ratio of the isothermal

processes and thus optimized the Carnot cycle by a more practical manner. Based on the assumption

that the temperature of a gas varies linearly with the temperature of the surface of the wall of the

cylinder that contains it, Klein [7] proposed the net output power and an optimized compression ratio

for designing an engine with the greatest possible power output. Wu and Kiang [8] performed a finite

heat transfer analysis to investigate the effects of a nonisentropic compression process, expansion

process, turbine efficiency and compressor efficiency and a heat exchanger on the output power

optimization. In 1993, Chen and Yan [9] evaluated the maximum output power with the consideration

of the irreversible factors, which are heat leakage, finite heat transfer between the heat reservoir and

the heat engine in the compression and expansion processes. In 1995, Ait-Ali [10] considered the range

of operating temperatures to optimize the output power of an endoreversible Carnot engine.

Angulo-Brown et al. [11] used the Clausius inequality to modify the parameters as well as a linear

time-temperature relationship, and took into account the power loss, to obtain analytic solutions for

both the output power and the thermal efficiency. In 1998, Chen et al. [12] noted that in an

Atkinson cycle, the thermal efficiency at the point of maximum power density is always better than

that at the point of maximum power. Chen et al. [13] also investigated the effect of heat transfer on an

Otto cycle. In 1999, [14] proposed a generalized Otto cycle and quantified the degree of irreversibility

in a study of performance optimization in various heat transfer modes. Lin et al. [15] also represented

an effective method for improving heat engine performance in practical operation for a dual

combustion cycle using the finite-time thermodynamics approach. In 2004, Hou [16] examined the

effect of heat transfer on a dual combustion cycle. Zhou et al. [17] studied the effect of a generalized

heat transfer law on the optimization of power output of a generalized Carnot engine with internal and

external irreversibility. In 2007, Hou [18] proved that the expansion ratio in an Atkinson cycle exceeds

the compressive ratio in an Otto cycle. In 2009, Lu et al. [19] applied the energy equilibrium equation

for a collecting plate to optimize a solar power generating system, in which the efficiency of the light

Energies 2012, 5

3575

collection unit of the solar concentrator was optimized at such operating temperature corresponding to

the working fluid temperature.

The finite-time thermodynamics, as its name indicates, applies in many range of fields, whenever

heat transfer occurs in a device or a system of finite extent or within a limited period. Under some

suitable assumptions, this work proposes a constrained model to optimize the performance of a

practical heat engines with irreversible thermodynamic process. In finite-time thermodynamics, the

optimization of performance indices involves the optimization of an overall system on the assumption

of irreversibility. Comparing with the classical thermodynamics, it is more adequate when applied to

practical applications, and is useful in studies of the best use of energy.

This paper investigates the maximum power output design problem for a solar powered Stirling

engine, where the thermal efficiency and solar collector temperature are considered as the design

parameters. The heat transfer between the solar collector to the engine and the surroundings is studied

such that the result can provide an adequate prediction for overall system thermal efficiency

in practice.

2. Maximum Power Analysis of Stirling Engine with Solar Collector

The thermal model for a Stirling engine with solar collector which mainly consists of a spherical

reflector and an absorber that acts as a heat source is shown in Figure 1 [20]. A collector, which is

connected to an expansion chamber of the heat engine, is directly heated, and the heat is released to the

ambient by radiation and natural convection as the surface temperature of the collector is increased.

Figure 1. Heat flows involved in a Stirling engine with solar collector.

Considering the heat loss on the collector surface and from the law of conservation of energy,

it yields:

?I sun Ac ? hn ?Tw ? Ts ?Ac ? ?? ?Tw4 ? Ts4 ?Ac ? qin

(1)

Energies 2012, 5

3576

The heat absorption efficiency on the collector surface is defined as:

? collector ?

q in

I sun ? Ac

(2)

In this model, heat is transferred to the expansion chamber, where a high conductive coefficient and

extremely thin surface of the collector is applied such that the temperature on the interior surface of the

expansion chamber is almost equal to that in the collector. The working fluid is considered as the ideal

gas and the heat transfer process follows Newton’s linear heat transfer law. In each cycle, Qin is the

heat absorbed by the working fluid, and Qout is the heat released to the ambient. The temperature of the

corrector is denoted as Tw and the ambient temperature is denoted as Ts . The heat from the collector to

the Stirling engine satisfies:

Qin ? ? 2 ?Tw ? T3 ?t 34 ? h f Ac ?Tw ? T3 ?t 34

(3)

and the heat released by the Stirling engine to the ambient is given by:

Qout ? ? 1 ?T1 ? Ts ?t12

(4)

where t12 and t34 are the times required for engine heat rejection and engine heat accumulation,

respectively. T1 and T3 are the temperature of the working fluid in the isothermal heat rejection and

addition process, respectively.

In the isothermal and endothermic process, the entropy terms in the thermodynamic relation yield:

? dP ?

TdS ? T ?

? dV

? dT ? v

(5)

PV ? nRT

(6)

The ideal gas equation is:

Substituting Equation (6) into Equation (5) yields:

TdS ?

nRT

dV

V

(7)

Integrating Equation (7) over states 3 and 4 yields:

4

4

nRT

dV

V

3

? TdS ? ?

3

(8)

Entropy is defined as:

? ?Q ?

S ??

?

? T ? int,rev

(9)

Substituting Equation (9) into Equation (8) yields:

Q34 ? Qin ? nRT3 ln

V4

V3

In each cycle, the incremental entropy in the working fluid inside a Stirling engine is given by:

(10)

Energies 2012, 5

3577

?S ? ?

?Q

T

? S gen

(11)

Since entropy behaves analogously to heat, it is independent of the integration path itself.

Therefore, the net change in entropy between the initial and final states of a complete cycle is

identically zero:

?S ? ?

?Q

T

? S gen ?

Qin Qout

?

? S gen ? 0

T3

T1

(12)

From the Second Law of thermodynamics:

S gen ? 0

(13)

The Clausius inequality yields:

?Q

?T

?0

(14)

From Equations (12) to (14):

Qin Qout

?

?0

T3

T1

(15)

Now, let:

Qin Qout

?

T3

?T1

? ?1

(16)

where ? is an irreversible property of a Stirling engine, such as thermal resistance, friction, heat loss.

Consistent with the second law of thermodynamics, such irreversible factors, which cannot be ignored,

bring about an increase in entropy in each cycle. Given an identical amount of heat transfer Qin , for an

endoreversible heat engine ? ? 1 , Qout is expressed as:

rev

Qin Qout

?

?0

T3

T1

(17)

rev

Qout ? ? ? Qout

(18)

Relating Equation (16) to (17) yields:

In a Stirling engine, the heat that is released to low-temperature thermal storage (cold chamber) in a

reversible isothermal process is:

rev

Qout

? nRT1 ln

V1

V2

(19)

Back-substituting Equation (19) into Equation (18) yields:

Qout ? ?nRT1 ln

Let compressive ratio rv be defined as:

V1

V2

(20)

Energies 2012, 5

3578

rv ?

V4 V1

?

V3 V2

(21)

Equating Equation (3) with Equation (10) and substituting Equation (21) into Equation (3) yields:

h f Ac ?Tw ? T3 ?t 34 ? nRT3 ln rv

(22)

The endothermic time of the heat engine is given by:

t 34 ?

nRT3 ln rv

h f Ac ?Tw ? T3 ?

(23)

Equating Equation (4) with Equation (20) and substituting Equation (21) into Equation (4) yields:

? ?T1 ? Ts ?t12 ? ?nRT1 ln rv

(24)

The time for which the heat engine is exothermic is:

t12 ?

?nRT1 ln rv

? 1 ?T1 ? Ts ?

(25)

Suppose that in the heat regenerating process, the temperature of the working fluid, Tfluid, varies

linearly with time:

dT fluid

dt

? ? K1

(26)

where K1 > 0, which is the average rate of change of temperature, and is independent of time but

dependent on the material of the heat regenerator, is called the heat regenerative time coefficient: a

positive or negative sign shows that the temperature increases or decreases with time, respectively.

The duration of the heat regenerative process from state 2 to state 3 is:

t23 ?

?T3 ? T2 ?

K1

(27)

Similarly, that of the heat regenerative process from state 4 to state 1 is:

t 41 ?

?T4 ? T1 ?

K1

(28)

Accordingly, time for a complete Stirling cycle is represented as:

tc ? t12 ? t 23 ? t34 ? t 41

(29)

Substituting Equations (23), (25), (27) and (28) into Equation (29) yields:

tc ?

nRT3 ln rv

?nRT1 ln rv ?T3 ? T2 ?

?T ? T ?

?

?

? 4 1

?1 ?T1 ? Ts ?

K1

h f Ac ?Tw ? T3 ?

K1

(30)

In each cycle, the effective energy in a collector is:

Qin ? qin ? tc

(31)

Energies 2012, 5

3579

Substituting Equations (10), (21) and (30) into Equation (31) yields the total amount of heat applied

to a Stirling engine in each cycle, which is:

qin ?

1

?

hf

h f ?T1

Ac ?Tw ? T3 ? ? 1T3 ?T1 ? Ts ?

?

2h f ?T3 ? T1 ?

(32)

nRT3 ln rv K1

The thermal efficiency of a heat engine is defined as:

P

Q ? Qout

T

? Strling ? Stirling ? in

? 1? 1 ?

Qin

Qin

T3

(33)

hf

2h f

into Equation (32) yields:

Substituting ? ? ? and K ?

nRK1

1

qin ?

hf

? 2 ?1 ? ? Stirling ?

K

?

?

Ac ?Tw ? T3 ? ? ?1 ? ? Stirling ?T3

? ln rv

? Ts ?

?

?

?

?

1

? ?1 ? ? Stirling ??

?

?1 ?

?

?

?

.

(34)

The heat that is released to the ambient in a Stirling cycle is given by:

qout ?

Qout

tc

(35)

Substituting Equations (20), (30) and (33) into Equation (35) yield:

qin ?

h f ?1 ? ? Stirling ?

? ?1 ? ? Stirling ?

K ? ?1 ? ? Stirling ??

?

?1 ?

?

?

? ?1 ? ? Stirling ?T3

? ln rv ?

?

? Ts ?

?

?

?

?

2

1

Ac ?Tw ? T3 ?

?

(36)

The output power that is provided by a Stirling engine in a Stirling cycle is:

PStirling ? ? Stirling ? qin

(37)

From the collector temperature and the thermal efficiency, the amount of heat that is applied to the

collector is determined using Equation (11); then, Equation (34) is solved for the endothermic

temperature of the heat engine, which is expressed in a quadratic form as:

T3 ?

? b ? b 2 ? 4ac

? f ?Tw ,? Stirling ?

2a

where:

a?

b?

?1 ? ?

Stirling

?

?

(38)

sAc ?1 ? ? Stirling ?

?

? ? 2 Ac ?1 ? ? Stirling ? ?

sAc Tw ?1 ? ? Stirling ?

?

? sAc Ts

Energies 2012, 5

3580

c ? ?Ts ? Ac? 2 ?1 ? ? Stirling ?Tw ? sAcTwTs

In the above three parameters, the parameter s is defined as:

s?

hf

qin

?

K

ln rv

? ?1 ? ? Stirling ??

?1 ?

?

?

?

?

From Equation (33), the rejection temperature of the heat engine is:

T1 ?

?1 ? ?

Stirling

?T

3

(39)

?

Given the collector temperature, the heat transferred from the collector to the heat engine is

determined. Partial differentiation of Equation (34) with respect to endothermic temperature yields the

optimized endothermic temperature:

?q in

?0

?T3

T3.opt ?

(40)

? ?Tw ?1 ? ? Stirling ? ? ?Ts

?1 ? ?

Stirling

??1 ?

??

?

(41)

Back-substituting Equation (41) into Equation (34) yields the amount of heat applied at the optimal

endothermic temperature:

qin ,opt ?

hf

? ?1 ? ? Stirling ?

K

1

?

?

Ac ?Tw ? T3,opt ? ? ?1 ? ? Stirling ?T3,opt

? ln rv

? Ts ?

?

?

?

?

2

? ?1 ? ? Stirling ??

?1 ?

?

?

?

?

.

(42)

3. Results and Discussion

For a given solar intensity, the relation between the thermal efficiency of a Stirling engine and the

collector temperature are studied in this section. Thermal efficiency of an engine can be improved by

either increasingg T3 of the isothermal heat addition process, or reducing T1 of the isothermal heat

rejection process. In general, the heat rejection temperature will not fall below the ambient temperature,

so the intended thermal efficiency can only be improved by elevating T3. Since the collector receives

limited energy for a given solar intensity and it loses heat to the surroundings by both radiation and

convection, a temperature upper bound will exist for the collector as well as the T3 of the thermal

process. Also, the temperature difference between them will determine the heat been accumulated by

the engine from the collector. In this study, a solar powered Stirling engine with the solar

intensity = 4000 W/m2 is considered with system parameters listed in Table 1.

Equation (1) is used to determine the heat loss from the collector to the surroundings which can be

used to determine a reasonable heat available for the engine. Figure 2 reveals the heat loss by the

collector with respect to collector temperature. When the collector temperature exceeds 698 K, the heat

loss from the collector exceeds the amount of solar power can possibly been accumulated.

Energies 2012, 5

3581

Table 1. System parameters for optimization study.

System Parameters

Solar intensity

The upper bound of thermal efficiency

The temperature of collector

The coefficient of convection in the expansion chamber

The coefficient of natural convection

The product of heat transfer coefficient and heat transfer area

between cold-end chamber to the surroundings

Surface radiation emission rate

Boltzmann constant

Collector wall absorption rate

Degree of irreversible factor in heat engine

Ambient temperature

Collector area

Values

Isun = 4000 W/m2

0.57

450–698 K

hf = 90 W/(m2·K)

hn = 5 W/(m2·K)

?1 = 50 W/K

? = 0.12

? = 5.67 ? 10?8 W/(m2·K4)

? = 0.9

? ?1

Ts = 293 K

Ac = 1 m2

Figure 2. Heat loss from collector to the surroundings.

Figure 3 plots the output power as a function of the collector temperature at thermal efficiencies of

0.1, 0.3 and 0.5. Figure 4 illustrates available heat for the engine for different isothermal heat addition

process temperature T3. The maximum power available according to the optimal hot-end temperature

determined by Equation (41) is also indicated in Figure 4. Accordingly, the output power generated by

the engine is shown in Figure 5. Figures 3–5 reveal that a higher T3 will result higher thermal

efficiency; however, it will also reduce the heat power been accumulated from the collector.

Therefore, the maximum power output of the engine requires a trade-off between thermal efficiency

and isothermal heat addition process temperature of the thermal process. Figures 6 and 7 illustrate the

maximum output power of the engine with respect to thermal efficiency and collector temperature,

respectively. They show that the solar powered Stirling engine will generate maximum power at a

thermal efficiency of between 0.3 and 0.4 and a collector temperature of between 550 K and 600 K.

Energies 2012, 5

3582

Figure 3. Engine output power.

Figure 4. Available power from the collector.

Figure 5. Engine output power.

Energies 2012, 5

3583

Max. engine

output power

(W)

Figure 6. Maximum engine output power with respect to thermal efficiency.

Figure 7. Maximum engine output power with respect to collector temperature.

Table 2. System parameters for maximum output power.

System Parameters

Solar intensity (Psolar)

Collector wall absorption rate

Power accumulated by collector

Collector temperature

Radiation heat loss

Convection heat loss

Collector efficiency

The input power of heat engine

Thermal efficiency

The temperature of the working fluid in the isothermal heat addition process (T3)

The temperature of the working fluid in the isothermal heat rejection process (T1)

The output heat of heat engine

The power generated by the heat engine (PStirling)

The total power loss of the system

Overall system thermal efficiency (PStirling/Psolar)

Values

4000 W/m2

0.9

3600 W/m2

560.4 K

621 W/m2

1337 W/m2

0.41

1642 W

0.363

517.4 K

329.7 K

1047 W

596 W

3404 W/m2

0.149

Energies 2012, 5

3584

Finally, the optimum design parameters are determined using a genetic algorithm. The maximum

output power of a Stirling engine and the corresponding parameters are determined using the Matlab

Toolbox Genetic Algorithm Toolbox, with the output power as an objective function, and the thermal

efficiency and the collector temperature as optimized parameters. As shown in Table 2, the simulation

yields a maximum output power of 596 W at a thermal efficiency of 0.363 and a collector temperature

of 560.4 K, which also confirms the expecting result from Figure 6 and 7.

4. Conclusions

This work adopts finite-time thermodynamics to optimize the performance of a solar powered

Stirling engine. The genetic algorithm is used to reveal the maximum output power of a Stirling engine

by determining the thermal efficiency, the collector temperature, and their corresponding values of the

optimized parameters. The results of the simulation herein described can be directly applied to

optimize the design of a practical solar powered Stirling engine. The proposed work investigates the

maximum power output of a Stirling engine under the solar concentrated heat of 4000 W/m2.

According to the optimal design, it reveals that maximum power can be generated if the engine is

designed with efficiency 0.363 and temperature of collector is kept to be about 560.4 K, which can be

achieved using a temperature regulating control loop.

Acknowledgments

Part of this research is financial supported by the National Science Council, Taiwan, under Grant

No. NSC 100-2221-E-006-104-MY2.

References

1.

2.

3.

4.

5.

6.

7.

8.

9.

Novikov, I.I. The efficiency of atomic power stations. J. Nucl. Energy II 1958, 7, 125–128.

Curzon, F.L.; Ahlborn, B. Efficiency of a Carnot engine at maximum power output. Am. J. Phys.

1975, 43, 22–24.

Ondrechen, M.J.; Rubin, M.H.; Band, Y.B. The generalized Carnot cycle-A working fluid

operating in finite time between finite heat source and sinks. J. Chem. Phys. 1983, 78, 4721–4727.

De Vos, A. Efficiency of some heat engines at maximum power conditions. Am. J. Phys. 1985,

53, 570–573.

Angulo, B.F. An ecological optimization criterion for finite time heat engines. J. Appl. Phys. 1991,

69, 7465–7469.

Ibrahim, O.M.; Klein, S.A.; Mitchell, J.W. Optimum heat power cycles for specified boundary

conditions. J. Eng. Gas Turb. Power 1991, 113, 514–521.

Klein, S.A. An explanation for observed compression ratios in internal combustion engines.

J. Eng. Gas Turb. Power 1991, 113, 511–513.

Wu, C.; Kiang, R.L. Power performance of a nonisentropic Brayton cycle. J. Eng. Gas Turb.

Power 1991, 113, 501–504.

Chen, J.C.; Yan, Z.J. Optimal performance of endoreversible cycles for another linear heat

transfer law. J. Phys. D Appl. Phys. 1993, 26, 1581–1586.

Energies 2012, 5

3585

10. Ait, A.M. Maximum power and thermal efficiency of an irreversible power cycle. J. Appl. Phys.

1995, 78, 4313–4318.

11. Angulo, B.F.; Rocha Martinez, J.A.; Navarrete Gonzalez, T.D. A non-endoreversible Otto cycle

model: Improving power output and efficiency. J. Phys. D Appl. Phys. 1996, 29, 80–83.

12. Chen, L.G.; Lin, J.X.; Sun, F.R.; Wu, C.I. Efficiency of an Atkinson engine at maximum power

density. Energy Convers. Manag. 1998, 39, 337–341.

13. Chen, L.G.; Wu, C.; Sun, F.R.; Cao, S. Heat transfer effects on the net work output and efficiency

characteristics for an air-standard Otto cycle. Energy Convers. Manag. 1998, 39, 643–648.

14. Chen, L.G.; Sun, F.R.; Wu, C. Effect of heat transfer law on the performance of a generalized

irreversible Carnot engine. J. Phys. D Appl. Phys. 1999, 32, 99–105.

15. Lin, J.X.; Chen, L.G.; Wu, C.; Sun, F.R. Finite time thermodynamic performance of a dual cycle.

Int. J. Energy Res. 1999, 23, 765–772.

16. Hou, S.S. Heat transfer effects on the performance of an air standard dual cycle.

Energy Convers. Manag. 2004, 45, 3003–3015.

17. Zhou, S.; Chen, L.; Sun, F.; Wu, C. Optimal performance of a generalized irreversible Carnot

engine. Appl. Energy 2005, 81, 376–387.

18. Hou, S.S. Comparison of performances of air standard Atkinson and Otto cycles with heat transfer

considerations. Energy Convers. Manag. 2007, 48, 1683–1690.

19. Lu, J.; Ding, J.; Yang, J. Optimal performance for solar thermal power system. Energy Power

Eng. 2009, 1,110–115.

20. Bejan, A. Advanced Engineering Thermodynamics; Wiley-Interscience: New York, NY, USA,

1997.

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article

distributed under the terms and conditions of the Creative Commons Attribution license

(http://creativecommons.org/licenses/by/3.0/).