Quantum degeneracy effect on the work output from a Stirling cycleHasan Saygin and Altuğ Şişman Citation: Journal of Applied Physics 90, 3086 (2001); doi: 10.1063/1.1396831 View online: http://dx.doi.org/10.1063/1.1396831 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/90/6?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Performance analysis of a micro-scaled quantum Stirling refrigeration cycle J. Appl. Phys. 112, 064908 (2012); 10.1063/1.4754420 Prospects for release-node quantum Monte Carlo J. Chem. Phys. 135, 184109 (2011); 10.1063/1.3659143 Statistical quantum probabilities in coherent bosonic and fermionic states J. Math. Phys. 47, 083302 (2006); 10.1063/1.2259583 Quantum statistics: Is there an effective fermion repulsion or boson attraction? Am. J. Phys. 71, 1223 (2003); 10.1119/1.1590658 Introduction to Relativistic Gases AIP Conf. Proc. 631, 3 (2002); 10.1063/1.1513675

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Quantum degeneracy effect on the work output from a Stirling cycleHasan Saygina) and Altug SismanIstanbul Technical University, Nuclear Energy Institute, 80626-Maslak, Istanbul, Turkey

~Received 15 January 2001; accepted for publication 30 June 2001!

The effect of quantum degeneracy on the work output from a Stirling cycle working at quantumdegeneracy conditions~QDCs! is analyzed. Expressions for net work outputs of Stirling powercycles working with monatomic ideal Bose and Fermi gases are derived by using the quantum idealgas equation of state. Ratios of net work outputs of Stirling cycles working with Bose and Fermigases to the net work output of a classical Stirling cycle (RW

B andRWF , respectively! are obtained.

Variations ofRWB andRW

F with TH are examined for a given temperature ratio (t5TL /TH) and aspecific volume ratio (r n5nH /nL). At QDC, it is seen thatRW

B has a maximum value, which isgreater than unity. On the other hand, there is no maximum or minimum point forRW

F and RWF

<1 for any values ofTH . Consequently, the use of Bose gas as a working fluid in a Stirling cycleprovides an advantage since it causes the net work output per cycle to increase by consuming moreheat energy. This fact is seen to be in the opposite direction for a Stirling cycle working with Fermigas. © 2001 American Institute of Physics.@DOI: 10.1063/1.1396831#

I. INTRODUCTION

It is well known that an ideal gas deviates from its clas-sical behavior under quantum degeneracy conditions~QDCs!at which the temperature is sufficiently low or the density issufficiently high. This fact results purely from the quantumdegeneracy of gas particles. Since the gases obey Bose–Einstein or Fermi–Dirac statistics at these conditions, theyare called ideal quantum gases.1 For these types of gases, thecorrected equation of state is valid instead of the classicalideal gas equation of state. Analyses of gas cycles workingwith ideal quantum gases allow us to investigate how ther-modynamic behaviors of gas cycles are effected by the quan-tum degeneracy of working fluid.

In recent years, analyses of quantum heat engines havebecome one of the interesting research subjects for peopleworking in thermodynamics and statistical physics. Finite-time thermodynamic analyses of some quantum heat engineshave been made.2–15 Some gas cycles working with idealquantum gases have also been thermodynamically analyzedin the literature.16–22

Efficiencies of Ericsson and Stirling power cycles work-ing with ideal quantum gases have been analyzed by S¸ ismanand Saygin.16,17 It has been shown that the thermodynamicbehaviors of these cycles are different from that of the clas-sical Ericsson and Stirling cycles. This is due to the appre-ciable dependence of specific heat capacities at constantvalue~or constant pressure! on specific volume~or pressure!as well as temperature at the QDC. As a result of this depen-dency, extra heat addition or rejection is required at interme-diate temperatures to maintain the energy balance betweenthe hot and cold stream in the regenerator. Because of theseextra processes, the efficiencies of these cycles are lowerthan the Carnot efficiency. A similar efficiency decrement

also appears in nonregenerative classical Stirling and Erics-son cycles.23,24 This efficiency decrement does not violatethe second principle of Carnot25 due to the existence of heatexchange at intermediate temperatures, as in ideal Otto andBrayton cycles.

Although the efficiencies of Stirling cycles working withideal quantum gases have been analyzed,17 the effect ofquantum degeneracy on the net work outputs of these cycleshas not been considered previously. The effect of quantumdegeneracy on the net work outputs of Stirling cycles is ana-lyzed in this work and some results, which cannot be pre-dicted by the classical ideal gas approximation, are obtained.As ideal Bose and Fermi gases,4He and3He gases are con-sidered here, respectively. Stirling cycles working with Boseand Fermi gases are called Bose and Fermi Stirling cycles.The expressions for net work outputs of Bose and FermiStirling cycles (WB and WF) are derived and the ratio ofthese quantities to that of the classical Stirling cycle (WC)are defined asRW

B 5WB /WC and RWF 5WF /WC . Variations

of RWB and RW

F with TH are examined. It is shown thatRWB

.1 for some values ofTH , althoughRWF ,1 for any values

of it. It is seen that there is some optimum value ofTH for aBose Stirling cycle andRW

B has a maximum for this optimumvalue. In a Stirling power cycle, consequently, the use ofBose gas as a working fluid provides an advantage since itcauses an increase of the net work output per cycle or adecrease of the number of cycles for a given work demand.On the other hand, when Fermi gas is used as a workingfluid, the opposed situation is seen to be valid. These resultscan be used to design a Stirling cycle working under quan-tum degeneracy conditions.

II. THERMODYNAMIC MODEL

In Fig. 1, theT-S diagram for a Stirling power cycle isgiven. As can be seen from Fig. 1, while the expansion work(We) is produced during the isothermal 3-4 process, the

a!Author to whom correspondence should be addressed; electronic mail:sayginh@itu.edu.tr

JOURNAL OF APPLIED PHYSICS VOLUME 90, NUMBER 6 15 SEPTEMBER 2001

30860021-8979/2001/90(6)/3086/4/$18.00 © 2001 American Institute of Physics

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compression work (Wc) is consumed during the isothermal1-2 process. For a classical Stirling cycle, expansion andcompression workWC

e and WCc are expressed by using the

classical ideal gas equation of state as follows:23,25

WCe 5E

VL

VHP~TH ,V!dV5mRTHLn~r n!, ~1!

WCc 5E

VL

VHP~TL ,V!dV5mRTLLn~r n!, ~2!

whereP is pressure,r n is a compression ratio defined asr n

5nH /nL5VH /VL , R is the gas constant, andm is the gasmass. By using Eqs.~1! and ~2!, the net work output of aclassical Stirling cycle is obtained as

WC5WCe 2WC

c 5mRTH~12t!Ln~r n!, ~3!

wheret is the temperature ratio defined ast5TL /TH . In aBose or Fermi Stirling cycle, expansion and compressionworks can be obtained by using the corrected equation ofstate, which considers the quantum degeneracy of gas par-ticles. The corrected equation of state is given as16

P5nkbTCF~m/kbT!, ~4a!

while the classical ideal gas equation of state is

P5nkbT, ~4b!

wheren is the particle density,kb is the Boltzmann’s con-stant, CF(m/kbT) is the correction factor, andm is chemicalpotential of the gas particles. For a Fermi gas, CF(m/kbT) is

CFF~m/kbT!5Li 5/2@2exp~m/kbT!#

Li 3/2@2exp~m/kbT!#. ~5!

For a Bose gas, it is defined as

T.T0⇒CFB~m/kbT!5Li 5/2@exp~m/kbT!#

Li 3/2@exp~m/kbT!#, ~6a!

T<T0⇒CFB~m/kbT!50.5135S T

T0D 3/2

, ~6b!

where Li (x) is the Polylogarithm function26 defined asLi i(x)5( l 51

` xl / l i andT0 is the Bose–Einstein condensationtemperature.T0 is also defined as

T0~n!5h2

2kbm0p S 1

2.612gD 2/3

n2/3, ~7!

whereg is the number of possible spin orientations of a gasparticle,h is the Planck’s constant, andm0 is the rest mass ofa gas particle.m(T,n) is implicitly calculated by using thefollowing equation:16

n56~2pm0!3/2g

h3~kbT!3/2Li 3/2@6exp~m/kbT!#, ~8!

where the upper signs are used for Bose gas and lower signsare used for Fermi gas. Hence, the correction factors arecalculated for given (T,n) or (T,n) parameters by Eqs.~5!–~8!.

By using Eq.~4a!, expressions of expansion and com-pression works for Bose and Fermi Stirling cycles can bederived as

Wie5mRTHE

VL

VHCFi~TH ,V!

VdV, ~9!

Wic5mRTLE

VL

VHCFi~TL ,V!

VdV, ~10!

wherei refers to the type of the working gas~Bose or Fermigas!. By using Eqs.~1!, ~2!, ~9!, and ~10!, expansion andcompression work ratios are, respectively, obtained as

a i5Wi

e

WCe 5

*VL

VH@CFi~TH ,V!/V#dV

Ln~r n!, ~11!

b i5Wi

c

WCc 5

*VL

VH@CFi~TL ,V!/V#dV

Ln~r n!. ~12!

Therefore, the ratio of net work output of Bose~or Fermi!Stirling cycle to that of the classical Stirling cycle is

RWi 5

a iWCe 2b iWC

c

WCe 2WC

c5

a i

12~WCc /WC

e !2

b i

~WCe /WC

c !21.

~13!

SinceWCc /WC

e 5t, Eq. ~13! can be simplified as

RWi 5

1

12t~a i2tb i !. ~14!

In Fig. 2, the variations ofa i andb i with TH are given.It is seen thataB and bB go to zero for low values ofTH

while aF and bF go to infinity for the same temperaturevalues. For high values ofTH , however, both of them go tounity because the quantum degeneracy becomes unimportantin this temperature range. For any values ofTH , aB.bB andaF.bF .

FIG. 1. Temperature-entropy diagram of the cycle. Paths 1→2 and 3→4 areisothermal processes taking place in compressor and turbine, respectively.Paths 2→3 and 4→1 are isochoric processes in the regenerator.

3087J. Appl. Phys., Vol. 90, No. 6, 15 September 2001 H. Saygin and A. Sisman

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III. ANALYTICALLY SOLVABLE CASES

For givent andr n values, there are three different caseswhich are analytically solvable. These cases are consideredas follows:

Case I: TH.TL@T0~nL!>T0~nH!

and

TH.TL@TF~nL!>TF~nH!.

Here, TF is the Fermi temperature and it is defined27,28 asTF5(3/4pg)2/3(h2/2m0kb)(1/n2/3). In this case, the workinggas is always a classical ideal gas throughout the cycle andcorrection factors are approximately equal to unity. There-fore, a i>b i>1 andRW

i >1. This can be seen in Figs. 2 and3 for high values ofTH .

Case II: TH@T0~nL!>T0~nH!>TL

and

TH@TF~nL!>TF~nH!@TL .

For this case, the working gas is a classical ideal gas duringthe expansion process while it is a completely degeneratequantum ideal gas during the compression process. SinceTH@TL , however, it is easily seen thatWB

e@WBc , WF

e

@WFc , andWC

e @WCc . Thus, the net work output is approxi-

mately equal to the expansion work. Because the workinggas is a classical ideal gas during the expansion process, it isobtained thatRW

B >RWF >1 for this case.

Case III: T0~nL!>T0~nH!>TH.TL

and

TF~nL!>TF~nH!@TH.TL .

In this case, the working gas is a completely degeneratequantum gas throughout the cycle.aB andbB are determinedby using Eqs.~6b!, ~11!, and~12!. aF andbF are also deter-mined by using the approximate expression of CFF

>2TF/5T given in the literature for completely degeneratemonatomic Fermi gas.16 Thus, it can be obtained thataB

>t23/2bB andaF>tbF . Consequently,RWB andRW

F are de-termined as follows, respectively:

RWB >aB

12t5/2

12t50.5135

r n21

r nLn~r n! F TH

T0~nH!G3/212t5/2

12t,

~15!

RWF >0. ~16!

The maximum value ofRWB is determined as

~RWB !Max5 Limit

t→1r n→1

TH→T0~nH!

~RWB !>1.2825. ~17!

IV. CONCLUSIONS

Although the efficiency and net work output of the cyclego to zero under the limitations given by Eq.~17!, it de-scribes an upper limit forRW

B . In the case oft,1 and r n

.1, however, the efficiency and net work of the cycle aregreater than zero and this case describes a meaningful cycle.As shown in Fig. 3,RW

B has a maximum greater than unityand lower than the upper limit given by Eq.~17!. From Fig.3, it is seen thatRW

F ,1 for all values ofTH andRWF goes to

zero under the conditions of Case III.For the range of temperatureTH in which RW

B .1, there-fore, the use of Bose gas as a working fluid provides anadvantage for a Stirling power cycle. On the other hand, theuse of Fermi gas is always a disadvantage for that powercycle. The advantage of a Stirling cycle working with a Bosegas arises from its capability of more heat energy consump-tion per cycle in comparison with that of the classicalStirling cycle. Thus, a Bose Stirling cycle produces more network output by consuming more heat energy. On the otherhand, a Fermi Stirling cycle consumes less heat energy percycle and produces less net work output in comparison with

FIG. 2. Variations ofa i andb i vs to TH . Figure parameterst, r n , andnH

are chosen ast50.6, r n52, andnH50.001 m3/kg.

FIG. 3. Variations ofRWB andRW

F vs to TH . Figure parameternH is chosenasnH50.001 m3/kg.

3088 J. Appl. Phys., Vol. 90, No. 6, 15 September 2001 H. Saygin and A. Sisman

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that of the classical Stirling cycle. For this reason, the use ofFermi gas as a working fluid constitutes a disadvantage forthe Stirling Cycle.

In a real cycle, machine life decreases with increasingnumber of cycles due to mechanical wear. Moreover, worklosses depending on friction also increases with increasingthe number of cycles. For these reasons, a Bose Stirlingcycle is advantageous since it needs less numbers of cyclesfor a given work demanded.

Consequently, it is found that quantum degeneracy canprovide an advantage for a Stirling cycle from the viewpointof net work output per cycle, although it constitutes a disad-vantage for cycle efficiency.17

ACKNOWLEDGMENT

This research was supported by the Research Fund ofIstanbul Technical University under Contract No. 1311.

1C. Garrod,Statistical Mechanics and Thermodynamics~Oxford UniversityPress, New York, 1995!, p. 170.

2R. A. Kosloff, J. Chem. Phys.80, 1625~1984!.3E. Geva and R. Kosloff, J. Chem. Phys.96, 3054~1992!.4E. Geva and R. Kosloff, J. Chem. Phys.97, 4396~1992!.5E. Geva and R. Kosloff, Phys. Rev. E49, 3903~1994!.6T. Feldmann, E. Geva, R. Kosloff, and P. Salamon, Am. J. Phys.64, 485~1996!.

7E. Geva and R. Kosloff, J. Chem. Phys.104, 7681~1996!.

8F. Wu, F. Sun, and L. Chen, J. Wuhan Institute Chem. Tech.19, 88 ~1997!~in Chinese!.

9X. Jin, F. Wu, F. Sun, and L. Chen, Pow. Sys. Eng.12, 53 ~1996! ~inChinese!.

10F. Wu, L. Chen, F. Sun, C. Wu, and P. Hua, Energy Convers. Manage.39,1161 ~1998!.

11F. Wu, L. Chen, F. Sun, and C. Wu, Int. J. Eng. Sci.38, 239 ~2000!.12X. Jin, F. Wu, and L. Chen, Cryogenics1, 41 ~1998! ~in Chinese!.13F. Wu, L. Chen, and F. Sun, Cryogenics1, 1 ~1996! ~in Chinese!.14F. Wu, L. Chen, and F. Sun, J. Appl. Sci.15, 223 ~1997! ~in Chinese!.15F. Wu, L. Chen, F. Sun, C. Wu, and Y. Zhu, Energy Convers. Manage.39,

733 ~1998!.16A. Sisman and H. Saygin, J. Phys. D32, 664 ~1999!.17A. Sisman and H. Saygin, Phys. Scr.63, 263 ~2001!.18A. Sisman and H. Saygin, Appl. Energy68, 367 ~2001!.19H. Saygin and A. S¸ isman, Appl. Energy69, 77 ~2001!.20H. Saygin, invited talk at the Conference on the Advances in Finite-Time

Thermodynamics, Jerusalem, November~2000!.21A. Sisman, invited talk at the Conference on the Advances in Finite-Time

Thermodynamics, Jerusalem, November~2000!.22A. Sisman and H. Saygin Phys. Scr.64, 108 ~2001!.23D. C. Look and H. J. Sauer,Engineering Thermodynamics, ~PWS Engi-

neering, Boston, MA, 1988!, p. 284.24F. W. Sears,Thermodynamics: An Introductory Text for Engineering Stu-

dents. Reading~Addison-Wesley, Reading, MA, 1963!, p. 460.25Y. A. Cengel and A. B. Boles,Thermodynamics, An Engineering Ap-

proach ~McGraw-Hill, New York, 1994!, pp. 104–108 and 468–472.26S. Wolfram,Mathematica: A System for Doing Mathematics by Computer

~Addison-Wesley, Redwood City, CA, 1991!, p. 575.27P. M. Morse,Thermal Physics~Benjamin, New York, 1965!, pp. 364–365.28L. D. Landau and E. M. Lifshitz,Statistical Physics~Pergamon, London,

1958!, Chap. 5.

3089J. Appl. Phys., Vol. 90, No. 6, 15 September 2001 H. Saygin and A. Sisman

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