2325c90d 04ab e011 86d5 b8ac6f85e9e8_article renewable energy 1

7/27/2019 2325c90d 04ab e011 86d5 b8ac6f85e9e8_Article Renewable Energy 1

http://slidepdf.com/reader/full/2325c90d-04ab-e011-86d5-b8ac6f85e9e8article-renewable-energy-1 1/11

Renewable Energy 33 (2008) 2134–2144

Technical Note

Performance optimization of Stirling engines

Youssef Timoumi Ã , Iskander Tlili, Sassi Ben NasrallahLaboratoire d’Etude des Syste `mes Thermiques et Energe ´ tiques, Ecole Nationale d’Inge ´ nieurs de Monastir, Rue Ibn El Jazzar, 5019 Monastir, Tunisie

Received 1 June 2007; accepted 16 December 2007Available online 13 February 2008

Abstract

The search for an engine cycle with high efciency, multi-sources of energy and less pollution has led to reconsideration of the Stirlingcycle. Several engine prototypes were designed but their performances remain relatively weak when compared with other types of combustion engines. In order to increase their performances and analyze their operations, a numerical simulation model taking intoaccount thermal losses has been developed and used, in this paper, to optimize the engine performance. This model has been tested usingthe experimental data obtained from the General Motor GPU-3 Stirling engine prototype. A good correlation between experimental dataand model prediction has been found. The model has also been used to investigate the inuence of geometrical and physical parameterson the Stirling engine performance and to determine the optimal parameters for an acceptable operational gas pressure.r 2007 Elsevier Ltd. All rights reserved.

Keywords: Stirling engines; Performance; Losses; Dynamic model; Regenerator; Thermal efciency

1. Introduction

The urgent need to preserve fossil fuels and use renew-able energies has led to the use of Stirling engines, whichhave an excellent theoretical efciency, equivalent to theCarnot one. They can use any source of energy (combus-tion energy, solar energy y ) and they are less pollutingthan the traditional engines.

Several prototypes were produced, Fig. 1 , but theiroutputs remain very weak compared to the excellenttheoretical yield, [1–5]. In fact, these engines haveextremely complex phenomena related to the compressibleuid mechanics, thermodynamics, and heat transfer. Anaccurate description and understanding of these highly

non-stationary phenomena is necessary so that differentengine losses, optimal performance and design parameterscan be determined.

Many investigators have studied the effect of some heatlosses and irreversibilities on the engine performanceindices. However, they have not calculated the optimalperformance and design parameters for maximum powerand efciency. Popescu et al. [6] show that the most

signicant reduction in performance is due to the non-adiabatic regenerator. Kaushik, Wu and co-workers [7,8]have found that heat conductance between the engine andthe reservoirs, the imperfect regenerator coefcient and therates of the two regenerating processes are the importantfactors affecting the performance of a Stirling engine.Kongtragool and Wongwises [9] investigated the effect of regenerator effectiveness and dead volume on the enginenetwork, heat input and efciency by using a theoreticalinvestigation on the thermodynamic analysis of a Stirlingengine. Costea et al. [10] studied the effect of irreversibilityon solar Stirling engine cycle performance; they includedthe effects of incomplete heat regeneration, internal andexternal irreversibility of the cycle as pressure losses due to

uid friction internal to the engine and mechanical frictionbetween the moving parts. Cun-quan et al. [11] haveestablished a dynamic simulation of an one-stage Oxfordsplit-Stirling cryocooler. The regenerator inefciency loss,the solid conduction loss, the shuttle loss, the pump lossand radiation loss are integrated into the mathematicalmodel. The regenerator inefciency loss and solid conduc-tion loss are the most important. An acceptable agreementbetween experiment and simulation has been achieved.Cinar et al. [12] manufactured a beta-type Stirling engineoperating at atmospheric pressure. The engine test

ARTICLE IN PRESS

www.elsevier.com/locate/renene

0960-1481/$- see front matter r 2007 Elsevier Ltd. All rights reserved.doi: 10.1016/j.renene.2007.12.012

Ã Corresponding author. Tel.: +21698 6762 54; fax: +21673 500514.E-mail address: [email protected] (Y. Timoumi).

http://www.elsevier.com/locate/renene

http://localhost/var/www/apps/conversion/tmp/scratch_12/dx.doi.org/10.1016/j.renene.2007.12.012

mailto:[email protected]

mailto:[email protected]

http://localhost/var/www/apps/conversion/tmp/scratch_12/dx.doi.org/10.1016/j.renene.2007.12.012

http://www.elsevier.com/locate/renene



indicated that the engine speed, engine torque and poweroutput increase proportionally with a rise in the hot sourcetemperature.

Walker [2] mentions other losses but without introducingthem in the models: the conduction losses in theexchangers, the load losses, the shuttle losses and the gasspring hysteresis losses. Furthermore, these losses are notusually studied in literature because of their complexity.Urieli and Berchowitz [13] developed an adiabatic modeland a quasi-stationary model where they introduced onlythe pressure drops into the exchangers. The results

obtained by this model are better than those of the othermodels, but remain different from the experimental results.Hence, the Stirling engine performance depends ongeometrical and physical parameters of the engine and onthe working uid gas properties such as regeneratorefciency and porosity, dead volume, swept volume,temperature of sources, pressure drop losses, shuttle losses,etc.

A dynamic model taking into account the different lossesis developed by the authors and tested using the GeneralMotor GPU-3 Stirling engine data, [14]. The results

obtained proved better than those obtained by othermodels and correlate more closely with experimental data.The model is used to determine the losses in differentengine compartments and to calculate the geometrical andphysical parameters corresponding to minimal losses [15].An optimization based on this model is presented in thisarticle. It will help study the inuence of geometrical andphysical parameters on the prototype performance of aStirling engine and therefore determine their optimalvalues.

2. Dynamic model including losses

A second-order adiabatic model has been initiallydeveloped. The estimated values of the engine parametersare obtained and for the sake of validation, the results arecompared with Berchowitz results under analogous condi-tions [14]. Afterward, a dynamic model, which takes intoaccount the losses in the different engine elements, wasdeveloped.

The losses considered in this model are the energydissipation by pressure drops in heat exchangers, energylost due to internal conduction through the exchangers,energy lost due to external conduction in the regenerator,energy lost due to the shuttle effect in the displacer andenergy lost due to gas spring hysteresis in the compression

ARTICLE IN PRESS

Nomenclature

A area, m 2

C p specic heat at constant pressure, J kgÀ1 K

À1

C pr heat capacity of each cell matrix, W KÀ1

e regenerator efciencyM mass of working gas in the engine, kg_m mass ow rate, kg s

À1

m mass of gas in different component, kgP pressure, PaQ heat, J_Q power, W

R gas constant, Jkg KÀ1

T temperature, KU convection heat transfer coefcient, W m

À2 KÀ1

V volume, m 3

W work, J

Subscripts

c compression spacediss dissipationd expansion space

E enteredf coolerh heaterirr irreversiblep lossPa wallr regeneratorr1 regenerator cell 1r2 regenerator cell 1S outletshtl shuttleT total

Fig. 1. Rhombic Stirling engine GPU-3 (built by General Motor [14]).

Y. Timoumi et al. / Renewable Energy 33 (2008) 2134–2144 2135



and expansion spaces [14]. The mechanical friction betweenthe moving parts is not considered.

The schematic model and the temperature distribution inthe various engine components are shown in Fig. 2 . Thedynamic model of the developed Stirling engine is based onthe following assumptions:

The gas temperature in the different engine elements isvariable.The cooler and the heater walls are maintainedisothermal at temperatures T paf and T pah .The gas temperature in the different components iscalculated using the perfect gas law.The regenerator is divided into two cells r1 and r2, eachcell has been associated with its respective mixed meangas temperature T r1 and T r2 expressed as follows:

T r1 ¼P r1 V r1

Rm r1

, (1)

T r2 ¼P r2 V r2Rm r2

. (2)

An extrapolated linear curve is drawn through temperaturevalues T r1 and T r2 , dening the regenerator interfacetemperature T f–r , T r–r and T r–h , as follows [15]:

T f Àr ¼3T r1 À T r2

2, (3)

T rÀr ¼T r1 þ T r2

2, (4)

T rÀh ¼3T r2 À T r1

2. (5)

According to the ow direction of the uid, the interface’stemperatures are dened as follows [14]:

T cÀf ¼ T c if _mcÀf 4 0; otherwise T cÀf ¼ T f .

T f Àr ¼ T f if _mf Àr4 0; otherwise T f Àr ¼ T rÀr .

T rÀh ¼ T rÀr if _mrÀh 4 0; otherwise T rÀh ¼ T h .

T hÀd ¼ T h if _mhÀd 4 0; otherwise T hÀd ¼ T d .

The regenerator matrix temperatures are thereforegiven by

dT par1

dt¼ À

dQ r1

C pr dt, (6)

dT par2

dt¼ À

dQ r2

C pr dt. (7)

Taking into account the losses by internal conduction inthe exchangers: d _Q pcdf , d _Qpcdr1 , d _Qpcdr2 , d _Qpcdh andexternal conduction in the regenerator, the power ex-changed in the different heat exchangers are given by

d _Q f ¼ U f Apaf ðT paf À T f Þ À d _Q pcdf , (8)

d _Q r1 ¼ U r1 Apar1 ðT par1 À T r1 Þ Àd _Qpcdr1

2, (9)

d _Q r2 ¼ U r2 Apar2 ðT par2 À T r2 Þ Àd _Qpcdr2

2, (10)

d _Qh ¼ U h Apah ðT pah À T h Þ À d _Q pcdh , (11)

where e is the regenerator efciency.The heat transfer coefcient of exchanges U f , U r1 , U r2

and U h are available only empirically [13]. The totalexchanged heat is

d _Q ¼ d _Q f þ d _Q r1 þ d _Q r2 þ d _Qh À d _Q pshtl , (12)

where d _Qpshtl is the shuttle loss in the displacer. Consider-ing the loss due to gas spring hysteresis in the compressionand expansion space: dW irrc =dt and dW irrd =dt, evaluated in[14], the work done in a cycle is

dW dt

¼ P cdV cdt

þ P ddV ddt

ÀdW irrc

dtÀ

dW irrddt

. (13)

The total engine volume is

V T ¼ V c þ V f þ V r1 þ V r2 þ V h þ V d . (14)

Since there is a variable pressure distribution throughoutthe engine, the compression space pressure P c has beenarbitrarily chosen as the baseline pressure. At eachincrement of the solution, P c is evaluated from the relevantdifferential equation and the pressure distribution isdetermined with respect to P c.

The other variables of the dynamic model are given bythe energy and mass conservation equation applied

ARTICLE IN PRESS

Fig. 2. Schematic model of the engine and various temperaturedistributions.

Y. Timoumi et al. / Renewable Energy 33 (2008) 2134–21442136



to a generalized cell:

d _Q þ C p T E _mE À C p T S _mS ¼ P dV dt

þ C VdðmT Þ

dt, (15)

M ¼ md þ mc þ mf þ mr þ mh . (16)

Applying the energy conservation equation to thedifferent engine cells and including energy dissipation bypressure drop in the exchangers, d _Qdiss , and the other lossesyields

ÀC p T cÀf _mcS ¼1R

C p P cdV cdt

þ C VV cdP cdt À d _W irrc ,

(17)

d _Q f À d _Qdissf þ C p T cÀf _mfE À C p T f Àr _mfS ¼C VV f

RdP cdt

,

(18)

d _Q r1 À d _Qdissr1 þ C p T f Àr _mr1E À C p T rÀr _mr1S ¼ C VV r1R

dP cdt

,

(19)

d _Q r2 À d _Qdissr2 þ C p T rÀr _mr2E À C p T rÀh _mr2S ¼C VV r2

RdP cdt

,

(20)

d _Q h À d _Qdissh þ C p T rÀh _mhE À C p T hÀe _mhS ¼C VV h

RdP cdt

,

(21)

C p T hÀ d _md À d _Qpshtl ¼ 1R

C p P d dV ddt

þ C VV d dP cdt À d _W irrd .

(22)

Summing Eqs. (22)–(27), the pressure variation wasobtained:

dP cdt

¼1

C vV TRðd _Q À d _Q dissT Þ À C p

dW dt , (23)

where d _QdissT ¼ d _Q dissf þ d _Q dissr1 þ d _Q dissr2 þ d _Qdissh , is thetotal heat dissipation generated by pressure drop.

The mass ow in the different engine components isgiven by the energy conservation equations (17)–(22):

_mcS ¼ À1

RT cÀf

P dV cdt

þ V cdP cg dt

þ

d _W irrC

PT

cÀf

, (24)

_mfS ¼1

cpT f Àrd _Q f À d _Qdissf þ cp T cÀf _mfE À

cVV f R

dP cdt ,

(25)

_mr1S ¼1

cp T rÀrd _Q r1 À d _Qdissr1 þ cp T f Àr _mr1E À

cVV r1R

dP cdt ,

(26)

_mr2S ¼1

cp T rÀhd _Q r2 À d _Qdissr2 þ cp T rÀr _mr2E À

cVV r2R

dP cdt ,

(27)

_mhS ¼1

cp T hÀdd _Qh À d _Qdissh þ cp T rÀ h _mhE

dmh

dtE À

cVV hR

dP cdt ,

(28)

with _mcS ¼ _mfE , _mfS ¼ _mr1E , _mr1S ¼ _mr2E , _mr2S ¼ _mhE and_mhS ¼ _mdE .

3. Prototype specications

The developed model has been tested using data from theStirling engine GPU-3 manufactured by General Motor;this engine has a rhombic motion transmission system, Fig.1. The geometrical parameters of this engine are given inTable 1 . The operating conditions are as follows: workinggas helium at a mean pressure of 4.13 MPa, frequency41.72 Hz, hot space temperature T pah ¼ 977 K and coldspace temperature T paf ¼ 288 K. The measured poweroutput was 3958 W, at a thermal efciency of 35%. Theindependent differential equations, obtained in paragraph

ARTICLE IN PRESS

Table 1Geometric parameter values of the GPU-3 Stirling engine

Parameters Values Parameters Values

Clearance volumes CoolerCompression space 28.68 cm 3 Tubes number/cylinder 312Expansion space 30.52 cm 3 Interns tube 1.08 mm

Swept volumes Diameter 46.1 mmCompression space 113.14 cm 3 Length of the tube 13.8 cm 3

Expansion space 120.82 cm 3 Void volumeExchanger piston conductivity 15 W/m K RegeneratorExchanger piston stroke 46 mm Diameter 22.6 mm

Length 22.6 mmHeater Wire diameter 40 mm

Tubes number 40 Porosity 0.697Tube inside diameter 3.02 mm Units numbers/cylinder 8Tube length 245.3 mm Thermal conductivity 15 W/m KVoid volume 70.88 cm 3 Void volume 50.55 cm 3




effect is about 3.1kW, which represents 13% of the totalenergy loss.

The energy lost due to external conduction in theregenerator is 27 kW, which represents 47% of the totallosses ( Fig. 5 ), and is very signicant and depends mainlyon the regenerator efciency. The energy lost due to

irreversibility in the compression and expansion spaces isvery low [15].

5. Optimization of the Stirling engine performance

The energy losses occur mainly in the regenerator. Theyare primarily due to the losses by external and internal

conduction and pressure drop through the heat exchangers.The energy lost due to the shuttle effect in the exchangerpiston is also signicant; it is about 13%. The other lossesare very weak [15].

Reduction of these losses can improve the engineperformance. Such losses depend mainly on the matrix

conductivity of the regenerator, its porosity, the inlettemperature variation, the working gas mass, the regen-erator volume and the geometrical characteristics of thedisplacer. To investigate the inuence of these parameterson the prototype performance, we have varied the studiedparameter in the model each time and have kept the othersunchanged, equal to the prototype parameters.

ARTICLE IN PRESS

Fig. 6. Effect of regenerator thermal conductivity on performance.

Fig. 7. Effect of regenerator heat capacity on performance.




5.1. Effect of the regenerator matrix conductivity and heatcapacity

The performance of the engine depends on the con-ductivity and heat capacity of the material constitutingthe regenerator matrix. Fig. 6 shows that with an increase

of the matrix regenerator thermal conductivity leadsto a reduction of the performance due to an increaseof internal conduction losses in the regenerator, [14,15].Fig. 7 shows that the engine performance improveswhen the heat capacity of the regenerator matrixincreases.

The matrix of the regenerator can be made from severalmaterials. The performances of the engine are givenaccording to the material type in Table 2 . The performanceof the engine depends on the regenerator matrix material.To increase heat exchange of the regenerator and to reducethe internal losses by conductivity, a material with highheat capacity and low conductivity must be chosen.Stainless steel and ordinary steel are the most suitablematerials to prepare the regenerator matrix.

5.2. Effect of the regenerator porosity

The porosity of the regenerator is an importantparameter for engine performance. It affects the hydraulicdiameter, dead volume, velocity of the gas, regeneratorheat transfer surface and regenerator effectiveness; and

thus affects the losses due to external and internalconduction and the dissipation by pressure drop [15].Engine performance decreases when the porosity in-

creases due to an increase in the external conduction lossesand a reduction of the exchanged energy between the gasand the regenerator, Q r (Fig. 8 ). A low porosity will give abetter result.

5.3. Effect of the regenerator temperature gradient(T f–r –T r–h )

Although engine losses increase when the temperaturegradient of the regenerator increases [15], the performanceof the engine also increases ( Fig. 9 ). In this case, theperformance enhancement is due to an increase of the

ARTICLE IN PRESS

Fig. 8. Effect of regenerator porosity on performance and exchanged energy.

Table 2Effect of nature of the regenerator metal on the engine performance

Regenerator matrixmetal

Volumetric capacityheat (J/m 3 K)

Conductivity(W/mK)

Engine power(W)

Engine effectiveness(%)

Exchanged energy inthe regenerator (J)

Steel 3.8465 Â 106 46 4258 38.84 441.25Stainless steel 3.545 Â 106 15 4273 39.29 448.72Copper 3.3972 Â 106 389 – – – Brass 3.145 Â 106 100 4080 34.6 415.67Aluminum 2.322 Â 106 200 3812 29.16 378.03Granite 2.262 Â 106 2.5 4091 34.51 430.75Glass 2.1252 Â 106 1.2 4062 33.85 427.8




When the stroke increases, the engine power decreases, butthe efciency reaches a maximum. The optimal perfor-mances are superior to that of the prototype. They areobtained when the area and the stroke are, respectively,equal to 3.8 10

À3 m 2 and 0.042 m, which correspond to apower of 4500 W and an efciency of 41%.

The thermal conductivity of the exchanging pistonconsiderably affects the engine performances ( Fig. 15 ). Aweak conductivity reduces the losses due to the shuttleeffect and consequently increases the engine power and theefciency.

6. Conclusion

The Stirling engine prototypes designed have lowoutputs because of the considerable losses in the regen-erator and the exchanger piston. These losses are primarilydue to external and internal conduction, pressure drops inthe regenerator and shuttle effect in the exchanger piston,which depend on the geometrical and physical parametersof the prototype design. An optimization of these para-meters has been carried out using the GPU-3 engine data,and has led to a reduction of the losses and to a notable

ARTICLE IN PRESS

Fig. 13. Effect of exchanger piston area on engine performance.

Fig. 14. Effect of exchanger piston stroke on engine performance.




improvement in the engine performance. The parameters of this prototype were rst applied on the developed model;the results were very close to the experimental data. Then,the inuence of each geometrical and physical parameteron the engine performance and the exchanged energy of theregenerator was studied.

A reduction of the matrix porosity and conductivity of the regenerator increase the performance. A rise of thetotal mass of gas leads to an increase of the engine power

and function pressure; however, the efciency reached amaximum. When the displacer section increases and thepiston stroke decreases, the engine power increases, and theefciency attains a maximum. A low conductivity of theexchanger piston reduces the losses due to the shuttle effectand consequently increases the engine power and efciency.

In future, it is hoped to introduce simultaneously in themodel all the optimal parameters obtained, determine theoptimal design parameters and consequently the interestingperformance.

References

[1] Stouffs P. Machines thermiques non conventionnelles: e tat de l’art,applications, proble mes a ` re soudre y , Thermodynamique des ma-chines thermiques non conventionnelles, Journe e d’e tude du 14octobre 1999, SFT, Paris.

[2] Walker G. Stirling engines. Oxford: Clarendon Press; 1980.[3] Kolin I. Stirling motor: history–theory–practice. Dubrovnik: Inter

University Center; 1991.

[4] Timoumi Y, Ben Nasrallah S. Design and fabrication of aStirling–Ringbom engine running at a low temperature. In: Proceed-ings of the TSS International conference in mechanics and engineer-ing, ICAME, March 2002, Hammamet-Tunisia.

[5] Halit K, Huseyin S, Atilla K. Manufacturing and testing of a V-typeStirling engine. Turk J Engine Environ Sci 2000;24:71–80.

[6] Popescu G, Radcenco V, Costea M, Feidt M. Optimisationthermodynamique en temps ni du moteur de Stirling endo- et exo-irre versible. Rev Ge n Therm 1996;35:656–61.

[7] Kaushik SC, Kumar S. Finite time thermodynamic analysis of endoreversible Stirling heat engine with regenerative losses. Energy2000;25:989–1003.

[8] Wu F, Chen L, Wu C, Sun F. Optimum performance of irreversibleStirling engine with imperfect regeneration. Energy Convers Manage1998;39:727–32.

[9] Kongtragool B, Wongwises S. Thermodynamic analysis of a Stirlingengine including dead volumes of hot space, cold space andregenerator. Renew Energy 2006;31:345–59.

[10] Costa M, Petrescu S, Harman C. The effect of irreversibilities onsolar Stirling engine cycle performance. Energy Convers Manage1999;40:1723–31.

[11] Cun-quan Z, Yi-nong W, Guo-lin J. Dynamic simulation of one-stageOxford split-Stirling cryocooler and comparison with experiment.Cryogenics 2002;42:377–586.

[12] Cinar C, Yucesu S, Topgul T, Okur M. Beta-type Stirlingengine operating at atmospheric pressure. Appl Energy 2005;81:351–7.

[13] Urieli I, Berchowitz D. Stirling cycle engine analysis. Bristol: AdamHilger Ltd.; 1984.

[14] Tlili I, Timoumi Y, Ben Nasrallah S. Numerical simulation and lossesanalysis in a Stirling engine. Heat Technol 2006;24:97–105.

[15] Timoumi Y, Tlili I, Ben Nasrallah S. Reduction of energy losses in aStirling engine. Heat Technol 2007; 27(1), in press.

ARTICLE IN PRESS

Fig. 15. Effect of displacer thermal conductivity on performance.


2325c90d 04ab e011 86d5 b8ac6f85e9e8_article renewable energy 1

Documents