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Renewable Energy 33 (2008) 1911–1921

www.elsevier.com/locate/renene

Analysis and design consideration of mean temperature differentialStirling engine for solar application

Iskander Tlili�, Youssef Timoumi, Sassi Ben Nasrallah

Laboratoire d’Etude des Systemes Thermiques et Energetiques Ecole Nationale d’Ingenieurs de Monastir, Rue Ibn El Jazzar, 5019 Monastir, Tunisie

Received 17 August 2006; accepted 21 September 2007

Available online 5 November 2007

Abstract

This article presents a technical innovation, study of solar power system based on the Stirling dish (SD) technology and design

considerations to be taken in designing of a mean temperature differential Stirling engine for solar application. The target power source

will be solar dish/Stirling with average concentration ratio, which will supply a constant source temperature of 320 1C. Hence, the system

design is based on a temperature difference of 300 1C, assuming that the sink is kept at 20 1C. During the preliminary design stage, the

critical parameters of the engine design are determined according to the dynamic model with losses energy and pressure drop in heat

exchangers was used during the design optimisation stage in order to establish a complete analytical model for the engine. The heat

exchangers are designed to be of high effectiveness and low pressure-drop. Upon optimisation, for given value of difference temperature,

operating frequency and dead volume there is a definite optimal value of swept volume at which the power is a maximum. The optimal

swept volume of 75 cm3 for operating frequency 75Hz with the power is 250W and the dead volume is of 370 cm3.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Solar-powered; Stirling engine; Design; Losses; Regenerator; Thermal efficiency

1. Introduction

The harmony between environmental protection andeconomic growth has become a worldwide concern; there isan urgent need to effectively reuse solar energy, this sourceof energy is one of the more attractive renewable energythat can be used as an input energy source for heat engines.In fact, any heat energy source can be used with the Stirlingengine. The solar radiation can be focused onto the heaterof Stirling engine as shown in Fig. 1(a), thereby creating asolar-powered prime mover. The direct conversion of solarpower into mechanical power reduces both the cost andcomplexity of the prime mover. In theory, the principaladvantages of Stirling engines are their use of an externalheat source and their high efficiency. Stirling engines areable to use solar energy that is a cheap source of energy.

Studies about high temperature Stirling engines have beenextensively reported in the literature [1] and commercial

e front matter r 2007 Elsevier Ltd. All rights reserved.

nene.2007.09.024

ing author. Tel.: +216 98 61 97 04; fax: +216 73 50 05 14.

ess: Iskander.Tlili@enim.rnu.tn (I. Tlili).

units have been in operation for many years. On the otherhand, low temperature Stirling engines are not as successfulas their high temperature counterparts. However, theformer have gained popularity in the last few decades dueto this potential to tap a variety of low concentrationenergy sources available, such as solar. The increasinginterest in Stirling engines is largely due to the fact theengine is more environmentally friendly than the widelyused internal combustion engine, and also to its non-explosive nature in converting energy into mechanical formand thus leading to silent and cleaner operation, which areessential for special applications, such as military opera-tions and medical uses.The systems with very strong concentration [2] call upon

an advanced and heavy technology, therefore are veryexpensive as they present, on the energy point of view, alimited interest. On the other hand, the systems withoutconcentration are not economically viable. The best systemsis with average concentration, leading to levels of temperatureabout 250–450 1C, but very few work seem to be devoted tothe installations with average concentration. The company

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Nomenclature

A area, m2

Cp specific heat at constant pressure, J kg�1K�1

Cpr heat capacity of each cell matrix, JK�1

Cv specific heat at constant volume, J kg�1K�1

d hydraulic diameter, mD diameter, mdm wire diameter, mfr friction factorFreq operating frequency, Hzh convection heat transfer coefficient,

Jm�2 s�1K�1

J annular gap between displacer and cylinder, mk thermal conductivity, Wm�1K�1

L length, mM mass of working gas in the engine, kg_m mass flow rate, kg s�1

m mass of gas in different component, kgNTU number of heat transfer unitP pressure, PaQ heat, J_Q power, W

R gas constant, J kgK�1

T temperature, KV volume, m3

W work, J

Subscripts

c compression spacech loadd expansion spaceE enteredext outsidef coolerh heatermoy meanP lossPa wallpis pistonr regeneratorr1 regenerator cell 1r2 regenerator cell 2S left

Greek symbols

y crank angle, rade effectivenessm Working GAS dynamic viscosity, kgm�1 s�1

r density, kgm�3

o angular frequency, rad s�1

c mesh porosity

I. Tlili et al. / Renewable Energy 33 (2008) 1911–19211912

BSR Solar Technologies GmbH, which developed theSUNPULSE, also works on a system intended to produceelectricity starting from solar energy fairly concentrated,which leads to levels of temperature about 450 1C.

Several analyses and simulation methods of the enginehave been established [3], as well as the procedures foroptimal design [4]. Most of the engines are fuel-fired andoperate at high temperature, which highlights the need forcareful material selection as well as good cooling system.For silent, light and portable equipment for leisure anddomestic uses, low power engines may be more appropriate.Nevertheless, research in Stirling engine technology has beenheavily masked by extensive and successful development ofinternal combustion engines, which have made Stirlingengines less competitive. Hence, in order to design a lowpower engine using solar, new design specifications andoptimisation criteria must be established [5–9]. This paperpresents design considerations which may be taken todevelop a solar Stirling engine with average concentrationoperating on mean temperature difference of 300 1C.

2. Losses in a Stirling engine

The energy losses in a Stirling engine are due to thethermodynamic and the mechanical processes. Compres-sion and expansion are not adiabatic. The exchangers arenot ideal since the pressure drops in the engine and the

losses of heat in the exchangers exist. To accurately predictpower and efficiency requires an understanding of theprinciple parasitic loss mechanisms.

2.1. Energy dissipation by pressure drops in heat exchangers

d _QPCh

Pressure drops due to friction and to area changes inheat exchangers is given by [10]

Dp ¼ �2f rmGV

Ad2r, (1)

where G is working gas mass flow (kgm�2 s�1), d is thehydraulic diameter, r is gas density (kgm�3), V is volume(m3) and fr is the Reynolds friction factor.The internal heat generation which occurs when the gas

is forced to flow against the frictional drag force, is givenby [10]:

d _QPch ¼ �Dp _m

r, (2)

_m is the mass flow rate (kg s�1).The total heat generated by pressure drop in the different

exchangers is

d _QPchT ¼ d _QPchf þ d _QPchr1 þ d _QPchr2 þ d _QPchh. (3)

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Fig. 1. (a) Schematic diagram of solar-powered Stirling engine. (b) Temperature distribution.

I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1913

2.2. Energy lost by the internal conduction d _QPcd

Energy lost due to the internal thermal conductivitybetween the hot parts and the cold parts of the enginethrough the exchangers are taken into account. Theselosses are directly proportional to the temperature differ-ence at the ends of the exchanger; they are given for thedifferent exchangers [11]:

d _QPcdr ¼ kcdrAr

LrðT r�h � T f�rÞ, (4)

d _QPcdf ¼ kcdfAf

LfðT f�r � T c�f Þ, (5)

d _QPcdh ¼ kcdhAh

LhðTh�d � T r�hÞ, (6)

kcd (Wm�1K�1) is the material thermal conductivity; A isthe effective area for conduction.So the total conduction loss is:

d _QPcdT ¼ d _QPcdr þ d _QPcdf þ d _QPcdh. (7)

2.3. Energy lost by external conduction d _QPext

Energy lost by external conduction is considered in theregenerator which is not adiabatic. These losses arespecified by the regenerator adiabatic coefficient, ep1,

ARTICLE IN PRESSI. Tlili et al. / Renewable Energy 33 (2008) 1911–19211914

definite as the report between the heat given up in theregenerator by the working gas at its passage towards thecompression space and the heat received in the regeneratorby the working gas at its passage towards the expansionspace [10]. So the energy stored by the regenerator at thetime of the passage of gas from the expansion space to thecompression space is not completely restored with this gasat the time of its return.

For the ideal case of the regenerator perfected insulation,e ¼ 1.

The energy lost by external conduction is

d _QPext ¼ ð1� �Þðd _Qr1 þ d _Qr2Þ. (8)

The effectiveness of the regenerator e is given starting fromthe equation below [8]

� ¼NTU

1þNTU, (9)

NTU is the number of heat transfer unit:

NTU ¼hAwg

Cp _m, (10)

where h is the overall heat transfer coefficient (hot stream/matrix/cold stream), Awg refers to the wall/gas, or ‘‘wetted’’area of the heat exchanger surface, Cp the specific heatcapacity at constant pressure, and _m (kg s�1) the mass flowrate through the regenerator.

2.4. Energy lost by Shuttle effect d _QPshtl

Shuttling the displacer between hot and cold spaceswithin a machine introduces another mechanism fortransferring heat from a hot to a cold space. Thus animportant thermal effect appears in Stirling engines called‘Shuttle heat transfer’ having the effect of increasing theapparent thermal conductance loss. The displacer absorbsa quantity of heat from the hot source and restores it to thecold source. This loss of energy is given by [11]:

d _QPshtl ¼0:4Z2kpisDd

JLdðTd � TcÞ, (11)

where J is the annular gap between displacer and cylinder(m), kpis is the piston thermal conductivity (Wm�1K�1),Dd is the displacer diameter (m), Ld is the displacer length(m), Z is the displacer stroke (m), Td and Tc are,respectively, the temperature in the expansion space andin the compression space (K).

3. Mathematical background

There are many different ways to degrade the powerproduced by an ideal machine and to accurately predictpower and efficiency requires an understanding of thedesign compartments.

Mathematical model takes into consideration differentlosses and pressure drop in heat exchangers.

Heat transfer and flow friction in the heat exchangers,i.e. the heater, the cooler and the regenerator, are evaluatedusing empirical equations under steady flow condition.No leakage is allowed either through the appendix gap

or through the seals of the connecting rods.The temperature distribution in the various engine

compartments is illustrated in Fig. 1(b).The gas temperature in the various engine compartments

is variable.The cooler and the heater walls are maintained

isothermally at temperatures Tpaf and Tpah.The pressure distribution is shown in Fig. 2.The gas temperature in the different compartments is

calculated according to the perfect gas law:

Tc ¼PcV c

Rmc, (12)

T f ¼PfV f

Rmf, (13)

Th ¼PhVh

Rmh, (14)

Td ¼PdVd

Rmd. (15)

The regenerator is divided into two cells r1 and r2, eachcell is been associated with its respective mixed mean gastemperature Tr1 and Tr2 expressed as follows:

T r1 ¼Pr1V r1

Rmr1, (16)

T r2 ¼Pr2V r2

Rmr2. (17)

An extrapolated linear curve is drawn through tempera-ture values Tr1 and Tr2 defining the regenerator interfacetemperature Tr–f , Tr–r and Tr–h, as follows [12]:

T r�f ¼3T r1 � T r2

2, (18)

T r�r ¼T r1 þ T r2

2, (19)

T r�h ¼3T r2 � T r1

2. (20)

According to the flow direction of the fluid, the interface’stemperatures: Tc–f , Tf–r , Tr–h and Th–d are defined asfollows [13]:

if _mc�f40; then T c�f ¼ T c; otherwise T c�f ¼ T f ,

if _mf�r40; then T f�r ¼ T f ; otherwise T f�r ¼ T r�f ,

if _mr�h40; then T r�h ¼ T r�h; otherwise T r�h ¼ Th,

if _mh�d40; then Th�d ¼ Th; otherwise Th�d ¼ Td,

ARTICLE IN PRESS

Fig. 2. Pressure distribution.

Fig. 3. Generalised cell.

I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1915

where Tc–f is the temperature of the interface between thecompression space and the cooler, Tf–r is the temperatureof the interface between the cooler and the regenerator,Tr�h is the temperature of the interface between theregenerator and the heater, Th�d is the temperature ofthe interface between the heater and the expansion space.

The matrix temperatures are so given by

dTpar1

dt¼ �

dQr1

Cpr dt, (21)

dTpar2

dt¼ �

dQr2

Cpr dt, (22)

where Cpr is the heat capacity of each cell matrix (JK�1),Qr1 is the quantity of heat exchanged to the regenerator r1(j), Qr2 is the quantity of heat exchanged to the regeneratorr2 (j), TPar1 is the matrix temperature in the regenerator r1(K) and TPar2 is the matrix temperature in the regeneratorr2 (K).

By taking into account the conduction loss in theexchangers and the regenerator effectiveness, the powerexchanged in the different exchangers is written

d _Qf ¼ hfApaf ðTpaf � T f Þ � d _QPcdf , (23)

d _Qr2 ¼ Ehr2Apar2ðTpar2 � T r2Þ �d _QPcdr2

2, (24)

d _Qr1 ¼ Ehr1Apar1ðTpar1 � T r1Þ �d _QPcdr1

2, (25)

d _Qh ¼ hhApahðTpah � ThÞ � d _QPcdh, (26)

where d _QPcdh is the conduction loss in the cooler (W),d _QPcdr1 is the conduction loss in the regenerator r1 (W),d _QPcdr2 is the conduction loss in the regenerator r2 (W) andd _QPcdh is the conduction loss in the heater (W).

The heat transfer coefficient of exchanges hf, hr1, hr2 andhh is only available empirically [14].

The total exchanged heat is

d _Q ¼ d _Qf þ d _Qr1 þ d _Qr2 þ d _Qh � d _QPshtl. (27)

The work given by the cycle is

dW

dt¼ Pc

dV c

dtþ Pd

dVd

dt. (28)

The thermal efficiency given by the cycle is:

Z ¼W

Qh

. (29)

The total engine volume is: VT ¼ V c þ V f þ V r1 þ V r2þ

Vh þ Vd.The other variables of the dynamic model are given by

energy and mass conservation equation, applied to ageneralised cell as follows (Fig. 3):

�

Energy conservation equation :

d _Qþ CpTE _mE � CpTS _mS ¼ PdV

dtþ Cv

dðmTÞ

dt. (30)

Since there is a variable pressure distribution throughoutthe engine, we have arbitrarily chosen the compressionspace pressure Pc as the baseline pressure. Thus, at eachincrement of the solution, Pc will be evaluated from therelevant differential equation and the pressure distributionis determined with respect to Pc. Thus it can be obtained

ARTICLE IN PRESSI. Tlili et al. / Renewable Energy 33 (2008) 1911–19211916

from the following expression:

Pf ¼ Pc þDPf

2, (31)

Pr1 ¼ Pf þðDPf þ DPr1Þ

2, (32)

Pr2 ¼ Pr1 þðDPr1 þ DPr2Þ

2, (33)

Ph ¼ Pr2 þðDPr1 þ DPhÞ

2, (34)

Pd ¼ Ph þDPh

2. (35)

Applying energy conservation equation to the differentengine cells, we obtain:

�CpTc�f _mcS ¼1

RCpPc

dV c

dtþ CvV c

dPc

dt

� �, (36)

d _Qf � d _Qpchf þ CpT c�f _mfE � CpT f�r _mfS ¼CvV f

R

dPc

dt,

(37)

d _Qr1 � d _QPchr1 þ CpT f�r _mr1E � CpT r�r _mr1S ¼CvV r1

R

dPc

dt,

(38)

d _Qr2 � d _QPchr2 þ CpT r�r _mr2E � CpT r�h _mr2S ¼CvV r2

R

dPc

dt,

(39)

d _Qh � d _QPchh þ CpT r�h _mhE � CpTh�e _mhS ¼CvVh

R

dPc

dt,

(40)

CpTh�d _md � d _QPshtl ¼1

RCpPd

dVd

dtþ CvVd

dPc

dt

� �. (41)

Summing Eqs. (36)–(41) we obtain the pressure variation:

dPc

dt¼

1

CvVTRðd _Q� d _QPchTÞ � Cp

dW

dt

� �. (42)

�

Mass conservation equation:

M ¼ md þmc þmf þmr þmh. (43)

The mass flow in the different engine compartments isgiven by the energy conservation Eqs. (36)–(41):

_mcS ¼ �1

RT c�fPdV c

dtþ V c

dPc

gdt

� �, (44)

_mfS ¼1

CpT f�rd _Qf � d _QPchf þ CpTc�f _mfE �

CvV f

R

dPc

dt

� �,

(45)

_mr1S ¼1

CpT r�rd _Qr1 � d _QPchr1 þ CpT f�r _mr1E �

CvVr1

R

dPc

dt

� �,

(46)

_mr2S ¼1

CpT r�hd _Qr2 � d _QPchr2 þ CpT r�r _mr2E �

CvV r2

R

dPc

dt

� �,

(47)

_mhS ¼1

CpTh�dd _Qh � d _QPchh þ CpT r�h _mhE

dmh

dtE�

CvVh

R

dPc

dt

� �,

(48)

where: _mcS ¼ _mfE; _mfS ¼ _mr1E; _mr1S ¼ _mr2E; _mr2S ¼ _mhE

and _mhS ¼ _mdE.

3.1. Solution method

The systems of differential equations are written asfollows:

dY ¼ F ðt; yÞ,

Y ðt0Þ ¼ Y 0,

Y is a vector representing the unknown of each system,Y(t0)=Y0 is the initial condition.These systems of equations are solved by the classical

fourth-order Runge–Kutta method, cycle after cycle untilsteady.

4. Design specification and concept

4.1. Engine specification

The engine parameters should be optimised [15] to avoidlosses and to obtain high thermal efficiency for all theengine components especially heat exchangers. While themain target of the engine is to produce sufficient power torun a connecting application, there are conditions whichpose critical constraints on the design, the working fluid ishydrogen and the temperature difference between theheater and the cooler is about 300 1C only.The engine presented in Fig. 4 uses a conventional crank

mechanism driving two pistons by means of yoke linkage.The major feature of this is that there is almost no lateralmovement of the connecting rods resulting in very smallside forces on the pistons. With the lack of lateralmovement of the connecting rods, there are relatively largeunbalanced lateral forces due to the crankshaft counter-weight. Ross has a patented gear mechanism whichbalances the lateral forces by splitting and counter-rotatingthe counterweight

4.2. Design concept

The yoke drive mechanism does not produce sinusoidalvolume variations and the exact piston displacementfunctions are extremely complex. The volume variations

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Table 1

Volumes variations

Geometrical parameters b1 ¼ sin j ¼ r cos y

I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1917

are derived from geometric considerations in Fig. 5 andTable 1.

The main Design concepts are listed in Table 2.

by ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib21 � ðr cos yÞ2

qX ¼ r sin yþ by

Displacements Y c ¼ r½sin y� cos yðb2=b1Þ� þ by

Y e ¼ r½sin yþ cos yðb2=b1Þ� þ by

Volume variations V c ¼ Vmc þ APðYmax � Y cÞ

V e ¼ Vme þ AdðYmax � Y yÞ

dV c

dy¼ Apr cos yþ sin y

b2

b1

� �þ

r sin y cos yby

� �

dV b� �

r sin y cos y� �

5. Design analyses

5.1. Relationship for engine power, swept volumes and dead

volumes

The purpose of this simulation is to estimate the mainvolumes of the engine spaces in terms of swept volumes and

Fig. 5. Geometric derivation of the Ross Yoke drive equation.

Fig. 4. The Ross Yoke drive engine—schematic cross section view.

e

dy¼ Adr cos y� sin y 2

b1þ

by

Table 2

Concepts and target performance

Parameters Values/type

Engine type Alpha

Working fluid Hydrogen

Crank length r ¼ 7.6mm

Yoke crank length b1 ¼ 29mm

Piston length b2 ¼ 29mm

Displacement extremities Ymin ¼ 17.75mm

Ymax ¼ 39.28mm

Mean phase angle advance a ¼ 901

Mass of gas in engine M ¼ 0.35 g

Hot space temperature Th ¼ 590K

Cold space temperature Tk ¼ 290K

Frequency Freq ¼ 41.72Hz

ARTICLE IN PRESS

Fig. 7. Relationship between swept volume and engine power (dead

volume 370 cm3).

I. Tlili et al. / Renewable Energy 33 (2008) 1911–19211918

dead volumes using Dynamic model with losses sincethese factors are essential in estimating the preliminaryconfiguration of the engine and will influence the sub-sequent optimisation process. Since it has been decided toadopt the successive alpha-type Ross Yoke configuration,the compression swept volume Vc should be equal to theexpansion swept volume Vd, and thus the swept volumeratio k ¼ Vc/Vd.

In addition, at this stage, it is assumed that the meanpressure of the engine during operation is of 8.7 bar, whichis the kind of pressure which normally occurs before theengine start-up. It is obvious from dynamic modelequations that the net cycle power and the thermal loadon the heat exchangers are direct linear functions of theengine speed (Operating rotation), the maximum pressureof the working fluid and the size of the engine, which isexpressed in term of the swept volume [16]. However, thedirect effects of the dead volume and swept volume to theengine power should be detailed. Figs. 6 and 7 illustratesthe variation of the power as a function of the sweptvolume, which was calculated on the dead volumes of 535and 370 cm3 under the fixed temperature differenceof 300 1C. It is shown that the power increases when theswept volume increases until an optimal value. Also, it isnoticeable that the power increases with the increase inspeed. These two remarks imply that we have an optimalvalue of swept volume for maximum engine power forseveral speeds. By comparing the two graphs in Figs. 6 and7, based on the same swept volume, it can be said that thedecrease in dead volume will lead to an increase in enginepower.

To illustrate the effect of the dead volume clearly, thevariation of the engine power as a function of dead volumeis calculated and the results are as shown in Figs. 8 and 9for the operating frequency of 75 and 35Hz.

Fig. 6. Relationship between swept volume and engine power (dead

volume 535 cm3).

Fig. 8. Relationship between dead volume and engine power (frequen-

cy ¼ 75Hz).

From Figs. 6–9, it can be seen that the increase in thedead volume produces an exponential drop in the netpower, which in turn decreases the maximum pressure.However, the calculation is performed under the assump-tion that the temperature difference is 300 1C, which can beobtained from the solar system with average concentration.

5.2. Relationship for heater and cooler parameters

An important factor in heat exchanger design is volume.Cooler and heater volumes contribute to large portions ofdead volume. Previous studies showed that the deadvolumes, which includes those in the heat exchangers, is

ARTICLE IN PRESS

Fig. 9. Relationship between dead volume and engine power (frequency ¼

35Hz). Fig. 10. Relationship of heater tube diameters with the friction losses

(swept volume ¼ 75 cm3, tube length ¼ 0.45m, cooler volume 165 cm3).

Fig. 11. Relationship of heater tube diameters with the friction losses

(swept volume ¼ 75 cm3, tube length ¼ 0.45m, cooler volume 80 cm3).

I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1919

an essential factor in the Stirling engine design, where itshould be small as possible [17]. To demonstrate therelationships for the heaters, specific conditions of 75 cm3

swept volume and 0.45m tube length are used.After carrying out thermodynamic simulation for the

heater, the variation of its tube diameter can be derived asa function of friction losses for several values of enginespeeds as being depicted in Fig. 10 for the heater volume of165 cm3. Similarly, Fig. 11 shows the graphs for the heatervolume of 80 cm3. Both graphs indicate an inverseproportionality between tube diameter and friction loss inthe heater. The explanation of this variation is that the tubewith smaller diameter having the same length delivers thesame mass flux, thus generates a shorter entrance lengthand a thicker viscous boundary layer, which then leads to ahigher friction factor of the flow. For the cooler, by usingthe same values for swept volume and tube length, theequivalent graphs for the cooler volumes of 165 and 80 cm3

are shown in Figs. 12 and 13, which indicates a similarpattern to that of the heater.

In designing heat exchangers, an important considera-tion for the heat exchangers is to have an ability to supplyor reject the required amount of heat to or from the engine.In this aspect, one crucial factor is the heat transfer area,which will decide the amount of heat energy to betransported. Hence, in order to achieve a high effectivenessfor the heater and the cooler, larger transfer areas, and thuslarger volumes, are needed.

5.3. Relationship for regenerator parameters

The effect of pressure drop in the regenerator of a meantemperature differential Stirling engine to thermal effi-ciency is very important since it can decrease the overallefficiency of the engine [16,17]. To analyse this effect and its

implication to the efficiency of the engine, six types ofmatrices has been selected and is being subjected to variouspressure drops and engine speeds. The configurations forthese six matrices are given in Table 3 for a standard totalwire length of 5m. The porosity of each matrix isimportant since it will have a direct impact on theperformance of the regenerator, and can be determinedby its geometry, namely, wire diameter, density of the meshand the void volume. Any changes in the porosity will alsochange the regenerator effectiveness and the pressure drop,which eventually affects the engine efficiency. Therefore,the best matrix for the regenerator should possess bothhigh efficiency and low-pressure drop.

ARTICLE IN PRESS

Fig. 12. Relationship of cooler tube diameters with the friction losses

(swept volume ¼ 75 cm3, tube length ¼ 0.45m, heater volume 165 cm3).

Fig. 13. Relationship of cooler tube diameters with the friction losses

(swept volume ¼ 75 cm3, tube length ¼ 0.45m, heater volume 80 cm3).

Table 3

Geometrical properties of wire mesh for regenerator

Matrix Wire diameter (m) Porosity (c)

M1 0.0035 0.9122

M2 0.0050 0.8359

M3 0.0065 0.7508

M4 0.0070 0.7221

M5 0.0080 0.6655

M6 0.0090 0.6112

Fig. 14. Relationship between pressure drop and operating frequency.

I. Tlili et al. / Renewable Energy 33 (2008) 1911–19211920

Fig. 14 shows the relationship between the operatingfrequency and the pressure drop for these matrices. Thepressure drop is found to be proportional to the frequencysince an increase in frequency increases the mass fluxthrough the regenerator as well as the pressure magnitudeup to the same proportion for the same matrices. On theother hand, the decrease in mesh porosity leads to thehigher friction factor as well as increases the pressure drop.Hence, it can be said that M1 has a lowest pressure drop incomparison to the others at a same speed because itsporosity is the highest. In order to obtain a higher porosity,and thus the lower pressure drop, the meshes should bemade from small wire diameter and should be as coarse as

possible. However, the pressure drop in the regeneratoralone is not sufficient in deciding the best regeneratorwithout considering its heat transfer behaviour. ButTable 4 shows the relationship between the thermalefficiency, power of the engine and matrix type. The bestmatrix should compromise between high effectiveness andlow-pressure drop in order to obtain minimal losses in theregenerator, and in this case, M6 with the porosity of0.6112 and wire diameter 0.009m has been chosen for thedesign.The decrease in mesh porosity leads to the higher friction

factor as well as increases the pressure drop. Hence, it canbe said that M1 has a lowest pressure drop in comparisonto the others at a same frequency because its porosity is thehighest. In order to obtain a higher porosity, and thus thelower pressure drop, the meshes should be made from smallwire diameter and should be as coarse as possible.However, the pressure drop in the regenerator alone is

ARTICLE IN PRESS

Table 4

Effect of matrix on power and thermal efficiency

Matrix Power (W) Thermal efficiency (%)

M1 159.39 10.79

M2 226.42 22.41

M3 249.20 33.90

M4 252.50 37.32

M5 255.92 43.32

M6 256.77 48.11

I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1921

not sufficient in deciding the best regenerator withoutconsidering its heat transfer behaviour this is the case ofM6 In spite of the higher pressure drop we have betterpower and thermal efficiency because we have better heattransfer.

The regenerator effectiveness e can be manipulated byvarying wire diameter and wire length, which in turnchanges the ‘‘wetted’’ surface area. It can be represented inform of the relationship between the porosity or thenumber of transfer units (NTU) and the thermal heatingefficiency of the engine. If the wetted surface area islarge, the resulting porosity should be low, and thisprovides the air or the work fluid with a large contactingsurface to achieve a high rate of heat transfer. Hence, theNTU, and thus e, are increased when the surface areaincreases. The effect of e on the thermal efficiency of theengine is that it represents the ability to reject the heat tothe working gas when the gas exits through the heater andthe ability to absorb the heat when the gas exits throughthe cooler.

6. Conclusion

In this paper, a number of technical considerations indesigning a mean temperature differential Stirling enginehave been proposed. These considerations have beenestablished through the use of the dynamic model withlosses energy and pressure drop in heat exchangers. As aresult, the optimal configuration for the design can besummarised as follow.

�

This studies show clearly that, for given value ofdifference temperature, operating frequency and deadvolume there is a definite optimal value of sweptvolume ratio at which the power is a maximum. Inthis paper, the optimal swept volume is 75 cm3 forfrequency ¼ 75Hz. � Upon optimisation, operating frequency has to be

limited between 35 and 75Hz at a temperaturedifference of 300 1C, where the best value is 75Hz.

� For the regenerator, its porosity plays a significant role

in controlling pressure drop of the regenerator, whichcan be manipulated by varying wire diameter and

length. For this engine, the selected parameters arethe wire diameter of 3.5mm with a total length of 5mand a porosity of 0.9122 to have low pressure drop butin our case M6 give the best thermal efficiency of theengine.

� The heat exchanger volumes should be evaluated by

considering both the pressure drop and the thermalefficiency of the engine. In our case the optimal heatexchanger volume has been found to be 165 cm3 for boththe cooler and the tube dimension is 0.011m in diameterand 0.450m in length.

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