1. ARTICLE IN PRESS
Renewable Energy 33 (2008) 1911–1921
www.elsevier.com/locate/renene
Analysis and design consideration of mean temperature differential
Stirling engine for solar application
Iskander TliliÃ, Youssef Timoumi, Sassi Ben Nasrallah
`mes Thermiques et Energe
Laboratoire d’Etude des Syste ´tiques Ecole Nationale d’Inge
´nieurs 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 75 Hz with the power is 250 W 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 units have been in operation for many years. On the other
hand, low temperature Stirling engines are not as successful
The harmony between environmental protection and as their high temperature counterparts. However, the
economic growth has become a worldwide concern; there is former have gained popularity in the last few decades due
an urgent need to effectively reuse solar energy, this source to this potential to tap a variety of low concentration
of energy is one of the more attractive renewable energy energy sources available, such as solar. The increasing
that can be used as an input energy source for heat engines. interest in Stirling engines is largely due to the fact the
In fact, any heat energy source can be used with the Stirling engine is more environmentally friendly than the widely
engine. The solar radiation can be focused onto the heater used internal combustion engine, and also to its non-
of Stirling engine as shown in Fig. 1(a), thereby creating a explosive nature in converting energy into mechanical form
solar-powered prime mover. The direct conversion of solar and thus leading to silent and cleaner operation, which are
power into mechanical power reduces both the cost and essential for special applications, such as military opera-
complexity of the prime mover. In theory, the principal tions and medical uses.
advantages of Stirling engines are their use of an external The systems with very strong concentration [2] call upon
heat source and their high efficiency. Stirling engines are an advanced and heavy technology, therefore are very
able to use solar energy that is a cheap source of energy. expensive as they present, on the energy point of view, a
Studies about high temperature Stirling engines have been limited interest. On the other hand, the systems without
extensively reported in the literature [1] and commercial concentration are not economically viable. The best systems
is with average concentration, leading to levels of temperature
ÃCorresponding author. Tel.: +216 98 61 97 04; fax: +216 73 50 05 14. about 250–450 1C, but very few work seem to be devoted to
E-mail address: Iskander.Tlili@enim.rnu.tn (I. Tlili). the installations with average concentration. The company
0960-1481/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.renene.2007.09.024

2. ARTICLE IN PRESS
1912 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921
Nomenclature Subscripts
A area, m2 c compression space
Cp specific heat at constant pressure, J kgÀ1 KÀ1 ch load
Cpr heat capacity of each cell matrix, J KÀ1 d expansion space
Cv specific heat at constant volume, J kgÀ1 KÀ1 E entered
d hydraulic diameter, m ext outside
D diameter, m f cooler
dm wire diameter, m h heater
fr friction factor moy mean
Freq operating frequency, Hz P loss
h convection heat transfer coefficient, Pa wall
J mÀ2 sÀ1 KÀ1 pis piston
J annular gap between displacer and cylinder, m r regenerator
k thermal conductivity, W mÀ1 KÀ1 r1 regenerator cell 1
L length, m r2 regenerator cell 2
M mass of working gas in the engine, kg S left
m_ mass flow rate, kg sÀ1
m mass of gas in different component, kg Greek symbols
NTU number of heat transfer unit
P pressure, Pa y crank angle, rad
Q heat, J e effectiveness
Q_ power, W m Working GAS dynamic viscosity, kg mÀ1 sÀ1
R gas constant, J kg KÀ1 r density, kg mÀ3
T temperature, K o angular frequency, rad sÀ1
V volume, m3 c mesh porosity
W work, J
BSR Solar Technologies GmbH, which developed the losses of heat in the exchangers exist. To accurately predict
SUNPULSE, also works on a system intended to produce power and efficiency requires an understanding of the
electricity starting from solar energy fairly concentrated, principle parasitic loss mechanisms.
which leads to levels of temperature about 450 1C.
Several analyses and simulation methods of the engine
have been established [3], as well as the procedures for 2.1. Energy dissipation by pressure drops in heat exchangers
optimal design [4]. Most of the engines are fuel-fired and _
d QPCh
operate at high temperature, which highlights the need for
careful material selection as well as good cooling system. Pressure drops due to friction and to area changes in
For silent, light and portable equipment for leisure and heat exchangers is given by [10]
domestic uses, low power engines may be more appropriate. 2f r mGV
Nevertheless, research in Stirling engine technology has been Dp ¼ À , (1)
heavily masked by extensive and successful development of Ad 2 r
internal combustion engines, which have made Stirling where G is working gas mass flow (kg mÀ2 sÀ1), d is the
engines less competitive. Hence, in order to design a low hydraulic diameter, r is gas density (kg mÀ3), V is volume
power engine using solar, new design specifications and (m3) and fr is the Reynolds friction factor.
optimisation criteria must be established [5–9]. This paper The internal heat generation which occurs when the gas
presents design considerations which may be taken to is forced to flow against the frictional drag force, is given
develop a solar Stirling engine with average concentration by [10]:
operating on mean temperature difference of 300 1C.
_ Dpm_
dQPch ¼ À , (2)
r
2. Losses in a Stirling engine
m is the mass flow rate (kg sÀ1).
_
The energy losses in a Stirling engine are due to the The total heat generated by pressure drop in the different
thermodynamic and the mechanical processes. Compres- exchangers is
sion and expansion are not adiabatic. The exchangers are
_ _ _ _ _
dQPchT ¼ dQPchf þ dQPchr1 þ dQPchr2 þ dQPchh . (3)
not ideal since the pressure drops in the engine and the

3. ARTICLE IN PRESS
I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1913
Fig. 1. (a) Schematic diagram of solar-powered Stirling engine. (b) Temperature distribution.
_
2.2. Energy lost by the internal conduction dQPcd Ah
_
dQPcdh ¼ kcdh ðT hÀd À T rÀh Þ, (6)
Lh
Energy lost due to the internal thermal conductivity
kcd (W mÀ1 KÀ1) is the material thermal conductivity; A is
between the hot parts and the cold parts of the engine
the effective area for conduction.
through the exchangers are taken into account. These
So the total conduction loss is:
losses are directly proportional to the temperature differ-
ence at the ends of the exchanger; they are given for the _ _ _ _
dQPcdT ¼ dQPcdr þ dQPcdf þ dQPcdh . (7)
different exchangers [11]:
_ Ar _
dQPcdr ¼ kcdr ðT rÀh À T fÀr Þ, (4) 2.3. Energy lost by external conduction dQPext
Lr
Energy lost by external conduction is considered in the
_ Af regenerator which is not adiabatic. These losses are
dQPcdf ¼ kcdf ðT fÀr À T cÀf Þ, (5)
Lf specified by the regenerator adiabatic coefficient, ep1,

4. ARTICLE IN PRESS
1914 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921
definite as the report between the heat given up in the Heat transfer and flow friction in the heat exchangers,
regenerator by the working gas at its passage towards the i.e. the heater, the cooler and the regenerator, are evaluated
compression space and the heat received in the regenerator using empirical equations under steady flow condition.
by the working gas at its passage towards the expansion No leakage is allowed either through the appendix gap
space [10]. So the energy stored by the regenerator at the or through the seals of the connecting rods.
time of the passage of gas from the expansion space to the The temperature distribution in the various engine
compression space is not completely restored with this gas compartments is illustrated in Fig. 1(b).
at the time of its return. The gas temperature in the various engine compartments
For the ideal case of the regenerator perfected insulation, is variable.
e ¼ 1. The cooler and the heater walls are maintained
The energy lost by external conduction is isothermally at temperatures Tpaf and Tpah.
_ _ _ The pressure distribution is shown in Fig. 2.
dQPext ¼ ð1 À ÞðdQr1 þ dQr2 Þ. (8) The gas temperature in the different compartments is
The effectiveness of the regenerator e is given starting from calculated according to the perfect gas law:
the equation below [8] Pc V c
Tc ¼ , (12)
NTU Rmc
¼ , (9)
1 þ NTU
Pf V f
NTU is the number of heat transfer unit: Tf ¼ , (13)
Rmf
hAwg
NTU ¼ , (10) Ph V h
Cpm_ Th ¼ , (14)
Rmh
where h is the overall heat transfer coefficient (hot stream/
matrix/cold stream), Awg refers to the wall/gas, or ‘‘wetted’’ Pd V d
Td ¼ . (15)
area of the heat exchanger surface, Cp the specific heat Rmd
capacity at constant pressure, and m (kg sÀ1) the mass flow
_ The regenerator is divided into two cells r1 and r2, each
rate through the regenerator. cell is been associated with its respective mixed mean gas
temperature Tr1 and Tr2 expressed as follows:
_
2.4. Energy lost by Shuttle effect dQPshtl Pr1 V r1
T r1 ¼ , (16)
Rmr1
Shuttling the displacer between hot and cold spaces
within a machine introduces another mechanism for Pr2 V r2
transferring heat from a hot to a cold space. Thus an T r2 ¼ . (17)
Rmr2
important thermal effect appears in Stirling engines called
An extrapolated linear curve is drawn through tempera-
‘Shuttle heat transfer’ having the effect of increasing the
ture values Tr1 and Tr2 defining the regenerator interface
apparent thermal conductance loss. The displacer absorbs
temperature Tr–f , Tr–r and Tr–h, as follows [12]:
a quantity of heat from the hot source and restores it to the
cold source. This loss of energy is given by [11]: 3T r1 À T r2
T rÀf ¼ , (18)
2 2
_ 0:4Z kpis Dd
dQPshtl ¼ ðT d À T c Þ, (11)
JLd T r1 þ T r2
T rÀr ¼ , (19)
where J is the annular gap between displacer and cylinder 2
(m), kpis is the piston thermal conductivity (W mÀ1 KÀ1), 3T r2 À T r1
Dd is the displacer diameter (m), Ld is the displacer length T rÀh ¼ . (20)
2
(m), Z is the displacer stroke (m), Td and Tc are,
respectively, the temperature in the expansion space and According to the flow direction of the fluid, the interface’s
in the compression space (K). temperatures: Tc–f , Tf–r , Tr–h and Th–d are defined as
follows [13]:
3. Mathematical background if _
mcÀf 40; then T cÀf ¼ T c ; otherwise T cÀf ¼ T f ,
There are many different ways to degrade the power if _
mfÀr 40; then T fÀr ¼ T f ; otherwise T fÀr ¼ T rÀf ,
produced by an ideal machine and to accurately predict
power and efficiency requires an understanding of the
if _
mrÀh 40; then T rÀh ¼ T rÀh ; otherwise T rÀh ¼ T h ,
design compartments.
Mathematical model takes into consideration different
losses and pressure drop in heat exchangers. if _
mhÀd 40; then T hÀd ¼ T h ; otherwise T hÀd ¼ T d ,

5. ARTICLE IN PRESS
I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1915
Fig. 2. Pressure distribution.
where Tc–f is the temperature of the interface between the
compression space and the cooler, Tf–r is the temperature
of the interface between the cooler and the regenerator,
TrÀh is the temperature of the interface between the
regenerator and the heater, ThÀd is the temperature of
the interface between the heater and the expansion space.
The matrix temperatures are so given by
dT par1 dQr1
¼À , (21)
dt C pr dt
dT par2 dQr2
¼À , (22) Fig. 3. Generalised cell.
dt C pr dt
where Cpr is the heat capacity of each cell matrix (J KÀ1),
Qr1 is the quantity of heat exchanged to the regenerator r1 The work given by the cycle is
(j), Qr2 is the quantity of heat exchanged to the regenerator
r2 (j), TPar1 is the matrix temperature in the regenerator r1 dW dV c dV d
¼ Pc þ Pd . (28)
(K) and TPar2 is the matrix temperature in the regenerator dt dt dt
r2 (K). The thermal efficiency given by the cycle is:
By taking into account the conduction loss in the
exchangers and the regenerator effectiveness, the power W
Z¼ . (29)
exchanged in the different exchangers is written Qh
_ _
d Qf ¼ hf Apaf ðT paf À T f Þ À dQPcdf , (23) 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.
_
dQPcdr2 The other variables of the dynamic model are given by
_
dQr2 ¼ Ehr2 Apar2 ðT par2 À T r2 Þ À , (24)
2 energy and mass conservation equation, applied to a
generalised cell as follows (Fig. 3):
_
dQPcdr1
_
dQr1 ¼ Ehr1 Apar1 ðT par1 À T r1 Þ À , (25)
2 Energy conservation equation :
_ _
dQh ¼ hh Apah ðT pah À T h Þ À dQPcdh , (26)
_ _ dV dðmTÞ
where dQPcdh is the conduction loss in the cooler (W), _ _
dQ þ C p T E mE À C p T S mS ¼ P þ Cv . (30)
_ dt dt
dQPcdr1 is the conduction loss in the regenerator r1 (W),
_
dQPcdr2 is the conduction loss in the regenerator r2 (W) and
_
dQPcdh is the conduction loss in the heater (W). Since there is a variable pressure distribution throughout
The heat transfer coefficient of exchanges hf, hr1, hr2 and the engine, we have arbitrarily chosen the compression
hh is only available empirically [14]. space pressure Pc as the baseline pressure. Thus, at each
The total exchanged heat is increment of the solution, Pc will be evaluated from the
relevant differential equation and the pressure distribution
_ _ _ _ _ _
dQ ¼ dQf þ d Qr1 þ dQr2 þ dQh À dQPshtl . (27) is determined with respect to Pc. Thus it can be obtained

6. ARTICLE IN PRESS
1916 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921
from the following expression: 1 _ _ C v V r1 dPc
_
mr1S ¼ _
dQr1 À dQPchr1 þ C p T fÀr mr1E À ,
C p T rÀr R dt
DPf
Pf ¼ Pc þ , (31) (46)
2
ðDPf þ DPr1 Þ _
mr2S ¼
1 _ r2 À dQPchr2 þ C p T rÀr mr2E À C v V r2 dPc ,
dQ _ _
Pr1 ¼ Pf þ , (32) C p T rÀh R dt
2
(47)
ðDPr1 þ DPr2 Þ
Pr2 ¼ Pr1 þ , (33)
2 1 _ À dQ
_ dmh C v V h dPc
_
mhS ¼ dQh _
Pchh þ C p T rÀh mhE À ,
C p T hÀd dtE R dt
ðDPr1 þ DPh Þ
Ph ¼ Pr2 þ , (34) (48)
2
_ _ _ _ _ _ _ _
where: mcS ¼ mfE ; mfS ¼ mr1E ; mr1S ¼ mr2E ; mr2S ¼ mhE
DPh _ _
and mhS ¼ mdE .
Pd ¼ Ph þ . (35)
2
Applying energy conservation equation to the different 3.1. Solution method
engine cells, we obtain:
The systems of differential equations are written as
1 dV c dPc
_
ÀC p T cÀf mcS ¼ C p Pc þ CvV c , (36) follows:
R dt dt
dY ¼ F ðt; yÞ,
_ _ C v V f dPc
dQf À dQpchf _ _
þ C p T cÀf mfE À C p T fÀr mfS ¼ , Y ðt0 Þ ¼ Y 0 ,
R dt
(37) Y is a vector representing the unknown of each system,
Y(t0)=Y0 is the initial condition.
_ _ C v V r1 dPc These systems of equations are solved by the classical
_ _
dQr1 À dQPchr1 þ C p T fÀr mr1E À C p T rÀr mr1S ¼ ,
R dt fourth-order Runge–Kutta method, cycle after cycle until
(38) steady.
_ _ C v V r2 dPc
_ _
dQr2 À dQPchr2 þ C p T rÀr mr2E À C p T rÀh mr2S ¼ , 4. Design specification and concept
R dt
(39) 4.1. Engine specification
_ _ C v V h dPc
_ _
dQh À dQPchh þ C p T rÀh mhE À C p T hÀe mhS ¼ , The engine parameters should be optimised [15] to avoid
R dt losses and to obtain high thermal efficiency for all the
(40) engine components especially heat exchangers. While the
main target of the engine is to produce sufficient power to
_ 1 dV d dPc
_
C p T hÀd md À dQPshtl ¼ C p Pd þ CvV d . (41) run a connecting application, there are conditions which
R dt dt pose critical constraints on the design, the working fluid is
Summing Eqs. (36)–(41) we obtain the pressure variation: hydrogen and the temperature difference between the
heater and the cooler is about 300 1C only.
dPc 1 _ _ dW The engine presented in Fig. 4 uses a conventional crank
¼ RðdQ À dQPchT Þ À C p . (42)
dt CvV T dt mechanism driving two pistons by means of yoke linkage.
The major feature of this is that there is almost no lateral
Mass conservation equation: movement of the connecting rods resulting in very small
side forces on the pistons. With the lack of lateral
movement of the connecting rods, there are relatively large
M ¼ md þ mc þ mf þ mr þ mh . (43)
unbalanced lateral forces due to the crankshaft counter-
The mass flow in the different engine compartments is weight. Ross has a patented gear mechanism which
given by the energy conservation Eqs. (36)–(41): balances the lateral forces by splitting and counter-rotating
the counterweight
1 dV c dPc
_
mcS ¼ À P þ Vc , (44)
RT cÀf dt gdt
4.2. Design concept
1 _ _ C v V f dPc
_
mfS ¼ _
dQf À dQPchf þ C p T cÀf mfE À , The yoke drive mechanism does not produce sinusoidal
C p T fÀr R dt
volume variations and the exact piston displacement
(45) functions are extremely complex. The volume variations

7. ARTICLE IN PRESS
I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1917
are derived from geometric considerations in Fig. 5 and Table 1
Table 1. Volumes variations
The main Design concepts are listed in Table 2. Geometrical parameters b1 ¼ sin j ¼ r cos y
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
by ¼ b2 À ðr cos yÞ2
1
X ¼ r sin y þ by
5. Design analyses
Displacements Y c ¼ r½sin y À cos yðb2 =b1 ÞŠ þ by
Y e ¼ r½sin y þ cos yðb2 =b1 ÞŠ þ by
5.1. Relationship for engine power, swept volumes and dead
volumes Volume variations V c ¼ V mc þ AP ðY max À Y c Þ
V e ¼ V me þ Ad ðY max À Y y Þ
dV c b2 r sin y cos y
¼ Ap r cos y þ sin y þ
The purpose of this simulation is to estimate the main dy b1 by
volumes of the engine spaces in terms of swept volumes and dV e
b2 r sin y cos y
¼ Ad r cos y À sin y þ
dy b1 by
Table 2
Concepts and target performance
Parameters Values/type
Engine type Alpha
Working fluid Hydrogen
Crank length r ¼ 7.6 mm
Yoke crank length b1 ¼ 29 mm
Piston length b2 ¼ 29 mm
Displacement extremities Ymin ¼ 17.75 mm
Ymax ¼ 39.28 mm
Mean phase angle advance a ¼ 901
Mass of gas in engine M ¼ 0.35 g
Hot space temperature Th ¼ 590 K
Cold space temperature Tk ¼ 290 K
Frequency Freq ¼ 41.72 Hz
Fig. 4. The Ross Yoke drive engine—schematic cross section view.
Fig. 5. Geometric derivation of the Ross Yoke drive equation.

8. ARTICLE IN PRESS
1918 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921
dead volumes using Dynamic model with losses since
these factors are essential in estimating the preliminary
configuration of the engine and will influence the sub-
sequent optimisation process. Since it has been decided to
adopt the successive alpha-type Ross Yoke configuration,
the compression swept volume Vc should be equal to the
expansion swept volume Vd, and thus the swept volume
ratio k ¼ Vc/Vd.
In addition, at this stage, it is assumed that the mean
pressure of the engine during operation is of 8.7 bar, which
is the kind of pressure which normally occurs before the
engine start-up. It is obvious from dynamic model
equations that the net cycle power and the thermal load
on the heat exchangers are direct linear functions of the
engine speed (Operating rotation), the maximum pressure
of the working fluid and the size of the engine, which is
expressed in term of the swept volume [16]. However, the
direct effects of the dead volume and swept volume to the
engine power should be detailed. Figs. 6 and 7 illustrates
Fig. 7. Relationship between swept volume and engine power (dead
the variation of the power as a function of the swept volume 370 cm3).
volume, which was calculated on the dead volumes of 535
and 370 cm3 under the fixed temperature difference
of 300 1C. It is shown that the power increases when the
swept volume increases until an optimal value. Also, it is
noticeable that the power increases with the increase in
speed. These two remarks imply that we have an optimal
value of swept volume for maximum engine power for
several speeds. By comparing the two graphs in Figs. 6 and
7, based on the same swept volume, it can be said that the
decrease in dead volume will lead to an increase in engine
power.
To illustrate the effect of the dead volume clearly, the
variation of the engine power as a function of dead volume
is calculated and the results are as shown in Figs. 8 and 9
for the operating frequency of 75 and 35 Hz.
Fig. 8. Relationship between dead volume and engine power (frequen-
cy ¼ 75 Hz).
From Figs. 6–9, it can be seen that the increase in the
dead volume produces an exponential drop in the net
power, 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 be
obtained 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 of
Fig. 6. Relationship between swept volume and engine power (dead dead volume. Previous studies showed that the dead
volume 535 cm3). volumes, which includes those in the heat exchangers, is

9. ARTICLE IN PRESS
I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1919
Fig. 9. Relationship between dead volume and engine power (frequency ¼
35 Hz). Fig. 10. Relationship of heater tube diameters with the friction losses
(swept volume ¼ 75 cm3, tube length ¼ 0.45 m, cooler volume 165 cm3).
an essential factor in the Stirling engine design, where it
should be small as possible [17]. To demonstrate the
relationships for the heaters, specific conditions of 75 cm3
swept volume and 0.45 m tube length are used.
After carrying out thermodynamic simulation for the
heater, the variation of its tube diameter can be derived as
a function of friction losses for several values of engine
speeds as being depicted in Fig. 10 for the heater volume of
165 cm3. Similarly, Fig. 11 shows the graphs for the heater
volume of 80 cm3. Both graphs indicate an inverse
proportionality between tube diameter and friction loss in
the heater. The explanation of this variation is that the tube
with smaller diameter having the same length delivers the
same mass flux, thus generates a shorter entrance length
and a thicker viscous boundary layer, which then leads to a
higher friction factor of the flow. For the cooler, by using
the same values for swept volume and tube length, the
equivalent graphs for the cooler volumes of 165 and 80 cm3
are shown in Figs. 12 and 13, which indicates a similar
pattern to that of the heater. Fig. 11. Relationship of heater tube diameters with the friction losses
In designing heat exchangers, an important considera- (swept volume ¼ 75 cm3, tube length ¼ 0.45 m, cooler volume 80 cm3).
tion for the heat exchangers is to have an ability to supply
or reject the required amount of heat to or from the engine. implication to the efficiency of the engine, six types of
In this aspect, one crucial factor is the heat transfer area, matrices has been selected and is being subjected to various
which will decide the amount of heat energy to be pressure drops and engine speeds. The configurations for
transported. Hence, in order to achieve a high effectiveness these six matrices are given in Table 3 for a standard total
for the heater and the cooler, larger transfer areas, and thus wire length of 5 m. The porosity of each matrix is
larger volumes, are needed. important since it will have a direct impact on the
performance of the regenerator, and can be determined
5.3. Relationship for regenerator parameters by its geometry, namely, wire diameter, density of the mesh
and the void volume. Any changes in the porosity will also
The effect of pressure drop in the regenerator of a mean change the regenerator effectiveness and the pressure drop,
temperature differential Stirling engine to thermal effi- which eventually affects the engine efficiency. Therefore,
ciency is very important since it can decrease the overall the best matrix for the regenerator should possess both
efficiency of the engine [16,17]. To analyse this effect and its high efficiency and low-pressure drop.

10. ARTICLE IN PRESS
1920 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921
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. 12. Relationship of cooler tube diameters with the friction losses
(swept volume ¼ 75 cm3, tube length ¼ 0.45 m, heater volume 165 cm3).
Fig. 14. Relationship between pressure drop and operating frequency.
possible. However, the pressure drop in the regenerator
alone is not sufficient in deciding the best regenerator
Fig. 13. Relationship of cooler tube diameters with the friction losses without considering its heat transfer behaviour. But
(swept volume ¼ 75 cm3, tube length ¼ 0.45 m, heater volume 80 cm3). Table 4 shows the relationship between the thermal
efficiency, power of the engine and matrix type. The best
Fig. 14 shows the relationship between the operating matrix should compromise between high effectiveness and
frequency and the pressure drop for these matrices. The low-pressure drop in order to obtain minimal losses in the
pressure drop is found to be proportional to the frequency regenerator, and in this case, M6 with the porosity of
since an increase in frequency increases the mass flux 0.6112 and wire diameter 0.009 m has been chosen for the
through the regenerator as well as the pressure magnitude design.
up to the same proportion for the same matrices. On the The decrease in mesh porosity leads to the higher friction
other hand, the decrease in mesh porosity leads to the factor as well as increases the pressure drop. Hence, it can
higher friction factor as well as increases the pressure drop. be said that M1 has a lowest pressure drop in comparison
Hence, it can be said that M1 has a lowest pressure drop in to the others at a same frequency because its porosity is the
comparison to the others at a same speed because its highest. In order to obtain a higher porosity, and thus the
porosity is the highest. In order to obtain a higher porosity, lower pressure drop, the meshes should be made from small
and thus the lower pressure drop, the meshes should be wire diameter and should be as coarse as possible.
made from small wire diameter and should be as coarse as However, the pressure drop in the regenerator alone is

11. ARTICLE IN PRESS
I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1921
Table 4 length. For this engine, the selected parameters are
Effect of matrix on power and thermal efficiency the wire diameter of 3.5 mm with a total length of 5 m
Matrix Power (W) Thermal efficiency (%)
and a porosity of 0.9122 to have low pressure drop but
in our case M6 give the best thermal efficiency of the
M1 159.39 10.79 engine.
M2 226.42 22.41 The heat exchanger volumes should be evaluated by
M3 249.20 33.90
considering both the pressure drop and the thermal
M4 252.50 37.32
M5 255.92 43.32 efficiency of the engine. In our case the optimal heat
M6 256.77 48.11 exchanger volume has been found to be 165 cm3 for both
the cooler and the tube dimension is 0.011 m in diameter
and 0.450 m in length.
not sufficient in deciding the best regenerator without
considering its heat transfer behaviour this is the case of
M6 In spite of the higher pressure drop we have better
power and thermal efficiency because we have better heat References
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