Analysis and design consideration of mean temperature differential
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 2007Abstract This article presents a technical innovation, study of solar power system based on the Stirling dish (SD) technology and designconsiderations to be taken in designing of a mean temperature differential Stirling engine for solar application. The target power sourcewill be solar dish/Stirling with average concentration ratio, which will supply a constant source temperature of 320 1C. Hence, the systemdesign is based on a temperature difference of 300 1C, assuming that the sink is kept at 20 1C. During the preliminary design stage, thecritical parameters of the engine design are determined according to the dynamic model with losses energy and pressure drop in heatexchangers was used during the design optimisation stage in order to establish a complete analytical model for the engine. The heatexchangers 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 optimalswept 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 efﬁciency1. 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, theeconomic growth has become a worldwide concern; there is former have gained popularity in the last few decades duean urgent need to effectively reuse solar energy, this source to this potential to tap a variety of low concentrationof energy is one of the more attractive renewable energy energy sources available, such as solar. The increasingthat can be used as an input energy source for heat engines. interest in Stirling engines is largely due to the fact theIn fact, any heat energy source can be used with the Stirling engine is more environmentally friendly than the widelyengine. 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 formsolar-powered prime mover. The direct conversion of solar and thus leading to silent and cleaner operation, which arepower 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  call uponheat source and their high efﬁciency. Stirling engines are an advanced and heavy technology, therefore are veryable 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 withoutextensively reported in the literature  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 company0960-1481/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.renene.2007.09.024
ARTICLE IN PRESS1912 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, JBSR Solar Technologies GmbH, which developed the losses of heat in the exchangers exist. To accurately predictSUNPULSE, also works on a system intended to produce power and efﬁciency requires an understanding of theelectricity 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 enginehave been established , as well as the procedures for 2.1. Energy dissipation by pressure drops in heat exchangersoptimal design . Most of the engines are fuel-ﬁred and _ d QPChoperate at high temperature, which highlights the need forcareful material selection as well as good cooling system. Pressure drops due to friction and to area changes inFor silent, light and portable equipment for leisure and heat exchangers is given by domestic uses, low power engines may be more appropriate. 2f r mGVNevertheless, research in Stirling engine technology has been Dp ¼ À , (1)heavily masked by extensive and successful development of Ad 2 rinternal combustion engines, which have made Stirling where G is working gas mass ﬂow (kg mÀ2 sÀ1), d is theengines less competitive. Hence, in order to design a low hydraulic diameter, r is gas density (kg mÀ3), V is volumepower 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 gaspresents design considerations which may be taken to is forced to ﬂow against the frictional drag force, is givendevelop a solar Stirling engine with average concentration by :operating on mean temperature difference of 300 1C. _ Dpm_ dQPch ¼ À , (2) r2. 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 differentthermodynamic and the mechanical processes. Compres- exchangers ission 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
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 isbetween 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 : _ 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 aredQPcdf ¼ kcdf ðT fÀr À T cÀf Þ, (5) Lf speciﬁed by the regenerator adiabatic coefﬁcient, ep1,
ARTICLE IN PRESS1914 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921deﬁnite 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 evaluatedcompression 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 gapspace . 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 enginecompression 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 isThe effectiveness of the regenerator e is given starting from calculated according to the perfect gas law:the equation below  Pc V c Tc ¼ , (12) NTU Rmc¼ , (9) 1 þ NTU Pf V fNTU is the number of heat transfer unit: Tf ¼ , (13) Rmf hAwgNTU ¼ , (10) Ph V h Cpm_ Th ¼ , (14) Rmhwhere 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 Rmdcapacity at constant pressure, and m (kg sÀ1) the mass ﬂow _ The regenerator is divided into two cells r1 and r2, eachrate 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 spaceswithin a machine introduces another mechanism for Pr2 V r2transferring heat from a hot to a cold space. Thus an T r2 ¼ . (17) Rmr2important 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 interfaceapparent thermal conductance loss. The displacer absorbs temperature Tr–f , Tr–r and Tr–h, as follows :a quantity of heat from the hot source and restores it to thecold source. This loss of energy is given by : 3T r1 À T r2 T rÀf ¼ , (18) 2 2 _ 0:4Z kpis DddQPshtl 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 r1Dd 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’sin the compression space (K). temperatures: Tc–f , Tf–r , Tr–h and Th–d are deﬁned as follows :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 predictpower 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 differentlosses and pressure drop in heat exchangers. if _ mhÀd 40; then T hÀd ¼ T h ; otherwise T hÀd ¼ T d ,
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 thecompression space and the cooler, Tf–r is the temperatureof the interface between the cooler and the regenerator,TrÀh is the temperature of the interface between theregenerator and the heater, ThÀd is the temperature ofthe interface between the heater and the expansion space. The matrix temperatures are so given bydT par1 dQr1 ¼À , (21) dt C pr dtdT par2 dQr2 ¼À , (22) Fig. 3. Generalised cell. dt C pr dtwhere 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 regeneratorr2 (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 dtr2 (K). The thermal efﬁciency given by the cycle is: By taking into account the conduction loss in theexchangers 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 dðmTÞwhere 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 dtdQPcdr1 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 compressionhh is only available empirically . 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
ARTICLE IN PRESS1916 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 DPfPf 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 ÞPr2 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 ÞPh 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 methodengine 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 dPcdQf 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  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
ARTICLE IN PRESS I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1917are derived from geometric considerations in Fig. 5 and Table 1Table 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 by5. Design analyses Displacements Y c ¼ r½sin y À cos yðb2 =b1 Þ þ by Y e ¼ r½sin y þ cos yðb2 =b1 Þ þ by5.1. Relationship for engine power, swept volumes and deadvolumes 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 byvolumes 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.
ARTICLE IN PRESS1918 I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921dead 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 . However, thedirect effects of the dead volume and swept volume to theengine power should be detailed. Figs. 6 and 7 illustrates Fig. 7. Relationship between swept volume and engine power (deadthe variation of the power as a function of the swept volume 370 cm3).volume, 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 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 ofFig. 6. Relationship between swept volume and engine power (dead dead volume. Previous studies showed that the deadvolume 535 cm3). volumes, which includes those in the heat exchangers, is
ARTICLE IN PRESS I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1919Fig. 9. Relationship between dead volume and engine power (frequency ¼35 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 itshould be small as possible . To demonstrate therelationships for the heaters, speciﬁc conditions of 75 cm3swept volume and 0.45 m tube length are used. After carrying out thermodynamic simulation for theheater, 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 cm3are shown in Figs. 12 and 13, which indicates a similarpattern 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 supplyor reject the required amount of heat to or from the engine. implication to the efﬁciency of the engine, six types ofIn this aspect, one crucial factor is the heat transfer area, matrices has been selected and is being subjected to variouswhich will decide the amount of heat energy to be pressure drops and engine speeds. The conﬁgurations fortransported. Hence, in order to achieve a high effectiveness these six matrices are given in Table 3 for a standard totalfor the heater and the cooler, larger transfer areas, and thus wire length of 5 m. The porosity of each matrix islarger volumes, are needed. important since it will have a direct impact on the performance of the regenerator, and can be determined5.3. 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 bothefﬁciency of the engine [16,17]. To analyse this effect and its high efﬁciency and low-pressure drop.
ARTICLE IN PRESS1920 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 0.6112Fig. 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 regeneratorFig. 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 andfrequency and the pressure drop for these matrices. The low-pressure drop in order to obtain minimal losses in thepressure drop is found to be proportional to the frequency regenerator, and in this case, M6 with the porosity ofsince an increase in frequency increases the mass ﬂux 0.6112 and wire diameter 0.009 m has been chosen for thethrough 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 frictionother hand, the decrease in mesh porosity leads to the factor as well as increases the pressure drop. Hence, it canhigher friction factor as well as increases the pressure drop. be said that M1 has a lowest pressure drop in comparisonHence, it can be said that M1 has a lowest pressure drop in to the others at a same frequency because its porosity is thecomparison to the others at a same speed because its highest. In order to obtain a higher porosity, and thus theporosity is the highest. In order to obtain a higher porosity, lower pressure drop, the meshes should be made from smalland 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
ARTICLE IN PRESS I. Tlili et al. / Renewable Energy 33 (2008) 1911–1921 1921Table 4 length. For this engine, the selected parameters areEffect of matrix on power and thermal efﬁciency the wire diameter of 3.5 mm with a total length of 5 mMatrix 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 theM1 159.39 10.79 engine.M2 226.42 22.41 The heat exchanger volumes should be evaluated byM3 249.20 33.90 considering both the pressure drop and the thermalM4 252.50 37.32M5 255.92 43.32 efﬁciency of the engine. In our case the optimal heatM6 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 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 heat Referencestransfer.  Kongtragool B, Wongwises S. A review of solar-powered Stirling The regenerator effectiveness e can be manipulated by engines and low temperature differential Stirling engines. Renewablevarying wire diameter and wire length, which in turn Sustainable Energy Rev 2003;7:131–54.changes the ‘‘wetted’’ surface area. It can be represented in  Bonnet S, Alaphilippe M, Stouffs P. Conversion thermodynamiqueform of the relationship between the porosity or the de l’energie solaire dans des installations de faible ou de ´ Moyenne Puissance: reﬂexions sur le Choix du Meilleur Degre de ´number of transfer units (NTU) and the thermal heating ` ´ concentration. Rev Energ Ren 11emes Journees Int Therm 2003;efﬁciency of the engine. If the wetted surface area is 73–80.large, the resulting porosity should be low, and this  Berchowitz DM, Urieli I. Stirling cycle engine analysis. Bristol: Adamprovides the air or the work ﬂuid with a large contacting Hilger Ltd; 1984.surface to achieve a high rate of heat transfer. Hence, the  Timoumi Y, Ben Nasrallah S. Design and fabrication of a Stirling–Ringbom engine running at a low temperature. In: TSSNTU, and thus e, are increased when the surface area international conference in mechanics and engineering (ICAME)increases. The effect of e on the thermal efﬁciency of the March, Hammamet, Tunisia; 2002.engine is that it represents the ability to reject the heat to  Park SJ, Hong YJ, Kim HB, Lee KB. An experimental study on thethe working gas when the gas exits through the heater and phase shift between piston and displacer in the Stirling cryocooler.the ability to absorb the heat when the gas exits through Curr Appl Phys 2003;3:449–55.the cooler.  Capeto C, Rispoli F. New drive mechanisms for the Stirling engine’ 8th international Stirling engine conference and exhibition, May 27–30, Ancona; 1997. p. 39–49.6. Conclusion  Bartczak L, Carlsen H, An optimization study of Stirling engines based on advanced simulation. In: Proceedings, 5th international In this paper, a number of technical considerations in Stirling engine conference. Dubrovnik; 1991. p. 161–6.designing a mean temperature differential Stirling engine  Kolin I. Stirling motor: history-theory-practice. Dubrovnik: Inter University Center; 1991.have been proposed. These considerations have been  James RS. Ringbom Stirling engines. New York: Oxford Universityestablished through the use of the dynamic model with Press; 1985.losses energy and pressure drop in heat exchangers. As a ´  Tlili S. Modelisation des moteurs Stirling, DEA, Ecole Nationalresult, the optimal conﬁguration for the design can be ´ d’Ingenieurs de Monastir. Tunisie; 2002.summarised as follow.  Tlili S, Timoumi Y. Numerical simulation and losses analysis in a Stirling engine. Int J Heat Tech 2006;24(1):97–103. ´  Poncet E, Nika P, Bereiziat D, Lanzetta F. Technique de This studies show clearly that, for given value of ´ ´ ´ ´ caracterisation d’un mini-regenerateur thermique pour mini-refroi- difference temperature, operating frequency and dead ` ´ ´ disseur Stirling ou tube a gaz pulse’. Mec Ind 2001;2(5):455–64. volume there is a deﬁnite optimal value of swept  Popescu G, Radcenco V, Costea M, Feidt M. Optimisation volume ratio at which the power is a maximum. In thermodynamique en temps ﬁni du moteur de Stirling endo- et exo- ´ ´ irreversible. Rev Gen Therm 1996;35:656–61. this paper, the optimal swept volume is 75 cm3 for  Walker G. Stirling engines. Oxford: Clarendon Press; 1980. frequency ¼ 75 Hz.  Halit K, Huseyin S, Atilla K. Manufacturing and testing of a V-type Upon optimisation, operating frequency has to be Stirling engine. Turk J Engine Environ Sci 2000;24:71–80. limited between 35 and 75 Hz at a temperature  Organ AJ. The regenerator and the Stirling engine. Mechanical Engineering Publications; 1997. difference of 300 1C, where the best value is 75 Hz.  Ataer OE. Numerical analysis of regenerators of piston type For the regenerator, its porosity plays a signiﬁcant role Stirling engines using Lagrangian formulation. Int J Refrig in controlling pressure drop of the regenerator, which 2002;25:640–52.p, cpr and cv has been changed to Cp, Cpr and Cv can be manipulated by varying wire diameter and respectively.4