Analytical model for predicting the effect of operating speed on shaft power
Energy 36 (2011) 5899e5908 Contents lists available at SciVerse ScienceDirect Energy journal homepage: www.elsevier.com/locate/energyAnalytical model for predicting the effect of operating speed on shaft poweroutput of Stirling enginesChin-Hsiang Cheng*, Hang-Suin YangInstitute of Aeronautics and Astronautics, National Cheng Kung University, No.1, Ta-Shieh Road, Tainan 70101, Taiwan, ROCa r t i c l e i n f o a b s t r a c tArticle history: This paper is concerned with numerical predictions of relationship between operating speed and shaftReceived 23 February 2011 power output of Stirling engines. Temperature variations in expansion and compression spaces as well asReceived in revised form the shaft power output corresponding to different operating speeds were investigated by using a lum-16 August 2011 ped-mass transient model. Effects of major operating parameters on power output were studied.Accepted 19 August 2011 Results show that as the operating speed increased, temperature difference between the expansion andAvailable online 21 September 2011 compression spaces was reduced and as a result, the shaft work output decreased. However, the shaft power output is determined in terms of the shaft work output and the operating speed. When theKeywords:Analytical model operating speed was elevated, the shaft power output reached a maximum at a critical operating speed.Stirling engine Over the critical operating speed, the shaft power output decreased in high-speed regime. In addition, asMechanism effectiveness air mass was reduced, either a decrease in thermal resistances or an increase in effectivenesses of theShaft power output regenerator leads to an increase in the engine power. Ó 2011 Elsevier Ltd. All rights reserved.1. Introduction is equal to the Carnot cycle efﬁciency under the same thermal reservoirs temperatures. As compared with other alternative power Owing to rapid depletion of fossil fuels and threat of global generation methods, the CSP-Stirling system features high efﬁ-warming, environmental problems have received great attention in ciency and modular construction. Therefore, cost of these systems isrecent years. A large number of researchers concentrate their focus expected to become more competitive in the distributed markets inon the electricity generation from solar energy which is treated as near future. However, it is noted that performance of the CSP-one of potential energy sources in near future. Practically, there are Stirling system for generating electrical power is dependent ontwo major effects to convert the solar energy into electricity: energy conversion and mechanical efﬁciencies of the Stirling engine.photovoltaic and thermodynamic. The photovoltaic (PV) effect is Therefore, predictions and improvement in the performance of theused to produce electricity when sunlight hits a PV panel and Stirling engine are of major concerns to related investigators.releases electrons from special layers of silicon and pushes them An ideal regenerative Stirling cycle consists of four processes inacross an electric ﬁeld. On the other hand, the thermodynamic a cycle. Firstly, working ﬂuid absorbs heat from high temperatureeffect is used in devices like concentrating solar power systems reservoir and experiences an isothermal expansion (1/2).with Stirling engines (CSP-Stirling) . The CSP-Stirling system is Secondly, the hot working ﬂuid ﬂows through a regenerator, and thetypically equipped with a solar energy receiver to absorb the solar regenerator absorbs heat from the hot working ﬂuid. Thus,radiation and the Stirling engine to convert the absorbed solar temperature of the working ﬂuid decreases in an isochoric processenergy to mechanical energy and then electricity [2,3]. In addition, (2/3). Thirdly, the working ﬂuid rejects heat to a low temperaturethe Stirling engines can also be applied in automobiles and reservoir and experiences an isothermal compression (3/4).submarines as power resources [4,5]. Finally, the cold working ﬂuid ﬂows back through the regenerator, The Stirling engines are referred to as external combustion heat and the regenerator rejects heat to the working ﬂuid. The temper-engines that are operated based on a regenerative closed power ature of work ﬂuid increases in the second isochoric process (4/1).cycle using air, nitrogen, helium, or hydrogen as working ﬂuid . In The p-V diagram of the ideal Stirling cycle (1/2/3/4/1) istheory, the thermal efﬁciency of an ideal regenerative Stirling cycle plotted in Fig. 1. As mentioned in Ref. , the Stirling engine was invented by * Corresponding author. Tel.: þ886 6 2757575x63627; fax: þ886 6 2389940. Robert Stirling in 1816, which was called ‘hot-air engine’ originally, E-mail address: firstname.lastname@example.org (C.-H. Cheng). and the ﬁrst successful reversible model for analysis of the Stirling0360-5442/$ e see front matter Ó 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.energy.2011.08.033
5900 C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 Nomenclature t time (sec) U gas velocity through regenerator (m/s) Cp speciﬁc heat (J=kg,K) Vc volume of compression chamber (m3 ) E mechanism effectiveness Vd dead volume (m3 ) eC effectiveness of regenerator in cooling process Ve volume of expansion chamber (m3 ) eH effectiveness of regenerator in heating process Vr volume of regenerator (m3 ) Lr length of regenerator (m) V total volume (m3 ) M molecular weight of working ﬂuid (kg/mol) W indicated work (J) m mass of working ﬂuid (kg) Ws shaft work (J) N number of moles WÀ forced work (J) Ps shaft power output (W) p pressure of work space (N=m2 ) Greek symbols Dp pressure difference between expansion and q crank angle (deg) compression spaces (N=m2 ) r density of working ﬂuid (kg=m3 ) Qgen:absorb heat absorbed by regenerator from working ﬂuid (J) s operating temperature ratio Qgen:reject heat rejected by regenerator to working ﬂuid (J) u operating speed (rad/s) R gas constant (J=K,mol) uo operating speed (rpm) RC thermal resistance of cold side (K=W) RH thermal resistance of hot side (K=W) Subscripts rh hydraulic radius of regenerator (m) C cold side TC cold side reservoir temperature (K) c compression space TRm mean ﬂuid temperature in regenerator channel (K) d dead space TH hot side reservoir temperature (K) e expansion space TL ﬂuid temperature in compression space (K) ex experimental TLm mean ﬂuid temperature in compression space (K) th theoretical TU ﬂuid temperature in expansion space (K) H hot side TUm mean temperature in expansion space (K) m mean TL0 initial temperature of cooling process (K) r regenerator TU0 initial temperature of heating process (K)engine was presented by Schmidt  in 1871. In the reversible to replace the isothermal processes and introduced a realisticmodel, a closed-form solution for indicted work output of the regenerator into the model. Wu et al.  also studied the irre-engine was obtained by assuming that the pressure of the working versible model of the Stirling engine with an imperfect regenerator.ﬂuid is uniform throughout the working space of the engine and The authors presented a criterion for optimizing the dead volumethat the variation in the pressure with time follows sinusoidal and studied the relationship between net power output andfunction. In the past several decades, the analysis has been thermal efﬁciency. Kaushik and Kumar [12,13] performed a ﬁniteextended to irreversible models. For examples, Blank and Wu  time thermodynamic analysis of the power output and the thermaloptimized the power of solar-radiant Stirling engine by using efﬁciency for the Stirling engine with internal heat loss in theﬁnite-time analysis. Erbay and Yavuz  used polytropic processes regenerator and with ﬁnite-heat-capacity external reservoirs. Senft [14,15] took into account internal heat losses and mechanical fric- tion losses and determined theoretical limits on the performance of Stirling engines as well as the maximum value of the mechanical p efﬁciency. Petrescu et al.  used a direct method to investigate the irreversible Stirling cycle model with ﬁnite speed. In this report, effects of the regenerator loss on the power output and the thermal TH 1 efﬁciency of the engines were evaluated. Recently, Kongtragool and Wongwises  extended the thermodynamic model to include the TU (t ) dead volumes inside the working spaces. Qin et al.  used a linear phenomenological heat-transfer law to analyze an ‘endoreversible’ Carnot heat engine. The authors found that the power output, the TU′ 2 thermal efﬁciency and the heat-conduction losses are dependent 4 on frequency and heat transfer time ratio. Karabulut et al.  TL′ compared different kinds of b-type Stirling engines and studied the effect of convective heat transfer coefﬁcient on the power 3′ output at different operating speeds. Meanwhile, experiments were TL (t ) 3 conducted by Kongtragool and Wongwises  for the low temperature differential g-type Stirling engines. In this study, TC engine torque, shaft work, and thermal efﬁciency of the engines were measured at different engine speeds. Most recently, Cheng and Yu  proposed an irreversible V thermodynamic model for the b-type Stirling engines. Periodic variation of pressures, volumes, temperatures, masses, and heat Fig. 1. p-V diagram of Stirling cycle with imperfect regenerator. transfers in expansion and compression chambers of the engines
C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 5901are predicted by taking into account heat transfer effectiveness ofregenerative channels as well as thermal resistances of heating andcooling heads. Lately, Cheng and Yu  ﬁrst performed a dynamicsimulation of the b-type Stirling engines with a cam-drive mech-anism by combining dynamic and thermodynamic models. Tran-sient variation in rotational speed of the engines during start-upperiod can then be predicted. Performance curves showing the shaft power output at differentoperating speeds of the Stirling engines may be obtained byanalytical or experimental methods based on the above literaturesurvey. However, unfortunately, in spite of the valuable analyticaland experimental studies performed in Refs. [19,20,23], so far onlya limited number of analytical models were presented that arecapable of predicting the dependence of the shaft power output onthe operating speed. In consequence, the effects of operatingparameters such as the operating temperature ratio, the mass ofworking ﬂuid, the thermal resistances of the heater and the cooler,the effectiveness of the regenerator, and the properties of workingﬂuids on the performance curves of the Stirling engines are notsufﬁciently understood. Fig. 2. Temperature changes of lumped mass in the processes. Under these circumstances, this study is aimed to develop ananalytical model that can be used to yield the relationship betweenthe shaft power output and the operating speed of the Stirling _ where qin denotes the rate of heat absorbed by the working ﬂuid,engines with imperfect regenerators. The analytical model includes andRH is the thermal resistance of the hot side. Magnitude of RH isa lumped-mass concept such that transient variation of the ther- dependent on the operating speed, properties of working ﬂuid,modynamic properties of the working ﬂuid in each process of the operating temperature, design of heater, and the heating method,cycle can be predicted. This approach ﬁrst allows the effects of the which can be determined by experiments. To simplify the analysis,operating parameters on the performance curve to be evaluated. here the value ofRH is ﬁxed. Based on the energy conservation law, _ qin is approximately equal to rate of change in enthalpy of the2. Analytical model working ﬂuid. That is, Let TH and TC represent the temperatures of the hot side and the dTU ðT À TU Þcold-side reservoirs, respectively, and TU and TL represent the ﬂuid mCp ¼ H (1) dt RHtemperatures in the expansion and the compression spaces,respectively. Note that TU < TH and TL > TC . Heat transfer between where m is the mass of the working ﬂuid, which is the product ofthe working ﬂuid and the thermal reservoirs is strongly dependent the molecular weight (M) and the molar number of working ﬂuidon the temperature differences, TH À TU and TL À TC , and time for (N) contained in the working space. Eq. (1) is a ﬁrst-order ordinaryheat transfer. Furthermore, because the time for heating in the differential equation. Thus, one can carry out the solution of theexpansion space and that for cooling in the compression space in equation asa cycle are decreased as the operating engine speed is increased; therefore, TU is reduced and TL is elevated by increasing the oper- t TU ðtÞ ¼ TH À C1 exp À (2)ating speed. As a result, the differences TH À TU and TL À TC both mCp RHincrease with the operating engine speed. It is noticed that in spiteof the constant thermal reservoirs temperatures, the working ﬂuid where C1 is a constant to determine. The working ﬂuid enters thetemperatures in the working spaces are varied with time in the expansion space at TU and there it is heated in process 10 -2. 0respective processes, which may be denoted by TU (t) and TL (t). 0 Therefore, introducing the initial condition TU(0) ¼ TU yields Temperature changes in the processes of the cycle must be 0 C1 ¼ TH À TU . The solution can be rewritten asexamined. Notice that due to the imperfection of the regenerator, 0the working ﬂuid enters the expansion space at TU , not TU , at state À 0 Á t1 0 after passing through the regenerator. Meanwhile, it enters the TU ðtÞ ¼ TH À TH À TU exp À (3) mCp RHcompression space at TL , not TL , at state 30 after ﬂowing back 0through the regenerator. The real cycle with an imperfect regen-erator is illustrated in Fig. 1. 2.2. Compression space In this study, a lumped-mass transient model is developed. Theworking ﬂuid of mass m is regarded as a lumped mass that travels Similarly, in the compression space, rate of heat rejected by thethrough four working spaces (expansion space, regenerator chan- working ﬂuid to the cold side reservoir is expressed asnels, compression space, and regenerator channels) and experi- _ _ qout ¼ ðTL À TC Þ=RC , where qout is the heat rejected by the workingences changes in temperature and pressure in the cycle. The ﬂuid, and RC is the thermal resistance of the cold side. The energytemperature changes of the working ﬂuid in the four working conservation law givesspaces in the cycle are indicated in Fig. 2. dTL ðT À TC Þ2.1. Expansion space mCp ¼ À L (4) dt RC In the expansion space, rate of heat absorbed by the working 0 Using the initial condition TL(0) ¼ TL , one has the solution for the _ﬂuid from the hot side reservoir is expressed as qin ¼ ðTH À TU Þ=RH , temperature of compression space
5902 C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 À 0 Á t In practices, the times consumed by the four processes are justTL ðtÞ ¼ TC þ TL À TC exp À (5) slightly different, and hence, the time each process needs is mCp RC approximated as a quarter of a cycle period. If the engine’s oper- ating speed is u, in rad/s, the time required in each process will be2.3. Regenerator p/2u [¼(1/4)(2p/u)]. Let TUm and TLm be mean temperatures of the working ﬂuid in the expansion space and the compression space, The effectivenesses of regenerator can be deﬁned as respectively, which are calculated by 0 TU À TLeH ¼ for process 4 À 10 ; (6a) p=2u Z TU À TL 1 TUm ¼ TU ðtÞdt ¼ TUm ðuÞ (9a)and p=2u 0 0 TU À TLeC ¼ for process 2 À 30 (6b) p=2u Z TU À TL 1 TLm ¼ TL ðtÞdt ¼ TLm ðuÞ (9b) Magnitudes of the regenerator effectivenesses are dependent p=2u 0on porosity, permeability and material of porous matrix.Temperature distribution in the regenerator can be measured by As suggested in [8,29], the operating temperature ratio isexperiments [7,24e26] to determine the regenerator effective- deﬁned bynesses. Furthermore, the magnitudes of the regenerator effec-tivenesses are also dependent on the operating speed. Actually, TLm ðuÞthey decrease with an increasing operating speed [27,28]. This is s¼ (10) TUm ðuÞbecause the time for heat exchange between the gas and theporous matrix of the regenerator is decreased by increasing the The mean temperature of the working ﬂuid contained in theoperating speed. The relation between the regenerator effective- regenerator channels can be evaluated as suggested in [17,29] withness and the operating speed is essential and worthy of furtherinvestigation. Note that with the imperfect regenerators, one has TUm ðuÞ þ TLm ðuÞeH 1 and eC 1. Heat transfer from the regenerator to the TRm ðuÞ ¼ (11)working ﬂuid of mass m moving through the regenerator channels 2in the direction from the compression space to the expansionspace (in process 4-10 ) is Qgen reject ¼ mCp(TU e TL), and the heat 0 2.5. Power outputtransfer to the regenerator from the working ﬂuid moving fromthe expansion space to the compression space (in process 2-30 ) is Before calculation of the power output of the engine, one needs 0Qgen absorb ¼ mCp(TU À TL ). to evaluate pressure difference between the expansion and the compression spaces. In terms of gas velocity ﬂowing through the2.4. Mean temperatures in working spaces regenerator, diameter of the regenerative channel, and kinematic 0 0 viscosity of the working ﬂuid at the average operating temperature,2.4.1. From Eqs. (6a) and (6b), TU and TL can be calculated by the magnitude of Reynolds number is estimated to be 102 approximately. According to the experimental data presented by 0 Kays and London , at this Reynolds number the magnitude ofTU ¼ TL þ eH ðTU À TL Þ (7a) friction factor is about 10À1 . The friction factor is deﬁned asand rh Dp 0TL ¼ TU À eC ðTU À TL Þ (7b) f ¼ (12) Lr rU 2 =2 Substituting Eqs. (7a) and (7b) into Eqs. (3) and (5) yields two where rh and Lr are hydraulic radius and length of the regenerativesimultaneous algebraic equations for TU(t) and TL(t). The solutions channels, respectively, r is density of the working ﬂuid, U is the gascan then be carried out as ﬂow velocity through the regenerator, and Dp is the pressure difference between the expansion and the compression spaces.TU ðtÞ ¼ DU =D (8a) Magnitude of Dp can then be evaluated asand Dp Lr U 2 f 10À1 Â 101 Â 10À1TL ðtÞ ¼ DL =D (8b) ¼ w À1 w10À3 (13) pm 2rh RTC 10 Â 101 Â 102where Thus, the pressure difference between the expansion and the Â À ÁD ¼ ð1 À eH Þð1 À eC Þ À exp t=nWcp RH compression spaces can be negligible. The pressure is assumed to ÃÂ À Á Ã be uniform throughout the engine and can be calculated in terms of À eH exp t=nWcp RC À eC ð8cÞ the volumes and their mean temperatures of the working spaces Â À Á ÃÂ À Á Ã [17,21,22,29] asDU ¼ À exp t=nWcp RH À 1 exp t=nWcp RC À eC TH Â À Á Ã mR À ð1 À eH Þ exp t=nWcp RC À 1 TC (8d) p ¼ (14) Ve Vc Vr þ þ Â À Á Ã TUm TLm TRmDL ¼ Àð1 À eC Þ exp t=nWcp RH À 1 TH Â À Á ÃÂ À Á Ã where R is gas constant; and Ve, Vc, and Vr represent instantaneous À exp t=nWcp RH À eH exp t=nWcp RC À 1 TC ð8eÞ volumes of the expansion space, the compression space, and the
C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 5903regenerator channels, respectively. In the derivation of the above power output can then be calculated from the obtained shaftequation, the working ﬂuid is assumed to be an ideal gas. Total work withvolume of the engine is V ¼ Ve þ Vc þ Vr and dead volume of theengine is Vd ¼ Vemin þ Vcmin þ Vr, where Vemin and Vcmin are Ps ¼ Ws ðu=2pÞ; in Watt (17)clearance volumes of the expansion and the compression spaces, In the present study, air is used as the working ﬂuid. For air, therespectively. molecular weight is M ¼ 2.897 Â 10À2 kg/mol and the speciﬁc heat The function p(V) is affected by the operating speed of engine cp ¼ 1087 J/(kg$K). Meanwhile, the hot side and the cold-side(u). Using Eq. (14), one can plot the p-V diagram at speciﬁc reservoir temperatures, TH and TC , are ﬁxed at 1200 K and 300 K,operating speed. Then, indicated work (W) and forced work (W_) respectively.of a cycle can be further calculated. The deﬁnitions of the indi-cated work and the forced work are already described in[14,15,29], which are illustrated in Fig. 3. The indicated work is 2.6. Experimental apparatus and testing procedurecalculated by integrating the area inside the cycle in the p-Vdiagram. That is, The purpose of the experiment is to measure the shaft power Z output of engine under different operating speed for validation.W ¼ pdV (15) Experimental apparatus is shown in Fig. 4. A non-contact infrared heater was used for heating the engine, and surface temperatures cycle at the hot and the cold sides are measured by using K-type ther- On the other hand, the forced work is represented by the shaded mocouples. In order to maintain constant heating temperature (TH),areas shown in the plot. pb is back pressure exerted on opposite side a PID controller was used to control the temperature by adjustingof the piston. The back pressure could usually be atmospheric the electrical voltage supplied to the infrared heater. The engine ispressure or an elevated pressure in case a pressurized space is an aetype Stirling engine with Ross mechanism, of which theplaced on the opposite side as a buffer zone. speciﬁcations are given in Tables 1 and 2. The engine shaft was The shaft work of engine can be determined in terms of theindicated work and the forced work, as suggested by Senft [15,29]as a Dynamic system 1 Hysteresis BrakeWs ¼ EW À À E WÀ (16) E Brake Controller Heating systemwhere E is referred to as mechanism effectiveness. The mecha-nism effectiveness is well known as a ratio measure of how well Heatera mechanism utilizes work if it is regarded as a work transmitter.It is a ratio of the work it transfers out through an actuator or Torque Controller Engineputs into internal storage to the work it receives through Sensoranother actuator or takes from its internal source. Typical value Thermocoupleof the mechanical effectiveness is in the range of 0.7 E 0.8. Inthis study, the mechanism effectiveness is assigned to be 0.7.Note that effects of friction are included in the mechanism Speed Monitoreffectiveness. Detailed information for the mechanism effec- Sensortiveness has been provided by Senft [15,29]; therefore, nofurther description is provided here to save space. The shaft Schematic of the performance test system b Photo of the performance test systemFig. 3. Deﬁnitions of indicated work (W) and forced work (W_). If the area of cycle is Fig. 4. Experimental apparatus. (a) Schematic of the performance test systemreduced, W is decreased while W_ is increased. (b) Photo of the performance test system.
5904 C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908Table 1Dimensions of test problem. a 1200 TUm 1100 TLm 1000 900 TUm 800 TLm N =0.01 mol N =0.005 mol N =0.0025 mol 700 (K) 600 500 400 300 0 500 1000 1500 ω0 (rpm) b 1200 TUm 1100 TLm 1000 900 TUmconnected to a speed sensor, a torque sensor and a hysteresis brake. RH=RC= 0.1 K/W 800The hysteresis brake can exert different constant loadings to the RH=RC= 0.3 K/W TLm RH=RC= 0.5 K/Wengine. Once the loading is applied, the engine speed will experi- 700ence a rapid decrease at the beginning, and then it reaches a stable (K)regime. The engine torque and the corresponding operating speed 600were recorded after the stable regime is reached at different 500loadings. The shaft power output at different operating speed canbe then obtained. 400 3003. Results and discussion 0 500 1000 1500 ω0 (rpm) Fig. 5 shows the dependence of the mean ﬂuid temperatures inthe expansion and compression spaces (TUM and TLM) on the c 1200operating speed (uo), in rpm. In this ﬁgure, effects of molar TUmnumber of air (N), thermal resistances of hot and cold sides (RH 1100 TLmand RC) and effectivenesses of regenerator in heating and cooling 1000processes (eH and eC) are displayed. It is found that as the oper-ating speed of the engine is increased, the mean ﬂuid temperature 900in the expansion space is decreased and that in the compression TUm eH=eC= 0.9space is increased. In other words, the temperature difference 800 eH=eC= 0.8between the expansion and the compression spaces is reduced as TLm eH=eC= 0.7 700 (K) 600Table 2Engine speciﬁcations and operating parameters. 500 Working ﬂuid air Charging pressure 1 atm 400 Displacer swept volume 82.85 cm3 Piston swept volume 64.49 cm3 300 Dead volume 24.61 cm3 0 500 1000 1500 ω0 (rpm) Compression ratio 3.37 Heating temperature, TH 1200 K Cooling temperature, TC 300 K Thermal resistance of hot side, RH 0.82 K/W Fig. 5. Mean ﬂuid temperatures vs. operating speed. (a) Effects of molar number at Thermal resistance of cold side, RC 0.61 K/W RH ¼ RC ¼ 0.3 K/W and eH ¼ eC ¼ 0.7. (b) Effects of thermal resistances at Effectiveness of regenerator, eH ¼ eC 0.69 N ¼ 0.0025 mol and eH ¼ eC ¼ 0.7. (c) Effects of regenerator effectiveness at The effectiveness of mechanism, E 0.7 N ¼ 0.0025 mol and RH ¼ RC ¼ 0.3 K/W. Heating method Infrared heater Cooling method Water jacket
C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 5905the operating speed is increased. This is reasonable since at a 1a higher speed, there is less time for the working ﬂuid to absorbheat from the hot side thermal reservoir in the expansion space orto reject heat to the cold side thermal reservoir in the compression 0.9space. It is straightforward to expect that the indicated work N =0.01 moloutput per cycle is decreased as the temperature difference is 0.8 N =0.005 molreduced. N =0.0025 mol In Fig. 5a, effects of the molar number of air at RH ¼ RC ¼ 0.3 K/W 0.7and eH ¼ eC ¼ 0.7 are evaluated. It is noted that as the mass of τworking ﬂuid is increased, the difference between TUM and TLM is 0.6further reduced. This may be attributed to the increased thermalcapacity when more air is charged. On the other hand, it is expected 0.5that when the thermal resistances are elevated or the regeneratoreffectiveness is decreased, the heat transfer is retarded so that the 0.4difference between TUM and TLM will be reduced. The results pre-sented in Fig. 5b and c clearly reﬂect this expectation. The cases 0.3shown in Fig. 5b are ﬁxed at N ¼ 0.0025 mol and eH ¼ eC ¼ 0.7, and 0 500 1000 1500Fig. 5c at N ¼ 0.0025 mol and RH ¼ RC ¼ 0.3 K/W. ω0 (rpm) Operating temperature ratio (s) is calculated by Eq. (10) based Effects of molar number at RH=RC=0.3 K/W and eH=eC=0.7.on the mean temperatures in the expansion and the compressionspaces obtained at different operating speeds, and the data are b 1provided in Fig. 6 for the same cases considered in Fig. 5. Themagnitude of the operating temperature ratio increases with the 0.9operating speed. In addition, it is observed that as the air mass iselevated, or the thermal resistances become higher, or the effec- 0.8 RH=RC= 0.3 K/Wtivenesses of the regenerator are lowered, the operating tempera-ture ratio is increased. 0.7 RH=RC= 0.5 K/W A photo of the test engine is shown in Fig. 7. The variations of Ve, τVc, V and Vd with the crank angle (q) are shown in Fig. 8. It is 0.6observed that the phase angle between the expansion and thecompression spaces is 100 . The total volume is varied between 48 0.5 RH=RC= 0.1 K/Wand 144 cm3, and the dead volume in the engine is 24 cm3.Numerical simulation of the test engine is carried out for the 0.4operating conditions at eH ¼ eC ¼ 0.69, N ¼ 0.0025 mol, andRH ¼ RC ¼ 0.3 K/W. 0.3 Fig. 9 shows the predicted p-V diagrams of the test engine atdifferent operating speeds. In this ﬁgure, it is clearly seen that an 0 500 1000 1500increase in the operating speed leads to a decrease in the area of ω0 (rpm)the p-V diagram, which implies that the indicated work output Effects of thermal resistances at N=0.0025 mol and eH=eC=0.7.(W) per cycle will be reduced as the engine speed is elevated. c 1The reduction of the indicated work can be attributed to thedecrease in the temperature difference between the expansion 0.9and the compression spaces as the operating speed is elevated,which is already observed in Figs. 5 and 6. Furthermore, the 0.8decrease in indicated work (W) per cycle due to an increase inthe operating speed is accompanied by an increase in forced 0.7work (W_). This can be illustrated in Fig. 3, in which the shaped eH=eC= 0.8 τarea representing W_ is increased when the area of cycle shrinks. 0.6By Eq. (16), shaft work of engine Ws is calculated in terms of W eH=eC= 0.7and W_. In this equation, it is found that as W is decreased and eH=eC= 0.9 0.5W_ is increased, the magnitude of Ws should be reduced obvi-ously. Plotted in Fig. 10 is the inﬂuence of the operating speed onW, W_, and Ws. 0.4 However, being aware that shaft power output is calculated interms of shaft work output and operating speed by using Eq. (17), 0.3one should recognize that shaft power output tends to increase 0 500 1000 1500with operating speed. As a result, one may obtain a maximum shaft ω0 (rpm)power output as the operating speed is varied in a certain range. Effects of regenerator effectiveness at N=0.0025 mol and RH=RC=0.3 K/W.Fig. 11 displays the dependence of shaft power output (Ps) of thetest engine on the operating speed. Again, the effects of the molar Fig. 6. Operating temperature ratio vs. operating speed. (a) Effects of molar number atnumber of air, thermal resistances of hot and cold sides and RH ¼ RC ¼ 0.3 K/W and eH ¼ eC ¼ 0.7. (b) Effects of thermal resistances at N ¼ 0.0025 mol and eH ¼ eC ¼ 0.7. (c) Effects of regenerator effectiveness ateffectivenesses of regenerator in heating and cooling processes are N ¼ 0.0025 mol and RH ¼ RC ¼ 0.3 K/W.evaluated. It is clearly seen that the shaft power output ﬁrstincreases with the operating speed in the low-speed regime. Whenthe operating speed is elevated, the shaft power output reaches
5906 C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 240 0 500rpm 220 1000rpm 200 1500rpm 180 p 160 (kPa) 140 120 pb =101.325 kPa 100 80 50 75 100 125 150 3 Vt (cm ) Fig. 7. Test engine. Fig. 9. Predicted p-V diagrams at different operating speeds for the test engine.a maximum at a critical operating speed. Over the critical operating [30,31] so that thermal resistances could be greatly reduced. Asspeed, the shaft power output gradually descends in the high- long as the related information is obtained, the information can bespeed regime. In addition, the peak shaft power output and its introduced into the present model to assess the effect of mass onaccompanying critical speed are strongly dependent on the molar the performance of the engine.number of the charged air, the thermal resistances, and the effec- A comparison between the experimental and the numericaltivenesses of the regenerator. In the plots of this ﬁgure, it is found data for the shaft power output as a function of the operatingthat as the air mass is elevated, or the thermal resistances are speed is shown in Fig. 12a, and the relative error between thesereduced, or the effectivenesses of the regenerator are increased, the two sets of data is shown in Fig. 12b as a function of the engineshaft power output can be remarkably increased. For example, in speed. The range of the engine speed in the experiments isFig. 11b, at RH ¼ RC ¼ 0.5 K/W the peak shaft power is around 6 W within 130 and 400 rpm, and the maximum shaft power outputand the critical speed is 200 rpm. As the thermal resistances are is found to be 4.4 W at 200 rpm. The experimental results closelyreduced to be RH ¼ RC ¼ 0.1 K/W, the maximum shaft power is agree with the numerical predictions at high-speed regime.increased to 29 W and the critical speed shifts to 900 rpm. Generally, the relative error is within 30%. However, in Fig. 12a, It is important to mention that for the cases considered in the absolute error between the experimental and numericalFig. 11a, the effect of the charged air mass is simply evaluated at results is remarkable when the engine speed is lower thanﬁxed thermal resistances (0.3 K/W). As a matter of fact, the effect of 200 rpm. This may probably be attributed to the greater uncer-the mass of charged air is rather involved. In general, when more air tainty of the measurement data at low engine speeds. For the testis charged into the working space of the engine, the thermal engine, the engine cannot be operated at a speed lower thanconductivity of the working ﬂuid and the heat transfer coefﬁcient 130 rpm.on the walls of the hot and cold sides may be signiﬁcantly increased 7 200 W Ve W- 180 Vc 6 Ws 160 Vd Vt 5 140 W Ve W- 120 4 Vc Ws Vd 100 (J) 3 Vt 80 (cm3) 2 60 40 1 20 0 0 500 1000 1500 2000 0 90 180 270 360 450 540 630 720 ω0 (rpm) θ (deg) Fig. 10. Predicted effects of operating speed on indicated work, forced work, and shaft Fig. 8. Volumes’ variation for the test engine. work for the test engine.
C.-H. Cheng, H.-S. Yang / Energy 36 (2011) 5899e5908 5907 a 10 a 6 9 Theoretical N =0.0025 mol Experimental 8 5 7 N =0.003 mol 6 4 Ps (W) 5 Ps 4 (W) 3 3 2 2 N =0.005 mol 1 1 0 0 100 200 300 400 500 600 700 ω0 (rpm) Effects of molar number at RH=RC=0.3 K/W and eH=eC=0.7. 0 0 100 200 300 400 b 30 ω0 (rpm) RH=RC= 0.1 K/W Shaft power output vs. engine speed. 25 b 100 20 90 Relative error ⎢Ps,th-Ps,ex⎢/Ps,th (%) Ps 80 (W) 15 70 RH=RC= 0.3 K/W 10 60 RH=RC= 0.5 K/W 50 5 40 0 30 0 500 1000 1500 2000 ω0 (rpm) 20 Effects of thermal resistances at N=0.0025 mol and eH=eC=0.7. c 25 10 0 eH=eC= 0.9 0 100 200 300 400 ω0 (rpm) 20 Relative error vs. engine speed. Fig. 12. Comparison between experimental and numerical data. (a) Shaft power 15 output vs. engine speed. (b) Relative error vs. engine speed. Ps eH=eC= 0.8 (W) 4. Conclusions 10 eH=eC= 0.7 In this study, an efﬁcient lumped-mass model has been devel- oped and used to predict the relationship between the operating speed and shaft power output of the Stirling engines. Temperature 5 variations in the expansion and the compression spaces as well as the shaft power output corresponding to different operating speeds are of major concerns. The concept of the mechanism effectiveness 0 initiated by Senft [15,29] is introduced in this study to calculate the 0 500 1000 1500 2000 ω0 (rpm) shaft work, in terms of the indicated work, the forced work, and the Effects of regenerator effectiveness at N=0.0025 mol and RH=RC=0.3 K/W. mechanism effectiveness. The effects of the operating parameters on the power output of the engine are also evaluated.Fig. 11. Predicted shaft power output vs. operating speed for the test engine. (a) Effects Results show that as the operating speed is increased, theof molar number at RH ¼ RC ¼ 0.3 K/W and eH ¼ eC ¼ 0.7. (b) Effects of thermal temperature difference between the expansion and theresistances at N ¼ 0.0025 mol and eH ¼ eC ¼ 0.7. (c) Effects of regenerator effectivenessat N ¼ 0.0025 mol and RH ¼ RC ¼ 0.3 K/W. compression spaces is reduced and as a result, the shaft work
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