A METHOD FOR DETERMINING THE PERFORMANCES OF STIRLING MACHINES BASED ON THE FIRST LAW FOR PROCESSES WITH FINITE SPEED AND USING A PV/PX DIAGRAM

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1 The 3rd International Conference on Coputational Mechanics and Virtual Enineerin COMEC OCTOBER 009 Brasov Roania A METHOD FOR DETERMINING THE PERFORMANCES OF STIRING MACHINES BASED ON THE FIRST AW FOR PROCESSES WITH FINITE SPEED AND USING A PV/PX DIAGRAM Prof.En. Traian FOREA PhD 1 Prof.En. Alexandru DRAGAINA PhD 1 En. Traian Vasile FOREA Assist.prof.en. Mihai BEJAN PhD 1 1 Departent of Theral Machines and Devices Naval Acadey of Constanta ROMANIA S.C. Delatreid S.R.. Constanta ROMANIA e-ail: prof_florea@yahoo.co; Abstract: A technique for calculatin the efficiency and power of Stirlin achines is presented. This technique is based on the First aw of Therodynaics for processes with finite Speed. A new and novel PV/Px diara is presented that shows the effects of pressure losses due to friction finite speed and throttlin processes in the reenerator of the Stirlin enine. The ethod used for the analysis of this irreversible cycle with finite speed involves the direct interation of equations based on the First aw for processes with finite speed to directly obtain the cycle efficiency and power. This technique is tered the Direct Method. The results predicted by this analysis were in ood areeent with the actual enine perforance data of tweve different Stirlin enines over a rane of output fro econoy to axiu power. This provides a solid verification that this analysis can accurately predict actual Stirlin enine perforance particulary with reard to efficiently and output power. Keywords: irreversibility losses reeneration Stirlin efficiency. 1. INTRODUCTION A new technique for calculatin the efficiency and power of actual operatin Stirlin achines is presented. This technique is based on the First aw of Therodynaics for processes with finite speed [1 14] and a new ethod for deterinin the iperfect reeneration coefficient [15].In the Direct Method for the Study and Optiization of Irreversible Cycles with Finite Speed [5 6 10] equations based on the first law of therodynaics for processes with finite speed are interated in such a anner that the expressions for efficiency and power are obtained directly for any irreversible cycle. This ethod has been applied to Stirlin achines previously [10 13]. The analytical results depend upon inclusion of two calibration coefficients (y and z) based on experiental data to accurately predict perforance. The theral efficiency is expressed as a product of the Carnot cycle efficiency and second law efficiency as has been suested by Bejan: 1 T = = T η + SE ηcc ηii irrev 1 1 TH S 1443 T 14 H S η η ( 1 T / T ) II irrev T 1 X H S 3µ Pi 1+ 1 ( γ 1) η P 3 1 TH S / T η II irrev X η II irrev CC η = 1 η ; µ = 1 1/ ( 3ε). T / with ( T H S ) II irrev X P i The pressure losses and their effect on efficiency and power of the enine depend on the piston speed and hence the speed of the enine. The power output of the enine is: 4 (1)

2 ( w / S) PowerSE = η SE RTH () The speed for axiu power ay be deterined since the power output is also a function of the enine speed. Therefore the operatin speed of a particular Stirlin enine can be selected for either axiu econoy or for axiu power. Also knowlede of the nature of these losses can be effectively used in enine desin. A ajor loss in Stirlin enines is caused by incoplete reeneration. This is expressed by the coefficient of reenerative losses X. An analysis for deterinin this loss has been ade by Petrescu et al. and Florea [15]: η = M + e X = 1+ M [ X 1 y+ X ( 1 y) ]( 1 T / TH S ) R / c ( T) II irrev X ; (3) v B 1 ; 0395 h= ( ) X M + e = 1+ M B ; 044 ( 4P / RT ) w c ( T ) v( T ) ( 1+ τ) 1 4 π [( b / d) + 1] P D 0576 R Pr / c M v = ; B= ( + M) C R R har S 1 ; (4-7) c w v (8) where y is one of the adjustin coefficients of the ethod with the value of 0.7. One objective of this paper is to ake a ore realistic analysis of the pressure losses throuh use of a PV/Px diara as will be described below and by Petrescu et al. [14]. Finally the technique for calculatin the efficiency and power of Stirlin enines is presented and the results predicted by this analysis are copared with perforance data taken on twelve actual Stirlin enines over a rane of operatin conditions.. AN INTUITIVE PV/Px DIAGRAM FOR DESCRIPTION OF THE STIRING CYCE PROCESS Oran akes a rearkable analysis of the different ethods for treatent of Stirlin cycle enines. He presents the existin ethods of analysis and points out the advantaes of each. He places each of these ethods between two extrees of coplexity. The siplest presents the Stirlin cycle as only a diara on PV coordinates that consists of two isotheral process lines cobined with two isoetric process lines. This is the typical introductory textbook representation. In discussin the liits of the analysis of the Stirlin cycle based on use of the PV diara Oran rearks that it is rerettable that the existin diaras do not describe the essential processes even for the ideal Stirlin cycle. The above entioned PV diara is rossly insufficient to provide a clear description of even the ideal cycle. It is even ore inadequate when considerin real achine. At the other extree of coplexity he considers the ost coplex ethods to be those based on the nuerical interation of differential equations written to account for as any of the actual variables in Stirlin cycle achines as possible. These coplex analyses (so called third order ethods ) are capable of a hih deree of atheatical precision but they have a severe disadvantae in that they tend to totally obscure the physical processes involved. They are not effective in aidin the understandin of the basic cycle nor are they effective aids to desiners in their efforts at optiization of actual Stirlin cycle achines. The objectives of this section are: (1) to present a new and novel PV/Px diara for the Stirlin cycle () to use this diara to provide a solid basis for understandin the processes of the actual Stirlin enine and (3) to provide an analysis and calculations based on the actual processes as elucidated by the diara useful for desin. Once the actual processes of the Stirlin cycle and the way the achine actually functions are clear and analysis is possible that predicts the perforance of the achines that in ters of coplexity lies soewhere between the extrees cited above. As is well known the ideal Stirlin cycle has no losses. However in the actual Stirlin cycle achine losses due to all heat transfer throuh a teperature difference pressure losses due to friction and finite speed and losses due to incoplete reeneration occur and need to be accounted for. Various aspects of these losses have been exained in previous papers [ ]. The nature and oriin of these losses or irreversibilities will be dealt in the analysis that follows. This analysis considers two of the arraneents that are characteristic to any operatin Stirlin enines [30] naely the split enine with displacer and piston (type 1) and two pistons enine (type ). The PV&Px diara for the enine type 1 is presented by Petrescu et al. [14] and that for type will be presented here. 43

3 Fiure 1: The PV/Px diara for description of the Stirlin cycle processes The ideal Stirlin cycle is shown as (a) in Fiure 1. In this idealization both pistons ove discontinuously as shown on the top of the fiure above each cylinder (1 C - C -3 C -4 C -1 C and 1 E - E -3 E -4 E -1 E ) with the speed w 0. The PV/Px diara correspondin to the theoretical Stirlin cycle with finite constant speed is shown as cycle (b) in Fiure 1. The piston speed is considered to be constant and equal to the averae speed. The pressure losses due to this finite piston speed are taken into account. This includes the pressure losses due to the interaction between the as and pistons the pressure losses due to throttlin in the reenerator and the pressure losses due to friction. In state 1 C at the beinnin of the copression process the copression piston in P C is position 1 C and the expansion piston P E is in position 1 E. Durin the isotheral copression process (1 C - C ) enerated by the piston P C the expansion piston P E is stationary. The pressure on the piston P C will be reater than averae pressure of the as in the syste so that the pressure difference P C occurs because of the finite speed of the piston P C [ ]. Durin the isotheral expansion process 3 E 4 E the conditions are reversed and the pressure on the piston P E will be less than the averae pressure of the as syste as a whole. The pressure differences produced in these processes will cause a reduction in the work produced in the cycle. Pressure differences also occur between the as and the pistons as they ove with finite speed in the process 3 and 41 in both cylinders as shown on PV/Px coordinates. For exaple the process -3 is accoplished by the siultaneous oveent of the pistons in the cylinders eneratin the lines C 3 C and E 3 E on Px coordinates. Due to this oveent the cold as passes throuh the reenerator R and is heated. The heat coes fro the porous atrix of the reenerator that have been heated durin 44

4 the previous cycle. The pressure differences produced in these processes also cause a reduction in the work produced by the Stirlin enine. Work losses are shown on the diara as shaded areas. Once this work is known the efficiency and power of the Stirlin enine cycle with finite speed ay be calculated. Therefore its evaluation was one of the ain objective of this study. The results of this evaluation are presented in a short for in the next section. 3. THE METHOD OF DETERMINING THE PERFORMANCE OF THE STIRING ENGINE Coputation of pressure losses work losses efficiency and power for the processes shown on the new PV/Px diaras [14 15] are ade usin the first law of therodynaics for processes with finite speed [1-13]. The first law written to specifically include these conditions is: aw b P f P du Q P thrott f = δ i 1± ± ± dv. (9) c P i P i The irreversible work is: aw b P P W P thrott f δ irrev = i 1± ± ± dv (10) c P i P i when applied to processes with finite speed as shown on the PV/Px diara. Coputin and suin all pressure losses of the Stirlin enine cycle presented above the ter η fro II irrev Σ Pi eq. (1) becoes [14 15]: w w 3 ( ) ( ) w γ 1+ τ 5 N + w w S S 4ρ4 ηii irrev Pi 1 Σ =. (11) τη τη The heat input durin the expansion process is also irreversible due to finite speed. In order to take account of this influence an calibration coefficient z is introduced: Q34 = z RT H. (1) Finally the real power output of the enine eq. () becoes: PowerSE irrev = ηse zrth ( w / S) (13) where the value of z was evaluated at 08 by coparison with available experiental data for twelve Stirlin enines. 4. DISCUSSIONS The variation of the coefficient of reenerative losses X with the piston speed for several values of the as averae pressure is shown in Fiure. It is an exaple of the results obtained fro the sensitivity study [15] for X. It shows an iportant increase of the reenerative losses with the as averae pressure. Then the results of coputations based on this analysis are copared to perforance data taken fro a nuber of operatin Stirlin enines in Fiures 3-5 and Table 1. The hih deree of correlation between the analytic and the operational data shown in these fiures indicate that the analysis is capable of accurately predictin Stirlin enine perforance under a wide rane of conditions. This capability should be of considerable value in Stirlin enine desin and in predictin the perforance of a particular Stirlin enine over a rane of operatin speed. Table 1: Coparison between the analytical results and actual enine perforance data Stirlin Enine Actual Power Calculated Power Calculated Actual Efficiency Efficiency NS-03M reie 1 (econoy) NS-03M reie 1 (ax. power) NS-03T reie 1 (econoy) NS-03T reie 1 (ax. power) NS-30A reie 1 (econoy) Table 1 (second part) 45

5 Stirlin Enine Actual Power Calculated Power Calculated Actual Efficiency Efficiency NS-30A reie 1 (ax. power) NS-30S reie 1 (econoy) NS-30S reie 1 (ax. power) STM V MKII GPU MP100 CA 00W 193.9W Free Piston Stirlin Enine RE Fiure : Coefficient of reenerative losses versus the piston speed for several values of the averae pressure of the workin as (D C = 60 b/d = 15 τ = N = 700) Fiure 3: Coparison of the analysis results with actual perforance data for the NS-30S Stirlin enine Fiure 4: Coparison of the analysis results with actual perforance data for the 4-75 Stirlin enine Fiure 5: Coparison of the analysis results with actual perforance data for the Free Piston Stirlin enine 46

6 5. CONCUSIONS The objective of this approach was to closely siulate the operation of actual Stirlin enines without losin insiht to the echaniss that enerate the irreversibilities. Pressure and work losses enerated by finite speed of the actual processes were coputed as were the power and efficiency of enines. The first law of therodynaics for processes with finite speed was used to copute the power losses enerated by the pressure losses. The analysis presented was applied to specific operatin Stirlin cycle enines and results were copared to the easured perforance of the enines. The stron correlation between the analytical results and actual enine perforance data indicates that the Direct Method of usin the First aw for Finite Speed is a valid ethod of analysis for irreversible cycles. REFERENCES [1] Stoicescu. Petrescu S.: The First aw of Therodynaics for Processes with Finite Speed in Closed Systes (Geran lanuae) Bulletin of Polytechnic Institute of Bucharest 1964a Vol. XXVI No. 5; [] Petrescu S. Florea T. Haran Ch. Costea M.: A Method for Calculatin the Coefficient for the Reenerative osses in Stirlin Machines European Stirlin Foru 000 Osnabrück Gerany February pa ; [3] Petrescu S. Florea T. Costea M. Haran Ch.: Application of the Direct ethod to Irreversible Stirlin cycles with Finite speed International Journal of Enery Research No.6 May 7 00 pa edited by John Wiley & Sons td. USA; [4] Florea T. Petrescu S. Costea M. Feidt M.: a ethode directe utilisée dans la therodynaique a vitesse finie pour l optiisation des achines therique Enerie environneent econoie et therodynaique Université Politehnica Bucarest avril 00 pa 7-79 ISBN ; [5] Florea T. Petrescu S. Feidt M. Haran Ch. Costea M.: Optiization of the Irreversible Carnot Cycle Enine for Maxiu Efficiency and Maxiu Power throuh Use of Finite Speed Therodynaic Analysis Int. ECOS 00 Conference G. Tsatsaronis M. Moran F. Cziesla and T. Bruckner eds. Berlin Gerany Vol. II pp ; [6] Florea T. Petrescu S. Costea M. Petre C. Feidt M.: A schee of Coputation Analysis Desin and Optiization of Solar Stirlin Enines ECOS-003 CopenhaenDenark Volue IEditors: Niels Houbac Brian Eleaard Bjorn Qvale Michael J. Moran June 30-July 003 pa isbn ; [7] Florea T. Draalina Al. Costiniuc C. Florea E. Florea T.-V.: A Method for Calculatin of the Coefficient for the Reenerative osses in Stirlin Machines COMEFIM 8 The 8 th International Conference on Mechatronics and Precision Enineerin Technical University of Cluj Napoca June 8 th - 10 th 006 pa ISBN ; [8] Florea T. Draalina Al. Costiniuc C. Florea E. Florea T.-V.: A Method for Deterini the Perforances of Stirlin Machines Based on the First aw for Processes with Finite Speed and usin a pv/px Diara COMEFIM 8 The 8 th International Conference on Mechatronics and Precision Enineerin Technical University of Cluj Napoca June 8 th - 10 th 006 pa ISBN ; [9] Draalina Al. Florea T. Costiniuc C. Dancu C.: Control Optiisation and oad Prediction for Marine Diesel Enines Usin a Mean Value Siulation Model Conferinta internationala NAV-MAR-EDU 007 Constanta noiebrie 007 ISBN ; [10] Florea T. Draalina Al. Florea T.-V. Pruiu A.: The study of the irreversibility of the operational process of the external cobustion enines with heat reenerators and increase of power and output for the internal cobustion enines usin the experiental and raphoanalytical ethods Proceedins of the internationally attended national conference on therodynaics Brasov Roania 1 May 009 Vol II Series I ISSN Special Issue No.1 Vol.1 009pa ISBN ; [11] Florea T.: Reenerarea cãldurii în asinile terice Editura eda&muntenia ConstanŃa pa. ISBN si ISBN X; [1] Florea T.: Reiurile optie de putere axiã şi rafice lobale sintetice pentru optiizarea otoarelor cu ardere externã Stirlin Editura Acadeiei Navale ConstanŃa pa. ISBN ; [13] Florea T. Petrescu S. Florea E.: Schee de calcul pentru studiul ireversibilitãńii proceselor funcńionale ale otoarelor cu ardere externã Stirlin Editura eda&muntenia ConstanŃa pa. ISBN şi ISBN ; [14] Petrescu S. Florea T. Haran Ch. Costea M.: Advanced Enery Conversion - volue I(ed.rev.) Bucknell University ewisbur PA USA January pa. MECH 4/6; [15] Petrescu S. Florea T. Zaiser J. Haran Ch. Petrescu V. Costea M. Petre C. Florea T.-V.: Advanced Enery Conversion - volue II (ed.rev.) Bucknell University ewisbur PA USA February pa. MECH 4/6. 47