Marsland Press Jurnal f American Science ;6( Perfrmance f an endreversible Atkinsn cycle with variable specific heat rati f wrking fluid Rahim Ebrahimi Department f Agriculture Machine Mechanics, Shahrekrd University, P.O. Bx 5, Shahrekrd, Iran Rahim.Ebrahimi@gmail.cm Abstract: The perfrmance f an air standard Atkinsn cycle is analyzed using finite-time thermdynamics. In the endreversible cycle mdel, the linear relatin between the specific heat rati f the wrking fluid and its temperature, and the heat transfer lss are cnsidered. The relatins between the net wrk utput, the thermal efficiency, and the cmpressin rati are indicated by numerical examples. Mrever, the effects f variable specific heats f the wrking fluid n the endreversible cycle perfrmance are analyzed. The results shw that the effect f the temperature dependent specific heat f the wrking fluid n the endreversible cycle perfrmance is significant. The cnclusins f this investigatin are f imprtance when cnsidering the designs f actual Atkinsn engines. [Jurnal f American Science ;6(:-7]. (ISSN: 55-. Key wrds: Atkinsn heat-engine; Finite-time prcesses; Heat lss; Perfrmance; Thermdynamics. Intrductin Recently, the analysis and ptimizatin f thermdynamic cycles fr different ptimizatin bjectives has made tremendus prgress by using finite-time thermdynamics (Aizenbud and Band, 98; Bejan, 996; Chen et al., 999; Wu et al., 999; Chen and Sun ; Aragn-Gnzalez et al., 6. Leff (987 determined the thermal efficiency f a reversible Atkinsn-engine cycle at maximum wrk utput. A pwer density maximizatin f a reversible Atkinsn cycle has been perfrmed by Chen et al. (998a. Their results shwed that the efficiency at maximum pwer density is always greater than that at maximum pwer, and the design parameters at maximum pwer density lead t smaller and mre efficient Atkinsn engines with larger pressure ratis. Al-Sarkhi et al. ( cmpared the perfrmance characteristic curves f the Atkinsn cycle with thse f the Miller and Jule Braytn cycles by using numerical examples, and utlined the effect f maximizing pwer density n the perfrmance f the cycle efficiency. Qin et al. ( derived the perfrmance characteristics f a universal generalized cycle mdel, which included the Atkinsn cycle, with heat-transfer lss. Wang and Hu (5 studied the perfrmance analysis and cmparisn f an Atkinsn cycle cupled t variable temperature heat reservirs under maximum wrk and maximum pwer density cnditins, assuming a cnstant specific heat, t. Their results shwed an engine design based n maximum pwer density is better than that based n maximum wrk cnditins, frm the view pints f engine size and thermal efficiency. Ge et al. (5a derived the perfrmance characteristics f a universal generalized cycle mdel, which included the Atkinsn cycle with heat transfer and frictin-like term lsses. Zha and Chen (6 perfrmed analysis and parametric ptimum criteria f an irreversible Atkinsn heat engine using finite time thermdynamics. Perfrmance analysis f an Atkinsn cycle with heat transfer, frictin and variable specific heats f the wrking fluid was studied by Ge et al. (6. Their results shwed that the effects f variable specific heats f wrking fluid and frictin-like term lsses n the irreversible cycle perfrmance are significant. Ge et al. (7 analyzed the effects f the heat transfer and variable specific heats f wrking fluid n the perfrmance f an endreversible Atkinsn cycle. Hu (7 cmpared the perfrmances f air standard Atkinsn and Ott cycles with heat transfer lss cnsideratins. Lin and Hu (7 investigated the effects f heat lss, as characterized by a percentage f fuel s energy, frictin and variable specific heats f the wrking fluid, n the perfrmance f an air standard Atkinsn cycle under the restrictin f the maximum cycle-temperature. Chen et al. (7 built a class f generalized irreversible universal steady flw heat engine cycle mdel cnsisting f tw heating branches, tw cling branches, and tw adiabatic branches with cnsideratin f the lsses f heat resistance, heat leakage, and internal irreversibility. The perfrmance characteristics f Diesel, Ott, Braytn, Atkinsn, Dual and Miller cycles were derived. Chen et al. (8 analyzed and http://www.americanscience.rg americansciencej@gmail.cm
Marsland Press Jurnal f American Science ;6( cmpared the perfrmance characteristics f endreversible and irreversible reciprcating Diesel, Ott, Atkinsn, Braytn, Brayssn, Carnt, dual, and Miller cycles with cnstant and variable specific heats f the wrking fluid. Thermdynamic analysis f an ideal air-standard Atkinsn cycle with temperature dependant specific heat is presented by Al-Sarkhi et al. (8. This paper utlines the effect f maximizing pwer density n the perfrmance f the cycle efficiency. Ge et al. (8a, 8b analyzed the perfrmance f an air standard Ott and Diesel cycles. In the irreversible cycle mdel, the nn-linear relatin between the specific heat f the wrking fluid and its temperature, the frictin lss cmputed accrding t the mean velcity f the pistn, the internal irreversibility described by using the cmpressin and expansin efficiencies, and the heat transfer lss are cnsidered. Ust (9 made a cmparative perfrmance analysis and ptimizatin f irreversible Atkinsn cycle under maximum pwer density and maximum pwer cnditins. All f the abve mentined research, the specific heats at cnstant pressure and vlume f wrking fluid are assumed t be cnstants r functins f temperature ae and have the linear and r the nn-linear frms. But when calculating the chemical heat released in cmbustin at each instant f time fr internal cmbustin engine, the specific heat rati is generally mdeled as a linear functin f mean charge temperature (Gatwski et al., 98; Ebrahimi, 6. The mdel has been widely used and the phenmena that it takes int accunt are well knws (Klein,. Hwever, since the specific heat rati has a great influence n the heat release peak and n the shape f the heat release curve (Brunt, 998, many researchers have elabrated different mathematical equatins t describe the dependence f specific heat rati frm temperature (Gatwski et al., 98; Brunt, 998; Egnell, 998; Klein, ; Klein and Eriksn, ; Ceviz and Kaymaz, 5. It shuld be mentined here that the mst imprtant thermdynamic prperty used in the heat release calculatins fr engines is the specific heat rati (Ceviz and Kaymaz, 5. Therefre, the bjective f this study is t examine the effect f variable specific heat rati n the net wrk utput and the thermal efficiency f air standard Atkinsn cycle.. Cycle mdel The Atkinsn cycle engine is a type f internal cmbustin engine, which was designed and built by James Atkinsn in 88 (Ge et al., 5a. The Atkinsn cycle, ne f the mst heat-efficient, high-expansin rati cycles, is designed t prvide efficiency at the expense f pwer. The Atkinsn cycle allws the intake, cmpressin, pwer, and exhaust strkes f the fur-strke cycle t ccur in a single turn f the crankshaft. By the use f clever mechanical linkages, the expansin rati is greater than the cmpressin rati, resulting in greater efficiency than with engines using the alternative Ott cycle. The cycle fr this engine is depicted in figure. The cycle is als called the Sargent cycle by several physics riented thermdynamic bks (Ge et al. 7. Figure presents pressure-vlume ( P V and temperature -entrpy ( T S diagrams fr the thermdynamic prcesses perfrmed by an air standard Atkinsn cycle. Prcess ( is an adiabatic (isentrpic cmpressin; prcess ( is a heat additin at a cnstant vlume; prcess ( adiabatic (isentrpic expansin; prcess ( is heat rejectin at a cnstant pressure. As already mentined in the previus sectin, it can be suppsed that the specific heat rati f the wrking fluid is functin f temperature ae and has the linear frms: γ = γ kt ( where γ is the specific heat rati and T is the abslute temperature. γ are cnstants. The heat added t the wrking fluid, during prcesses ( is T T R Qin = M ( cvdt = M T dt = T γ kt ( MR γ kt k γ kt where M is the mlar number f the wrking fluid. R and c p are mlar gas cnstant and mlar specific heat at cnstant vlume fr the wrking fluid, respectively. The heat rejected in the isbaric heat rejectin prcess ( may be written as T T ( γ kt R Qut = M cpdt = M T dt = T γ kt ( γ kt MR T T + k γ kt where cp is the mlar specific heat at cnstant http://www.americanscience.rg americansciencej@gmail.cm
Marsland Press Jurnal f American Science ;6( P T Pressure Temperature (a Vlume V (b Entrpy S Figure. (a P V diagram;(b T S diagram fr the air standard Atkinsn cycle pressure fr the wrking fluid. Accrding t references (Ge et al., 6; Ebrahimi 9, the equatin fr a reversible adiabatic prcess with variable specific heat rati can be writing as fllws: γ TV ( T dt ( V dv γ = + + ( Re-arranging equatins ( and (, we get the fllwing equatin Ti ( γ kt j Tj ( γ kt i ( V j / Vi γ = (5 The specific cmpressin, r c, and cmpressin, r c, ratis are defined as rc = V V (6 and V T r c = = rc (7 V T Therefre, the equatins fr prcesses ( and ( are shwn, respectively, by the fllwing: γ ( γ ( = ( γ T k T r T k T (8 c γ T T ( γ kt = T ( γ kt rc (9 T The energy transferred t the wrking fluid during cmbustin is given by the fllwing linear relatin (Chen et al. 998b; Ge et al., 8a. in ( Q = M A B T + T ( where A and B are tw cnstants related t cmbustin and heat transfer which are functin f engine speed. Frm equatin (, it can be seen that Q cntained tw parts: the first part is MA, the in released heat by cmbustin per secnd, and the secnd part is the heat leak lss per secnd, Qleak = MB ( T + T. Frm equatins ( and (, the net wrk utput f the Atkinsn cycle engine is given by: Wut = Qin Qut = MR ( γ kt ( γ kt 5 ( MR ( T T k + ( γ kt ( γ kt The thermal efficiency f the Atkinsn cycle engine is expressed by: ( γ kt ( γ kt 5 T T k ( γ kt ( γ kt + η th = ( γ kt + T T k γ kt When the values f rc and T are given, T can be btained frm equatin (8, then, substituting equatin ( int equatin ( yields T. The last unknwn is T, which can be deduced frm equatin (9. Finally, by substituting T, T, T and T int equatins ( and (, respectively, the net wrk utput and thermal efficiency f the Atkinsn cycle engine can be btained. Therefre, the relatins between the net wrk utput, the thermal efficiency and the cmpressin rati can be derived.. Numerical examples and discussin Accrding t references (Ebrahimi, 9, Ghatak and Chakrabrty, 7; Ge et al., 7, Chen et al., 6; Ge et al., 5b the fllwing cnstants and ranges f parameters are used in the calculatins: T = 6 K, γ =.., A = 6 J. ml, http://www.americanscience.rg americansciencej@gmail.cm
Marsland Press Jurnal f American Science ;6( 5 M =.57 kml, k =..9 and B = 8 J. ml K. Numerical examples are shwn as fllws. Figures -5 shw the effect f the parameters γ and k related t the variable specific heat rati f the wrking fluid n the Atkinsn cycle perfrmance with cnsideratins f heat transfer. Frm these figures, it can be fund that γ play a key rle n the wrk utput and the thermal efficiency. It is clearly seen that the effects f γ n the wrk utput and thermal efficiency are related t cmpressin rati. They reflect the perfrmance characteristics f an endreversible Atkinsn cycle engine. It shuld be nted that the heat added and the heat rejected by the wrking fluid decrease with increases f γ, while increase with increasing k. (see Eqs. ( and (. It can be seen that the effect f γ is mre than that f k n the net wrk utput and thermal efficiency. It shuld be mentined here that fr a fixed k, a larger γ crrespnds t a greater value f the specific heat rati and fr a given γ, a larger k crrespnds t a lwer value f the specific heat rati....9.8 γ =. γ =.6 γ =..9.8 k =. k =.6 k =.9 Net wrk utput (kj.7.6.5.. Net wrk utput (kj.7.6.5...... 6 Cmpressin rati Cmpressin rati Figure. Effect f γ n the variatin f the net wrk Figure. Effect f k n the variatin f the net k =.6 wrk utput with cmpressin rati ( γ =.6 utput with cmpressin rati ( The effects f γ n the net wrk utput are shwn in Figures and. It can be fund frm these figures that the net wrk utput versus cmpressin rati characteristic is apprximately parablic like curves. In ther wrds, the net wrk utput increases with increasing cmpressin rati, reach their maximum values and then decreases with further increase in cmpressin rati. It can als be fund frm the figures and that if cmpressin rati is less than certain value, the increase (decrease f γ ( k will make the net wrk utput bigger, due t the increase in the rati f the heat added t the heat rejected. In cntrast, if cmpressin rati exceeds certain value, the increase (decrease f γ ( k will make the net wrk utput less, because f decrease in the rati f the heat added t the heat rejected. One can see that the maximum net wrk utput, the wrking range f the cycle and the ptimal cmpressin rati crrespnding t maximum net wrk utput decrease (increase abut % (.8% and 66.5% (5.7%,.6% (5% when γ ( k increases 7.6% (%. This is due t the fact that the rati f heat added t heat rejected increases (decreases with increasing γ ( k in this case. It shuld be nted here that bth the heat added and the heat rejected by the wrking fluid decrease with increasing γ (see Eq. (, and increase with increase f k (see Eq. (5. The effects f γ n the thermal efficiency are shwn in Figures and 5. It can be fund that the thermal efficiency increases with the increase f γ and the decrease f k thrughut the cmpressin http://www.americanscience.rg 5 americansciencej@gmail.cm
Marsland Press Jurnal f American Science ;6( 7 7 6 6 Thermal efficiency (% 5 γ =..6 γ = γ =. Thermal efficiency (% 5 k =. k =.6 k =.9 6 Cmpressin rati Cmpressin rati Figure. Effect f γ n the variatin f the thermal Figure. Effect f k n the variatin f the thermal k =.6 efficiency with cmpressin rati ( γ =.6 efficiency with cmpressin rati ( rati range. On average, the thermal efficiency increases (decreases by abut 9.7% (9.% when γ ( k increases (increases 7.6% (% ver a range f cmpressin ratis frm. (. t 8.9 (.6.. Cnclusin In this paper, an endreversible air standard Atkinsn cycle mdel taking cnsideratins f heat transfer lss and the variable specific heat rati f wrking fluid is presented. The relatins between the net wrk utput and the cmpressin rati and between the thermal efficiency and the cmpressin rati f the cycle are derived. The effects f the cycle parameters, such as γ, n the net wrk utput and the efficiency were analyzed by detailed numerical examples. The results btained may prvide a theretical basis fr bth the ptimal design and peratin f real Atkinsn heat engines. Crrespndence t: Rahim Ebrahimi Department f Agriculture Machine Mechanics Shahrekrd University, P O Bx 5, Shahrekrd, Iran Tel/Fax: 98-8- Email: Rahim.Ebrahimi@gmail.cm Reference [] Aizenbud BM, Band YB. Optimizatin f mdel internal cmbustin engine. Jurnal f Applied Physics 98; 5: 77 8. [] Al-Sarkhi A, Akash BA. Efficiency f Miller engine at maximum pwer-density. Internatinal Cmmunicatins in Heat and Mass Transfer ;9: 59 67. [] Al-Sarkhi A, Akash B, Abu-Nada E, Al-Hinti I. Efficiency f Atkinsn engine at maximum pwer density using temperature dependent specific heats, Jrdan Jurnal f Mechanical and Industrial Engineering 8;(:7 75. [] Aragn-Gnzalez G, Canales-Palma A, Len-Galicia A, Mrales-Gmez JR. Optimizatin f an irreversible Carnt engine in finite time and finite size. Revista Mexicana de Fisica 6;5 (:9. [5] Bejan A. Entrpy-generatin minimizatin: the new thermdynamics f finite-size device and finite-time prcesses. Jurnal f Applied Physics 996;79(:9 8. [6] Brunt MFJ, Rai H, Emtage AL. The calculatin f heat release energy frm engine cylinder pressure data. SAE Paper N. 985, 998. [7] Ceviz MA, Kaymaz I. Temperature and air fuel rati dependent specific heat rati functins fr lean burned and unburned mixture. Energy Cnversin and Management 5;6:87. [8] Chen L, Lin J, Wu C, Sun F. Efficiency f an Atkinsn engine http://www.americanscience.rg 6 americansciencej@gmail.cm
Marsland Press Jurnal f American Science ;6( at maximum pwer-density. Energy Cnversin Management 998a;9(/:7. [9] Chen L, Wu C, Sun F, Ca S. Heat-transfer effects n the net wrk-utput and efficiency characteristics fr an air standard Ott cycle. Energy Cnversin Management 998b; 9(7:6 68. [] Chen L, Wu C, Sun F. Finite-time thermdynamic ptimizatin r entrpy-generatin minimizatin f energy systems. J Nn-Equilib Thermdyn 999;(:7 59. [] Chen L, Sun F. Advances in finite-time thermdynamics:analysis and ptimizatin. New Yrk, Nva Science Publishers.. [] Chen L, Ge Y, Sun F, Wu C. Effects f heat transfer, frictin and variable specific-heats f a wrking fluid n perfrmance f an irreversible Dual cycle. Energy Cnversin and Management 6;7(8/9:. [] Chen L, Zhang W, Sun F. Pwer, efficiency, entrpy generatin rate and eclgical ptimizatin fr a class f generalized irreversible universal heat engine cycles. Appllied Energy 7;8(5:5 55. [] Chen L, Ge Y, Sun F. Unified thermdynamic descriptin and ptimizatin fr a class f irreversible reciprcating heat engine cycles. Prceedings f the Institutin f Mechanical Engineers, Part D: Jurnal f Autmbile Engineering. 8;:89-5. [5] Ebrahimi, R., Experimental study n the aut ignitin in HCCI engine. Ph.D. Thesis, Valenciennes et du Hainaut-Cambrésis, France, (In French. 6. [6] Ebrahimi R. Effects f cut-ff rati n perfrmance f an irreversible Dual cycle. Jurnal f American Science 9;5(:8-9. [7] Egnell R. Cmbustin diagnstics by means f multizne heat release analysis and NO calculatin. SAE Paper N. 98, 998. [8] Hu, S.S.,. Heat transfer effects n the perfrmance f an air standard Dual cycle. Energy Cnversin and Management. 5(8/9, -5. [9] Gatwski JA, Balles EN, Chun KM, Nelsn F, Ekchian JA, Heywd FB. A heat release analysis f engine pressure data. SAE Paper N. 859. 98. [] Ghatak A, Chakrabrty S. Effect f external irreversibilities and variable thermal prperties f wrking fluid n thermal perfrmance f a Dual internal cmbustin engine cycle. Strjnicky Caspis (Jurnal f Mechanical Energy 7;58:. [] Ge Y, Chen L, Sun F, Wu C. Perfrmance f an Atkinsn cycle with heat transfer, frictin and variable specific heats f wrking fluid. Applied Energy 6;8(:. [] Ge Y, Chen L, Sun F, Wu C. Perfrmance f an endreversible Atkinsn cycle. Jurnal f the Energy Institute 7;8(:5 5. [] Ge Y, Chen L, Sun F, Wu C. Reciprcating heat-engine cycles. Applied Energy 5a;8(, 97 8. [] Ge Y, Chen L, Sun F, Wu C. Thermdynamic simulatin f perfrmance f an Ott cycle with heat transfer and variable specific heats f wrking fluid. Internatinal Jurnal f Thermal Sciences 5b;(5:56 5. [5] Ge Y, Chen L, Sun F. Finite time thermdynamic mdeling and analysis f an irreversible Ott cycle. Applied Energy 8a;85(7:68-6. [6] Ge Y, Chen L, Sun F. Finite time thermdynamic mdeling and analysis fr an irreversible Diesel cycle. Prceedings IMechE, Part D: Jurnal f Autmbile Engineering 8b;(D5:887-89. [7] Klein M. A Specific Heat Rati Mdel and Cmpressin Rati Estimatin. Department f Electrical Engineering, Ph.D. Thesis, Linköping University. Sweden.. [8] Klein M, Eriksn L. A specific heat rati mdel fr single-zne heat release mdels. SAE Paper N. --6.. [9] Lanzafame R, Messina M. ICE grss heat release strngly influenced by specific heat rati values. Internatinal Jurnal f Autmtive Technlgy ;:5. [] Leff HS. Thermal efficiency at maximum pwer utput: new results fr ld heat engines. American Jurnal f Physics 987;55:6 6. [] Lin JC, Hu SS. Influence f heat lss n the perfrmance f an air-standard Atkinsn cycle. Applied Energy 7;8:9 9. [] Qin X, Chen L, Sun F. The universal pwer and efficiency characteristics fr irreversible reciprcating heat engine cycles. Eurpean Jurnal f Physics ;(:59 66. [] Ust Y. A cmparative perfrmance analysis and ptimizatin f irreversible Atkinsn cycle under maximum pwer density and maximum pwer cnditins. Internatinal Jurnal f Thermphysics, in press, 9 [] Wang P, Hu SS. Perfrmance analysis and cmparisn f an Atkinsn cycle cupled t variable temperature heat reservirs under maximum pwer and maximum pwer density cnditins. Energy Cnversin and Management 5;6:67 655. [5] Wu C, Chen L, Chen C. Recent Advances in Finite Time Thermdynamics. New Yrk: Nva Science Publishers. 999. [6] Zha Y, Chen J, An irreversible heat engine mdel including three typical thermdynamic cycles and the ptimum perfrmance analysis. Internatinal Jurnal f Thermal Sciences 7;6(6: 65 6. 9//9 http://www.americanscience.rg 7 americansciencej@gmail.cm