FINITE TIME THERMODYNAMIC MODELING AND ANALYSIS FOR AN IRREVERSIBLE ATKINSON CYCLE

Transcription

1 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp FINIE IME HERMODYNAMIC MODELING AND ANALYSIS FOR AN IRREVERSIBLE AKINSON CYCLE by Yanlin GE, Lingen CHEN *, and Fengrui SUN Postgraduate School, Naval University of Engineering, Wuhan, P. R. China Orig i nal sci en tific pa per UDC: 36.27:17.97 DOI: /SCI G Per for mance of an air-stan dard Atkinson cy cle is an a lyzed by us ing fi nite-time ther mo dy nam ics. he ir re vers ible cy cle model which is more close to prac tice is founded. In this model, the non-lin ear re la tion be tween the spe cific heats of work - ing fluid and its tem per a ture, the fric tion loss com puted ac cord ing to the mean ve - loc ity of the pis ton, the in ter nal ir re vers ibil ity de scribed by us ing the com pres sion and ex pan sion ef fi cien cies, and heat trans fer loss are con sid ered. he re la tions be - tween the power out put and the com pres sion ra tio, be tween the ther mal ef fi ciency and the com pres sion ra tio, as well as the op ti mal re la tion be tween power out put and the ef fi ciency of the cy cle are de rived by de tailed nu mer i cal ex am ples. More - over, the ef fects of in ter nal ir re vers ibil ity, heat trans fer loss and fric tion loss on the cy cle per for mance are an a lyzed. he re sults ob tained in this pa per may pro vide guide lines for the de sign of prac ti cal in ter nal com bus tion en gines. Key words: finite-time thermodynamics, Atkinson cycles, heat resistance, friction, internal irreversibility, performance optimization Introduction Fi nite time ther mo dy nam ics can an swer some global ques tions which clas si cal ther - mo dy nam ics does not try to an swer and con ven tional ir re vers ible ther mo dy nam ics can not an - swer be cause of its mi cro, dif fer en tial view point. Ex am ples of such ques tions are: (1) What is the least en ergy re quired by a given ma chine to pro duce a given work in a given time? (2) What is the most work that can be pro duced by a given ma chine in given time, uti liz ing a given en - ergy? (3) What is the most ef fi cient way to run a given ther mo dy namic pro cess (op ti mal path) in fi nite time? (4) What is the op ti mal time-de pend ent (on and off) pro cess? () What is the op ti mal dis tri bu tion be tween heat exchanger heat trans fer sur face ar eas or heat con duc tances cor re - spond ing to the op ti mal per for mance of the ther mo dy namic de vices for the fixed to tal heat exchanger heat trans fer sur face area or to tal heat con duc tance? (6) What are the quan ti ta tive and qual i ta tive fea tures of the ef fects of heat re sis tance, in ter nal ir re vers ibil ity and heat leak age on the per for mance of real ther mo dy namic pro cesses and de vices? A se ries of achieve ments have been made since fi nite-time ther mo dy nam ics was used to an a lyze and op ti mize per for mance of real heat en gines [1-10]. Chen et al. [11] stud ied the ef fi ciency of an Atkinson cy cle at max i - * Corresponding author; s: lgchenna@yahoo.com, lingenchen@hotmail.com

2 888 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp mum power den sity with out any loss. Qin et al. [12] and Ge et al. [13] de rived the per for mance char ac ter is tics of Atkinson cy cle with heat trans fer loss [12] and with heat trans fer and fric - tion-like term losses [13], re spec tively. Ge et al. [14, 1] con sid ered the ef fect of vari able spe - cific heats on the cy cle pro cess and stud ied the per for mance char ac ter is tic of endoreversible and ir re vers ible Atkinson cy cles when vari able spe cific heats of work ing fluid are lin ear func tions of its tem per a ture and the max i mum tem per a ture of the cy cle is not fixed. Wang et al. [16] an a - lyzed and com pared the per for mance of an Atkinson cy cle cou pled to vari able-tem per a ture heat res er voirs un der max i mum power and max i mum power den sity con di tions. Zhao et al. [17] an a - lyzed the per for mance and op ti mized the para met ric cri te ria of an ir re vers ible Atkinson heat en - gine. Hou et al. [18] com pared the per for mance of air stan dard Atkinson and Otto cy cles with heat trans fer con sid er ations. Lin et al. [19] an a lyzed the in flu ence of heat loss, as char ac ter ized by a per cent age of fuel s en ergy, fric tion, and vari able spe cific heats of work ing fluid on the per - for mance of an air-stan dard Atkinson cy cle when vari able spe cific heats of work ing fluid are lin ear func tions of its tem per a ture and the max i mum tem per a ture of the cy cle is not fixed. Al-Sarkhi et al. [20] out lined the ef fect of max i mum power den sity on the per for mance of the Atkinson cy cle ef fi ciency when the vari able spe cific heats of work ing fluid are lin ear func tions of its tem per a ture. Abu-Nada et al. [21] and Al-Sarkhi et al. [22] ad vanced a non-lin ear re la tion be tween the spe cific heats of work ing fluid and its tem per a ture and com pared the per for mance of the cy cle with con stant and vari able spe cific heats. Parlak et al. [23] de fined the in ter nal ir re - vers ibil ity by us ing en tropy pro duc tion, and an a lyzed the ef fect of the in ter nal ir re vers ibil ity on the per for mance of ir re vers ible re cip ro cat ing heat en gine cy cles. Zhao et al. [24-26] de fined the in ter nal ir re vers ibil ity by us ing com pres sion and ex pan sion ef fi cien cies and an a lyzed the per - for mance of die sel, Otto, dual, and Miller cy cles when the max i mum tem per a ture of the cy cle is fixed and the ef fi ciency has a new def i ni tion. Zhao et al. [27, 28] used the model of spe cific heats ad vanced in refs. [14, 1], the in ter nal ir re vers ibil ity de fined in [24-26], and stud ied the op ti mum per for mance of Otto and die sel cy cles when the max i mum tem per a ture of the cy cles is fixed. Ge et al. [29-30] adopted the spe cific heat model ad vanced in refs. [21, 22], the in ter nal ir - re vers ibil ity de fined in refs. [24-28], and the fric tion loss de fined in ref. [31], and stud ied the per for mance of an ir re vers ible Otto, die sel, and dual cy cles when heat trans fer, fric tion, and in - ter nal ir re vers ibil ity losses are con sid ered. his pa per will adopt the spe cific heats model ad - vanced in refs. [21, 22, 29, 30], the in ter nal ir re vers ibil ity and ef fi ciency de fined in refs. [24-30] and the fric tion loss de fined in refs. [29-31], and study the per for mance of an ir re vers ible Atkinson cy cle when heat trans fer, fric tion, and in ter nal ir re vers ibil ity losses and non-lin ear vari able spe cific heats of the work ing fluid are con sid ered. Cy cle model and anal y sis Fig ure 1. -S di a gram for the cy cle model An air stan dard Atkinson cy cle model is shown in fig. 1. Process 1 2S is a re vers ible adi a batic com pres sion, while process 1 2 is an ir re vers ible adi a batic pro cess that takes into account the internal irreversibility in the real compression pro cess. he heat ad di tion is an isochoric pro cess 2 3. Pro - cess 3 4S is a re vers ible adi a batic ex pan sion, while 3 4 is an ir re vers ible adi a batic pro cess that takes into ac count the in - ternal irreversibility in the real expansion process. he heat re jec tion is an iso baric pro cess 4 1.

3 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp In most cy cle model, the work ing fluid is as sumed to be have as an ideal gas with con - stant spe cific heats. But this as sump tion can be valid only for small tem per a ture dif fer ence. For the large tem per a ture dif fer ence en coun tered in prac ti cal cy cle, this as sump tion can not be ap - plied. Ac cord ing to ref. [21], for the tem per a ture range of K, the spe cific heat ca pac ity with con stant pres sure can be writ ten as: C p = ( )R g (1) For the tem per a ture range of K, the equation is writ ten as: C ( p ) Rg (2) Equa tions (1) and (2) can be ap plied to a tem per a ture range of K which is too wide for the tem per a ture range ( K) of prac ti cal en gine. So a sin gle equa tion was used to de scribe the spe cific heat model which is based on the as sump tion that air is an ideal gas mix - ture con tain ing 78.1% ni tro gen, 20.9% ox y gen, 0.92% ar gon, and 0.03% car bon di ox ide. C p (3) Ac cord ing to the re la tion be tween spe cific heat with con stant pres sure and spe cific heat with con stant vol ume: C v = C p R g (4) the spe cific heat with con stant vol ume can be writ ten as: C C R v p g () 0. where R g = kj/kgk is the gas con stant of the work ing fluid. he unit of C v and C p is [kjkg 1 K 1 ]. he heat added to the work ing fluid dur ing pro cess 2 3 is: 3 3 Qin M C vd = M ( ) d 7 M[ ] he heat re jected by the work ing fluid dur ing pro cess is: 3 2 (6) 4 Qout M C pd = M ( ) d M[ ] 4 (7) 1

4 890 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp where M is the mass flow rate of the work ing fluid, 1, 2, 3, and 4 [K] are the tem per a tures at states 1, 2, 3, and 4. For the two adi a batic pro cesses 1 2 and 3 4, the com pres sion and ex pan sion ef fi - cien cies can be de fined as [24-30]: 2S 1 h c (8) h c (9) 4S 3 hese two ef fi cien cies can be used to de scribe the in ter nal ir re vers ibil ity of the pro - cesses. Since C p and C v are de pend ent on tem per a ture, adi a batic ex po nent k = C p /C v will vary with tem per a ture as well. here fore, the equa tion of ten used in re vers ible adi a batic pro cess with con stant k can not be used in re vers ible adi a batic pro cess with vari able k. How ever, ac cord ing to refs. [14, 1, 29-40], a suit able en gi neer ing ap prox i ma tion for re vers ible adi a batic pro cess with vari able k can be made, i. e. this pro cess can be bro ken up into a large num ber of in fin i tes i mally small pro cesses and for each of these pro cesses, adi a batic ex po nent k can be re garded as a con - stant. For ex am ple, for any re vers ible adi a batic pro cess be tween states i and j can be re garded as con sist ing of nu mer ous in fin i tes i mally small pro cesses with con stant k. For any of these pro - cesses, when an in fin i tes i mally small change in tem per a ture d, and vol ume dv of the work ing fluid takes place, the equa tion for re vers ible adi a batic pro cess with vari able k can be writ ten as fol lows. V k 1 ( d )( V d V ) k 1 (10) For an isochoric heat ad di tion pro cess i j, the heat added is Q in = C v ( j i ) = DS ij = = C v ln( j / i ). So one has = ( j i )/ln( j / i ), where is the equiv a lent tem per a ture of heat ab - sorption process. When C v is the func tion of tem per a ture, the C v () can be re garded as mean spe - cific heat with con stant vol ume. From eq. (10), one gets j C R Vi v ln g ln (11) V where the tem per a ture in the equa tion of C v is = ( j i )/ln( j / i ). he com pres sion ra tio is de fined as: i g V V1 2 here fore, equa tions for re vers ible adi a batic pro cesses 1 2S and 3 4S are: C v C v 2 ln 1 S j (12) R ln g (13) g 4S 1 ln Rg ln Rg ln g (14) 3 For an ideal Atkinson cy cle model, there are no heat trans fer losses. How ever, for a real Atkinson cy cle, heat trans fer ir re vers ibil ity be tween work ing fluid and the cyl in der wall is 4S

5 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp not neg li gi ble. One can as sume that the heat trans fer loss through the cyl in der wall (i. e. the heat leak age loss) is pro por tional to av er age tem per a ture of both the work ing fluid and the cyl in der wall and that the wall tem per a ture is con stant, 0 [K]. If the re leased heat by com bus tion per sec - ond is A 1 [kw] the heat leak age co ef fi cient of the cyl in der wall is B 1 [kjkg 1 K 1 ] which has con - sid ered the heat trans fer co ef fi cient and the heat ex change sur face, one has the heat added to the work ing fluid per sec ond by com bus tion in the fol low ing lin ear re la tion [12-1]: Q A MB in From eq. (1), one can see that Q in con tained two parts, the first part is A 1, the re leased heat by com bus tion per sec ond, and the sec ond part is the heat leak loss per sec ond, it can be writ ten as: Q leak = MB( ) (16) where B = B 1 /2. ak ing into ac count the fric tion loss of the pis ton as rec om mended by Chen et al. [31] for the Dual cy cle and as sum ing a dis si pa tion term rep re sented by a fric tion force which in a liner func tion of the ve loc ity gives x f m mv m d (17) dt where m [Nsm 1 ] is a co ef fi cient of fric tion which takes into ac count the global losses and x is the pis ton dis place ment. hen, the lost power is: P m 0 (1) dwm dx dx m mv 2 (18) dt dt dt If one spec i fies the en gine is a four stroke cy cle en gine, the to tal dis tance the pis ton trav els per cy cle is: 4L = 4(x 1 x 2 ) (19) For a four stroke cy cle en gine, run ning at N cy cles per sec ond, the mean ve loc ity of the pis ton is: v 4 LN (20) where x 1 and x 2 [m] are the pis ton po si tion at max i mum and min i mum vol ume and L [m] is the dis tance that the pis ton trav els per stroke, re spec tively. hus, the power out put is: P Q Q P at in out M[ ( ) ( ) ( ) ( ) ( ) ( ) (.... ) m ( ) ( )] m v 2 (21) he ef fi ciency of the cy cle is:

6 892 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp Pat h at Qin Qleak M[ ( ) (.... ) ( ) ( ) ( ) ( ) (.... ) ( ) ( )] mn M[ ( ) ( ) ( ) (.. ) )] MB( ) ( ) ( ) ( ) ( When g, 1, 3, h c, and h e are given, 2S can be ob tained from eq. (13), then, sub sti tut - ing 2S into eq. (8) yields 2, 4S can be ob tained from eq. (14), and the last, 4 can be solved out by sub sti tut ing 4S into eq. (9). Sub sti tut ing 2 and 4 into eqs. (21) and (22) yields the power and ef fi ciency. hen, the re la tions be tween the power out put and the com pres sion ra tio, be - tween the ther mal ef fi ciency and the com pres sion ra tio, as well as the op ti mal re la tion be tween power out put and the ef fi ciency of the cy cle can be ob tained. Nu mer i cal ex am ples and dis cus sion Ac cord ing to ref. [29-31], the fol low ing pa ram e ters are used: 1 = 30 K, 3 = 2200 K, x 1 = , x 2 = m, N = 30, and M = kg/s. Fig ures 2-4 show the ef fects of the in ter nal ir re vers ibil ity, heat trans fer loss, and fric tion loss on the per for mance of the cy cle. One can see that when the above three irreversibilities are not in cluded, the power out put vs. com - pres sion ra tio char ac ter is tic and the power out put vs. ef fi ciency char ac ter is tic are par a bolic-like curves, while the ef fi ciency will in crease with the in crease of the com pres sion ra tio. When more than one irreversibilities are in cluded, the power out put vs. com pres sion ra tio char ac ter is tic and the ef fi ciency vs. com pres sion ra tio char ac ter is tic are par a bolic like curves and the power out put vs. ef fi ciency curve is loop-shaped one. Ac cord ing to eq. (21), the def i ni tions of the power out put, the heat trans fer loss has no ef fect on the power out put of the cy cle. So fig. 2 only shows the ef fects of the in ter nal ir re vers ibil ity and fric tion loss on the power out put of the cy cle. Com par ing curves 1 with 1' and 2 with 2', one can see that the power out put in creases with the de - crease of in ter nal ir re vers ibil ity. Com par ing curves 1 with 2 and 1' with 2', one can see that the power out put de creases with the in crease of fric - tion loss. Fig ure 3 shows the ef fects of the in ter nal ir re - vers ibil ity, heat trans fer loss and fric tion loss on Figure 2. he influences of the internal irreversibility and friction loss on the power output (22) the ef fi ciency of the cy cle. Curve 1 is the ef fi - ciency vs. compression ratio characteristic without irreversibility. Under this circumstance, the effi-

7 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp ciency in creases with the in crease of com pres sion ra tio. Other curves are ef fi - ciency vs. compression ratio characteristic with one or more irreversibilities and these curves are par a bolic-like ones. Com par ing curves 1 with 1', 2 with 2', 3 with 3', and 4 with 4', one can see that the ef fi ciency in creases with the de crease of internal irreversibility. Comparing curves 1 with 3, 2 with 4, 1' with 3', and 2' with 4', one can see that the ef fi ciency de creases with the in crease of heat trans fer loss. Com par ing curves 1 with 2, 3 with 4, 1' with 2', and 3' with 4', one can see that the ef fi ciency de creases with the in crease of friction loss. Fig ure 4 shows the ef fects of the in - ter nal ir re vers ibil ity, heat trans fer loss, and fric tion loss on the power out put vs. the ef fi ciency char ac ter is tic. Curve 1 which is a par a bolic like curve is the power out put vs. ef fi ciency char ac ter is - tic of the cy cle with out ir re vers ibil ity, while else curves are loop-shaped ones with one or more irreversibilities. Com - par ing curves 1 with 1', 2 with 2', 3 with 3', and 4 with 4', one can see that the max i mum power out put and the ef fi - ciency at the max i mum power out put de - crease with the in crease of in ter nal ir re - vers ibil ity. Com par ing curves 1 with 3, 2 with 4, 1' with 3', and 2' with 4', one can see that the max i mum power out put is not in flu enced by heat trans fer loss, while the ef fi ciency at the max i mum power out put de creases with the in crease of heat trans fer loss. Com par ing curves 1 with 2, 3 with 4, 1' with 2', and 3' with 4', one can see that both the max i mum power out put and the cor re spond ing ef fi ciency de crease with the in - crease of fric tion loss. Conclusions Figure 3. he influences of internal irreversibility, heat transfer loss, and friction loss on the efficiency Figure 4. he influences of internal irreversibility, heat transfer loss, and friction loss on the power output vs. efficiency characteristic In this pa per, an ir re vers ible air stan dard Atkinson cy cle model which is more close to prac tice is founded. In this model, the non-lin ear re la tion be tween the spe cific heats of work ing fluid and its tem per a ture, the fric tion loss com puted ac cord ing to the mean ve loc ity of the pis ton, the in ter nal ir re vers ibil ity de scribed by us ing the com pres sion and ex pan sion ef fi ciency, and heat trans fer loss are pre sented. he per for mance char ac ter is tics of the cy cle were ob tained by de tailed nu mer i cal ex am ples. he ef fects of in ter nal ir re vers ibil ity, heat trans fer loss and fric -

8 894 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp tion loss on the per for mance of the cy cle were an a lyzed. he re sults ob tained herein may pro - vide guide lines for the de sign of prac ti cal in ter nal com bus tion en gines. Acknowledgments his pa per is sup ported by Pro gram for New Cen tury Ex cel lent al ents in Uni ver sity of P. R. China (Pro ject No. NCE ) and he Foun da tion for the Au thor of Na tional Ex - cel lent Doc toral Dis ser ta tion of P. R. China (Pro ject No ). he au thors wish to thank the re view ers for their care ful, un bi ased, and con struc tive sug ges tions, which led to this re vised manu script. Nomenclature A 1 heat released by combustion per second, [kw] B constant related to heat transfer, [kjkg 1 K 1 ] C p specific heat with constant pressure, [kjkg 1 K 1 ] C v specific heat with constant volume, [kjkg 1 K 1 ] k ratio of specific heats, [ ] L total distance of the piston traveling per cycle, [m] M mass flow rate of the working fluid, [kgs 1 ] N number of the cycle operating in a second, [ ] P 2,3 pressure at different states 2 and 3, [Pa] power output of the cycle, [kw] P m Q in heat added to the working fluid in a second, [kw] Q out heat rejected by the working fluid in a second, [kw] R g air constant of the working fluid, [kjkg 1 K 1 ] 1-,2S,S temperature at different states, [K] V 1,2 volume at different states 1 and 2, [m 3 ] v velocity of the piston, [ms 1 ] x 1 the piston position at maximum volume, [m] the piston position at minimum volume, [m] x 2 Greek let ters h c efficiency of the cycle, [ ] h e expansion efficiency, [ ] m coefficient of friction, [Nsm 1 ] g compression ratio, [ ] References [1] Andresen, B., Salamon, P., Berry, R. S., her mo dy nam ics in Fi nite ime, Phys. o day, 37 (1984), 9, pp [2] Sieniutycz, S., Salamon, P., Ad vances in her mo dy nam ics, Vol. 4, 1990, Fi nite ime her mo dy nam ics and hermoeconomics, ay lor & Fran cis, New York, USA [3] Radcenco, V., Gen er al ized her mo dy nam ics, Editura echica, Bucharest, 1994 [4] Bejan, A., En tropy Gen er a tion Minimization: he New her mo dy nam ics of Fi nite-size De vice and Fi - nite-ime Pro cesses, J. Appl. Phys., 79 (1996), 3, pp [] Hoffmann, K. H., Burzler, J. M., Schu bert S., Endoreversible hermodynamics, J. Non-Equilib. hermodyn., 22 (1997), 4, pp [6] Berry, R. S., et al., her mo dy namic Op ti mi za tion of Fi nite ime Pro cesses, John Wiley and Sons, Chichester, UK, 1999 [7] Chen, L., Wu, C., Sun, F., Fi nite ime her mo dy namic Op ti mi za tion or En tropy Gen er a tion Minimization of En ergy Sys tems, J. Non-Equilib. hermodyn., 24 (1999), 4, pp [8] Chen, L., Sun, F., Ad vances in Fi nite ime her mo dy nam ics: Anal y sis and Op ti mi za tion, Nova Sci ence Pub lish ers, New York, USA, 2004 [9] Radcenco, V., et al., New Ap proach to her mal Power Plants Op er a tion Re gimes Max i mum Power ver sus Max i mum Ef fi ciency, Int. J. her mal Sci ences, 46 (2007), 12, pp [10] Feidt, M., Op ti mal Use of En ergy Sys tems and Pro cesses, Int. J. Exergy, (2008), /6, pp

9 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp [11] Chen, L., et al., Ef fi ciency of an Atkinson En gine at Max i mum Power Den sity, Energy Convers. Mgnt., 39 (1998), 3/4, pp [12] Qin, X., Chen, L., Sun, F., he Uni ver sal Power and Ef fi ciency Char ac ter is tics for Ir re vers ible Re cip ro cat - ing Heat En gine Cy cles, Eur. J. Phys., 24 (2003), 4, pp [13] Ge, Y., et al., Reciprocating Heat-Engine Cycles, Appl. En ergy, 81 (200), 4, pp [14] Ge, Y., et al., Per for mance of an Endoreversible Atkinson Cy cle, J. En ergy In sti tute, 80 (2007), 1, pp. 2-4 [1] Ge, Y., et al., Per for mance of Atkinson Cy cle with Heat rans fer, Fric tion and Vari able Spe cific Heats of Work ing Fluid, Appl. En ergy, 83 (2006), 11, pp [16] Wang, P., Hou, S., Per for mance Anal y sis and Com par i son of an Atkinson Cy cle Cou pled to Vari able em per a ture Heat Res er voirs un der Maximum Power and Maximum Power Density Conditions, Energy Convers. Mgnt., 46 (200), 1-16, pp [17] Zhao, Y., Chen, J., Per for mance Analysis and Parametric Optimum Criteria of an Irreversible Atkinson Heat-En gine, Appl. En ergy, 83 (2006), 8, pp [18] Hou, S., Com par i son of Per for mances of Air Stan dard Atkinson and Otto Cy cles with Heat rans fer Con - sid er ations, En ergy Convers. Mgnt., 48 (2007),, pp [19] Lin, J., Hou, S., In flu ence of Heat Loss on the Per for mance of an Air-Stan dard Atkinson Cy cle, Appl. En - ergy, 84 (2007), 9, pp [20] Al-Sarkhi, A., et al., Ef fi ciency of Atkinson En gine at Max i mum Power Den sity Us ing em per a ture De - pend ent Spe cific Heats, Jor dan J. Mech. Industrial Engineering, 2 (2008), 2, pp [21] Abu-Nada, E., et al., her mo dy namic Modeling of Spark-Ignition Engine: Effect of emperature Depend - ent Spe cific Heats, Int. Comm. Heat Mass rans fer, 32 (200), 8, pp [22] Al-Sarkhi, A., et al., Per for mance Evaluation of Irreversible Miller Engine under Various Specific Heat Mod els, Int. Comm. Heat Mass rans fer, 34 (2007), 7, pp [23] Parlak, A., Sahin, B., Per for mance Optimization of Reciprocating Heat Engine Cycles with Internal Irre - vers ibil ity, J. En ergy In sti tute, 79 (2006), 4, pp [24] Zhao, Y., et al., Per for mance Anal y sis and Para met ric Op ti mum De sign of an Ir re vers ible Die sel Heat En - gine, En ergy Convers. Mgnt., 47 (2006), 18-19, pp [2] Zhao, Y., Chen, J., An Ir re vers ible Heat En gine Model In clud ing hree yp i cal her mo dy namic Cy cles and the Op ti mum Per for mance Anal y sis, Int. J. her mal Sci ence, 46 (2007), 6, pp [26] Zhao, Y., Chen, J., Per for mance Anal y sis of an Ir re vers ible Miller Heat En gine and Its Op ti mal Cri te ria, Appl. her mal Engng., 27 (2007), 11-12, pp [27] Zhao, Y., Lin, B., Chen, J., Op ti mum Cri te ria on the Im por tant Pa ram e ters of an Ir re vers ible Otto Heat En - gine with the em per a ture-de pend ent Heat Capacities of the Working Fluid, ASME rans. J. En ergy Res. ech., 129 (2007), 4, pp [28] Zhao, Y., Chen, J., Op ti mum Per for mance Anal y sis of an Ir re vers ible Die sel Heat En gine Af fected by Vari able Heat Ca pac i ties of Work ing Fluid, Energy Convers. Mgnt., 48 (2007), 9, pp [29] Ge, Y., Chen, L., Sun, F., Fi nite ime her mo dy namic Mod el ing and Anal y sis for an Ir re vers ible Die sel Cy cle, Pro ceed ings, IMechE, Part D: J. Automobile Engineering, 222 (2008), D, pp [30] Ge, Y., Chen, L., Sun, F., Fi nite ime her mo dy namic Mod el ing and Anal y sis of an Ir re vers ible Otto Cy - cle, Appl. En ergy, 8 (2008), 7, pp [31] Chen, L., et al., Ef fects of Heat rans fer, Fric tion and Vari able Spe cific Heats of Work ing Fluid on Per for - mance of an Ir re vers ible Dual Cy cle, Energy Convers. Mgnt., 47 (2006), 18-19, pp [32] Ge, Y., et al., her mo dy namic Sim u la tion of Per for mance of an Otto Cy cle with Heat rans fer and Vari - able Spe cific Heats of Work ing Fluid, Int. J. her mal Sci ence, 44 (200),, pp [33] Ge, Y., et al., he ef fects of Vari able Spe cific Heats of Work ing Fluid on the Per for mance of an Ir re vers - ible Otto Cy cle, Int. J. Exergy, 2 (200), 3, pp [34] Ge, Y., et al., Per for mance of an Endoreversible Die sel Cy cle with Vari able Spe cific Heats of Work ing Fluid, Int. J. Am bi ent En ergy, 29 (2008), 3, pp [3] Ge, Y., et al., Per for mance of Die sel Cy cle with Heat rans fer, Fric tion and Vari able Spe cific Heats of Work ing Fluid, J. En ergy In sti tute, 80 (2007), 4, pp [36] Ge, Y., et al., Per for mance of Re cip ro cat ing Brayton Cy cle with Heat rans fer, Fric tion and Vari able Spe - cific Heats of Work ing Fluid, Int. J. Am bi ent En ergy, 29 (2008), 2, pp [37] Ge, Y., et al., Ef fects of Heat rans fer and Vari able Spe cific Heats of Work ing Fluid on Per for mance of a Miller Cy cle, Int. J. Am bi ent En ergy, 26 (200), 4, pp

10 896 HERMAL SCIENCE: Year 2010, Vol. 14, No. 4, pp [38] Chen, L., Ge, Y., Sun, F., Uni fied her mo dy namic De scrip tion and Op ti mi za tion for a Class of Ir re vers - ible Re cip ro cat ing Heat En gine Cy cles, Pro ceed ings, IMechE, Part D: J. Au to mo bile En gi neer ing, 222 (2008), D8, pp [39] Al-Sarkhi, A., et al., Ef fects of Fric tion and em per a ture-de pend ent Spe cific-heat of the Work ing Fluid on the Per for mance of a Die sel-en gine, Appl. En ergy, 83 (2006), 2, pp [40] Al-Sarkhi, A., Jaber, J. O., Probert, S. D., Ef fi ciency of a Miller En gine, Appl. En ergy, 83 (2006), 4, pp Paper submitted: January 28, 2009 Paper revised: April 14, 2009 Paper accepted: April 30, 2009