JJMIE Jordan Journal of Mechanical and Industrial Engineering

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1 JJMIE Jordan Journal of Mechanical and Industrial Engineering Volume, Number, Jun. 8 ISSN Pages 7-75 Efficiency of Atkinson Engine at Maximum Power Density using emerature Deendent Secific Heats A. Al-Sarkhi, B. Akash *, E. Abu-Nada and I. Al-Hinti Deartment of Mechanical Engineering, Hashemite Uniersity, Zarqa, 5, Jordan Abstract hermodynamic analysis of an ideal air-standard Atkinson cycle with temerature deendant secific heat is resented in this aer. he aer outlines the effect of maximizing ower density on the erformance of the cycle efficiency. he ower density is defined as the ratio of the ower outut to the maximum cycle secific olume. It showed significant effect on the erformance of the cycle oer the constant secific heat model. he results obtained from this work can be helful in the thermodynamic modeling and in the ealuation of real Atkinson engines oer other engines. 8 Jordan Journal of Mechanical and Industrial Engineering. All rights resered Keywords: Atkinson engine; ower density; temerature deendant secific heat;. Introduction he Atkinson cycle (the comlete exansion cycle) is named after its inentor James Atkinson in 88 []. As shown in Fig., the cycle inoles isentroic comression followed by isochoric heat addition. he exansion occurs isentroically, and finally the cycle has isobaric heat rejection. P S=const. S =constant V=constant q out q in P=constant Figure : Sketch of P-V and -S diagrams of Atkinson cycle Otimization and erformance analysis can be alied using finite time thermodynamic techniques. hey are used to study erformance of arious air-standard ower cycles [-8]. For examle, Chen et al. [6] and Sahin et al. [7] examined Atkinson cycle and Joule-Brayton cycle, resectiely, at maximum ower density. Both studies found that the efficiency at the maximum ower density is greater than that at the maximum ower outut. Constant S secific heats of air (the working fluid) were used. he maximization of the ower density is defined as the ratio of the maximum ower to the maximum secific olume in the cycle. It takes into consideration the engine size instead of just maximizing its ower outut. he inclusion of the engine size in the calculation of its erformance is a ery imortant factor from an economical oint of iew. Many researchers hae erformed finite-time thermodynamic analysis on Atkinson engines. Most of them assumed constant secific heats for the working fluid in their studies. Performance analysis of an Atkinson cycle with heat transfer, friction and ariable secific-heats of the working fluid was studied by Ge et al. [9]. heir results showed that the effects of ariable secific-heats of working fluid and friction-like term losses on the irreersible cycle erformance are significant. Zhao and Chen [] erformed analysis and arametric otimum criteria of an irreersible Atkinson heat-engine using finite-time thermodynamics rocesses. he otimum criteria of some imortant arameters, such as the ower outut, efficiency and ressure ratio were gien in their study. Constant secific heat in their model was assumed in their model. Wang and Hou [] studied the erformance analysis and comarison of an Atkinson cycle couled to ariable temerature heat reseroirs under maximum ower and maximum ower density conditions, assuming a constant secific heat, too. heir results showed an engine design based on maximum ower density is better than that based on maximum ower conditions, from the iew oints of engine size and thermal efficiency. Howeer, due to the higher comression ratio and maximum temerature in the cycle, * Corresonding author. bakash@hu.edu.jo

2 7 8 Jordan Journal of Mechanical and Industrial Engineering. All rights resered - Volume, Number (ISSN ) an engine design based on maximum ower density conditions requires tougher materials for engine construction than one based on maximum ower conditions. Performance analyses under maximum ower and ower density hae been also erformed on Brayton cycle and an endo-reersible Braysson cycle, resectiely [,]. Both studies assumed constant secific heat of the working fluid. Recent studies were inoled in erforming thermodynamic analysis using temerature deendent secific heats on arious conditions. It was found that temerature deendent secific heat gies better aroximation to actual cycles than using constant temerature secific heat [-9]. his aer examines the significance of using the temerature deendant secific heat on the erformance of an Atkinson engine under maximum ower density. he results are comared to those obtained from recent study by Chen et al. using constant secific heat [6]. Different arameters affecting cycle erformance and net work outut at maximum ower density will be considered. he obtained results will be resented as erformance characteristic cures for the Atkinson cycle using numerical examles. hermodynamic Analysis Figure resents ressure-olume (P-V) and temerature-entroy (-S) diagrams for the thermodynamic rocesses erformed by an ideal airstandard Atkinson cycle. All four rocesses are reersible. Process - is an adiabatic (isentroic) comression; rocess - is a heat addition at a constant olume; rocess - adiabatic (isentroic) exansion; rocess - is heat rejection at a constant ressure. he emloyed ariable secific heat model assumes ariation of secific heat with temerature in a linear fashion. he best straight line fit was determined. It is lotted in Fig.. C, kj/kg-k Ref. [] Linear fit y =.x +.95 R = emerature, K Figure : he best linear fit for temerature deendant secific heat of air in the temerature range of to 5 K. he ariations of secific heats with temeratures are resented in equations () and (), as C + C C = R = a a () () where a, a are constants [9] and R is the gas constant. Equation () is taken as the best straight line that fit the ariable C as in Çengel []; he heat inut of Atkinson cycle is gien as Q = m C d () in and the heat rejected from the cycle during the rocess is Q out = m C d hen by substituting the ariable secific heat equation and erforming the integration the heat addition and rejection will be comuted as in equations (5) and (6), resectiely. a Q in = m ( a R)( ) + ( ) (5) a Q out = m a ( ) + ( ) (6) he ower outut, W, of this cycle can be written as a ( a R)( ) + ( ) W = m a a ( ) ( ) he ower density, P, is defined as the ower er maximum secific olume in the cycle W P = (8) where,, is the maximum secific olume a ( a R)( ) ( ) m + P = a a ( ) ( ) for simlicity and since most of the temerature changes occur during the isochoric ( to ) and isobaric rocess ( to ) a constant secific heat can be assumed during the isentroic rocesses (i.e. from to and from to ). he total entroy change in the cycle is equal to zero, therefore, = where, / k () (7) (9) () C k = C () Let

3 8 Jordan Journal of Mechanical and Industrial Engineering. All rights resered - Volume, Number (ISSN ) 7 = θ () and = () hen the ower density (equation (6)) in terms of θ and, becomes m P = a /k a /k a ( ) ( ) θ θ ( a R)( θ) + ( θ ) () where is the temerature of the working fluid at sate (atmosheric condition) in the cycle.. Power density maximization For a gien, the ower density will be differentiated with resect to θ and the result will be equating to zero (dp/dθ = ) this gies a R a a θ + θ k a /k ( k)/k (/k) ( k)/k + θ = k (5) the root of equation (5) is ( θ ) is the ( θ ) at which the ower density is maximum. Substituting ( θ ) in equation () gies the maximum ower density P max. he cycle efficiency ( η ) at maximum ower density oint ( θ ) is / k a / k ( / θ ) ) + ( / θ ) ) a η = a (6) ( a )( ) + ( ) R θ θ for the numerical calculation in the resent study the following alues will be used a =.95 kj/(kg-k), a =. kj/kg, and R =.87 kj/(kg-k). Performance Comarison he deried formula aboe is used and lotted in order to comare Atkinson engine with ariable secific heat with those results assuming a constant secific heat as shown in Figures through 9. he following constants and range of arameters are selected: k =., = to 6 and = 98 K. By arying isentroic temerature ratio (θ ) or thermal efficiency (η ) and for a gien alue of cycle temerature ratio ( ) the normalized ower density (P/P max ) is lotted. he normalized ower density is the ratio of ower density to the maximum ower density at (θ = θ ). he results are resented in figure through 9. θ Figure : Variations of the isentroic temerature ratios for the ariable and constant secific heats with cycle temerature ratio Figure shows the ariations of the isentroic temerature ratios at maximum ower density ( θ ), for ariable and constant secific heats with temerature ratio ( ). he deiation between ( θ ) using the ariable and constant secific heat increases with increasing the cycle temerature ratio ( ). he alue of ( θ ) using constant secific is larger than that using ariable secific heat. Figure shows the ariation of the thermal efficiency with cycle temerature ratio ( ). he thermal efficiency at maximum ower density of Atkinson cycle using the constant secific heat is oer redicted as comared to the ariable secific heat. η Carnot 6 8 Figure : ariation of the cycle thermal efficiency at maximum ower density oint comared to Carnot cycle. Figure 5 shows the ariation of the ercentage difference in ( η ) between constant and ariable secific heats with cycle temerature ratio ( ). he ercentage difference is defined as η ariable c η constant c η Δ = % (7) η ariable c he maximum ercentage difference occurs at low alues of. he ercentage difference remains constant at around 6% beyond =. he minimum difference occurs at =.7. he behaior of this cure suggests that the efficiency at maximum ower density is underestimated at low alues of and it is oer estimated for high alues of (aboe.7).

4 7 8 Jordan Journal of Mechanical and Industrial Engineering. All rights resered - Volume, Number (ISSN ) Δ η, % Figure 5: % error in thermal efficiency at max. ower density oint-ariable theta; Variation of the ercentage difference in η between constant and ariable secific heat Normalized ower density ariation with isentroic temerature ratio at = is shown in Figure 6. he normalized ower density is different at same temerature ratio θ. Both constant and ariable secific heat cures of normalized ower density hae arabolic trend. Similar trend for = is shown in Figure 7 but greater difference aears at higher alues of. P/P max = θ Figure 6: Normalized ower density ariation with isentroic temerature ratio at = P/P max = 5 θ Figure 7: Normalized ower density ariation with isentroic temerature ratio at = Figure 8 resents the ariation of the ercentage difference in the normalized ower density between constant and ariable secific heat for three cases, =., and. All cures start with high ercentage difference at low alues of θ (i.e. P/Pmax is underestimated) and the then the ercentage difference decreases until it reaches a minimum oint and then increases again with increasing θ (i.e. P/Pmax is oerestimated). According to this figure, there is a oint where both ariable and constant secific heat will gie the aroximately the same result. Δ(P/Pmax), % θ Figure 8: ariation of the ercentage difference in the normalized ower density between constant and ariable secific heat Figure 9 shows the ariation of normalized ower density with thermal efficiency at =. he maximum oint of the normalized ower density is not at the same oint of maximum efficiency. Both cares are not identical. At a small range of cycle thermal efficiency aroximately from η =.58 to.66 where both constant and ariable secific heat will gie aroximately the same result but most of the range of the cure will gie different alue. P/P max = η Figure 9: Variation of normalized ower density with thermal efficiency at = Figure shows the ariation of ercentage difference of the normalized ower density using the constant and ariable secific with thermal efficiency at =. he show small difference at low alue of thermal efficiency and the difference then increases and the decreases then increases in harmonic like behaior. his figure clearly indicates the needs for using ariable secific heat in calculations of cycle erformance. Δ(P/Pmax), % = η Figure : Variation of ercentage difference of the normalized ower density with thermal efficiency at =..

5 8 Jordan Journal of Mechanical and Industrial Engineering. All rights resered - Volume, Number (ISSN ) Conclusion Using constant or temerature deendant secific heats will affect the Atkinson maximum ower density calculation. he differences in the results using the two methods are significant. herefore, temerature deendant secific heat must be used in modeling the erformance of Atkinson engine at maximum ower density. he efficiency at maximum ower density oint when using the constant secific heat is oer redicted. he maximum ower density occurs at higher isentroic temerature ratios for the case of constant secific heat which also leads to incorrect otimum ower density. It is recommended that other arameters be considered for future work. References [] W.W. Pulkrabek. Engineering Fundamentals of the Internal Combustion Engines. Second Edition. Uer Saddle Rier, New Jersey: Prentice-Hall.;. [] E.F. Obert. Internal Combustion Engines & Air Pollution. hird Edition. New York, NY: Harer and Row Publishers; 97. [] J.B. Haywood, Internal Combustion Engine Fundamentals, New York, McGraw-Hill, 988. [] C.R. Ferguson, A.. Kirkatrick. Internal Combustion Engines: Alied hermosciences. John Wiley & Sons: New York, NY;. [5] D.A. Blank, C. Wu. he effect of combustion on a ower otimized endoreersible diesel cycle. Energy Conersion & Management, (99),, [6] L. Chen, J. Lin, F. Sun and C. Wu. Efficiency of an Atkinson engine at maximum ower density. Energy Conersion & Management, (998), 9, 7-. [7] B. Sahin, A. Kodal, and H. Yauz. Efficiency of a Joule- Brayton engine at maximum ower density. J. Phys. D: Al. Phys., (995), 8, 9-. [8] L. Chen, J. Zheng, F. Sun, and C. Wu. Performance comarison of an endoreersible closed ariable temerature heat reseroir Brayton cycle under maximum ower density and maximum ower conditions. Energy Conersion & Management, (),, -. [9] Y. Ge, L. Chen,, F. Sun, C. Wu. Performance of an Atkinson cycle with heat transfer, friction and ariable secific-heats of the working fluid. Alied Energy, (6) 8,. [] Y. Zhao, J. Chen. Performance analysis and arametric otimum criteria of an irreersible Atkinson heat-engine. Alied Energy, (6), 8, [] P. Wang, and S.-S. Hou. Performance analysis and comarison of an Atkinson cycle couled to ariable temerature heat reseroirs under maximum ower and maximum ower density conditions. Energy Conersion & Management, (5), 6, [] L. Chen, J. Zheng, F. Sun, C. Wu. Performance comarison of an irreersible closed Brayton cycle under maximum ower density and maximum ower conditions. Exergy, (),, 5 5. []. Zheng, L. Chen, F. Sun, C. Wu. Power, ower density and efficiency otimization of an endoreersible, Braysson cycle. Exergy, (),, [] E. Abu-Nada, I. Al-Hinti, A. Al-Sarkhi, B. Akash, Effect of iston friction on the erformance of SI engine: A new thermodynamic aroach. ASME Journal of Engineering for Gas urbines and Power, (8), (), Paer No. 8. [5] E. Abu-Nada, I. Al-Hinti, B. Akash, A. Al-Sarkhi, hermodynamic analysis of sark-ignition engine using a gas mixture model for the working fluid. Int. J. Energy Research, (7),, -6. [6] Al-Sarkhi, E. Abu-Nada, I. Al-Hinti, B. Akash, Performance ealuation of a miller engine under arious secific heat models. Int. Comm. Heat Mass ransfer, (7),, [7] E. Abu-Nada, I. Al-Hinti, A. Al-Sarkhi, B. Akash, hermodynamic modeling of sark-ignition engine: Effect of temerature deendent secific heats. Int. Comm. Heat Mass ransfer, (6),, 6-7. [8] Al-Sarkhi, J.O. Jaber, S.D. Probert, Efficiency of a Millar engine, Alied Energy, (6), 8,5-65. [9] Al-Sarkhi, J.O. Jaber, M. Abu-Qudais, S.D. Probert, Effect of friction and temerature-deendent secific-heat of the working fluid on the erformance of a diesel engine, Alied Energy, (6), 8, -5. [] Y. Cengel and R. urner, Fundamental of hermal-fluid Sciences. Second Edition, New York, McGraw-Hill; 5.

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