FINITE TIME THERMODYNAMIC MODELING AND ANALYSIS FOR AN IRREVERSIBLE ATKINSON CYCLE. By Yanlin GE, Lingen CHEN, and Fengrui SUN

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1 FINIE IME HERMODYNAMIC MODELING AND ANALYSIS FOR AN IRREVERSIBLE AKINSON CYCLE By Yanlin GE, Lingen CHEN, and Fengrui SUN Performance of an air-standard Atkinson cycle is analyzed by using finite-time thermodynamics. he irreversible cycle model which is more close to ractice is founded. In this model, the nonlinear relation between the secific heats of working fluid and its temerature, the friction loss comuted according to the mean velocity of the iston, the internal irreversibility described by using the comression and exansion efficiencies, and heat transfer loss are considered. he relations between the ower outut and the comression ratio, between the thermal efficiency and the comression ratio, as well as the otimal relation between ower outut and the efficiency of the cycle are derived by detailed numerical examles. Moreover, the effects of internal irreversibility, heat transfer loss and friction loss on the cycle erformance are analyzed. he results obtained in this aer may rovide guidelines for the design of ractical internal combustion engines. Key words: finite-time thermodynamics, Atkinson cycles, heat resistance, friction, internal irreversibility, erformance otimization.. Introduction Finite time thermodynamics can answer some global questions which classical thermodynamics does not try to answer and conventional irreversible thermodynamics can not answer because of its micro, differential viewoint. Examles of such questions are: ( What is the least energy required by a given machine to roduce a given work in a given time? (2 What is the most work that can be roduced by a given machine in given time, utilizing a given energy? (3 What is the most efficient way to run a given thermodynamic rocess (otimal ath in finite time? (4 What is the otimal time-deendent (on & off rocess? (5 What is the otimal distribution between heat exchanger heat transfer surface areas or heat conductances corresonding to the otimal erformance of the thermodynamic devices for the fixed total heat exchanger heat transfer surface area or total heat conductance? (6 What are the quantitative and qualitative features of the effects of heat resistance, internal irreversibility and heat leakage on the erformance of real thermodynamic rocesses and devices? A series of achievements have been made since finite-time thermodynamics was used to analyze and otimize erformance of real heat engines [-0]. Chen et al [] studied the efficiency of an Atkinson cycle at maximum ower density without any loss. Qin et al [2] and Ge et al [3] derived the erformance characteristics of Atkinson cycle with heat transfer loss [2] and with heat transfer and friction-like term losses [3], resectively. Ge et al [4, 5] considered the effect of variable secific heats on the cycle rocess and studied the erformance characteristic of

2 endoreversible and irreversible Atkinson cycles when variable secific heats of working fluid are linear functions of its temerature and the maximum temerature of the cycle is not fixed. Wang et al [7] analyzed and comared the erformance of an Atkinson cycle couled to variable-temerature heat reservoirs under maximum ower and maximum ower density conditions. Zhao et al [7] analyzed the erformance and otimized the arametric criteria of an irreversible Atkinson heat engine. Hou et al [8] comared the erformance of air standard Atkinson and Otto cycles with heat transfer considerations. Lin et al [9] analyzed the influence of heat loss, as characterized by a ercentage of fuel s energy, friction and variable secific heats of working fluid on the erformance of an air-standard Atkinson cycle when variable secific heats of working fluid are linear functions of its temerature and the maximum temerature of the cycle is not fixed. Al-Sarkhi et al [20] outlined the effect of maximum ower density on the erformance of the Atkinson cycle efficiency when the variable secific heats of working fluid are linear functions of its temerature. Abu-nada et al [2] and Al-Sarkhi et al [22] advanced a nonlinear relation between the secific heats of working fluid and its temerature and comared the erformance of the cycle with constant and variable secific heats. Parlak et al [23] defined the internal irreversibility by using entroy roduction, and analyzed the effect of the internal irreversibility on the erformance of irreversible recirocating heat engine cycles. Zhao et al [24-26] defined the internal irreversibility by using comression and exansion efficiencies and analyzed the erformance of Diesel, Otto, Dual and Miller cycles when the maximum temerature of the cycle is fixed and the efficiency has a new definition. Zhao et al [27, 28] used the model of secific heats advanced in Refs. [4, 5], the internal irreversibility defined in [24-26], and studied the otimum erformance of Otto and Diesel cycles when the maximum temerature of the cycles is fixed. Ge et al [29-30] adoted the secific heat model advanced in Ref. [2, 22], the internal irreversibility defined in Refs. [24-28] and the friction loss defined in Ref. [3], and studied the erformance of an irreversible Otto, Diesel and Dual cycles when heat transfer, friction and internal irreversibility losses are considered. his aer will adot the secific heats model advanced in Ref. [2, 22, 29-30], the internal irreversibility and efficiency defined in Refs. [24-30] and the friction loss defined in Refs. [29-3], and study the erformance of an irreversible Atkinson cycle when heat transfer, friction and internal irreversibility losses and nonlinear variable secific heats of the working fluid are considered. 2. Cycle model and analysis An air standard Atkinson cycle model is shown in fig. Process 2S is a reversible adiabatic comression, while rocess 2is an irreversible adiabatic rocess that takes into account the internal irreversibility in the real comression rocess. he heat addition is an isochoric rocess 2 3. Process 3 4S is a reversible adiabatic exansion, while 3 4is an irreversible adiabatic rocess that takes into account the internal irreversibility in the real exansion rocess. he heat rejection is an isobaric rocess 4. In most cycle model, the working fluid is assumed to behave as an ideal gas with constant secific heats. But this assumtion can be valid only for small temerature difference. For the large temerature difference encountered in ractical cycle, this assumtion can not be alied. According to Ref. [2], for the temerature range of ressure can be written as [K], the secific heat caacity with constant

3 C = ( R g ( Figure. -s diagram for the cycle model. For the temerature range of [K], the above equation is written as C = ( R g (2 Equations ( and (2 can be alied to a temerature range of for the temerature range ( [K] which is too wide [K] of ractical engine. So a single equation was used to describe the secific heat model which is based on the assumtion that air is an ideal gas mixture containing 78.% nitrogen, 20.95% oxygen, 0.92% argon, and 0.03% carbon dioxide. C = (3 According to the relation between secific heat with constant ressure and secific heat with constant volume Cv C R g = (4 the secific heat with constant volume can be written as 2

4 C = C R = v g (5 - - where R g = [kjkg K ] is the gas constant of the working fluid. he unit of Cv and C is - - [kjkg K ]. he heat added to the working fluid during rocess 2 3 is in 3 3 Q = M C d = M ( v [ d = M ] 2 (6 he heat rejected by the working fluid during rocess 4 is out 4 4 Q = M C d = M ( = M[ d ] (7 where the M [kg / s ]is the mass flow rate of the working fluid,, 2, 3 and 4 [K ] are the temeratures at states, 2, 3 and 4. For the two adiabatic rocesses 2 and 3 4, the comression and exansion efficiencies can be defined as [24-30] η = ( ( (8 c 2S 2 η = ( ( (9 e 4 3 4S 3 hese two efficiencies can be used to describe the internal irreversibility of the rocesses. Since C and are deendent on temerature, adiabatic exonent Cv k C Cv = will vary with temerature as well. herefore, the equation often used in reversible adiabatic rocess with constant k can t be used in reversible adiabatic rocess with variable k. However, according to Refs. [4, 5, 29-40], a suitable engineering aroximation for reversible adiabatic rocess with variable k can be made, i.e. this rocess can be broken u into a large number of infinitesimally small rocesses and for each of these rocesses, adiabatic exonent k can be regarded as a constant. For examle, for any reversible adiabatic rocess between states i and j can be regarded as consisting of numerous infinitesimally small rocesses with constant k. For any of these rocesses, when an infinitesimally small change in temerature d, and volume dv of the working fluid takes lace, the equation for reversible adiabatic rocess with variable k can be written as follows 3

5 V = ( + d ( V + d V k- k- (0 For an isochoric heat addition rocess i j, the heat added is Qin = Cv ( j i = Δ Si j = Cv ln( j / i. So one has j i = ( /ln( j i, where is the equivalent temerature of heat absortion rocess. When C is the function of temerature, the C ( v can be regarded as v mean secific heat with constant volume. From eq. (0, one gets j Cvln i V i = Rg ln ( Vj where the temerature in the equation of C v is j i =. ln( j i he comression ratio is defined as γ = V V2 (2 herefore, equations for reversible adiabatic rocesses 2S and 3 4S are as follows 2S Cvln = R ln γ (3 g C ln R ln = R ln γ (4 4S v g g 3 4S For an ideal Atkinson cycle model, there are no heat transfer losses. However, for a real Atkinson cycle, heat transfer irreversibility between working fluid and the cylinder wall is not negligible. One can assume that the heat transfer loss through the cylinder wall (i. e. the heat leakage loss is roortional to average temerature of both the working fluid and the cylinder wall and that the wall temerature is constant the heat leakage coefficient of the cylinder wall is [ 0 B kjkg K [K ]. If the released heat by combustion er second is A [ kw ], ] which has considered the heat transfer coefficient and the heat exchange surface, one has the heat added to the working fluid er second by combustion in the following linear relation [2-5] Q = A MB[( + / 2 ] in (5 From eq. (5, one can see that contained two arts, the first art is Qin A, the released heat by combustion er second, and the second art is the heat leak loss er second, it can be written as 4

6 Q = MB( + 2 (6 leak where B = B /2. aking into account the friction loss of the iston as recommended by Chen et al [3] for the Dual cycle and assuming a dissiation term reresented by a friction force which in a liner function of the velocity gives f μ dx = μv = μ (7 dt where μ [N sm] is a coefficient of friction which takes into account the global losses and x is the iston dislacement. hen, the lost ower is dw dx dx dt dt dt μ 2 μ = = μ = μν (8 P If one secifies the engine is a four stroke cycle engine, the total distance the iston travels er cycle is 4L = 4( x x (9 2 For a four stroke cycle engine, running at iston is N cycles er second, the mean velocity of the ν = 4LN (20 where x and x 2 [ m ] are the iston osition at maximum and minimum volume and L [m ] is the distance that the iston travels er stroke, resectively. hus, the ower outut is P = Q Q P at in out μ = M[ ( ( ( ( (.3303( ( ( ( + ] μv (2 he efficiency of the cycle is 5

7 ηat = Q in P at + Q leak M[ ( ( ( ( (.3303( ( ( ( ] μv = M[ ( ( ( ( ( ( ( ( ] + MB( + 2 (22 When γ,, 3, η c and η e are given, 2S can be obtained from eq. (3, then, substituting 2S into eq. (8 yields 2, 4S can be obtained from eq. (4, and the last, 4 can be solved out by substituting 4S into eq. (9. Substituting 2 and 4 into eqs. (2 and (22 yields the ower and efficiency. hen, the relations between the ower outut and the comression ratio, between the thermal efficiency and the comression ratio, as well as the otimal relation between ower outut and the efficiency of the cycle can be obtained. 3. Numerical examles and discussion According to Ref. [29-3], the following arameters are used: = 350[K], 3 = 2200[K], x = 8 0 m, x 2 = 0 m, N = 30, and M = [kg/s]. Figures 2-4 show the effects of the internal irreversibility, heat transfer loss and friction loss on the erformance of the cycle. One can see that when the above three irreversibilities are not included, the ower outut versus comression ratio characteristic and the ower outut versus efficiency characteristic are arabolic-like curves, while the efficiency will increase with the increase of the comression ratio. When more than one irreversibilities are included, the ower outut versus comression ratio characteristic and the efficiency versus comression ratio characteristic are arabolic like curves and the ower outut versus efficiency curve is loo-shaed one. Figure 2. he influences of the internal irreversibility and friction loss on the ower outut 6

8 According to eq. (2, the definitions of the ower outut, the heat transfer loss has no effect on the ower outut of the cycle. So fig. 2 only shows the effects of the internal irreversibility and friction loss on the ower outut of the cycle. Comaring curves with, 2 with 2, one can see that the ower outut increases with the decrease of internal irreversibility. Comaring curves with 2 with 4, with 2, one can see that the ower outut decreases with the increase of friction loss. Figure 3 shows the effects of the internal irreversibility, heat transfer loss and friction loss on the efficiency of the cycle. Curve is the efficiency versus comression ratio characteristic without irreversibility. Under this circumstance, the efficiency increases with the increase of comression ratio. Other curves are efficiency versus comression ratio characteristic with one or more irreversibilities and these curves are arabolic-like ones. Comaring curves with, 2 with 2, 3 with 3 and 4 with 4, one can see that the efficiency increases with the decrease of internal irreversibility. Comaring curves with 3, 2 with 4, with 3 and 2 with 4, one can see that the efficiency decreases with the increase of heat transfer loss. Comaring curves with 2, 3 with 4, with 2 and 3 with 4, one can see that the efficiency decreases with the increase of friction loss. Figure 3. he influences of internal irreversibility, heat transfer loss and friction loss on the efficiency Figure 4 shows the effects of the internal irreversibility, heat transfer loss and friction loss on the ower outut versus the efficiency characteristic. Curve which is a arabolic like curve is the ower outut versus efficiency characteristic of the cycle without irreversibility, while else curves are loo-shaed ones with one or more irreversibilities. Comaring curves with, 2 with 2, 3 with 3, 4 with 4, one can see that the maximum ower outut and the efficiency at the maximum ower outut decrease with the increase of internal irreversibility. Comaring curves with 3, 2 with 4, with 3 and 2 with 4, one can see that the maximum ower outut is not influenced by heat transfer loss, while the efficiency at the maximum ower outut decreases with the increase of heat transfer loss. Comaring curves with 2, 3 with 4, with 2 and 3 with 4, one can 7

9 see that both the maximum ower outut and the corresonding efficiency decrease with the increase of friction loss. Figure 4. he influences of internal irreversibility, heat transfer loss and friction loss on the ower outut versus efficiency characteristic 4. Conclusion In this aer, an irreversible air standard Atkinson cycle model which is more close to ractice is founded. In this model, the nonlinear relation between the secific heats of working fluid and its temerature, the friction loss comuted according to the mean velocity of the iston, the internal irreversibility described by using the comression and exansion efficiency, and heat transfer loss are resented. he erformance characteristics of the cycle were obtained by detailed numerical examles. he effects of internal irreversibility, heat transfer loss and friction loss on the erformance of the cycle were analyzed. he results obtained herein may rovide guidelines for the design of ractical internal combustion engines. Acknowledgments his aer is suorted by Program for New Century Excellent alents in University of P. R. China (Project No. NCE and he Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (Project No he authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscrit. Nomenclature A -heat released by combustion er second, [ kw ] - - B -constant related to heat transfer, [ kjkg K ] 8

10 - - C -secific heat with constant ressure, [ kjkg K ] - - C -secific heat with constant volume, [ kjkg K ] v k -ratio of secific heats,[-] L -total distance of the iston traveling er cycle, [ m ] M -mass flow rate of the working fluid, [ kg s ] N -number of the cycle oerating in a second,[-] P 2,3 -ressure at different states 2 and 3, [ Pa ] P du -ower outut of the cycle, [ kw ] P μ -lost ower due to friction, [ kw ] 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 -air constant of the working fluid, [ kjkg K ] g 5,2 S,5S -temerature at different states, [ K ] V 3,2 -volume at different states and 2,[ m ] v -mean velocity of the iston, [ ms] x -the iston osition at maximum volume, [ m ] x 2 -the iston osition at minimum volume, [ m ] Greek symbols η du -efficiency of the cycle,[-] η c -comression efficiency,[-] η e -exansion efficiency,[-] μ -coefficient of friction, [ N sm] 9

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