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Vol. 124 (2013) ACTA PHYSICA POLONICA A No. 1 Thermodynamic Modeling of an with respect to Relative AirFuel Ratio, Fuel Mass Flow Rate and Residual Gases R. Ebrahimi∗ Department of Agriculture Machine Mechanics, Shahrekord University, P.O. Box 115, Shahrekord, Iran (Received March 26, 2013; in nal form April 17, 2013) The performance of an air standard Atkinson cycle is analyzed using nite-time . The results show that if the compression ratio is less than a certain value, the output increases with increasing relative airfuel ratio, while if the compression ratio exceeds a certain value, the power output rst increases and then starts to decrease with increase of relative airfuel ratio. With a further increase in compression ratio, the increase in relative airfuel ratio results in decrease of the power output. Throughout the compression ratio range, the power output increases with increase of fuel mass ow rate. The results also show that if the compression ratio is less than a certain value, the power output increases with increase of residual gases, on the contrast, if the compression ratio exceeds a certain value, the power output decreases with increase of residual gases. The results obtained herein can provide guidance for the design of practical Atkinson . DOI: 10.12693/APhysPolA.124.29 PACS: 05.70.Ln, 82.60.Fa, 88.05.−b 1. Introduction formance of an air standard Atkinson cycle under the re- striction of the maximum cycle . The results The Atkinson cycle is a type of internal combustion show that the power output as well as the eciency, for , which was designed and built by a British engi- which the maximum power-output occurs, will rise with neer James Atkinson in 1882. The cycle is also called the the increase of maximum cycle temperature. Sargent cycle by several physics-oriented thermodynamic The temperature dependent specic of the - books. The Atkinson cycle engine mainly works at part ing uid have a signicant inuence on the performance. load operating conditions especially in the middle to high Chen et al. [17] built a class of generalized irreversible load range when it is used in a [1]. For ex- universal steady ow engine cycle model consist- ample, Toyota in its Prius II uses a spark ignition engine ing of two heating branches, two cooling branches, and which achieves high eciency by using a Atkinson cycle two adiabatic branches with consideration of the losses of based on management [2]. heat resistance, heat leakage, and internal irreversibility. In the last two decades there have been many eorts in The performance characteristics of Diesel, Otto, Brayton, developing nite-time thermodynamics [37]. Several au- Atkinson, dual and Miller cycles were derived. thors have examined the nite-time thermodynamic per- Thermodynamic analysis of an ideal air standard formance of the reciprocating heat engines [811]. Le Atkinson cycle with temperature dependent specic heat [12] determined the thermal eciency of a reversible is presented by Al-Sarkhi et al. [18]. This paper outlines Atkinson engine cycle at maximum work output. Ge the eect of maximizing power density on the perfor- et al. [13, 14] studied the eect of variable specic heat mance of the cycle eciency. Chen et al. [19] analyzed of the working uid on the performance of endoreversible the performance characteristics of endoreversible and ir- and irreversible Atkinson cycles. Hou [15] compared the reversible reciprocating Diesel, Otto, Atkinson, Brayton, performances of air standard Atkinson and Otto cycles Braysson, Carnot, dual, and Miller cycles with constant with heat transfer loss considerations. The results show and variable specic heats of the working uid. Ust [20] that the Atkinson cycle has a greater work output and a made a comparative performance analysis and optimiza- higher thermal eciency than the at the same tion of irreversible Atkinson cycle under maximum power operating condition. density and maximum power conditions. It is shown that The compression ratios that maximize the work of the for the Atkinson cycle, a design based on the maximum Otto cycle are always found to be higher than those for power density conditions is more advantageous from the the Atkinson cycle at the same operating conditions. Lin point of view of engine sizes and thermal eciency. Liu and Hou [16] investigated the eects of heat loss, as char- and Chen [21] investigated the inuence of variable heat acterized by a percentage of fuel's , and capacities of the working uid, heat leak losses, com- variable specic heats of the working uid, on the per- pression and expansion eciencies and other parame- ters on the optimal performance of the Otto, Brayton, Miller, Diesel and Atkinson cycles. Ebrahimi [22] ana- lyzed the eects of the variable residual gases of working ∗e-mail: [email protected] uid on the performance of an endoreversible Atkinson cycle. Ebrahimi [23] also investigated the eects of the

(29) 30 R. Ebrahimi

h i h i mean piston speed, the equivalence ratio and the cylinder ma  ma  (1−xr) λ m sCva + Cvf + xrCvr 1+λ m s wall temperature on the performance of an irreversible C = f f vt m  Atkinson cycle. On the basis of research workers [1223], 1 + λ a mf s (4) eects of relative airfuel ratio, fuel mass ow rate and where , and are the mass specic heat at con- residual gases on the performance of the air standard Cva Cvf Cvr stant for air, fuel and residual gases, respectively. Atkinson cycle are derived in this paper. The mass specic heat at constant for the working uid ( ) is dened as: 2. Thermodynamic analysis Cpt h i h i ma  ma  (1−xr) λ m sCpa + Cpf + xrCpr 1+λ m s The pressurevolume (P V ) and the temperature C = f f pt m  (  ) diagrams of an irreversible Atkinson heat 1+λ a T S mf s (5) engine is shown in Fig. 1, where T , T , T , T , T , 1 2s 2 3 4 where , and are the mass specic heat at and T are the of the working substance Cpa Cpf Cpr 4s constant pressure for air, fuel and residual gases, respec- in state points 1, 2s, 2, 3, 4, and 4s. Process 1 → 2s is tively. a reversible adiabatic compression, while process 1 → 2 is an irreversible that takes into ac- The heat added per second in the isochoric (2 → 3) count the internal irreversibility in the real compression heat addition process may be written as process. Heat addition is an 2 → 3. Q˙ in =m ˙ tCvt(T3 − T2) Process s is a reversible adiabatic expansion, while 3 → 4 n h i is an irreversible adiabatic process that takes into ma  3 → 4 =m ˙ f (1−xr) λ m sCva + Cvf account the internal irreversibility in the real expansion f h i oT − T ma  3 2 process. The heat-removing process is the reversible con- +xrCvr 1+λ (6) mf s stant pressure 4 → 1. 1−xr where T is the absolute temperature. The heat rejected in the isobaric heat rejection process (4 → 1) may be written as n  ma  Q˙ out =m ˙ tCpt(T4 − T1) =m ˙ f (1−xr) λ Cpa mf s

oT4 − T1   ma   (7) +Cpf + xrCpr 1+λ m s . f 1−xr The total energy of the fuel per second input into the engine, of a gasoline type fuel, such as octane, can be expressed in terms of equivalence ratio factor from mea- sured data as [25, 26]:

Q˙ fuel = ηcomm˙ f QLHV = (−1.44738 + 4.18581λ Fig. 1. (a)  diagram, (b)  diagram for the air P V T S 2 standard Atkinson cycle. −1.86876λ )m ˙ f QLHV, (8) where ηcom is the combustion eciency, QLHV is the The relations between the mass ow rate of the fuel lower caloric value of the fuel.

(m˙ f ) and the mass ow rate of the air (m˙ a), between m˙ f Besides the irreversibility of the adiabatic processes, and the mass ow rate of the airfuel mixture (m˙ t) are the heat transfer irreversibility between the working uid dened as [24, 25]: and the cylinder wall is not negligible.

ma  For an ideal Atkinson cycle model, there are no heat m˙ a =m ˙ f λ (1) mf s and transfer losses. However, for a real Atkinson cycle, heat h i transfer irreversibility between the working uid and the ma  m˙ f 1+λ cylinder wall is not negligible. If the heat leakage coe- mf s (2) m˙ t = , cient of the cylinder wall is , the heat loss through the 1−xr B cylinder wall is given in the following linear expression where ma/mf is the airfuel ratio and the subscript s denotes stoichiometric conditions, λ is the relative air [19, 25]: fuel ratio and is the residual fraction from the previous h i xr ma  Q˙ ht =m ˙ tB(T2 + T3 − 2T0) =m ˙ f B 1+λ cycle, and one can nd it as mf s mr T2 + T3 − 2T0 xr = , (3) × . (9) ma + mf + mr 1−xr where mr is the mass of residual gases. It should be noted It is assumed that the heat loss through the cylinder wall here that the residual gases were assumed to consist of is proportional to the average temperature of both the

CO2,H2O and N2. working uid and the cylinder wall and that the wall tem- The mass specic heat at constant volume for the perature is constant at T0. Equation (9) implies the fact working uid (Cvt) is dened as: that there is a heat leak loss in the combustion process. Thermodynamic Modeling of an Atkinson Cycle . . . 31

Since the total energy of the delivered fuel ˙ is as- Qfuel   ma   sumed to be the sum of the heat added to the working +xrCpr 1+λ (T4 − T1) mf s uid Q˙ in and the heat leakage Q˙ ht, Q˙ = Q˙ − Q˙ = (−1.44738 + 4.18581λ L∗N(T /(T r∗) − 1)2 in fuel ht −4µ 4 1 c . (20) h i r∗ − 1 2 ma  c −1.86876λ )m ˙ f QLHV − m˙ f B 1+λ mf s The eciency of the cycle is  T2 + T3 − 2T0 Pout m˙ f (T3 − T2)n × . (10) ηth = × 100 = Pout (1−xr) 1−xr Qin 1−xr For the two adiabatic processes, the compression and ex- h i h i o pansion eciencies [20, 21]: ma  ma  × λ Cva + Cvf + xrCvr 1+λ mf s mf s ηc = (T2s − T1)/(T2 − T1) (11) and ×100. (21) When ∗, , , ma , , , , , , , , rc ηc ηe ( )s Cva Cvf Cvr Cpa Cpf Cpr B η = (T − T )/(T − T ). (12) mf e 4 3 4s 3 N, T , µ, L, Q , m˙ , x , λ, and T are given, T can These two eciencies can be used to describe the internal 0 LHV f r 1 2s be obtained from Eq. (15), then, substituting T into irreversibility of the processes. 2s Eq. (11) yields T2. T3 can be deduced by substituting Eq. (6) into Eq. (10). T4s can be found from Eq. (16), The specic compression, ∗, and compression, , ra- rc rc and T4 can be deduced by substituting T4s into Eq. (12). tios are dened as: Substituting T1, T2, T3, and T4 into Eqs. (20) and (21), ∗ (13) respectively, the power output and the thermal eciency rc = V1/V2 and of the Atkinson cycle engine can be obtained. Therefore, the relations between the power output, the thermal ef- ∗ (14) rc = V4/V2 = T4/(T1rc ). ciency and the compression ratio can be derived. Therefore ma ma [1−xr][λ( )sCpa+Cpf ]+xrCpr[1+λ( )s] mf mf 3. Results and discussion ma ma −1 ∗ [1−xr][λ( )sCva+Cvf ]+xrCvr[1+λ( )s] mf mf (15) T2s = T1(rc ) According to Refs. [1724], the following parame- and ters are used in the calculations: η = 0.97, η = ma ma c e [1−xr][λ( )sCva+Cvf ]+xrCvr[1+λ( )s] mf mf 0.97, (ma/mf )s = 15.1, Cva = 0.717 kJ/kg K, ma ma T [1−xr][λ( )sCpa+Cpf ]+xrCpr[1+λ( )s] T = T ( 3 ) mf mf . (16) Cvf = 1.638 kJ/kg K, Cvr = 0.8268 kJ/kg K 4s 1 T2 Taking into account the friction loss of the piston and as- Cpa = 1.004 kJ/kg K, Cpf = 1.711 kJ/kg K, Cpr = −1 −1 suming a term represented by a friction force 1.1335 kJ/kg K, T1 = 350 K, B = 1.3 kJ kg K , N = −1 that is a linear function of the piston velocity gives [19] 3000 rpm, T0 = 430 K, µ = 12.9 N s m , L = 80 mm, kJ kg−1, kg s−1, dx QLHV = 44347 m˙ f = 0.0015 → 0.003 fµ = −µv = −µ , (17) xr = 5% → 20% and λ = 0.9 → 1.2. dt The variations in the temperatures , and with where is the coecient of friction, which takes into T2 T3 T4 µ the compression ratio are shown in Fig. 2. It is found account the global losses, is the piston's displacement x that increases with the increase of compression ratio, and is the piston's velocity. Therefore, the lost power T2 v while and decrease with the increase of compression due to friction is T3 T4 ratio. dW  dx2 P = µ = −µ = −µv2. (18) µ dt dt Running at N revolutions per second, the mean velocity of the piston is [25]:  ∗  ∗ T4/(T1rc ) − 1 (19) v¯ = 2LN = 2L ∗ N, rc − 1 where L is the total distance the piston travels per cycle and L∗ is the length of the 1 → 2.

The power output of the Atkinson cycle engine is ex- pressed by W m˙  f   ma  Fig. 2. The temperature versus compression ratio Pout = − |Pµ| = (1−xr) λ Cva mf s τ 1−xr (xr = 20%, m˙ f = 0.0025 kg/s and λ = 1.0).

  ma   +Cvf + xrCvr 1+λ (T3 − T2) mf s As it can be clearly seen from above equations, the power output and the thermal eciency of the Atkin-   ma   − (1−xr) λ Cpa + Cpf son cycle are dependent on the relative airfuel ratio, mf s 32 R. Ebrahimi the fuel mass ow rate and the residual gases. In order to illustrate the eects of these parameters, the relations between the power output and the compression ratio, be- tween the power output and the thermal eciency of the cycle are presented. It can be seen from Figs. 38 that an increase in compression ratio rst leads to an increase in power output, and after reaching a peak, the power output decreases dramatically with further increases in compression ratio. It is found that the relative airfuel ratio, the fuel mass ow rate and the residual gases play important roles on the performance cycle. It is clearly seen from these gures that the eects of the relative Fig. 5. Eect of fuel mass ow rate on the variation airfuel ratio, the fuel mass ow rate and the residual of the power output with compression ratio (xr = 20% gases on the power output is related to the compression and λ = 1). ratio. They show that the maximum power output point and the maximum eciency point are very adjacent. It should be noted that the heat added and the heat re- jected by the working uid decrease as the compression ratio increases.

Fig. 6. Eect of fuel mass ow rate on the variation

of the power output with thermal eciency (xr = 20% and λ = 1).

Fig. 3. Eect of relative airfuel ratio on the variation

of the power output with compression ratio (xr = 20% and m˙ f = 0.0025 kg/s).

Fig. 7. Eect of residual gases on the variation of the

power output with compression ratio (λ = 1.0 and m˙ f = 0.0025 kg/s).

Fig. 4. Eect of relative airfuel ratio on the variation

of the power output with thermal eciency (xr = 20% and m˙ f = 0.0025 kg/s).

It can be concluded from Fig. 3 that if the compres- sion ratio is less than a certain value, the power output increases with increase of relative airfuel ratio. This can be attributed to the fact that the ratio of the heat rejected by the working uid to the heat added by the Fig. 8. Eect of residual gases on the variation of the working uid increase with increase of relative airfuel power output with thermal eciency (λ = 1.0 and m˙ f = ratio. While if the compression ratio exceeds a certain 0.0025 kg/s). Thermodynamic Modeling of an Atkinson Cycle . . . 33 value, the power output rst increases and then starts to the maximum power output will increase by about 165%, decrease with increase of relative airfuel ratio. This can the maximum thermal eciency will increase by about be attributed to the fact that the dierence between heat 25.7%, the working range of the cycle will increase by added and heat rejected rst increase and then starts to about 15.8%, the power output at maximum thermal ef- decrease with increase of relative airfuel ratio. This re- ciency will increase by about 135.3%, the thermal e- sult is consistent with the experimental results in the ciency at maximum power output will increase by about internal combustion engine [27]. 21%, and the optimal compression ratio corresponding With a further increase in compression ratio, the power to maximum power output point will decrease by about output decreases with increase of relative airfuel ratio. 17.4%. It can be concluded that the eect of the fuel This is due to the increase of the heat rejected by the mass ow rate on the power output is more than the working uid being more than the increase of the heat thermal eciency. added by the working uid. In other words, the dier- Figures 7 and 8 indicate the eects of the parameter ence between heat added and heat rejected decreases with residual gases on the power output and the thermal ef- increase of relative airfuel ratio. ciency of cycle for dierent values of the compression Referring to Figs. 3 and 4, it can be revealed that ratio. It can be found from Figs. 7 and 8 that if the com- the maximum power output, the maximum thermal e- pression ratio is less than a certain value, the increase of ciency, the power output at maximum thermal eciency, residual gases will increase the power output, due to the and the thermal eciency at maximum power output increase in the ratio of the heat added by the working rst increase and then decrease as the relative airfuel uid to the heat rejected by the working uid. In con- ratio increases. While the working range of the cycle trast, if the compression ratio exceeds a certain value, and the optimal compression ratio corresponding to max- the increase of residual gases will reduce the power out- imum power output point decreases as the relative air put, because of the decrease in the dierence between fuel ratio increases. Numerical calculation shows that for heat added and heat rejected. This result is consistent any same compression ratio, the smallest power output with the experimental results in the internal combustion engine [28]. is for λ = 0.9 when r < 8.2 and is for λ = 1.2 when c From these gures, it can be concluded that the maxi- r ≥ 8.2, and also the largest power output is for λ = 1.2 c mum power output, the maximum thermal eciency, the when r < 8.4, is for λ = 1.1 when 8.4 ≤ r < 9, is c c compression ratio at the maximum power output, the for λ = 1 when 9 ≤ r < 16.5 and is for λ = 0.9 when c power output at maximum thermal eciency, the ther- rc ≥ 16.5. In order to illustrate the eect of the fuel mass ow mal eciency at maximum power output, and the work- rate on the performance of the Atkinson cycle, the re- ing range of the cycle decrease with increasing residual lations between the power output and the compression gases. For this case, when residual gases increase 15%, ratio and between the power output and the thermal ef- the maximum power output, the compression ratio at ciency of the cycle are presented in Figs. 5 and 6. It the maximum power output, the maximum thermal ef- must be noted here that when the fuel rate increases, ciency, the working range of the cycle, the power out- the amount of heat added to the working uid increases, put at maximum thermal eciency, and the thermal ef- too. In general, the increase in the added heat is faster ciency at maximum power output decrease by about 6, than the increase in the lost heat. It can be concluded 18, 7.1, 40.9, 5.4, and 8%, respectively. from Figs. 5 and 6 that, throughout the compression ra- According to above analysis, it can be found that the tio range, the power output increases with increase of eects of the relative airfuel ratio, the fuel mass ow fuel mass ow rate. This can be attributed to the fact rate, and the residual gases on the cycle performance are that the dierence between heat added and heat rejected, obvious, and they should be considered in practice cycle throughout the compression ratio range, increases with analysis in order to make the cycle model more close to increase of fuel mass ow rate. This result is consistent the practice. with the experimental results in the internal combustion engine [24]. It should be noted here that the heat added 4. Conclusions and the heat rejected by the working uid, throughout The eects of the relative airfuel ratio, the fuel mass the compression ratio range, increases as the fuel mass ow rate and the residual gases on the performance of ow rate increases. an Atkinson cycle are investigated in this study. The From these gures, it can be concluded that the maxi- general conclusions drawn from the results of this work mum power output, the maximum thermal eciency, the are as follows: working range of the cycle, the power output at max- imum thermal eciency, and the thermal eciency at • If the compression ratio is less than a certain value, maximum power output improved when the fuel mass the power output increases with increase of relative ow rate increased. However, the optimal compression airfuel ratio, while if the compression ratio exceeds ratio corresponding to maximum power output point de- a certain value, the power output rst increases and creases with increase of fuel mass ow rate. In this case, then starts to decrease with increase of relative air when the fuel mass ow rate increases by about 100%, fuel ratio. With a further increase in compression 34 R. Ebrahimi

ratio, the increase in relative airfuel ratio results References in decrease of the power output. [1] http://en.wikipedia.org/wiki/Atkinson_cycle . The increasing relative airfuel ratio increases the • [2] K. Nobuki, N. Kiyoshi, K. Toshihiro, Development of maximum power output, the maximum thermal ef- new 1.8-l engine for hybrid vehicles, SAE technical ciency, the power output at maximum thermal paper no. 2009-01-1061, 2009. eciency and the thermal eciency at maximum [3] P.Y. Wang, S.S. Hou, Energy Convers. Manag. 46, power output at rst and then decreases. While the 2637 (2005). working range of the cycle and the optimal com- [4] Y. Ust, B. Sahin, A. Safa, Acta Phys. Pol. A 120, pression ratio corresponding to maximum power 412 (2011). output point decreases as the relative airfuel ratio [5] R. Ebrahimi, J. Energy Inst. 84, 38 (2011). increases. [6] A. Parlak, Energy Convers. Manag. 46, 351 (2005). • Throughout the compression ratio range, the power [7] R. Ebrahimi, Acta Phys. Pol. A 120, 384 (2011). output increases with increase of fuel mass ow [8] L. Chen, F. Meng, F. Sun, Scientia Iranica 19, 1337 rate. (2012). [9] S.S. Hou, J.C. Lin, Acta Phys. Pol. A 120, 979 • The maximum power output, the maximum ther- (2011). mal eciency, the working range of the cycle, the [10] R. Ebrahimi, Comput. Math. Appl. 62, 2169 (2011). power output at maximum thermal eciency and [11] R. Ebrahimi, Acta Phys. Pol. A 117, 887 (2010). the thermal eciency at maximum power output increase as the fuel mass ow rate increases. While [12] H.S. Le, Am. J. Phys. 55, 602 (1987). the optimal compression ratio corresponding to [13] Y. Ge, L. Chen, F. Sun, C. Wu, J. Energy Inst. 80, maximum power output point decreases with in- 52 (2007). crease of fuel mass ow rate. [14] Y. Ge, L. Chen, F. Sun, C. Wu, Appl. Energy 83, 1210 (2006). • If the compression ratio is less than a certain value, [15] S.S. Hou, Energy Convers. Manag. 48, 1683 (2007). the increase of residual gases will increase the power [16] J.C. Lin, S.S. Hou, Appl. Energy 84, 904 (2007). output. In contrast, if the compression ratio ex- [17] L. Chen, W. Zhang, F. Sun, Appl. Energy 84, 512 ceeds a certain value, the increase of residual gases (2007). will reduce the power output. [18] A. Al-Sarkhi, B. Akash, E. Abu-Nada, I. Al-Hinti, Jordan J. Mech. Industr. Eng. 2, 71 (2008). • The maximum power output, the maximum ther- mal eciency, the compression ratio at the max- [19] L.G. Chen, Y.L. Ge, F.R. Sun, Proc. Inst. Mech. Eng. Part D: J. Automob. Eng. 222, 1489 (2008). imum power output, the compression ratio at the maximum thermal eciency, and the working range [20] Y. Ust, Int. J. Thermophys 30, 1001 (2009). of the cycle decrease with increase of residual gases. [21] J. Liu, J. Chen, Int. J. Ambient Energy 31, 59 (2010). The analysis helps us to understand the strong eects [22] R. Ebrahimi, J. Am. Sci. 6, (2) 12 (2010). of relative airfuel ratio, fuel mass ow rate, and residual [23] R. Ebrahimi, Math. Comput. Model. 53, 1289 gases on the performance of an Atkinson cycle. There- (2011). fore, the results are of great signicance to provide good [24] R. Ebrahimi, Ph.D. Thesis, Université de Valen- guidance for the performance evaluation and improve- ciennes et du Hainaut-Cambrésis, France 2006 (in ment of real Atkinson engines. French). [25] J.B. Heywood, Internal Combustion Engine Funda- mentals, Mc-Graw Hill, New York 1988. Acknowledgments [26] G.H. Abd Alla, Energy Convers. Manag. 43, 1043 (2002). The author would like to thank the Shahrekord University for the nancial support of this work. [27] M. Mercier, Ph.D. Thesis, Université de Valenciennes et du Hainaut Cambrésis, France 2006 (in French). [28] A. Hocine, PhD thesis, Université de Valenciennes et du Hainaut-Cambrésis, France 2003 (in French).