Vol. 124 (2013) ACTA PHYSICA POLONICA A No. 1 Thermodynamic Modeling of an Atkinson Cycle 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 thermodynamics. The results show that if the compression ratio is less than a certain value, the power 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 engines. 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 temperature. The results The Atkinson cycle is a type of internal combustion show that the power output as well as the eciency, for engine, 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 heats of the work- 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 hybrid vehicle [1]. For ex- universal steady ow heat 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 variable valve timing 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 Otto cycle 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 energy, friction 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 volume for air, fuel and residual gases, respectively. Atkinson cycle are derived in this paper. The mass specic heat at constant pressure 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 entropy ( ) 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 temperatures 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 adiabatic process 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 isochoric process 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 )_mf 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.