Performance of Universal Reciprocating Heat-Engine Cycle with Variable Specific Heats Ratio of Working Fluid

Considering the finite time characteristic, heat transfer loss, friction loss and internal irreversibility loss, an air standard reciprocating heat-engine cycle model is founded by using finite time thermodynamics. The cycle model, which consists of two endothermic processes, two exothermic processes and two adiabatic processes, is well generalized. The performance parameters, including the power output and efficiency (PAE), are obtained. The PAE versus compression ratio relations are obtained by numerical computation. The impacts of variable specific heats ratio (SHR) of working fluid (WF) on universal cycle performances are analyzed and various special cycles are also discussed. The results include the PAE performance characteristics of various special cycles (including Miller, Dual, Atkinson, Brayton, Diesel and Otto cycles) when the SHR of WF is constant and variable (including the SHR varied with linear function (LF) and nonlinear function (NLF) of WF temperature). The maximum power outputs and the corresponding optimal compression ratios, as well as the maximum efficiencies and the corresponding optimal compression ratios for various special cycles with three SHR models are compared.

Assuming the SHR of WF varied with temperature with NLF, the SHR  can be written as where a , b and c are constants and T is WF temperature. It can be supposed that the four thermal capacities of the cycle are 22 It can be supposed that two adiabatic processes are instantaneous, and the temperature of WF changes at a constant speed. 1 k , 2 k , 3 k and 4 k are constants, then the time spent on each cycle is The heat addition in the processes 23 → and 34 → can be written as ( ) The heat rejection in the processes 56 → and 61 → can be written as When the above four SH are different values, this universal cycle model will be simplified to all kinds of special cycle models. The two irreversible adiabatic processes are shown as 1 → 2 and 4 → 5 ; the two heating processes are shown as 2 → 3 and; and the two cooling processes are shown as 5 → 6 and 6 → 1 .
Assuming the SHR of WF varied with temperature with NLF, the SHR γ can be written as where a, b and c are constants and T is WF temperature. It can be supposed that the four thermal capacities of the cycle are where m in1 , m in2 , m out1 , m out2 , n in1 , n in2 , n out1 and n out2 are constants. When m in1 is 1, n in1 is c, and when m in1 is 0, n in1 is 1, so do m in2 and n in2 , m out1 and n out1 and m out2 and n out2 . When the constants have different values, the four thermal capacities can change into the SH with constant pressure and constant volume It can be supposed that two adiabatic processes are instantaneous, and the temperature of WF changes at a constant speed. k 1 , k 2 , k 3 and k 4 are constants, then the time spent on each cycle is The heat addition in the processes 2 → 3 and 3 → 4 can be written as The heat rejection in the processes 5 → 6 and 6 → 1 can be written as R is gas constant and M is mole number of WF. The following parameters are defined as For the two irreversible adiabatic processes, the IIL are defined as the expansion and compression efficiencies [86][87][88][89] η According to references [75][76][77][78][79][80][81], the expression for reversible adiabatic process when SHR is varied is From Equation (10), one has For the endoreversible adiabatic process 1 → 2S , one has The special cycles m in1 , m in2 , m out1 , m out2 , n in1 , n in2 , n out1 and n out2 are fixed, and Equation (12) becomes the expression of the adiabatic process for the various special cycles.
After a cycle, the entropy change of the WF is zero, so one has For a practical cycle, there exists HTL and FL. According reference [67], the heat addition rate to the WF by combustion is: where A is the heat released by fuel, B is heat leakage coefficient, T 0 is the ambient temperature and A = A + B T 0 and B = B /2 are two constants.
According to reference [70], the lost power due to FL is The mean velocity of piston motion is v = x(r − 1)/∆t 12 (16) where µ is the friction coefficient in exhaust stroke, v is the mean velocity of piston, X is the piston displacement, x is the piston position at upper dead point and ∆t 12 is the time of the power stroke.

Power Output and Thermal Efficiency
The net power output is In order to make the cycle run normally, State 3 must be between States 2 and 4. When States 2 and 3 coincide, it gives r p = (r p ) min = 1, and when States 3 and 4 coincide, it gives r p = (r p ) max and T 3 = (r p ) max T 2 = T 4 . T 2 can be gotten by Equations (12) and (8). Substituting T 3 = (r p ) max T 2 = T 4 into Equations (5) and (14) gives (r p ) max . So the range of r p is State 6 must be between States 1 and 5. When States 1 and 6 coincide, it gives r c = (r c ) min = 1, and when States 3 and 4 coincide, it gives r c = (r c ) max and T 6 = (r c ) max T 1 = T 5 . Substituting T 2 into Equation (7) gives T 3 , substituting T 3 into Equations (5) and (14) gives T 4 and substituting T 6 = (r c ) max T 1 = T 5 and the temperatures above into Equation (13) gives (r p ) max . So the range of r c is

Discussions
Equations (17) and (18) are the PAE characteristics of the universal cycle which include all kinds of RHEC with different loss items.
(4) When m in1 = m in2 = m out1 = m out2 = 1 and n in1 = n in2 = n out1 = n out2 = c, the expressions can be simplified into the PAE for an AS Brayton cycle.
(7) When a 0, the expressions can be simplified into the PAE for the universal cycle with variable SHR of WF with the NLF of temperature; when a = 0 and b 0, the expressions can be turned into those with variable SHR with the LF of temperature; and when a = b = 0, the expressions can be simplified into those for the constant SHR.
(8) When η c 1 and η e 1, the expressions can be simplified into the PAE for the universal cycle with IIL, and when η c = η e = 1, the expressions can be simplified into that for the cycle without IIL.
(9) When b f 0, the expressions can be simplified into the PAE for the cycle with FL, and when b f = 0, the expressions can be simplified into those for the cycle without FFL.
(10) When B 0, the expressions can be simplified into the PAE for the cycle with HTL, and when B = 0, the expressions can be simplified into those for the cycle without HTL.

Numerical Examples
According to references [81,[95][96][97][98], the following constants are used in the computations: A = 60000 J/mol, B = 25 J/(mol · K), T 1 = 300 K, b = −9.7617 × 10 −5 K −1 , a = 1.6928 × 10 −8 K −1 , c = 1.4235, M = 0.0157 mol, R = 8.314, γ p = 1.2, γ c = 1.2, b f = 32.5 W, K 3 = K 4 = 18.67 × 10 −6 s · K −1 and K 1 = K 2 = 8.128 × 10 −6 s · K −1 . Figures 2 and 3 illustrate the relations of power output versus CR and efficiency versus CR for various special cycles with the constant SHR and variable SHR with LF and NLF of temperature. Figure 2 shows that, compared with the constant SHR, the ranges of the CRs of various special cycles with the variable SHR with the LF of temperature increase from 13.5 to about 16, the power output decreases (0.5% decrease for Miller cycle; 3% decrease for Otto and Atkinson cycles; 8% decrease for Diesel, Brayton and Dual cycles). Compared with the variable SHR with the LF of temperature, the ranges of CR of various special cycles with the variable SHR with NLF of temperature decrease from 16 to about 15; the changes of power output are unobvious (0.5% increase for Diesel cycle; 0.1% increase for Dual cycle; about 0.5-2% decrease for Otto, Atkinson, Brayton and Miller cycles). The order of the maximum power outputs (MPO) of various special cycles is P br > P at > P mi > P du > P di > P ot with every one of the three SHR models.        For the Otto cycle, under three SHR models, the orders of the MPOs and the corresponding optimal CRs are (P ot ) C > (P ot ) L > (P ot ) N and (r ot ) L > (r ot ) N > (r ot ) C . For the Diesel cycle, under three SHR models, the orders of the MPOs and the corresponding optimal CRs are (P di ) C > (P di ) N > (P di ) L and (r di ) C > (r di ) L > (r di ) N . For Atkinson cycle, under three SHR models, the orders of the MPOs and the corresponding optimal CRs are (P at ) C > (P at ) L > (P at ) N and (r at ) L > (r at ) N > (r at ) C . For the Brayton cycle, under three SHR models, the orders of the MPOs and the corresponding optimal CRs are (P br ) C > (P br ) L > (P br ) N and (r br ) L > (r br ) C > (r br ) N . For the Dual cycle, under three SHR models, the orders of the MPOs and the corresponding optimal CRs are (P du ) C > (p du ) L > (P du ) N and (r du ) C > (r du ) L > (r du ) N . For the Miller cycle, under three SHR models, the orders of the MPOs and the corresponding optimal CRs are (P mi ) C > (P mi ) L > (P mi ) N and (r mi ) L > (r mi ) N > (r mi ) C . Under constant SHR model, the order of the corresponding optimal CRs at the MPO points of various special cycles is r di > r du > r br > r mi > r ot > r at . Under variable SHR with the LF of the temperature model, the order of the corresponding optimal CRs at the MPO points of various special cycles is r di > r br > r mi > r du > r ot > r at . Under variable SHR with NLF of temperature model, the order of the corresponding optimal CRs at the MPO points of various special cycles is r di > r du > r mi > r br > r ot > r at . Figure 3 shows that, compared with the constant SHR, the efficiencies of various special cycles with the variable SHR with the LF of temperature decrease by about 12%. Compared with the variable SHR with the LF of temperature, the efficiencies of various special cycles with variable SHR with NLF of temperature increase by about 0.7-1.7%. The order of the maximum efficiency of various special cycles is η at > η mi > η br > η ot > η du > η di with every one of the three SHR models.
For the Otto cycle, under three SHR models, the orders of the maximum efficiencies and corresponding optimal CRs are (η ot ) C > (η ot ) N > (η ot ) L and (r ot ) C > (r ot ) L > (r ot ) N . For the Diesel cycle, under three SHR models, the orders of the maximum efficiencies and corresponding optimal CRs are For the Atkinson cycle, under three SHR models, the orders of the maximum efficiencies and corresponding optimal CRs are (η at ) C > (η at ) N > (η at ) L and (r at ) L > (r at ) N > (r at ) C . For the Brayton cycle, under three SHR models, the orders of the maximum efficiencies and corresponding optimal CRs are (η br ) C > (η br ) N > (η br ) L and (r br ) C > (r br ) N > (r br ) L . For the Dual cycle, under three SHR models, the orders of the maximum efficiencies and corresponding optimal CRs are (η du ) C > (η du ) N > (η du ) L and (r du ) C > (r du ) N > (r du ) L . For the Miller cycle, under three SHR models, the orders of the maximum efficiencies and corresponding optimal CRs are (η mi ) C > (η mi ) N > (η mi ) L and (r mi ) L > (r mi ) N > (r mi ) C .
Under the constant SHR model, the order of the corresponding optimal CRs at the maximum efficiency points of various special cycles is r di > r du > r br > r ot > r mi > r at . Under variable SHR with the LF of temperature model, the order of the corresponding optimal CRs at the maximum efficiency points of various special cycles is r di > r du > r br > r mi > r ot > r at . Under variable SHR with NLF of temperature model, the order of the corresponding optimal CRs at the maximum efficiency points of various special cycles is r di > r du > r br > r ot > r mi > r at .
In general, the optimal CR at MPO point is not the same as the optimal CR at maximum efficiency point, for all discussed cycles with three SHR models. The reasonable design range for all of the discussed cycles with three SHR models should be between the optimal CR at MPO point and the optimal CR at maximum efficiency point from the point of view of compromised optimization of the PAE.
From what was mentioned above, one can see that there are influences of the variable SHR model on the performance of every special cycle; and the performances of Miller, Brayton and Atkinson cycles are more excellent than those of Otto, Diesel and Dual cycles with every one of the three SHR models.

Conclusions
The AS RHEC model considering HTL, FL and IIL is established in this paper. The cycle performances with various SHR are analyzed. The performance parameters including the PAE are derived. The performances of all kinds of special cycles are discussed and the MPO and the maximum efficiency of each special cycle and the corresponding optimal CRs are compared. The results show that the orders of the MPO and the maximum efficiency remain the same with every one of the three SHR models, but the PAE changes, which suggests that the various SHRs have influences on cycle performance. When the model of variable SHR is more complicated, the distance between the cycle model and the practice one is closer. The reasonable design range for various cycles should be between the optimal CR at MPO point and the optimal CR at maximum efficiency point for the compromise optimization of the PAE.
Author Contributions: L.C., Y.G., C.L., H.F. and G.L. equally contributed to the manuscript. All authors have read and agreed to the published version of the manuscript.
Acknowledgments: This paper is supported by the National Natural Science Foundation of China (project number 51779262). The author wishes to thank four reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.