Four-Objective Optimization for an Irreversible Porous Medium Cycle with Linear Variation in Working Fluid’s Specific Heat

Considering that the specific heat of the working fluid varies linearly with its temperature, this paper applies finite time thermodynamic theory and NSGA-II to conduct thermodynamic analysis and multi-objective optimization for irreversible porous medium cycle. The effects of working fluid’s variable-specific heat characteristics, heat transfer, friction and internal irreversibility losses on cycle power density and ecological function characteristics are analyzed. The relationship between power density and ecological function versus compression ratio or thermal efficiency are obtained. When operating in the circumstances of maximum power density, the thermal efficiency of the porous medium cycle engine is higher and its size is less than when operating in the circumstances of maximum power output, and it is also more efficient when operating in the circumstances of maximum ecological function. The four objectives of dimensionless power density, dimensionless power output, thermal efficiency and dimensionless ecological function are optimized simultaneously, and the Pareto front with a set of solutions is obtained. The best results are obtained in two-objective optimization, targeting power output and thermal efficiency, which indicates that the optimal results of the multi-objective are better than that of one-objective.

Many scholars have studied the P, η and Ep objective functions of the heat engine cycles. Diskin and Tartakovsky [46] combined electrochemical and Otto cycles, and studied the η characteristic relationship in the circumstances of maximum P. Wang et al. [47] established the PM engine model with classical thermodynamic theory, and calculated the influence of compression ratio, pre-expansion ratio, pre-pressure ratio on the η and work output of the PM engine. Zhao et al. [84] investigated the effects of initial temperature, structure and injection duration on engine compression ignition in a methane-powered PM engine.
As one of the thermodynamic cycles, the PM cycle has constant volume processes in both endothermic and exothermic processes, similar to the Otto cycle. Liu et al. [85] first applied FTT theory to investigate P and η of an endoreversible PM cycle. Ge et al. [86] studied the P and η of an irreversible PM cycle. The PM cycle can be changed to the Otto cycle when the pre-expansion ratio is 1. Zang et al. [87] studied the P d performance and performed MOO of the P, η, P d and E of an irreversible PM cycle.
The previous research of PM cycles assumed that the SH of the WF remained constant during the cycle, but in the actual cycle, the SH of the WF is constantly changing during the functioning of the heat engine. In this paper, based on Ref. [86], an irreversible PM cycle model will be established based on the linear change in SH of the working fluid with its temperature [88], and the FTT theory will be applied to further study the performance of P d and E. The η, P, P d and E of the irreversible PM cycle will be optimized by MOO, and the optimal result with the smallest deviation index (DI) will be obtained.

Model of an Irreversible PM Cycle
The working process of the PM engine is shown in Figure 1a, and the PM combustion chamber is installed on the top of the cylinder. Fresh air enters the cylinder, at this time the PM chamber is isolated from the cylinder, and the PM chamber is fuel vapor. At the end of the intake process, the starter continues to drive the crankshaft to rotate, and the piston moves from bottom to top. At the same time, the PM chamber is closed, and the gas sucked into the cylinder by the intake stroke is enclosed in a closed space. The gas in the cylinder is compressed and the temperature and pressure are getting higher and higher At the end of the compression process, the valve of the PM chamber is opened, and the compressed air enters the PM chamber for instant recuperation, and the recuperation process is approximately a constant volume process. Air and fuel vapor are rapidly mixed in the PM chamber and self-ignited. The heat released during the combustion process is partly stored in the PM chamber and partly driven by the piston to do work, and the combustion process is approximately an isothermal endothermic process. At the end of the adiabatic expansion stroke, the PM chamber valve is closed. After the constant volume exhaust stroke, the intake stroke of a new cycle begins.
An irreversible PM cycle shown in Figure 1b,c: 1-2s is a reversible adiabatic compression process, 1-2 is an irreversible adiabatic compression process; 2 -3 is a constant volume heat recovery process; 3-4 is an isothermal endothermic process; 4-5s is an reversible process of adiabatic expansion, 4-5 is an irreversible process of adiabatic expansion; and 5-1 is constant volume exothermic process.
In the actual cycle, the SH of the WF is constantly changing during the functioning of the heat engine. According to Ref. [88], when the working temperature of the heat engine is between 300-2200 K, the SH of the WF changes linearly with its temperature, and the constant volume SH of the WF is where b v and K are constants. The cycle temperature ratio (τ), pre-expansion ratio (ρ) and compression ratio (γ) are defined as For processes 1-2 and 4-5, the IIL due to friction, turbulence and viscous stress of the cycle is represented by the compression and expansion efficiency: Because the WF's SH fluctuates with temperature, according to Ref. [88], it is assumed that the process can be decomposed into an infinite number of infinitesimal processes. For each infinitesimal process, it can be approximated that the SH is constant, adding all the infinitesimal processes together constitutes the entire adiabatic process, and any reversible adiabatic process between states i and j may be considered a reversible adiabatic process with infinitely small adiabatic exponent k as a constant. When the temperature and specific volume of the WF change by dT and dV, the following formula can be obtained According to Equation (7), one has According to the processes 1 → 2s and 4 → 5s , one has The heat absorption rate of WF is The heat release rate of WF is where M is the mass flow rate.
In an actual PM cycle, there is HTL between the WF and the cylinder. According to Ref. [13], the HTL rate is defined as where A represents the fuel exothermic rate, T 0 represents ambient temperature and B = 2B 1 represents the HTL coefficient. The FL needs to be considered in an actual PM cycle. According to Ref. [35], the FL is a linear function of speed. The power dissipated by FL is where n represents the rotational speed and L represents the stroke length. The cycle P and η are According to Ref. [89], the volume of total cycle, stroke and clearance are, respectively, as follows: According to Ref. [60], the P d is defined as The entropy generation rates due to FL, HTL, IIL and exhaust stroke are, respectively: The total entropy generation rate is In Equation (26), the temperature in constant volume SH In the actual cycle, the state 3 must be between states 2 and 4, so ρ should satisfy: According to Ref. [86], PM cycle converts to the Otto cycle when ρ = 1, and the P, η, P d , and E expressions of the Otto cycle can be derived from Equations (15), (16), (20) and (27).
The P, P d and E after dimensionless treatment are, respectively: Given the γ, the initial temperature T 1 , the ρ, the maximum cycle temperature T 4 , the η c and η e , the Equation (9) can be used to solve T 2S . Then solve T 2 from Equation (5), solve T 5S from Equation (10), and finally solve T 5 from Equation (6). By substituting the solved T 2 and T 5 into Equations (15), (16), (20) and (27), you can obtain the corresponding P, η, P d and E.

Power Density and Ecological Functions Analyses and Optimizations
The parameters are determined according to Refs. [75,86]: ρ = 1.2, τ= 5.78 ∼ 6.78, b v = 19.868-23.868 J/mol·K, k 1 = 0.003844-0.009844 J/mol·K 2 , T 0 = 300 K, T 1 = 350 K, µ = 1.2 kg/s, . M = 1 mol/s, B= 2.2 W/K, L = 0.07 m and n = 30 s −1 . Figure 2 shows the effects of τ and ρ on the P d and γ (P d − γ) as well as the P d and η (P d − η) characteristics. The curve of P d − γ is parabolic-like one, and the (P d ) max corresponds to a optimal γ (γ P d ). The curve of P d − η is loop-shaped one which starts from the origin and back to the origin, and there are operating points of (P d ) max and maximum η (η max ) in the cycle.

Power Density Analyses and Optimizations
As seen in Figure 2a,b, as τ grows, both γ P d and η P d get larger. When τ grows from 5.78 to 6.78, γ P d grows from 16.5 to 22.3, η P d grows from 0.4809 to 0.5139 and η P d grows by about 6.86%. As seen in Figure 2c,d, as ρ grows, both γ P d and η P d get larger. When ρ grows from 1.2 to 1.6, γ P d grows from 19.3 to 21.9, η P d grows from 0.4986 to 0.5154 and η P d grows by about 3.37%. With the increase in the temperature ratio and pre-expansion ratio, the compression ratio and thermal efficiency in the circumstances of maximum dimensionless power density increase. In Figure 2, ρ = 1 is the performance characteristics of the Otto cycle. Obviously, the PM cycle has a higher η than the Otto cycle. Figure 3 shows the P d − γ and P d − η curves with varying losses and SH characteristics.
The degree of variation in the SH of the WF with temperature is represented by k 1 . The larger the k 1 , the larger the variation range of the SH. As k 1 grows, γ P d grows and η P d declines.
When k 1 = 0, the cycle WF is constant SH. When k 1 grows from 0.003844 J/mol·K 2 to 0.009844 J/mol·K 2 , γ P d grows from 15.8 to 28.4, η P d declines from 0.4992 to 0.4949, a decline of 0.86%. Figure 3c,d show the effects of b v on P d − γ and P d − η characteristics. As b v grows, both γ P d and η P d will become larger. When b v grows from 19.868 J/mol·K to 23.868 J/mol·K, γ P d grows from 19.3 to 28.4, η P d grows from 0.4986 to 0.4993 and η P d grows by about 0.14%. As seen in Figure 3e,f, when only FL exists, comparing curves 1 and 2, as µ grows from 0 kg/s to 1.2 kg/s, γ P d is nearly unchanged, and η P d declines from 62.95% to 62.03%, a decline of 1.46%. When IIL exists only, comparing curves 1 and 1 , as η c and η e declines from 1 to 0.94, γ P d declines from 22.9 to 19.3, η P d declines from 62.95% to 54.65%, a decline of 13.19%. When only HTL exists, comparing curves 1 and 3, as B grows from 0 W/K to 2.2 W/K, η P d declines from 62.95% to 58.34%, a decline of 7.32%. When µ, η c and η e exist, comparing curves 1 and 2 , as µ grows from 0 kg/s to 1.2 kg/s, and the η c and η e decline from 1 to 0.94, γ P d declines from 22.9 to 19.3, η P d declines from 62.95% to 53.74%, a decline of 14.63%. When FL and HTL exist, comparing curves 1 and 4, as µ grows from 0 kg/s to 1.2 kg/s, and B grows from 0 W/K to 2.2 W/K, η P d declines from 62.95% to 57.49%, a decline of 8.67%. When IIL and HTL exist, comparing curves 1 and 3 , as η c and η e decline from 1 to 0.94, the B grows from 0 W/K to 2.2 W/K, γ P d declines from 22.9 to 19.3, η P d declines from 62.95% to 50.71%, a decline of 19.44%. When FL, HTL and IIL exist, comparing curves 1 and 4 , as µ grows from 0 kg/s to 1.2 kg/s, the B grows from 0 W/K to 2.2 W/K, and the η c and η e decline from 1 to 0.94, γ P d declines from 22.9 to 19.3, η P d declines from 62.95% to 49.86%, a decline of 20.79%. As the specific heat of the working fluid changes more violently with temperature and the three losses increase, the thermal efficiency in the circumstances of maximum dimensionless power density decreases. Figure 4 shows the variation in maximum-specific volume ratio (v 1 /v s ), η and maximum pressure ratio (p 3 /p 1 ) with τ in the circumstances of P max and (P d ) max . Figure 4a shows the v 1 /v s , where v 1 is the maximum-specific volume, v s is the stroke volume, and the larger the v 1 /v s , the larger the volume of the engine. Figure 4c shows the p 3 /p 1 , p 3 is the maximum pressure of the cycle, p 1 is the minimum pressure of the cycle, the larger the p 3 /p 1 , the higher the internal pressure of the engine, and the higher the requirements for engine materials. The v 1 /v s corresponding to P max is always larger than v 1 /v s corresponding to (P d ) max , the p 3 /p 1 corresponding to (P d ) max is always larger than the p 3 /p 1 ratio corresponding to P max and η P d is always higher than η P . Compared with P max , the cycle in the circumstances of (P d ) max is smaller and more efficient. Figure 5 shows the effects of cycle parameters on the E and γ (E − γ) as well as the E and η (E − η) characteristics. It can be seen that the E − γ is parabolic-like one, and the maximum ecological function (E max ) corresponds to a γ of γ E .The E − η is loop-shaped one, and there is an E max operating point and an η max operating point in the cycle As seen in Figure 5a,b, as τ grows, both γ E and η E get larger. When τ grows from 5.78 to 6.78, γ E grows from 25.8 to 37.1, η E grows from 0.5086 to 0.5450 and η E grows by about 7.16%. As seen in Figure 5c,d, as ρ grows, both γ E and η E get larger. When ρ grows from 1.2 to 1.6, γ E grows from 33.5 to 43.6, η E grows from 0.5303 to 0.5634 and η E grows by about 3.37%. With the increase in the temperature ratio and pre-expansion ratio, the compression ratio and thermal efficiency in the circumstances of maximum dimensionless ecological function increase. Figure 6 shows the E − P and E − η curves with varying losses and SH characteristics. Figure 6a,c and e show that, except at the P max point, corresponding to any E of the cycle, the P has two different values. The E of the cycle decreases with increasing µ, B, η c and η e . Curve 1 in Figure 6f is reversible without any loss, and the curve is a parabolic-like one, whereas the others are loop-shaped. Each E value (except the maximum value point) corresponds to two η values. The heat engine should be run in the circumstances with a higher η during actual operation. Figure 6a-d show the effects of SH of WF characteristics on cycle performance. Among them, curve 1 is the E − P of the heat engine and the E − η under the conditions of constant SH of WF. Under certain conditions of ecological function, the PM heat engine should be run at a larger power output during actual operation. As the specific heat of the working fluid changes more violently with temperature and the three losses decrease, the ecological function, power output and thermal efficiency will all increase. Figure 7 shows the relationship between P and η characteristics under different OOs. Through numerical calculations, the P max , η max , P in the circumstances of η max (P η ), P in the circumstances of E max (P E ), P in the circumstances of (P d ) max (P p d ), η in the circumstances of P max (η p ), η in the circumstances of (P d ) max (η p d ), and η in the circumstances of E max (η E ) can be obtained. Both P and η decline with the increases of µ, and P max >P p d >P E >P η , η max >η E >η p d >η p . Numerical calculations show that when the µ is 1.2kg/s, P max is 20162 W, P p d is 20049 W, P E is 18904 W, P η is 16725 W, η max is 0.5383 W, η p d is 0.4986 W, η E is 0.5280, and η p is 0.4811. Compared with P max , P p d decreased by about 0.56%, P E decreased by about 6.23%, and P η decreased by about 17.05%. Compared with η max , η p d decreased by about 7.38%, η E decreased by about 1.91%, η p decreased by about 10.63%. Compared with E max , P p d decreased by about 5.71%, η p d increased by about 5.57%. P p d and P E are higher than P η , η E and η p d are higher than η p , P p d is higher than P E and η E is higher than η p d . The ecological function objective function reflects the compromise between power output and efficiency.

Multi-Objective Optimizations
With the increase in cycle OOs, the optimization of the cycle sometimes needs to take into account MOO. However, MOO cannot make many OOs achieve the highest value simultaneously. The finest compromise can be obtained by weighing the advantages and disadvantages of MOO. The NSGA-II ( Figure 8 is the flow chart of the arithmetic) is applied herein, γ is taken as the optimization variables, and the P d , P, η and E are taken as OOs, and one-, two-, three-and four-objective optimizations are performed. Three decision-making methods, LINMAP [90], TOPSIS [91,92] and Shannon Entropy [93], are used to select the reasonable solution, and the average distances (i.e., deviation index) [94] between Pareto frontier and positive or negative ideal point are compared, and the reasonable solution is obtained.
The deviation index is [94] where G j is the j-th optimization objective, G positive j is the j-th optimization objective of the positive ideal point and G negative j is the j-th optimization objective of the negative ideal point. Figure 9 shows the Pareto fronts for MOO, including six two-objective optimizations, four three-objective optimizations, and one four-objective optimization. Table 1 lists the numerical results. As seen in Figure 9a-f, as P grows, η, E, and P d decline. As η grows, P d and E decline. As E grows, P d declines. It can be seen from Table 1 that when E and P d serve as the OOs, the DI obtained by the LINMAP is smaller. When P and η or P and E or η and P d serve as the OOs, the DI obtained by the TOPSIS is smaller. When P and P d or η and E serve as OOs, the DI obtained by the Shannon Entropy is smaller. In the two-objective optimization, when P and η serve as OOs, the DI obtained is the smallest. Figure 10a shows the average spread and generation number of P − η in the circumstances of two-objective optimization. The arithmetic converged at generation 395, and the DI is 0.128. Table 1. Results of one-, two-, three-and four-objective optimizations.    As seen in Figure 9g,h, as P grows, η declines, E and P d first grow and then decline. As seen in Figure 9i, as P grows, E declines, and P d first grows and then declines. As seen in Figure 9j, as η grows, P d declines, and E grows first and then declines. It can be seen from Table 1 that when P, η and P d serve as OOs, the DI obtained by LINMAP is smaller. When P, E and P d serve as OOs, the DI obtained by TOPSIS is smaller. When P, η and E or η, E and P d serve as OOs, the DI obtained by the LINMAP and TOPSIS are the same, and both are smaller than the DI obtained by the Shannon Entropy.

Optimization Schemes Solutions
In the three-objective optimization, when P, E and P d serve OOs, the DI is the smallest. Figure 10b shows the average spread and generation number of P − E − P d in the circumstances of three-objective optimization. The arithmetic converged at generation 344 and the DI is 0.1353.
As seen in Figure 9k, as P grows, η declines, P d grows, and E grows first and then declines. The DI obtained by the LINMAP is smaller. Figure 10c shows the average spread and generation number of P − η − E − P d in the circumstances of four-objective optimization. The arithmetic converged at generation 304, and the DI is 0.1367.
It can be seen from Table 1 that when single-objective optimizations are carried out in the circumstances of P max , η max , E max and (P d ) max , respectively, the DI are 0.5448, 0.2897, 0.1960 and 0.2108, respectively, which are all larger than the best DI 0.1419 obtained in the four-objective optimization, which indicates that MOO produces better results.

Conclusions
Considering the linear variable SH characteristics of the WF, the optimal performance of irreversible PM cycle is studied with P d and E as the OOs in this paper. The effects of the parameters of the cycle on the P d and the E are analyzed; the corresponding η, v 1 /v s and p 3 /p 1 of the cycle under the conditions of (P d ) max and P max are compared; and the corresponding P and η of the cycle under the conditions of P max , η max , (P d ) max , and E max are compared. The four OOs of the irreversible PM cycle are optimized with one-, two-, three-and four-objectives, respectively. The results show that: 1.
The P d − γ and P d − η curves of the cycle are parabolic-like and loop-shaped, respectively. As the temperature ratio and pre-expansion ratio increase, three losses decrease and the specific heat of the working fluid changes more violently with temperature, the compression ratio and thermal efficiency in the circumstances of maximum dimensionless power density increase.

2.
The E − γ and E − P curves of the cycle are parabolic-like and the E − η curves of the cycle are loop-shaped. As the temperature ratio and pre-expansion ratio increase, the compression ratio and thermal efficiency in the circumstances of maximum dimensionless ecological function increase. As three losses decrease and the specific heat of the working fluid changes more violently with temperature, the ecological function, power output and thermal efficiency increase.

3.
Compared with the P max condition, the cycle in the circumstances of (P d ) max is smaller and more efficient.

4.
The DI obtained in one-objective optimization is larger than the optimal DI obtained in MOO, indicating that the MOO results are better. Comparing the results obtained by one-, two-, three-and four-objective optimization, the MOO corresponding to the double-objective optimization P − η is the smallest, and its design scheme is the most ideal.

5.
Variable SH characteristics of the WF always exist. It is necessary to study its effects on the MOO performances of irreversible PM cycles.

Acknowledgments:
The authors wish to thank the reviewers for their careful, unbiased and constructive suggestions, which led to this revised manuscript.

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