Performance Optimizations with Single-, Bi-, Tri-, and Quadru-Objective for Irreversible Atkinson Cycle with Nonlinear Variation of Working Fluid’s Speciﬁc Heat

: Considering nonlinear variation of working ﬂuid’s speciﬁc heat with its temperature, ﬁnite-time thermodynamic theory is applied to analyze and optimize the characteristics of an irreversible Atkinson cycle. Through numerical calculations, performance relationships between cycle dimensionless power density versus compression ratio and dimensionless power density versus thermal efﬁciency are obtained, respectively. When the design parameters take certain speciﬁc values, the performance differences of reversible, endoreversible and irreversible Atkinson cycles are compared. The maximum speciﬁc volume ratio, maximum pressure ratio, and thermal efﬁciency under the conditions of the maximum power output and maximum power density are compared. Based on NSGA-II, the single-, bi-, tri-, and quadru-objective optimizations are performed when the compression ratio is used as the optimization variable, and the cycle dimensionless power output, thermal efﬁciency, dimensionless ecological function, and dimensionless power density are used as the optimization objectives. The deviation indexes are obtained based on LINMAP, TOPSIS, and Shannon entropy solutions under different combinations of optimization objectives. By comparing the deviation indexes of bi-, tri- and quadru-objective optimization and the deviation indexes of single-objective optimizations based on maximum power output, maximum thermal efﬁciency, maximum ecological function and maximum power density, it is found that the deviation indexes of multi-objective optimization are smaller, and the solution of multi-objective optimization is desirable. The comparison results show that when the LINMAP solution is optimized with the dimensionless power output, thermal efﬁciency, and dimensionless power density as the objective functions, the deviation index is 0.1247, and this optimization objective combination is the most ideal.

cycles. References [90,91] carried out MOO of chemical reactor by considering the entropy generation and production rate as optimization objectives. Sadeghi et al. [92] performed MOO of solar hydrogen production plant by taking into account exergy efficiency and exergy cost of product as optimization objectives. References [93,94] carried out MOO on the total pumping power and entropy generation rate in ocean thermal energy conversion system [93] and surrogate models [94]. Shi et al. [64,95] optimized the AC [64] and Diesel cycle [95] PC under the condition of constant WF's SHs, and obtained four-objective optimization results based on NSGA-II.
From the references mentioned above, there is no report about the P d performance of an irreversible AC with nonlinear variable WF's SHs with its temperature, and MOO for AC is also rarely presented. Based on the model established in references [62,70], this paper further analyzes the maximum P d PC of an irreversible AC under the condition of nonlinear variable WF's SHs with its temperature and compare the results with those obtained under the condition of the maximum P. Based on NSGA-II, the single-, bi-, tri-, and quadruobjective optimization results will be obtained when the compression ratio is used as the optimization variable and the P, η, E, and P d are used as the objective functions. Three decision-making methods are selected to analyze the optimization results and the best choices under different conditions are obtained. Compared with reference [62], a further step made in this paper is to perform single-, bi-, tri-, and quadru-objective optimization of different optimization objective combinations for an irreversible AC when the WF's SHs are nonlinear variable with its temperature. Figure 1 shows the T − s diagram (a) [62] and p − v diagram (b) of the irreversible AC. An irreversible AC contains an adiabatic compression process 1 → 2 , an isometric process 2 → 3 , an adiabatic expansion process 3 → 4 , and an isobaric process 4 → 1 . The processes 1 → 2 s and 3 → 4 s are reversible processes without considering the IIL. et al. [82] used P , d P , E density and exergy loss rate as objective functions to perform MOO of Brayton cycle hybrid system. Ghasemkhani et al. [83] performed MOO of endoreversible combined cycles under different heat exchangers. References [84][85][86][87] performed MOO on the performance of thermal and economic investment cost of organic Rankine cycle. References [88,89] performed MOO on the dimensionless P ( P ),  , dimensionless E ( E ), and dimensionless d P ( d P ) of endoreversible [88] and irreversible [89] closed modified Brayton cycles. References [90,91] carried out MOO of chemical reactor by considering the entropy generation and production rate as optimization objectives. Sadeghi et al. [92] performed MOO of solar hydrogen production plant by taking into account exergy efficiency and exergy cost of product as optimization objectives. References [93,94] carried out MOO on the total pumping power and entropy generation rate in ocean thermal energy conversion system [93] and surrogate models [94]. Shi et al. [64,95] optimized the AC [64] and Diesel cycle [95] PC under the condition of constant WF's SHs, and obtained four-objective optimization results based on NSGA-II.

Cycle Model and Performance Parameters
From the references mentioned above, there is no report about the d P performance of an irreversible AC with nonlinear variable WF's SHs with its temperature, and MOO for AC is also rarely presented. Based on the model established in references [62,70], this paper further analyzes the maximum d P PC of an irreversible AC under the condition of nonlinear variable WF's SHs with its temperature and compare the results with those obtained under the condition of the maximum P . Based on NSGA-II, the single-, bi-, tri-, and quadru-objective optimization results will be obtained when the compression ratio is used as the optimization variable and the P ,  , E , and d P are used as the objective functions. Three decision-making methods are selected to analyze the optimization results and the best choices under different conditions are obtained. Compared with reference [62], a further step made in this paper is to perform single-, bi-, tri-, and quadru-objective optimization of different optimization objective combinations for an irreversible AC when the WF's SHs are nonlinear variable with its temperature. Figure 1 shows the T s  diagram (a) [62] and p v

Cycle Model and Performance Parameters
AC. An irreversible AC contains an adiabatic compression process 1 2  , an isometric process 2 3  , an adiabatic expansion process 3 4  , and an isobaric process 4 1  . The processes 1 2s  and 3 4s  are reversible processes without considering the IIL.  In the early research [62], the WF's SHs were assumed to be constants, but in the actual cycle, accompanying with the combustion reaction, the nature and composition of WF will change. For this reason, the variable SH model can be used to obtain more accurate results. When the cycle-working-temperature range is 300 K − 3500 K, the nonlinear variable SH model is defined as [85] C p = 7.2674 × 10 −10 T 2 + 4.2166 × 10 −6 T 1.5 − 1.23134 × 10 −5 T + 9.1698 × 10 −4 T 0.5 + 38.5787 − 4.3848 × 10 5 T −1.5 + 8.8827 × 10 6 T −2 − 6.4148 × 10 8 T −3 (1) According to the relationship between constant pressure SH and constant volume SH, one has where R = 8.3145J/(mol·K) is the WF's gas constant. The heat flux rate supplied to the AC is T 2 (7.2674 × 10 −10 T 2 + 4.2166 × 10 −6 T 1.5 − 1.23134 × 10 −5 T + 9.1698 ×10 −4 T 0.5 + 30.2642 − 4.3848 × 10 5 T −1.5 + 8.8827 × 10 6 T −2 − 6.4148 × 10 8 T −3 dT = . m[2.422 × 10 −10 T 3 + 1.6866 × 10 −6 T 2.5 − 6.1567 × 10 −6 T 2 + 6.1132 × 10 −4 T 1.5 where . m is the molar flow rate of the WF. The heat flux rate transferred to the environment is For the two irreversible adiabatic processes 1 → 2 and 3 → 4 , the IIL is defined as the irreversible compression and expansion efficiencies [62,70] According to reference [70], the adiabatic process can be decomposed into numerous infinitely small processes. It is approximately considered that each infinitely small process has a constant adiabatic index. When the temperature of the WF changes dT and the specific volume changes dV, one has Changing Equation (8) one can obtain: where the temperature in C v is the logarithmic average temperature between states i and j, and The cycle compression ratio γ and maximum temperature ratio τ are defined as Therefore, for the two adiabatic processes 1 → 2 s and 3 → 4 s of an irreversible AC with WF's SHs as nonlinear variable with its temperature, one has C v ln(T 2s /T 1 ) = R ln γ (12) According to the reference [62], the HTL rate and the power loss due to FL are expressed as Energies 2021, 14, 4175 5 of 23 where b = µx 2 2 /(∆t 12 ) 2 , the heat transfer coefficient is expressed as B, the ambient temperature is expressed as T 0 , the work consumed by friction loss is expressed as W µ , the friction coefficient is expressed as µ, the piston position at the minimum volume is expressed as x 2 , and the power stroke time is expressed as ∆t 12 .
The P and η of the AC are, respectively According to the definition of P d in references [55,62], one has The entropy production rates resulting from the HTL, FL, and IIL are defined as The entropy production rate produced by the exhaust stroke is The total entropy production rate due to HTL, FL, IIL and exhaust process is According to the definition of E in references [96][97][98], one has According to the treatment method in the references [55,62], the P, η, E, and P d are defined as P = P/P max (26) When the compression ratio γ, the cycle initial temperature T 1 , and the maximum temperature ratio τ are given, the numerical solutions of temperatures at each state point and cycle performances can be obtained.

Performance Optimization with the Maximum Power Density Criterion
According to reference [62], the values of the cycle parameters can be determined: Figures 2 and 3 show the effect of cycle maximum temperature ratio (τ) on cycle dimensionless power density versus compression ratio (P d − γ) and cycle dimensionless power density versus thermal efficiency (P d − η), respectively. It can be noticed that there is an optimal compression ratio (γ P d ) to make P d reach the maximum. As the τ increases from 5.78 to 6.78, the γ P d increases from 8.3 to 9.0, and increases by about 8.434%. The η P d corresponding to the cycle maximum P d increases from 0.4330 to 0.4579, and increases by 5.75%. It shows that under the maximum P d criterion, the increases of the γ P d and η P d of the cycle are accompanied with the increase of the τ.   is an optimal compression ratio (    Figure 4, when only considering the FL, comparing curves 1 and 2, the FL increases from 0 to 20 W, the γ P d will decrease from 31.4 to 10.9 and decrease by 65.287%. When only considering the IIL, comparing curves 1 and 1 , the IIL increases (from 1 to 0.94), the γ P d will decrease from 31.4 to 18.5, and decrease by 41.083%. When considering both FL and IIL, comparing curves 1 and 2 , the IIL increases (from 1 to 0.94) the FL increases from 0 to 20 W, the γ P d will decrease from 31.4 to 9.1 and decrease by 71.019%. It can be seen that the decrease of the γ P d of the cycle is accompanied with the increases of the cycle losses.     In Figure 5, when only considering the FL, comparing curves 1 and 2, the FL increases from 0 to 20 W, the η P d will decrease from 0.7114 to 0.5611 and decrease by 21.13%. When only considering the HTL, comparing curves 1 and 3, the HTL increases from 0 to 2.2 W/K, the η P d will decrease from 0.7114 to 0.6371 and decrease by 10.44%. When only considering the IIL, comparing curves 1 and 1 , the IIL increases (from 1 to 0.94), the η P d will decrease from 0.7114 to 0.5618 and decrease by 21.03%. When considering HTL and FL at the same time, comparing curves 1 and 4, the HTL increases from 0 to 2.2 W/K and the FL increases from 0 to 20 W, the η P d will decrease from 0.7114 to 0.5213 and decrease by 23.72%. When considering HTL and IIL, comparing curves 1 and 3 , the HTL increases from 0 to 2.2 W/K and the IIL increases (from 1 to 0.94), the η P d will decrease from 0.7114 to 0.5122 and decrease by 28.00%. When considering FL and IIL, comparing curves 1 and 2 , the FL increases from 0 to 20 W and the IIL increases (from 1 to 0.94), the η P d will decrease from Energies 2021, 14, 4175 9 of 23 0.7114 to 0.4791 and decrease by 32.65%. When considering FL, HTL, and IIL at the same time, comparing curves 1 and 4 , the FL increases from 0 to 20 W, the HTL increases from 0 to 2.2 W/K, and the IIL increases from 1 to 0.94, the η P d will decrease from 0.7114 to 0.4459 and decrease by 37.32%. It can be seen that the decrease of the η P d of the cycle is accompanied with the increases of the cycle losses. Figures 6-8 show the compared results of the cycle η, maximum specific volume ratio, and pressure ratio under the maximum P and maximum P d criterions when there are three losses. Comparing with the maximum P criterion, the η and maximum pressure ratio under the maximum P d criterion are higher, while the maximum specific volume ratio under the maximum P d criterion is smaller. Therefore, the engine designed based on the maximum P d criterion is smaller in size and more efficient.  Figures 6-8 show the compared results of the cycle η , maximum specific volume ratio, and pressure ratio under the maximum P and maximum d P criterions when there are three losses. Comparing with the maximum P criterion, the η and maximum pressure ratio under the maximum d P criterion are higher, while the maximum specific volume ratio under the maximum d P criterion is smaller. Therefore, the engine designed based on the maximum d P criterion is smaller in size and more efficient.

Multi-Objective Optimization
In actual cycle, there is no point at which the P , η , E , and d P are optimized at the same time. Therefore, when solving the MOO problem, it is very important to take into account the trade-offs between the interests of different objectives, and obtain the Pareto optimal solution that simultaneously satisfies multiple different or even contradictory goals. The Pareto frontier is defined as the solution set of the optimization objectives. Figure 9 shows the algorithm diagram of NSGA-II [62]. When taking the compression ratio as the optimization variable and taking the P , η , E , and d P as the optimization objectives, the single-, bi-, tri-, and quadru-objective optimization results are obtained. The optimal solution is obtained by comparing the magnitude of the deviation indexes obtained by LINMAP, TOPSIS, and Shannon entropy solutions.

Multi-Objective Optimization
In actual cycle, there is no point at which the P, η, E, and P d are optimized at the same time. Therefore, when solving the MOO problem, it is very important to take into account the trade-offs between the interests of different objectives, and obtain the Pareto optimal solution that simultaneously satisfies multiple different or even contradictory goals. The Pareto frontier is defined as the solution set of the optimization objectives. Figure 9 shows the algorithm diagram of NSGA-II [62]. When taking the compression ratio as the optimization variable and taking the P, η, E, and P d as the optimization objectives, the single-, bi-, tri-, and quadru-objective optimization results are obtained. The optimal solution is obtained by comparing the magnitude of the deviation indexes obtained by LINMAP, TOPSIS, and Shannon entropy solutions.
The optimization problems are solved with different optimization objective combinations, which forms different MOO problems. The one quadru-objective optimization problem is as follows: The four tri-objective optimization problems are as follows: The six bi-objective optimization problems are as follows:  The optimization problems are solved with different optimization objective combinations, which forms different MOO problems. The one quadru-objective optimization problem is as follows: The four tri-objective optimization problems are as follows: The six bi-objective optimization problems are as follows:                     The deviation indexes of maximum P, maximum η, maximum E, and maximum P d are 0.7326, 0.2752, 0.1260, and 0.5120, respectively. For the quadrur-objective optimization, Figure 10 shows the Pareto frontier for P − η − E − P d . It can be noticed that with the increase of η, the P and P d increase, and the E first increases and then decreases. The deviation indexes (0.1250, 0.1250, 0.5266) obtained by the LINMAP, TOPSIS, and Shannon entropy solutions are smaller than those of single-objective optimization. It means that the results obtained by four-objective optimization are more perfect than single-objective optimization. In addition, when taking P, η, E, and P d as the objective functions, the deviation indexes obtained by LINMAP and TOPSIS solutions are the same, and the optimization results are more desirable than those obtained by the Shannon entropy solution. For tri-objective optimization, Figures 11-14 show the Pareto frontiers for P − η − E, P − η − P d , P − E − P d , and η − E − P d . It can be noticed that with the increases of η, the P and P d increase, while the E first increases and then decreases. With the increase of P d , the P decreases, the η increases, and the E first increases and then decreases. The deviation indexes obtained by different decision-makings are smaller than those obtained by single-objective optimization, and which is the same as that obtained by quadru-objective optimization. When taking P, η, and E as the objective functions, the deviation index obtained by the Shannon entropy solution is smaller. When taking P, η, and P d as the objective functions, the deviation index obtained by the LINMAP solution is smaller. When taking P, E, and P d as the objective functions, the LINMAP and TOPSIS solutions get the same deviation indexes, and the optimization results are more desirable than those obtained by the Shannon entropy solution. When taking η, E, and P d as the objective functions, the deviation index obtained by the TOPSIS solution is smaller and the result is better.
For bi-objective optimization, Figures 15-20 show the Pareto frontiers for P − η, P − E . , , η − P d , and E − P d . It can be noticed that the η, E, and P d decrease with the increase of P, the E and P d decrease with the increase of η, and the P d decreases with the increase of E. The indexes obtained by different decision-makings are smaller than those obtained by single-objective optimization, and the conclusion is the same as that obtained by tri-and quadru-objective optimization. When taking P and η or taking P d and η as the objective functions, the deviation index obtained by the LINMAP solution is smaller. When taking P and E or taking η and E as the objective functions, the deviation index obtained by the Shannon entropy solution is smaller. When taking P and P d or taking E and P d as the objective functions, the deviation index obtained by the TOPSIS solution is smaller and the result is better. By comparing the deviation indexes obtained under various conditions, the results show that the solution obtained by MOO is more desirable, and the deviation indexes are smaller. In addition, when the LINMAP solution is optimized with P, η, and P d as the objective functions, the deviation index is 0.1247, the contradiction obtained is the smallest, and the result is the best. In practical applications, the optimal plan can be selected from the Pareto frontier, and the design can be optimized according to the actual requirements of the decision-maker.

Conclusions
Through FTT analysis, this paper performs the performance analyses of the irreversible AC under the maximum P d criterion when the WF's SHs are nonlinear variable with its temperature. The results of the η, maximum specific volume ratio and pressure ratio obtained under the maximum P d criterion are compared with those under the maximum P criterion. Based on NSGA-II, when the compression ratio is the optimization variable and the P, η, E, and P d are the optimization objectives, the single-, bi-, tri-, and quadru-objective optimization results are obtained. The optimal solution is obtained by comparing the deviation indexes of LINMAP, TOPSIS, and Shannon entropy solutions. It can be noticed that: (1) There is an γ P d to maximize the P d . With the cycle maximum temperature ratio increases, the γ P d and η P d corresponding to the P d will increase. With the increases of HTL, FL, IIL, the γ P d and η P d corresponding to the cycle maximum P d will decrease. (2) Under the maximum P d criterion, the η will be higher and the size will be smaller.
(3) Compared with single-objective optimization, MOO has less contradictions and conflicts. Comparing the results of single-, bi-, tri-, and quadru-objective optimization, when the LINMAP solution is optimized with P, η, and P d as the objective functions, the contradiction is smaller and the result is more perfect. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

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.