Performance Analysis and Four-Objective Optimization of an Irreversible Rectangular Cycle

Based on the established model of the irreversible rectangular cycle in the previous literature, in this paper, finite time thermodynamics theory is applied to analyze the performance characteristics of an irreversible rectangular cycle by firstly taking power density and effective power as the objective functions. Then, four performance indicators of the cycle, that is, the thermal efficiency, dimensionless power output, dimensionless effective power, and dimensionless power density, are optimized with the cycle expansion ratio as the optimization variable by applying the nondominated sorting genetic algorithm II (NSGA-II) and considering four-objective, three-objective, and two-objective optimization combinations. Finally, optimal results are selected through three decision-making methods. The results show that although the efficiency of the irreversible rectangular cycle under the maximum power density point is less than that at the maximum power output point, the cycle under the maximum power density point can acquire a smaller size parameter. The efficiency at the maximum effective power point is always larger than that at the maximum power output point. When multi-objective optimization is performed on dimensionless power output, dimensionless effective power, and dimensionless power density, the deviation index obtained from the technique for order preference by similarity to an ideal solution (TOPSIS) decision-making method is the smallest value, which means the result is the best.


Introduction
After decades of development, a series of instructive and practical achievements have been obtained in finite time thermodynamics [1][2][3][4][5][6][7][8], the research objects of which include heat engines [9][10][11][12][13][14][15], refrigerators [16,17], heat pumps [18], chemical cycles [19], and quantum cycles [20,21]. The rectangular cycle (RC) is composed of four thermodynamic processes, its endothermic processes are closed to the endothermic processes of the dual cycle, and its exothermic processes are closed to the exothermic processes of the Miller cycle. Compared with the common internal combustion engine cycles, the cycle has no adiabatic process, so it is easier to realize in practical engineering. Since the cycle p − v diagram is rectangular, it is called the RC. Ferreira [22] first applied the classical thermodynamic theory to study the performance of the RC and derived the work output and thermal efficiency (TEF). Some scholars have introduced finite time thermodynamics theory into the performance analyses of the RC on this basis. Considering that the specific heats (SHs) of the working fluid (WF) were constants, Liu et al. [18] derived the work output and TEF of an endoreversible RC. Based on [23], Liu et al. [24] analyzed the influences of heat transfer loss and friction loss on the power output (POW) and TEF of an irreversible On the basis of the irreversible RC model with constant SH of WF built in [24], this paper will firstly obtain the analytical expressions of cycle PD and effective power by using finite time thermodynamics theory and compare the performance differences of the cycle under the conditions of the maximum PD, maximum effective power, and maximum POW; secondly carry out MOO for the dimensionless POW, TEF, dimensionless PD, and dimensionless effective power by taking the cycle expansion ratio as the optimization variable and applying the NSGA-II; and finally obtain the optimal solutions of four-objective, three-objective, and two-objective optimizations by using three decision-making methods. Figures 1 and 2 show the irreversible RC model [24]. The cycle consists of two constant volume processes, 1 → 2 and 3 → 4 , as well as two constant pressure processes, 2 → 3 and 4 → 1 . Processes 1 → 2 and 2 → 3 are endothermic ones, and processes 3 → 4 and 4 → 1 are exothermic ones. for future soft robots due to their easy integration into soft structures and low-voltage power requirements. It is useful to further study the performance of the RC. On the basis of the irreversible RC model with constant SH of WF built in [24], this paper will firstly obtain the analytical expressions of cycle PD and effective power by using finite time thermodynamics theory and compare the performance differences of the cycle under the conditions of the maximum PD, maximum effective power, and maximum POW; secondly carry out MOO for the dimensionless POW, TEF, dimensionless PD, and dimensionless effective power by taking the cycle expansion ratio as the optimization variable and applying the NSGA-II; and finally obtain the optimal solutions of four-objective, three-objective, and two-objective optimizations by using three decision-making methods. Figures 1 and 2 show the irreversible RC model [24]. The cycle consists of two constant volume processes, 1 2  and 3 4  , as well as two constant pressure processes, 2 3  and 4 1  . Processes 1 2  and 2 3  are endothermic ones, and processes 3 4  and 4 1  are exothermic ones.  According to the state equation of ideal gas, one has  for future soft robots due to their easy integration into soft structures and low-voltage power requirements. It is useful to further study the performance of the RC. On the basis of the irreversible RC model with constant SH of WF built in [24], this paper will firstly obtain the analytical expressions of cycle PD and effective power by using finite time thermodynamics theory and compare the performance differences of the cycle under the conditions of the maximum PD, maximum effective power, and maximum POW; secondly carry out MOO for the dimensionless POW, TEF, dimensionless PD, and dimensionless effective power by taking the cycle expansion ratio as the optimization variable and applying the NSGA-II; and finally obtain the optimal solutions of four-objective, three-objective, and two-objective optimizations by using three decision-making methods. Figures 1 and 2 show the irreversible RC model [24]. The cycle consists of two constant volume processes, 1 2  and 3 4  , as well as two constant pressure processes, 2 3  and 4 1  . Processes 1 2  and 2 3  are endothermic ones, and processes 3 4  and 4 1  are exothermic ones.  According to the state equation of ideal gas, one has According to the state equation of ideal gas, one has

Model and Performance Indicators of an Irreversible RC
The expansion ratio of the RC is defined as γ = v 3 /v 2 , and the cycle temperature ratio (TR) is defined as τ = T 3 /T 1 . The following equations can be obtained from Equations (1) and (2): The heat absorption rate of the WF is The heat release rate of the WF is where . m is the mass flow rate of the WF, and C p and C v are the SHs at constant pressure and constant volume, respectively.
There is no loss in the ideal RC, but for the actual RC, it is necessary to consider the heat transfer loss between the WF and the external cylinder wall because the cylinder wall is not adiabatic, and the large temperature difference between the high-temperature WF and the outside makes this heat transfer loss not negligible. According to [63,64], it is assumed that the heat transfer loss through the cylinder wall is proportional to the temperature difference between the average temperature of the WF and the ambient temperature, and the heat transfer loss rate can be expressed as where T 0 is the ambient temperature, and B = 3B 1 is the heat transfer loss coefficient between the external cylinder wall and the WF. For the irreversible RC, it is also necessary to consider the friction loss between the piston and the cylinder wall. According to [65], the POW consumed by friction loss can be written as where µ is the friction loss coefficient. The average piston speed can be expressed as where x 1 and x 2 are the position of the piston at the maximum and minimum volume, and ∆t 12 is the time consumed in the power stroke. Substituting the average piston speed v with v in Equation (9), the cycle POW and TEF can be obtained as According to the definition of PD in [34], the PD of the RC is According to [35], the expressions of standardized dimensionless PD and dimensionless POW are written as According to the definition of effective power (W ep ) in [27,28], the W ep of the RC is According to [30], the expression of the dimensionless W ep is written as When γ, T 1 , τ, and T 0 are given, the temperatures of each state point in the cycle can be calculated, and then the cycle POW, TEF, PD, and effective power can be obtained by substituting the calculation results into Equations (11)- (13) and (16).

Power Density and Effective Power Performance Analyses
According to [22,24,[63][64][65], the parameters are as follows: C p = 29.092 J/(mol · K), C v = 20.78 J/(mol · K), γ = 1.0 ∼ 10.0, T 0 = 300 K, T 1 = 350 K, τ = 4.2 ∼ 6.2, b = 32.5 W, B = 2.2 W/K, and . m= 1 mol/s. Figure 3 shows the relationships between the maximum specific volume ratio (v 4 /v 1 ) and τ under the conditions of P max and (P d ) max . It can be seen from the figure that, at the same TR (τ), the (v 4 /v 1 ) P corresponding to P max is always bigger than the (v 4 /v 1 ) P d corresponding to (P d ) max . When τ is 6.2, (v 4 /v 1 ) P is 2.451, and (v 4 /v 1 ) P d is 1.717, which decreases by about 29.95%, which means that the size of the heat engine is smaller when it works at the maximum PD point.       Figure 4 shows the relationships between the maximum pressure ratio (p 3 /p 1 ) and τ under the conditions of P max and (P d ) max . It can be seen from the figure that the (p 3 /p 1 ) P corresponding to P max is always less than the (p 3 /p 1 ) P d corresponding to (P d ) max . When τ is 6.2, (p 3 /p 1 ) P is 2.53, and (p 3 /p 1 ) P d is 3.61, which increases by about 42.69%, which means that although the heat engine has a smaller size under the condition of maximum PD, it is accompanied by a larger pressure ratio.       Figure 5 shows the relationships between the TEF versus τ under the conditions of P max and (P d ) max . It can be seen from the figure that the η P corresponding to P max is always bigger than the η P d corresponding to (P d ) max . When τ is 6.2, η P is 0.1269, and η P d is 0.1144, which decreases by about 9.8%. / v v decreases greatly. This shows that when taking the maximum PD as the goal, although part of the TEF of the heat engine is sacrificed, the size of the heat engine decreases greatly.

Multi-Objective Optimization
In this section, the  is used as the optimization variable; the d P , P ,  , and ep W are taken as the optimization goals; and the irreversible RC is optimized by using the According to Figures 3 and 5, when τ is 6.2, η P d decreases by 9.8% compared with η P , and the TEF decreases slightly. However, (v 4 /v 1 ) P d decreases by 28.95% compared with (v 4 /v 1 ) P , and v 4 /v 1 decreases greatly. This shows that when taking the maximum PD as the goal, although part of the TEF of the heat engine is sacrificed, the size of the heat engine decreases greatly. Figure 6 shows the relationships of W ep − γ and W ep − η when τ = 6.2. When τ = 6.2, the dimensionless POW corresponding to the (W ep ) max is 0.9990, the TEF corresponding to (W ep ) max is 0.1273, and the TEF corresponding to P max is 0.1269. Compared with the maximum POW condition point, the POW corresponding to the (W ep ) max decreases by 0.1%, and the TEF corresponding to the (W ep ) max increases by 0.32%. Therefore, when the W ep is taken as the objective function, the cycle TEF can increase with sacrificing part of the POW, and the W ep reflects the compromise between the POW and TEF. / v v decreases greatly. This shows that when taking the maximum PD as the goal, although part of the TEF of the heat engine is sacrificed, the size of the heat engine decreases greatly.

Multi-Objective Optimization
In this section, the  is used as the optimization variable; the d P , P ,  , and ep W are taken as the optimization goals; and the irreversible RC is optimized by using the "gamultiobj" algorithm that comes from the MATLAB software. Then, the corresponding

Multi-Objective Optimization
In this section, the γ is used as the optimization variable; the P d , P, η, and W ep are taken as the optimization goals; and the irreversible RC is optimized by using the "gamultiobj" algorithm that comes from the MATLAB software. Then, the corresponding Pareto frontiers are obtained, and the optimal solutions can be picked out by applying three decision-making methods of LINMAP, TOPSIS, and Shannon Entropy. Then, the three results obtained are compared.
Several MOO problems will arise when solving different combinations of optimization objectives.
Any combination of two objective functions can obtain six two-objective optimization expressions: Any combination of three objective functions can obtain four three-objective optimization expressions: Any combination of four objective functions can obtain one four-objective optimization expression: Figure 7 shows the complete process of NSGA-II [47]. Compared with the previous generation MOO evolutionary algorithm NSGA, the NSGA-II mainly makes the following three improvements: (1) A new algorithm for fast non-dominant sorting is added, which greatly reduces the computational complexity. (2) Elite strategy is introduced, and a new population is formed which is composed of two populations, the parent and the offspring populations, selecting superior individuals in the new population instead of selecting only in the offspring population, which not only expands the range of options but also reduces the selection loss of excellent individuals in the parent population (3) Canceling the artificial designation of the shared parameters, which has been replaced by the congestion degree and the congestion degree comparison operator. Table 1 lists the optimal solutions obtained by four-objective, three-objective, two-objective, and one-objective optimizations. The deviation index indicates the degree of deviation between the optimization result and the positive ideal point; the smaller the deviation index is, the closer the obtained optimization result is to the positive ideal point. Comparing the results of four-objective, three-objective, two-objective, and one-objective optimizations listed in Table 1, it can be found that when the MOO of P, P d , and W ep is performed, the deviation index under the TOPSIS method is the minimum, which means that its result is the best, and the MOO solution is better than the single-objective optimal solutions. Figures 8-13 show the Pareto optimal frontiers obtained by taking two objectives (P − η, P − P d , P − W ep , η − P d , η − W ep , P d − W ep ) as the optimization goals. It can be seen from the curves that as P increases, η, P d , and W ep will decrease; as η increases, P d and W ep will decrease; and as W ep increases, P d will decrease. Comparing the results of the two-objective optimizations in Table 1, it can be found that when the MOO of P d and W ep is performed, the deviation index under the TOPSIS method is the minimum.  Table 1 lists the optimal solutions obtained by four-objective, three-objective, twoobjective, and one-objective optimizations. The deviation index indicates the degree of deviation between the optimization result and the positive ideal point; the smaller the deviation index is, the closer the obtained optimization result is to the positive ideal point. Comparing the results of four-objective, three-objective, two-objective, and one-objective optimizations listed in Table 1, it can be found that when the MOO of P , d P , and ep W is performed, the deviation index under the TOPSIS method is the minimum, which means that its result is the best, and the MOO solution is better than the single-objective optimal solutions. Figures 8-13 show the Pareto optimal frontiers obtained by taking two objectives (               Figure 13. Two-objective optimization on P d − W ep . Figures 14-17 show the Pareto optimal frontiers obtained by taking three objectives as the optimization goals. It can be seen from the curves that as P and η increase, P d will decrease; as P increases and η decreases, W ep will first increase and then decrease; as P and W ep increase, P d will decrease; and as η and W ep increase, P d will decrease. Comparing the results of the three-objective optimizations in Table 1, it can be found that when the MOO of P, P d , and W ep is performed, the deviation index under the TOPSIS method is the minimum.    Figure 14. Three-objective optimization on P − η − P d .    Figure 18 shows the Pareto optimal frontier obtained by taking P , d P ,  , and ep W of the RC as the optimization goals. In this figure, the positive ideal point in the figure means that d P , P ,  , and ep W can all reach the maximum, and the negative ideal point means that d P , P ,  , and ep W can all reach the minimum. It can be seen from the figure that both the positive and negative ideal points are all located outside the Pareto frontier, so there is no optimal  which makes d P , P ,  , and ep W all reach the maximum or minimum. Additionally, for the four-objective optimization, the optimization result obtained by using the TOPSIS method has a larger P ,  , and ep W , while the Shannon Entropy decision method has a larger d P ; in other words, the corresponding heat engine size is smaller. Comparing the results of the four-objective optimizations in Table 1, it can be found that the deviation index under the TOPSIS method is the minimum.  Figure 18 shows the Pareto optimal frontier obtained by taking P , d P ,  , and ep W of the RC as the optimization goals. In this figure, the positive ideal point in the figure means that d P , P ,  , and ep W can all reach the maximum, and the negative ideal point means that d P , P ,  , and ep W can all reach the minimum. It can be seen from the figure that both the positive and negative ideal points are all located outside the Pareto frontier, so there is no optimal  which makes d P , P ,  , and ep W all reach the maximum or minimum. Additionally, for the four-objective optimization, the optimization result obtained by using the TOPSIS method has a larger P ,  , and ep W , while the Shannon Entropy decision method has a larger d P ; in other words, the corresponding heat engine size is smaller. Comparing the results of the four-objective optimizations in Table 1, it can be found that the deviation index under the TOPSIS method is the minimum.  Figure 18 shows the Pareto optimal frontier obtained by taking P, P d , η, and W ep of the RC as the optimization goals. In this figure, the positive ideal point in the figure means that P d , P, η, and W ep can all reach the maximum, and the negative ideal point means that P d , P, η, and W ep can all reach the minimum. It can be seen from the figure that both the positive and negative ideal points are all located outside the Pareto frontier, so there is no optimal γ which makes P d , P, η, and W ep all reach the maximum or minimum. Additionally, for the four-objective optimization, the optimization result obtained by using the TOPSIS method has a larger P, η, and W ep , while the Shannon Entropy decision method has a larger P d ; in other words, the corresponding heat engine size is smaller. Comparing the results of the four-objective optimizations in Table 1, it can be found that the deviation index under the TOPSIS method is the minimum. Figures 19 and 20 show the average distance and average spread versus generations for two different multi-objective optimizations. It can be seen from the figures that the genetic algorithm stops when convergence is attained, and it can be observed that this occurs at 626 and 666 generations for the four-objective optimization and three-objective optimization on P, P d , and W ep .

Conclusions
Based on the established irreversible RC model, the analytical expressions of cycle PD and effective power are obtained, and the performance comparisons of the cycle under the conditions of the maximum PD, maximum effective power, and maximum POW are compared. The four objectives of the dimensionless POW, TEF, dimensionless PD, and dimensionless effective power are optimized, and the optimal solutions of four-objective, three-objective, and two-objective optimizations are compared. The results show that: (1) Compared with the maximum POW condition, although part of the TEF is sacrificed when the heat engine works under the maximum PD condition, the heat engine's size reduces greatly, which has certain guidance for the actual design of the heat engine. (2) Compared with the maximum POW condition, the TEF is higher when the cycle works under the maximum effective power condition, the TEF can increase with sacrificing part of the POW under the maximum effective power condition, and the effective power reflects the compromise between the POW and TEF. (3) Comparing the results of four-objective, three-objective, two-objective, and oneobjective optimizations, when MOO is performed on dimensionless POW, dimensionless PD, and dimensionless effective power, the deviation index obtained from the TOPSIS decision-making method is the smallest value. At this time, the deviation index is 0.2348, and the optimal compression ratio is 2.1077, which means that the result is the best, and the multi-objective optimization solution is better than the single objective optimal solutions. Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.

Acknowledgments:
The authors wish to thank the academic editor and 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.

B
Heat transfer loss coefficient (W/K) C p Specific heat at constant pressure (J/(mol · K)) C v Specific heat at constant volume (J/(mol · K)) P Power