Performance Optimizations with Single-, Bi-, Tri-, and Quadru-Objective for Irreversible Diesel Cycle

Applying finite time thermodynamics theory and the non-dominated sorting genetic algorithm-II (NSGA-II), thermodynamic analysis and multi-objective optimization of an irreversible Diesel cycle are performed. Through numerical calculations, the impact of the cycle temperature ratio on the power density of the cycle is analyzed. The characteristic relationships among the cycle power density versus the compression ratio and thermal efficiency are obtained with three different loss issues. The thermal efficiency, the maximum specific volume (the size of the total volume of the cylinder), and the maximum pressure ratio are compared under the maximum power output and the maximum power density criteria. Using NSGA-II, single-, bi-, tri-, and quadru-objective optimizations are performed for an irreversible Diesel cycle by introducing dimensionless power output, thermal efficiency, dimensionless ecological function, and dimensionless power density as objectives, respectively. The optimal design plan is obtained by using three solution methods, that is, the linear programming technique for multidimensional analysis of preference (LINMAP), the technique for order preferences by similarity to ideal solution (TOPSIS), and Shannon entropy, to compare the results under different objective function combinations. The comparison results indicate that the deviation index of multi-objective optimization is small. When taking the dimensionless power output, dimensionless ecological function, and dimensionless power density as the objective function to perform tri-objective optimization, the LINMAP solution is used to obtain the minimum deviation index. The deviation index at this time is 0.1333, and the design scheme is closer to the ideal scheme.

In addition to the above objective functions, Sahin et al. [33,34] took power density (P d , defined as the ratio of the cycle P to the maximum specific volume) as a new optimization criterion to optimize Joule-Brayton engines and found that the heat engine designed under the P d criterion has higher η and a smaller size when no loss is considered. Chen et al. [35]   The heat absorption and release rates are, respectively, 3 2 ( ) where m  is the mass flow rate, and v C and p C are the specific heats under constant volume and pressure, respectively. Some internal irreversibility loss (IIL) is caused by friction, turbulence, and viscous stress. The irreversible compression and expansion internal efficiencies are expressed as [16,19,20,30] The cycle compression ratio γ and temperature ratio τ are  The heat absorption and release rates are, respectively, 3 2 ( ) where m  is the mass flow rate, and v C and p C are the specific heats under constant volume and pressure, respectively. Some internal irreversibility loss (IIL) is caused by friction, turbulence, and viscous stress. The irreversible compression and expansion internal efficiencies are expressed as [16,19,20,30] The cycle compression ratio γ and temperature ratio τ are The heat absorption and release rates are, respectively, .
where . m is the mass flow rate, and C v and C p are the specific heats under constant volume and pressure, respectively. Some internal irreversibility loss (IIL) is caused by friction, turbulence, and viscous stress. The irreversible compression and expansion internal efficiencies are expressed as [16,19,20,30] The cycle compression ratio γ and temperature ratio τ are According to the property of isentropic process, one has Entropy 2021, 23, 826 4 of 18 (T 3 /T 2s ) k = T 4s /T 1 (8) According to Equations (3)-(8), one has For the actual heat engine, there is heat transfer loss (HTL) between the working fluid and the cylinder. According to Refs. [14,24,27], it is known that the fuel exothermic rate is equal to the sum of the total endothermic rate and the HTL rate; one has where A is the fuel exothermic rate and B is the HTL coefficient. Similarly, as the piston generates friction with the cylinder wall when running at high speed, the friction loss (FL) of the cycle cannot be ignored. As a four-stroke heat engine, a Diesel heat engine has four strokes of intake, compression, expansion, and exhaust, and all of them produce FL. According to Refs. [24,32], for the treatment of FL in each stroke, the FL during compression and expansion is included in internal irreversible losses. According to Refs. [57][58][59], the piston motion resistance in the intake process is greater than that in the exhaust process. If the friction coefficient in the exhaust process is µ, the equivalent friction coefficient, which includes the pressure drop loss in the intake process, is 3µ. The friction coefficients on the exhaust and intake stroke are µ and 3µ, respectively. There is a linear relationship between friction force and speed: fµ = −µv = −µdx/dt, where x is the piston displacement and µ is the FL coefficient. The power consumed due to FL during the exhaust and intake strokes can be derived as For a Diesel cycle, the average speed of the piston in four reciprocating motions is v = 4Ln (14) where n is the rotating speed and L is the stroke length. Therefore, the power consumed by cycle FL is The cycle P and η are, respectively, According to the definition of P d in Refs. [33][34][35], the P d is expressed as According to Refs. [38,39], the total volume v t , stroke volume v s , and gap volume v c of the cycle are defined v c = πd 2 L/4(γ − 1) In the Diesel cycle, v t = v max = v 1 , v c = v 2 . According to Equations (5) and (17)- (19), one has P d = P/v max = P/v t = 4(γ − 1)P/πd 2 Lγ According to Ref. [24], an irreversible Diesel cycle has four kinds of entropy generation due to FL, HTL, IIL, and exhaust stroke to the environment. The four entropy generation rates are expressed as mC v ln(T 4 /T 4s ) Therefore, the total entropy generation rate is According to the definition of E in Ref. [24], the E is expressed as According to the processing method of Refs. [35,44], P, P d , and E are respectively defined as P = P/P max (30) According to Equations (4), (9) and (11) and given the compression ratio γ, the initial cycle temperature T 1 , and the cycle temperature ratio τ, by solving the temperatures at the 2, 3, and 4 state points, the corresponding numerical solutions of P, η, P d , and E can be obtained.

Maximum Power Density Optimization
The working fluid is assumed to be an ideal gas. According to the nature of the air, , and τ = 5.78 − 6.78. According to Refs. [24,44], the cycle parameters are determined: The relationships between the objective functions (P d and η) of an irreversible Diesel cycle and the cycle design parameters (the cycle temperature ratio, HTL, FL, and IIL) are shown in Figures 3-6. It can be noticed that the relationship between P d and γ (P d − γ) is a parabolic-like one. When no loss is considered, the relationship between P d and η (P d − η) is a parabolic-like one, and when there is loss, the relationship curve of P d − η is a loop-shaped one.   According to Figure 3, it can be seen that there is an optimal compression ratio (           22.55% compared to that without any losses. According to Figure 5, it can be seen that as the compression ratio increases, d P first increases  Figure 3, it can be seen that there is an optimal compression ratio (γ P d ), which makes P d reach the maximum. As τ increases, γ P d increases; when τ increases from 5.78 to 6.78, γ P d increases from 12.7 to 16 (an increase of 25.98%). According to Figure 4, there is thermal efficiency (η P d ) corresponding to the maximum P d . As τ increases, η P d increases; when τ increases from 5.78 to 6.78, η P d increases from 45.82% to 49.29% (an increase of 7.40%). It can be seen that with the increase in τ, γ P d , and η P d corresponding to the maximum P d also increases. Figures 5 and 6 show the P d − γ and P d − η curves of the cycle when there are three different losses. Table 1 lists η P d when considering different losses and the percentage of the decrease in η P d compared with when no loss is considered. It can be seen that, with the increase in the losses considered, η P d decreases. When the three losses are considered at the same time, η P d decreases by 22.55% compared to that without any losses. According to Figure 5, it can be seen that as the compression ratio increases, P d first increases and then decreases. According to Figure 6, it can be seen that when there are increases in HFL, FL, and IIL, η P d corresponding to the maximum P d decreases.  Figures 7-9 show the change trends of the corresponding maximum specific volume, maximum pressure ratio, and η with the τ under the maximum P and maximum P d criteria of an irreversible Diesel cycle. According to Figures 7 and 8, compared with the corresponding results under the maximum P criterion, the maximum specific volume is smaller and the maximum pressure ratio is larger under the maximum P d criterion. It is observed that the Diesel heat engine designed under the maximum P d criterion has a smaller size. According to Figure 9, the η of the cycle under the maximum d P criterion is higher. When 6.28 τ = , the η obtained under the maximum P and maximum d P criterion are 46.04% and 47.64%, respectively. The latter is an increase of 3.54% over the former. Therefore, compared with the maximum P criterion, the engine designed under the maximum d P criterion has a smaller size and a higher η .

Multi-Objective Optimization with Power Output, Thermal Efficiency, Ecological Function, and Power Density
MOO cannot make multiple objective functions reach the optimal value at the same time. The best compromise is achieved by comparing the pros and cons of each objective function. Therefore, the MOO solution set is not unique, and a series of feasible alternatives can be obtained, which are called Pareto frontiers. In this section, P , η , E , and

Multi-Objective Optimization with Power Output, Thermal Efficiency, Ecological Function, and Power Density
MOO cannot make multiple objective functions reach the optimal value at the same time. The best compromise is achieved by comparing the pros and cons of each objective function. Therefore, the MOO solution set is not unique, and a series of feasible alternatives can be obtained, which are called Pareto frontiers. In this section, P , η , E , and According to Figure 9, the η of the cycle under the maximum P d criterion is higher. When τ = 6.28, the η obtained under the maximum P and maximum P d criterion are 46.04% Entropy 2021, 23, 826 9 of 18 and 47.64%, respectively. The latter is an increase of 3.54% over the former. Therefore, compared with the maximum P criterion, the engine designed under the maximum P d criterion has a smaller size and a higher η.

Multi-Objective Optimization with Power Output, Thermal Efficiency, Ecological Function, and Power Density
MOO cannot make multiple objective functions reach the optimal value at the same time. The best compromise is achieved by comparing the pros and cons of each objective function. Therefore, the MOO solution set is not unique, and a series of feasible alternatives can be obtained, which are called Pareto frontiers. In this section, P, η, E, and P d are used as objective functions; the compression ratio (γ) is used as an optimization variable; and NSGA-II [44][45][46][47][48][49][50][51][52] is used to perform bi-, tri-, and quadru-objective optimizations for an irreversible Diesel cycle. Through three different solutions, that is, LINMAP, TOPSIS, and Shannon entropy, the optimization results under different objective function combinations are obtained.
In the LINMAP solution, a minimum spatial distance from the ideal point is selected as the desired final optimal solution. In the TOPSIS solution, a maximum distance from the non-ideal point and a minimum distance from the ideal point are selected as the desired final optimal solution. In the Shannon entropy solution, a maximum value corresponding to a certain objective function is selected as the desired final optimal solution.
The optimization problems are solved with different optimization objective combinations, which form different MOO problems.
The six bi-objective optimization problems are as follows: The four tri-objective optimization problems are as follows: The one quadru-objective optimization problem is as follows: The evolution flow chart of NSGA-II is shown in Figure 10. The optimization results obtained by the combination of different objective functions in the three solutions are listed in Table 2. It can be seen that when single-objective optimization is performed under the criterions of maximum P,η, E, and P d , the deviation indexes (0.5828, 0.5210, 0.2086, and 0.4122, respectively) obtained are much larger than the result obtained by MOO. This indicates that the design scheme of MOO is more ideal. When taking P, E, and P d as the optimization objectives to perform tri-objective optimization, the deviation index obtained by the LINMAP solution is smaller, and the design scheme is closer to the ideal scheme. the criterions of maximum P , η , E , and d P , the deviation indexes (0.5828, 0.5210, 0.2086, and 0.4122, respectively) obtained are much larger than the result obtained by MOO. This indicates that the design scheme of MOO is more ideal. When taking P , E , and d P as the optimization objectives to perform tri-objective optimization, the deviation index obtained by the LINMAP solution is smaller, and the design scheme is closer to the ideal scheme.  , , η − P d , and E − P d ). When P increases, η, E, and P d all decrease; when η increases, E and P d both decrease; when E increases, P d decreases. According to Table 1, when P and η or P and E are the objective functions, the deviation index obtained by the LINMAP solution is smaller. When P and P d or η and E are the optimization objectives, the deviation index obtained by the Shannon entropy solution is smaller. When E and P d are the optimization objectives, the deviation indexes obtained by the LINMAP and TOPSIS solutions are smaller than those obtained by the Shannon entropy solution. When η and P d are the objective functions, the deviation index obtained by the TOPSIS solution is smaller.
When P increases, η decreases, and E and P d first increase and then decrease. When η increases, P d decreases, and E first increases and then decreases. When η, E, and P d are the optimization objectives, the deviation indexes obtained by the LINMAP and TOPSIS solutions are smaller than those obtained by the Shannon entropy solution. When the combination of the other three objective functions are the optimization objectives, the deviation index obtained by the LINMAP solution is smaller, and the result is better. Figure 21 shows the Pareto frontier of the quadru-objective optimization (P − η − E − P d ). With the increase in P, η increases, P d decreases, and E first increases and then decreases. When P, η, E, and P d are the optimization objectives, the deviation index obtained by the LINMAP solution is the smallest, and the result is the best.  Figure 21 shows the Pareto frontier of the quadru-objective optimization ( With the increase in P , η increases, d P decreases, and E first increases and then decreases. When P , η , E , and d P are the optimization objectives, the deviation index obtained by the LINMAP solution is the smallest, and the result is the best.

Conclusions
The expression of the P d of an irreversible Diesel cycle was derived in this paper, and the impacts of τ and three loss issues on the cycle of P d versus γ and η characteristics were analyzed. The performance parameters (maximum specific volume, maximum pressure ratio, and η) of an irreversible Diesel cycle based on the criteria of maximum P and P d were compared. Using three different solutions, including LINMAP, TOPSIS, and Shannon entropy, the results of single-, bi-, tri-, and quadru-objective optimization for an irreversible Diesel cycle were analyzed and compared. Comparing the deviation indexes obtained under different objective function combinations, the optimal design scheme was selected. The results showed the following:

1.
The relationship curves of the cycles P d − γ and P d − η were a parabolic-like one and a loop-shaped one, respectively. With the increases in the cycle temperature ratio, the γ P d and η P d corresponding to the maximum P d increased. With the increases in HFL, FL, and IIL, the γ P d and η P d corresponding to the maximum P d decreased.

2.
Under the maximum P d criterion, a smaller size and higher efficiency engine will be designed.

3.
The deviation index of MOO was smaller. When taking P, E, and P d as the optimization objectives to perform tri-objective optimization, the deviation index obtained by the LINMAP solution was smaller, and the design scheme was closer to the ideal scheme. 4.
The next step will be to use exergy efficiency optimization to further reinforce the results of MOO.

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)) E Dimensionless ecological function P Dimensionless power output P d Dimensionless power density Q Heat transfer rate (W) T Temperature (K) Greek symbols γ Compression ratio (-)