Multi- objective performance optimization of irreversible molten carbonate fuel cell–Stirling heat engine–reverse osmosis and thermodynamic assessment with ecological objective approach

1Faculty of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran 2Faculty of Engineering and Computer Sciences (ENCS), Concordia University, Montreal, Quebec, Canada 3Department of Renewable Energy and Environmental Engineering, University of Tehran, Tehran, Iran 4School of Civil Engineering and Architecture, Wuhan University of Technology, Wuhan, China 5Mechanical Engineering Department, Prince Sultan Endowment for Energy and Environment, Prince Mohammad Bin Fahd University, Al-Khobar, Saudi Arabia 6RAK Research and Innovation Center, American University of Ras Al Khaimah, Ras Al Khaimah, United Arab Emirates 7Department of Mechanical Engineering, Technology Faculty, Fırat University, Elazig, Turkey


| INTRODUCTION
Stirling cycle is one of the most common air cycles for heat engines. The advantages of the cycle include the specific efficiency and the ability to use different types of fuels for different heating purposes. In theory, the Stirling engine can transform heat into work with a high efficiency when isothermal expansion and compression and ideal regeneration are applied (Carnot efficiency). The cold and hot temperatures of the cycle are determinative for the thermal limitation of the operational condition in Stirling engine. Generally, the cold and hot temperatures of the Stirling engine are 338 and 925 K, respectively. 1 Stirling engine performance could be varied from 30% to 40% corresponding to a temperature range of 923-1073 K. 2,3 Combination of solar collectors and Stirling engines is a modern technique which provides ease of power generation.
There are four various categories of concentrated solar power (CSP) plants: Linear Fresnel Reflectors, Parabolic Troughs, Power Tower, and Stirling Dish systems. Although the aforementioned systems have different setups, their primary apparatus are similar; Being based on concentration of solar radiation onto a small area. In CSP plants systems, the Stirling dish has the maximum efficiency about 27%. 4 Dish-Stirling system arrangements are represented in  In the past 20 years, eight different dish-Stirling configurations were developed by companies in Germany, the United States, Russia, and Japan. 8 A 1.5 MW capacity installation is currently under construction at Peoria, AZ, and multi-hundred megawatts capacity plants have been designed and are in progress. 9 There are some 9-10 MW capacity plants in India and additional high capacity plants in the United States (750-850 MW). 10 Recently, many studies have been carried out to determine the optimal performance of solar-driven energy systems, using the finite-time thermodynamic (FTT) analysis. [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] The answers obtained by the FTT are more consistent than other thermodynamic methods. Lund 11 defined parametric equations to describe these conditions for a solar thermal power plant in terms of rate of reversible finite heat transfer. Ibrahim and Ladas 14 obtained a finite-time parameter to enhance the heat transfer specifications of a Stirling engine. Blank et al 15 studied and optimized an actual endoreversible Stirling engine. Trukhow et al 17 investigated the equivalence of energy of a solar power plant coupled with a Stirling engine and showed that the generated electricity is subjected to the solar radiation direction. Chen et al 18 studied the efficiency of a solar Stirling engine at the maximum electricity generation rate. Yilmaz et al 24 have obtained the aforementioned conditions for a Carnot-type irreversible solar-powered heat engine. Yaqi et al 26 applied FTT to regulate optimum operational condition of a solar-powered Stirling heat engine.
Earlier, fuel cells (FCs) have been investigated in numerous studies due to growing environmental pollution, climate changes, and reducing fossil resources. As well as being zero emission, having high efficiency, and consistent facilities, FCs could be combined with various thermal cycles. Different models of FCs are studied and investigated in the literature. Still, these setups can be evaluated from different aspects. Thermodynamic assessment and numerical study are known as main efficiency analysis tools for these thermal cycles. [28][29][30][31][32][33][34] Zhao et al 28 represented a thermodynamic approach to investigate an irreversible solid oxide fuel cell (SOFC), by the mean of nonequilibrium thermodynamics. Zhang et al 30 studied molten carbonate fuel cell (MCFC) and employed a multi-optimization method aiming the energy efficiency and power output simultaneously.
In line with the progression of FC technology, various approaches have been developed. Hybridization of FCs is one of the aforementioned approaches which recovers rejected heat of high temperature to enhance the overall performance. Various arrangements of FC-heat engines setups are feasible. [35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] SOFC and MCFC which operate at high temperatures are more suitable for these purposes. Zhang et al 37 designed a model for a hybrid MCFC-heat engine setup, and evaluated the effective parameters on the system efficiency. Moreover, a MCFC-GT arrangement was optimized and investigated by defining the performance and efficiency. 39 Chen et al 41 investigated a MCFC-Stirling engine and studied the performance enhancement due to coupling. Similarly, they studied the affective operational parameters of the system and their impact on the performance. To recover the waste heat produced by MCFCs, Yang et al 47 proposed a new model to investigate an MCFC-themophotovotaic cell (TPVC) hybrid system which revealed the advantages of the potential cycle. The current density of the MCFC, voltage output of the TPVC, electrode area ratio of the MCFC to the TPVC, and energy gap of the material in the photovoltaic cell were considered as the decision variables to optimize the power output and the efficiency of the hybrid cycle. For a similar purpose, Ye et al 51 proposed a hybrid system consisting of an MCFC, a regenerator and a thermoradiative cell (TRC) with similar objective functions. They showed that the TRC can be coupled with other hightemperature fuel cells (HTFCs), like SOFC and direct carbon fuel cell, to utilize the waste heat released by the HTFCs more effectively.
Evolutionary algorithms (EAs) were developed since the mid-1980s in order to help clarify this wide-ranging type of multi-objective issues. 52 Generally, EA specify a collection of satisfying answers, with no overlaps. 53 Generally, in multi-objective optimization problems, a numerous set of answers called Pareto frontier is examined, to represent the closest match in the objective function area. On this subject, researchers have evaluated multi-objective optimization for different energy systems in order to determine the best optimal operation conditions of the system.  In the present paper, an irreversible thermodynamicelectrochemical model of a MCFC-Stirling heat engine-RO hybrid system is established, in which not only the irreversible losses in the MCFC but also the heat-leak from the FC to the environment as well as heat transfer between the FC and the heat engine are considered.

| MCFC model
The discussion of different aspects of the MCFC is beyond the scope of this research. However, a brief description of the cycle in thermodynamics point of view is provided here. Natural gas is converted to hydrogen in a reformer. Then a mixture of water vapor and natural gas is preheated in a heat exchanger and fed to the reformer. The required heat for the reforming reactions is supplied by the catalytic burner which burns the anode outlet in an air stream, which is preheated by part of cathode exhaust gases. The air stream provides the required amount of oxygen for the electrochemical reaction at the cathode side and further combustion of the fuel. The reformer outlet stream preheats the inlet steam and is cooled to the desired temperature, and enters the anode where it takes part in an electrochemical reaction. The anode outlet gas mixes with air and enters the catalytic burner. The high temperature outlet stream of the burner is a significant source of heat and provides the heat needed to allow the bottom cycle to attain the required performance. After the stream is fed to cathode, the carbonate ion produced at the cathode side is transferred to the anode side. The cathode exhaust gases contain a large amount of thermal energy which is used for preheating the inlet air stream and supplying the steam needed by the reformer, because the overall temperature of the FC should maintain constant and also the temperature of the reactants is lower than the needed working temperature by the FC. The performance of MCFCs from different points of view and in more detail are studied in Refs 30,39,41,47,48,51. Anode potential (U an ), cathode potential (U cat ), ohm overpotential (U ohm ), and theoretical maximum potential (U t ) of MCFC can be calculated as follows, respectively 91 : where, j is the current density, p H 2 , an is the partial pressure of hydrogen at the anode, p CO 2 , an is the partial pressure of carbon dioxide at the anode, p H 2 O, an is the partial pressure of water at the anode, p O 2 , cat is the partial pressure of oxygen at the cathode, p CO 2 , cat is the partial pressure of carbon dioxide at the cathode, R is the universal gas constant, T is the operating temperature of the MCFC, En act is the activation energy, F is the Faraday constant, n e is the number of electrons, and U i is the ideal standard potential. where −ΔḢ is the maximum possible power from the FC and it can be described as: where, A is the area of the interconnect plate and Δh is the molar enthalpy change. Exergy destruction rate of FC is: Regenerative loss in the regenerator can be described as: where, K r is the heat conductance of the regenerator, ε r is the regenerator efficiency, and T 0 is the environment temperature. Heat input to the bottom cycle is written as Equation (12)

| Analysis of the Stirling engine
Ideal Stirling cycle consists of four processes including two isothermal and two isochoric processes in the regenerator. In a real cycle, it is impractical to have an ideal heat transfer in the regenerator, in which the entire amount of absorbed heat is transferred to the working fluid in the isochoric heating process. Therefore, a heat transfer loss occurs in the regenerator. In addition, a conductive heat transfer occurs between the heat source and sink, namely conductive thermal bridge loss. More details on the Stirling engines and how they work can be found in previous works. 1,34,58,59,[62][63][64][65]69,77,83,89,92 In the next step, the thermodynamic parameters of this system must be calculated. Equations (11) and (12) are representing the heat transfer in the cycle. Heat inputs are: Heat outputs are: where θ is the heat conductance (W m −2 K −1 ), n is the mole number, x is the compression ratio (x = V max /V min ), T 1 and T 2 are average temperatures of the compression and expansion processes, and t 1 and t 2 are the times spent on the two isothermal branches. Applying Equations (13) and (14) will provide t 1 and t 2 . T 1 is calculated by applying Equations (12) and (13a).
Regenerative loss is: where C is the heat capacity (J mol −1 ), α is the imperfect regeneration coefficient. Time spent on regeneration time is: Heat rate can be obtained as: Using Equations (12) and (17), x can be defined. Energy efficiency of the irreversible Stirling cycle can be described as: where δ is the ratio of the specific heats, and heat rejection from the irreversible Stirling engine is obtained by using Equation (18) Power output, energy efficiency, and exergy destruction rate density of the hybrid system are described in Equations (23)(24), respectively: In the next step the environmental criteria are investigated. The ecological function is defined by the difference of power output and exergy destruction created by the entropy generation. This function can enhance the power output so long as decreasing exergy destruction which leads to lower environmental side-effects.

| Reserve osmosis desalination subsystem
Reverse Osmosis, commonly referred to as RO is one of the leading methods for fresh water production which is based on the selective permeability. Reverse osmosis process involves the movement of water through the membrane, opposed to the concentration gradient (from low to high). Furthermore, a pressure is applied on the concentrated side in order to prevail osmotic pressure. Dessouky 93 have presented a numerical model for the RO arrangement.
The mass and salt balances are given by where ṁ f , ṁ p , and ṁ b are the feed flow rate, the permeate flow rate, the brine flow rate, respectively and x f , x p , x b are the feed salinity, the permeate salinity, the brine salinity.
The feed water mass flow rate ṁ f based on the recovery ratio R R , whose value we chosen is 0.3, and the fresh water mass flow rate ṁ p is The mass flow rate of water passage through a semipermeable membrane is given by: where A m is the area of reserve osmosis membrane, K w is the water permeability coefficient, which is expressed as where T f is the temperature of feed water. Δp is the permeate hydraulic and Δπ is the osmotic pressure, and they can be expressed by where p p and π p are the hydraulic and osmotic pressures of the permeate stream, respectively. p and ̄ are the average feed water pressure and average osmotic pressures on the feed side and brine side, which are given by where p f and p b are the hydraulic pressure of feed stream and reject stream. π f and π b is the osmotic pressure of feed stream and reject stream.
Osmotic pressures can be expressed by where R is the universal gas constant and T is the water temperature.
The mass flow rate of fresh water output driven by the mechanical power P H is estimated as where f is the feed flow rate density, η pump is the driving pump mechanical efficiency, and Δp net is the net pressure difference across the high pressure pump.

| OPTIMIZATION ALGORITHM
Generally, in multi-objective optimization problems, a numerous set of answers called Pareto frontier is provided, to represent the closest match in the objective function area. 53 Multi-objective evolutionary algorithms (MOEAs), which provide a number of solutions to estimate the Pareto optimal class, have been employed widely by researchers. It is approved that MOEAs can overcome the difficulties of regular methods. 52 In this study, the Pareto frontier is provided by applying NSGA-II approach. The principles of this method are explained in Refs 94,95. In order to select the optimum solution from the outputs in multi-objective issues, a procedure called "decision making" is used. Several decision-making methods are available for selecting final optimum answer from Pareto front. On account of the probable dissimilar scales of objectives, the dimensions must be unified. Moreover, in decision-making procedure objectives' vectors have to be nondimensionalized. Most common nondimensionalization methods are: Fuzzy, Linear, and Euclidian. In this paper, three decision-making methods including LINMAP, TOPSIS, and Fuzzy are utilized. In this simulation, the energy efficiency (η H ), fresh water production rate (ṁ p ), exergy destruction rate density (exd H ), and the ecological function density (e H ) are considered as the objective functions in order to optimize the performance of the proposed cycle. Three decision variables are used for this optimization. Table 1 lists these variables and their ranges which are determined by the suggestions provided in the literature. 91,96-100 They include the current density (j) which is the most important controllable variable of the FC, working temperature of the MCFC (T), which is the maximum temperature of the hybrid cycle, and the thermal conductance between the working substance and the heat reservoirs at temperatures T l and T (θ). Other parameters are listed in Table 2.
The energy efficiency, fresh water production rate, and the exergy destruction rate density are the three objective functions for the first scenario, evaluated via Equations (23,30,24).
The energy efficiency, fresh water production rate, and the ecological function density are the three objective functions for the second scenario, evaluated via Equations (23,30,26).
To determine the optimal design variables of the system, based on the genetic algorithm approach, a simulation program was coded through Matlab software. Specifications of GA for optimization puzzle are reported through Table 3.

| Parametric evaluation of the hybrid cycle
In the first part of this analysis, the characteristic curves of the hybrid cycles are plotted (Figure 1f). This stage is of high importance because it helps understand the behavior of the cycle and also enable one to interpret the outcomes of the next part more clearly. Figure 1a shows the variation of the energy efficiency with the dimensionless exergy destruction rate ( * = exd H ∕ max (exd H )). This plot starts at j = 1000 (A m −2 ), on the bottom right corner, and by increasing the current density, the energy efficiency first increases and then decreases, while the exergy destruction rate always increases. Therefore, if one wants to optimize these two targets, the answer is somewhere between the starting point and the point where the energy efficiency is maximum. Another point here is that the influence of changing decision variables is less noticeable in this region. Figure 1b depicts the variation of the energy efficiency with the dimensionless fresh water production rate (m * = m p ∕ max (m p )). The curve starts on the bottom right hand corner. With an increase in the current density, the energy efficiency and the fresh water production rate both increase at first. But, then the energy efficiency starts to decrease while the fresh water production rate still increases. Finally, when the fresh water production rate passes its maximum, both targets decrease. Therefore, if the goal is to optimize these objective functions, the selected point will be somewhere between the point which maximizes the energy efficiency and the other point which maximizes the fresh water production rate. This figure also shows the fact observed before which is by increasing the current density, the influence of changing other decision variables, including the working temperature of the MCFC and the thermal conductance, will increase. Figure 1c illustrates the variation of the dimensionless fresh water production rate with the dimensionless exergy destruction rate. These curves start on the bottom left hand corner where the exergy destruction rate is minimized. Then by increasing the current density, the exergy destruction rate increases along with a temporary increase in the fresh water production rate. After maximizing the fresh water production rate, the curves start to go down. Figure 1d shows the variation of the energy efficiency with the dimensionless ecological function (e * = e H ∕abs(min(e H ))) . The curves start on the top right hand corner where it seems both the targets are maximized. But it should be noticed that at first, the energy efficiency and the ecological function are not exactly maximized. By increasing the current density, they reach their maximum points. Another point here is that at first, increasing the working temperature of the MCFC and the thermal conductance help the ecological function and the energy efficiency but at higher current densities, it has a negative effect on the ecological function. The effect of increasing these two decision variables always has a positive effect on the energy efficiency. Figure 1e depicts the variation of the dimensionless ecological function with the dimensionless exergy destruction rate. It starts on the top right hand corner. The exergy destruction rate always increases with an increase in the current density while the ecological function first increases and then decreases. Therefore, if one aims to optimize these targets together, the answer is somewhere between the starting point and the point where the ecological function is optimized. The influence of changing other decision variables on the targets again increases along with an increase in the current density. Figure 1f illustrates the variation of the dimensionless ecological function with the dimensionless fresh water production rate. The ecological function first increases slightly along with an increase in the fresh water production rate but then decreases until the end. The figure also shows that the maximization of the fresh water production rate occurs at higher current densities compared to the energy efficiency.

| Multi-objective optimization
As shown in the proceeding section, there is a conflict between the objective functions and there is no single solution to optimize all of them concurrently. Therefore, in this step two multi-objective optimization problems are defined to try to optimize different combinations of the targets and compare the results. The first scenario is to maximize the energy efficiency and the fresh water production rate while minimizing the exergy destruction rate.
In the second scenario, the exergy destruction rate density is replaced with the ecological function density. Three robust decision-making methods are employed to select the final answers between the Pareto fronts obtained by the evolution algorithm. The Pareto optimal fronts for the first and second scenarios are shown in Figures 2 and 3, respectively. Tables 4 and 5 address the optimal outputs achieved by executing LINMAP, TOPSIS, and Fuzzy decision-making methods.
As these results show, the final outputs are located at higher working temperatures of the MCFC and also higher values of the thermal conductance. This accord with the results obtained in the last section and validates the optimization algorithm. However, different values of the current density in the outputs are due to the differences between the decision-making methods. There is no other means to judge which one is better. However, the results can be compared, as in the next section.

| Sensitivity analysis
Finally, to compare the results of the different decisionmaking methods and the different proposed scenarios, and also to make sure the answers are the best possible ones, in this section, the sensitivity of the objective functions to the decision variables is computed. Tables 6 and 7 address the sensitivity of the objective functions at the optimum points at the operating temperature of the MCFC and the thermal conductance, respectively. Four current densities are selected. They include the current densities of 1000, 1359, 1602, and 4653 (A m −2 ). These points are those which approximately optimize one of the objective functions individually.
According to Table 6, with 2.21% increase in the working temperature of the MCFC (from 905 to 925 K), the energy efficiency increases by 0.73%, 1.39%, 1.86%, and 15.42% at current densities of 1000, 1359, 1602, and 4653 (A m −2 ), respectively; The fresh water production rate increases by 1.06%, 1.55%, 1.99%, and 15.41% at the different current densities mentioned above, respectively; The exergy destruction rate density decreases by 25.47%, 14.10%, 12.37%, and 12.88%, respectively; Finally, the ecological function density increases by 5.80%, 9.17%, 12.62%, and 39.43%, respectively. This shows that the most influenced objective function is the ecological function density and the sensitivity also increases with the increase in the current density in the three objective functions including the energy efficiency, fresh water production rate, and the ecological function density. However, the sensitivity of the exergy destruction rate density has the opposite behavior. It is maximum at first (at j = 1000 (A m −2 )) where the function itself is also optimum. Then it decreases until where the ecological function is nearly maximized and finally increases again at j = 4653 (A m −2 ). According to Table 7, with an increase of 22.22% in the thermal conductance (from 900 to 1100 (W K −1 )), the energy efficiency virtually stays constant at j = 1000 (A m −2 ), and increases by 0.06%, 0.12%, and 4.14% at j = 1359, 1602, and 4653 (A m −2 ), respectively; The fresh water production rate, also, does not change at j = 1000 and 1359 (A m −2 ), but increases by 0.33 and 4.13% at j = 1602 and 4653 (A m −2 ), respectively. The exergy destruction rate always increases by the increase in the thermal conductance and these increases augment with the increase in the current density. It increases by 0.06%, 0.21%, 0.31%, and 2.26% at j = 1000, 1359, 1602, and 4653, respectively, with an increase of 22.22% in the thermal conductance. Finally, the ecological function sensitivity to the thermal conductance increases with the increase in the current density and it increases by 0.02%, 0.18%, 0.42%, and 9.89% with an increase of 22.22% in the thermal conductance at the current densities mentioned above, respectively. These results also show that the most influential decision variable is the current density of the FC. Figure 4 shows what percent each decision variable changes along with an increase of 1% in the current density in different scenarios, at selected points by the different decision-making methods.

| CONCLUSIONS
A detailed thermodynamic analysis is conducted in this study in trying to study the performance of a hybrid cycle consisting of an MCFC and a Stirling engine which drives high pressure pumps of a reverse osmosis desalination system to produce fresh water. After studying the characteristic curves of the cycle, two triple-objective optimization approaches are implemented; each considering a combination of the desired objective functions. Three well-known and powerful decision-making methods, including TOPSIS, LINMAP, and Fuzzy, are employed to select the final outcomes from the Pareto fronts obtained by the Genetic algorithm. In the first scenario which considers the energy efficiency, fresh water production rate and the exergy destruction rate density, the energy efficiency has a better condition in the TOPSIS approach. It meets its maximum, 0.7299, which is 99.96% of its ideal value; the fresh water production rate reaches its optimum value in the Fuzzy approach, 5.07 (m 3 s −1 ) which is 74.98% of its ideal value; and the exergy destruction rate density is at its optimum value in the TOPSIS approach, 0.48 (kW m −2 ) which is 341.35% more than its ideal value which is 0.11 (kW m −2 ). In the second scenario which include the energy efficiency, fresh water production rate and the ecological function density, the energy efficiency meets its maximum again in the TOPSIS results, 0.6962, which is 95.34% of its ideal value; the fresh water production rate is at its peak in the Fuzzy results, again, which is 5.51 (m 3 s −1 ), 81.58% of its ideal value; the ecological function density reaches its maximum in the TOPSIS results, 0.77 (kW m −2 ), which is 80.93% of its ideal value. Finally, a sensitivity analysis is performed indicating what percent the final outputs of each decision-making method changes due to changes in the decision variables, in different conditions.