Efficiency and Size Optimization of a General Solid Oxide Fuel Cell System

CHEN Peng (陈 鹏)， HUANG Yue-wu (黄跃武)， 2， LIU Si-xu (刘思煦)

1 College of Environmental Science and Engineering, Donghua University, Shanghai 201620, China2 State Environmental Protection Engineering Center for Pollution Treatment and Control in Textile Industry, Donghua University, Shanghai 201620, China

Efficiency and Size Optimization of a General Solid Oxide Fuel Cell System

CHEN Peng (陈 鹏)1， HUANG Yue-wu (黄跃武)1， 2， LIU Si-xu (刘思煦)1

1CollegeofEnvironmentalScienceandEngineering,DonghuaUniversity,Shanghai201620,China2StateEnvironmentalProtectionEngineeringCenterforPollutionTreatmentandControlinTextileIndustry,DonghuaUniversity,Shanghai201620,China

An irreversible model of high temperature solid oxide fuel cells (SOFCs) working at steady-state is developed, devoted to performing the optimization with regard to two objectives: minimization of the fuel cell size and maximization of the system efficiency. The performance characteristics of the system are analyzed in details, illustrated by the curves of power density, efficiency and voltage. Genetic algorithm is used to perform the multi-objective optimization with four decision variables: the operating pressure, the fuel stoichiometric ratio, the air stoichiometric ratio and the current density. A Pareto set giving a quantative description of the trade-off between the two objectives is used to analyze the results. Optimization results prove the existence of optimal designs region for a 50 kW system with efficiency from 43% corresponding to a 14.6 m2electrolyte area to 48% corresponding to a 25.4 m2electrolyte area. The SOFC model used is general and the optimization results could be applied to the practical SOFC design.

solidoxidefuelcell(SOFC);multi-objective;optimization;efficiency;size

Introduction

High temperature solid oxide fuel cells (SOFCs)have gained plenty of interest from the worldwide researcher, because of its high efficiency and the capacity of working with a wide range of fuels[1]. Many studies have done on the optimization of SOFC system designs. Although they made significant contributions to the exploration of SOFC performance characteristics, most of them were limited to a single design objective. Wenetal.[2]considered the net power of SOFC as the design objective and obtained the optimal geometric and operating parameters through a 4-way-optimization procedure with the total volume fixed. Odukoyaetal.[3]investigated the optimal operating conditions of a cogeneration power plant combined with SOFC. Their study obtained the maximun fuel cell net work output with the optimal pressure ratio and the operating temperature determined. Zamfirescu and Dincer[4]investigated the performance of a power and heating system for vehicular applications using proton-conducting SOFC and its optimization for power output and efficiency. In their study, the two objectives were optimized individually and the interaction between them was not considered. Actually it was a single-objective optimisation problem. However, the interaction between the multiple objectives has not been considered in those papers, and the potentially conflicting nature of different objectives makes the determination of the optimal solution more challenging.

There have been a few studies dealing with thermo-economic optimization of SOFC system reported in Refs.[5-9]. Autissieretal.[6]delt with the multi-objective optimization for the investment cost and efficiency of an SOFC-μ gas turbine (GT) hydrid system and obtained Pareto solutions representing the trade-off of the two objectives. Cheddie[7]developed a thermo-economic model of a 10 MW GT power plant indirectly coupled with SOFC system. The system efficiency and energy cost were the design objectives, whilst the SOFC size, molar flow rate of fuel and oxygen and the SOFC anode recycle fraction were the decision variables. Most of those papers concerned on the cost and the thermodynamic efficiency of the system. However, as the size of the SOFC system is inherently related to its economics, the economic issue associated with the system also can be considered from the point of size. A compact design can help to achieve a lower material cost and is required for both man-portable and stationary powerhouse applications[10].

This paper presents a model suitable for multi-objective optimization for the size and efficiency of an SOFC system. The size of SOFC system is denoted by the effective area of electrolyte and the decision variables affecting the performance of the system are the operating pressure, the current density, the fuel stoichiometric ratio and the air stoichiometric ratio. We aim to investigate the size and efficiency trade-off involved in the designs of SOFC system and determine a set of optimal solutions. In addition, we will discuss the comparison between the optimal results and the reference case, and the relationships between the optimal decision variables and the efficiency.

1 SOFC Mathematic Models

The working principle of the SOFC system is illustrated in Fig.1. The fuel cell operates as an electronchemical energy conversion device, fed by pressurized humid hidrogen as the fuel and pressurized air (79%, 21%) as the oxidant. At the cathode, oxygen is reduced by the electrons supplied by the external circuit to produce oxygen ions. Then the produced oxygen ions are conducted through the electrolyte to the anode to electrochemically combine with the hydrogen to form water and release electrons to the external circuit. The electrochemical reactions can be summarized as follows:

at the anode:H2+O2-→H2O+2e-;

at the cathode:1/2O2+ 2e-→O2-;

overall reaction:H2+1/2 O2→ H2O + Heat + Electricity.

Fig.1 A single-cell SOFC system fed by hydrogen and air

To simplify the mathematic model to be suited for multi-objective optimization within acceptable accuracy, this paper makes several assumptions as follows. (1) The fuel cell operates under steady-state conditions. (2) The spatial variations of the reactants, products, operating temperature and operating pressure are out of consideration in this model. (3) Reactants are compressible ideal gases. (4) No side reactions or electrode reactions occur and no gas leakage is considered.

1.1 Mass balances

For a given current density, the mass balance equations at respective electrodes can be expressed as

MH2, in=λH2iA/(2F),

(1)

MO2, in=λairiA/(4F),

(2)

whereλH2andλairare the respective stoichiometric ratios of hydrogen and air,Ais the effective area of the electrolyte,Fis the Faraday’constant, andiis the current density.

MH2, in=MH2, out+Ai/(2F),

(3)

MO2, in=MO2, out+Ai/(4F),

(4)

MH2O, out=MH2O, in+Ai/(2F),

(5)

MN2, in=MN2, out.

(6)

As operating temperatureTreaches 1 073 K in the model, the water exists in gaseous state in the anode channel. Nitrogen does not participate in the reaction, and its quantity keeps unchanged as it flows through the channel.

1.2 Electrochemical models

The fuel cell achieves its maximum reversible voltage when the external load is open, in which case that the forward reaction and the reverse reaction are in balance at respective electrodes. According to Nernst equation, the reversible voltage is given by[11]

(7)

whereRis the universal gas constant;Tis the operating temperature;pH2,pO2andpH2O are the respective partial pressures of hydrogen, oxygen and water;neis the number of electrons participating the reaction. ThenE0is the standard reversible voltage, derived by

(8)

where Δg0is the standard Gibbs free-energy change for the chemical reaction atp=1.01×105Pa, which also depends on temperature.

The actual operating potential of the fuel cell is derived by taking into account various kinds of overpotential, activation overpotential due to reaction kinetics, ohmic overpotential from ionic and electronic conduction, concentration overpotential due to mass transport. The output voltage of the fuel cell is computed as a function of the operating temperature, the operating pressure, the fuel stoichiometric ratio, the air stoichiometric ratio and the current density, which can be written by:

V=E-Vact-Vohm-Vconc,

(9)

whereVactis the activation overpotential,Vohmis the ohmic overpotential, andVconcis the concentration overpotential.

Activation overpotential is the voltage overpotential required to overcome the activation energy of the electrochemical reaction on the catalytic surface, derived by[12]

(10)

(11)

(12)

(13)

Herei0, aandi0, care the respective exchange current densities of the anode and the cathode;p0is the reference pressure;γaandγcare the respective pre-exponential coefficients of the anode and the cathode;Eact, aandEact, care the respective activation energies of the anode and the cathode. The values ofγa,γc,Eact, aandEact, care listed in Table 1.

Table 1 Related parameters in the model

ParametersValuesNumberofreactionelectronsne2Pre-factorforanodeexchangecurrentdensityγa/(A·m-2)5.5×108ActivationenergyofanodeEact,c/(J·mol-1)115781ActivationenergyofcathodeEact,c/(J·mol-1)157659ElectrolytethicknessLe/μm20Pre-factorforcathodeexchangecurrentdensityγc/(A·m-2)7.0×108ReferenceelectryleionicconductivityσO/(S·m-1)3.6×107ActivationenergyofO2-Ee/(J·mol-1)8.0×104AnodelimitingcurrentdensityiL,a/(A·m-2)2.99×104CathodelimitingcurrentdensityiL,c/(A·m-2)2.16×104FaradaysconstantF/(C·mol-1)96485UniversalgasconstantR/[J·(mol·K)-1]8.314Lowerheatingvalueofhydrogen/(J·mol-1)2.4×105

Compared to the electolyte resistance, the resistance from the anode and the cathode is negligible. Therefore, only the electolyte resistance is taken into account in the model. The ohmic overpotential can be derived by[13]

Vohm=IRohm,

(14)

(15)

(16)

whereRohmis the electolyte resistance,Leis the thickness of electrolyte,σeis the ionic conductivity of electrolyte,Eeis the electrolyte activation energy for oxygen ion transport, andσois the reference ionic conductivity of electrolyte. The values ofLe,σoandEeare also listed in Table 1, andIis the current, namely, the product of the current densityiand the effective area of the electrolyteA.

Concentration loss is caused by the reduction of reactant concentration on the catalyst surface, which reduces the thermodynamic voltage from the Nernst equation, derived by[14]

(17)

(18)

whereiL, candiL, aare the limiting current densities of the cathode and the anode, the values of which are listed in Table 1[12-16].

1.3 System efficiency

Efficiency is an important performance parameter to evaluate the performance of a fuel cell system, which is defined by[17]

(19)

Wstack=nVI,

(20)

whereWstackis the electrical power output of the stack,Wcompressoris the power consumed by the compressors[18],Wfuelis the energy in the inlet fuel, which is computed on the lower heating value of hydrogen[19]， andnis the number of the cells in series. As this model is based on single-cell SOFC, the value ofnis 1.

2 General Performance Characteristics of the Reference Case

A reference case with 30 m2electrolyte area is set to validate the model, in which case the operating pressure is 3.03×105Pa, the operating temperature is 1 073 K and the stoichiometric ratios of air and hydrogen are 2 and 1.5, respectively. The fuel is consisted of 97% hydrogen and 3% water, and the air is taken from the ambient air (21% O2+79%N2). Numerical calculations are performed, and some performance characteristic curves are plotted to investigate the effect of the operating temperature and pressure on the system.

Figures 2 (a)-(c) show the effect of operating temperature on the polarization curves, the efficiency and power density curves. Figure 2 (a) shows that the reversible voltage decreases as operating temperature increases. However, there would be a higher cell voltage for higher operating temperature when the current density reaches a point. It can be explained by that higher operating temperature would reduce the Gibbs free-energy change of the reaction, whilst reduce the polarization losses and make the polarization cure go down slower. Thus, the polarization curves of higher operating temperature would take higher position, eventually. The system efficiency curves are similar to the voltage curves as Fig.2 (b) shows, explanations for the polarization curves also can be applied here. It also can be seen from Fig.2(c) that the power density and its maximum increase as operating temperature increases. To summarize, the operating temperature enhances the performance of the system, thus, the limiting operating temperature is given by the properties of the high temperature materials employed; in our case the operating temperature is fixed at 1 073 K.

(a)

(b)

(c)

(a)

(b)

(c)

(d)

Figures 3(a)-(d) show the effect of operating pressure on polarization curves, the efficiency and power density curves. It can be seen from Figs. 3(a) and (d) that the cell voltage and power density increase as the operating pressure increases. However, the effect of the operating pressure on the efficiency is not distinguished as Figs. 3(b) and (c) show. For the system efficiency, the benefit from increases of cell voltage and power density is offset by the increasing energy consumption by compressors.

For a given output powerWstack, the effective areaAof electrolyte varies inversely with the power density. And one can know from Figs. 1 (b), (c) and Figs. 2 (b), (c) that, as the current density increases, the system efficiency decreases and the power density increases. Figure 4 shows that the effective electrolyte area increases with the system efficiency increasing. Therefore, the optimization for two objectives: maximizing the system efficiency and minimizing the fuel cell size (area of the electrolyte), is conflicting.

Fig.4 Electrolyte area varying with system efficiency within valid range of current density

3 Optimization

The multi-objective optimization minimizes several objectives simultaneously under a number of constraints. The objectives in the model can be defined by

(21)

X=[i， p, λair, λH2],

whereXis the vector of the decision variables,Ais derived from

(22)

andηis given by Eq. (19).

In this paper, the resolution of the optimization is performed using an evolutionary algorithm multi-objective optimizer (MOO) which has been developed at MATLAB. It mimics the process of natural evolution, which creates initial population to explore the search space and evolves new population through the process of selection, crossover and mutation, and repeats above process until a termination condition reaches[20]. We set the population size to 75, the population type to double vector, and the crossover fraction to 0.8. Feasible population is selected as the creation function, and scattered function is selected as the crossover function[21].

The aim is to get a set of optimal solutions, called Pareto set which reflects the compromise of maximizingηand minimizingA. The bounds of decision variables are given in Table 2[6, 22-23].

Table 2 Bounds of decision variables

DecisionvariableRangeCurrentdensityi0.10.5(×104A/m2)Operatingpressurep1.26(1.01×105Pa)Airstoichiometricratioλair1.15HydrogenstoichiometricratioλH21.12

4 Results and Discussion

For three given stack output powers, namely 50, 75, and 100 kW, optimization is performed and three sets of Pareto points are obtained as Fig.5 shows. It’s clearly found that the optimal system efficiency and the size of cell is conflicting, namely, to achieve a higher system efficiency, fuel cell size (area of electrolyte) has to be outspreaded, or the system efficiency would drop for a smaller size. Hence, in practical application, as one objective is optimized, another must be taken into account to be ensured reasonable.

Fig.5 Pareto sets for 50, 75, and 100 kW

As Fig.5 shows, the Pareto set of a 50 kW system has a topmost point with the maximum system efficiency and size, and the lowest point with the minimum system efficiency and size, which represent single-objective optimization for the system efficiency and size, respectively. Going down along the Pareto set in the figure, the optimization is more prone to the size optimization. At the highest system efficiency (52%), the value of electrolyte area is 62 m2, whilst at the minimum size (11.7 m2)， the value of efficiency is 16%. Obviously, the solutions of single-objective optimization for system design aren’t practical.

Furthermore, in the flat region of the figure, where the value of efficiency from 43% to 17%, the system efficiency is decreased by 60%, while the size is just reduced by 3 m2. For economic reason, this flat region isn’t ideal. Similarly, above efficiency 48%, the upward trend of Pareto set is steep, namely, the efficiency is risen from 48% to 52% (by 8%), whilst the size is expanded from 25.4 m2to 62.4 m2(by 146%). Obviously, the price of size expanded for efficiency raised is heavy. Hence, the ideal optimal region should be from 43% efficiency with 14.6 m2electrolyte area to 48% efficiency with 25.4 m2electrolyte area. Compared to the reference case, which has 38.8% system efficiency with a 30 m2electrolyte area, at least, the efficiency is elevated by 10% and the size is shrunk by 15.33%.

Figure 5 also shows the comparison of Pareto sets for differentWstack, 50, 75 and 100 kW. It’s seen that the shapes of different output powers’ Pareto sets are similar and as the out power increases, bigger size (electrolyte area) is needed.

Figures 6-9 show links between the decision variables and efficiency. It’s noted that the values of decision variables are non-increasing with the system efficiency, and that the values of decision variables，which in the ideal optimal region mentioned above, are far away from their upper bounds.

Fig.6 Operating pressure vs. efficiency

Fig.7 Current density vs. efficiency

Fig.8 Air stoichiometric ratio vs. efficiency

Fig.9 Fuel stoichiometric ratio vs. efficiency

5 Conclusions

A multi-objective optimization approach has been applied to the optimization of an SOFC system based on the mathematical model. The approach generates Pareto sets that represent the trade-off between the optimal system efficiency and size. The results prove that the ideal optimal region, where efficiency from 43% corresponding to 14.6 m2electrolyte area to 48% corresponding to 25.4 m2electrolyte area, at least, has efficiency elevated by 10% and size shrunk by 15.33%, compared to the reference case. High efficiency can be obtained with low fuel stoichiometric ratio, and the air stoichiometric ratio about 2-3 has outstanding advantage. The results obtained here can contribute to the improvement of performance and the optimization design of the SOFC system.

[1] O’Hayre R, Cha S, Colella W,etal. Fuel Cell Fundamentals [M]. 2nd ed. New York: John Wiley & Sons, Inc., 2006: 1-546.

[2] Wen H, Ordonez J C, Vargas J V C. Optimization of Single SOFC Structural Design for Maximum Power [J].AppliedThermalEngineering, 2013, 50 (1): 12-25.

[3] Odukoya A, Carretero J A, Reddy B V. Thermodynamic Optimization of Solid Oxide Fuel Cell-Based Combined Cycle Cogeneration Plant [J].InternationalJournalofEnergyResearch, 2011, 35(15): 1399-1411.

[4] Zamfirescu C, Dincer I. Thermodynamic Performance Analysis and Optimizatiom of an SOFC-H+System [J].ThermochimicaActa, 2009, 486 (1/2): 32-40.

[5] Marechal F, Palazzi F, Godat F. Thermo-Economic Modelling and Optimization of Fuel Cell System [J].FuelCells, 2005, 5(1): 5-24.

[6] Autissier N, Palazzi F, Herle J V. Thermo-Economic Optimization of a Solid Oxide Fuel Cell, Gas Turbine Hybrid System [J].JournalofFuelCellScienceandTechnology, 2007, 4(2): 123-129.

[7] Cheddie D F. Thermo-Economic Optimization of an Indirectly Coupled Solid Oxide Fuel Cell/Gas Turbine Hybrid Power Plant [J].InternationalJournalofHydrogenEnergy, 2010, 36(2): 1702-1709.

[8] Shirazi A, Aminyavari M, Najafi B. Thermal-Economic-Environmental Analysis and Multi-objective Optimization of an Internal-Reforming Solid Oxide Fuel Cell-Gas Turbine Hybrid System [J].InternationalJournalofHydrogenEnergy, 2012, 37(24): 19111-19124.

[9] Morandin M, Maréchal F, Giacomini S. Synthesis and Thermo-Economic Design Optimization of Wood-Gasifier-SOFC Systems for Small Scale Applications [J].BiomassandBioenergy, 2013, 49: 299-314.

[10] Fraga S. A Multi-objective Optimisation Model for a General Polymer Electrolyte Membrane Fuel Cell System [J].JournalofPowerSources, 2010, 195(9): 2754-2763.

[11] Zhao Y R, Ou C J, Chen J C. A New Analytical Approach to Model and Evaluate the Performance of a Class of Irreversible Fuel Cells [J].InternationalJournalofHydrogenEnergy, 2008, 33(15): 4161-4170.

[12] Kakac S, Pramuanjaroenkij A, Zhou X Y. A Review of Numerical Modeling of Solid Oxide Fuel Cells [J].InternationalJournalofHydrogenEnergy, 2007, 32(7): 761-786.

[13] Aguiar P, Adjiman C S, Brandon N P. Anode-Supported Intermediate Temperature Direct Internal Reforming Solid Oxide Fuel Cell. I: Model-Based Steady-State Performance [J].JournalofPowerSources, 2004, 138(1/2): 120-136.

[14] Petruzzi L, Cocchi S, Fineschi F. A Global Thermo-Electrochemical Model for SOFC Systems Design and Engineering [J].JournalofPowerSources, 2003, 118(1/2): 96-107.

[15] Ho T X, Kosinski P, Hoffmann A C. Numerical Modeling of Solid Oxide Fuel Cells [J].ChemicalEngineeringScience, 2008, 63(21): 5356-5365.

[16] Wen H, Ordonez J C, Vargas J V C. Single Solid Oxide Fuel Cell Modeling and Optimization [J].JournalofPowerSources, 2011, 196 (18): 7519-7532.

[17] Ro S T, Sohn J L. Some Issues on Performance Analysis of Fuel Cells in Thermodynamic Point of View [J].JournalofPowerSources, 2007, 167(2): 295-301.

[18] Chan S H, Low C F, Ding O L. Energy and Exergy Analysis of Simple Soli-Oxide Fuel-Cell Power Systems [J].JournalofPowerSources, 2002, 103(2): 188-200.

[19] Pei P C, Yang W, Li P. Numerical Prediction on an Automotive Fuel Cell Driving System [J].InternationalJournalofHydrogenEnergy, 2006, 31(3): 361-369.

[20] Reeves C R, Rowe J E. Genetic Algorithms-Principles and Perspectives: a Guide to GA Theory [M]. Berlin: Springer, 2002.

[21] Palazzi F, Autissier N, Marechal F M A. A Methodology for Thermo-Economic Modeling and Optimization of Solid Oxide Fuel Cell Systems [J].AppliedThermalEngineering, 2007, 27(16): 2703-2712.

[22] Herle J V, Marechal F, S. Leuenberger S, D,etal. Energy Balance Model of an SOFC Cogenerator Operated with Biogas [J].JournalofPowerSources, 2003, 118(1/2): 375-383.

[23] Nehter P. A High Fuel Utilizing Solid Oxide Fuel Cell Cycle with Regard to the Formation of Nickel Oxide and Power Density [J].JournalofPowerSources, 2007, 164(1): 252-259.

Foundation items: National Natural Science Foundation of China (No. 51078068); the Fundamental Research Funds for the Central Universities, China (No. 11D11314)Correspondence should be addressed to HUANG Yue-wu, E-mail: huangyuewu@dhu.edu.cn

TK91 Document code: A

1672-5220(2015)01-0113-06

Received date: 2013-10-09