Modelling Of Proton Exchange Membrane Fuel Cell Performance Based On Semi-empirical Equations

Renewable Energy 30 (2005) 1587–1599 www.elsevier.com/locate/renene

Technical Note

Modelling of proton exchange membrane fuel cell performance based on semi-empirical equations Maher A.R. Sadiq Al-Baghdadi1* Department of Mechanical Engineering, College of Engineering, University of Babylon, Babylon, Iraq Received 8 August 2004; accepted 29 November 2004 Available online 19 January 2005

Abstract Using semi-empirical equations for modeling a proton exchange membrane fuel cell is proposed for providing a tool for the design and analysis of fuel cell total systems. The focus of this study is to derive an empirical model including process variations to estimate the performance of fuel cell without extensive calculations. The model take into account not only the current density but also the process variations, such as the gas pressure, temperature, humidity, and utilization to cover operating processes, which are important factors in determining the real performance of fuel cell. The modelling results are compared well with known experimental results. The comparison shows good agreements between the modeling results and the experimental data. The model can be used to investigate the influence of process variables for design optimization of fuel cells, stacks, and complete fuel cell power system. q 2005 Elsevier Ltd. All rights reserved. Keywords: PEM fuel cell; Electrochemistry; Modelling; Energy conversion

1. Introduction The proton exchange membrane fuel cells (PEMFCs) are suitable for portable, mobile and residential applications, due to their inherent advantages, such as high-power density, * Fax: C218 41 632249. E-mail address: [email protected]. 1 Present address: Mechanical and Energy Department, The Higher Institute for Engineering Comprehensive Vocations, P.O. Box 65943, Yefren, Libya. 0960-1481/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.renene.2004.11.015

1588

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

Nomenclature A ac CH
C CH
2 CH
2 O CO
2 E Efc F i I ka0 ; kc0 l LHVH2 m, n m_ H2 MWH2 P a, P c P
H2 ; P
O2 Psat H2 O R Rinternal rM T Vcell Wgross DGe DGec

active cell area (cm2) chemical activity parameter for the cathode proton concentration at the cathode membrane/gas interface (mol/cm3) liquid phase concentration of hydrogen at anode/gas interface (mol/cm3) water concentration at the cathode membrane/gas interface (mol/cm3) oxygen concentration at the cathode membrane/gas interface (mol/cm3) thermodynamic potential (V) thermodynamic efficiency Faraday’s constant (96,487 8C/mol) current density (A/cm2) current (A) intrinsic rate constant for the anode and cathode reactions, respectively (cm/s) thickness of the polymer membrane (cm) lower heating value of hydrogen (J/kg) mass transfer coefficients hydrogen mass flow rate (kg/s) molecular mass of hydrogen (kg/mol) total pressure of anode and cathode, respectively (atm) partial pressure of hydrogen and oxygen at the anode catalyst/gas interface and cathode catalyst/gas interface, respectively (atm) water saturation pressure (atm for Eqs. (12) and (14)), (bar for Eq. (15)) gas constant (8.314 J/mol K) total internal resistance (U cm2) membrane specific resistivity for the flow of hydrated protons (U cm) cell temperature (K) cell voltage (V) gross output power (W) standard state free energy of the cathode reaction (J/mol) standard state free energy of chemisorption from the gas state (J/mol)

Greek letters hact activation over potential hdiff diffusion over potential hohmic ohmic over potential x1,x2,x3,x4 semi-empirical coefficients for calculation of activation overpotential

simple and safe construction and quick startup even at low operating temperatures. Rapid development recently has brought the PEMFC significantly closer to commercial reality. Although prototypes of fuel cell vehicles and residential fuel cell systems have already been introduced, it remains to reduce the cost and enhance their efficiencies. To improve the system performance, design optimization and analysis of fuel cell systems are

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

1589

important. Mathematical models and simulation are needed as tools for design optimization of fuel cells, stacks, and fuel cell power systems. In order to understand and improve the performance of PEMFC systems, several different mathematical models have been proposed to estimate the behavior of voltage variation with discharge current of a PEMFC. Recently, numerical modeling and computer simulation have been performed for understanding better the fuel cell itself [1–19]. Numerical models are useful to simulate the inner details of PEMFC, but the calculation required for these models is too extensive to be used for system models. In system studies, it is important to have an adequate model to estimate overall performance of a PEMFC in terms of operating conditions without extensive calculations. But, few studies have focused on the simple models, which can be used to investigate the impact of cell operating conditions on the cell performance and can be used to design practical fuel cell total systems. In this paper, a lumped model is presented. Empirical equations are useful for estimating the performance of PEMFC stacks and optimization of fuel cell system integration and operation than numerical models. The aim of this research is to develop a model for investigating the performance of a PEM fuel cell at different operation variables using semi-empirical equations. Model validation against the experimental data of Chahine et al. [11] is presented.

2. Background The fundamental structure of a PEM fuel cell can be described as two electrodes (anode and cathode) separated by a solid membrane acting as an electrolyte (Fig. 1). Hydrogen fuel flows through a network of channels to the anode, where it dissociates into protons that, in turn, flow through the membrane to the cathode and electrons that are collected as electrical current by an external circuit linking the two electrodes. The oxidant (air in this study) flows through a similar network of channels to the cathode where oxygen combines with the electrons in the external circuit and the protons flowing through the membrane,

Fig. 1. Schematic of a single typical proton exchange membrane fuel cells.

1590

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

thus producing water. The chemical reactions occurring at the anode and cathode electrode of a PEM fuel cell are as follows: Anode reaction: 2H2 / 4HCC 4eK Cathode reaction: O2 C 4HCC 4eK/ 2H2 O Total cell reaction: 2H2 C O2 / 2H2 OC electricityC heat The products of this process are water, DC electricity and heat.

3. Mathematical model Useful work (electrical energy) is obtained from a fuel cell only when a current is drawn, but the actual cell potential (Vcell) is decreased from its equilibrium thermodynamic potential (E) because of irreversible losses. When current flows, a deviation from the thermodynamic potential occurs corresponding to the electrical work performed by the cell. The deviation from the equilibrium value is called the over potential and has been given the symbol (h). The over potentials originate primary from activation over potential (hact), ohmic over potential (hohmic) and diffusion over potential (hdiff). Therefore, the expression of the voltage of a single cell is: Vcell Z E C hact C hohmic C hdiff

(1)

The reversible thermodynamic potential of the H2CO2 reaction previously described is given by the Nernst equation: E Z E0 C

RT ln½P
H2 ðP
O2 Þ0:5  zF

(2)

where E0 is a reference potential and the partial pressure terms are related to the hydrogen and oxygen concentrations at the anode and cathode. Further expansion of this equation return [11,12]:
 1 K3 K5

E Z 1:229 K 0:85 !10 ðT K 298:15Þ C 4:3085 !10 T lnðPH2 Þ C lnðPO2 Þ 2 (3) Activation overpotential arises from the kinetics of charge transfer reaction across the electrode–electrolyte interface. In other words, a portion of the electrode potential is lost in driving the electron transfer reaction. Activation overpotential is directly related to the nature of the electrochemical reactions and represents the magnitude of activation energy, when the reaction propagates at the rate demanded by the current. The activation overpotential can be divided into the anode and cathode overpotentials. The equation for the anode overpotential is [11–13]: hact;a Z K

DGec RT RT lnð4FAka0 CH
2 Þ K lnðiÞ C 2F 2F 2F

(4)

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

1591

The respective equation used for calculating the cathode overpotential is:
     RT KDGe 0
ð1Kac Þ
ð1Kac Þ
ac ln zFAkc exp ðCHC Þ ðCH2 O Þ hact;c Z K lnðiÞ ðCO2 Þ ac zF RT (5) where zZ1 is the number of equivalents involved in the cathode reaction. In order to have a single expression of the activation overpotential, Eqs. (4) and (5) can be combined and written in a parametric form as follows: hact Z x1 C x2 T C x3 T½lnðCO
2 Þ C x4 T½lnðiÞ

(6)

where the terms xi are semi-empirical coefficients, defined by the following equations:     KDGe KDGec x1 Z C (7) ac zF 2F x2 Z

  R R ½lnð4FAka0 CH
2 Þ ln zFAkc0 ðCH
C Þð1Kac Þ ðCH
2 O Þac C ac zF 2F

(8)

x3 Z

R ð1 K ac Þ ac zF

(9)

  R R C x4 Z K ac zF 2F

(10)

The use of such semi-empirical coefficients gives a significant degree of flexibility when the model is applied to simulate a specific fuel cell stack, as the terms xi can be obtained by a fitting procedure based on the measured polarization curve of the stack. At the same time, these coefficients have a significant mechanistic background. The values used here for the coefficients xi are the ones proposed in Ref. [11] and also with the works of Maxoulis et al. [12] and Fowler et al. [13] and are shown as x1 x2 x3 x4

K0.9514 0.00312 7.4!10K5 K0.000187

The water concentration levels at the anode and cathode may play an important role in the activation losses of a fuel cell, as the reactants H2 at the anode and O2 at the cathode side must diffuse through a water film to reach the catalyst active sites. The anode side of the membrane is expected to present a quite low water concentration, while the water film at the cathode side is expected to be significantly thicker due to the production of water there. The effective hydrogen concentration at the anode catalyst sites, which can be approximated, by the hydrogen concentration at the anode water–gas interface, is

1592

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

expressed as [12]; CH
2 Z

P
H2   1:09 !106 exp 77 T

(11)

The partial pressure of hydrogen, in turn, depends on the water content in the anode channel (which is assumed constant, due to the zero-dimension approach) based on the expression; 2 3   1 sat 4 1

P P
H2 Z K 15 (12) 2 H2 O exp 1:653!i x 1:334 T

H2 O

The effective oxygen concentration at the cathode catalyst sites, which can be approximated, by the oxygen concentration at the cathode water–gas interface, is expressed as [12]; CO
2 Z

P
O2   5:08 !106 exp K498 T

(13)

The partial pressure of oxygen at the water–gas interface is related to the water concentration at the cathode channel with the expression; 2 3 1 4

P
O2 Z ðPsat K 15 (14) H2 O Þ exp 4:192!i x H2 O T 1:334 The saturation pressure of water vapor can be computed from Berning et al. [14]; K5 log10 Psat H2 O Z K2:1794 C 0:02953ðT K 273:15Þ K 9:1837 !10

!ðT K 273:15Þ2 C 1:4454 !10K7 ðT K 273:15Þ3

(15)

Ohmic overpotential result from electrical resistance losses in the cell. These resistances can be found in practically all fuel cell components: ionic resistance in the membrane, ionic and electronic resistance in the electrodes, and electronic resistance in the gas diffusion backings, bipolar plates and terminal connections. This could be expressed using Ohm’s Law equations such as: hohmic Z KiRinternal

(16)

The total internal resistance is a complex function of temperature and current. A general expression for resistance is defined to include all the important membrane parameters [13]; Rinternal Z

rM :l A

(17)

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

1593

The following empirical expression for Nafion membrane resistivity is proposed [13]; h    T 2  i 2:5 i 181:6 1 C 0:03 Ai C 0:062 303 A  i    TK303  rM Z  14 K 0:634 K 3 A exp 4:18 T

(18)

Diffusion overpotential is caused by mass transfer limitations on the availability of the reactants near the electrodes. The electrode reactions require a constant supply of reactants in order to sustain the current flow. When the diffusion limitations reduce the availability of a reactant, part of the available reaction energy is used to drive the mass transfer, thus creating a corresponding loss in output voltage. Similar problems can develop if a reaction product accumulates near the electrode surface and obstructs the diffusion paths or dilutes the reactants. As proposed by Berning et al. [14], Chahine et al. [15], and Hamelin et al. [16], the total diffusion overpotential can be represented by the following expression: hdiff Z m expðniÞ

(19)

The diffusion overpotential is directly related to the concentration drop of reactant gases, and thus inversely to the growth rate n of byproducts of the electrochemical reaction in the catalyst layers, flow fields, and across the electrode. A physical interpretation for the parameters m and n was not given, but Berning et al. [14] found in their study that m correlates to the electrolyte conductivity and n to the porosity of the gas diffusion layer. Both m and n relate to water management issues. A partially dehydrated electrolyte membrane leads to a decrease in conductivity, which can be represented by m, whereas an excess in liquid water leads to a reduction in porosity and hence to an early onset of mass transport limitations, which can be captured by the parameter n. The mass transfer coefficient m decreases linearly with cell temperature but it has two dramatically different slopes as shown by the following expressions [15]. m Z 1:1 !10K4 K 1:2 !10K6 ðT K 273:15Þ

for T R 312:15 K ð39 8CÞ

(20)

m Z 3:3 !10K3 K 8:2 !10K5 ðT K 273:15Þ

for T ! 312:15 K ð39 8CÞ

(21)

The thermodynamic efficiency of the fuel cell Efc can be determined as the ratio of output work rate Wgross to the product of the hydrogen consumption rate m_ H2 and the lower heating value of hydrogen LHVH2 [17]. Efc Z

Wgross m_ H2 $LHVH2

(22)

1594

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

Once the output voltage of the stack is determined for a given output current, the gross output power is found as: Wgross Z IVcell

(23)

The output current is correlated with the hydrogen mass flow rate by the equation [17]; m_ H2 Z

IMWH2 2F

(24)

Thus, the thermodynamic efficiency of the fuel cell can simplifies as follows; Efc Z

2Vcell F MWH2 $LHVH2

(25)

4. Results and discussion Model validation involves the comparison of model results with experimental data, primarily for the purpose of establishing confidence in the model. To validation the mathematical model presented in the preceding section, comparisons were made to

Fig. 2. Comparison between the model predictions and experimental results of Chahine et al. [15].

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

1595

Fig. 3. Predicted polarization curve, efficiency and power density at 31 8C cell temperature and 1 atm reactant pressures.

the experimental data of Chahine et al. [15] for a single cell operated at different temperatures. Fig. 2 compares the computed polarization curves with the measured ones. The calculated curves show good agreement with the experimental data for all temperatures. Polarization curves with different cell temperature and different operating pressures are shown in Figs. 3–5. The pressures of anode and cathode sides were kept the same. The performance of the fuel cell increases with the increase of the cell temperature, as will be shown in Figs. 3 and 4. The exchange current density increases with the increase of fuel cell temperature, which reduces activation losses. Another reason for the improved performances is that higher temperatures improve mass transfer within the fuel cells and results in a net decrease in cell resistance (as the temperature increases the electronic conduction in metals decreases but the ionic conduction in the electrolyte increases). This may explain the improvement of the performance [18,19]. The shifting of the polarization curves towards higher voltage at higher current densities when increasing the cell temperature is due to the increase of conductivity of the membrane. Also, Figs. 3 and 4 show that the maximum power density shifts towards higher current density with an increasing temperature as a result of reduced ohmic loss [14].

1596

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

Fig. 4. Predicted polarization curve, efficiency and power density at 72 8C cell temperature and 1 atm reactant pressures.

Figs. 4 and 5 show that the performance of the fuel cell improves with the increase of pressure. The higher open circuit voltage at the higher pressures can be explained by the Nernst equation. The overall polarization curves shift positively as the pressure increases. Another reason for the improved performances is the partial pressure increase of the reactant gases with increasing operating pressure. Changes in operating pressure have a large impact on the inlet composition and, hence, on the power density, as will be shown in Figs. 4 and 5. The maximum power density shifts positively with an increasing pressure because of the rate of the chemical reaction is proportional to the partial pressures of the hydrogen and the oxygen. Thus, the effect of increased pressure is most prominent when using air. In essence, higher pressures help to force the hydrogen and oxygen into contact with the electrolyte. This sensitivity to pressure is greater at high currents. Return to Figs. 3–5, the maximum power corresponds to relatively high current. At the peak point, the internal resistance of the cell is equal to the electrical resistance of the external circuit. However, since efficiency drops with increasing current, there is a tradeoff between high power and high efficiency. Fuel cell system designers must select the desired operating range according to whether efficiency or power is paramount for the given application.

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

1597

Fig. 5. Predicted polarization curve, efficiency and power density at 72 8C cell temperature and 5 atm reactant pressures.

The influence of cathode/anode pressure on the performance of PEMFC at 72 8C is shown in Fig. 6. It is clearly shown that cathode pressure is more effective to the fuel cell performance than anode pressure. This result demonstrates that an increase in the cathode pressure results in a significant reduction in polarization at the cathode (Eqs. (3) and (6)). Performance improvements due to increased pressure must be balanced against the energy required to pressurize the reactant gases. The overall system must be optimized according to power, efficiency, cost, and size.

5. Conclusion An empirical model of a PEM fuel cell has been developed and the effect of operation conditions on the cell performance has been investigated. The objective was to develop an empirical model that would simulate the performance of fuel cells without extensive calculations. The results of the present study indicate the operating temperature and pressure can be optimized, based on cell performance, for given design and other operating conditions. For most applications, and particularly for steady operation, a fuel cell does not

1598

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

Fig. 6. Effect of cathode/anode pressure on the fuel cell performance.

have to be operated at its maximum power, where the efficiency is the lowest. When higher nominal cell potential is selected, savings on fuel cost offset the cost of additional cells. The empirical model of electrochemical reactions and current distribution as presented herein is shown to be provide a computer-aided tool for design and optimization of future fuel cell engines with much higher power density and lower cost.

References [1] Hu M, Gu A, Wang M, Zhu X, Yu L. Three dimensional, two phase flow mathematical model for PEM fuel cell: part I. model development. Energy Conversion Manage 2004;45(11–12):1861–82. [2] Hu M, Gu A, Wang M, Zhu X, Yu L. Three dimensional, two phase flow mathematical model for PEM fuel cell: part II. Analysis and discussion of the internal transport mechanisms. Energy Conversion Manage 2004;45(11–12):1883–916. [3] Jung H-M, Lee W-Y, Park J-S, Kim C-S. Numerical analysis of a polymer electrolyte fuel cell. Int J Hydrogen Energy 2004;29(9):945–54. [4] Nguyen PT, Berning T, Djilali N. Computational model of a PEM fuel cell with serpentine gas flow channels. J Power Sources 2004;130(1–2):149–57. [5] Siegel NP, Ellis MW, Nelson DJ, von Spakovsky MR. A two-dimensional computational model of a PEMFC with liquid water transport. J Power Sources 2004;128(2):173–84.

M.A.R.S. Al-Baghdadi / Renewable Energy 30 (2005) 1587–1599

1599

[6] Amphlett JC, Mann RF, Peppley BA, Roberge PR, Rodrigues A. A model predicting transient responses of proton exchange membrane fuel cells. J Power Sources 1996;61(1–2):183–8. [7] Lee JH, Lalk TR, Appleby AJ. Modeling electrochemical performance in large scale proton exchange membrane fuel cell stacks. J Power Sources 1998;70(2):258–68. [8] Ceraolo M, Miulli C, Pozio A. Modelling static and dynamic behaviour of proton exchange membrane fuel cells on the basis of electro-chemical description. J Power Sources 2003;113(1):131–44. [9] von Spakovsky MR, Olsommer B. Fuel cell systems and system modeling and analysis perspectives for fuel cell development. Energy Conversion Manage 2002;43(9–12):1249–57. [10] Siegel NP, Ellis MW, Nelson DJ, von Spakovsky MR. Single domain PEMFC model based on agglomerate catalyst geometry. J Power Sources 2003;115(1):81–9. [11] Mann RF, Amphlett JC, Hooper MAI, Jensen HM, Peppley BA, Roberge PR. Development and application of a generalised steady-state electrochemical model for a PEM fuel cell. J Power Sources 2000;86(1–2): 173–80. [12] Maxoulis CN, Tsinoglou DN, Koltsakis GC. Modeling of automotive fuel cell operation in driving cycles. Energy Conversion Manage 2004;45(4):559–73. [13] Fowler MW, Mann RF, Amphlett JC, Peppley BA, Roberge PR. Incorporation of voltage degradation into a generalised steady state electrochemical model for a PEM fuel cell. J Power Sources 2002;106(1–2): 274–83. [14] Berning T, Djilali N. Three-dimensional computational analysis of transport phenomena in a PEM fuel cell—a parametric study. J Power Sources 2003;124(2):440–52. [15] Chahine R, Laurencelle F, Hamelin J, Agbossou K, Fournier M, Bose TK, et al. Characterization of a Ballard MK5-E proton exchange membrane fuel cell stack. Fuel cells 2001;1(1):66–71. [16] Hamelin J, Agbossou K, Laperriere A, Bose TK. Dynamic behavior of a PEM fuel cell stack for stationary applications. Int J Hydrogen Energy 2001;26(6):625–9. [17] Cownden R, Nahon M, Rosen MA. Modelling and analysis of a solid polymer fuel cell system for transportation applications. Int J Hydrogen Energy 2001;26(6):615–23. [18] Rowea A, Li X. Mathematical modeling of proton exchange membrane fuel cells. J Power Sources 2001; 102(1–2):82–96. [19] Yerramalla S, Davari A, Feliachi A, Biswas T. Modeling and simulation of the dynamic behavior of a polymer electrolyte membrane fuel cell. J Power Sources 2003;124(1):104–13.