Wei Zidong, Tan Jun, Zhang Shengtao, Chen Changguo, Huang Wenzhang , Wang Quntao, Tang Zhiyuan^# , Guo Hetong^#
(Department of Applied Chemistry, Chongqing University, Chongqing , 400044;  ^#Department of Applied Chemistry, Tianjin University, Tianjin , 300072 , China)

Received January 16, 2000; Supported by the National Natural Science Foundation of China. (Grant No.29976047)

Abstract A model of the air electrode used in PAFC was developed. The model can characterize the behavior of a PAFC air electrode at a wide range of current densities and at any operation gas pressure. The two parameters AB and AI were introduced into the model, The parameter AB, mainly reflecting the effects of the interfacial surface area between gas and liquid phases on the air electrode performance, was proved to has a different linear relationship with the current density at a different gas pressure. With the increase of operation pressure, AB gets to become less dependent of current density I. The parameter AI, mainly reflecting the effects of the interfacial surface area between liquid and solid phases on the air electrode performance, was proved to keep nearly unchanging at any current density and gas pressure. The study also showed that the reaction rate was not uniform across the reaction layer.
Keywords air electrode, PAFC, mathematical model.

The phosphoric acid fuel cell (PAFC) is capable of providing a clean, efficient, and high-powered source of electric energy. Its relative mature in technique has made PAFC an attractive commercial prospect. PAFC has undergone many advances in past years due to improved electrocatalysts and electrode structure, and also due to optimization of operating conditions and cell design. For example, improved electrocatalysts has led to the dramatic reduction of platinum loading from early 10 mg/cm² to current 0.5 mg/cm². The PAFC operating under a high pressure and a high temperature has displayed a higher power output. After several decades of effort PAFC is approaching commercialization. The founding of an 11 MW PAFC plant at Goi. (Japan) in 1991 for a prototype demonstrator marked the beginning of this kind of fuel cells commercialization ^[1,2]. However, for its complete commercialization, further capital cost reduction is certainly required. The performance of PAFC also needs to be improved.
    Mathematical modeling of fuel cells can supply a convenient approach to assess the influence of electrode structure parameters and operating conditions on the various types of polarization, which consequently affect the performance of a fuel cell. The 'uniform electrolyte film' concept, a widely accepted physical model of porous electrodes was first proposed by Cutlip ^[3]. Afterwards, Tian proposed a 'uneven-uniform electrolyte film' model ^[4]. Cha and his coworkers also established a formula reflecting the relationship between the current density and overpotential ^[5]. Recently, Vidts and White tried to give a general governing mathematical model for any porous electrode in an imaginary binary solution based on so-called 'volume-averaging technique'^[6]. Almost all models established to date took the surface areas between two phases i.e., gas/liquid and liquid/solid as two constants. However, Zhou's experiments ^[7] showed that the electrolyte films in Teflon-bonded porous electrodes got thinning with the increase of overpotential. The experiments in the authors' laboratory also confirmed Zhou's conclusion. This phenomenon indicates that the conventional view on the interfacial surface area between two phases is questionable .The present model is distinct from the previous ones in that the interfacial surface area between gas and liquid is no longer taken as a constant for different current densities. The relationship between the parameter AB and current densities is formulated in the present paper. The reaction rate throughout the reaction layer of PAFC air electrode is analyzed using the proposed model. The conclusions from the numerical analysis of the model are novel and worth deep consideration.

1. Mathematical Model for PAFC Cathode
An improved model based on Cutlip's^[3] was presented by the authors previously ^[8,9] .The parameters of the air electrode structure and operation conditions on which Cutlip's model was established deviated from the actual state of the current PAFC. For instance, the pore size in the Teflon-bonded catalyst layer is much larger than that adopted by Cutlip. Even though the pore size of catalyst layer has a wide distribution, the micro-pores in the catalyst layer are thought to be filled by electrolyte because they have much stronger capillary force than macro-pores, which are tunnel of gas diffusion. The size of macro-pores is so large that the Knudsen diffusion can be ignored ^[10]. Besides Cutlip's model didn't involve the operation condition in which the gas pressure was over one atmosphere. Therefore, Cutlip's model did not correctly demonstrate the real situation of today's PAFC. The latest developments in oxygen reduction research and the structure parameters of PAFC in operation were taken into accounts in construction of the present model.
    The main assumptions on which the model was based are as follows:
    (a) Kundsen diffusion of multi-component molecular diffusion is not considered either in the gas diffusion layer or in the reaction layer.
    (b) The drop of multi-component pressure within gas diffusion layer is negligible according to the calculation of Ref [8].
    (c) The gas-diffusion layer is described by a homogeneous continuum consisting of gaseous-filled layer and solid electrode framework. Similarly the reaction layer can be described by superimposing the gaseous-filled and liquid-filled pores with solid electrode framework.
    (d) The potential distribution within the air electrode is not included into the model because it is almost unchangeable throughout the reaction layer ^[3].
    A schematic of the PAFC cathode is shown in Fig .1, which also displays the process occurring in the cathode.

Fig.1 Schematic of air electrode of PAFC

    A one-dimensional mathematical model of the reaction layer of PAFC cathode can be written in the following differential group ^[8,9].
dP₁ /dZ = ( RTN₁ / P ) ( -2 P₁ / D₁₂ - P₂ / D₁₂ P₃ / D₁₃ )                                    (1)
dP₂ /dZ = ( RTN₁ / P ) ( 2 P₁ / D₁₂ - P₂ / D₁₂ + 2 P₃ / D₂₃ )                               (2)
dP₃ /dZ = ( RTN₁ / P ) ( 1 / D₁₃ 2 / D₂₃ )                                                            (3)
dN₁/dZ = - P₁ / ( 4F / ( a ^l i⁰ exp ( 2.3 (E_r E ) /T_s )) + d/a^g D^l C₁⁰ )                    (4)
The boundary conditions for the above the above each equation are
P_i = P_i ( 0 ) (i = 1 , 2 , 3)                                                                                     (5)
Ni (0 ) =0                                                                                                            (6)
Where                                                                                          (7)
D_ij=D_ij^r ( T / T ^r ) ^1.75 ( P ^r / P) e_c^g / t_c ^[10]                                                           (8)

2. SOLUTION OF THE MODEL  
The critical structure parameters which determine the performance of porous electrodes are the interfacial surface area between gas and liquid phase, i.e. a^g , and the area between liquid and solid phrase , i.e. a^l . They were thought unchangeable previously. For example , Kimble and White took a^l times exchange currents ( i⁰ ) as 0.60 A cm^-3 , and a^g divided by diffusion layer thickness of dissolved oxygen in electrolyte film (d) as 5^.10⁸ cm^-2[11]. Actually the above-mentioned parameters cannot be independently determined from experiments and therefore must be obtained from the other experimental data. The curve fitting or more advanced optimization method is a common approach to solve the uncertain parameters. In order to minimize the number of fitted parameters and avoid using questionable values of some parameters, several uncertain parameters are merged into the following two parameters:
AB = a ^g D ^l C₁⁰ / d                                                                                             (10)
AI = a ^l i ⁰                                                                                                             (11)
Then the equation (4) has a new form (12)
dN₁/dZ = - P₁ / ( 4F / ( AI exp ( 2.3 (E_r E ) /T_s )) + 1 / AB )                                (12)
Finally, the four governing equations for the reaction layer of PAFC cathode are Eq. (1), (2), (3) and (12).

Fig.2 Phosphoric acid fuel cell-performance for a range of pressures ( at start of test )
(1)Atmospheric pressure, 190°C, 0.5mg/cm² Pt on carbon  (2) 3^.10⁵Pa, 190°C, 0.5mg/cm² Pt on carbon
(3) 6^.10⁵Pa, 210°C, 0.5mg/cm² Pt on carbon  (4) 8^.10⁵Pa, 220°C, 0.5mg/cm² Pt on carbon

    A computer program TMSTB purchased from the Computer Center of Academia Sinica was used to solve the differential equations group. An accelerated method called Simplex [12] was used to search the optimum values of parameter AB and AI. This optimum method can automatically expand and contract the searching step then greatly accelerates the searching rate of the Simplex. The objective function G is defined as
                                                                                 (13)
Where, I_k and I_{k, model} are the current densities from the experiments as shown in Fig.2 and from the calculation of the model respectively. The current density I_{k , model} is calculated from N₁, i.e. ,
I_{k , model} =N₁ (0) / 4F                                                                                        (14)
The magnitude of G also reflects the deviation of the model from the experiment .The less of G in value, the closer the model to experiments.
    The subroutine program for optimization of AB and AI is named SUBROUTINE OPT. The block diagram of computing is shown in Fig .3

Fig .3 The block figure of the model calculating program

3. RESULTS AND DISCUSSION
The operating conditions and the cathode potential E were substituted into the mathematical model to calculate the current densities I_{k , model} .
    If AB and AI were taken as a single value respectively in all range of current densities to meet the demand of Eq. (13), there was a great deviation between I_{k , model} and I_k although the results predicted by the model followed the tendency of experiments by and large as shown in Table 1.
    Table 1 Comparison between the experimental I_k   and the model predicted I_{k , model} as AB and AI keep constant in all range of current densities

    Based on the fact that the electrode behavior is always highly dependent on the current density, the scale of current densities was divided into two sections to fit the Eq. (13), one from zero to 100 mA^.cm^-2 and the other from 100 mA^.cm^-2 to 300 mA^.cm^-2. The parameters AB and AI were respectively optimized at the two sections. The difference between I_{k , model} and I_k was markedly reduced. Meanwhile, the value of AB was quite different at the two sections. Based on this fact, parameters AB and AI were optimized point by point along experiment E-I curve. It was surprised to learn that the parameter AB was completely linear with the current density at each operating gas pressure. While parameter AI was nearly free of the current density. In fact, parameter AI oscillated only in the range of (4.2 ± 5%) ^.10 ^-2 (A^.cm^-3).
    As parameters AB and AI were required only to satisfy the minimum of G for one point, i.e.. m=1 in Eq. (13) . The Eq. (13) can be easily satisfied with a very low value of G. Mostly, G is less than 1^.10 ^-14 . It means that the current density from experiments is nearly equal to that from the model. Consequently, the foregoing assumption resulting in this result is probably reasonable.
It had been found that parameter AB and current density I had the following relationship:
AB = l + w I          (15)
The relationships between AB and I are shown in Fig.4.

Fig4. The relation of AB and AI at a range of the pressure (1) 1^.10⁵Pa (2) 3^.10⁵Pa (3) 6^.10⁵Pa (4) 8^.10⁵Pa
    From the definition of parameter AB, i.e. AB = a^g DC₁⁰ /d , the variation of AB with the current density actually reveals the relationship between a^g /d and the current density. The phenomenon that the parameter AB increases with the current density means an increase of the interfacial surface area between gas and liquid phase ( a^g ) and/or a decrease of the thickness of the electrolyte film (d) . This novel conclusion had been already contested by Zhou and Poorten^[7] . In light of Kimble and White's research^[11], the three-phase electrode should contain a large number of gas-liquid sites. It can be easily drawn from Fig.4 that the high current density is benefit for a high-energy output of a porous electrode. The data related to gas pressure is not sufficient enough for deducing a certain relationship between parameters AB and the gas pressure. However, the decreasing tendency of AB at a given current density with the gas pressure can also be disclosed in Fig.4. This phenomenon suggests that the high gas pressure will reduce the interfacial surface area between gas and liquid phase, although it can increase the reactant gas concentration in catalyst surface. The gas pressure has a contrary effect on the electrode performance. The extra-high pressure is not always positive for achieving a high-energy output. It can also get a conclusion from Fig.4 that AB gets to become less dependent of current densities with the increase of operation pressure. This is coincident with the conventional view, i.e., the interfacial surface area between gas and liquid is a parameter free of current densities.

Fig5. The relative reaction rate on the catalystlayer at the current density of 100mA^.cm^-2 at the pressure of 3^.10⁵Pa

    The flux distribution of oxygen across the catalyst layer can be converted into an index indicating the electrochemical reaction rate across the dimensionless reaction layer. The reduction of N₁ in magnitude from Z_i to Z_i + DZ, i.e. DN₁ (Z_i) equals the electrochemical reaction rate in the thickness of DZ. The relative reaction rate (RRR) i.e., the ratio of DN₁ (Z_i) to the Max {DN₁ (Z_i)} vs. (Z/L_c) are demonstrated in Fig .5, from which it can be observed that the reaction rate is not uniform throughout the reaction layer. The results also display that the two maximum points appear at Z / L _c = 0.3 and Z / L _c = 0.9 respectively.

3. CONCLUSION  
A mathematical model of an air electrode used in PAFC has been developed. The parameters AB and AI characterizing the three-phase electrode are introduced into the model. When AB and AI are taken as constants from zero to 300 mA^.cm ^-2 of current density, the relative error of the current density predicted by the model will be up to 17 % relative to that by the experiments. When parameters AB and AI are fitted below and above 100 mA^.cm ^-2 respectively, this error will drop to less than 5 %. This means that the interfacial area between gas and liquid phases is highly relative to current densities. Furthermore, the AB and AI are fitted at each single current density. The above-mentioned error is nearly negligible. It is also found that parameter AB is proportional to the current densities at a given operation pressure, but it gets to become less dependent on current densities with the increase of operation pressure. Parameter AI just oscillates in a very narrow range for all current densities. At a lower operation pressure, the increase of parameter AB with the current density indicates that the high current density leads to more efficient utilization of air electrodes. At a higher operation pressure, however, the increase of current densities dose not play the same role as it dose at a lower operation pressure

LIST OF SYMBOLS
a^g       interfacial surface area between gas and liquid phase , cm^2.cm^-3
a^l       interfacial surface area between liquid and solid phase , cm^2.cm^-3
AB    parameter defined by Eq .(10) , mol^.cm^-3. s^-1
AI     parameter defined by Eq .(11) , A^. cm^-3
C₁⁰   solubility of oxygen in the electrolyte at 1.103×10⁵ Pa ,mol^. m^-3. Pa^-1
D_l     diffusion coefficient of oxygen dissolved in the electrolyte , m^2. s ^-1
D_ij    effective gas diffusion coefficient for component i and j at the given pressure , m^2. s ^-1
D_ij^r   effective gas diffusion coefficient for component i and j at the reference state, m^2. s ^-1
E      electrode potential , V
E⁰   open circuit potential for oxygen electrode , V
F      faraday's constant , 96,487 C^. mol^-1
G     function defined by Eq.(13)
I      current density , A^. cm ^-2
I_k     current density from experiments , A^.cm ^-2
I_k,model     current density calculated from the model , A^.cm ^-2
i⁰      exchange current density , A^.cm^-2
N₁    flux of oxygen , mol^.cm ^-2.s ^-1
P      total pressure in the gas stream and porous backing , Pa
P_i     partial pressure of component i , Pa
r      relevant coefficient
R    gas constant , 8,314 J^. mol ^-1. K ^-1
RRR     relative reaction rate, i.e., the ratio of DN₁ (Z_i) to the Max{DN₁ (Z_i) }
T     temperature , K
T^r    reference temperature , K
T_s   tafel slope , V/decade
Z     spatial coordinate , cm
d     thickness of electrolyte film , cm
e     porosity
t     tortuosity
l     parameter in Eq . (15) , mol^.cm ^-3. s ^-1
w    parameter in Eq . (15) , cm ^-1
    Superscripts
g     gas phase
l      liquid phase
r     reference
    Subscripts
c     catalyst layer
d     gas diffusion layer
i      species i , i = 1 ,2 ,3 corresponding to O₂ , H₂O (g) , N₂ respectively

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