Figure 1 illustrates the multi-plate single cell of PEFC module (1D) used in this study. In the module, the separator’s back is the opposite side of the surface contacting the GDL. The separator’s back surface temperature T_surf,c and T_surf,a were measured using thermograph.

The heat transfer across the module is assumed to be in 1D direction only. In the module, the cell is divided into a gas channel and a rib part. In Figure 1, the upper and the lower parts represents rib part and channel part, respectively. For

both parts, the heat transfer was assumed to be in the through-plane direction. The reaction heat generated on the reaction surface is transferred to the cathode and anode sides separately. Although the gas flowing through the gas channel from the inlet to the outlet of the cell carries away some heat, the amount of heat taken is less than 1% of the estimated reaction heat of approximately 20 W [25] . Therefore, the heat carried away by the gas flow was neglected in this model. Additionally, the mass flow rate of gas flowing through the gas channel is very small ranging from 10⁻⁸ to 10⁻⁶ kg/s, resulting that the thermal conduction of gas in the gas channel is assumed since the gas is thought to be static.

The heat generation rate H_react as a reaction product is calculated as follows:

H react = E i − W E (1)

where E_i is the ideal (total) energy generation rate by the water formation from H₂ and O₂ based on higher heating value except the initial temperature of cell (T_ini) = 373 K. The lower heating value is adopted for T_ini = 373 K. W_E is the electric work generated by PEFC. E_i and W_E are expressed as follows:

E i = m H 2 × q HHV   or   q LHV (2)

W E = I × V (3)

where I is the load current obtained by the experiment (=20 A). In this study, the power generation data from a load current of 20 A (=0.80 A/cm²) were used for the heat transfer modeling. m H 2 is the molar flow rate of supplied H₂, which is equal to the ideal reaction consumption rate of H₂ required for the generation of 20 A, i.e., the stoichiometric ratio of 1.0. Here, the stoichiometric ratio is the ratio of the feed amount of H₂ or O₂ to that required to generate a current of 20 A. The flow rate of the supply gas (H₂) at the stoichiometric ratio of 1.0 is defined as follows.

m H 2 = I / n F (4)

where m H 2 is the molar flow rate of the supplied H₂ [mol/s], n is the valence of the ion (=2 for H₂) [−], and F is the Faraday constant (=96500) [C/mol]. m O 2 is the molar flow rate of the supplied O₂ [mol/s] and is calculated as follows:

H 2 + 1 / 2 O 2 = H 2 O (5)

The actual stoichiometric ratio of the supply gas was confirmed, using the mass flow controller installed at the inlet of the single cell and the mass flow meter installed at the outlet of the cell in the power generation experiment [26] .

The reaction heat at rib and channel are expressed by the following equations:

H rib , c = K rib , c A ( T react , rib − T surf , c ) / 2 (6)

H chan , c = K chan , c A ( T react , chan − T surf , c ) / 2 (7)

H rib , a = K rib , a A ( T react , rib − T surf , a ) / 2 (8)

H chan , a = K chan , a A ( T react , chan − T surf , a ) / 2 (9)

H react = H rib , c + H chan , c + H rib , a + H chan , a (10)

where A is the heat transfer area, which is the active are of MEA (i.e., power generation area = 0.0025 m²). The overall heat transfer coefficients K_rib,c, K_chan,c, K_rib,a and K_chan,a are defined as follows:

1 / K rib , c = d cat / k cat + d GDL / k GDL + d rib / k rib + d sep / k sep (11)

1 / K chan , c = d cat / k cat + d GDL / k GDL + d chan / k chan , c + d sep / k sep (12)

1 / K rib , a = d PEM / k PEM + d cat / k cat + d GDL / k GDL + d rib / k rib + d sep / k sep (13)

1 / K chan , a = d PEM / k PEM + d cat / k cat + d GDL / k GDL + d chan / k chan , a + d sep / k sep (14)

Table 1 lists the specification of cell components used in the model. In Table 1, the effective thermal conductivity of porous media k, are the values of the cell components used in the present experiment and in references [26] [27] . Since the effective thermal conductivities given in Table 1 are obtained when the cell component pores are filled with the air at room temperature, the corrected effective thermal conductivities are calculated for the cell components pores filled with H₂ or O₂ at 353 K or 363 K or 373 K, which were the T_ini value assumed in this study. In this calculation, the thermal conductivity of each gas is from The Japan Society of Mechanical Engineers [28] .

In order to solve Equations (6)-(9), the temperatures measured using the thermograph were substituted into these equations as T_surf,c and T_surf,a. The operation conditions used for power generation in order to measure temperatures with thermograph are given in Table 2.

Regarding a cathode gas, this study selects O₂. It can be expected that H₂, which is produced from a renewable energy via H₂O electrolyzer, will be used as a fuel for PEFC in order to realize a zero-CO₂-emission society in the near future. When H₂ is produced by H₂O electrolysis, O₂ is also produced as a by-product. This study suggests that not only H₂ but also O₂ produced from H₂O electrolysis are used for PEFC. This study also proposes that the total system consisting of renewable energy, H₂O electrolyzer, and PEFC system operated using H₂ and O₂ produced by H₂O elecrtrolyzer. Therefore, in this study, O₂ is adopted as the cathode gas for the numerical simulation. If O₂ was adopted as a cathode gas, a higher current density on the interface between PEM and the catalyst layer could be expected, especially under the rib, compared to the case using an air [29] .

Analysis using 1D model as well as 3D model is carried out by means of the data obtained under the conditions listed in Table 2. The experimental procedure for measuring temperature during the power generation has been explained in the reference [26] . In the heat transfer analysis, it was assumed that T_surf,c on the rib side was equal to T_surf,c on the channel side as well as T_surf,a because the difference between them could not be recognized by the measured data.

By the comparison of temperature distribution between in-plane and through-plane, the difference between T_react,rib and T_react,chan was found to be small, i.e., less than 1 K [30] [31] [32] . Consequently, it is believed that the heat flow in the through-plane direction dominates the heat transfer in the cell.

Considering the above described assumptions and Equations (6)-(14), the reaction surface temperature T_react is expressed as follows:

T react = T react , rib = T react , chan = { 2 H react / A + ( K rib , c + K chan , c ) T surf , c + ( K rib , a + K chan , a ) T surf , a }             / ( K rib , c + K chan , c + K rib , a + K chan , a ) (15)

In this study, the 3D numerical simulation has been conducted using a multi-physics software COMSOL Multiphysics. This software has the simulation code for PEFC composed of the continuity equation, the Brinkman equation for a momentum transfer, the Maxwell-Stefan equation for a diffusion transfer and Butler-Volmer equation for an electrochemical reaction. This simulation code for PEFC has been validated well by many previous studies [18] [33] [34] .

The continuity equation which considers the gas species in porous media, e.g. catalyst layer, MPL and GDL as well as the gas channel is expressed as follows:

∂ ∂ t ( ε p ρ ) + ∇ ⋅ ( ρ u ) = Q m (16)

where ε_p indicates the porosity [−], ρ indicates the density [kg/m³], u indicates the gas velocity vector [m/s], Q_m indicates the mass source term [kg/(m³∙s)] and t indicates the time [s].

Brinkman equation considering the relationship between the pressure and gas flow velocity, which is solved in porous media, e.g. catalyst layer, MPL and GDL as well as in gas channel, is expressed as follows:

ρ ε p { ∂ u ∂ t + ( u ⋅ ∇ ) u ε p } = − ∇ p + ∇ ⋅ [ 1 ε p { μ ( ∇ u + ( ∇ u ) T ) − 2 3 μ ( ∇ ⋅ u ) I } ] − ( κ − 1 μ + Q m ε p 2 ) u + F (17)

where p indicates the pressure [Pa], μ indicates the viscosity [Pa·s], I indicates the unit vector [−], κ indicates the permeability [m²] and F indicates the force vector [kg/(m²∙s)], e.g., gravity.

Maxwell-Stefan equation which considers the mass transfer such as the diffusion, ion transfer and convection transfer is expressed as follows:

N i = − D i ∇ C i − z i u m , i F C i ∇ φ l + C i u = J i + C i u (18)

∂ C i ∂ t + ∇ ⋅ N i = R i , tot (19)

where N i indicates the vector of the molar flow rate on the interface between PEM and electrode [mol/(m²·s)], D_i indicates the diffusion coefficient [m²/s], C_i indicates the concentration of ion i [mol/m³], z_i indicates the valence of ion [−], u_m,i indicates the mobility of ion i [(s∙mol)/kg], F indicates the Faraday constant [C/mol], φ_l indicates the electrical potential of liquid [35] [V], J i indicates the molar flow rate of the convection transfer [mol/(m²∙s)], and R_i,tot indicates the reaction rate of species [mol/(m³∙s)].

Butler-Volmer equation calculates the electrochemical reaction as follows:

i = i 0 { exp ( α a F η R T ) − exp ( − α c F η R T ) } (20)

η = φ s − φ l − E eq (21)

where i indicates the current density [A/m²], i₀ indicates the exchange current density [A/m²], α_a indicates the charge transfer coefficient at the anode [−], η indicates the activation over-potential [35] [V], R indicates the gas constant [J/(mol·K)], T indicates the temperature [K], α_c indicates the charge transfer coefficient at the cathode [−], φ_s indicates the electrical potential of solid [35] [V], E_eq indicates the equilibrium electric potential [35] [V].

Heat transfer equation considering electrical reaction is expressed as follows:

ρ C p u ⋅ ∇ T = ∇ ⋅ ( k ∇ T ) + Q jh + ∑ m a v Q e (22)

Q jh = − ( i s ⋅ ∇ φ s + i l ⋅ ∇ φ l ) (23)

Q e = ( η + T δ E eq δ T ) i (24)

where C_p indicates the specific heat [J/(kg∙K)], u is gas velocity [m/s], k indicates the thermal conductivity [W/(m∙K)], a_v indicates the activation specific area [1/m], i s indicates the current density vector in electrode [A/m²] and i l indicates the current density in electrolyte [A/m²].

Figure 2 illustrates 3D model of single cell of PEFC for the numerical simulation used in this study [20] [34] . This structure follows the commercial single cell used in the experimental studies carried out by the authors [36] [37] . The roof of the gas separator at anode side and cathode side is omitted in this model. The cell has a gas separator with a serpentine flow channel consisting of five gas channels with the width of 1.0 mm and the depth of 1.0 mm as well as a rib with the width of 1.0 mm. The size of cell components listed in Table 1 is adopted for this numerical simulation. Operation conditions listed in Table 2 are also adopted for this numerical simulation. Table 3 lists physical parameters adopted for this numerical simulation. To investigate and compare the distribution of T_react between 1D heat transfer model and 3D numerical simulation, this study selects the analysis points of A to K as shown in Figure 3. The average value on the cross sectional area of the interface between PEM and cathode catalyst layer at each point, which covers the part under gas channel and that under rib, is calculated.

This study set the following assumptions [20] [34] .

1) The distributions of the inlet gas flow rate at the anode side and the cathode side are uniform, respectively.

2) The pressure of the outlet of the gas channel is the atmospheric pressure.

3) No slip on the gas channel wall excluding the inlet and the outlet of the gas channel is considered.

4) The cell voltage obtained by the power generation experiment is set at the cathode electrode and the earth ground in set at the anode electrode. The in-plane distribution of cell voltage at the cathode electrode is uniform.

5) Reactant gases are treated as an ideal gas and incompressible Newton fluid.

6) H₂O is treated as a vapour.

7) The cell temperature is uniform and the outside boundary of the 3D model is set at T_ini.

8) The effective porosity and the permeability of the porous media are isotropic. The conductivity in the porous media is also isotropic.

The impacts of T_ini and RH of supply gas on the distribution of T_react have been investigated by 1D model and 3D model. The impacts of T_ini and RH of supply gas on the distributions of H₂, O₂, H₂O and current density have been also investigated by 3D model. In this paper, we focus on the distributions of O₂, H₂O and current density as well as distribution of T_react exhibited on the reaction surface, which are shown later.

Figure 4 and Figure 5 show distributions of T_react calculated by 1D heat transfer

model and 3D numerical simulation at T_ini = 353 K, respectively. Figure 6 and Figure 7 show distributions of T_react calculated by 1D heat transfer model and 3D numerical simulation at T_ini = 363 K, respectively. Figure 8 and Figure 9 show distributions of T_react calculated by 1D heat transfer model and 3D numerical simulation at T_ini = 373 K, respectively. In these figures, RH of supply gas is changed.

According to Figures 4-9, it is seen that the change in T_react from the inlet to the outlet is more even with the increase in T_ini irrespective of the investigated model. Since the saturation pressure of H₂O vapour increase with the temperature exponentially [44] , it is easy to dehydrate PEM at higher temperature. In other words, it is easy to decrease the proton conductivity of PEM at higher temperature. As a result, the power generation performance is dropped at higher temperature due to large ohmic loss, resulting in the lower generated heat. Therefore, the change in T_react from the inlet to the outlet is more even with the increase in T_ini. Since this study has set the excess gas which is over s.r. = 1.0 as the inlet gas flow rate, the generated heat is accumulated along with the excess gas flow through the gas channel. Consequently, it is thought T_react increases from the inlet to the outlet largely, especially at T_ini = 353 K since the power generation performance is better.

Comparing the results obtained by 1D heat transfer model with those obtained by 3D numerical simulation, the temperature gas between them is below approximately 0.5 K. Therefore, it can be claimed that 1D heat transfer model predicts the distribution of T_react well even though we think the heat transfer in single cell of PEFC only. This study has calculated the amount of heat taken by the gas flow along through the gas channel from the inlet to the outlet of the cell from the results obtained by 3D numerical simulation, resulting that it is approximately 0.01% of the heat generated. Therefore, it is thought that 1D heat transfer model can predict the distribution of T_react well. However, the conditions validated by this study are 353 K, 363 K and 373 K only. In the near future, this study will validate under the other operation condition to verify the accuracy of 1D Heat Transfer Model proposed by the authors. In the following section, we discuss the other distributions to clarify the phenomena.

Figure 10 shows distributions of molar concentration of O₂ calculated by 3D

numerical simulation at T_ini = 353 K, 363 K and 373 K, respectively. In this figure, RH of supply gas is changed.

According to Figure 10, it is seen that the change in the molar concentration of O₂ from the inlet to the outlet is more even with the increase in T_ini. It is known from the previous studies [45] [46] that the proton conductivity of PEM increases with the increase in temperature as well as the increase in RH. On the other hand, the saturation pressure of H₂O vapour increases with the temperature exponentially [44] , resulting that it is easy to dehydrate PEM at T_ini = 373 K compared with T_ini = 353 K. As a result, the proton conductivity of PEM decreases at T_ini = 373 K. If the proton conductivity of PEM decreases, the performance of the O₂ reduction reaction drops by the lack of proton. Since the hydration of PEM is not enough at T_ini = 373 K, the high O₂ partial pressure is needed to progress the O₂ reduction reaction [44] . Therefore, the change in the molar concentration of O₂ from the inlet to the outlet decreases with the increase in T_ini.

Figure 11 shows distributions of molar concentration of H₂O calculated by 3D numerical simulation at T_ini = 353 K, 363 K and 373 K, respectively. In this figure, RH of supply gas is changed.

According to Figure 11, it is seen that the change in the molar concentration of H₂O from the inlet to the outlet is more even with the increase in T_ini. As discussed above, the proton conductivity of PEM increases with the increase in temperature as well as the increase in RH [45] [46] . On the other hand, since the saturation pressure of H₂O vapour increases with the temperature exponentially [44] , it is easy to dehydrate PEM at T_ini = 373 K compared with T_ini = 353 K. Therefore, the proton conductivity of PEM decreases at T_ini = 373 K. If the proton conductivity of PEM decreases, the performance of O₂ reduction reaction drops by the lack of proton. Moreover, the dehydration of PEM is not enough at T_ini = 373 K, resulting that high O₂ partial pressure is needed to progress the O₂ reduction reaction [44] . Consequently, it can be claimed that the change in the molar concentration of H₂O from the inlet to the outlet becomes more even with the increase in T_ini. According to Figures 4-9, it is observed that the change in T_react from the inlet to the outlet is more even with the increase in T_ini. Since the generated heat from the power generation is lower with the increase in T_ini due to the lower performance of O₂ reduction reaction [47] , it can be thought that the change in T_react from the inlet to the outlet is more even with the increase in T_ini.

It can be seen from Figure 11 that the change in the molar concentration of H₂O from the inlet to the outlet for A40%RH, C40%RH is more even compared with the other RH conditions. Since A40%RH, C40%RH is the dry condition, PEM and catalyst layer are dehydrated easily [48] . The proton conductivity of PEM is smaller under a dry condition [49] . In addition, the RH influences the performance of O₂ reduction reaction carrying out on the ionomer in the catalyst layer at the cathode [40] . Therefore, there is the optimum H₂O saturation for ionomer in the catalyst layer at cathode [21] , indicating that the performance of O₂ reduction reaction which produces H₂O is lower for A40% RH, C40% RH. On the other hand, it is seen from Figures 4-9 that the distribution of T_react for A40%RH, C40%RH is relatively even compared to the other RH conditions.

Since the generated heat from the power generation is lower for A40%RH, C40%RH due to the lower performance of O₂ reduction reaction [47] , it can be thought that the distribution of T_react for A40%RH, C40%RH is more even.

Figure 12 shows distributions of current density calculated by 3D numerical

simulation at T_ini = 353 K, 363 K and 373 K, respectively. In this figure, RH of supply gas is changed.

According to Figure 12, it is seen that the change in the current density from the inlet to the outlet is more even with the increase in T_ini and the value of current density is smaller with the increase in T_ini. As discussed above, the proton conductivity of PEM decreases with the increase in T_ini, resulting that the performance of O₂ reduction reaction drops due to the lack of proton. In addition, the hydration of PEM is not enough at high temperature, high O₂ partial pressure is needed to progress the O₂ reduction reaction [44] . Consequently, it can be claimed that the current density decreases with the increase in T_ini due to the increase in the ohmic over-potential and the concentration over-potential [44] .

According to Figures 4-9, it is observed that the change in T_react from the inlet to the outlet is more even with the increase in T_ini. Since the generated heat from the power generation is lower with the increase in T_ini due to the lower performance of O₂ reduction reaction [47] , it can be thought that the change in T_react from the inlet to the outlet is more even with the increase in T_ini.

From the investigation of this study, we can claim that it is necessary to control the hydration of PEM and catalyst layer in order to obtain the high power generation performance at higher temperature such as 363 K and 373 K. As a procedure to control the hydration of PEM and catalyst layer under higher temperature operation condition, this study suggests recirculating the H₂O which is emitted from the cell and promoting the heat transfer in order to cool the cell. This study would like to investigate these trials in the next step.