A Dynamic Tanks-in-Series Model for a High-Temperature PEM Fuel Cell

Abstract

A dynamic tanks-in-series model has been developed for the coupled heat, mass, and charge transfer processes in a high-temperature proton exchange membrane fuel cell. The semi-empirical model includes the heat and mass balance equations in the gas channels and the membrane electrode assembly together with the charge balance at the electrode/membrane interfaces. The outputs of the tanks-in-series model are the concentration, the temperature, and the current density with a step change from tank to tank. The dynamic non-isothermal model is capable of predicting both the transient and steady-state behavior of the fuel cell and reproducing impedance data under harmonic perturbations of the cell potential together with a comprehensive interpretation of experimental data.

1. Introduction

A fuel cell system with a high-temperature proton exchange membrane fuel cell (HT-PEMFC) is identified as a system with high system efficiency, simplified water management, high intensity of electrochemical reactions, simplified system flowsheets, and high tolerance to carbon monoxide (CO) content in the reformate feed [1,2,3]. Fuel processing is a component of a fuel cell system, which is used for the conversion of fuel (hydrocarbons) into hydrogen-containing gas mixtures and further electricity generation in fuel cells. The application of computational fluid dynamics (CFD) models can significantly accelerate the design and commercialization of fuel cell systems. Modeling is a valuable computational tool for the in situ evaluation of transfer processes and the investigation of the impact of different parameters on fuel cell performance. A considerable body of literature has been published on the development of the HT-PEMFC model, with different degrees of complexity, including zero-dimensional and three-dimensional case studies. A zero-dimensional process model is used for the evaluation of the fuel cell performance, utilizing a voltage equation [1,2,3]. A one-dimensional model [4,5] is used for describing the coupled mass and charge transfer within the gas diffusion and catalyst layer in an HT-PEMFC. A typical three-dimensional model for an HT-PEMFC describes the coupled momentum, mass, heat, and charge transfer processes in gas channels, the gas diffusion layer, and the membrane electrode assembly [6,7,8,9,10,11,12]. Kvesić and colleagues [13] experimentally and numerically investigated HT-PEFC stack operation. They validated the predicted current density and temperature profiles within the stack using the measured local current and temperature profiles.

Using a dynamic fuel cell model, impedance analysis provides a major advance in interpreting experimental measurements and linking equivalent circuit resistances and coupled mass and charge transfer processes. A pseudo-two-dimensional isothermal dynamic model for an HT-PEM fuel cell incorporates charge and mass transfer within the catalyst layer, gas diffusion layer, and gas channel on the anode and cathode sides [14,15]. Vivona et al. [16] obtained an analytical solution for the impedance of the HT-PEMFC cathode with pseudo-two-dimensional dynamic charge and mass transfer in the gas channel, diffusion layer, and cathode catalyst agglomerates. A simplified two-dimensional isothermal dynamic model [17] is applicable for the impedance of an HT-PEMFC due to capturing the coupled charge and mass transfer processes on the cathode.

The models cited above offer a wide range of transfer process descriptions using three-dimensional, two-dimensional, and one-dimensional approaches, apart from a semi-empirical model. There are several tanks-in-series models for low-temperature PEM fuel cells (LT-PEMFCs), electrochemical hydrogen pumps, and solid oxide fuel cells (SOFCs) in the literature [18,19,20,21,22,23]. The objective of this research is to develop a tanks-in-series model for predicting the concentration, temperature, and current density profiles in an HT-PEMFC under potentiostatic mode. The main advantage of the semi-empirical model is that it can adequately predict fuel cell performance with much less computational effort than other models described in the literature. In addition, the dynamic model is also suitable for the analysis of electrochemical impedance spectra.

2. Tanks-in-Series Model for HT-PEMFC

The continuous operation of tanks-in-series reactors (TSRs) in a network configuration enables the approximation of the flow patterns within a planar high-temperature polymer electrolyte fuel cell (HT-PEMFC) with a parallel flow field design. For the co-current fuel and oxidant flow, the tanks-in-series model is composed according to the diagram in Figure 1. The anode stream is represented as a pseudo-three-component reformate flow, including hydrogen (H₂), carbon dioxide (CO₂), and vapor water (H₂O). The cathode feed stream consists of oxygen (O₂), nitrogen (N₂), and vapor water (H₂O). Each tank contains a gas channel and a gas diffusion layer volume on the cathode and anode sides. The following assumptions have been used for the HT PEMFC tanks-in-series model development:

(i)

There is complete mixing in a tank with a step change in variables from tank to tank;

(ii)

Anode electrochemical reaction occurs at the anode electrode/membrane interface;

(iii)

Cathode electrochemical reaction occurs at the cathode electrode/membrane interface;

(iv)

There is uniform gas flow distribution in the anode and cathode channels and there are pseudo-three-component reformate gas flows in the anode channels (H₂ + CO₂ + H₂O);

(v)

There is a Chilton–Colburn analogy between heat and mass transfer.

For the control volume of the tank (V_j), dynamic material and heat balances are defined as follows [24]:

$$V_{\mathrm{j}}\frac{\mathrm{d}{C}_{\mathrm{j}}}{\mathrm{d}t}={\sum \left(m_{\mathrm{j}}\right)}_{\mathrm{i}\mathrm{n}}-{\sum \left(m_{\mathrm{j}}\right)}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$

(1)

$$\rho C_{\mathrm{p}}{V}_{\mathrm{j}}\frac{\mathrm{d}{T}_j}{\mathrm{d}t}={\sum \left(q_{\mathrm{j}}\right)}_{\mathrm{i}\mathrm{n}}-{\sum \left(q_{\mathrm{j}}\right)}_{\mathrm{o}\mathrm{u}\mathrm{t}}$$

(2)

Case studies of balance equations are listed in [18,19,20,21,22] for different PEM fuel cells. The Appendix A illustrates the development of the heat and mass balance equations for the tanks-in-series model of an HT-PEMFC.

2.1. Charge Balance

The charge balances are defined at the anode electrode/membrane and the cathode electrode/membrane interfaces within a membrane electrode assembly (MEA). Due to the different partial pressures of the species on the anode and cathode sides, there is a crossflow of the species across the membrane during the fuel cell operation. The crossflow of hydrogen creates a parasitic current on the cathode side, while the crossflow of oxygen across the membrane creates a parasitic current on the anode side. In consideration of the parasitic current, the charge balance equations are defined as follows:

$$C_{\mathrm{d}\mathrm{l}}^{\mathrm{A}}\frac{\mathrm{d}{\eta }_{\mathrm{j}}^{\mathrm{A}}}{\mathrm{d}t}=I_{\mathrm{m}\mathrm{e}\mathrm{m},\mathrm{j}}-(I_{\mathrm{j}}^{\mathrm{A}}+I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{O}}_2})$$

(3)

$$C_{\mathrm{d}\mathrm{l}}^{\mathrm{C}}\frac{\mathrm{d}{\eta }_{\mathrm{j}}^{\mathrm{C}}}{\mathrm{d}t}=-I_{\mathrm{m}\mathrm{e}\mathrm{m},\mathrm{j}}-(I_{\mathrm{j}}^{\mathrm{C}}+I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{H}}_2})$$

(4)

where I_mem,j—the membrane current density; $I_{\mathrm{j}}^{\mathrm{A}}$—the anode electrode current density; $I_{\mathrm{j}}^{\mathrm{C}}$—the cathode electrode current density; $I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{O}}_2}$—the anode parasitic crossover current density; $I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{H}}_2}$—the cathode parasitic crossover current density; ${\eta }_{\mathrm{j}}^{\mathrm{A}}$—the anode potential difference at the anode electrode/membrane interface; ${\eta }_{\mathrm{j}}^{\mathrm{C}}$—the cathode potential difference at the cathode electrode/membrane interface; C_dl—the double-layer capacitance. For steady-state operation, dη/dt = 0, as illustrated by transient profiles in Figure 2.

2.2. Electrode Current

The Butler–Volmer equation defines the overall rate of the anode electrochemical reaction:

$$I_{\mathrm{j}}^{\mathrm{A}}=(1-{\theta }_j^{\mathrm{C}\mathrm{O}}{)}^2{I}_0^{\mathrm{A}}\exp \left(-\frac{E_{\mathrm{a}}^{\mathrm{A}}}{R{T}_{\mathrm{j}}^{\mathrm{A}}}(1-\frac{T_{\mathrm{j}}^{\mathrm{A}}}{T_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{A}}})\right){\left(\frac{C_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{L},\mathrm{j}}^{{\mathrm{H}}_2}}{C_{\mathrm{r}\mathrm{e}\mathrm{f}}^{{\mathrm{H}}_2}}\right)}^{0.5}\left(\exp (\frac{{\alpha }_{\mathrm{A}}^{\mathrm{A}}F}{R{T}_{\mathrm{j}}^{\mathrm{A}}}{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{A}})-\exp (-\frac{{\alpha }_{\mathrm{C}}^{\mathrm{A}}F}{R{T}_{\mathrm{j}}^{\mathrm{A}}}{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{A}})\right)$$

(5)

where ${\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{A}}$—the anode activation overpotential, ${\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{A}}={\eta }_{\mathrm{j}}^{\mathrm{A}}-{\eta }_{\mathrm{e}\mathrm{q}}^{\mathrm{A}}$; $C_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{L},\mathrm{j}}^{{\mathrm{H}}_2}$—the molar concentration of hydrogen dissolved in liquid phase within the anode catalyst layer; $F$—Faraday’s constant; $R$—ideal gas constant; ${\theta }_{\mathrm{j}}^{\mathrm{C}\mathrm{O}}$—the surface coverage of carbon monoxide; $I_0^{\mathrm{A}}$—the anode exchange current density; $E_{\mathrm{a}}^{\mathrm{A}}$—the anode activation energy; $T_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{A}}$—the reference anode temperature; ${\alpha }_{\mathrm{A}}^{\mathrm{A}}$—the anodic charge transfer coefficients for the anode; ${\alpha }_{\mathrm{C}}^{\mathrm{A}}$—the cathodic charge transfer coefficients for the anode. The presence of carbon monoxide (CO) indicates the catalyst poisoning in the anode electrochemical reaction’s overall rate. An empirical expression for calculating the surface coverage (θ^CO) is provided in Appendix A. The anode parasitic crossover current ($I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{O}}_2}$) is calculated as a molar flux of oxygen from the cathode catalyst layer to the anode catalyst layer with a linear concentration profile in the membrane:

$$I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{O}}_2}=\frac{n_{\mathrm{e}}^{{\mathrm{O}}_2}}{{\nu }^{{\mathrm{O}}_2}}F\frac{K_{\mathrm{p}}^{{\mathrm{P}}_2}}{{\delta }_{\mathrm{m}\mathrm{e}\mathrm{m}}}(P_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{j}}^{{\mathrm{O}}_2}-0)$$

(6)

where $K_{\mathrm{p}}^{{\mathrm{O}}_2}$—the permeability coefficient of oxygen in the membrane; ${\nu }^{{\mathrm{O}}_2}$—the stoichiometry coefficient; $n_{\mathrm{e}}^{{\mathrm{O}}_2}$—the number of electrons; $P_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{j}}^{{\mathrm{O}}_2}$—the partial pressure of oxygen in the cathode catalyst layer. The overall rate of the cathode electrochemical reaction rate is

$$I_{\mathrm{j}}^{\mathrm{C}}=I_0^{\mathrm{C}}\exp \left(-\frac{E_{\mathrm{a}}^{\mathrm{C}}}{R{T}_{\mathrm{j}}^{\mathrm{C}}}(1-\frac{T_{\mathrm{j}}^{\mathrm{C}}}{T_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{C}}})\right)\frac{C_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{L},\mathrm{j}}^{{\mathrm{O}}_2}}{C_{\mathrm{r}\mathrm{e}\mathrm{f}}^{{\mathrm{O}}_2}}\left(\exp (\frac{{\alpha }_{\mathrm{A}}^{\mathrm{C}}F}{R{T}_{\mathrm{j}}^{\mathrm{C}}}{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{C}})-\exp (-\frac{{\alpha }_{\mathrm{C}}^{\mathrm{C}}F}{R{T}_{\mathrm{j}}^{\mathrm{C}}}{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{C}})\right)$$

(7)

where $I_0^{\mathrm{C}}$—the cathode exchange current density; $E_{\mathrm{a}}^{\mathrm{C}}$—the cathode activation energy; $T_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{C}}$—the reference cathode temperature; ${\alpha }_{\mathrm{A}}^{\mathrm{C}}$—the anodic charge transfer coefficients for the cathode; ${\alpha }_{\mathrm{C}}^{\mathrm{C}}$—the cathodic charge transfer coefficients for cathode; ${\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{C}}$—the cathode activation overpotential, ${\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{C}}={\eta }_{\mathrm{j}}^{\mathrm{C}}-{\eta }_{\mathrm{e}\mathrm{q}}^{\mathrm{C}}$; $C_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{L},\mathrm{j}}^{{\mathrm{O}}_2}$—the molar concentration of oxygen dissolved in liquid phase within the cathode catalyst layer. The cathode parasitic crossover current ($I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{H}}_2}$) is calculated as a molar flux of hydrogen from the anode catalyst layer to the cathode catalyst layer with a linear concentration profile in the membrane:

$$I_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s},\mathrm{j}}^{{\mathrm{H}}_2}=\frac{n_{\mathrm{e}}^{{\mathrm{H}}_2}}{{\nu }^{{\mathrm{H}}_2}}F\frac{K_{\mathrm{p}}^{{\mathrm{H}}_2}}{{\delta }_{\mathrm{m}\mathrm{e}\mathrm{m}}}(P_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{j}}^{{\mathrm{H}}_2}-0)$$

(8)

where $K_{\mathrm{p}}^{{\mathrm{H}}_2}$—the permeability coefficient of hydrogen in the membrane; ${\nu }^{{\mathrm{H}}_2}$—the stoichiometry coefficient; $n_{\mathrm{e}}^{{\mathrm{H}}_2}$—the number of electrons; $P_{\mathrm{c}\mathrm{a}\mathrm{t},\mathrm{j}}^{{\mathrm{H}}_2}$—the partial pressure of hydrogen in the anode catalyst layer. Siegel et al. [25] have provided empirical correlations for calculating the permeability coefficient in an HT-MEA.

2.3. Electrolyte Current

The ionic current is transferred from the anode electrode to the cathode electrode through the proton-conducting membrane due to the membrane potential difference. Assuming a linear membrane phase potential across the proton exchange membrane, the electrolyte current density is calculated using the following voltage equation:

$$I_{\mathrm{m}\mathrm{e}\mathrm{m},\mathrm{j}}=(E_{0,\mathrm{j}}-E_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}-{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{A}}+{\eta }_{\mathrm{a}\mathrm{c}\mathrm{t},\mathrm{j}}^{\mathrm{C}})/R_{\mathrm{m}\mathrm{e}\mathrm{m},\mathrm{j}}$$

(9)

where $R_{\mathrm{m}\mathrm{e}\mathrm{m},\mathrm{j}}$—the ohmic resistance, $R_{\mathrm{m}\mathrm{e}\mathrm{m},\mathrm{j}}={\delta }_{\mathrm{m}\mathrm{e}\mathrm{m}}/k_{\mathrm{m}\mathrm{e}\mathrm{m}}$. The theoretical potential (E_o,j) is a function of component activities and temperature at the catalyst layer, as shown in Equation (A10) in Appendix A. For the potentiostatic mode, the cell voltage (E_cell) is an input parameter. In their studies, Olapade et al. [25,26] presented the following empirical correlation for the ionic conductivity of the PBI membrane:

$$k_{\mathrm{m}\mathrm{e}\mathrm{m}}=\frac{a b}{T}\exp \left(-\frac{E_{\mathrm{a}}}{RT}\right)$$

(10)

where a, b—constants.

2.4. Electrochemical Reactions

The source term of the balance equation reflects the consumption or production of species by electrochemical reaction or mass transfer. The electrochemical reaction rate in the catalyst layer is

$$r_{\mathrm{j}}^{\left(\mathrm{k}\right)}=S_{\mathrm{j}}\frac{{\nu }^{\left(\mathrm{k}\right)}}{n_{\mathrm{e}}F}{I}_{\mathrm{j}} \mathrm{k}={\mathrm{H}}_2, {\mathrm{O}}_2, {\mathrm{H}}_2\mathrm{O}$$

(11)

where S_j—electrode area in j-tank; I_j—electrode current density in j-tank. Appendix A provides the rate expressions for the anode and cathode electrochemical reactions.

3. Electrochemical Impedance Model

Electrochemical impedance spectroscopy is an important technique for gathering information about the coupled charge and mass transfer processes in the fuel cell. Potentiostatic electrochemical impedance is defined as the current density response to a potential variation:

$$\mathsf{\delta }E=\mathsf{\delta }{E}_{\mathrm{m}\mathrm{a}\mathrm{x}}\sin \left(2\pi ft\right)$$

(12)

$$\mathsf{\delta }I=\mathsf{\delta }{I}_{\mathrm{m}\mathrm{a}\mathrm{x}}\sin (2\pi ft+\theta )$$

(13)

where f—the frequency; θ—the phase shift. Complex impedance is defined as

$$Z=\frac{\mathsf{\delta }{E}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathsf{\delta }{I}_{\mathrm{m}\mathrm{a}\mathrm{x}}}(\cos \left(\theta \right)+i \mathrm{s}\mathrm{i}\mathrm{n}(\theta ))$$

(14)

The above studies outline the critical role of describing the dynamic transfer processes taking into account the charge balance. A non-isothermal tanks-in-series model (1)–(11) can be used to simulate the harmonic potential perturbation and predict the cell current response as impedance data over a wide range of frequencies. For a time-dependent simulation, the harmonic potential perturbation is simulated using a sinusoidal cell voltage signal (12) with a small amplitude (δE_max = 0.005 V). Following Häffelin’s [27] time-domain technique, the signal is discretized with 100 timesteps per period. The final two periods are employed in the fitting of Equation (13) to the response current density signal (δI) and calculation of the impedance (Z). The steady-state solution of the model Equations (1)–(11) is used to specify the initial values for the time-dependent simulation. The visualization of the transient profiles is the main advantage of the time-domain technique, together with the supporting physical interpretation of the impedance spectrum. Figure 3 demonstrates the simulated cell voltage signal and the resulting time-domain response of the model (1)–(11). At the specified frequency of 25 Hz, there is a phase shift between the variables due to the coupled charge, mass, and heat transfer processes.

4. Results

A large number of experimental polarization curves are available in the literature for HT-PEMFCs operated with different fuel compositions, including pure hydrogen and reformate. A typical reformate is a multicomponent gas mixture containing hydrogen, water, carbon dioxide, carbon monoxide, methane, and nitrogen. The next step in the interpretation of experimental data is the simulation of the coupled heat, mass, and charge transfer processes. In the context of fuel cell operation under reformate, the gas mixture in the anode channel is represented as a pseudo-three-component system, including hydrogen, water, and inert species (as subsidies for carbon dioxide, carbon monoxide, and methane).

4.1. Modeling of Transfer Processes

For a typical high-temperature proton exchange membrane fuel cell with parallel channels (Figure 1), the input data for the tanks-in-series model include three categories of parameters: (i) geometric parameters, (ii) operating conditions, and (iii) electrochemical and mass transfer parameters. To validate the predictive capability of the developed tanks-in-series model, we utilized experimental data reported by Pinar et al. [28] and Rau et al. [29] for an HT-PEMFC operating with reformate at the anode and air at the cathode, with the specified operational conditions presented in

Table 1. Operating conditions for the HT-PEMFC.

Parameter	G1	G2	G3	G4	C1	C2	S1
H2 stoichiometric ratio	1.5	1.5	1.5	1.5	1.5	1.5	1.5
O2 stoichiometric ratio	2	2	2	2	2	2	6
Temperature (K)	433	433	433	433	433	433	433
Pressure (atm)	1	1	1	1	1	1	1
H2 mole fraction	1.00	0.78	0.54	0.54	1.00	0.33	1.00
CO2 mole fraction	0	0.22	0.15	0.14	0	0.2	0
H2O mole fraction	0	0	0.31	0.31	0	0.46	0
CO mole fraction	0	0	0	0.01	0	0.01	0
Reference	[28]	[28]	[28]	[28]	[29]	[29]	[17]

. Authors [28,29,30] successfully demonstrated the operation of a high-temperature proton exchange membrane fuel cell with a commercial high-temperature membrane electrode assembly (HT-MEA) under wet reformate feed conditions.

Table 2. Geometric and electrochemical parameters for the HT-PEMFC.

Parameter	Units	Symbol	G1–G4	C1,C2	S1
Reference exchange current	(A m−2)	$I_0^{\mathrm{A}}$	1.44 × 103	1.44 × 103	1.44 × 103
Activation energy	(J mol−1)	$E_{\mathrm{a}}^{\mathrm{A}}$	1.69 × 104	1.69 × 104	1.69 × 104
Reference temperature	(K)	$T_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{A}}$	433	433	433
Reference concentration	(mol m−3)	$C_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{A}}$	3.93 × 10−3	3.93 × 10−3	3.93 × 10−3
Charge transfer coefficient		${\alpha }_{\mathrm{A}}^{\mathrm{A}}$	1	1	1
Charge transfer coefficient		${\alpha }_{\mathrm{A}}^{\mathrm{C}}$	1	1	1
Reference exchange current	(A m−2)	$I_0^{\mathrm{C}}$	7.89 × 10−4	7.89 × 10−4	7.89 × 10−4
Activation energy	(J mol−1)	$E_{\mathrm{a}}^{\mathrm{C}}$	7.24 × 104	7.24 × 104	7.24 × 104
Reference temperature	(K)	$T_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{C}}$	423	423	423
Reference concentration	(mol m−3)	$C_{\mathrm{r}\mathrm{e}\mathrm{f}}^{\mathrm{C}}$	1.5 × 10−3	1.5 × 10−3	1.5 × 10−3
Charge transfer coefficient		${\alpha }_{\mathrm{C}}^{\mathrm{A}}$	1	1	1
Charge transfer coefficient		${\alpha }_{\mathrm{C}}^{\mathrm{C}}$	1	1	1
Channel height	(m)	h	1 × 10−3	1 × 10−3	1 × 10−3
Electrode area	(m2)	S	25 × 10−4	21 × 10−4	5 × 10−4
Membrane width	(m)	${\delta }_{\mathrm{m}\mathrm{e}\mathrm{m}}$	60 × 10−6	40 × 10−6	60 × 10−6
Catalyst width	(m)	${\delta }_{\mathrm{c}\mathrm{a}\mathrm{t}}$	20 × 10−6	20 × 10−6	50 × 10−6
Double layer capacitance	(mF cm2)	Cdl	150	150	150

presents the geometric and electrochemical characteristics of an HT-PEMFC utilizing a commercially available PBI membrane.

The number of tanks (N_TSR) is defined by gas flow patterns within the channels. Boillot et al. [31] experimentally studied the gas flow patterns in a parallel flow field design under a laminar regime and determined the number of tanks from the experimental residence time distribution function. For the specified operating conditions in Table 1, the number of tanks, N_TSR, calculated from the empirical correlation presented in Appendix A is consistent with the experimental value (N_TSR = 4) reported by Boillot et al. [31]. Using the Chilton–Colburn analogy between heat and mass transfer, the same number of tanks can be used in the development of mass and heat balance equations for the laminar gas flow in the anode and cathode channels.

A system of ordinary differential equations (ODEs), Equations (A1)–(A8), was implemented in the MATLAB 6.0 environment using the ode15s solver under the specified initial conditions. The initial conditions were defined following the operating conditions in the experimental setup. The outputs of the semi-empirical tanks-in-series HT-PEMFC model are concentration, temperature, and current density profiles in the channel and catalyst layer. Figure 4 compares the experimental and calculated polarization curves for the HT-PEMFC with a parallel flow field under co-current flow mode.

The solid line represents the fuel cell performance predicted by the tanks-in-series model. The model Equations (1)–(11) are adequate for describing the performance of the HT-PEMFC operating under reformate with the gas compositions listed in Table 1 and the electrochemical parameters listed in Table 2.

Figure 5 and Figure 6 illustrate the steady-state current density, temperature, and concentration profiles predicted by the model Equations (1)–(11) for the HT-PEMFC with co-current flow of fuel and oxidant. Complete mixing within the tank results in a step change in the concentration, temperature, and current density profiles from tank to tank. The charge balance Equations (3) and (4) indicate that an increase in the electrolyte current results in an increase in the electrochemical reaction rate via an increase in the potential difference. The anode parasitic crossover current (6) causes a difference between the membrane (I_mem) and anode (I^A) current densities, as illustrated in Figure 5a. The discrepancy in bulk and catalyst concentrations in Figure 6 provides insight into the degree of mass transfer intensity at the anode and cathode sides of the HT-PEMFC, as well as the corresponding mass driving forces with potential for further mass transfer intensification. It should be noted that the HT-PEMFC is capable of operating with either dry or humidified anode feed.

The experimental findings demonstrate that humidifying the anode gases results in a slight enhancement in fuel cell performance [32,33,34]. As reported by Reimer et al. [35], the liquid water flow is generated by the corresponding crossover current on the anode side. In their study, Bezmalinović et al. [36] employed a two-dimensional computational fluid dynamics (CFD) model to numerically investigate the impact of inlet flow rate on concentration profiles along the gas channels. They noted that a considerable quantity of water can permeate the membrane electrode assembly (MEA) during fuel cell operation. The results of their study indicate that the transport of water across the membrane is influenced by current density and cathode stoichiometry. The component balance equations define the water concentration profiles in Figure 6, taking into account the additional water flux on the anode side due to the crossflow of water across the membrane and the parasitic currents, as shown in the Supplementary Materials provided. The difference between the component concentrations in the channel bulk and the catalyst layer indicates the mass transfer resistance as defined by the mass transfer Equation (A3) in the Appendix A.

Waller et al. [37] investigated the influence of operating conditions (temperature and pressure) on the performance of HT-PEMFCs. They found that the decrease in fuel cell performance at high fuel dilution can be compensated by increasing the operating temperature and pressure. Søndergaard et al. [38] measured local temperature profiles within a high-temperature membrane electrode assembly (HT-MEA) during a single fuel cell test. Their findings indicated that the local temperature increased within the HT-MEA due to the ohmic resistance. The energy balance Equations (A6)–(A8) define the temperature distribution in the gas channels and HT-MEA, as illustrated in Figure 4. The coupled heat, mass, and charge transfer processes in the catalyst layer include the heat generated by the electrochemical reactions and the heat transfer from the catalyst layer to the channel bulk.

In their investigation, Søndergaard et al. [39] examined the impact of varying oxygen concentrations on the performance of an HT-PEMFC operated with different cathode feed compositions. Their research findings indicate that an increase in oxygen concentration results in an improvement in fuel cell performance. In their experimental investigation, Thomas et al. [40] demonstrated that mass transfer resistance is a function of the current density and acid doping of an HT-MEA. Lobato et al. [41] used a 49 cm² HT-PEM fuel cell with equally distributed sensors for the purpose of measuring the local current density profile. They found that feeding the cathode with pure oxygen resulted in a 25% increase in performance compared to feeding the cathode with air. The predicted current density, temperature, and concentration profiles in Figure 5 and Figure 6 illustrate a relationship between heat, mass, and charge transfer processes. The high current density and temperature predicted at the inlet region correlate with high component concentration. The tanks-in-series model is applicable for evaluating the influence of various parameters on fuel cell performance and for interpreting impedance data under harmonic perturbations of the potential.

4.2. Simulation of HT-PEMFC Impedance Spectrum

A time-dependent tanks-in-series model has been developed for transient transfer processes and impedance analysis in an HT-PEMFC considering the following phenomena:

• Electrochemical oxidation of hydrogen in the anode catalyst layer;
• Electrochemical reduction of oxygen in the cathode catalyst layer;
• Charge balances at the electrode/membrane interfaces;
• Energy balances in the gas channels and catalyst layer (HT-MEA);
• Anode and cathode parasitic currents.

The dynamic non-isothermal tanks-in-series model (1)–(11) is a flexible tool for analyzing the effects of various parameters on the transient behavior and impedance of an HT-PEMFC. The sinusoidal cell voltage signal (12) causes harmonic changes in the mean gas-phase component concentration (δC), mean temperature (δT), mean membrane current (δI_mem), mean anode overpotential (δη_A), and mean cathode overpotential (δη_C), indicating the coupled heat, mass, and charge transfer in the membrane electrode assembly (MEA), as shown in Figure 3. The EIS response incorporates valuable information on transfer processes in the fuel cell, including diffusivity, kinetics, mass transfer, and other relevant data. The analysis of HT-PEMFC impedance spectra, as presented in [42,43,44], includes the interpretation of transfer processes in the membrane (high frequency), kinetics (intermediate frequency), and mass transfer (low frequency). The phase shift difference between overvoltage and current (Figure 7) is due to the capacitive behavior defined by the mass and charge balance equations. The phase shift (θ) between cell voltage and current density decreases with increasing frequency. The simulation results provided by the tanks-in-series model are valuable for interpreting the effects of different parameters and operating conditions on the performance and impedance spectra of HT-PEMFCs.

5. Conclusions

A dynamic tanks-in-series model has been developed for the numerical analysis of the coupled heat, mass, and charge transfer processes in an HT-PEMFC, taking into account parasitic currents. The scale-independent semi-empirical model provides an adequate description of the transient and steady-state behavior of the HT-PEMFC with reduced computational effort in comparison with scale-dependent CFD models. The predicted concentration and current density profiles indicate the influence of components crossover through the membrane electrode assembly on the intensity of the transfer processes. The harmonic change in cell voltage produces a corresponding change in component concentration, temperature, membrane current, anode overpotential, and cathode overpotential, indicating the coupled heat, mass, and charge transfer in the fuel cell.