Numerical Simulation of Solid Oxide Fuel Cell Energy Production Processes †

International Conference on Electronics, Engineering Physics and


Introduction
Solid oxide fuel cells (SOFCs) are a promising energy and resource-efficient technology [1]. However, carrying out experimental studies of SOFCs under various operating conditions is not always possible due to technical and economic reasons. It is sometimes not possible to measure all flow characteristics such as the temperature, pressure, or distribution of flow rates through a fuel cell using experimental methods. SOFC numerical simulations reveal much more possibilities for researchers [2][3][4]. The accurate modeling of SOFCs is a complex task. The equations of mass, energy, charge, electron transport, and electrochemistry should be calculated simultaneously at the boundaries of liquid, solids, and porous media. This physico-electrochemical task with a large number of involved parameters can be solved by numerical simulations using commercial software packages [5,6].
The finite difference method (FDM), the finite volume method (FVM), and the finite element method (FEM) are the main applied numerical approaches for SOFC modeling [7]. There are many commercial CFD packages based on the FVM and the FEM that are used for simulations of fluid and gas flows. The ANSYS and COMSOL software systems are the most frequently presented in the scientific literature among all available programs for CFD, multiphysics, chemistry, and electrochemistry. There are fundamental differences between these software packages. For example, ANSYS is based on the FVM, while COMSOL is based on the FEM [8][9][10]. Table 1. Current SOFC modeling areas.

Modeling Area Simulation Parameters
Simulation of specific processes Electrochemical reaction, electrode microstructure Alternative fuels Special fuel processing options, typical impurities, and purification systems Reforming Internal, external, and partial preliminary preparation Operation parameters Influence of temperature, pressure, and fuel composition Thermal effects Evaluation of various heat transfer phenomena in collectors, stacks, and single cells Design Planar and tubular structures, layer thicknesses, and materials Degradation Degradation mechanisms due to impurities and the thermal cycle System level research Evaluation of the influence of the properties of a single cell on the entire stack An important advantage of SOFCs is the possibility of internal conversion of a hydrocarbon fuel into hydrogen. Therefore, it is possible to use methane and synthesis gas as fuel without a pre-reformer system. In the case of external fuel reforming, SOFCs can use complex hydrocarbons, biofuels, and industrial and social waste, but it is necessary to carry out purification from sulfur. For example, during oil refining, a hydrogen-containing gas is formed with hydrocarbons from C1 to C7, which can be converted by steam reforming to synthesis gas and then fed to the SOFC.
The issues of improving the environmental friendliness of industrial enterprises and reducing the negative impact of gas waste on the environment are relevant today. Therefore, the purpose of the paper is to study the possibility of SOFC operation on various fuels and to compare the parameters of operation on hydrogen and oil refining waste.

Materials and Methods
Numerical simulations were carried out on a personal computer with an Intel Xeon Gold processor, 512 GB of total RAM and a 1 TB SSD for more accurate and faster mathematical calculations.
The calculations were carried out in the COMSOL Multiphysics 2021 universal software system for finite element analyses. The geometric 3D model was built according to the specification of the actual stack design. The model is based on a 1 kW anode-supported planar SOFC developed in China. The cell size of the SOFC is 16 × 16 cm 2 , with an active area of 10 × 10 cm 2 . Flows enter/exit the stack (fuel cells) through gas inlets/outlets (manifolds). Each fuel cell module block consists of a membrane-electrode complex cathode (positive electrode), an electrolyte, an anode (negative electrode), air and fuel channels, and interconnections. Thus, the electrochemical active area of the 30-element SOFC stack consists of 900 identical block modules. Due to the same geometry, a numerical simulation of one block of the SOFC cell was carried out. The dimensions and geometry of the block are presented in Table 2. The input parameters and composition of the simulated gas flows are presented in Table 3. Hydrogen, methane, and synthesis gas were used as fuel. The composition of the synthesis gas, which was obtained from the gas waste of oil refining by steam reforming, was modeled in the Ansys Fluent 2021 software package at a temperature of 873 K and a pressure of 10 bar on a nickel catalyst. The real processes in a fuel cell are a combination of matter transfer, heat transfer, electron transfer, and electrochemistry.
The velocity and pressure in stationary hydrodynamics do not depend on time and space. Therefore, for the distribution of velocity and pressure in the cell, the momentum conservation equations (Navier-Stokes equations) were used. The diffusion of chemicals in a laminar flow was modeled according to Fick's law.
When modeling gas flows, the laws of conservation of mass, momentum, and energy of incompressible laminar flows were used. For a finite volume dV, the fundamental equations apply under stationary conditions ∂/∂t = 0.
The mass of all chemicals is constant, but the composition changes due to electrochemical reactions.
m i,in -mass flow rate of chemicals, kg/m 3 ·s; c i,k -stoichiometric coefficient of the i-th component in the k-th reaction; v k -k-th reaction rate.

Mass conservation equation:
ε-porosity and ρ-density, kg/m 3 . The total consumption of substances can be calculated using the following formula: where the consumption of each i-th component is calculated as follows: M i -molar mass. Due to the low Reynolds number and steady state, the conservation equation can be written as: where k g is the permeability of the gas phase in m 2 and µ is the gas viscosity in kg/(m·s) The equation of substance transfer in a SOFC can be written as: where y i is the mass fraction of the substance; D ef,ij is the effective diffusion coefficient between the i-th and j-th substances (m 2 /s); and x j is the mole fraction of the j-th substance. The energy conservation equation can be formulated as follows: where c p is the specific heat capacity, J/kg·K; k ef is the thermal conductivity coefficient, W/m·K; and Q v is the heat flow, W/m 3 . To solve the charge conservation equation, the transport of electrons and ions must be taken into account. An electronic charge occurs in the electrodes and connecting element, while an ionic charge exists only in the electrodes and electrolyte.
According to Ohm's law, the electronic charge balance is calculated as follows: • On the anode: ∇(σ a ∇∅ e ) = −J a A V ; • On the cathode: ∇(σ c ∇∅ e ) = −J c A V . According to Ohm's law, the ionic charge balance is calculated as follows: where ∅ is the exchange potential (i-ionic, e-electronic, el-electrolyte) in V; σ a and σ c are the electrical conductivities of the anode and cathode, respectively, in S/m; J a and J c are the current densities of the anode and cathode, respectively, in A/m 2 ; and A V is the reaction area per unit volume in m 2 /m 3 .
The cell voltage can be calculated using the following equation: where ЕN is the Nernst voltage (open circuit voltage) in V and ŋohm, ŋact, ŋcon are the ohmic, activation and concentration overvoltages, respectively. The value of the Nernst voltage is related to the composition of the gas, operating pressure, and operating temperature and is determined by the equation: where T is the temperature in K; Р is the pressure in Pa; R is the gas constant in J/mol·K; F is the Faraday constant in C/mol; and ΔG is the Gibbs energy in J.
The activation voltage loss is calculated using the Butler-Volmer equation: where is the current density on the electrodes in A/m 2 ; -exchange current density, A/m 2 ; -charge transfer coefficient; z-number of electrons.
The concentration voltage loss is determined by the equation: where is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where , , and γ are the SOFC constant coefficients; δ and γ are the internal resistances; and r is the total resistance of the SOFC.
When using synthesis gas that contains methane or carbon monoxide, a rather low electrochemical CH4 conversion rate and a fast steam shift reaction at the SOFC operating temperature with the formation of CO/H2 and CO2/H2O are noted, which almost instantly come into equilibrium. Thus, the electrochemical conversion of CO is indirectly modeled by the additional electrochemical conversion of hydrogen and includes the rate of the carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic reaction and the electrochemical oxidation of hydrogen is an exothermic reaction, this is the main reason for the temperature gradient within the stack. The model should include the corresponding homogeneous gaseous methane steam reforming and steam shift reactions. The steam reforming reaction proceeds as a direct bulk gas reaction with a reaction rate of ϑ 4.
where is the reaction rate constant and are the partial pressures of CH4, H2O, H2, and CO.
The Nernst potential in a gas mixture of equilibrium composition is equivalent for each oxidation reaction under consideration, since only the gradient matters for the partial pressure of oxygen between the electrodes.
As a consequence, the hydrogen oxidation reaction provides the same Nernst voltage as the oxidation of carbon monoxide in a gas mixture at equilibrium: where ЕN is the Nernst voltage (open circuit voltage) in V and ŋohm, ŋact, ŋcon are the ohmic, activation and concentration overvoltages, respectively.
The value of the Nernst voltage is related to the composition of the gas, operating pressure, and operating temperature and is determined by the equation: where T is the temperature in K; Р is the pressure in Pa; R is the gas constant in J/mol·K; F is the Faraday constant in C/mol; and ΔG is the Gibbs energy in J.
The activation voltage loss is calculated using the Butler-Volmer equation: where is the current density on the electrodes in A/m 2 ; -exchange current density, A/m 2 ; -charge transfer coefficient; z-number of electrons.
The concentration voltage loss is determined by the equation: where is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where , , and γ are the SOFC constant coefficients; δ and γ are the internal resistances; and r is the total resistance of the SOFC.
When using synthesis gas that contains methane or carbon monoxide, a rather low electrochemical CH4 conversion rate and a fast steam shift reaction at the SOFC operating temperature with the formation of CO/H2 and CO2/H2O are noted, which almost instantly come into equilibrium. Thus, the electrochemical conversion of CO is indirectly modeled by the additional electrochemical conversion of hydrogen and includes the rate of the carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic reaction and the electrochemical oxidation of hydrogen is an exothermic reaction, this is the main reason for the temperature gradient within the stack. The model should include the corresponding homogeneous gaseous methane steam reforming and steam shift reactions. The steam reforming reaction proceeds as a direct bulk gas reaction with a reaction rate of ϑ where is the reaction rate constant and are the partial pressures of CH4, H2O, H2, and CO.
The Nernst potential in a gas mixture of equilibrium composition is equivalent for each oxidation reaction under consideration, since only the gradient matters for the partial pressure of oxygen between the electrodes.
As a consequence, the hydrogen oxidation reaction provides the same Nernst voltage as the oxidation of carbon monoxide in a gas mixture at equilibrium: where ЕN is the Nernst voltage (open circuit voltage) in V and ŋohm, ŋact, ŋcon are the ohm activation and concentration overvoltages, respectively.
The value of the Nernst voltage is related to the composition of the gas, operat pressure, and operating temperature and is determined by the equation: where T is the temperature in K; Р is the pressure in Pa; R is the gas constant in J/mo F is the Faraday constant in C/mol; and ΔG is the Gibbs energy in J.
The activation voltage loss is calculated using the Butler-Volmer equation: where is the current density on the electrodes in A/m 2 ; -exchange current density, A/m 2 ; -charge transfer coefficient; z-number of electrons.
The concentration voltage loss is determined by the equation: where is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where , , and γ are the SOFC constant coefficients; δ and γ are the inter resistances; and r is the total resistance of the SOFC.
When using synthesis gas that contains methane or carbon monoxide, a rather electrochemical CH4 conversion rate and a fast steam shift reaction at the SOFC operat temperature with the formation of CO/H2 and CO2/H2O are noted, which almost instan come into equilibrium. Thus, the electrochemical conversion of CO is indirectly mode by the additional electrochemical conversion of hydrogen and includes the rate of carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic reaction and electrochemical oxidation of hydrogen is an exothermic reaction, this is the main rea for the temperature gradient within the stack. The model should include corresponding homogeneous gaseous methane steam reforming and steam s reactions. The steam reforming reaction proceeds as a direct bulk gas reaction wit reaction rate of ϑ where is the reaction rate constant and are the partial pressures of CH4, H2O, and CO.
The Nernst potential in a gas mixture of equilibrium composition is equivalent each oxidation reaction under consideration, since only the gradient matters for the par pressure of oxygen between the electrodes. = − (T,p , ) As a consequence, the hydrogen oxidation reaction provides the same Nernst volt as the oxidation of carbon monoxide in a gas mixture at equilibrium: The value of the Nernst voltage is related to the composition of the gas, o pressure, and operating temperature and is determined by the equation: where T is the temperature in K; Р is the pressure in Pa; R is the gas constant in F is the Faraday constant in C/mol; and ΔG is the Gibbs energy in J.
The activation voltage loss is calculated using the Butler-Volmer equation: = exp The concentration voltage loss is determined by the equation: where is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where , , and γ are the SOFC constant coefficients; δ and γ are the resistances; and r is the total resistance of the SOFC.
When using synthesis gas that contains methane or carbon monoxide, a rat electrochemical CH4 conversion rate and a fast steam shift reaction at the SOFC o temperature with the formation of CO/H2 and CO2/H2O are noted, which almost i come into equilibrium. Thus, the electrochemical conversion of CO is indirectly m by the additional electrochemical conversion of hydrogen and includes the rat carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic reaction a electrochemical oxidation of hydrogen is an exothermic reaction, this is the main for the temperature gradient within the stack. The model should inclu corresponding homogeneous gaseous methane steam reforming and stea reactions. The steam reforming reaction proceeds as a direct bulk gas reaction reaction rate of ϑ 4.

= × ∏ ,
where is the reaction rate constant and are the partial pressures of CH4, H and CO.
The Nernst potential in a gas mixture of equilibrium composition is equiva each oxidation reaction under consideration, since only the gradient matters for th pressure of oxygen between the electrodes. The value of the Nernst voltage is related to the composition of the pressure, and operating temperature and is determined by the equation: where T is the temperature in K; Р is the pressure in Pa; R is the gas consta F is the Faraday constant in C/mol; and ΔG is the Gibbs energy in J. The activation voltage loss is calculated using the Butler-Volmer equa The concentration voltage loss is determined by the equation: where is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where , , and γ are the SOFC constant coefficients; δ and γ are resistances; and r is the total resistance of the SOFC.
When using synthesis gas that contains methane or carbon monoxide electrochemical CH4 conversion rate and a fast steam shift reaction at the SO temperature with the formation of CO/H2 and CO2/H2O are noted, which al come into equilibrium. Thus, the electrochemical conversion of CO is indir by the additional electrochemical conversion of hydrogen and includes t carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic reac electrochemical oxidation of hydrogen is an exothermic reaction, this is th for the temperature gradient within the stack. The model should corresponding homogeneous gaseous methane steam reforming and reactions. The steam reforming reaction proceeds as a direct bulk gas re reaction rate of ϑ 4.

= × ∏ ,
where is the reaction rate constant and are the partial pressures of and CO.
The Nernst potential in a gas mixture of equilibrium composition is each oxidation reaction under consideration, since only the gradient matters pressure of oxygen between the electrodes. The value of the Nernst voltage is related to the composition of pressure, and operating temperature and is determined by the equatio where T is the temperature in K; Р is the pressure in Pa; R is the gas c F is the Faraday constant in C/mol; and ΔG is the Gibbs energy in J. The activation voltage loss is calculated using the Butler-Volmer The concentration voltage loss is determined by the equation: where is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where , , and γ are the SOFC constant coefficients; δ and γ resistances; and r is the total resistance of the SOFC.
When using synthesis gas that contains methane or carbon mono electrochemical CH4 conversion rate and a fast steam shift reaction at t temperature with the formation of CO/H2 and CO2/H2O are noted, whi come into equilibrium. Thus, the electrochemical conversion of CO is by the additional electrochemical conversion of hydrogen and inclu carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic electrochemical oxidation of hydrogen is an exothermic reaction, this for the temperature gradient within the stack. The model sh corresponding homogeneous gaseous methane steam reforming reactions. The steam reforming reaction proceeds as a direct bulk g reaction rate of ϑ 4.

= × ∏ ,
where is the reaction rate constant and are the partial pressure and CO.
The Nernst potential in a gas mixture of equilibrium compositio each oxidation reaction under consideration, since only the gradient ma pressure of oxygen between the electrodes.

RT 4F
As a consequence, the hydrogen oxidation reaction provides the sa as the oxidation of carbon monoxide in a gas mixture at equilibrium con are the ohmic, activation and concentration overvoltages, respectively.
The value of the Nernst voltage is related to the composition of the gas, operating pressure, and operating temperature and is determined by the equation: where T is the temperature in K; P is the pressure in Pa; R is the gas constant in J/mol·K; F is the Faraday constant in C/mol; and ∆G is the Gibbs energy in J. The activation voltage loss is calculated using the Butler-Volmer equation: where J is the current density on the electrodes in A/m 2 ; J 0 -exchange current density, A/m 2 ; a-charge transfer coefficient; z-number of electrons.
The concentration voltage loss is determined by the equation: where I L is the limiting current density in A/m 2 Finally, the ohmic voltage loss is described as follows: where T 0 , δ, and γ are the SOFC constant coefficients; δ and γ are the internal resistances; and r is the total resistance of the SOFC. When using synthesis gas that contains methane or carbon monoxide, a rather low electrochemical CH 4 conversion rate and a fast steam shift reaction at the SOFC operating temperature with the formation of CO/H 2 and CO 2 /H 2 O are noted, which almost instantly come into equilibrium. Thus, the electrochemical conversion of CO is indirectly modeled by the additional electrochemical conversion of hydrogen and includes the rate of the carbon monoxide conversion reaction.
Since the steam reforming of methane is an endothermic reaction and the electrochemical oxidation of hydrogen is an exothermic reaction, this is the main reason for the temperature gradient within the stack. The model should include the corresponding homogeneous gaseous methane steam reforming and steam shift reactions. The steam reforming reaction proceeds as a direct bulk gas reaction with a reaction rate of ϑCH 4 .
where k is the reaction rate constant and p i are the partial pressures of CH 4 , H 2 O, H 2 , and CO.
The Nernst potential in a gas mixture of equilibrium composition is equivalent for each oxidation reaction under consideration, since only the gradient matters for the partial pressure of oxygen between the electrodes.
As a consequence, the hydrogen oxidation reaction provides the same Nernst voltage V N as the oxidation of carbon monoxide in a gas mixture at equilibrium:

Results and Discussion
The predicted current density versus voltage (J-V) for the numerical model showed an acceptable accuracy with the manufacturer's data, as shown in Figure 1. However, the simulations exhibited different current densities at lower voltages.

Results and Discussion
The predicted current density versus voltage (J-V) for the numerical model showed an acceptable accuracy with the manufacturer's data, as shown in Figure 1. However, the simulations exhibited different current densities at lower voltages. The volt-ampere characteristics for methane and synthesis gas are shown in Figure  2a,b. At the same voltage, the maximum current density is reached for hydrogen fuel (0.22 A/cm 2 at 0.6V), then it decreases slightly for synthesis gas (0.2 A/cm 2 at 0.6V), and the lowest current density is observed for methane (0.17 A/cm 2 at 0.6V). These obtained results are consistent with the data on the net calorific value of the fuel (Table 4).   The volt-ampere characteristics for methane and synthesis gas are shown in Figure 2a,b. At the same voltage, the maximum current density is reached for hydrogen fuel (0.22 A/cm 2 at 0.6V), then it decreases slightly for synthesis gas (0.2 A/cm 2 at 0.6V), and the lowest current density is observed for methane (0.17 A/cm 2 at 0.6V). These obtained results are consistent with the data on the net calorific value of the fuel (Table 4).

Results and Discussion
The predicted current density versus voltage (J-V) for the numerical model showed an acceptable accuracy with the manufacturer's data, as shown in Figure 1. However, the simulations exhibited different current densities at lower voltages. The volt-ampere characteristics for methane and synthesis gas are shown in Figure  2a,b. At the same voltage, the maximum current density is reached for hydrogen fuel (0.22 A/cm 2 at 0.6V), then it decreases slightly for synthesis gas (0.2 A/cm 2 at 0.6V), and the lowest current density is observed for methane (0.17 A/cm 2 at 0.6V). These obtained results are consistent with the data on the net calorific value of the fuel (Table 4).    The distribution of chemicals along the channel is shown in Figures 3-5. When methane is supplied as a fuel, a steam reforming reaction occurs. Hydrogen is formed at the anode, which increases towards the center of the channel and then is consumed in an electrochemical reaction with oxygen ( Figure 4). Water is supplied together with methane. Therefore, there is a large amount of water at the beginning of the channel and at the end, as a product of the electrochemical reaction.
When synthesis gas is used as a fuel, the reactions of steam reforming, carbon monoxide oxidation, and the interaction of hydrogen with oxygen occur simultaneously.  The distribution of chemicals along the channel is shown in Figures 3-5.  When methane is supplied as a fuel, a steam reforming reaction occurs. Hydrogen is formed at the anode, which increases towards the center of the channel and then is consumed in an electrochemical reaction with oxygen ( Figure 4). Water is supplied together with methane. Therefore, there is a large amount of water at the beginning of the channel and at the end, as a product of the electrochemical reaction.
When synthesis gas is used as a fuel, the reactions of steam reforming, carbon mon-  The distribution of chemicals along the channel is shown in Figures 3-5.  When methane is supplied as a fuel, a steam reforming reaction occurs. Hydrogen is formed at the anode, which increases towards the center of the channel and then is consumed in an electrochemical reaction with oxygen ( Figure 4). Water is supplied together with methane. Therefore, there is a large amount of water at the beginning of the channel and at the end, as a product of the electrochemical reaction. When methane is supplied as a fuel, a steam reforming reaction occurs. Hydrogen is formed at the anode, which increases towards the center of the channel and then is consumed in an electrochemical reaction with oxygen ( Figure 4). Water is supplied together with methane. Therefore, there is a large amount of water at the beginning of the channel and at the end, as a product of the electrochemical reaction.
When synthesis gas is used as a fuel, the reactions of steam reforming, carbon monoxide oxidation, and the interaction of hydrogen with oxygen occur simultaneously.
Hydrogen is supplied in a sufficiently large amount in the fuel mixture, and is also formed as a result of steam reforming. Therefore, a large amount of hydrogen is observed up to the middle of the channel, then it begins to be consumed in the electrochemical reaction ( Figure 5). Carbon monoxide also enters the fuel mixture and reacts with oxygen to form carbon dioxide. Table 4 presents the calculated parameters of the SOFC operation efficiency on various types of fuel.
According to the results of the study, the highest electrical efficiency can be obtained when operating the SOFC with hydrogen. Synthesis gas obtained by reforming industrial waste also shows good results in terms of efficiency, fuel consumption, and reagent utilization due to its high content of hydrogen, methane, and carbon monoxide.
The ratio of steam to methane is 1.7. The obtained value is within the range of acceptable values indicated in the literature (1.7-2.5) in which no carbon deposits are observed. In the case of using synthesis gas, additional water is formed in the reaction of hydrogen with oxygen, which can also be used for methane reforming.
The heat generated by the fuel cell can be used for pre-reforming or for hybridization of the cycle with other power plants. The largest amount of heat can be obtained by using methane as a fuel.

Conclusions
The authors present a model of the SOFC membrane-electrode module, which takes into account the influence of hydro-gas dynamics and mass transfer and the effects of electrochemical and thermal transfers in the cell structure. The mathematical model includes interconnected equations of momentum, mass, heat, and charge transfers; electrochemical reactions; and reforming reactions. Model verification and validation are provided by experimental data. The numerical stability and simulation capabilities of this multiphysics model are provided by the computational grid, which consists of 2.2 million cells. The paper provides detailed information on the distribution of chemical flows, temperature, current density, etc. With the help of numerical simulations, the influence of various types of fuel on the efficiency of the SOFC was studied. It is shown that fuel from petrochemical waste has high electrical and thermal efficiencies with an average consumption of reagents and a high fuel utilization factor.