The mathematical, multiphysic, multidimensional, and electrochemical modelation of a high temperature solid oxide fuel cell system (planar electrolytesupported configuration) is discussed in the present paper. The mass transport within the cell is studied using the StefanMaxwel model, and the momentum balance is solved by means of NavierStokes and Brinkman equations, respectively. On the other hand, the energy balance includes the generation term coupled with the convection and conduction equations. It was demonstrated that the diffusion resistances play an important role in the cell performance, and the oxidant concentration is enough high to work at fuel utilization coefficient of 0.8. The current density suffers a reduction (10 A/m^{2} to
The use of fossil fuels is a one of the major concerns due to its effect on the environment pollution and global warming. The fuel cells (FCs) are electrochemical devices which can produce electric power and heat, by chemical combination among a fuel (generally H_{2}) and an oxidizer (O_{2}) [
A simple SOFC system is composed of two electrodes (anode and cathode), separated by an electrolyte through the one to which the ions O^{2−} are transferred. The current is generated fundamentally in the anode by means of the direct conversion of the chemical energy of the fuel [
As can be corroborated the low negative effect on the environment and higher energy efficiency than traditional generating technologies made the fuel cells one of the candidates for the near energy future [
Nowadays this devices are built of ceramic and cermet materials (metallicceramic) [
The aim of the present communication is to develop a comprehensive model which can be used to describe the processes mentioned above and to be used as design tool. The mathematical solution of the partial differential equations was performed using the finite element method to a microsolid oxide fuel cell with a planar geometry.
Since a lot of complicated models are needed to simulate the whole cell performance in the present paper, a single unit was taken as reference case. The domain includes the electrodes, electrolyte, and the channels for the fuel and oxidizer, in an SOFC system working with H_{2} like fuel and O_{2} (air), as oxidizer. The following processes are described starting from the multiphysics combination of the phenomenological models as follows.
Species transport in channels and solid elements.
Momentum balance in channels and electrodes.
Dissociation and ionization of the oxidizer in the interface cathode/electrolyte.
Fuel reaction at interface anode/electrolyte.
Heat balance for overall domain.
The conservation laws are applied to the geometry represented in Figure
Model geometry.
The mathematical formulation and the solution procedure were simplified assuming isotropic and homogeneous electrodes, with uniform morphological properties (porosity and permeability), parallel flow between the fuel and the oxidizer, negligible ohmic drop in electronically conductive solid womb of porous electrodes and ideal gas mixtures.
Weight fractions of the fuel and oxidizer in the regions where the balances of mass are applied, potential in any point of the solid surfaces, the temperature in the whole cell and the falls of pressure in the channels and electrodes, respectively, are the main incognito in the modelation. The general operation conditions are summarized in Table
General operational conditions.
Parameters  Value  Dimension 

Anode reference current density (io^{+}) 

A/m^{2} 
Cathode reference current density (io^{}) 

A/m^{2} 
Transfer coefficients anode and cathode ( 
0.5  
Cathode viscosity ( 
3.48e^{5}  Pas 
Anode viscosity ( 
2.65e^{5}  Pas 
Electrodes and electrolyte porosity ( 
0.5  
Anode and cathode permeability ( 


Temperature ( 
1273 

Pressure  101325  Pa 
 
Molar concentration (mol/m^{3})  
Species  Anode  Cathode 
 
Oxygen  0  1.914 
Hydrogen  8.616  0 
Water  0.917  0.917 
Nitrogen  0  7.2 
The mass balance is based on the StefanMaxwell equation (
The momentum balance in the channels and the porous elements is described by the NavierStokes (
The potential variation in the solid elements of the cell is subject to fluctuations within the interfaces electrode/electrolyte, due to finite discontinuities that appear because of variations of species concentrations in those places [
The exchange current densities at anode and cathode (
The activation overpotential is related to the electrode kinetics at the reaction site and the relationship between overpotentialcurrent density can be expressed by the ButlerVolmer equation [
This expression is valid when 2 electrons are transferred in the electrochemical reaction, the symmetric factor of the SOFC (alpha) is 0.5 and could vary if the oxidations of CH_{4} and CO are considered [
On the other hand, the concentration overpotential is evaluated considering the limit current density, defined by Wang [
The effect of the Ohmic overpotential on the cell voltage is calculated using the equation presented by Ni et al. [
The general heat balance equation for convection and conduction is used to calculate the heat effects and temperatures profiles along the channels and electrodes, adding a source term defined as
To solve the complex system of partial differential equations, the finite element method was implemented. The solution region was divided into several subregions, and the generated mesh in the workspace was refined to increase the effectiveness in all the calculations at the electrode/electrolyte boundary. The iterative procedure is chosen to solve the problem, establishing a minimum error between iterations of 10^{−6} and a total of 500 iterations.
The simulation of the system is carried out for a fuel utilization coefficient of 80% and oxygen around 30%. At these conditions and a fuel cell temperature of 923 K the species concentration, current density curve, and the convective flux influence on the mass balance within the device were studied. Figures
Oxidizer concentration in an SOFC channel.
Fuel concentration in an SOFC channel.
All previously outlined depends on the species migration and the diffusive flux of hydrogen and oxygen in the channels which can be corroborated analyzing Figures
Change in H_{2} concentration versus diffusive flux on channels and electrodes.
Diffusive flux and gas composition in an SOFC fuel channel.
In Figure
The current density along the cell is represented in Figure
Current density at SOFC anode.
Velocity profile at the fuel channel.
The developed model allows describing the phenomenon that take place inside an SOFC. The coupling of a multiphysics system of complex mathematical equations has been created taking into account the transport phenomena that describe the process and the kinetic aspects of the electrochemical reactions inside the device. Although the study is even preliminary, it is considered an advance step in the development of useful tools in the reliable and good design of fuel cells. If other fuel than hydrogen is used, the conversion reaction of methane, carbon monoxide, or other compounds should be included in the generation terms defined in (
Dimension
Density (Kg/
Weight fraction of component
Velocity (m/s)
Permeability
Pressure (Pa)
Temperature (
Viscosity (Pa s)
Constant of graveness (9.8 m/
Heat capacity P cte (J/Kg
Gaseous mixture thermal conductivity (W/mK)
Gaseous mixture binary diffusivity (
Reference exchange current density for anode and cathode (A/
Current limit density (A/
Over potential (V)
Current density at anode and cathode (A/
Coefficient
Electric conductivity (
Fuel flow (
Electric potential from Nerst equation(V)
Faraday constant (96487 C/mol)
Electronic transfer coefficient
Constant universal of the gases (8.32 J/
Film thickness (m)
Concentration of species
Molecular weight species
Porosity
Effective diffusivity (
Tortuosity
Potential (V)
Thermal conductivity for gas mixtures and solids (W/mK).