A 3-dimensional planar SOFC stack model Michel Bernier, James Ferguson and Raphaèle Herbin 1 Introduction A fuel cell converts chemical energy to electrical energy through electro-chemical reactions at the anode and cathode in much the same way as an ordinary battery. However, a fuel cell is continuously replenished with fuel and can therefore never go dead the way conventional batteries do. Like a battery, the voltage produced by the chemical reactions is typically on the order of a volt. To produce an appreciable amount of power to operate household devices or industrial equipment, the electrochemical cells must be connected in series and parallel. Such collections of cells are called stacks. Solid oxide fuel cells (SOFC) differ from other fuel cell technologies in that the electrolyte is a solid. This provides many advantages. There is no corrosive liquid to contain as in molten carbonate or phosphoric acid fuel cells nor is there any consumption of the electrolyte eleminating the need for electrolyte replenishment. Since fuel cells typically operate with gases as fuel and oxidant, SOFCs can be constructed without moving parts which often leads to high reliability. The principal disadvantage of SOFCs is their operating temperature. Since these cells typically operate from to selection of materials which are stabile, durable and compatible is a difficult task. Differing coefficients of thermal expansion, interdiffusion of materials used for the anode, electrolyte and cathode, and stresses caused by temperature gradients are the major engineering problems faced in the design of SOFCs. Towards this end the prediction of temperature distribution within SOFC stacks is crucial to the successful design of SOFC systems. The number of cathode-electrolyte-anode structures representing a positive electrode, electrolyte and negative electrode (PEN) in a planar stack may number as many as with up to channels on each side of apen structure (i.e fuel channels). The calculation of the three dimensional distributions which resolve the Université de Provence, 1353 Marseille, France, work sponsored by Gaz de France and ADEME Boise State University, Idaho, 83725, U.S.A. Université de Provence, 1353 Marseille, France 1
detail of the channels and PEN structures is beyond the capability of present computers. As a consequence the influence of the channels and PEN structure on the distribution of, voltage, current, fuel and air distribution is achieved through modelling. Interconnection Plate Anode Stack Electrolyte Cathode Interconnection channels Figure 1 : Stack geometry The approach adopted here is to model a sub-unit of the stack named a unit cell using conventional conservation equations and constitutive laws resolving geometric details of the PEN structure and channel geometry and use the resulting model to determine parameters which are required to model the full stack, so as to determine the behaviour of the temperature, voltage, fuel and air distributions at the stack scale. A unit cell is taken to be the smallest geometrically repeating unit which can be used to construct a stack. Figure 2 illustrates unit cells for the co-, counter-, and cross-flow geometries and gives their typical dimensions. CO-COUNTER FLOW GEOMETRY CROSS FLOW GEOMETRY PEN Electrolyte Anode "! %$ "! %& "! Cathode Figure 2 : Unit cell geometries for the co-, counter-, and cross- flow THe next sections develop the governing equations for the temperature, voltage, current, fuel species and oxidant species distributions in a three dimensional stack,
%$ examine the limiting behaviour asthe number of unit cells increases and describe a solution procedure. 2 The SOFC stack model 2.1 Thermal Energy Conservation 2.1.1 Unit cell effective properties approach In this section the thermal energy equation describing the temperature distribution of an SOFC stack is developed from the unit cell model (see also [6]). Following this approach provides relationships and definitions establishing a connection between parameters in the stack model and results from unit cell computations. Figure 3 : Thermal heat flux on the unit cell Conservation of thermal energy states that the sum of fluxes over a control volume surface equals the heat produced in the control volume at steady state (see figure 3). Considering the control volume to be a unit cell, thermal energy conservation can be written as : eff Considering a unit cell for which &, the average heat flux across a face of the cell must equal the heat flux at the stack level. Therefore, using Fourier s law of conduction for the cell flux, the effective conductivity ' at the stack level to the parent material conductivity )( in the unit cell is found to be -.+/1032,87 +* 0165 (, -+9:;032 * 01 5! " (1)
The ratio ( represents a shape factor which accounts for the curvature and concentration of adiabats (heat flows paths) in the unit cell geometry for fluxes flowing in the direction. Similar computations are performed for the and directions. Applying the constitutive equation (1) to the thermal energy conservation for the 2 stack yields : 5 eff &. 2.1.2 Homogeneization techniques Homogeneization techniques have been used over the past twenty years to try and model composite materials made out of fine microstructures as homogeneous materials. The set of equations which describes the behaviour of a stack at the cell level is nonlinear and therefore not so easy to homogenize. However, the homogeneization techniques may be used to solve the linear systems which arise at one point of a given iterative procedure. we shall give here as an example the homogeneized model for the temperature equation. As seen previously, the conservation of thermal energy states that 5 &, where. Instead of computing an effective conductivity as was proposed above, we now try to analyse the behaviour of the temperature as grows, where the subscript denotes the number of cells of the stack. Assume here, for simplicity, that the channel temperatures are known and denoted by and, (depending on the space variable). Then the thermal equations written at the cell level write : 5 & "!$&% ' on air channel wall "!$&% ' on fuel channel wall T N homogenized solution (from unit cell computations) T fine grid solution on the whole stack 6 00 1600 3600 Number of unit cells Figure : comparison between thermal distributions calculated by both methods 600
A mathematical result which was shown in [1] proves that at point may be approached by the algebraic expression: where, and are computed from unit cell calcultations (see [1]). Figure compares the temperature distribution computed by the homogeneization technique and by solving the exact model on a very fine discretisation grid on the whole stack. Note that these distributions were both computed by part of the software which is described in section 3.2. The error between the two solutions decreases when grows. This method is adapted to solve the thermal problem in the 3D stack model. 2.2 Thermal sinks and sources The heat source & at the stack level incorporates each of the heat producing or heat absorbing mechanisms observed at the unit cell level ; in the SOFC stack, two of the source terms produce heat (& ohm and & el ) while the other two absorb heat (& ref and & ch ). The source term is decomposed as & & ohm & el & ref & ch Ohmic heating (& ohm ) : is the major source of heat in a fuel cell. Ohmic heat is generated throughout the solid structure since electrical current flows thoughout this region. The two regions which dominate the heat production are the interconnect and electrolyte. The electrolyte ohmic heat production is included in el. Only the interconnect is considered here. As in the unit cell model, the ohmic heat production is given by ohm eff Electro-chemical Heat generation (& el ) : includes effects of ohmic heat dissipation in the electrolyte plus heat generated by electro-chemical reactions at the electrode-electrolyte interfaces. At the stack-level of modelling these heat sources are 2-dimensional in nature (assuming a thin electrolyte for the ohmic heating). Channel cooling (& ch ) : the air and fuel channels which permeate the SOFC stack provide the principal means by which the temperature of the stack is maintained. To model the heat transfer between the fluid and channel walls an expression of the form ch is assumed. Expressions for and is found by equating these expressions to the actual heat transfered in a unit cell (see [2]), similar to the computation of the effective conductivities. Heat Reforming loss( & ref ) : methane is a component of fuel used to drive SOFCs but it is not oxidized directly. Therefore methane is converted catalytically with to and through the reactions :
7 7 Reaction R3 is heat absorbing and is kinetically controlled. Reaction is slightly exothermic and is sufficiently fast that it can be assumed to be at chemical equilibrium (see []). 2.3 Electrical Problem The electrical potential equation, which describes the variation of electrical potential in a SOFC stack model is developed in the same way as in the thermal problem. The equation is developed by considering the current fluxes and potential distribution on the unit cell. In this way essential parameters required in the stack model can be related to the unit cell and computed. Finally the conservation of electrical energy principle states : 5. The electric potential is assumed to be proportional to the gradient of the stack potential, namely : eff, where eff is an effective conductivity related to the material and geometry of the unit cell. Electrochemicals reactions and potential jump At electrolyte/electrodes interfaces, two reactions are taken into account : Across electrolyte layers, the potential increases due to the electrochemical reactions occuring at the electrode-electrolyte interfaces. The magnitude of the potential jump is given by : jump Nernst loss The Nernst potential Nernst is a function only of the reactant distributions, on one side, on the other. For both reactions and, the Nernst Voltage is given by : $% Nernst 6 where for! ", and for! $ ". If both reactions occur at the anode-electrolyte interface then &%(' Nernst )% Nernst. The loss term loss, is composed of : the ohmic loss across electrolyte, the anode and cathode activation polarization losses, and the anode and cathode diffusion losses. 2. Channel transport mechanism Conservation law for the mass flux ( 7 ) writes : 5 +* 7, where * 7 is the production rate of species ; The species concentrations,, are studied in the channels. The diffusion of the reactants in the porous anode and cathode was carefully studied at the cell level in []. At the stack level,
/ / 5 5 use is made of these results by stating that the concentration of a given species at an interface is proportional to the concentration on the same section in the channel, where the proportion is computed by the cell model results. The conservation of the electrons must be written at the interfaces where the electrochemical reactions occur. Hence the mass flux is related to the electric current by Faraday s law. (see [] for more details). In the gas channels, the thermal flux is mainly convective in the gas flow direction, and conductive from the channel to solid parts. Hence, in the two channels, the expression for the flux is : species where is the heat conductivity and 7 is the heat capacity of gas. 3 Solution procedure 3.1 Finite Volume discretization Following [5] we choose here the finite volume method to discretize the set of non linear equations obtained from the above model. The advantage of a finite volume method over a finite difference method is that the discretization of the flux is conservative and consistent even in the case of discontinuous diffusion coefficients (see [5]). The finite volume method is also somewhat simpler to implement than the finite element method in the case of jump conditions such as those encountered in the SOFC model (see [5]). Here we use a three dimensional non uniform parallelepipedic mesh. For the sake of simplicity, we shall present here the finite volume method in the two-dimensional case. Let us consider the following thermal problem : 5 & in solid parts "! &% ' (2) on boundaries where is the solid component of stack temperature, and is the gas temperature, % the heat transfert coefficient, and & is the source term. The principle of the finite volume method is to write the flux balance over a discretization cell (or control volume) 7 ; for equation (2) this yields : - / - /! &
% where n is a outward pointing normal vector. M L u I u I M u M L u L Figure 5: local notations With the notations shown in figure 5, the interface flux can be formulated on each side of the interface between the control volumes and by : Imposing the continuity on the approximation value of the flux yields I By similar computations, Fourier boundaries conditions are discretized by : &% 3.2 Software configuration Three computer codes are developed for the simulation software Heol2D which performs a two dimensional simulation of a simplified model in order to compare the numerical results obtained when using homogeneisation results against those obtain from a cell model. Stack3D which is the computing kernel for the three dimensional simulation. XHeol which is the graphic interface used for the analysis of the results from Heol2D and Stack3D. All three parts of the software were written in C under the Unix system. The software is portable on any Unix machine and was already tested on Irix (SGI), SunOS (Sun Microsystems) and HP-Unix (Hewlett Packard). The graphical interface uses the Motif and X11 libraries. The computer programs are written in a modular way so that a computational method of a subsystem for the stack model may be replaced by another without any effect on the rest of the code. Similarly, all geometrical data (width and thickness of the various parts of the unit cell, number of cells
), physical parameters (material conductivites, reaction rates and operating conditions (temperature and mass fractions of the reactants at the inlet, gas velocities ) are defined in external files, which are directly accessible to the user, without recompiling the code itself. The Stack3D part contains the treatment of all electrochemical, thermal and mass equations. It also takes into account the reforming and shift reactions, the diffusion losses and the overpotentials. It contains a unit cell module which allows the computation of the effective parameters when this approach is chosen, and of the homogeneisation coefficients otherwise. Heol Packages INPUT DATA Operating conditions Unit cell geometry Stack geometry Gas parameters Voltage parameters Model data Materials model Electrochemical reactions model LibMath Package Linear system treatment STACK3D Package Thermal problem treatment Electrical problem treatment Mass problem treatment LibMesh Package Mesh treatment XHeol Package Figure 6 : Heol architecture OUTPUT DATA XHeol format data TASF format The use of the software requires the definition of the various parameters through the various input files, as shown in Figure. Once the code is run, the results are stored in output files which are directly accessed by the graphical interface XHeol. Conclusions A three dimensional planar stack simulation code is developed using modelling techniques and recent mathematical and numerical techniques. The software is modular and portable, and is designed to be a user-friendly tool for the optimization of SOFC configurations. Références [1] Ph. BATOUX and M. BERNIER and R. HERBIN, Numerical simulation of heat and current conduction using homogeneisation techniques, in preparation.
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