Energy Conversion and Management

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1 Energy Conversion and Management 5 (29) Contents lists available at ScienceDirect Energy Conversion and Management journal homepage: From D to 1D modeling of tubular solid oxide fuel cell Michał Karcz Energy Conversion Department, Institute of Fluid Flow Machinery, Polish Academy of Sciences, Fiszera 14, 8952 Gdańsk, Poland article info abstract Article history: Received 6 October 28 Accepted 16 May 29 Available online 17 June 29 Keywords: Tubular fuel cell SOFC Current density vector An analysis of tubular solid oxide fuel cell fuelled with humidified hydrogen is presented in the paper. Numerical calculations have been performed by means of D and 1D levels of the electrical current density vector prediction together with 3D level of continuity, momentum, energy and species transport modeling. It has been shown that constant current density assumption throughout the fuel cell, as in D model, gives some overestimation of voltage, power and temperature in comparison with 1D model. However, obtained characteristics are similar for both cases. Additionally, an influence of discretization segments number on the model prediction has been analyzed. The local changes of current density along the fuel cell tube are also presented. It is shown that for increasing values of average current density throughout the cell, relevant local values of current density decrease linearly with tube length. Ó 29 Elsevier Ltd. All rights reserved. 1. Introduction Solid oxide fuel cells (SOFC) are still considered as a potential technology especially for low-power installations (up to 2 MW [1]) as in the modern distributed power generation systems. Numerical modeling by means of computational fluid dynamics technique (CFD) can assist in improving of fuel cell designs toward higher efficiency, fuel diversity and low cost technology [2]. There has been a significant progress in the fuel cells modeling technique. Increasing number of operating industrial installations and experimental test rigs will provide data for an extended verification in the nearest future. For the last 15 years an unprecedented growth of publication number on this subject has also been noticed. A comprehensive review of solid oxide fuel cell modeling issues one can find in several papers given by Kakaç et al. [3], Ma et al. [4] and also Kee et al. [5]. Presented numerical models have been used mainly for temperature field prediction. As far as solid-state electrolyte is made from YSZ Yttria Stabilized Zirconia which is rather brittle material it is important to avoid dangerous level of thermal stresses [6]. It can be done via optimization technique that bases on computed temperature distribution inside the cell. Usually numerical models are three-dimensional by means of implemented geometry. As a consequence the solution of mass, momentum and energy balance equations is also three-dimensional. However, this is not always necessary for the current density vector prediction in tubular fuel cell where axial direction is a dominated one. In the most popular one-dimensional model tube is divided into several slices by planes perpendicular to its axis. Then electric circuit is solved individually for every single slice. address: michal.karcz@imp.gda.pl Similar technique for the local current prediction has been employed by Bharadwaj et al. [2], Li and Suzuki [7], Jia et al. [8], Campanari and Iora [9], Sanchez et al. [1], Suwaranwangkul et al. [1,11], Zhu and Kee [12] and also Song et al. [13]. There are also full three-dimensional level of modeling but only a few papers give an information on results obtained with this technique. Some data relating to 3D electrochemical model for tubular fuel cell can be found in the paper by Zhang et al. [14] and Klein et al. [15]. In the present paper two basic levels of electrical performance prediction are discussed, i.e. D and 1D. Both have been incorporated via CFD technique together with 3D geometrical models. For the sake of simplicity only one geometrical configuration has been analyzed a tubular one of Siemens Westinghouse design [8,9], which is still more developed than a competing planar concept. Presented models are verified against the experimental data that come from literature. As a base for the numerical implementation, a commercial CFD solver Fluent has been chosen [16]. This code provides a solution for standard fluid-flow processes with mass and energy transfer, and can be extended by means of user defined subroutines to cope with more sophisticated cases. This paper is some continuation of previous author s investigations concerning numerical modeling of tubular solid oxide fuel cells fueled with hydrogen [17], reformed natural gas [18] and reformed synthesis gas [19]. Present model however has been extended toward local current density estimation by means of 1D technique for the hydrogen fueled cell. 2. Tubular fuel cell modeling techniques Fuel cell operation principle is inevitable connected with electrochemical reactions which result in ions/electrons transfer /$ - see front matter Ó 29 Elsevier Ltd. All rights reserved. doi:1.116/j.enconman

2 238 M. Karcz / Energy Conversion and Management 5 (29) In the simplest case of humidified hydrogen fuel, these reactions involve only reduction at the cathode: 1 2 O 2 þ 2e! O 2 ; ð1þ and oxidation reaction at anode: H 2 þ O 2! H 2 O þ 2e : ð2þ The charge transfer per time is known as electric current I. However, more fundamental seems to be current density expressed as current per surface area [2]: j ~ ij¼ I A : ð3þ In other part of the paper symbol i is used instead of expression j ~ ij for the sake of simplicity. From the classical theory of electrostatics comes the statement that the electric field vector ~ E is the gradient of scalar potential / [21]: ~ E ¼ r/: ð4þ Usually the voltage U is used instead of gradient of electric field potential and then relation (4) becomes: U ¼ D/: Classical Ohm law that relates potential drop D/ to current I and material resistance R can be extended for ions/electrons transport in 3D domain. Then local current density vector ~ i has to be applied [3,22]: ~ 1 i ¼ r/ ¼ rr/; q ð6þ where resistivity q and conductivity r ¼ 1=q are inherent material properties that generally depend on temperature. The level of electricity field prediction is directly coupled with the level of current density vector consideration. When all three current density vector components are computed using Eq. (6) on the same numerical grid as for the rest balance equations then the model can be referred as a fully three-dimensional. When only one or two components of current density vector are considered then the model can be described as one- or two-dimensional, respectively. In the case of constant average value of current density applied for a whole fuel cell the model can be treated only as a zero-dimensional one. ð5þ Fig. 1. Fuel cell discretization scheme for D electrical field prediction Model 1D One-dimensional model gives a possibility of computing a local changes of current density in axial direction of tubular fuel cell. Radial and tangential directions are not explicitly solved then, and instead of an area-averaging technique is being applied. The model used in the present study employs potentiostatic conditions, i.e. assumption of cell voltage or in other words terminal voltage as a potential difference between cathode and anode interconnectors [22]. The cell terminal voltage can be assumed as constant because of sufficiently low electrical resistance of interconnectors and nickel felts [8]. Only in few papers an ohmic drop in current collection systems has been explicitly included, as in [12]. In the present analysis current density is not considered on the same numerical grid as for the main governing equations. Instead of a tube is being additionally discretized into several segments n, as in Fig. 2, where all parameters necessary for local current density prediction can be assumed as locally uniform. Obviously 1D modeling can be simplified to D case when only single discretization segment is employed ðn ¼ 1Þ. An influence of 2.1. Model D The simplest way of computing electrical behavior of fuel cell by means of CFD is a zero dimensional assumption of current density vector j ~ ij as an area-average value being constant throughout the cell. The model is not complicated but on the other hand seems to be very robust and can be easily applied for any SOFC geometry without any special numerical treatment. Such a model allows one to predict temperature field, species distribution and also a potential field throughout the fuel cell by means of employing the galvanostatic conditions, i.e. mean current density across the cell [22] see also Fig. 1. This kind of modeling has been successively applied in earlier authors papers [17 19]. First one was addressed to tubular fuel cell fueled by humidified hydrogen the same as in current analysis. Second paper considered the fuel cell with internal methane reforming. In the last article a lignite derived syngas was applied as a fuel. In the case of syngas or natural gas feeding, some fuel preparation works should be also modeling. This can be achieved by means of pure geometrical as well as mathematical zero-dimensional analysis given in details by Badur and Lemański [23]. Fig. 2. Fuel cell discretization scheme for 1D electrical field prediction.

3 M. Karcz / Energy Conversion and Management 5 (29) the discretization segments number n on the results will be discussed later. 3. CFD modeling of SOFC A mathematical model of SOFC is based on the elementary balance equations solved for fluids: continuity, momentum, energy, and species transport in the frame of finite volume method. The fuel cell generally consists of two porous electrodes which are separated by thin solid electrolyte. Then porosity should be explicitly considered in the governing equations. The porosity factor e is defined as a parameter that indicates the amounts of fluid volume V f within the finite volume V [16]: e ¼ V f V : The governing equations for the whole fuel cell model can be given in the following compact with CFD form [17,24]: eq eq~t >= >< eqe þð1 eþq s e s þ 8 >< ¼ div eqy k $ e s c $ e s c ~t þ ~q c e ~ J k ~ i 9 8 >= >< þ eq~t eq~t ~t eqe~t eqy k ~t eqs m eqs v 9 >= >< þ div eqs f e þð1 eþqss e eqs k S / 9 8 ep $ I ep~t 9 >= >= ; ð7þ where q density, ~v velocity vector, Y k mass fraction of gas $ component k; e total energy, p pressure, s c total diffusive momentum flux, ~q c total diffusive heat flux, ~ J k diffusive flux of specie k; S m mass source, S v momentum source, S e energy source, S k creation/destruction source of specie k; S / source of ionic/electronic potential. Indices f and s refer to fluid and solid, respectively. Set of equations (7) is solved in similar manner as for standard fluid-flow processes with chemical reactions. Main differences arise due to special treatment of diffusive transport of species inside porous structure of fuel cell electrodes and due to solving of charge conservation equation at different numerical grid Diffusive transport modeling Transport of species through the cell is dependent on three different mechanisms, i.e. ordinary molecular diffusion in gas channels, Knudsen diffusion and Darcy s pressure-driven flow in porous electrodes [5,25]. In the porous electrode there are three characteristic microstructure parameters that can strongly influence diffusive transport, i.e. porosity e, tortuosity s and mean radii of pores r [15,26]. The diffusive flux ~ J k can be calculated on the base of gradient hypothesis employing Fick s law [3]: ~ Jk ¼ qd eff k ry k: ð8þ It is important however to include multicomponent character of diffusion inside porous electrodes of fuel cell. The effective diffusion coefficient of given constituent k in the mixture that account influence of molecular D m k and Knudsen diffusion DK k, and also includes the porous structure characteristic parameters, can be written as in [27]: 1 D eff k ¼ s e! 1 a k X k D m þ 1 ; ð9þ k D K k where coefficient a k is defined as: a k ¼ 1 M 1 2 k : ð1þ M Knudsen diffusivity has the standard form as in [28,29]: sffiffiffiffiffiffiffiffiffiffi D K k ¼ 2 3 r 8RT : ð11þ pm k Molecular diffusion of species k is calculated on the base of binary diffusivities obtained via Eq. (13) through the simplified formula [9,12,27,28]: D m k ¼ 1 X k P : X l k l D kl ð12þ Species binary diffusivities D kl are usually calculated via modified Chapman Enskog theory that base on Lennard Jones potential model for collision integral estimation [27,3]. Todd and Young [31] reported however that in the higher temperature of operation ðt > 1 KÞ the Fuller et al. method is more accurate so it has been incorporated also in the present study as in [9,1,29]. The relevant equation for binary diffusivities estimation has the following form [32]: 1 1 þ 1 2 D kl ¼ :1T 1:75 M k M l h p ð P vþ 1 3 þ P i k ð vþ 1 2 ; ð13þ 3 l where p is a total pressure in atm, M k molecular weight and v is atomic diffusion volume that one can find in [32] Electrical field prediction The theoretical voltage generated in the atmospheric fuel cell due to elementary reactions (1) and (2) can be estimated from classical Nernst formulation [3,5,9,1,33]: DE ¼ DG H 2 O þ RT 2F 2F ln X H 2 X :5 O 2 : ð14þ X H2 O The species mole fractions employed in (14) are estimated in bulk channel flow rather than directly at electrode/electrolyte interface. When the cell starts to operate and current is supplying to an external load, the operating voltage always drops due to some irreversibilities that are commonly called polarizations [33]. There are three main loss mechanisms usually included during modeling, namely activation, concentration and ohmic. Sometimes also interface-contact and leakage losses are considered [34]. Then the operating cell potential can be written as: U cell ¼ E X gðiþ; ð15þ where the last term at right is a sum of all polarizations affecting fuel cell performance, i.e.: X gðiþ ¼gact þ g conc þ g ohm þ g interface þ g leakage : ð16þ Usually in the range of nominal loading of fuel cell the most important are activation and ohmic losses, while concentration polarization is negligibly small [9,33]. Apparently at the high reaction rates concentration polarization governs the cell operation. Information on interface and leakage polarization are rather scarce so they have not been included in the present study Activation polarization Activation polarization is implicitly given by Butler Volmer equation: i ¼ i e ð1 aþfg act i e afg act ; ð17þ

4 231 M. Karcz / Energy Conversion and Management 5 (29) where a is a symmetry factor, and f is a relation that combines Faraday constant F, universal gas constant R and temperature T: f ¼ F RT : ð18þ This equation is usually transformed into much simpler explicit formulas. Analysis of different equations for g act conducted by Noren and Hoffman [33] proved that the same result as original Butler Volmer formula gave the following hyperbolic sine approximation: g act ¼ 2 i nf sinh 1 ; ð19þ i The exchange current density i at anode and cathode are often assumed as constant and equal 53 A=m 2 and 2 A=m 2, respectively [35]. On the other hand, some changes of its value should be considered at the electrolyte electrode interface since its value depends on temperature and partial pressures of reacting species [5,14], and influence kinetics of electrochemical reactions [1]. The exchange current density can be effectively calculated from an Arrhenius-type formula [8,9,33]: i a ¼ c p m H2 p n H2 O a exp Ea act : ð2þ i c ¼ c c p a p a p l O2 exp Ec act p c RT RT : ð21þ In the present study the value E a act ¼ 1 kj=mol and Ec act ¼ 12 kj=mol has been assumed for anode and cathode activation energy, respectively, as in the paper by Zhang et al. [14]. It should be remembered however that there is a strong dependancy of modelled fuel cell operation on activation energy changes. The relevant sensitivity studies have been performed by Campanari and Iora [9]. Value of the pre-exponential factors c a and c c have been also adopted from [14] Concentration polarization Despite of gas-phase transport modeling within the electrode, in the present analysis also the concentration polarizations has been added to the sum of voltage losses. There are two main reasons of that. At first the Nernst potential is calculated from bulk values of species mole fractions not at electrolyte interfaces it is opposite to Zhu and Kee [12], Cordiner et al. [36], but similar to Campanari and Iora [9]. Additionally, the fuel and oxidizer mass flow rates have been coupled with average value of current density via fuel and oxidizer utilization factors U f and U a, respectively. Increasing value of i avg leads then to relevant mass flow changes that in summary conserve species mole fractions despite of simultaneous increasing of electrochemical reaction rate. Some limiting current is needed then to predict a fuel cell species transport blockage due to an increase of mass flow rate. On the base of predicted species diffusivities a limiting current densities for anode and cathode can be calculated via simplified formulae [37]: i a L ¼ 2fDa eff p H 2 d a ; ð22þ! i c L ¼ 4fDc p eff op d c ln ; ð23þ p op p O2 where p op is operating pressure, p k partial pressure of particular species k ¼ H 2 ; O 2 and d a;c value is an electrode thickness. The average value of limiting current density is about i L ¼ 6 7 A=m 2 [38] for the tubular fuel cell. Derivation of effective diffusion coefficients at anode D a eff and cathode Dc eff is described by Chan et al. [3]. Limiting current densities given by Eqs. (22) and (23) allow us to estimate concentration polarizations at electrodes that arise due to diluting effect of substrates and products that block species transport to/from reaction site. Concentration polarizations can be estimated from general formula for both electrodes: g conc ¼ 2 ln 1 i nf i a;c L ð24þ Ohmic polarization Ohmic losses estimation bases directly on formula (6) valid for both ionic and electronic flows in electrolyte and electrodes, respectively. General formulation for ohmic polarization can be written as: g ohm ¼ iq q d q ; Fig. 3. Electric circuit in single fuel cell section [13]. ð25þ where q = (a, c, e, int), i.e. anode, cathode, electrolyte and interconnector, respectively. Because the current density equation is not solved in radial and tangential direction in zero-dimensional or one-dimensional formulations an additional information about real current pathway should be included. A substitutional current pathway length d for the electrodes, electrolyte and interconnector can be calculated via formula given by Song et al. [13] for the circuit similar as in Fig. 3. The specific lengths for particular fuel cell elements can be defined as follow: d a ¼ ðapdþ2 8d a ; ð26þ d c ¼ ðapdþ2 A½A þ 2ð1 A BÞŠ; ð27þ 8d c d e ¼ d e ; ð28þ d int d int ¼ ðpdþ : ð29þ w int Parameters Ap and Bp that have appeared in Eqs. (26) (29) are directly related to the geometry of tubular fuel cell, i.e. to the circumferential length of electrical contact between anode/electrolyte and interconnector/electrolyte. In the present study values.84 and.13, as in original paper by Song et al. [13], have been assumed, respectively. 4. Model implementation and solution strategy Presented model have been implemented into commercial Fluent code using UDF (User Defined Function) technique [16]. Model

5 M. Karcz / Energy Conversion and Management 5 (29) Fig. 4. Geometry and the numerical grid (not in scale). employs standard balance equations for mass, momentum, energy, and species transport. All equations have been changed via relevant source terms and diffusion fluxes. The numerical grid for tubular fuel cell has been created on the base of Siemens Westinghouse concept. The detailed geometry one can be found in [1,8,9,39] as in the previous author s papers [18,19] and is not fully repeated here. Due to system symmetry only half of the tube has been considered. Fuel cell tube discretized by means of finite volumes method (FVM) is shown in Fig. 4. The density of the grid has been assumed based on sensitivity studies for pure flow problem without any electrochemical reactions. Material properties have been adopted from different sources. Resistivity q of fuel cells elements came from Zhang et al. [14]. Thermal conductivities k of electrodes, electrolyte and interconnector have been assumed the same as in Lin and Beale [22], and for air injection tube a formula from Campanari and Iora [9] has been employed. Porosities of electrodes generally range from e ¼ :2 toe ¼ :6, tortuosity factor s vary between 2 and 1 for various porous bodies, and typical mean pore size lies in the range r ¼ :5 1 lm [18]. In the present analysis, the porosity value e ¼ :5, tortuosity factor s ¼ 3 and pore radius r ¼ 1 lm for both cathode and anode electrodes have been assumed as in [9]. The permeability of electrodes has been determined by Kozeny Carman relationship [1,12,35]: e 2 B ¼ 72sð1 eþ 2 ð2rþ2 ð3þ This parameter is necessary for estimation of Darcy s force which acts as an additional source S v in momentum Eq. (7) [3,4]. Additionally, the DO Discrete Ordinate radiation model has been employed for temperature prediction inside the fuel cell tube [16]. It is important to include a radiative heat transfer, especially from the fuel cell supporting tube to the air-supply pipe due to the large temperature difference between these two surfaces [4]. Emissivity for both surfaces has been assumed from [8,1] as equal.9. Solution procedure for D and 1D modeling levels has four common steps as follows: setting the number of longitudinal segments n ¼ 1 as in Fig. 1, setting of an initial value of current density at the electrode/ electrolyte interfaces, temperature and species transport equations are solved for constant i avg, from simplified D calculations a constant voltage can be obtained and stored. For 1D problem additional steps have to be proceeded: setting the number of longitudinal segments n > 1asinFig. 2 (in the present case n ¼ 2), using stored value of constant voltage a local current density can be calculated for every segments, temperature and species distributions through the cell are estimated for varying i axial, average value of current density is obtained by integration of its local distribution, voltage value is iteratively adjusted to fit the average value of current density and stored, process is iteratively repeated until convergence of current density, temperature and species distribution is obtained, The adjusting of voltage is underrelaxed to increase the speed of convergence. Updated value of voltage is based on the value from previous ðt 1Þ and actual iteration t according to formulae: U ¼ð1 bþu ðt 1Þ þ bu t ; where < b 6 1 is an underrelaxation factor. 5. Model results ð31þ Analysis presented here is partly similar to the one described by Song et al. [13] where also a 1D model of Siemens Westinghouse tubular fuel cell was employed. Similar techniques have been

6 2312 M. Karcz / Energy Conversion and Management 5 (29) adopted for tube discretization and ohmic losses estimation. On the other hand there are many differences between previous and present computations. First of all present 1D model concerns electrical field only, and opposite to Song et al. [13] is not applied for other balance equations, which are being solved in purely three dimensional manner. Additionally, an analysis for a wide range of current densities, i.e. 1 6 A=m 2, not for single value only, have been performed. It has forced a careful treatment in low and high fuel cell loading, where activation and concentration losses play a significant role, respectively. As a fuel the humidified hydrogen has been employed. It should be underlined however that the model presented here can cope also with reformed natural and synthesis gas see previous works [18,19] Fuel cell basic characteristics There is no proper data for full model validation. Partially it can be done through the current voltage characteristics which are often available for particular cells. Beside this only scarce data exists for model performance estimation. Results presented in the paper by Zhang et al. [14] have been employed for validation purposes. Specific inlet boundary conditions data are presented in Table 1. Comparisons between numerical and experimental data are shown in Fig. 5 and also in Fig. 6 for current voltage and current power characteristics, respectively. The numerical data have been obtained by means of 1D analysis with n ¼ 2 discretization segments dedicated for electrochemical model prediction. As it can be seen, the computed values of voltage and power are higher than experimental ones in the range of A=m 2. Similar trend is observed in the paper by Zhang et al. [14]. It should be underlined however that the leakage and interface-contact polarizations have not been included in the present analysis. These polarizations could possibly lower the overall voltage generated by the cell. On the other hand the numerical results fit well experimental data in the range below 2 A=m 2. For higher current densities (above 5 A=m 2 ) some more pronounced differences Table 1 Mass fraction and temperature of fuel and oxidizer assumed at the fuel cell inlet. Fuel Oxidizer Y H2 ¼ :89 Y O2 ¼ :23 Y H2O ¼ :11 Y N2 ¼ :77 T fuel ¼ 1123 K T ox ¼ 873 K Fig. 6. Average current density power characteristic. Comparison of 1D ðn ¼ 2Þ analysis of fuel cell with experimental data. between numerical and experimental data can be observed. There is a visible voltage drop that can be related to the increasing concentration polarizations, that are computed via Eq. (24) as a function of diffusion coefficients and gas partial pressure. Estimated average value of limiting current density were close to the single value employed in the previous author s analysis i L ¼ 67 A=m 2 [17]. In the Fig. 7 a current density distribution along the tube axis obtained with 1D model is presented. In every single diagram an average value of current density is additionally marked. As it can be observed the local changes of current density are more pronounced for lower average values. However, for higher average values of current density the relevant local changes tend to be linear with the tube length. For whole range of average current density there is always a drop of its local value with an increasing distance from the fuel inlet. It is a result of species concentration at both electrolyte sides which are obviously highest at the fuel entrance area. From Nernst Eq. (14) potential differences are also highest then. The local value of current density is however computed from losses via Eq. (15). Then assumption of constant voltage at interconnector for whole cell results in higher value of current density near fuel inlet. Presented results are similar to those given by Li and Suzuki [7], where also a steady current density drop in the downstream direction were observed. On the other hand there are some differences with data presented for example by Song et al. [13] and Suwaranwangkul et al. [1]. In those papers decrease of current density is shown along the tube, starting from.3 m distance from the fuel inlet, while in the entrance region (below.3 m) there is apparently some rise of its value. However, it should be pointed that results presented in [13] were not obtained for hydrogen but for reformed gas as a fuel. In such a case, due to exothermic reforming and water gas shift endothermic reactions, rather different temperature and current density distribution throughout the cell is usually expected Model performance in D and 1D mode Fig. 5. Average current density voltage characteristic. Comparison of 1D ðn ¼ 2Þ analysis of fuel cell with experimental data. The number of discretization segments n for electrochemical model obviously influences the fuel cell operation prediction. Results by means of changing voltage, power and temperature as a current density functions are presented in subsequent diagrams Figs As it can be seen from Fig. 8 the increasing number of discretization segments n results in lowering voltage prediction by

7 M. Karcz / Energy Conversion and Management 5 (29) Fig. 7. Changes of local current density along fuel cell tube for different average current density values. employed numerical model. Results between n ¼ 1 and n ¼ 2 are almost identical. There is no sense then to employ a higher number of segments for such geometry this is consistent with results presented by Song et al. [13]. However, when only single segment n ¼ 1 is being used, then voltage is overestimated from.2 to.5 V in relation to 1D model with n ¼ 1 and n ¼ 2. Similar trends are observed also in Fig. 9 for power changes due to increasing current density. Lower voltage obtained for the same current density in the case of increasing number of discretization segments results in power lowering. The differences between n ¼ 1 and n ¼ 2 are more pronounced in the range located near current power characteristic maximum while difference between D and 1D models is almost constant and equal 2 W for single fuel cell tube. Interesting results concern average temperature throughout the tubular fuel cell for different segments number as in Fig. 1. Average temperature is steadily lowered for higher current density values. There are practically no differences between n ¼ 1 and n ¼ 2 cases. On the other hand when D model is employed then for lower values of current density a higher average temperature is predicted while for higher values ð> 2 A=m 2 Þ results are identical with 1D model. Temperature distribution along fuel cell tube at a line located in the anode electrode is given in Fig. 11. Two values of average current density have been considered, namely i avg ¼ 3 A=m 2 and i avg ¼ 5 A=m 2. Accordingly to the results presented in Fig. 13 there is an average temperature drop with increasing current density. Then curves addressed to i avg ¼ 5 A=m 2 case, are located below these related to i avg ¼ 3 A=m 2. There is an Fig. 8. Fuel cell current voltage characteristic obtained by means of D (single discretization segment n ¼ 1) and 1D (n ¼ 1 and n ¼ 2) model. Fig. 9. Fuel cell current power characteristic obtained by means of D (single discretization segment n ¼ 1) and 1D (n ¼ 1 and n ¼ 2) model.

8 2314 M. Karcz / Energy Conversion and Management 5 (29) Fig. 1. Fuel cell current temperature characteristic obtained by means of D (single discretization segment n ¼ 1) and 1D (n ¼ 1 and n ¼ 2) model. interesting issue concerning temperature distribution for the case of 1D and D models with n ¼ 2 and n ¼ 1, respectively. For the distance about 1 m from the fuel entry, local temperatures predicted with D model are higher than those obtained with 1D model. Differences are more pronounced for lower current densities, and reach about DT ¼ 2 K for i avg ¼ 3 A=m 2, while for i avg ¼ 5 A=m 2 it is less than DT < 1 K. Further downstream, starting from the distance about 1 m from the fuel entry, there is an opposite situation where 1D model gives higher temperatures, however the differences between D and 1D models are much smaller then. Temperature distribution at the anode external surface is presented in Fig. 12 for the case i avg ¼ 4 A=m 2. Changes in temperature field are similar to the results presented by different authors [1,8,13,14]. Temperature maximum is located in the middle of tube, while its lowering near inlet and exit areas is related directly to the cooling effects of fuel and air streams, respectively. Similar results are obtained with both D and 1D model see also Fig. 11. Fig. 11. Distribution of static temperature along external anode surface of fuel cell tube. Fig. 13. Fuel cell average static temperature computed with 1D model ðn ¼ 2Þ, with and without employing DO radiation model. Fig. 12. Distribution of static temperature at the external anode surface of fuel cell tube for i avg ¼ 4 A=m 2 without radiation.

9 M. Karcz / Energy Conversion and Management 5 (29) As it was mentioned earlier the temperature distribution is a critical element as far as it governs level of thermal stresses throughout the cell. For proper heat transfer modeling a DO radiation model has been additionally employed in the present analysis [16]. Radiation should lead to overall tube temperature lowering as it was presented by Suwanwarangkul et al. [1]. In the present case the difference between results obtained with and without radiation model reaches 3 4 C in average, as it is shown in Fig. 13. These results are consistent with the data in [1]. 6. Conclusions Different methods of electrical field treatment are discussed in the paper, i.e. D and 1D. These methods have been employed together with 3D reactive fluid-flow model. As a computational example a tubular fuel cell has been considered. Despite of considerable simplifications the simplest zero-dimensional model gives a rather good view inside fuel cell processes. Such a method can compete with more sophisticated models by means of voltage, current density, generated power and temperature prediction, even if some differences in results between particular methods are observed. Despite of dramatic changes of current density distribution along the tube with 1D modeling, a rather similar characteristics of voltage, power and temperature as a current density functions, are obtained also for D model, where apparently constant average current density has been specified. Generally speaking for higher current densities the temperature level predicted via D and 1D models are quite similar. It is practically identical when an average cell temperature was considered. Difference between curves of local temperature values obtained with D and 1D models, is less than <1 K in the same range of fuel cell loading. For lower current density, an average temperature of the fuel cell obtained with D model is higher than computed with 1D model. Also the dissimilarity between local temperatures along the tube estimated with both methods is more visible then. Due to high temperature of solid-oxide fuel cell operation radiative heat transfer seems to be an important factor affecting its performance. Temperature is lowered for whole current density range when radiation model is employed. Results of numerical analysis suggest that proper prediction of local temperature changes are more important than related local current density values. Temperature is an important factor during species source terms estimation, losses prediction and also for material and transport properties modeling. On the other hand current density is explicitly involving only in source terms and losses prediction. 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