CHAPTER 1 Multiple transport processes in solid oxide fuel cells P.-W. Li, L. Schaefer & M.K.. Chyu Department of Mechanical Engineering, University of Pittsburgh, USA. Abstract In this topic, three important issues are discussed which concern the theoretical fundamentals and practical operation of a solid oxide fuel cell. The thermodynamic and electrochemical fundamentals of a fuel cell are reviewed in the Section 2. These fundamentals concern the ideal efficiency and energy distribution of a fuel cell’s conversion of chemical energy directly into electrical energy through the oxidation of a fuel. Issues of the chemical equilibrium for a solid oxide fuel cell with internal reforming and shift reactions (in case of methane or natural gas being used as the fuel), are also discussed in detail in this section. The losses of electrical potential in the practical operation of a fuel cell are elucidated in the third section, which includes a discussion about activation polarization, Ohmic loss, and the losses due to mass transport resistance. In the fourth section, the coupled processes of flow, heat/mass transfer, chemical reaction, and electrochemistry, which influence the performance of a fuel cell, are analyzed, and modeling and numerical computation for the fields of flow, temperature, and species concentration, which collectively determine the local and overall electromotive force in a solid oxide fuel cell, are examined in detail. 1 Introduction A fuel cell is a device that converts the chemical energy of a fuel oxidation reaction directly into electricity. It is substantially different from a conventional thermal power plant, where the fuel is oxidized in a combustion process and a thermalmechanical-electrical energy conversion process is employed. Therefore, unlike heat engines that are subjected to the Carnot cycle efficiency limitation, fuel cells can have energy conversion efficiencies generally higher than that of heat engines [1]. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/1-85312-840-6/01 2 Transport Phenomena in Fuel Cells Figure 1: Principle of operation of a SOFC. Ideally, the Gibbs free energy change of fuel oxidation is directly converted into electricity [1, 2] in a fuel cell. As is common in many kinds of fuel cells, the core component of a solid oxide fuel cell (SOFC) is a thin gas-tight ion conducting electrolyte layer sandwiched by a porous anode and cathode, as shown in Fig. 1. For a SOFC, this electrolyte is a solid oxide material that only allows the passage of charge-carrying oxide ions. To produce useful electrical work, free electrons released in the oxidation of a fuel at the anode must travel to the cathode through an external load/circuit. Therefore, the electrolyte must conduct ions while preventing electrons released at the anode from returning back to the cathode by the same route. The oxide ions are driven across the electrolyte by the chemical potential difference on the two sides of the electrolyte, which is due to the oxidation of fuel at the anode. This difference in the chemical potential is proportional to the electromotive force across the electrolyte, which, therefore, sets up a terminal voltage across the external load/circuit. The solid oxide electrolyte has sufficient ion conductivity only at high temperatures (from 600–1000 ◦ C). The high operating temperature of a SOFC also ensures rapid fuel-side reaction kinetics without requiring an expensive catalyst. In addition, the high temperature exhaust from a SOFC can be directed to a gas turbine (GT); thus, using a SOFC-GT hybrid system, one can achieve an efficiency of at least 66.3% based on the lower heating value (LHV, which means that the electrochemical product, water, is in a gaseous state) of the SOFC [3–6]. Since it operates via transport of oxide ions rather than that of fuel-derived ions, in principle, a SOFC can be used to oxidize a number of gaseous fuels. In particular, a SOFC can consume CO as well as hydrogen as its fuel, and therefore can be fueled with reformer gas containing a mix of CO and H2 [7, 8]. Recently, ammonia has also been reported as a fuel for SOFCs [9]. Since a SOFC operates under high temperatures, its energy conversion efficiency and component safety are both of concern to industry. In the following sections, the issues to be discussed will include: (1) the thermodynamic and electrochemical fundamentals of the energy conversion and species variation, (2) the potential losses in practical operation, (3) the influence of fluid flow and heat and mass transfer on operational efficiency and safety, and (4) the creation of a numerical model to simulate the performance and the fields of flow, temperature, and species concentration. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 3 Multiple transport processes in solid oxide fuel cells 2 Thermodynamic and electrochemical fundamentals for solid oxide fuel cells To study the energy conversion efficiency and distribution of the conversion processes in a fuel cell, one must understand the basic principles. The chemical potential and, thereof, electromotive force across the electrolyte involve the interrelation of thermodynamics, electrochemistry, ion/electron conduction, and heat/mass transfer. In this section, the fundamentals of thermodynamics and the electrochemistry for a solid oxide fuel cell system are reviewed. The isothermal oxidation of a fuel A with oxidant B can be expressed by the following equation: aA + bB + · · · → xX + yY + · · ·. (1) The systematic changes of enthalpy, Gibbs free energy, and entropy production in the reaction are related by H = T S + G. (2) In a solid oxide fuel cell, the operating temperature is from 600 ◦ C to 1000 ◦ C and the pressure of gases is relatively not high. Thus, the gas species of reactants and products can be treated as ideal gases, which allows the chemical enthalpy change to be expressed as: H = (xhX + yhY + · · ·) − (ahA + bhB + · · ·), (3) where the h is the specific enthalpy. When a gas is pure, ideal, and at 1 atm, it is said to be in its standard state. The standard state is designated by writing a superscript 0 after the symbol of interest [10]. The Gibbs free energy which pertains to one mole of a chemical species is called the chemical potential. For an ideal gas at temperature of T and pressure of p, the chemical potential is expressed as: g = g 0 + RT ln p , p0 (4) where R is the gas constant and p0 is the standard pressure of 1 atm. One may omit the p0 in the denominator of the logarithm in eqn (4), but in such a case, the pressure p must be measured in atm. The systematic change of the Gibbs free energy in eqn (1) can be expressed in terms of the standard state Gibbs free energy and the partial pressures of the reactants and products: G = (xgX + ygY + · · ·) − (agA + bgB + · · ·) = [xgX0 + ygY0 + · · ·] − [agA0 + bgB0 + · · ·] + RT ln = G 0 + RT ln (pX /p0 )x (pY /p0 )y · · · (pA /p0 )a (pB /p0 )b · · · (pX /p0 )x (pY /p0 )y · · · , (pA /p0 )a (pB /p0 )b · · · WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) (5) 4 Transport Phenomena in Fuel Cells where G 0 = (xgX0 + ygY0 + · · ·) − (agA0 + bgB0 + · · ·), (6) which is the Gibbs free energy change of the standard reaction at temperature T (i.e., with each reactant supplied and each product removed at the standard atmospheric pressure, p0 = 1 atm). The theoretical electromotive force (EMF) induced from the chemical potential (G) is the Nernst potential: E= −G −G 0 RT (pA /p0 )a (pB /p0 )b · · · , = + ln ne F ne F ne F (pX /p0 )x (pY /p0 )y · · · (7) where F(=96486.7 C/mol) is Faraday’s constant. The first part of the righthand side of the standard reaction is also called the ideal potential, which is denoted by: −G 0 , (8) E0 = ne F where ne is the number of electrons derived from a molecules of the fuel, when the fuel is oxidized in the reaction of eqn (1). While the Gibbs free energy change, −G, converts to electrical power, the entropy production, −T S, is the thermal energy that is released in the electrochemical oxidation of the fuel. Both the h and g 0 are solely functions of temperature for ideal gases, which are given in Tables 1(a) and 1(b) for the gas species involved in the reactions of a SOFC. While the electromotive force in a fuel cell is determinable from the chemical potentials as discussed above, the current to be withdrawn from a fuel cell, denoted by I , is directly related to the molar consumption rate of fuel and oxidant through the following expressions: mfuel = I nfuel e F ; mO2 = I 2 nO e F , (9) 2 where nO e is for oxygen, and is the number of electrons per b molecules of oxygen is the number of obtained in the electrochemical reaction (in eqn (1)), and nfuel e electrons derived per a molecules of the fuel. 2.1 Operation with hydrogen fuel If a SOFC operates on hydrogen gas, the oxidation of hydrogen is the only electrochemical reaction in the fuel cell, which may be expressed by the following chemical equation: 1 H2 + O2 = H2 O (gas). (10) 2 WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 5 Table 1(a): Enthalpy and standard state Gibbs free energy of species. Formula weight CO 28.01 CO2 44.01 H2 2.016 T (K) h (kJ/k mol) g◦ (kJ/k mol) h (kJ/k mol) g◦ (kJ/k mol) h (kJ/k mol) g◦ (kJ/k mol) 298.15 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 −110530 −110476 −109892 −109309 −108725 −108141 −107555 −106967 −106377 −105887 −105195 −104599 −103102 −101588 −100053 −98507 −96938 −95353 −93749 −92129 −90499 −88840 −85495 −82100 −78662 −75187 −71680 −68145 −64585 −61004 −57404 −169474 −169816 −173796 −177819 −181877 −185927 −190035 −194201 −198337 −202671 −206763 −210999 −221737 −232568 −243573 −254677 −265838 −277193 −288569 −300119 −311659 −323340 −346965 −370940 −395082 −419587 −444280 −469265 −494515 −519824 −545324 −393510 −393441 −392687 −391916 −391128 −390326 −389507 −388675 −387827 −386966 −386094 −385205 −382938 −380603 −378207 −375756 −373250 −370704 −368112 −365480 −362821 −360113 −354626 −349037 −343362 −337614 −331805 −325941 −320030 −314079 −308091 −457254 −457641 −461967 −466308 −470688 −475142 −479627 −484141 −488719 −493318 −497982 −502655 −514498 −526583 −538822 −551316 −564800 −576704 −589622 −602720 −615996 −629413 −656576 −684317 −712432 −741094 −770105 −799541 −829350 −859479 −889871 0 53 630 1209 1791 2373 2959 3544 4131 4715 5298 5882 6760 8811 10278 11749 13223 14702 16186 17676 19175 20680 23719 26797 29918 33082 36290 39541 42835 46169 49541 −38968 −39217 −41866 −44521 −47241 −49953 −52721 −55508 −58305 −61157 −64062 −66968 −74970 −81849 −89432 −97171 −104977 −112898 −121004 −129114 −137290 −145520 −162291 −179363 −196672 −214158 −231910 −249899 −268095 −286471 −305189 The Nernst potential from this electrochemical reaction will be: E(H2 +1/2O2 =H2 O) = RT ln( pH2 /p0 )anode 2F 2F 0 + ln( pO2 /p0 )0.5 cathode − ln( pH2 O /p )anode . 0 −G(H 2 +1/2O 2 =H2 O) + (11) The ideal chemical potentials at the temperature T (K) can be calculated from the data given by handbooks [11]. As a convenient reference, Table 1(c) gives the WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 6 Transport Phenomena in Fuel Cells Table 1(b): Enthalpy and standard state Gibbs free energy of species. O2 31.999 Formula weight T (K) 298.15 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 H2 O (Gas) 18.015 CH4 16.043 h (kJ/k mol) g◦ (kJ/k mol) h (kJ/k mol) g◦ (kJ/k mol) h (kJ/k mol) g◦ (kJ/k mol) 0 54 643 1234 1828 2425 3025 3629 4236 4847 5463 6084 7653 9244 10859 12499 14158 15835 17531 19241 20965 22703 26212 29761 33344 36957 40599 44266 47958 51673 55413 −61151 −61536 −65661 −69826 −74024 −78325 −82495 −86797 −91112 −95433 −99849 −104266 −115382 −126656 −138056 −149551 −161117 −172885 −184684 −196669 −208745 −220897 −245378 −270239 −295426 −320883 −346551 −372374 −398632 −424967 −451507 −241814 −241752 −241079 −240404 −239726 −239045 −238362 −237675 −236985 −236291 −235592 −234889 −233115 −231313 −229493 −227622 −225732 −223812 −221860 −219876 −217860 −215814 −211623 −207308 −202872 −198321 −193663 −188906 −184056 −179121 −174108 −298105 −298452 −302263 −306126 −309998 −313943 −317882 −321885 −325865 −329947 −334040 −338139 −348890 −359173 −369828 −380712 −391707 −402852 −414130 −425526 −436930 −448514 −471993 −495908 −520072 −544681 −569563 −594826 −620276 −646221 −672288 – −74448 −73718 −72974 −72213 −71432 −70631 −69808 −68962 −68094 −67202 −66287 −63892 −61356 −58671 −55853 −52897 −49818 −46613 −43296 −39866 −36336 −28981 −21274 −13254 −4956 3587 12347 21295 30406 39658 – −130398 −134166 −137948 −141801 −145684 −149591 −153598 −157578 −161658 −165698 −169837 −180327 −191016 −201899 −213073 −224347 −235898 −247638 −259566 −271666 −283936 −309041 −334834 −361264 −388416 −416113 −444293 −473235 −502574 −532432 ideal chemical potentials and enthalpies for the gas species that are typically utilized in a SOFC. Recognizing the electrochemical equilibrium in the anode gas mixture: 0 + RT ln( pH2 /p0 )anode + ln( pO2 /p0 )0.5 G = −G(H anode 2 +1/2O2 =H2 O) − ln( pH2 O /p0 )anode = 0. (12) WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 7 Table 1(c): Change of enthalpy and standard state Gibbs free energy of reactions. Reaction T (K) 298.15 300 320 340 360 380 400 420 440 460 480 500 550 600 650 700 750 800 850 900 950 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 H2 + 1/2O2 = H2 O (gas) CH4 + H2 O = 3H2 + CO CO + H2 O = H2 + CO2 H (kJ/mol) G 0 (kJ/mol) E0 (V) H (kJ/mol) G 0 (kJ/mol) H (kJ/mol) G 0 (kJ/mol) −241.814 −241.832 −242.031 −242.230 −242.431 −242.631 −242.834 −243.034 −243.234 −243.430 −243.622 −243.813 −243.702 −244.746 −245.201 −245.621 −246.034 −246.432 −246.812 −247.173 −247.518 −247.846 −248.448 −248.986 −249.462 −249.882 −250.253 −250.580 −250.870 −251.127 −251.356 −228.561 −228.467 −227.567 −226.692 −225.745 −224.828 −223.914 −222.979 −222.004 −221.074 −220.054 −219.038 −216.229 −213.996 −211.368 −208.766 −206.172 −203.512 −200.784 −198.078 −195.268 −192.546 −187.013 −181.426 −175.687 −170.082 −164.378 −158.740 −152.865 −147.267 −141.346 1.184 1.184 1.179 1.175 1.170 1.165 1.160 1.155 1.150 1.146 1.140 1.135 1.121 1.109 1.095 1.082 1.068 1.055 1.040 1.026 1.012 0.998 0.969 0.940 0.910 0.881 0.852 0.823 0.792 0.763 0.732 – 205.883 206.795 207.696 208.587 209.455 210.315 211.148 211.963 212.643 213.493 214.223 214.185 217.514 218.945 220.215 221.360 222.383 223.282 224.071 224.752 225.350 226.266 226.873 227.218 227.336 227.266 227.037 226.681 226.218 225.669 – 141.383 137.035 132.692 128.199 123.841 119.275 114.758 110.191 105.463 100.789 96.073 82.570 72.074 59.858 47.595 35.285 22.863 10.187 −2.369 −14.934 −27.450 −52.804 −78.287 −103.762 −128.964 −154.334 −179.843 −205.289 −230.442 −256.171 – −41.160 −41.086 −40.994 −40.886 −40.767 −40.631 −40.489 −40.334 −40.073 −40.009 −39.835 −39.961 −38.891 −38.383 −37.878 −37.357 −36.837 −36.317 −35.799 −35.287 −34.779 −33.789 −32.832 −31.910 −31.024 −30.172 −29.349 −28.554 −27.785 −27.038 – −28.590 −27.774 −26.884 −26.054 −25.225 −24.431 −23.563 −22.822 −21.857 −21.241 −20.485 −18.841 −16.691 −14.853 −13.098 −12.232 −9.557 −7.927 −6.189 −4.697 −3.079 0.091 3.168 6.050 9.016 11.828 14.651 17.346 20.095 22.552 Substituting eqn (12) into eqn (11), the Nernst potential in another form for the electrochemical reaction of eqn (10) is obtained: RT 0 0.5 E(H2 +1/2O2 =H2 O) = (13) ln( pO2 /p0 )0.5 cathode − ln( pO2 /p )anode . 2F Since the oxygen partial pressure at the anode is very low (on the order of 10−22 bar) due to the anode reaction [2], it does not cause an appreciable effect on the partial pressures of the other major species in the anode flow. Therefore, when calculating the partial pressures of hydrogen and water vapor for determining the Nernst potential of the electrochemical reaction of eqn (10), the oxygen partial pressure in anode flow stream is ignorable. Hereafter, for an electrochemical reaction WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 8 Transport Phenomena in Fuel Cells as in eqn (1), the common practice in determining the Nernst potential will be to use eqn (7), in which the partial pressure of oxygen on the cathode side and those of the fuel and product species on the anode side are used. The molar consumption rate of hydrogen and oxygen in the electrochemical reaction of eqn (10) can be easily derived from eqn (9) as: mH2 = I ; 2F mO2 = I . 4F (14) 2.2 Operation with methane through internal reforming and shift reactions As previously mentioned, it is necessary to have a high operating temperature in a solid oxide fuel cell in order to maintain sufficient ionic conductivity for the solid oxide electrolyte [2, 4]. This provides a favorable environment for the reforming of hydrocarbon fuels like methane. In fact, since a solid oxide fuel cell operates based on the transport of oxide ions through the electrolyte layer from the cathode side to the anode side, the reforming products of hydrogen and carbon monoxide in the fuel channel can both serve as fuels. Given this advantage, solid oxide fuel cells can directly utilize hydrocarbon fuels or, at least, methane as a pre-reformed or partly reformed gas with components of CH4 , CO, CO2 , H2 and H2 O. Therefore, the fuel reforming and shift reactions will occur in the fuel channel in a solid oxide fuel cell. The anode is, in fact, a good material to serve as the catalyst for such chemical reactions, since the high temperature in a SOFC means that no noble metals are needed for a catalyst [12]. If there are five gas species, CH4 , CO, CO2 , H2 , and H2 O, in the fuel channel, the solid oxide fuel cell will operate with internal reforming and shift reactions. Therefore, the electrochemical reaction and the coexisting chemical reactions of reforming and shift need to be considered for determining the species’mole fractions (which are crucial to the electromotive forces in the fuel cell). Reforming : Shift : CH 4 + H2 O ↔ CO + 3H2 . CO + H2 O ↔ CO2 + H2 . (15) (16) Since the high operating temperature of a SOFC ensures rapid fuel reaction kinetics, it is a common practice to assume that the reforming and shift reactions are in chemical equilibrium [4] when determining the mole fractions of the species, which makes the computation significantly convenient. From the concept of chemical equilibrium, the reactants and products must satisfy the condition of G = 0. Therefore, the mole fractions or partial pressures of the five gas species in the fuel stream are related through the following two simultaneous equations [13]: p 3 KPR = pCO p0 p p CH4 H2 O p0 p0 H2 p0 = exp − 0 Greforming RT , WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) (17) Multiple transport processes in solid oxide fuel cells p KPS CO2 p0 p H2 0 Gshift p = exp − = H2 O pCO RT p0 p0 9 . (18) p0 The dominant electrochemical reaction has been reported to be the oxidation of H2 [12], which is primarily responsible for the electromotive force. However, at the same time, the electrochemical oxidation of the CO is also possible, and likely occurs to some extent in the solid oxide fuel cell. It has been reported that fuel cells operated by using mixtures of CO and CO2 have shown that the electrochemical oxidation of CO is an order of magnitude slower than that of hydrogen [14]. Nevertheless, there is no necessity to distinguish whether the electrochemical oxidation process involves H2 or CO in order to formulate the electromotive force. The following discussion will clarify this point. When the shift reaction of eqn (16) in the anodic gas is in chemical equilibrium, there is pCO2 pH2 0 0 G = gCO2 + RT ln + gH2 + RT ln p0 p0 pCO pH2 O 0 0 − gCO + g = 0. (19) + RT ln + RT ln H2 O p0 p0 Rearranging this equation gives: pCO2 pCO 0 0 gCO − g + RT ln + RT ln CO 2 p0 p0 pH2 O pH2 0 0 = gH2 O + RT ln − gH2 + RT ln . p0 p0 (20) 1/2 Subtracting a term of [(1/2)gO0 2 +RT ln( pO2 /p0 )cathode ] from both sides of eqn (20), results in: pCO pO2 1/2 1 0 pCO2 0 0 gCO2 − gCO − gO2 + RT ln − RT ln − RT ln 2 p0 p0 p0 cathode 1 pH2 O = gH0 2 O − gH0 2 − gO0 2 + RT ln 2 p0 pH2 pO2 1/2 − RT ln − RT ln . (21) p0 p0 cathode It is easy to see that the left-hand side of eqn (17) is the Gibbs free energy change of the electrochemical oxidation of CO, and the right-hand side is that for H2 . Dividing by (2F) on both sides, eqn (17) is further reduced to: E(H2 +1/2O2 =H2 O) = E(CO+1/2O2 =CO2 ) , WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) (22) 10 Transport Phenomena in Fuel Cells where E(H2 +1/2O2 =H2 O) is given in eqn (11), while the Nernst potential for the electrochemical oxidation of CO is E(CO+1/2O2 =CO2 ) = RT ln (pCO /p0 )anode 2F 2F 0 + ln (pO2 /p0 )0.5 cathode − ln (pCO2 /p )anode . 0 −G(CO+1/2O 2 =CO2 ) + (23) It is preferable that the EMF of an internal reforming SOFC be calculated from the electrochemical oxidation of H2 ; however, the species’consumption and production are the results collectively determined from the reactions of eqns (10), (15) and (16). The above discussion clearly indicates that the electrochemical reaction can be assumed to be driven by the hydrogen, and the electrochemical fuel value of CO is readily exchanged for hydrogen by the shift reaction under chemical equilibrium. Therefore, only H2 is considered as the electrochemical fuel in the following analysis, and CO only takes part in the shift reaction. For convenience, the mole flow rates of CH4 , CO and H2 are denoted by their formulae. Assuming that, x̄, ȳ, and z̄ are the mole flow rates, respectively, for the CH4 , CO, and H2 that are consumed in the three reactions given by eqns (15), (16) and (10) in the fuel channel, the coupled variations of the five species between the inlet and the outlet of an interested section of fuel channel are in the following forms [8, 15]: CH4 out = CH4 in − x̄, (24) COout = COin + x̄ − ȳ, (25) = CO2 + ȳ, in (26) H2 out = H2 in + 3x̄ + ȳ − z̄, (27) H2 Oout = H2 Oin − x̄ − ȳ + z̄. (28) CO2 out The overall mole flow rate of fuel, denoted by Mf , will vary from the inlet to the outlet of the section of interest in the fuel channel in the form of Mfout = Mfin + 2x̄. (29) Meanwhile, the partial pressures of the species, proportional to the mole fractions, must satisfy eqns (15) and (16) at the outlet of the section, which thus gives: KPR = COin +x̄−ȳ Mfin +2x̄ CH4 in −x̄ Mfin +2x̄ H2 in +3x̄+ȳ−z̄ Mfin +2x̄ 3 2 H2 Oin −x̄−ȳ+z̄ Mfin +2x̄ p p0 , WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) (30) Multiple transport processes in solid oxide fuel cells KPS = H2 in +3x̄+ȳ−z̄ Mfin +2x̄ COin +x̄−ȳ Mfin +2x̄ CO2 in +ȳ Mfin +2x̄ 11 H2 Oin −x̄−ȳ+z̄ Mfin +2x̄ , (31) where the p is the overall pressure of the fuel flow in the section of interest. Since, as discussed in the preceding section, the oxidation of H2 is responsible for the electrochemical reaction, the consumption of hydrogen is directly related to the charge transfer rate, or current, I , across the electrolyte layer: z̄ = I /(2F). (32) From the electrochemical reaction, the molar consumption of oxygen on the cathode side can be calculated by using eqn (14). By finding a simultaneous solution for eqns (30)–(32), the species variations, x̄, ȳ and z̄, can be determined. Finally, with the reacted mole numbers of CH4 and CO determined, the heat absorbed in the reforming reaction and released from the shift reaction can be obtained: QReforming = H Reforming · x̄, (33) QShift = H Shift · ȳ. (34) Nevertheless, prior to finding a solution for eqns (30) and (31), the electric current of the fuel cell in eqn (32) must be known. This demonstrates that the processes in a SOFC feature a strong coupling of the species molar variation and the electromotive force, as well as interdependency of the ion conduction and current flow. The ion transfer rate or current conduction in a SOFC will be discussed in Section 3. 3 Electrical potential losses The ideal efficiency is never attained in practical operation for any fuel cell. In fact, there are three potential drops in a fuel cell that cause the actual output potential to be lower than the ideal electromotive forces of the electrochemical reaction. The nature of the fuel cell performance in response to loading condition can be realized by its polarization curve, typically shown as in Fig. 2. With an increase in current density, the cell potential experiences three kinds of potential losses due to different dominant resistances. The potential drop due to the activation resistance, which is the activation polarization, is associated with the electrochemical reactions in the system. Another potential drop comes from the ohmic resistance in the fuel cell components, when the ions and electrons are conducted in the electrolyte and electrodes, respectively. The third drop, which can be sharp at high current densities, is attributable to the mass transport resistance, or concentration polarization, in the flow of the fuel and oxidant. It is known from observing the Nernst equation that the electromotive force of a fuel cell is a function of the temperature and the gas species’ partial pressures at the electrolyte/electrode WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 12 Transport Phenomena in Fuel Cells Figure 2: Over-potential in the operation of a fuel cell. interfaces, which are directly proportional to their mole fractions. It is important to note that, in the fuel stream, fuel must be transported or diffused from the core region of the stream to the anode surface, and, also, the product of the electrochemical reaction must conversely be transported or diffused from the reaction site to the core region of the fuel flow. On the cathode side, oxygen must be transported and diffused from the core region of airflow to the cathode surface. Along with the fuel and air streams, the consumption of reactants or development of products will make the mole fractions of reactants decrease and those of the product increase. Due to these resistances in the mass transport process, the feeding of reactants and removing of products to/from the reaction site can only proceed under a large concentration gradient between the bulk flow and the electrode surface when the current density is high, which therefore induces a sharp drop in the fuel cell potential. As a consequence of all the above-mentioned potential drops, extra thermal energy will be released together with the heat (−T S) from systematic entropy production. The heat transfer issues in a solid oxide fuel cell will be considered later in Section 4. 3.1 Activation polarization The activation polarization is the electronic barrier that must be overcome prior to current and ion flow in the fuel cell. Chemical reactions, including electrochemical reactions, also involve energy barriers, which must be overcome by the reacting species. The activation polarization may also be viewed as the extra potential necessary to overcome the energy barrier of the rate-determining step of the reaction to a value such that the electrode reaction proceeds at a desired rate [16–18]. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 13 The Butler-Volmer equation is a well-known expression for the activation polarization, ηAct : ne FηAct ne FηAct i = i0 exp β − exp −(1 − β) , (35) RT RT where β, which is usually 0.5 for the fuel cell application [16], is the transfer coefficient; i is the actual current density in the fuel cell; and i0 is the exchange current density. The transfer coefficient is considered to be the fraction of the change in polarization that leads to a change in the reaction-rate constant. The exchange current density, i0 , is the forward and reverse electrode reaction rate at the equilibrium potential. A high exchange current density means that a high electrochemical reaction rate and good fuel cell performance can be expected. The ne in eqn (35) is the number of electrons transferred per reaction, which is 2 for the reaction of eqn (10). Substituting the value of β = 0.5 into eqn (35), one can obtain a new expression as follows: ne FηAct i = 2i0 sinh (36) 2RT from which the activation polarization can be expressed as: 2 i i 2RT 2RT i ηAct = or ηAct = + + 1. sinh−1 ln ne F 2i0 ne F 2i0 2i0 (37) For a high activation polarization, eqn (37) can be approximated as the simple and well-known Tafel equation [16]: 2RT i ηAct = . (38) ln ne F i0 On the other hand, if the activation polarization is small, eqn (37) can be approximated as the linear current-potential expression [16]: ηAct = 2RT i . ne F i0 (39) Nevertheless, eqn (37) is recommended for its integrity and accuracy in calculating the activation polarization. The value of the exchange current density (i0 ) is different for the anode and cathode, and is also dependent on the electrochemical reaction temperature, the partial pressures of the gases [18, 19], and the electrode materials. The determination for i0 shows diversity in different literature [16–22]. There are formulations available in the literature [18–20], but some parameters used in the formulation are not well documented. On the other hand, empirical estimation of i0 is also WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 14 Transport Phenomena in Fuel Cells made in literature, like the work of Chan et al. [16]. An i0 of 5300 A/m2 for the anode and 2000 A/m2 for the cathode for a SOFC were used; however there was no comment on the selection methodology for setting these values in the paper [16]. Keegan et al. [17] also adjusted the i0 so as to obtain a simulation result to satisfy their experimental data; no report, however, is given about the adjusted i0 values in their paper. The present authors used a slightly higher value of 6300 A/m2 and 3000 A/m2 [23, 24], respectively, for the i0 of the anode and cathode, which resulted in very good agreement between the numerical simulated cell terminal voltage and experimental results from different researchers [25–28]. Nevertheless, i0 varies according to the temperature and pressures of the electrochemical reaction. For SOFCs working at temperature from 800 ◦ C–1000 ◦ C and pressures up to 15 atm, an i0 of 5300–6300 A/m2 for the anode and 2000–3000 A/m2 for the cathode are recommended from the study by the present authors [23]. 3.2 Ohmic loss The ohmic loss comes from the electric resistances of the electrodes and the current collecting components, as well as the ionic conduction resistance of the electrolyte layer. Therefore, the conductivity of the materials for the cell components and the current collecting pathway are the two factors most influential to the overall ohmic loss of a SOFC. In state-of-the-art SOFC technology, lanthanum manganite suitably doped with alkaline and rare earth elements is used for the cathode (air electrode) [20, 27], yttria stabilized zirconia (YSZ) has been most successfully employed for electrolyte, and nickel/YSZ is applied over the electrolyte to form the anode. Temperature could significantly affect the conductivity of SOFC materials. Especially for the electrolyte, for example, the resistivity could be two orders of magnitude smaller if its temperature increases from 600 ◦ C to 1000 ◦ C. The equations of resistivity for SOFC components suggested in literature are collected in Table 2. Table 2: Data and equations for resistivity of SOFC components. Cathode ( · cm) Bessette et al. [29] Ahmed et al. [30] Nagata et al. [18] Ferguson et al. [31] Electrolyte ( · cm) Anode ( · cm) Interconnect ( · cm) 0.008114e500/T 0.00294e10350/T 0.00298e−1392/T – ∗ 0.0014 0.3685 + 0.002838e10300/T ∗ 0.0186 ∗ 0.5 ∗ 0.1 10.0e[10092(1/T −1/1273)] ∗ 0.013 ∗ 0.5 T e1200/T 4.2×105 1 e10300/T 3.34×102 T e1150/T 9.5×105 T e1100/T 9.3×104 ∗At temperature of 1000 ◦ C. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 15 A careful check for the equations in Table 2 was conducted. The expressions by Bessette et al. [29] were found to be reliable, and to give nearly identical predictions as those by Ahmed et al. [30] and Nagata et al. [18]. The predicted data for anode resistivity by the equation of Ferguson et al. [31] shows significant discrepancies with the predictions by other equations. It is rational to assume that the passage of the charge-carrying species through the electrolyte, or the ion conduction through the electrolyte, is a charge transfer, like a current flow. In a planar type SOFC, as shown in Fig. 3, the current collects through the channel walls, also called ribs, after it moves perpendicularly across the electrolyte layer. The network circuit for current flow modeled by Iwata et al. [19] considers the channel walls as current collection pathways in a planar SOFC. However, the height and the width of the gas channel are both small (less than 3 mm), and the electric resistance through the channel wall might be negligible [30]. This simplification leads to the consideration that the current is almost exclusively perpendicularly collected, which means that the current flows normally to the trilayer of the cathode, electrolyte and anode. When calculating the local current density, the ohmic loss is thus simply accounted for in the following way [30]: I = A · (E − ηaAct − ηcAct ) − Vcell , (δa ρea + δe ρee + δc ρec ) (40) where A is a unit area on the anode/electrolyte/cathode tri-layer, through which the current I passes; δ is the thickness of the individual layers; ρe is the resistivity of the electrodes and electrolyte; Vcell is the cell terminal voltage; and the denominator of the right-hand side term is the summation of the resistance of the tri-layer. The Joule heating due to current flow in the volume of A×δ is expressed for the anode in the form of a QOhmic = I 2 · (δa ρea /A). (41) This is also applicable to the electrolyte and cathode by replacing the thickness and resistivity accordingly. Figure 3: Schematic of a planar type SOFC. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 16 Transport Phenomena in Fuel Cells Figure 4: Schematic of a tubular SOFC. Figure 5: Ion/electron conduction network in a tubular SOFC. In case the current pathway is relatively long in a fuel cell, as, for example, in a tubular type SOFC (shown in Fig. 4), the current collects circumferentially, which leads to a much longer pathway [32] compared to that of a planar type SOFC. In order to account for the ohmic loss and the Joule heating of the current flow in the circumferential pathway, a network circuit [23, 25, 33, 34] for current flow may be adopted, as shown in Fig. 5. Because the current collection is symmetric in the peripheral direction in the cell components, only half of the tube shell is deployed with a mesh in the analysis. The local current routing from the anode to cathode through the electrolyte is determinable based on the local electromotive force, EMF, the local potentials in the anode and cathode, and the ionic resistance of the electrolyte layer, which yields the expression: I= E − ηaAct − ηcAct − (V c − V a ) , Re (42) where V a and V c are the potentials in the anode and cathode, respectively. Re is the ionic resistance of the electrolyte layer given a thickness of δe and a unit WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 17 area of A: Re = ρee · δe /A, (43) ρee where the is the ionic resistivity of the electrolyte. In order to obtain the local current across the electrolyte by using eqn (35), supplemental equations for V a and V c are necessary. Applying Kirchhoff’s law of current to any grid located in the anode, the equation associating the potential of the central grid point P with the potentials of its neighboring points (east, west, north, south) and the corresponding grid P in the cathode can be obtained: a VSa − VPa VEa − VPa VN − VPa VWa − VPa + + + Rae Raw Ran Ras VPc − VPa − (EP − ηPAct ) + = 0. ReP (44) In the same way for a grid point P in the cathode: c VSc − VPc VN − VPc VWc − VPc VEc − VPc + + + Rce Rcw Rcn Rcs VPa − VPc + (EP − ηPAct ) + = 0, ReP (45) where Ra and Rc are the discretized resistances in the anode and cathode respectively, which are determined according to the resistivity, the length of the current path, and the area upon which the current acts; ηPAct is the total activation polarization, including from both the anode side and the cathode side. With all of the equations for the discretized grids in both the cathode and anode given, a matrix representing the pair of eqns (37) and (38) can be created. When finding a solution for such a matrix equation for the potentials, the following approximations are useful: 1. At the two ends of the cell tube there is no longitudinal current flow, and, therefore, an insulation condition is applicable. 2. At the symmetric plane A–A, as shown in Figs 4 and 5, there is no peripheral current in the cathode and anode, unless the cathode or anode is in contact with nickel felt, through which the current flows in or out. 3. The potentials of the nickel felts are assumed to be uniform due to their high electric conductivities. 4. Since the potential difference between the two nickel felts is the cell terminal voltage, the potential at the nickel felt in contact with the anode layer can be assumed to be zero. Thus, the potential at the nickel felt in contact with the cathode will be the terminal voltage of the fuel cell. Once all the local electromotive forces are obtained, the only unknown condition for the equation matrix is either the total current flowing out from the cell or WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 18 Transport Phenomena in Fuel Cells the potential at the nickel felt in contact with the cathode. This highlights two approaches that can be taken when predicting the performance of a SOFC. If the total current taken out from the cell is prescribed as the initial condition, the terminal voltage can be predicted. On the other hand, one can prescribe the terminal voltage and predict the total current, i.e., the summation of the local current I across the entire electrolyte layer. Once the potentials are obtained in the electrode layer, the volumetric Joule heating in the electrode for a volume centered about P will be: (VWa − VPa )2 (VNa − VPa )2 1 (VEa − VPa )2 a q̇P = + + 2 Rae Raw Ran (V a − V a )2 (xP · r a · θP · δa ), + S a P (46) Rs (VWc − VPc )2 (VNc − VPc )2 1 (VEc − VPc )2 c q̇P = + + 2 Rce Rcw Rcn (VSc − VPc )2 (xP · r c · θP · δc ), + (47) Rcs P − V c + V a )2 (E − η P P P Act q̇Pe = (xP · r e · θP · δe ), (48) ReP where the r and δ with the corresponding superscripts of a , c , and e are the average, radius and thickness, respectively, for the anode, cathode and electrolyte, and xP and θP are the P-controlled mesh size in the axial and peripheral directions, as shown in Fig. 5. The volumetric heat induced from the activation polarization in the anode and cathode is: P,a a a q̇Act = IP · ηP,a Act /(xP · r · θP · δ ), (49) P,c c c = IP · ηP,c q̇Act Act /(xP · r · θP · δ ). (50) The thermodynamic heat generation occurring at the anode/electrolyte interface in the area around P is: QPR = (H − G) · IP /(2F). (51) 3.3 Mass transport and concentration polarization Due to their gradual consumption, the fractions of the reactants and oxidant will decrease, in the fuel and air streams, respectively, which will cause the electromotive force to decrease gradually along the flow stream. On the other hand, due to the mass transport resistance, the concentration of the gas species will encounter polarization in between the core flow region and the electrode surface, which will result in lower partial pressures for the reactants, but higher partial pressures for the products at the WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 19 Figure 6: The interrelation amongst concentration and other parameters. electrode surfaces. Therefore, the fuel cell terminal voltage will be lower than the ideal value that is indicated by the Nernst equation. At high cell current density, the increased requirements for the feeding of the reactants and removal of the products can make the concentration polarization higher, and, thus, the cell output potential will sharply decrease. In order to take the concentration polarization into account when calculating the electromotive force, the local partial pressures of the reactants and products at the electrode surface are used. However, this requires the solution of the concentration fields for the gas species in the fuel and oxidizer channels, which might be either simply based on a one-dimensional [35–37] or else based on a complicated twoor three-dimensional solution for the mass conservation governing equations [23, 38–40]. In fact, the concentration fields are strongly coupled with the gas flow, temperature, and the distribution of the electromotive force in the ways indicated in Fig. 6. First, the gas species mass fraction determines the gas properties in the flow field, while the flow fields affect the gas species concentration distribution and temperature. Second, the gas species concentration field and temperature distribution determines the electromotive forces, while the ion/electron conduction due to the electromotive force determines the mass variation and heat generations in the fuel cell. The inter-dependency of these parameters will be discussed in detail in the following section when modeling a SOFC in order to predict both the fuel cell performance and the detailed distributions of the temperature, gas species concentration, and flow fields. 4 Computer modeling of a tubular SOFC An operation curve for a SOFC that characterizes the average current density versus the terminal voltage is very important when designing a SOFC system or a hybrid WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 20 Transport Phenomena in Fuel Cells SOFC/GT system [41–44]. Other information, like the temperature and concentration fields in a SOFC, is also of high concern for the safe operation of both the SOFC itself and the downstream facilities if a hybrid system is under consideration. Although there have been some experimental data generated about the operational performance and temperature of SOFCs [25–28], rigorous experimental testing for a SOFC is still rather tough because of its high operating temperature. Therefore, numerical modeling of SOFCs is very necessary. The purpose of computer simulation for a SOFC is to predict the operational characteristics in terms of the average current density versus the terminal voltage (based on prescribed operating conditions). The operating conditions of a SOFC are solely determined by fixing the flow rates and the thermodynamic state of the fuel and oxidant, as well as a load condition such as terminal voltage, the current being withdrawn, and external load [45]. The flow rates and thermodynamic conditions of the fuel and oxidant may be called internal conditions, and the terminal voltage, current to be withdrawn, and external load may be designated external conditions. Like any kind of “battery,” the external load condition of a SOFC determines the consumption of the fuel/oxidant and the generation of products in the electrochemical reaction [46]; the only difference in a fuel cell is its continuous feeding of fuel/oxidant and removal of products and waste species. According to the different ways of prescribing the external parameters, the following three schemes might be designed in order to predict the other unknown parameters when constructing a numerical model for a SOFC: (1) Use the internal conditions and terminal voltage to predict the total current to be withdrawn. (2) Use the internal conditions and current to be withdrawn to predict the terminal voltage. (3) Use the internal conditions and external load to predict the terminal voltage and current density. The cost of iterative computation using the three schemes is quite different. In the first scheme, the cell terminal voltage is known, and thus the local current can be obtained, for example, by using eqn (42) and solving eqns (44) and (45), for a planar and tubular type SOFC, respectively, once the temperature and partial pressure fields of the gas species are available. The integrated value from the local current will be the total current to be withdrawn from the SOFC. In the second scheme, however, the terminal voltage needs to be assumed, and then checked by integrating the total current from the local current until the calculated total current agrees with the prescribed value. In this computation process, a proper method is needed to find the best-fit terminal voltage iteratively. The third scheme resembles the second scheme, in that one needs to assume a terminal voltage to find the total current. The computation will be stopped only when the voltage-current ratio equals the prescribed load. With an understanding of the principles of the energy conversion, chemical equilibrium, potential loss, and the operation of a SOFC, a computer model for a SOFC can now be constructed. Generally speaking, the modeling and computation for a tubular SOFC and a planar SOFC share rather common features except for the Ohmic losses and Joule heating, for which differences result from the different structures variation in the current pathway in the electrodes. Relatively speaking, WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 21 the tubular SOFC has a more complex current pathway in the electrodes [27] and will be discussed in the following analysis. Modeling works on planar type SOFCs are available in both the current author’s work and in the literature [17, 19, 30, 31, 39, 47–49]. In the following subsections, there are three issues that address the construction of a numerical model. 4.1 Outline of a computation domain In a practical tubular SOFC stack, multiple tubular cells are mounted in a container to form a cell bundle, as shown in Fig. 7. A pre-reformer might be put adjacent to the cell bundles [50, 51]. In order to conduct a modeling study with relatively less complexity, it is assumed that most of the single tubular SOFCs operate under the same environment of temperature and concentrations of gas species. This allows the definition of a controllable domain in the cross-section, which pertains to one single cell, as outlined by the dashed-line square in Fig. 7. It is then specified that there must be no flow velocity and fluxes of heat and mass across the outline. This will significantly simplify the analysis for a cell stack. Through analysis of the heat/mass transfer and the chemical/electrochemical performance for the single cell and its controllable area, one can obtain results very useful for evaluating the performance of an entire cell stack. Also considering the longitudinal direction, the heat and mass transfer in the above outlined square area enclosing the tubular SOFC are three-dimensional in nature. For a solution of the three-dimensional governing equations of momentum, energy, and species conservation, a large number of discretized mesh points are necessary, which results in an unacceptably heavy computational load. In order to reduce computational cost, the square area enclosing the tubular SOFC is approximated to be an equivalent circular area; therefore, the domain enclosing the single tubular SOFC is viewed as a 2-dimensional axi-symmetric one, as seen in Fig. 7. Figure 7: Schematic of a tubular SOFC in a cell stack. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 22 Transport Phenomena in Fuel Cells Figure 8: Computation domain for a tubular SOFC. It should be noted, though, that the zero-flux, or insulation of heat and mass transfer at the boundary remains unchanged, even given this geometric approximation. From the preceding discussions, an axi-symmetrical two-dimensional (x − r) computation domain is profiled as shown in Fig. 8, which includes two flow streams and a solid area of the cell tube and air-inducing tube. 4.2 Governing equations and boundary conditions Since the mass fractions of the species vary in the flow field, all of the thermal and transport properties of the fluids are functions of the local species concentration, temperature, and pressure; therefore, the governing equations for momentum, energy, and species conservation (based on mass fraction) have variable thermal and transport properties: ∂(ρu) 1 ∂(rρv) + = 0, ∂x r ∂r ∂(ρuu) 1 ∂(rρvu) ∂p ∂ ∂u 1 ∂ ∂u + =− + µ + rµ ∂x r ∂r ∂x ∂x ∂x r ∂r ∂r ∂ ∂u 1 ∂ ∂v + µ + rµ , ∂x ∂x r ∂r ∂x ∂(ρuv) 1 ∂(rρvv) ∂p ∂ ∂v 1 ∂ ∂v + =− + µ + rµ ∂x r ∂r ∂r ∂x ∂x r ∂r ∂r ∂ ∂u 1 ∂ ∂v 2µv + µ + rµ − 2 , ∂x ∂r r ∂r ∂r r ∂(ρCpuT ) 1 ∂(rρCpvT ) ∂ ∂T 1 ∂ ∂T + = λ + rλ + q̇, ∂x r ∂r ∂x ∂x r ∂r ∂r WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) (52) (53) (54) (55) Multiple transport processes in solid oxide fuel cells ∂ ∂(ρuYJ ) 1 ∂(rρvYJ ) ∂YJ + = ρDJ ,m ∂x r ∂r ∂x ∂x 23 1 ∂ ∂YJ + rρDJ ,m + SJ . (56) r ∂r ∂r These equations are applied universally to the entire computation domain; however, zero velocities will be assigned to the solid area in the numerical computation. In the energy conservation equation, thermal energy from the chemical and electrochemical reactions (expressed by eqns (33), (34), (51)) and the Joule heating in electrodes and electrolyte (expressed by eqns (46)–(50)), represented by q̇, are introduced as source terms in the proper locations in the fuel cell. Some terms due to energy diffusion driven by the concentration diffusion of the gas species are very small, and thus neglected [52, 53]. The boundary conditions for the momentum, heat and mass conservation equations are as follows: 1. On the symmetrical axis, or at r = 0: v = 0, and ∂φ/∂r = 0, where φ represents general variables except for v. 2. At the outmost boundary of r = rfo : there are thermally adiabatic conditions; impermeability for species and non-chemical reaction are also assumed, which gives v = 0, and ∂φ/∂r = 0, where φ represents general variables except for v. 3. At x = 0: the fuel inlet has a prescribed uniform velocity, temperature, and species mass fraction; the solid part has u = 0, v = 0, ∂T /∂x = 0, and ∂YJ /∂x = 0. 4. At x = L: the air inlet has a prescribed uniform velocity, temperature and species mass fraction; the gas exit has v = 0, ∂u/∂x = 0, ∂T /∂x = 0, and ∂YJ /∂x = 0; the tube-end solid part has u = 0, v = 0, ∂T /∂x = 0, and ∂YJ /∂x = 0. 5. At the interfaces of the air/solid, r = rair , and fuel/anode, r = rf : u = 0 is assumed. In the fuel flow passage, the mass flow rate increases along the x direction due to the transferring in of oxide ions. Similarly, a reduction of the air flow rate occurs in the air flow passage, due to the ionization of oxygen and the transferring of the oxide ions to the fuel side. Therefore, radial velocities at r = rair and r = rf are: fuel,species ṁx vf = (57) r=rf , fuel ρx air,species ṁx r=r , (58) vair = air air ρx where ṁ [kg/(m2 s)] is mass flux of the gas species at the interface of the electrodes and fluid, which arises from the electrochemical reaction in the fuel cell. The mass fractions of all participating chemical components at the boundaries of r = rair and r = rf are calculated with consideration of both diffusion and convection effects [54, 55]: ∂YJ (59) + ρxair YJ vair , ṁJx ,air = −DJ ,air ρxair ∂r ∂YJ ṁJx , fuel = −DJ , fuel ρxfuel + ρxfuel YJ vf . (60) ∂r WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 24 Transport Phenomena in Fuel Cells Table 3: Properties of SOFC materials. Thermal conductivity (W/(m K)) Cathode Electrolyte Anode Support tube Air-inducing tube Interconnector aAhmed Cp (J/(kg K)) Density (kg/m3 ) d 11; c 2.0; b 2.0 b 623 a 4930 d 2.7; c 2.7; b 2.0 b 623 a 5710 c 11.0; d 6.0; b 2.0 b 623 a 4460 b 800 a 6320; b 7700 c 1.0 c 1.0 b 13; c 2.0; d 6.0 et al. [30]; b Recknagle et al. [39]; c Nagata et al. [18]; d Iwata et al. [19]. It is worth noting that the mass fluxes for the species in the above equations, eqns (57)–(60), strongly relate to the ion/electron conduction; the determination of mass variation and related mass flux that arise from the electrochemical reaction has been discussed (as expressed by eqns (30)–(32)) in Section 2. As a consequence, the mass/mole fraction at the solid/fluid interface, derived from eqns (59) and (60), will be used for the determination of the partial pressures and, thereof, the local electromotive forces by eqn (11). The properties of solid materials in a SOFC are given in Table 3, which show some variation based on the different literature sources. The single gas properties are available from references [11] and [56]. For gas mixtures, equations from references [11, 57] are available, and some selected equations from reference [11] for calculating the properties are listed in the following section. The mixing rule for the viscosity is: 1/2 1/4 2 n Mj µi X i µi Mi −1/2 1 n µm = 1+ ; φij = 1/2 1 + , M µ Mi X φ 8 j j j ij j=1 i=1 (61) where µm (Pa · sec) is the viscosity for the mixture, and µi or µj are the viscosities of individual species (Pa · sec); Mi or Mj is the molecular weight of a species; Xi or Xj is the mole fraction; and when i = j, φij = 1. The mixing rule for the thermal conductivity of gases at atmospheric pressure or less is: n X i ki n ; km = j=1 Xj Aij i=1 (62) 1/2 2 3/4 T + Sij µ i Mj T + Si 1 Aij = 1+ , 4 µj M i T + Sj T + Si WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 25 where km [W/(m · K)] and ki [W/(m · K)] are the thermal conductivities of the mixture and species; Sij = C(Si Sj )1/2 , and C = 1.0, but when either or both components i and j are very polar, C = 0.73; for helium, hydrogen, and neon, Si or Sj is 79 K; otherwise, Si = 1.5Tbi and Sj = 1.5Tbj , where Tb is the boiling point temperature of species; and the unit of T is K. When the gas mixture is above atmospheric pressure, the following correction is applied to the km obtained above: k = k + A × 10−4 (eBρr + C) , 1/6 Tc M 1/2 5 Z 2/3 c (63) Pc ρr < 0.5, 0.5 < ρr < 2.0, 2.0 < ρr < 2.8, A = 2.702, A = 2.528, A = 0.574, B = 0.535, B = 0.670, B = 1.155, C = −1.000, C = −1.069, C = 2.016, where k [W/(m · K)] is the gas thermal conductivity at the temperature T (K) and pressure P of interest in the mixture; k [W/(m · K)] is the thermal conductivity at T and atmospheric pressure obtained by eqn (62); ρr = Vc /V is the reduced density; Vc (m3 /kmol) is the critical molar volume; V (m3 /kmol) is the molar volume at T and P; Tc (K) is the critical temperature; M is the molecular weight; Pc (MPa) is the critical pressure; Zc = Pc Vc /(RTc ) is the critical compressibility factor; and R is the gas constant, which is 0.008314 MPa · m3 /(kmol · K). The mixture critical properties are obtained via the following equations: n Tcm − Tpc , (64) Xi ω i Pcm = Ppc + Ppc 5.808 + 4.93 Tpc i=1 Tcm = n j=1 Vcm = Xj Vcj n Tcj , i=1 Xi Vci i (65) φi φj vij (i = j), (66) j where 2/3 Xj Vcj ; φj = n 2/3 i=1 Xi Vci vij = Vij (Vci + Vcj ) ; 2.0 Vci − Vcj + C, Vij = −1.4684 V +V ci cj and C is zero for hydrocarbon systems and is 0.1559 for systems containing a non-hydrocarbon gas. In all the above equations, Xi or Xj is the mole fraction of a species in the mixture; ωi is the acentric factor of a species; Pcm , Tcm and Vcm are the mixture critical properties; and Ppc and Tpc are the pseudocritical properties of the mixture, which are expressed as: Tpc = n i=1 Xi Tci ; Ppc = n Xi Pci . i=1 WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) (67) 26 Transport Phenomena in Fuel Cells Table 4: Atomic diffusion volumes for use in eqn (68). Atomic and structural diffusion-volume increments v [11] C H O N Aromatic ring Heterocyclic ring 16.50 1.98 5.481 5.69 −20.2 −20.2 H2 N2 O2 CO CO2 H2 O 7.07 17.9 16.6 18.9 26.9 12.7 The gas diffusivity of one species against the remaining species of a mixture is expressed in the form of: Dim 1 − Xi = , jj=i Xj /Dij 0.5 0.01013T 1.75 M1i + M1j Dij = , P[( vi )1/3 + ( vj )1/3 ]2 (68) where units of T , P, and D are K, Pa, and m2 /sec, respectively; Mi or Mj is the molecular weight; and all vi or vj are group contribution values for the subscript component summed over atoms, groups and structural features, which are listed in Table 4. 4.3 Numerical computation In order to conduct a numerical computation for flow, temperature, and concentration fields in a SOFC, a mesh system with a sufficient grid number both in the r and x directions must be deployed at the computational domain. All the governing equations may be discretized by using the finite volume approach, and the SIMPLE algorithm can be adopted to treat the coupling of the velocity and pressure fields [58, 59]. The temperature difference between the cell tube and the air-inducing tube might be large enough to have radiation heat transfer; therefore, a numerical treatment based on the method introduced in the literature [60] can be used to consider the radiation heat exchange. As has been discussed at the beginning of Section 4, the computation may be based on the internal conditions and the current to be withdrawn; and, as a consequence of the simulation, the terminal voltage will be given as an output along with other operational details. The convenience of using this procedure in the simulation is discussed next. It is quite common in practice that the total current is prescribed in terms of the average current density of the fuel cell. Also, instead of the flow rates of fuel and air, the stoichiometric data are prescribed in terms of the utilization percentage of hydrogen and oxygen. This kind of designation of the operating conditions results in a convenient comparison of the fuel cell performance based on the same level of WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 27 average current density and the hydrogen and oxygen utilization percentage. The inlet velocities of fuel and air are, then, obtainable in the forms of: RTf Acell icell , (69) ufuel = 2FUH2 XH2 Afuel Pf RTair Acell icell , (70) uair = 4FUO2 XO2 Aair Pair where icell is the cell current density; Acell is the outside surface area of the fuel cell; Afuel and Aair are the cross-sectional inlet areas of the fuel and air; Pf , Pair and Tf , Tair are the inlet pressure and temperature of the fuel and air flows respectively; XH2 and XO2 are the mole fractions of hydrogen in the fuel and oxygen in the air, respectively; and UH2 and UO2 are the utilization percentage for hydrogen and oxygen. The computation process is highly iterative and coupled in nature. As the first step, the latest local temperature, pressure, and species’ mass fractions are used in the network circuit analysis to obtain the cell terminal voltage and local current across the electrolyte, and thus the local species’ transfer fluxes and local heat sources. In the second step, the local temperature, pressure and species’ mass fractions are, in turn, obtained through solution of the governing equations under the new boundary conditions determined by the latest-available species’fluxes and heat sources. The two steps iterate until convergence is obtained. 4.4 Typical results from numerical computation for tubular SOFCs The present authors have conducted numerical computations for three different single tubular SOFCs [23], which have been tested by Hagiwara et al. [26], Hirano et al. [25], Singhal [27], and Tomlins et al. [28]. The fuel tested by Hirano et al. [25] had components of H2 , H2 O, CO and CO2 ; therefore, there is a water-shift reaction of the carbon monoxide in the fuel cell to be considered together with the electrochemical reaction. The fuel used by the other researchers [26–28] had components of H2 and H2 O, where there is no chemical reaction except for the electrochemical reaction in the fuel channel. The dimensions of the three different solid oxide fuel cells tested in their studies are summarized in Table 5, in which the mesh size adopted in our numerical computation is also given. The operating conditions are listed in Table 6, including the species mole fractions and the temperature of the fuel and air in those tests, which are the prescribed conditions for the numerical computation. In the experimental work by Singhal [27], a test of the pressure effect was also conducted by varying the fuel and air pressure from 1 atm to 15 atm. It is expected that the experimental data for these SOFCs in different dimensions and operating conditions will facilitate a wide benchmark range for validation of the numerical modeling work. 4.4.1 The SOFC terminal voltage The computer calculated and the experimentally obtained cell terminal voltages under different cell current densities are shown in Fig. 9. The relative deviation of WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 28 Transport Phenomena in Fuel Cells Table 5: Example SOFCs with test data available. Data sequence: Outer diameter (mm)/Thickness (mm)/Length (mm) Singhal [27] Hagiwara et al. [26] Hirano et al. [25] Tomlins et al. [28] Air-inducing tube Support tube Cathode Electrolyte Anode Fuel boundary Grid number (r × x) 7.00/1.00/485 – 15.72/2.20/500 15.80/0.04/500 16.00/0.10/500 18.10/ – /500 66×602 6.00/1.00/290 13.00/1.50/300 14.40/0.70/300 14.48/0.04/300 14.68/0.10/300 16.61/ – /300 66×602 12.00/1.00/1450 – 21.72/2.20/1500 21.80/0.04/1500 22.00/0.10/1500 24.87/ – /1500 66×1602 Table 6: Species’ mole fractions, utilization percentages, and temperatures. Air O2 %−UO2 /N2 %/T (◦ C) I II III fuel H2 %–UH2 /H2 O%/CH4 %/CO%/CO2 %/T (◦ C) 21.00–17.00/79.00/600.0 98.64–85.00/1.36 /0/ 0 /0 /900.0 55.70–80.00/27.70/0/10.80/5.80/800.0 ∗∗ 21.00–25.00/79.00/400.0 55.70–80.00/27.70/0/10.80/5.80/800.0 21.00–17.00/79.00/600.0 98.64–85.00/1.36 /0/ 0 /0 /800.0 ∗ 21.00–25.00/79.00/600.0 ∗ Current density = 185 mA/cm2 ; ∗∗ Current density = 370 mA/cm2 . I: Tested by Hagiwara et al. [26]. II: Tested by Hirano et al. [25]. III: Tested by Singhal [27] and Tomlins et al.[28]. the model-predicted data from the experimental data is no larger than 1.0% for the SOFC tested by Hirano et al. [25], 5.6% for that by Hagiwara et al. [26], and 6.0% for that by Tomlins et al. [28]. It is interesting to observe from Fig. 9 that, under the same cell current density, the cell voltage of the SOFC tested by Hagiwara et al. [26] is the highest and that by Hirano et al. [25] is the lowest. The mole fraction of hydrogen in the fuel for the SOFC tested by Hirano et al. [25] is low, which might be the major reason that this cell has the lowest cell voltage. Because the current must be collected circumferentially in a tubular type fuel cell, the large diameter of the cell tube investigated by Singhal [27] and Tomlins et al. [28] will lead to a longer current pathway. Thus, the cell voltages of these cells are lower than those found by Hagiwara et al. [26], even though the former investigators tested the SOFCs at a pressurized operation of 5 atm, which, in fact, helps to improve the cell voltage. Under a current density of 300 mA/cm2 , the cell voltage and power increase with the increasing operating pressure, as seen in Fig. 10. The agreement between our WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 29 Figure 9: Results of prediction and testing for cell voltage versus current density. (The operating pressure of the cell tested by Hagiwara et al. [26] and Hirano et al. [25] is 1.0 atm, and that by Tomlins et al. [28] is 5 atm.) Figure 10: Effect of operating pressure on the terminal voltage and power. model-predicted results and the experimental ones by Singhal [27] is quite good, showing a maximum deviation of 7.4% at a low operating pressure. When the operating pressure increases from 1 atm to 5 atm, the cell output power shows a significant improvement of 9%. However, raising the operating pressure becomes less effective for improving the output power when the operating pressure is high. For example, the cell output power shows an increase of only 6% when the operating pressure increases from 5 atm to 15 atm. The reason for this is that the operating pressure contributes to the cell voltage in a logarithmic manner. Nevertheless, WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 30 Transport Phenomena in Fuel Cells pressurized operation of the fuel cell can improve the output power significantly. For example, when increasing the operating pressure from 1 atm to 15 atm, the cell output power can have an increment of 15.8%. There is no doubt from the above investigation that the investigators can satisfactorily predict the overall performance of a SOFC through numerical modeling and computation. On the basis of this good agreement with the overall fuel cell performance, the internal details of the flow, temperature, and concentration fields from numerical prediction can also be reliably presented. 4.4.2 Cell temperature distribution Because the measurement of temperature in a SOFC is very difficult, only three experimental data points, the temperature at the two ends and in the middle of the cell tube, were available from the work on Hirano et al. [25]. Figure 11 shows the simulated cell temperature distribution for the SOFC, for which Hirano et al. [25] provided the test data. The agreement of the simulated data and the experimental results is good in the middle, where the hotspot is located; relatively larger deviations between the predicted and experimental values appear at the two ends of the cell. Nevertheless, such a discrepancy is acceptable when designing a SOFC with respect to concerns about the prevention of excessive heat in the cell materials. The predicted temperature distributions for the fuel cells tested by Hagiwara et al. [26] and Tomlins et al. [28] are given in Fig. 12. Unfortunately, there was no experimental data on the cell temperature. Generally, the two ends of the cell tube have lower temperatures than the middle of the cell tube. However, at low current densities, the hotspot is located closer to the closed end of the cell. With an increase in current density, the hotspot shifts to the open-end side, and the hotspot temperature also decreases, which improves the uniformity of the temperature distribution along the fuel cell. It should be observed that the heat transfer between the cooling air and the cell tube at the closed-end region is dominated by laminar jet impingement, since the exit velocity from the air-inducing tube is quite low. However, the velocity of the exit air from the air-inducing tube affects the heat Figure 11: Longitudinal temperature distribution in the fuel cell. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 31 Figure 12: Predicted longitudinal temperature distribution for two SOFCs. transfer coefficient significantly. For the high current density case, the flow rate of air also becomes large accordingly. Thus, the heat transfer coefficient between the air and the fuel cell closed-end region is increased. This can suppress the temperature level of the closed-end region of the fuel cell significantly. Since the air receives a large amount of heat at the closed-end region, its cooling to the fuel cell in the downstream region becomes weak, and the uniformity of the cell temperature distribution becomes much better when the fuel cell operates at high current densities. 4.4.3 Flow, temperature and concentration fields Figure 13 shows the flow and temperature fields for the SOFC tested by Hirano et al. [25] at a current density of 185 mA/cm2 . The air speed in the air-inducing tube has a slight acceleration because the air absorbs heat and expands in this flow passage. After leaving the air-inducing tube, the air impinges on the closed end of the fuel cell, and then flows backwards to the outside. In this pathway, the air obtains heat from the heat-generating fuel cell tube and transfers the heat to the cold air in the air-feeding tube. It is easy to understand that the electrochemical reaction at the closed end of the fuel cell is strong because the concentrations of fuel and air are both high there. Therefore, the heat generation due to Joule heating and the entropy change of the electrochemical reaction is high at the upstream area of the fuel path. However, it is known from both experiments and computation that the closed-end region of the fuel cell does not demonstrate the highest temperature; therefore, it is believed that the cooling of the air in the closed-end area of the fuel cell is responsible for this. After being heated at the closed-end region, air exhibits a higher temperature, and its cooling ability to the cell tube is low when it is in the annulus between the air-inducing tube and the cell tube. At the cell open-end region, the air in the annulus can transfer heat to the incoming cold fresh air in the air-inducing tube, and this will help it to cool the fuel cell tube. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 32 Transport Phenomena in Fuel Cells Figure 13: Predicted flow and temperature fields for the SOFC reported by Hirano et al. [25] at a current density of 185 mA/cm2 . From this airflow arrangement, the hotspot temperature of the cell tube may mostly occur in the center region in the longitudinal direction of the cell tube. The airflow has two passes, incoming in the air-inducing tube and outgoing in the annulus between the air-inducing tube and the cell tube. The heat exchange in between the two passes allows the air to mitigate its temperature fluctuation in the whole air path, and thus the temperature field in the fuel cell might be maintained as relatively uniform. Nevertheless, the heat generation, air and fuel temperature, and air-cooling to the fuel cell will collectively affect the temperature field in the fuel cell. Therefore, the hot spot position in a cell tube might shift more or less away from the center region depending on the operating condition of the fuel cell. Figure 14 shows the gas species’mole fraction contours for the same SOFC under the same operating conditions as discussed with respect to Fig. 13. In the air path, oxygen consumption at the closed-end region is relatively large, which leads to more densely distributed contour lines. The contour shape of oxygen also indicates a relatively larger difference of the mole fraction between the bulk flow and the wall of the cathode/air interface. This implies that the mass transport resistance on the air side might be dominant in lowering the cell performance if the stoichiometry of the oxygen is low. Feeding more air than is needed is already well applied in operational fuel cell technology. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 33 Figure 14: Predicted fields of the mole fraction of the species for the SOFC reported by Hirano et al. [25] at a current density of 185 mA/cm2 . The hydrogen budget is collectively determined by the consumption by the electrochemical reaction and the generation from the water-shift reaction of CO. Since the consumption dominates, the hydrogen mole fraction decreases along the fuel stream. Corresponding to this hydrogen variation, consumption due to the water-shift reaction and production due to the electrochemical reaction cause the water vapor to increase gradually along the fuel stream. The water-shift of CO proceeds gradually in the fuel path, and thus the mole fraction of CO decreases but the CO2 increases. The shape of the contour lines of the species in the fuel path is WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 34 Transport Phenomena in Fuel Cells Figure 15: Predicted streamwise molar flow rate variation for species in the fuel channel for the SOFC reported by Hirano et al. [25]. relatively flat from the cell wall to the bulk flow. This indicates that mass diffusion in the fuel channel is relatively stronger than that in the airflow. For a further illustration of the variation of the gas species, Fig. 15 shows the molar flow rate variation along the fuel path. In one third of the length from the fuel inlet, the hydrogen flow rate shows a faster decrease and the water flow rate shows a faster increase, indicating a strong reaction in the upstream region. The flow rate of CO and CO2 vary roughly in a linear style, and a small amount of CO still exists in the waste gas. 5 Concluding remarks Fuel cell technology is currently under rapid development. To improve SOFC performance, for high power density and efficiency, efforts have been made to reduce the three over-potentials: activation polarization, ohmic loss, and concentration polarization. Better understanding of these three over-potentials is also very important in developing accurate computer models for predicting the overall performance and internal details of a SOFC. The activation polarization relates to the porous structure of the electrode and electrocatalyst materials. The state-of-the-art in material and manufacturing processes for the electrodes and electrolyte has been reported by Singhal [27]. The reduction of ohmic losses also heavily relies on the reduction of electronic and ionic resistances in the electrodes and electrolyte. A shorter current collection pathway also helps to reduce ohmic loss. A new design, referred to as a high power density solid oxide fuel cell (HPD-SOFC), has been developed by Siemens Westinghouse Power Corporation [27, 32], and has a significantly shorter current pathway, and WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 35 thus improves the power density significantly. A planar structure also promises to have a shorter current pathway and thus a higher power density, and measures for reducing the ohmic loss in a planar type SOFC have been reported by Tanner and Virkar [61]. The reduction of mass transport resistance, or the concentration polarization, has not been given much attention. Mass transfer enhancement has been reported to be effective in polymer electrolyte membrane fuel cells (PEMFCs) for obtaining a higher cell current density [62] before a sharp drop in cell voltage (which is due to excessive concentration polarization). It might also be possible for SOFCs to obtain a higher current density by means of mass transfer enhancement. In a numerical model of a SOFC, the precise calculation of the over-potentials is very important in order to accurately predict the overall current-voltage performance. The heat generation from the over-potentials is also significant in computing the temperature, flow, and species concentration fields. With respect to the activation polarization, studies elucidating the data and equations for the exchange current density are still needed. For the prediction of ohmic losses, reliable property data for electrodes are required. Additionally, a method for analyzing a complex network circuit in a SOFC needs to be developed. The concentration polarization is considered in the numerical computation by using the local mole fractions of the species at the interface of the electrode and fluid when calculating the electromotive force by the Nernst equation. Because the porous electrodes also serve as the reaction site, there is no well-described model for the mass transport resistance in the electrodes. Adopting a lower exchange current density, which induces a larger over-potential of the activation polarization, may be a way to incorporate the mass transport resistance in the electrodes into the activation polarization. The method given by Hirano et al. [25] for the consideration of mass transport resistances in the electrodes is convenient, but may be too simple and needs more investigation. With the progress being made in computer modeling of SOFCs, it is expected that costs for research and development of SOFCs will be significantly reduced by using computer simulations in the future. 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Nguyen T.V., A gas distributor design for proton-exchange-membrane fuel cells. J. Electrochem. Soc., 413(5), pp. L103–L105, 1996. Nomenclature a A Acell Aair , Afuel b B C Cp DJ ,m Stoichiometric coefficient of chemical species. Chemical species. Area (m2 ). General variable. Outer surface area of fuel cell (m2 ). Inlet flow area of air and fuel, respectively (m2 ). Stoichiometric coefficient of chemical species. Chemical species. General variable. General variable. Specific heat capacity at constant pressure [J/(kg K)]. Diffusion coefficient of jth species into the left gases of a mixture (m2 /s). WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) 40 Transport Phenomena in Fuel Cells E F g h H i i0 I k KPR , KPS L m ṁ M Mf ne p, P q̇ Q r ra , rc, re R Ra , Rc , Re S T u U v V Vcell V a, V c W x X x̄, ȳ, z̄ y Y z Z Electromotive force or electric potential (V). Faraday’s constant [96486.7 (C/mol)]. Gibbs free energy (J/mol). Chemical enthalpy (kJ/kmol) or (J/mol). Height (m). Current density (A/m2 ). Exchange current density (A/m2 ). Current (A). Thermal conductivity (W/m K). Chemical equilibrium constant for reforming and shift reactions, respectively. Length (m). Mass transfer rate or mass consumption/production rate (mol/s). Mass flux [mol/(m2 s)]. Molecular weight (g/mol). Total mole rate of fuel flow (mol/s). Number of electrons involved in per fuel molecule in oxidation reaction. Pressure (Pa) or position. Volumetric heat source ( W/m3 ). Heat energy (W). Radial coordinate (m). Average radius of anode, cathode, and electrolyte layers (m). Universal gas constant [8.31434 J/(mol K)]. Discretized resistance in anode, cathode, and electrolyte (). Source term of gas species (kg/m3 ); General variable. Temperature (K). Velocity in axial direction (m/s). Utilization percentage (0–1). Velocity in radial direction (m/s); Diffusion volume in eqn (68). Specific volume (m3 /kmol). Cell terminal voltage (V). Potentials in anode and cathode, respectively (V). Width (m). Stoichiometric coefficient of chemical species; Axial coordinate (m). Chemical species. Mole fraction. Reacted mole rate of CH4 , CO and H2 , respectively in a section of interest in flow channel (mol/s). Stoichiometric coefficient of chemical species; Coordinate (m). Chemical species. Mass fraction. Coordinate (m). Compressibility factor. WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Multiple transport processes in solid oxide fuel cells 41 Greek symbols θ δ G G 0 H S x λ µ ρ ρea , ρec ρee ρr ηAct Circumferential position. Angle. Thickness of electrodes and electrolyte layers (m). Gibbs free energy change of a chemical reaction (J/mol). Standard state Gibbs free energy change of a chemical reaction (J/mol). Enthalpy change of a chemical reaction (J/mol). Entropy production [J/(mol K)]. One axial section of fuel cell centered at x position (m). Thermal conductivity [W/(m ◦ C)]. Dynamic viscosity (Pa s). Density (kg/m3 ). Electronic resistivity of anode and cathode respectively ( · cm). Ionic resistivity of electrolyte ( · cm). Reduced density. Activation polarization (V ). Subscripts a c cell e, w, n, s E, W , N , S f i j m P R x X,Y x Anode. Cathode. Overall parameter of fuel cell. East, west, north, and south interfaces between grid P and it neighboring grids. East, west, north, and south neighboring grids of grid P. Fuel. Subscript variable. Gas species; Subscript variable. Mixture. Variables at grid P. Electrochemical reaction. Axial position. Chemical species. Variation in the channel section of x. Superscripts a b c e in out P R x Anode. Sequence. Sequence. Cathode. Sequence. Electrolyte. Inlet of a channel section of interest. Outlet of a channel section of interest. Variables at grid P. Reaction. Axial position. 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