chapter 1 multiple transport processes in solid oxide fuel cells · 2014-05-10 · oxide fuel cells...

CHAPTER 1

Multiple transport processes in solidoxide fuel cells

P.-W. Li, L. Schaefer & M.K. . C hyuDepartment of Mechanical Engineering, University of Pittsburgh, USA.

Abstract

In this topic, three important issues are discussed which concern the theoreticalfundamentals and practical operation of a solid oxide fuel cell. The thermodynamicand electrochemical fundamentals of a fuel cell are reviewed in the Section 2. Thesefundamentals concern the ideal efficiency and energy distribution of a fuel cell’sconversion of chemical energy directly into electrical energy through the oxidationof a fuel. Issues of the chemical equilibrium for a solid oxide fuel cell with internalreforming and shift reactions (in case of methane or natural gas being used as thefuel), are also discussed in detail in this section. The losses of electrical potentialin the practical operation of a fuel cell are elucidated in the third section, whichincludes a discussion about activation polarization, Ohmic loss, and the losses dueto mass transport resistance. In the fourth section, the coupled processes of flow,heat/mass transfer, chemical reaction, and electrochemistry, which influence theperformance of a fuel cell, are analyzed, and modeling and numerical computationfor the fields of flow, temperature, and species concentration, which collectivelydetermine the local and overall electromotive force in a solid oxide fuel cell, areexamined in detail.

1 Introduction

A fuel cell is a device that converts the chemical energy of a fuel oxidation reactiondirectly into electricity. It is substantially different from a conventional thermalpower plant, where the fuel is oxidized in a combustion process and a thermal-mechanical-electrical energy conversion process is employed. Therefore, unlikeheat engines that are subjected to the Carnot cycle efficiency limitation, fuel cells canhave energy conversion efficiencies generally higher than that of heat engines [1].

www.witpress.com, ISSN 1755-8336 (on-line) WIT Transactions on State of the Art in Science and Engineering, Vol 10, © 2005 WIT Press

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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 intoelectricity [1, 2] in a fuel cell.

As is common in many kinds of fuel cells, the core component of a solid oxidefuel cell (SOFC) is a thin gas-tight ion conducting electrolyte layer sandwiched bya porous anode and cathode, as shown in Fig. 1. For a SOFC, this electrolyte is asolid oxide material that only allows the passage of charge-carrying oxide ions. Toproduce useful electrical work, free electrons released in the oxidation of a fuel atthe anode must travel to the cathode through an external load/circuit. Therefore,the electrolyte must conduct ions while preventing electrons released at the anodefrom returning back to the cathode by the same route. The oxide ions are drivenacross the electrolyte by the chemical potential difference on the two sides of theelectrolyte, which is due to the oxidation of fuel at the anode. This difference in thechemical 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 tempera-tures (from 600–1000 ◦C). The high operating temperature of a SOFC also ensuresrapid fuel-side reaction kinetics without requiring an expensive catalyst. In addi-tion, 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 atleast 66.3% based on the lower heating value (LHV, which means that the electro-chemical product, water, is in a gaseous state) of the SOFC [3–6]. Since it operatesvia transport of oxide ions rather than that of fuel-derived ions, in principle, a SOFCcan be used to oxidize a number of gaseous fuels. In particular, a SOFC can consumeCO as well as hydrogen as its fuel, and therefore can be fueled with reformer gascontaining a mix of CO and H2 [7, 8]. Recently, ammonia has also been reportedas a fuel for SOFCs [9].

Since a SOFC operates under high temperatures, its energy conversion efficiencyand component safety are both of concern to industry. In the following sections,the issues to be discussed will include: (1) the thermodynamic and electrochemi-cal fundamentals of the energy conversion and species variation, (2) the potentiallosses in practical operation, (3) the influence of fluid flow and heat and masstransfer on operational efficiency and safety, and (4) the creation of a numericalmodel to simulate the performance and the fields of flow, temperature, and speciesconcentration.


Multiple transport processes in solid oxide fuel cells 3

2 Thermodynamic and electrochemical fundamentals forsolid oxide fuel cells

To study the energy conversion efficiency and distribution of the conversion pro-cesses in a fuel cell, one must understand the basic principles.The chemical potentialand, thereof, electromotive force across the electrolyte involve the interrelation ofthermodynamics, electrochemistry, ion/electron conduction, and heat/mass trans-fer. In this section, the fundamentals of thermodynamics and the electrochemistryfor a solid oxide fuel cell system are reviewed.

The isothermal oxidation of a fuel A with oxidant B can be expressed by thefollowing equation:

aA + bB + · · · → xX + yY + · · ·. (1)

The systematic changes of enthalpy, Gibbs free energy, and entropy production inthe 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 andthe pressure of gases is relatively not high. Thus, the gas species of reactants andproducts can be treated as ideal gases, which allows the chemical enthalpy changeto 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 saidto 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 onemole of a chemical species is called the chemical potential. For an ideal gas attemperature of T and pressure of p, the chemical potential is expressed as:

g = g0 + RT lnp

p0, (4)

where R is the gas constant and p0 is the standard pressure of 1 atm. One may omit thep0 in the denominator of the logarithm in eqn (4), but in such a case, the pressurep must be measured in atm. The systematic change of the Gibbs free energy ineqn (1) can be expressed in terms of the standard state Gibbs free energy and thepartial pressures of the reactants and products:

�G = (xgX + ygY + · · ·) − (agA + bgB + · · ·)

= [xg0X + yg0

Y + · · ·] − [ag0A + bg0

B + · · ·] + RT ln(pX /p0)x(pY /p0)y · · ·(pA/p0)a(pB/p0)b · · ·

= �G0 + RT ln(pX /p0)x(pY /p0)y · · ·(pA/p0)a(pB/p0)b · · · , (5)



where

�G0 = (xg0X + yg0

Y + · · ·) − (ag0A + bg0

B + · · ·), (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 atmosphericpressure, p0 = 1 atm).

The theoretical electromotive force (EMF) induced from the chemical potential(�G) is the Nernst potential:

E = −�G

neF= −�G0

neF+ RT

neFln

(pA/p0)a(pB/p0)b · · ·(pX /p0)x(pY /p0)y · · · , (7)

where F(=96486.7 C/mol) is Faraday’s constant. The first part of the right-hand side of the standard reaction is also called the ideal potential, which isdenoted by:

E0 = −�G0

neF, (8)

where ne is the number of electrons derived from a molecules of the fuel, whenthe 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 thermalenergy that is released in the electrochemical oxidation of the fuel. Both the h andg0 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 chemicalpotentials as discussed above, the current to be withdrawn from a fuel cell, denotedby I , is directly related to the molar consumption rate of fuel and oxidant throughthe following expressions:

mfuel = I

nfuele F

; mO2 = I

nO2e F

, (9)

where nO2e is for oxygen, and is the number of electrons per b molecules of oxygen

obtained in the electrochemical reaction (in eqn (1)), and nfuele is the number of

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 electro-chemical reaction in the fuel cell, which may be expressed by the following chemicalequation:

H2 + 1

2O2 = H2O (gas). (10)



Table 1(a): Enthalpy and standard state Gibbs free energy of species.

Formula CO CO2 H2weight 28.01 44.01 2.016

T h g◦ h g◦ h g◦(K) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol)

298.15 −110530 −169474 −393510 −457254 0 −38968300 −110476 −169816 −393441 −457641 53 −39217320 −109892 −173796 −392687 −461967 630 −41866340 −109309 −177819 −391916 −466308 1209 −44521360 −108725 −181877 −391128 −470688 1791 −47241380 −108141 −185927 −390326 −475142 2373 −49953400 −107555 −190035 −389507 −479627 2959 −52721420 −106967 −194201 −388675 −484141 3544 −55508440 −106377 −198337 −387827 −488719 4131 −58305460 −105887 −202671 −386966 −493318 4715 −61157480 −105195 −206763 −386094 −497982 5298 −64062500 −104599 −210999 −385205 −502655 5882 −66968550 −103102 −221737 −382938 −514498 6760 −74970600 −101588 −232568 −380603 −526583 8811 −81849650 −100053 −243573 −378207 −538822 10278 −89432700 −98507 −254677 −375756 −551316 11749 −97171750 −96938 −265838 −373250 −564800 13223 −104977800 −95353 −277193 −370704 −576704 14702 −112898850 −93749 −288569 −368112 −589622 16186 −121004900 −92129 −300119 −365480 −602720 17676 −129114950 −90499 −311659 −362821 −615996 19175 −137290

1000 −88840 −323340 −360113 −629413 20680 −1455201100 −85495 −346965 −354626 −656576 23719 −1622911200 −82100 −370940 −349037 −684317 26797 −1793631300 −78662 −395082 −343362 −712432 29918 −1966721400 −75187 −419587 −337614 −741094 33082 −2141581500 −71680 −444280 −331805 −770105 36290 −2319101600 −68145 −469265 −325941 −799541 39541 −2498991700 −64585 −494515 −320030 −829350 42835 −2680951800 −61004 −519824 −314079 −859479 46169 −2864711900 −57404 −545324 −308091 −889871 49541 −305189

The Nernst potential from this electrochemical reaction will be:

E(H2+1/2O2=H2O) =−�G0

(H2+1/2O2=H2O)

2F+ RT

2F

[ln(pH2/p0)anode

+ ln(pO2/p0)0.5cathode − ln(pH2O/p0)anode

]. (11)

The ideal chemical potentials at the temperature T (K) can be calculated fromthe data given by handbooks [11]. As a convenient reference, Table 1(c) gives the



Table 1(b): Enthalpy and standard state Gibbs free energy of species.

Formula O2 H2O (Gas) CH4weight 31.999 18.015 16.043

T h g◦ h g◦ h g◦(K) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol) (kJ/kmol)

298.15 0 −61151 −241814 −298105 – –300 54 −61536 −241752 −298452 −74448 −130398320 643 −65661 −241079 −302263 −73718 −134166340 1234 −69826 −240404 −306126 −72974 −137948360 1828 −74024 −239726 −309998 −72213 −141801380 2425 −78325 −239045 −313943 −71432 −145684400 3025 −82495 −238362 −317882 −70631 −149591420 3629 −86797 −237675 −321885 −69808 −153598440 4236 −91112 −236985 −325865 −68962 −157578460 4847 −95433 −236291 −329947 −68094 −161658480 5463 −99849 −235592 −334040 −67202 −165698500 6084 −104266 −234889 −338139 −66287 −169837550 7653 −115382 −233115 −348890 −63892 −180327600 9244 −126656 −231313 −359173 −61356 −191016650 10859 −138056 −229493 −369828 −58671 −201899700 12499 −149551 −227622 −380712 −55853 −213073750 14158 −161117 −225732 −391707 −52897 −224347800 15835 −172885 −223812 −402852 −49818 −235898850 17531 −184684 −221860 −414130 −46613 −247638900 19241 −196669 −219876 −425526 −43296 −259566950 20965 −208745 −217860 −436930 −39866 −271666

1000 22703 −220897 −215814 −448514 −36336 −2839361100 26212 −245378 −211623 −471993 −28981 −3090411200 29761 −270239 −207308 −495908 −21274 −3348341300 33344 −295426 −202872 −520072 −13254 −3612641400 36957 −320883 −198321 −544681 −4956 −3884161500 40599 −346551 −193663 −569563 3587 −4161131600 44266 −372374 −188906 −594826 12347 −4442931700 47958 −398632 −184056 −620276 21295 −4732351800 51673 −424967 −179121 −646221 30406 −5025741900 55413 −451507 −174108 −672288 39658 −532432

ideal chemical potentials and enthalpies for the gas species that are typically utili-zed in a SOFC. Recognizing the electrochemical equilibrium in the anode gasmixture:

�G = −�G0(H2+1/2O2=H2O) + RT

[ln(pH2/p0)anode + ln(pO2/p0)0.5

anode

− ln(pH2O/p0)anode

]= 0. (12)



Table 1(c): Change of enthalpy and standard state Gibbs free energy of reactions.

Reaction H2 + 1/2O2 = H2O (gas) CH4 + H2O = 3H2 + CO CO + H2O = H2 + CO2

T �H �G0 E0 �H �G0 �H �G0

(K) (kJ/mol) (kJ/mol) (V) (kJ/mol) (kJ/mol) (kJ/mol) (kJ/mol)

298.15 −241.814 −228.561 1.184 – – – –300 −241.832 −228.467 1.184 205.883 141.383 −41.160 −28.590320 −242.031 −227.567 1.179 206.795 137.035 −41.086 −27.774340 −242.230 −226.692 1.175 207.696 132.692 −40.994 −26.884360 −242.431 −225.745 1.170 208.587 128.199 −40.886 −26.054380 −242.631 −224.828 1.165 209.455 123.841 −40.767 −25.225400 −242.834 −223.914 1.160 210.315 119.275 −40.631 −24.431420 −243.034 −222.979 1.155 211.148 114.758 −40.489 −23.563440 −243.234 −222.004 1.150 211.963 110.191 −40.334 −22.822460 −243.430 −221.074 1.146 212.643 105.463 −40.073 −21.857480 −243.622 −220.054 1.140 213.493 100.789 −40.009 −21.241500 −243.813 −219.038 1.135 214.223 96.073 −39.835 −20.485550 −243.702 −216.229 1.121 214.185 82.570 −39.961 −18.841600 −244.746 −213.996 1.109 217.514 72.074 −38.891 −16.691650 −245.201 −211.368 1.095 218.945 59.858 −38.383 −14.853700 −245.621 −208.766 1.082 220.215 47.595 −37.878 −13.098750 −246.034 −206.172 1.068 221.360 35.285 −37.357 −12.232800 −246.432 −203.512 1.055 222.383 22.863 −36.837 −9.557850 −246.812 −200.784 1.040 223.282 10.187 −36.317 −7.927900 −247.173 −198.078 1.026 224.071 −2.369 −35.799 −6.189950 −247.518 −195.268 1.012 224.752 −14.934 −35.287 −4.697

1000 −247.846 −192.546 0.998 225.350 −27.450 −34.779 −3.0791100 −248.448 −187.013 0.969 226.266 −52.804 −33.789 0.0911200 −248.986 −181.426 0.940 226.873 −78.287 −32.832 3.1681300 −249.462 −175.687 0.910 227.218 −103.762 −31.910 6.0501400 −249.882 −170.082 0.881 227.336 −128.964 −31.024 9.0161500 −250.253 −164.378 0.852 227.266 −154.334 −30.172 11.8281600 −250.580 −158.740 0.823 227.037 −179.843 −29.349 14.6511700 −250.870 −152.865 0.792 226.681 −205.289 −28.554 17.3461800 −251.127 −147.267 0.763 226.218 −230.442 −27.785 20.0951900 −251.356 −141.346 0.732 225.669 −256.171 −27.038 22.552

Substituting eqn (12) into eqn (11), the Nernst potential in another form for theelectrochemical reaction of eqn (10) is obtained:

E(H2+1/2O2=H2O) = RT

2F

[ln(pO2/p0)0.5

cathode − ln(pO2/p0)0.5anode

]. (13)

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 thepartial pressures of the other major species in the anode flow. Therefore, whencalculating the partial pressures of hydrogen and water vapor for determining theNernst potential of the electrochemical reaction of eqn (10), the oxygen partial pres-sure in anode flow stream is ignorable. Hereafter, for an electrochemical reaction



as in eqn (1), the common practice in determining the Nernst potential will be touse eqn (7), in which the partial pressure of oxygen on the cathode side and thoseof the fuel and product species on the anode side are used.

The molar consumption rate of hydrogen and oxygen in the electrochemicalreaction 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 asolid oxide fuel cell in order to maintain sufficient ionic conductivity for the solidoxide electrolyte [2, 4]. This provides a favorable environment for the reforming ofhydrocarbon fuels like methane. In fact, since a solid oxide fuel cell operates basedon the transport of oxide ions through the electrolyte layer from the cathode sideto the anode side, the reforming products of hydrogen and carbon monoxide in thefuel channel can both serve as fuels. Given this advantage, solid oxide fuel cells candirectly utilize hydrocarbon fuels or, at least, methane as a pre-reformed or partlyreformed gas with components of CH4, CO, CO2, H2 and H2O. Therefore, the fuelreforming and shift reactions will occur in the fuel channel in a solid oxide fuelcell. The anode is, in fact, a good material to serve as the catalyst for such chemicalreactions, since the high temperature in a SOFC means that no noble metals areneeded for a catalyst [12].

If there are five gas species, CH4, CO, CO2, H2, and H2O, 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 ofreforming and shift need to be considered for determining the species’mole fractions(which are crucial to the electromotive forces in the fuel cell).

Reforming : CH 4 + H2O ↔ CO + 3H2. (15)

Shift : CO + H2O ↔ CO2 + H2. (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 inchemical equilibrium [4] when determining the mole fractions of the species, whichmakes the computation significantly convenient. From the concept of chemicalequilibrium, 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 fuelstream are related through the following two simultaneous equations [13]:

KPR =(

pH2p0

)3 (pCOp0

)(

pCH4p0

) (pH2O

p0

) = exp

(−�G0

reforming

RT

), (17)



KPS =(

pCO2p0

) (pH2p0

)(

pCOp0

) (pH2O

p0

) = exp

(−�G0

shift

RT

). (18)

The dominant electrochemical reaction has been reported to be the oxidationof H2 [12], which is primarily responsible for the electromotive force. However,at the same time, the electrochemical oxidation of the CO is also possible, andlikely occurs to some extent in the solid oxide fuel cell. It has been reported thatfuel cells operated by using mixtures of CO and CO2 have shown that the electro-chemical oxidation of CO is an order of magnitude slower than that of hydrogen[14]. Nevertheless, there is no necessity to distinguish whether the electrochemicaloxidation 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

�G =[

g0CO2

+ RT ln

(pCO2

p0

)+ g0

H2+ RT ln

(pH2

p0

)]

−[

g0CO + RT ln

(pCO

p0

)+ g0

H2O + RT ln

(pH2O

p0

)]= 0. (19)

Rearranging this equation gives:

g0CO2

+ RT ln

(pCO2

p0

)−

[g0

CO + RT ln

(pCO

p0

)]

= g0H2O + RT ln

(pH2O

p0

)−

[g0

H2+ RT ln

(pH2

p0

)]. (20)

Subtracting a term of [(1/2)g0O2

+RT ln(pO2/p0)1/2cathode] from both sides of eqn (20),

results in:

g0CO2

− g0CO − 1

2g0

O2+ RT ln

(pCO2

p0

)− RT ln

(pCO

p0

)− RT ln

(pO2

p0

)1/2

cathode

= g0H2O − g0

H2− 1

2g0

O2+ RT ln

(pH2O

p0

)

− RT ln

(pH2

p0

)− RT ln

(pO2

p0

)1/2

cathode. (21)

It is easy to see that the left-hand side of eqn (17) is the Gibbs free energy changeof 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=H2O) = E(CO+1/2O2=CO2), (22)



where E(H2+1/2O2=H2O) is given in eqn (11), while the Nernst potential for theelectrochemical oxidation of CO is

E(CO+1/2O2=CO2) = −�G0(CO+1/2O2=CO2)

2F+ RT

2F

[ln (pCO/p0)anode

+ ln (pO2/p0)0.5cathode − ln (pCO2/p0)anode

]. (23)

It is preferable that the EMF of an internal reforming SOFC be calculated from theelectrochemical oxidation of H2; however, the species’consumption and productionare the results collectively determined from the reactions of eqns (10), (15) and (16).

The above discussion clearly indicates that the electrochemical reaction can beassumed to be driven by the hydrogen, and the electrochemical fuel value of COis readily exchanged for hydrogen by the shift reaction under chemical equilib-rium. Therefore, only H2 is considered as the electrochemical fuel in the followinganalysis, and CO only takes part in the shift reaction.

For convenience, the mole flow rates of CH4, CO and H2 are denoted by theirformulae. Assuming that, x, y, and z are the mole flow rates, respectively, for theCH4, 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 theinlet and the outlet of an interested section of fuel channel are in the followingforms [8, 15]:

CH4out = CH4

in − x, (24)

COout = COin + x − y, (25)

CO2out = CO2

in + y, (26)

H2out = H2

in + 3x + y − z, (27)

H2Oout = H2Oin − x − y + z. (28)

The overall mole flow rate of fuel, denoted by Mf , will vary from the inlet to theoutlet of the section of interest in the fuel channel in the form of

M outf = M in

f + 2x. (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+x−y

M inf +2x

)(H2

in+3x+y−zM in

f +2x

)3 (pp0

)2

(CH4

in−xM in

f +2x

)(H2Oin−x−y+z

M inf +2x

) , (30)



KPS =

(H2

in+3x+y−zM in

f +2x

)(CO2

in+yM in

f +2x

)(

COin+x−yM in

f +2x

)(H2Oin−x−y+z

M inf +2x

) , (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 tothe charge transfer rate, or current, I , across the electrolyte layer:

z = I/(2F). (32)

From the electrochemical reaction, the molar consumption of oxygen on thecathode side can be calculated by using eqn (14). By finding a simultaneous solutionfor eqns (30)–(32), the species variations, x, y and z, can be determined. Finally,with the reacted mole numbers of CH4 and CO determined, the heat absorbed inthe reforming reaction and released from the shift reaction can be obtained:

QReforming = �H Reforming · x, (33)

QShift = �H Shift · y. (34)

Nevertheless, prior to finding a solution for eqns (30) and (31), the electric currentof the fuel cell in eqn (32) must be known. This demonstrates that the processes in aSOFC feature a strong coupling of the species molar variation and the electromotiveforce, as well as interdependency of the ion conduction and current flow. The iontransfer 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 tobe lower than the ideal electromotive forces of the electrochemical reaction. Thenature of the fuel cell performance in response to loading condition can be realizedby its polarization curve, typically shown as in Fig. 2.

With an increase in current density, the cell potential experiences three kindsof potential losses due to different dominant resistances. The potential drop dueto the activation resistance, which is the activation polarization, is associated withthe electrochemical reactions in the system. Another potential drop comes fromthe ohmic resistance in the fuel cell components, when the ions and electrons areconducted in the electrolyte and electrodes, respectively. The third drop, which canbe 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 fromobserving the Nernst equation that the electromotive force of a fuel cell is a functionof the temperature and the gas species’ partial pressures at the electrolyte/electrode



Figure 2: Over-potential in the operation of a fuel cell.

interfaces, which are directly proportional to their mole fractions. It is importantto note that, in the fuel stream, fuel must be transported or diffused from the coreregion of the stream to the anode surface, and, also, the product of the electrochem-ical reaction must conversely be transported or diffused from the reaction site tothe core region of the fuel flow. On the cathode side, oxygen must be transportedand diffused from the core region of airflow to the cathode surface. Along withthe fuel and air streams, the consumption of reactants or development of prod-ucts will make the mole fractions of reactants decrease and those of the productincrease. Due to these resistances in the mass transport process, the feeding of reac-tants and removing of products to/from the reaction site can only proceed under alarge concentration gradient between the bulk flow and the electrode surface whenthe current density is high, which therefore induces a sharp drop in the fuel cellpotential.

As a consequence of all the above-mentioned potential drops, extra thermalenergy will be released together with the heat (−T�S) from systematic entropyproduction. The heat transfer issues in a solid oxide fuel cell will be consideredlater in Section 4.

3.1 Activation polarization

The activation polarization is the electronic barrier that must be overcome prior tocurrent and ion flow in the fuel cell. Chemical reactions, including electrochemicalreactions, also involve energy barriers, which must be overcome by the reactingspecies. The activation polarization may also be viewed as the extra potential nec-essary to overcome the energy barrier of the rate-determining step of the reactionto a value such that the electrode reaction proceeds at a desired rate [16–18].



The Butler-Volmer equation is a well-known expression for the activation polar-ization, ηAct:

i = i0

{exp

(β

neFηAct

RT

)− exp

[−(1 − β)

neFηAct

RT

]}, (35)

where β, which is usually 0.5 for the fuel cell application [16], is the transfercoefficient; i is the actual current density in the fuel cell; and i0 is the exchangecurrent density. The transfer coefficient is considered to be the fraction of thechange in polarization that leads to a change in the reaction-rate constant. Theexchange current density, i0, is the forward and reverse electrode reaction rateat the equilibrium potential. A high exchange current density means that a highelectrochemical reaction rate and good fuel cell performance can be expected. Thene in eqn (35) is the number of electrons transferred per reaction, which is 2 forthe reaction of eqn (10). Substituting the value of β = 0.5 into eqn (35), one canobtain a new expression as follows:

i = 2i0 sinh

(neFηAct

2RT

)(36)

from which the activation polarization can be expressed as:

ηAct = 2RT

neFsinh−1

(i

2i0

)or ηAct = 2RT

neFln

(

i

2i0

)+

√(i

2i0

)2

+ 1

.

(37)

For a high activation polarization, eqn (37) can be approximated as the simple andwell-known Tafel equation [16]:

ηAct = 2RT

neFln

(i

i0

). (38)

On the other hand, if the activation polarization is small, eqn (37) can be approxi-mated as the linear current-potential expression [16]:

ηAct = 2RT

neF

i

i0. (39)

Nevertheless, eqn (37) is recommended for its integrity and accuracy in calculatingthe activation polarization.

The value of the exchange current density (i0) is different for the anode andcathode, and is also dependent on the electrochemical reaction temperature, thepartial pressures of the gases [18, 19], and the electrode materials. The determina-tion for i0 shows diversity in different literature [16–22]. There are formulationsavailable in the literature [18–20], but some parameters used in the formulationare not well documented. On the other hand, empirical estimation of i0 is also



made in literature, like the work of Chan et al. [16]. An i0 of 5300A/m2 for theanode and 2000A/m2 for the cathode for a SOFC were used; however there wasno 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 tosatisfy their experimental data; no report, however, is given about the adjusted i0values in their paper. The present authors used a slightly higher value of 6300A/m2

and 3000A/m2 [23, 24], respectively, for the i0 of the anode and cathode, whichresulted in very good agreement between the numerical simulated cell terminalvoltage and experimental results from different researchers [25–28]. Nevertheless,i0 varies according to the temperature and pressures of the electrochemical reac-tion. For SOFCs working at temperature from 800 ◦C–1000 ◦C and pressures upto 15 atm, an i0 of 5300–6300A/m2 for the anode and 2000–3000A/m2 for thecathode 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 currentcollecting components, as well as the ionic conduction resistance of the electrolytelayer. Therefore, the conductivity of the materials for the cell components and thecurrent collecting pathway are the two factors most influential to the overall ohmicloss of a SOFC.

In state-of-the-art SOFC technology, lanthanum manganite suitably doped withalkaline and rare earth elements is used for the cathode (air electrode) [20, 27], yttriastabilized zirconia (YSZ) has been most successfully employed for electrolyte,and nickel/YSZ is applied over the electrolyte to form the anode. Temperaturecould significantly affect the conductivity of SOFC materials. Especially for theelectrolyte, for example, the resistivity could be two orders of magnitude smaller ifits temperature increases from 600 ◦C to 1000 ◦C. The equations of resistivity forSOFC components suggested in literature are collected in Table 2.

Table 2: Data and equations for resistivity of SOFC components.

Cathode Electrolyte Anode Interconnect(� · cm) (� · cm) (� · cm) (� · cm)

Bessette 0.008114e500/T 0.00294e10350/T 0.00298e−1392/T –et al. [29]

Ahmed ∗0.0014 0.3685 + 0.002838e10300/T ∗0.0186 ∗0.5et al. [30]

Nagata ∗0.1 10.0e[10092(1/T−1/1273)] ∗0.013 ∗0.5et al. [18]

Ferguson T4.2×105 e1200/T 1

3.34×102 e10300/T T9.5×105 e1150/T T

9.3×104 e1100/T

et al. [31]

∗At temperature of 1000 ◦C.



A careful check for the equations in Table 2 was conducted. The expressions byBessette et al. [29] were found to be reliable, and to give nearly identical predictionsas those by Ahmed et al. [30] and Nagata et al. [18]. The predicted data for anoderesistivity by the equation of Ferguson et al. [31] shows significant discrepancieswith the predictions by other equations.

It is rational to assume that the passage of the charge-carrying species throughthe 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 collectsthrough the channel walls, also called ribs, after it moves perpendicularly acrossthe 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 than3 mm), and the electric resistance through the channel wall might be negligible [30].This simplification leads to the consideration that the current is almost exclusivelyperpendicularly collected, which means that the current flows normally to the tri-layer of the cathode, electrolyte and anode. When calculating the local currentdensity, the ohmic loss is thus simply accounted for in the following way [30]:

I = �A · (E − ηaAct − ηc

Act) − Vcell

(δaρae + δeρe

e + δcρce)

, (40)

where �A is a unit area on the anode/electrolyte/cathode tri-layer, through whichthe current I passes; δ is the thickness of the individual layers; ρe is the resistivity ofthe electrodes and electrolyte; Vcell is the cell terminal voltage; and the denominatorof the right-hand side term is the summation of the resistance of the tri-layer. TheJoule heating due to current flow in the volume of �A×δ is expressed for the anodein the form of

QaOhmic = I2 · (δaρa

e/�A). (41)

This is also applicable to the electrolyte and cathode by replacing the thickness andresistivity accordingly.

Figure 3: Schematic of a planar type SOFC.



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 atubular type SOFC (shown in Fig. 4), the current collects circumferentially, whichleads 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 flowin the circumferential pathway, a network circuit [23, 25, 33, 34] for current flowmay be adopted, as shown in Fig. 5. Because the current collection is symmetricin the peripheral direction in the cell components, only half of the tube shell isdeployed with a mesh in the analysis. The local current routing from the anode tocathode through the electrolyte is determinable based on the local electromotiveforce, EMF, the local potentials in the anode and cathode, and the ionic resistanceof the electrolyte layer, which yields the expression:

I = E − ηaAct − ηc

Act − (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



area of �A:Re = ρe

e · δe/�A, (43)

where the ρee 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 ofcurrent to any grid located in the anode, the equation associating the potential ofthe 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:

(V a

E − V aP

Rae

+ V aW − V a

P

Raw

)+

(V a

N − V aP

Ran

+ V aS − V a

P

Ras

)

+[

V cP − V a

P − (EP − ηPAct)

ReP

]= 0. (44)

In the same way for a grid point P in the cathode:

(V c

E − V cP

Rce

+ V cW − V c

P

Rcw

)+

(V c

N − V cP

Rcn

+ V cS − V c

P

Rcs

)

+[

V aP − V c

P + (EP − ηPAct)

ReP

]= 0, (45)

where Ra and Rc are the discretized resistances in the anode and cathode respec-tively, which are determined according to the resistivity, the length of the currentpath, and the area upon which the current acts; ηP

Act is the total activation polariza-tion, including from both the anode side and the cathode side.

With all of the equations for the discretized grids in both the cathode and anodegiven, a matrix representing the pair of eqns (37) and (38) can be created. Whenfinding a solution for such a matrix equation for the potentials, the following approx-imations 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 peripheralcurrent in the cathode and anode, unless the cathode or anode is in contact withnickel felt, through which the current flows in or out.

3. The potentials of the nickel felts are assumed to be uniform due to their highelectric conductivities.

4. Since the potential difference between the two nickel felts is the cell terminalvoltage, the potential at the nickel felt in contact with the anode layer can beassumed to be zero. Thus, the potential at the nickel felt in contact with thecathode will be the terminal voltage of the fuel cell.

Once all the local electromotive forces are obtained, the only unknown conditionfor the equation matrix is either the total current flowing out from the cell or



the potential at the nickel felt in contact with the cathode. This highlights twoapproaches that can be taken when predicting the performance of a SOFC. If thetotal current taken out from the cell is prescribed as the initial condition, the terminalvoltage can be predicted. On the other hand, one can prescribe the terminal voltageand predict the total current, i.e., the summation of the local current I across theentire electrolyte layer.

Once the potentials are obtained in the electrode layer, the volumetric Jouleheating in the electrode for a volume centered about P will be:

qaP = 1

2

[(V a

E − V aP )2

Rae

+ (V aW − V a

P )2

Raw

+ (V aN − V a

P )2

Ran

+ (V aS − V a

P )2

Ras

]/(�xP · ra · �θP · δa), (46)

qcP = 1

2

[(V c

E − V cP)2

Rce

+ (V cW − V c

P)2

Rcw

+ (V cN − V c

P)2

Rcn

+ (V cS − V c

P)2

Rcs

]/(�xP · rc · �θP · δc), (47)

qeP =

[(EP − ηP

Act − V cP + V a

P )2

ReP

]/(�xP · re · �θP · δe), (48)

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, asshown in Fig. 5. The volumetric heat induced from the activation polarization inthe anode and cathode is:

qP,aAct = IP · η

P,aAct/(�xP · ra · �θP · δa), (49)

qP,cAct = IP · η

P,cAct/(�xP · rc · �θP · δc). (50)

The thermodynamic heat generation occurring at the anode/electrolyte interface inthe area around P is:

QRP = (�H − �G) · IP/(2F). (51)

3.3 Mass transport and concentration polarization

Due to their gradual consumption, the fractions of the reactants and oxidant willdecrease, in the fuel and air streams, respectively, which will cause the electromotiveforce to decrease gradually along the flow stream. On the other hand, due to the masstransport resistance, the concentration of the gas species will encounter polarizationin between the core flow region and the electrode surface, which will result in lowerpartial pressures for the reactants, but higher partial pressures for the products at the



Figure 6: The interrelation amongst concentration and other parameters.

electrode surfaces. Therefore, the fuel cell terminal voltage will be lower than theideal value that is indicated by the Nernst equation. At high cell current density, theincreased requirements for the feeding of the reactants and removal of the productscan make the concentration polarization higher, and, thus, the cell output potentialwill sharply decrease.

In order to take the concentration polarization into account when calculating theelectromotive force, the local partial pressures of the reactants and products at theelectrode surface are used. However, this requires the solution of the concentrationfields for the gas species in the fuel and oxidizer channels, which might be eithersimply based on a one-dimensional [35–37] or else based on a complicated two-or 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 indicatedin Fig. 6. First, the gas species mass fraction determines the gas properties in theflow field, while the flow fields affect the gas species concentration distributionand temperature. Second, the gas species concentration field and temperature dis-tribution determines the electromotive forces, while the ion/electron conductiondue to the electromotive force determines the mass variation and heat generationsin the fuel cell. The inter-dependency of these parameters will be discussed indetail in the following section when modeling a SOFC in order to predict both thefuel cell performance and the detailed distributions of the temperature, gas speciesconcentration, and flow fields.

4 Computer modeling of a tubular SOFC

An operation curve for a SOFC that characterizes the average current density versusthe terminal voltage is very important when designing a SOFC system or a hybrid



SOFC/GT system [41–44]. Other information, like the temperature and concen-tration fields in a SOFC, is also of high concern for the safe operation of both theSOFC itself and the downstream facilities if a hybrid system is under consideration.Although there have been some experimental data generated about the operationalperformance and temperature of SOFCs [25–28], rigorous experimental testing fora 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 operationalcharacteristics in terms of the average current density versus the terminal voltage(based on prescribed operating conditions). The operating conditions of a SOFCare solely determined by fixing the flow rates and the thermodynamic state ofthe fuel and oxidant, as well as a load condition such as terminal voltage, thecurrent being withdrawn, and external load [45]. The flow rates and thermodynamicconditions of the fuel and oxidant may be called internal conditions, and the terminalvoltage, current to be withdrawn, and external load may be designated externalconditions. Like any kind of “battery,” the external load condition of a SOFCdetermines the consumption of the fuel/oxidant and the generation of products inthe electrochemical reaction [46]; the only difference in a fuel cell is its continuousfeeding of fuel/oxidant and removal of products and waste species.

According to the different ways of prescribing the external parameters, the fol-lowing three schemes might be designed in order to predict the other unknownparameters when constructing a numerical model for a SOFC: (1) Use the internalconditions and terminal voltage to predict the total current to be withdrawn. (2) Usethe 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 andcurrent density.

The cost of iterative computation using the three schemes is quite different. Inthe first scheme, the cell terminal voltage is known, and thus the local current canbe obtained, for example, by using eqn (42) and solving eqns (44) and (45), fora planar and tubular type SOFC, respectively, once the temperature and partialpressure fields of the gas species are available. The integrated value from the localcurrent will be the total current to be withdrawn from the SOFC. In the secondscheme, however, the terminal voltage needs to be assumed, and then checked byintegrating the total current from the local current until the calculated total currentagrees with the prescribed value. In this computation process, a proper method isneeded to find the best-fit terminal voltage iteratively. The third scheme resemblesthe second scheme, in that one needs to assume a terminal voltage to find the totalcurrent. The computation will be stopped only when the voltage-current ratio equalsthe prescribed load.

With an understanding of the principles of the energy conversion, chemical equi-librium, potential loss, and the operation of a SOFC, a computer model for a SOFCcan now be constructed. Generally speaking, the modeling and computation fora tubular SOFC and a planar SOFC share rather common features except for theOhmic losses and Joule heating, for which differences result from the differentstructures variation in the current pathway in the electrodes. Relatively speaking,



the tubular SOFC has a more complex current pathway in the electrodes [27] andwill be discussed in the following analysis. Modeling works on planar type SOFCsare 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 constructionof a numerical model.

4.1 Outline of a computation domain

In a practical tubular SOFC stack, multiple tubular cells are mounted in a containerto form a cell bundle, as shown in Fig. 7. A pre-reformer might be put adjacent tothe cell bundles [50, 51]. In order to conduct a modeling study with relatively lesscomplexity, it is assumed that most of the single tubular SOFCs operate under thesame environment of temperature and concentrations of gas species. This allowsthe definition of a controllable domain in the cross-section, which pertains to onesingle cell, as outlined by the dashed-line square in Fig. 7. It is then specified thatthere 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 ofthe heat/mass transfer and the chemical/electrochemical performance for the singlecell and its controllable area, one can obtain results very useful for evaluating theperformance of an entire cell stack.

Also considering the longitudinal direction, the heat and mass transfer in theabove outlined square area enclosing the tubular SOFC are three-dimensional innature. For a solution of the three-dimensional governing equations of momentum,energy, and species conservation, a large number of discretized mesh points arenecessary, which results in an unacceptably heavy computational load. In order toreduce computational cost, the square area enclosing the tubular SOFC is approxi-mated to be an equivalent circular area; therefore, the domain enclosing the singletubular 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.



Figure 8: Computation domain for a tubular SOFC.

It should be noted, though, that the zero-flux, or insulation of heat and mass transferat 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 streamsand 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 thermaland transport properties of the fluids are functions of the local species concentra-tion, temperature, and pressure; therefore, the governing equations for momentum,energy, and species conservation (based on mass fraction) have variable thermaland transport properties:

∂(ρu)

∂x+ 1

r

∂(rρv)

∂r= 0, (52)

∂(ρuu)

∂x+ 1

r

∂(rρvu)

∂r= −∂p

∂x+ ∂

∂x

(µ

∂u

∂x

)+ 1

r

∂

∂r

(rµ

∂u

∂r

)

+ ∂

∂x

(µ

∂u

∂x

)+ 1

r

∂

∂r

(rµ

∂v

∂x

), (53)

∂(ρuv)

∂x+ 1

r

∂(rρvv)

∂r= −∂p

∂r+ ∂

∂x

(µ

∂v

∂x

)+ 1

r

∂

∂r

(rµ

∂v

∂r

)

+ ∂

∂x

(µ

∂u

∂r

)+ 1

r

∂

∂r

(rµ

∂v

∂r

)− 2µv

r2, (54)

∂(ρCpuT )

∂x+ 1

r

∂(rρCpvT )

∂r= ∂

∂x

(λ

∂T

∂x

)+ 1

r

∂

∂r

(rλ

∂T

∂r

)+ q, (55)



∂(ρuYJ )

∂x+ 1

r

∂(rρvYJ )

∂r= ∂

∂x

(ρDJ ,m

∂YJ

∂x

)+ 1

r

∂

∂r

(rρDJ ,m

∂YJ

∂r

)+ SJ . (56)

These equations are applied universally to the entire computation domain; how-ever, zero velocities will be assigned to the solid area in the numerical computation.In the energy conservation equation, thermal energy from the chemical and elec-trochemical reactions (expressed by eqns (33), (34), (51)) and the Joule heatingin electrodes and electrolyte (expressed by eqns (46)–(50)), represented by q, areintroduced as source terms in the proper locations in the fuel cell. Some terms dueto energy diffusion driven by the concentration diffusion of the gas species are verysmall, 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 representsgeneral 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, whichgives 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, andspecies 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 speciesmass 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 isassumed.

In the fuel flow passage, the mass flow rate increases along the x direction dueto the transferring in of oxide ions. Similarly, a reduction of the air flow rate occursin the air flow passage, due to the ionization of oxygen and the transferring of theoxide ions to the fuel side. Therefore, radial velocities at r = rair and r = rf are:

vf =∑

mfuel,speciesx

ρfuelx

∣∣∣r=rf , (57)

vair =∑

mair,speciesx

ρairx

∣∣r=rair , (58)

where m [kg/(m2s)] is mass flux of the gas species at the interface of the electrodesand fluid, which arises from the electrochemical reaction in the fuel cell. The massfractions of all participating chemical components at the boundaries of r = rair andr = rf are calculated with consideration of both diffusion and convection effects[54, 55]:

mJ ,airx = −DJ ,airρ

airx

∂YJ

∂r+ ρair

x YJ vair , (59)

mJ , fuelx = −DJ , fuelρ

fuelx

∂YJ

∂r+ ρ fuel

x YJ vf . (60)



Table 3: Properties of SOFC materials.

Thermal conductivity Cp Density(W/(m K)) (J/(kg K)) (kg/m3)

Cathode d 11; c2.0; b2.0 b623 a4930Electrolyte d 2.7; c2.7; b2.0 b623 a5710Anode c11.0; d 6.0; b2.0 b623 a4460Support tube c1.0Air-inducing tube c1.0Interconnector b13; c2.0; d 6.0 b800 a6320; b7700

aAhmed et al. [30]; bRecknagle et al. [39]; cNagata 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 massvariation and related mass flux that arise from the electrochemical reaction hasbeen 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 localelectromotive forces by eqn (11).

The properties of solid materials in a SOFC are given in Table 3, which showsome variation based on the different literature sources. The single gas propertiesare available from references [11] and [56]. For gas mixtures, equations from ref-erences [11, 57] are available, and some selected equations from reference [11] forcalculating the properties are listed in the following section.

The mixing rule for the viscosity is:

µm =n∑

i=1

Xiµi∑nj=1 Xjφij

; φij = 1

81/2

(1 + Mi

Mj

)−1/2[

1 +(

µi

µj

)1/2 (Mj

Mi

)1/4]2

,

(61)

where µm (Pa · sec) is the viscosity for the mixture, and µi or µj are the viscositiesof individual species (Pa · sec); Mi or Mj is the molecular weight of a species; Xi orXj is the mole fraction; and when i = j, φij = 1.

The mixing rule for the thermal conductivity of gases at atmospheric pressure orless is:

km =n∑

i=1

Xiki∑nj=1XjAij

;

(62)

Aij = 1

4

1 +

[µi

µj

(Mj

Mi

)3/4 (T + Si

T + Sj

)]1/2

2 (T + Sij

T + Si

),



where km [W/(m · K)] and ki [W/(m · K)] are the thermal conductivities of the mix-ture and species; Sij = C(SiSj)1/2, and C = 1.0, but when either or both componentsi 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 temperatureof species; and the unit of T is K .

When the gas mixture is above atmospheric pressure, the following correctionis applied to the km obtained above:

k = k ′ + A × 10−4(eBρr + C)(T 1/6

c M 1/2

P2/3c

)Z5

c

, (63)

ρr < 0.5, A = 2.702, B = 0.535, C = −1.000,0.5 < ρr < 2.0, A = 2.528, B = 0.670, C = −1.069,2.0 < ρr < 2.8, A = 0.574, B = 1.155, C = 2.016,

where k [W/(m · K)] is the gas thermal conductivity at the temperature T (K) andpressure P of interest in the mixture; k ′[W/(m · K)] is the thermal conductivity at Tand 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 Tand P; Tc (K) is the critical temperature; M is the molecular weight; Pc (MPa) isthe critical pressure; Zc = PcVc/(RTc) is the critical compressibility factor; andR is the gas constant, which is 0.008314 MPa · m3/(kmol · K). The mixture criticalproperties are obtained via the following equations:

Pcm = Ppc + Ppc

[5.808 + 4.93

(n∑

i=1

Xiωi

)][Tcm − Tpc

Tpc

], (64)

Tcm =n∑

j=1

(XjVcj∑ni=1 XiVci

Tcj

), (65)

Vcm =∑

i

∑j

φiφjvij(i �= j), (66)

where

φj = XjV2/3cj∑n

i=1 XiV2/3ci

; vij = Vij(Vci + Vcj)

2.0; Vij = −1.4684

(∣∣∣∣Vci − Vcj

Vci + Vcj

∣∣∣∣)

+ C,

and C is zero for hydrocarbon systems and is 0.1559 for systems containing anon-hydrocarbon gas. In all the above equations, Xi or Xj is the mole fraction of aspecies in the mixture; ωi is the acentric factor of a species; Pcm, Tcm and Vcm arethe mixture critical properties; and Ppc and Tpc are the pseudocritical properties ofthe mixture, which are expressed as:

Tpc =n∑

i=1

XiTci; Ppc =n∑

i=1

XiPci. (67)



Table 4: Atomic diffusion volumes for use in eqn (68).

Atomic and structural diffusion-volume increments v [11]

C 16.50 H2 7.07H 1.98 N2 17.9O 5.481 O2 16.6N 5.69 CO 18.9Aromatic ring −20.2 CO2 26.9Heterocyclic ring −20.2 H2O 12.7

The gas diffusivity of one species against the remaining species of a mixture isexpressed in the form of:

Dim = 1 − Xi∑jj �=i

Xj/Dij, Dij =

0.01013T 1.75(

1Mi

+ 1Mj

)0.5

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 themolecular weight; and all vi or vj are group contribution values for the subscriptcomponent summed over atoms, groups and structural features, which are listed inTable 4.

4.3 Numerical computation

In order to conduct a numerical computation for flow, temperature, and concentra-tion fields in a SOFC, a mesh system with a sufficient grid number both in the rand x directions must be deployed at the computational domain. All the governingequations may be discretized by using the finite volume approach, and the SIMPLEalgorithm 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 mightbe large enough to have radiation heat transfer; therefore, a numerical treatmentbased on the method introduced in the literature [60] can be used to consider theradiation heat exchange.

As has been discussed at the beginning of Section 4, the computation may bebased on the internal conditions and the current to be withdrawn; and, as a conse-quence of the simulation, the terminal voltage will be given as an output along withother operational details. The convenience of using this procedure in the simulationis discussed next.

It is quite common in practice that the total current is prescribed in terms of theaverage current density of the fuel cell. Also, instead of the flow rates of fuel andair, the stoichiometric data are prescribed in terms of the utilization percentage ofhydrogen and oxygen. This kind of designation of the operating conditions resultsin a convenient comparison of the fuel cell performance based on the same level of



average current density and the hydrogen and oxygen utilization percentage. Theinlet velocities of fuel and air are, then, obtainable in the forms of:

ufuel =(

Acellicell

2FUH2 XH2 Afuel

)RTf

Pf, (69)

uair =(

Acellicell

4FUO2 XO2 Aair

)RTair

Pair, (70)

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, respec-tively; and UH2 and UO2 are the utilization percentage for hydrogen and oxygen.

The computation process is highly iterative and coupled in nature. As the firststep, the latest local temperature, pressure, and species’ mass fractions are used inthe network circuit analysis to obtain the cell terminal voltage and local currentacross the electrolyte, and thus the local species’ transfer fluxes and local heatsources. In the second step, the local temperature, pressure and species’ mass frac-tions are, in turn, obtained through solution of the governing equations under thenew boundary conditions determined by the latest-available species’fluxes and heatsources. 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 sin-gle tubular SOFCs [23], which have been tested by Hagiwara et al. [26], Hiranoet al. [25], Singhal [27], and Tomlins et al. [28]. The fuel tested by Hirano et al.[25] had components of H2, H2O, CO and CO2; therefore, there is a water-shiftreaction of the carbon monoxide in the fuel cell to be considered together withthe electrochemical reaction. The fuel used by the other researchers [26–28] hadcomponents of H2 and H2O, where there is no chemical reaction except for theelectrochemical reaction in the fuel channel. The dimensions of the three differentsolid oxide fuel cells tested in their studies are summarized in Table 5, in whichthe mesh size adopted in our numerical computation is also given. The operatingconditions are listed in Table 6, including the species mole fractions and the tem-perature of the fuel and air in those tests, which are the prescribed conditions forthe numerical computation. In the experimental work by Singhal [27], a test of thepressure effect was also conducted by varying the fuel and air pressure from 1 atmto 15 atm. It is expected that the experimental data for these SOFCs in differentdimensions and operating conditions will facilitate a wide benchmark range forvalidation of the numerical modeling work.

4.4.1 The SOFC terminal voltageThe computer calculated and the experimentally obtained cell terminal voltagesunder different cell current densities are shown in Fig. 9. The relative deviation of



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 7.00/1.00/485 6.00/1.00/290 12.00/1.00/1450Support tube – 13.00/1.50/300 –Cathode 15.72/2.20/500 14.40/0.70/300 21.72/2.20/1500Electrolyte 15.80/0.04/500 14.48/0.04/300 21.80/0.04/1500Anode 16.00/0.10/500 14.68/0.10/300 22.00/0.10/1500Fuel boundary 18.10/ – /500 16.61/ – /300 24.87/ – /1500Grid number (r × x) 66×602 66×602 66×1602

Table 6: Species’ mole fractions, utilization percentages, and temperatures.

Air fuelO2%−UO2 /N2%/T (◦C) H2%–UH2 /H2O%/CH4%/CO%/CO2%/T (◦C)

I 21.00–17.00/79.00/600.0 98.64–85.00/1.36 /0/ 0 /0 /900.0II ∗21.00–25.00/79.00/600.0 55.70–80.00/27.70/0/10.80/5.80/800.0

∗∗21.00–25.00/79.00/400.0 55.70–80.00/27.70/0/10.80/5.80/800.0III 21.00–17.00/79.00/600.0 98.64–85.00/1.36 /0/ 0 /0 /800.0

∗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 theSOFC 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 thatby Hirano et al. [25] is the lowest. The mole fraction of hydrogen in the fuel for theSOFC tested by Hirano et al. [25] is low, which might be the major reason that thiscell has the lowest cell voltage. Because the current must be collected circumfer-entially in a tubular type fuel cell, the large diameter of the cell tube investigatedby 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 operationof 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 withthe increasing operating pressure, as seen in Fig. 10. The agreement between our



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] andHirano 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 theoperating pressure increases from 1 atm to 5 atm, the cell output power shows asignificant improvement of 9%. However, raising the operating pressure becomesless 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 operat-ing pressure increases from 5 atm to 15 atm. The reason for this is that the operat-ing pressure contributes to the cell voltage in a logarithmic manner. Nevertheless,



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, thecell output power can have an increment of 15.8%. There is no doubt from theabove investigation that the investigators can satisfactorily predict the overall per-formance of a SOFC through numerical modeling and computation. On the basisof this good agreement with the overall fuel cell performance, the internal detailsof the flow, temperature, and concentration fields from numerical prediction canalso be reliably presented.

4.4.2 Cell temperature distributionBecause the measurement of temperature in a SOFC is very difficult, only threeexperimental data points, the temperature at the two ends and in the middle of thecell tube, were available from the work on Hirano et al. [25]. Figure 11 shows thesimulated cell temperature distribution for the SOFC, for which Hirano et al. [25]provided the test data. The agreement of the simulated data and the experimentalresults is good in the middle, where the hotspot is located; relatively larger devia-tions between the predicted and experimental values appear at the two ends of thecell. Nevertheless, such a discrepancy is acceptable when designing a SOFC withrespect to concerns about the prevention of excessive heat in the cell materials.

The predicted temperature distributions for the fuel cells tested by Hagiwaraet al. [26] and Tomlins et al. [28] are given in Fig. 12. Unfortunately, there wasno experimental data on the cell temperature. Generally, the two ends of the celltube have lower temperatures than the middle of the cell tube. However, at lowcurrent densities, the hotspot is located closer to the closed end of the cell. With anincrease in current density, the hotspot shifts to the open-end side, and the hotspottemperature also decreases, which improves the uniformity of the temperature dis-tribution along the fuel cell. It should be observed that the heat transfer betweenthe cooling air and the cell tube at the closed-end region is dominated by laminarjet 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.



Figure 12: Predicted longitudinal temperature distribution for two SOFCs.

transfer coefficient significantly. For the high current density case, the flow rate ofair also becomes large accordingly. Thus, the heat transfer coefficient between theair and the fuel cell closed-end region is increased. This can suppress the tempera-ture level of the closed-end region of the fuel cell significantly. Since the air receivesa large amount of heat at the closed-end region, its cooling to the fuel cell in thedownstream region becomes weak, and the uniformity of the cell temperature dis-tribution becomes much better when the fuel cell operates at high current densities.

4.4.3 Flow, temperature and concentration fieldsFigure 13 shows the flow and temperature fields for the SOFC tested by Hiranoet al. [25] at a current density of 185 mA/cm2. The air speed in the air-inducingtube has a slight acceleration because the air absorbs heat and expands in this flowpassage. After leaving the air-inducing tube, the air impinges on the closed endof the fuel cell, and then flows backwards to the outside. In this pathway, the airobtains heat from the heat-generating fuel cell tube and transfers the heat to thecold air in the air-feeding tube. It is easy to understand that the electrochemicalreaction at the closed end of the fuel cell is strong because the concentrations offuel and air are both high there. Therefore, the heat generation due to Joule heatingand the entropy change of the electrochemical reaction is high at the upstream areaof the fuel path. However, it is known from both experiments and computation thatthe 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 fuelcell is responsible for this. After being heated at the closed-end region, air exhibitsa higher temperature, and its cooling ability to the cell tube is low when it is inthe annulus between the air-inducing tube and the cell tube. At the cell open-endregion, the air in the annulus can transfer heat to the incoming cold fresh air in theair-inducing tube, and this will help it to cool the fuel cell tube.



Figure 13: Predicted flow and temperature fields for the SOFC reported by Hiranoet al. [25] at a current density of 185 mA/cm2.

From this airflow arrangement, the hotspot temperature of the cell tube maymostly 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 inthe annulus between the air-inducing tube and the cell tube. The heat exchange inbetween the two passes allows the air to mitigate its temperature fluctuation in thewhole air path, and thus the temperature field in the fuel cell might be maintainedas relatively uniform. Nevertheless, the heat generation, air and fuel temperature,and air-cooling to the fuel cell will collectively affect the temperature field in thefuel cell. Therefore, the hot spot position in a cell tube might shift more or lessaway 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 underthe 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 tomore densely distributed contour lines. The contour shape of oxygen also indicatesa relatively larger difference of the mole fraction between the bulk flow and thewall of the cathode/air interface. This implies that the mass transport resistance onthe air side might be dominant in lowering the cell performance if the stoichiometryof the oxygen is low. Feeding more air than is needed is already well applied inoperational fuel cell technology.



Figure 14: Predicted fields of the mole fraction of the species for the SOFC reportedby Hirano et al. [25] at a current density of 185 mA/cm2.

The hydrogen budget is collectively determined by the consumption by theelectrochemical reaction and the generation from the water-shift reaction of CO.Since the consumption dominates, the hydrogen mole fraction decreases alongthe fuel stream. Corresponding to this hydrogen variation, consumption due to thewater-shift reaction and production due to the electrochemical reaction cause thewater vapor to increase gradually along the fuel stream. The water-shift of COproceeds gradually in the fuel path, and thus the mole fraction of CO decreases butthe CO2 increases. The shape of the contour lines of the species in the fuel path is



Figure 15: Predicted streamwise molar flow rate variation for species in the fuelchannel for the SOFC reported by Hirano et al. [25].

relatively flat from the cell wall to the bulk flow. This indicates that mass diffusionin 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 themolar flow rate variation along the fuel path. In one third of the length from the fuelinlet, the hydrogen flow rate shows a faster decrease and the water flow rate showsa faster increase, indicating a strong reaction in the upstream region. The flow rateof CO and CO2 vary roughly in a linear style, and a small amount of CO still existsin the waste gas.

5 Concluding remarks

Fuel cell technology is currently under rapid development. To improve SOFC per-formance, for high power density and efficiency, efforts have been made to reducethe three over-potentials: activation polarization, ohmic loss, and concentrationpolarization. Better understanding of these three over-potentials is also very impor-tant in developing accurate computer models for predicting the overall performanceand internal details of a SOFC.

The activation polarization relates to the porous structure of the electrode andelectrocatalyst materials. The state-of-the-art in material and manufacturing pro-cesses for the electrodes and electrolyte has been reported by Singhal [27]. Thereduction of ohmic losses also heavily relies on the reduction of electronic and ionicresistances in the electrodes and electrolyte. A shorter current collection pathwayalso helps to reduce ohmic loss. A new design, referred to as a high power densitysolid oxide fuel cell (HPD-SOFC), has been developed by Siemens WestinghousePower Corporation [27, 32], and has a significantly shorter current pathway, and



thus improves the power density significantly. A planar structure also promisesto have a shorter current pathway and thus a higher power density, and measuresfor reducing the ohmic loss in a planar type SOFC have been reported by Tannerand Virkar [61]. The reduction of mass transport resistance, or the concentrationpolarization, has not been given much attention. Mass transfer enhancement hasbeen 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 forSOFCs 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-potentialsis very important in order to accurately predict the overall current-voltage perfor-mance. The heat generation from the over-potentials is also significant in com-puting the temperature, flow, and species concentration fields. With respect to theactivation polarization, studies elucidating the data and equations for the exchangecurrent density are still needed. For the prediction of ohmic losses, reliable propertydata for electrodes are required. Additionally, a method for analyzing a complexnetwork circuit in a SOFC needs to be developed. The concentration polarizationis considered in the numerical computation by using the local mole fractions ofthe species at the interface of the electrode and fluid when calculating the electro-motive force by the Nernst equation. Because the porous electrodes also serve asthe reaction site, there is no well-described model for the mass transport resistancein the electrodes.Adopting a lower exchange current density, which induces a largerover-potential of the activation polarization, may be a way to incorporate the masstransport resistance in the electrodes into the activation polarization. The methodgiven by Hirano et al. [25] for the consideration of mass transport resistances inthe electrodes is convenient, but may be too simple and needs more investigation.

With the progress being made in computer modeling of SOFCs, it is expectedthat costs for research and development of SOFCs will be significantly reduced byusing computer simulations in the future.

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Nomenclature

a Stoichiometric coefficient of chemical species.A Chemical species. Area (m2). General variable.Acell Outer surface area of fuel cell (m2).Aair , Afuel Inlet flow area of air and fuel, respectively (m2).b Stoichiometric coefficient of chemical species.B Chemical species. General variable.C General variable.Cp Specific heat capacity at constant pressure [J/(kg K)].DJ ,m Diffusion coefficient of jth species into the left gases

of a mixture (m2/s).



E Electromotive force or electric potential (V).F Faraday’s constant [96486.7 (C/mol)].g Gibbs free energy (J/mol).h Chemical enthalpy (kJ/kmol) or (J/mol).H Height (m).i Current density (A/m2).i0 Exchange current density (A/m2).I Current (A).k Thermal conductivity (W/m K).KPR, KPS Chemical equilibrium constant for reforming and shift

reactions, respectively.L Length (m).m Mass transfer rate or mass consumption/production rate (mol/s).m Mass flux [mol/(m2s)].M Molecular weight (g/mol).Mf Total mole rate of fuel flow (mol/s).ne Number of electrons involved in per fuel molecule in oxidation

reaction.p, P Pressure (Pa) or position.q Volumetric heat source ( W/m3).Q Heat energy (W).r Radial coordinate (m).ra, rc, re Average radius of anode, cathode, and electrolyte layers (m).R Universal gas constant [8.31434 J/(mol K)].Ra, Rc, Re Discretized resistance in anode, cathode, and electrolyte (�).S Source term of gas species (kg/m3); General variable.T Temperature (K).u Velocity in axial direction (m/s).U Utilization percentage (0–1).v Velocity in radial direction (m/s); Diffusion volume in eqn (68).V Specific volume (m3/kmol).Vcell Cell terminal voltage (V).V a, V c Potentials in anode and cathode, respectively (V).W Width (m).x Stoichiometric coefficient of chemical species;

Axial coordinate (m).X Chemical species. Mole fraction.x, y, z Reacted mole rate of CH4, CO and H2, respectively in a section

of interest in flow channel (mol/s).y Stoichiometric coefficient of chemical species; Coordinate (m).Y Chemical species. Mass fraction.z Coordinate (m).Z Compressibility factor.



Greek symbols

θ Circumferential position. Angle.δ Thickness of electrodes and electrolyte layers (m).�G Gibbs free energy change of a chemical reaction (J/mol).�G0 Standard state Gibbs free energy change of a chemical

reaction (J/mol).�H Enthalpy change of a chemical reaction (J/mol).�S Entropy production [J/(mol K)].�x One axial section of fuel cell centered at x position (m).λ Thermal conductivity [W/(m ◦C)].µ Dynamic viscosity (Pa s).ρ Density (kg/m3).ρa

e , ρce Electronic resistivity of anode and cathode respectively (� · cm).

ρee Ionic resistivity of electrolyte (� · cm).

ρr Reduced density.ηAct Activation polarization (V ).

Subscripts

a Anode.c Cathode.cell Overall parameter of fuel cell.e, w, n, s East, west, north, and south interfaces between grid P and it

neighboring grids.E, W , N , S East, west, north, and south neighboring grids of grid P.f Fuel.i Subscript variable.j Gas species; Subscript variable.m Mixture.P Variables at grid P.R Electrochemical reaction.x Axial position.X , Y Chemical species.�x Variation in the channel section of �x.

Superscripts

a Anode. Sequence.b Sequence.c Cathode. Sequence.e Electrolyte.in Inlet of a channel section of interest.out Outlet of a channel section of interest.P Variables at grid P.R Reaction.x Axial position.


chapter 1 multiple transport processes in solid oxide fuel cells · 2014-05-10 · oxide fuel cells...

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