Energy Conversion and Management 71 (2013) 172–185

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Energy Conversion and Management

journal homepage: www.elsevier .com/locate /enconman

Transient dynamic and modeling parameter sensitivity analysisof 1D solid oxide fuel cell model

0196-8904/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.enconman.2013.03.029

⇑ Corresponding author at: IRTES-SET Laboratory, University of Technology ofBelfort-Montbéliard, 90010 Belfort, France. Tel.: +33 384583801; fax: +3384583413.

E-mail addresses: yigeng@nwpu.edu.cn (Y. Huangfu), fei.gao@utbm.fr (F. Gao),abdeljalil.abbas-turki@utbm.fr (A. Abbas-Turki), david.bouquain@utbm.fr (D. Bou-quain), abdellatif.miraoui@utbm.fr (A. Miraoui).

Yigeng Huangfu a, Fei Gao b,c,⇑, Abdeljalil Abbas-Turki b, David Bouquain b,c, Abdellatif Miraoui b,c

a Institute of REPM Electrical Machines and Control Technology, Northwestern Polytechnical University, 710072 Xi’an, Chinab IRTES-SET Laboratory, University of Technology of Belfort-Montbéliard, 90010 Belfort, Francec Fclab Research Federation, CNRS 3539, 90010 Belfort, France

a r t i c l e i n f o

Article history:Received 3 January 2013Accepted 31 March 2013Available online 30 April 2013

Keywords:Solid oxide fuel cellModelingParameter sensitivityTime constant

a b s t r a c t

In this paper, a multiphysics solid oxide fuel cell (SOFC) dynamic model is developed by using a onedimensional (1D) modeling approach. The dynamic effects of double layer capacitance on the electro-chemical domain and the dynamic effect of thermal capacity on thermal domain are thoroughly consid-ered. The 1D approach allows the model to predict the non-uniform distributions of current density, gaspressure and temperature in SOFC during its operation. The developed model has been experimentallyvalidated, under different conditions of temperature and gas pressure. Based on the proposed model,the explicit time constant expressions for different dynamic phenomena in SOFC have been given and dis-cussed in detail. A parameters sensitivity study has also been performed and discussed by using statisti-cal Multi Parameter Sensitivity Analysis (MPSA) method, in order to investigate the impact of parameterson the modeling accuracy.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Solid oxide fuel cell (SOFC) is considered as one of potential can-didates for the renewable energy applications in the future. TheSOFC based energy production systems are well known for theirhigh energy efficiency, environmentally friendly by-products andwide power range (from some kW to some MW) in different appli-cations. The SOFC uses solid ceramic materials (usually YSZ, Yttria-Stabilized Zirconia) as electrolyte, to operate at very high temper-ature (typically 700–1000 �C). The structures of SOFC can bemainly distinguished into two types: tubular SOFC structure andplanar SOFC structure. During the operation of an anode supportedtubular SOFC, hydrogen is supplied to the inner tube of cell (e.g. an-ode) and air is supplied to the outer surface of cell (e.g. cathode).The reactant gases diffuse through a porous ceramic electrode toreach the catalytic zone (usually Ni is used as catalyst). At the cath-ode, oxygen is reduced to oxygen–ion (O2�). At the anode, hydro-gen is oxidized to proton (H+). The oxygen–ion passes throughthe solid electrolyte and forms water with the proton at the anode.The ceramic electrolyte is impermeable to electrons. Thus, the elec-

tron takes the path of external circuit from anode to cathode, sup-plying electrical power to an external load.

The SOFC operation involves some complicate multiphysicsphenomena, including electrochemical reactions, gas diffusionsand heat generations. In order to explain the operation charac-teristics and predict its performance, an accurate mathematicalmodel needs to be developed. However, due to the complexityof physical phenomena in SOFC as well as the lack of detailsabout the used materials, some empirical parameters are usuallyused in SOFC models. The choice of the numeric values of theseempirical parameters may significantly impacts the accuracy ofthe developed model. Thus, performing the parameters sensitiv-ity analysis is required for providing a reliable model. Such anal-ysis could reveal the different degrees of sensitivity of theparameters in a model. The analysis results could be used as ref-erences for determining empirical parameters from experimentaltests.

Furthermore, SOFC transient behaviors could be observed dur-ing the load power variations. The dynamic behaviors of SOFC havea direct impact on the fuel cell system component design, forexample, the design of the power converter connected to the fuelcells. A better understanding of the dynamic behaviors in a fuel cellcould facilitate the design and conditioning of the system.

In the literature, an electrical model that includes simple ther-mal dynamic for a 5-cells SOFC stack with experimental validationshas been given in [1]. The proposed model has been used to inves-tigate the temperature and output voltage variation between dif-ferent cells in the stack. Only the large time scale dynamic (up to

Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185 173

some hours) has been considered in the model. A simple 0D SOFCmodel that focuses on the influence of cell heat capacity on thethermal dynamic response has been presented in [2]. The largetime scale dynamic response of the cell temperature and voltagedue to the cell current change has been discussed in detail. A moredetailed planar SOFC electrochemical model has been introducedin [3]. The model takes into account the hydrogen consumptionacross the gas channels. A power density analysis in SOFC has alsobeen given for different cell geometries. However, the thermal dy-namic has not been included in the model. A 1D tubular SOFC mod-el has been developed in [4]. The model considers the local currentdensity and a prediction accuracy comparison between 0D and 1Dmodel has been made. Furthermore, the thermal modeling hasbeen considered in a SOFC-micro gas turbine hybrid power systemmodel in [5]. The model has been developed by using a leastsquares support vector machine (LS-SVM) identification method.Even though the thermal responses can be successfully predictedby the proposed modeling method, the mathematical fitting-modeldoes not include any physical thermal phenomena equation. Asimple physical SOFC model coupled with a micro gas turbinehas been introduced in [6], however the presented model takesinto account only the electrochemical phenomena. A similar sim-ple electrochemical SOFC model has also been reported in [7], asa part of a fuel cell powered vehicle system model. A parametricstudy of a SOFC electrochemical model has been presented in [8].The influence of the cell geometric and operating parameters onthe SOFC performance has been discussed in great detail. however,the thermal domain has not been modeled. in addition to singlecell modeling, The 1D IP-SOFC stack model presented in [9] dem-onstrates the non-uniform effect of SOFC parameters in the caseof a 15 cells stack. A more complete model for a combined fuelreforming and SOFC system has been developed in [10]. The energyefficiency of such a system has been studied. A planar SOFC 2Dmodel has been developed in [11] in COMSOL modeling environ-ment. A dusty-gas model has been used to model the diffusionphenomena in the SOFC electrodes. The non-isothermal effecthas also been considered. However, the model in [11] has beenexperimentally validated only for one operating condition at thesteady-state. Another steady-state planar SOFC 2D stack modelhas been proposed in [12] with a model reduction method. Themodel predicted the current density and temperature distributionin the planar SOFC under steady-state conditions has been dis-cussed. A 2D SOFC which covers electrochemical, thermal and flu-idic domains has been presented in [13], with an investigation ofcell operating parameters, such as inlet gas composition and elec-trode porosity. In [14], a 2D SOFC model with the consideration ofpartially pre-reformed inlet gas has been introduced. The cell per-formance has been evaluated by considering internal reformingprocess and water gas shift phenomena. Some other 2D SOFC mod-els can also be found in the literature for cell performance predic-tion and fuel composition investigation [15–17]. A detailed 3Dcomputational fluid dynamic model for a planar SOFC has beenintroduced in [18] for optimization purpose. Although the 3-Dmodel shows a great interest for SOFC design improvement, thecomplexity of a 3-D model could limit its use in different applica-tions. An investigation of the electrode thickness on the SOFC per-formance has also been made by proposing a two finite layersmodel of SOFC electrode in [19]. The modeling focuses are onlymade on the electrochemical phenomena in the SOFC. A 3D multi-physics model for a planar SOFC has been reported in [20]. Thetemperature and current density distributions of four differentflow fields have been shown and discussed. A study in the effectsof radiative heat transfer of SOFC has been introduced in [21] byusing a detailed finite volume SOFC model with internal gasreforming. The factors that can lead to a decrease of SOFC efficiencyhave been discussed. In [22], the thermal time constant for SOFC

has been investigated and discussed. The results show that thethermal processes of the fuel cell is the most important in model-ing systems due to their influence on other parameters. Similarconclusion can also be found in [23] for a micro-tubular SOFC. Inaddition, SOFC material and geometry studies for the dynamicoperation have also been presented in [24]. Based on a steady-statefinite volume SOFC model, the influence of operating parameterson the SOFC performance has been discussed in [25]. The influenceof the finite volume section number on the model output has alsobeen shown. A sensitivity analysis for the geometry parameters ofa planar SOFC using a 3-D model and a discrete adjoint method hasbeen presented in [26]. An electrochemical parameter sensitivityanalysis has also been introduced in [27] and [28] for low temper-ature proton exchange membrane fuel cells (PEMFC). The resultsshow that, the electrode activation-related parameters have themost impact on PEMFC performance.

In this paper, a dynamic multiphysics model of a tubular solidoxide fuel cell is presented by using one dimensional modeling ap-proach. The model is validated experimentally in different operat-ing conditions. The non-uniformity effect, such as current densitydistribution, has also been shown and discussed. The fuel cell dy-namic effects due to the double layer capacitances and cell thermalcapacity are discussed later. The first order time constant expres-sion for each dynamic phenomenon is calculated and discussedin detail. At last, a modeling parameters sensitivity analysis is per-formed for different semi-empirical parameters used in the SOFCmodel. The different degrees of sensitivity for semi-empiricalparameters are shown and carefully discussed.

The main objectives of this paper are then: (1) Providing a dy-namic, multiphysics 1D tubular SOFC model with experimentalvalidations; (2) Discussing the dynamic phenomena in SOFC andgiving the mathematical estimations of dynamic transient time;(3) Performing modeling parameters sensitivity analysis in differ-ent physical domains to show the different influences of parame-ters on the SOFC model output. The remainder of the paper isorganized as follows. The next section presents in detail the pro-posed mathematical dynamic model for an anode supported hightemperature tubular SOFC. The third section experimentally vali-dates the 1D multiphysics SOFC dynamic model through an SOFCprototype. In the fourth section, the dynamic phenomena areinvestigated in detail in electrochemical and thermal domains.The last section gives a detailed parameters sensitivity analysisfor semi-empirical parameters in the SOFC model, beforeconcluding.

2. Solid–oxide fuel cell modeling

The multiphysics model presented in this section considerselectrochemical, fluidic and thermal domains. Besides, the modelincludes electrochemical and thermal dynamic phenomena. Theelectrochemical dynamic phenomenon is due to the double layercapacitances, whereas the thermal dynamic phenomenon is dueto the cell thermal capacity. The governing equations of the modelare presented hereafter.

2.1. Electrochemical domain governing equations

The SOFC output voltage can be described by the followingequation:

Vcell ¼ EEMF � Vact;C � Vact;A � Vohm ð1Þ

where EEMF is the cell thermodynamic voltage (V), Vohm is the cellohmic losses (V) and Vact,C, Vact,A are the cell activation losses (V)of the cathode and the anode side, respectively.

174 Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185

2.1.1. Thermodynamic voltageThe thermodynamic voltage is defined by the Nernst equation

as a function of the Gibbs free energy change during the electro-chemical reaction:

EEMF ¼ �DG0

2Fþ R � T

2Fln

PH2 �ffiffiffiffiffiffiffiPO2

pPH2O

!ð2Þ

where DG0is the Gibbs free energy change from reactants to prod-ucts (J mol�1), F = 96485.3 is the Faraday constant (C mol�1),R = 8.314 is the universal gas constant (J mol�1 K�1), T is the electro-chemical reaction temperature (K) and PH2 ; PO2 , PH2O are the reactinggas pressures (atm) of hydrogen, oxygen and vapor, respectively.

The expression of DG0 can be obtained from the followingequations:

DG0 ¼ G0H2O � G0

H2þ 1

2G0

O2

� �ð3Þ

G0specie ¼ H0

specie � T � S0specie ð4Þ

where G0H2O;G

0H2;G0

O2are the Gibbs free energies (J mol�1) of vapor,

hydrogen, oxygen, respectively, H0specie is the corresponding enthalpy

(J mol�1) and S0specie is the corresponding entropy (J mol�1 K�1).

The values of enthalpy H0 and entropy S0 for H2, O2 and H2O inSOFC temperature range (750–850 �C) can be found in [29].

2.1.2. Activation lossesThe steady-state activation losses in the fuel cell can be ex-

pressed by the electrochemical Butler–Volmer equation:

i ¼ Ael � j0 ea�ne �F

R�T gact � e�ð1�aÞ�ne �F

R�T gact

� �ð5Þ

where i is the cell electrical current (A), Ael is the electrode area(m2), j0 is the exchange current density (A m�2) at the cathode orthe anode side, gact is the corresponding steady-state activationlosses (V) at the cathode or the anode, a is the electrochemical sym-metry factor and ne = 4 or 2 is the number of electrons involved inthe half reactions of the cathode or the anode, respectively.

The cathode and the anode exchange current density equationsfor a typical SOFC are proposed in [30]:

j0;C ¼ cC � P0:25O2� e �EC

RT

� �ð6Þ

j0;A ¼ cA � PH2 � PH2O � e �EART

� �ð7Þ

where P(specie) is the partial pressure (atm) of the corresponding gas,EC and EA are the activation energies (J mol�1) at the cathode andthe anode, respectively, cC and cA are the reference exchange cur-rent density: two semi-empirical parameters (A m�2) for the cath-ode and the anode respectively that should be identified fromexperimental tests.

It should be noted that, the steady-state activation loss in theButler–Volmer equation has an implicit form (e.g. two exponentialterms at right side of the equation). In order to solve this equationdirectly in a simulation model, an iterative method needs to be ap-plied. A simple dichotomy iterative method has been proposed in[31] to resolve the non-linear Butler–Volmer equation with a de-sired precision.

Due to the electrochemical double layer capacitance effects onboth anode side and cathode side in SOFC, the dynamic activationloss expressions for cathode side and anode side can be modeledby first order differential equations as presented in [32–34]:

dVact;C

dt¼ i

Cdl;C1� 1

gact;CVact;C

!ð8Þ

dVact;A

dt¼ i

Cdl;A1� 1

gact;AVact;A

!ð9Þ

where Vact,C and Vact,A are the dynamic activation losses (V) at cath-ode side and anode side, respectively, Cdl,C and Cdl,A are the doublelayer capacitances (F) at cathode side and anode side, respectively.The value of the double layer capacitances for both cathode and an-ode in a fuel cell can be identified experimentally by using EIS (Elec-trochemical Impedance Spectroscopy) method [35–37].

2.1.3. Ohmic lossIn SOFCs, the Yttria-Stabilized Zirconia (YSZ) electrolyte electri-

cal conductivity rYSZ (S m�1) can be calculated from [38]:

rYSZ ¼r0

Te �EYSZ

RT

� �ð10Þ

where r0 is the reference YSZ conductivity (S K m�1) and EYSZ is theYSZ membrane activation energy (J mol�1).

Thus, the ohmic losses can be obtained:

Vohm ¼i � del

Ael � rYSZð11Þ

where del is the YSZ electrolyte thickness (m) and Ael is the electro-lyte area (m2).

2.2. Fluidic domain governing equations

The modeled SOFC cell has a tubular anode support structure.The mixed-gas fuel (H2, H2O and Ar) flows into the cell inner

tube channel (anode) and the preheated air is supplied to cellouter surface (cathode). The presence of argon gas (Ar) is usedto adjust the H2 and H2O molar faction supplied to the cell whileconserving the constant inlet mixed-gas pressure and the molarflow rate.

2.2.1. Anode fuel channelFrom the fundamental law of mass conservation, the relations

between the inlet gas and outlet gas flow rate can be obtained:

qH2 ;in ¼ qin � XH2 ;in ð12ÞqH2O;in ¼ qin � XH2O;in ð13ÞqAr;in ¼ qin � XAr;in ð14Þ

qH2 ;out ¼ qH2 ;in �i

2Fð15Þ

qH2O;out ¼ qH2O;in þi

2Fð16Þ

qAr;out ¼ qAr;in ð17Þqout ¼ qH2 ;out þ qH2O;out þ qAr;out ð18Þ

where qin and qout are the inlet and outlet mixed-gas molarflow rates (mol s�1), respectively, X(specie), in is the inlet molar frac-tion of one gas specie and q(specie),in, q(specie), out are the corre-sponding species inlet and outlet molar flow rates (mol s�1),respectively. The argon gas is not involved into the electrochem-ical reaction.

The mixed-gas pressure drop DP (Pa) through the channel tubecan be modeled using the Darcy–Weisbach equation [39]:

DP ¼32 � lgas � Lch � kgas

D2ch

ð19Þ

where lgas is the mixed-gas viscosity (Pa s), kgas is the mixed-gasvelocity in the channel (m s�1), Lch is the channel length (m) andDch is the channel tube diameter (m).

Assuming ideal gas conditions, the mixed-gas velocity in thechannel can be obtained from the channel mixed-gas molar flowrate kgas (mol s�1), the mixed-gas average molar mass Mgas

(kg mol�1), the mixed-gas average density qgas (kg m�3) and thechannel diameter Dch (m):

Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185 175

kgas ¼4 � qgas �Mgas

p � qgas � D2ch

ð20Þ

with the expression of the mixed-gas average density:

qgas ¼Pch �Mgas

R � Tchð21Þ

The hydrogen and vapor pressure in the channel can be also cal-culated using the previous determined quantities:

PH2 ;ch ¼ XH2 ;ch � Pch ð22ÞPH2O;ch ¼ XH2O;ch � Pch ð23Þ

2.2.2. Gas diffusion layers (GDL electrodes)The H2, H2O and O2 molar flow rates (mol s�1) through the gas

diffusion layer can be calculated from the cell current i (A):

qH2O;GDL ¼ �i

2Fð24Þ

qO2 ;GDL ¼i

4Fð25Þ

The individual gas pressure variation through the gas diffusionlayer can be expressed from the modified Fick diffusion law:

Pch � Pcata ¼qsp;GDL � dGDL � R � T

Dsp�GDL;eff � AGDLð26Þ

where Dsp�GDL,eff is the gas effective diffusion coefficient to other gasthrough the porous gas diffusion layer (m2 s�1), Pcata is the gas pres-sure in the catalyst sites (Pa), dGDL is the gas diffusion layer thick-ness (m) and AGDL is the gas diffusion layer area (m2).

The effective gas diffusion coefficients can be obtained using thebinary gas diffusion coefficient with the Bruggemann correction:

Dsp�GDL;eff ¼ Dsp�GDL � es ð27Þ

where Dsp�GDL is the binary gas diffusion coefficient (m2 s�1) in gasdiffusion layers, e is the corresponding gas diffusion layer porosityand s is the corresponding gas diffusion layer tortuosity, whichare empirical parameters that can be identified experimentally.

The pressure and temperature-dependent binary gas diffusioncoefficients of gases can be calculated from Slattery–Bird formulain [39,40]:

Di�j ¼1

Pgas� a � Tgasffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Tc;i � Tc;j

p !b

� Pc;i � Pc;j� �1=3 � ðTc;i � Tc;jÞ5=12

� 10�3

Miþ 10�3

Mj

!1=2

ð28Þ

The critical temperature Tc and pressures Pcof the differentgases and the values of a and b are given in [40].

2.2.3. Cathode air sideBased on the presented cell configuration, the preheated air

cathode side is assumed to have a uniform gas pressure distribu-tion equal to the cathode operating pressure. The O2 molar fractionin the preheated air is assumed to be constant and equal to 20.9%.

2.3. Thermal domain governing equations

In the radial direction, the tubular SOFC cell is divided radiallyby two thermal control volumes in the present model: electro-lyte–electrode and anode gas channel.

2.3.1. Electrolyte–electrodeThe dynamic energy conservation equation for the electrolyte–

electrode is given as follow:

qEEVEECp;EEdTEE

dt¼ Q cd þ Q fc þ Qm þ Q int ð29Þ

where qEE is the electrolyte–electrode density (kg m�3), VEE is thevolume (m3), Cp,EE is the thermal capacity (J kg�1 K�1), TEE is theelectrolyte–electrode temperature (K), Qcd is the conduction heatflow (J s�1), Qfc is the forced convection heat flow (J s�1), Qm is theconvective mass energy flow (J s�1) and Qint is the internal heatsources in the electrolyte–electrode (J s�1).

The conduction heat flow in the electrolyte–electrode is due tothe contact between the cathode constant temperature area andthe cell tube outer surface, which can be modeled by:

Qcd ¼kEE � Aouter

dEEðTcathode � TEEÞ ð30Þ

where kEE is the electrolyte–electrode thermal conductivity(W m�1 K�1), Aouter is the cell tube outer area (m2), dEE is the elec-trode–electrolyte thickness (m) and Tcathode is the cathode side tem-perature (K).

The forced convection heat flow is due to the heat exchange be-tween the forced gas flow through the fuel channel and the celltube inner surface. The Newton cooling law is used here to describethis phenomenon:Qfc ¼ hfc � AinnerðTch � TEEÞ ð31Þ

where hfc is the forced convection coefficient (W m�2 K�1) in thetube channel, Ainner is the tube inner area (m2) and Tch is the channeltemperature (K).

The expression of hfc can be obtained as follow:

hfc ¼Nu � kgas

Dchð32Þ

where Nu is the Nusselt number in anode channel and kgas is themixed-gas average thermal conductivity (W m�1 K�1). In a SOFC,the Nusselt number can be calculated as [41,42]:

Nu ¼ 1:86 �Cp;gas � qgas � Lch � kgas

kgas

� �1=3

� Dch

Lch

� �1=3

ð33Þ

where Cp,gas is the mixed-gas thermal capacity (J kg�1 K�1), qgas isthe mixed-gas density (kg m�3) and kgas is the mixed-gas velocityin the channel (m s�1).

The convective mass energy flow is due to the gas species of dif-ferent temperatures flowing in and out of the electrolyte–elec-trode. This heat flow can be expressed as:

Qm ¼X

species

qsp;GDL �Msp � Cp;sp � ðTflowin � TEEÞ ð34Þ

where Cp,sp is the gas specie thermal capacity (J kg�1 K�1).The internal heat sources in the electrolyte–electrode are due to

different phenomena: resistive losses, activation losses and irre-versible electrochemical losses. These sources can be calculatedas follow:

Qint ¼ i � Vohm þ Vact;C þ Vact;Að Þ � i � TEE � DS0

2Fð35Þ

where DS0 is the entropy change (J mol�1 K�1) during the electro-chemical reaction:

DS0 ¼ S0H2O � S0

H2þ 1

2S0

O2

� �ð36Þ

2.3.2. Anode gas channelThe energy conservation equation for the anode gas channel can

be described as below:

qgasVchCp;gasdTch

dt¼ Qfc þ Qm ð37Þ

176 Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185

where Cp,gas is the mixed-gas average thermal capacity (J kg�1 K�1),Vch is the channel tube volume (m3), Tch is the channel temperature(K), Qfc is the forced convection heat flow (J s�1) and Qm is the con-vective mass energy flow (J s�1).

The forced convection heat flow can be calculated by:

Q fc ¼ hfc � AinnerðTEE � TchÞ ð38Þ

The value of hfc is the same as described in (32).The convective mass energy flow can be calculated from the

same formula in (34).

2.4. One dimensional modeling grid approach

Using the physical equations presented in the previous section,the anode-supported tubular SOFC is modeled in 1D as shown inFig. 1.

The two modeling axis have been considered as axial and radiallengths. In the radial direction, four control volumes can be distin-guished: (1) anode tubular channel; (2) anode electrode (GDL); (3)YSZ electrolyte and (4) cathode electrode (GDL). In axial direction,the tube is divided into adjustable N segments (control volumesgrid) in order to investigate the axial distribution of current den-sity, temperature, and so on. Each segment in axial direction hasthe same geometry and material properties, but the differentboundary conditions during simulation.

2.5. Numerical current density solver scheme

In order to solve the current density distribution of the pre-sented 1D SOFC model under Matlab–Simulink, two boundary con-ditions at cell level are known:

1. The electrical potentials Vcell from Eq. (1) of each axial segmentare the same:

Vcell ¼ Vseg;n for n ¼ 1 to N ð39Þ

2. The total cell current i (known) is equal to the sum of the cur-rent in each axial segment:

i ¼XN

n¼1

iseg;n ð40Þ

where N is the segment number in axial direction, iseg,n and gseg,n

are, respectively, current and voltage of each segment.

The numerical solution challenge is to solve simultaneously thecurrent density in each segment. The model only input is the celltotal current. The current in each segment is not known, becausethe electro-chemical control volumes present a parallel connection

Fig. 1. 1D tubular SOFC model control volumes.

(Fig. 1) with only two known boundary conditions as describedabove.

The inter-coupled segment currents cannot be resolved explic-itly due to the complex implicit form of the cell potential (Eqs. (1)–(11)). Thus, an iterative method to calculate current in each seg-ment has been proposed here. The major steps of this proposedmethod are:

1. Estimate an initial cell potential Vcell value.2. Estimate in each segment a current value iseg,n based on this Vcell

value.3. Calculate in each segment, the corresponding potential Vseg,n by

using the estimated iseg,n.4. Compare in each segment, the Vseg,n calculated from iseg,n with

the Vcell. If Vseg,n is smaller than Vcell, estimate a new smaller iseg,n,and vice versa, in order to let the Vseg,n converge to Vcell.

5. When all iseg,n are calculated using Vcell, compare the sum of theiseg,n with the real cell total current value. If the real cell totalcurrent is smaller than the sum of iseg,n, estimate a new biggerVcell, and vice versa, in order to let the sum of iseg,n converge toreal cell total current.

6. Return to step 2), until the sum of the obtained iseg,n equal toreal cell total current. Thus, the corresponding Vcell and iseg,n val-ues are found.

2.6. Grid-independence check result

Before performing the model simulation, a modeling grid-inde-pendence check must be done to determine the influence of themodeling segments number (grid) to the simulation results. Sucha grid-independence check can help the choice of the segmentsnumber for simulation between the model accuracy and the modelsimulation speed.

From the Table 1, it can be concluded that, from the segmentnumber N = 15, the simulation result can be considered grid-independent.

3. Model experimental validation

3.1. Experimental setup and conditions

The developed multiphysics SOFC model has been validatedexperimentally with a prototype SOFC single cell manufacturedin the laboratory. The experimental setup for SOFC testing is pre-sented in Fig. 2. During experimental tests, the dry gas composition(e.g. mass flowrate) is controlled by 2 mass flow controllers. Thewater is injected from a controlled water pump and mixed withthe dry gas in a heated vaporizer cylinder. The gas line after thevaporizer is remain heated to prevent water condensation beforeflow into the furnace. The prototype cell is placed in an industrialfurnace and connected to an electronic load before testing. Duringthe tests, the fuel is supplied to the inner tube (anode) of the celland the cathode of the cell is exposed to the heated ambient airwithin the furnace. The cell electrical current is applied by the elec-tronic load and the voltage is measured from the load sensor.

SOFC test start-up process: after the connection of the fuel supplyand electrical wires, the temperature in the furnace is augmentedat 30 �C h�1 until the desired operating temperature is reached.During the heat up, hydrogen is continuously supplied to the anodecell tube. Once the temperature is reached, the cell open circuitvoltage is checked with the theoretical value before the beginningof the test sequences.

Experimental testing sequences: for each specific gas compositionand temperature test, the gas flow is regulated and supplied bydifferent mass flow controllers of the test bench. Once the gas

Table 1SOFC model grid-independence check result.

Axial segmentnumber

Model outputs difference (to previous segment numberresults) (%)

1 –2 5.543 1.785 1.34

10 0.0915 0.0220 0.0005

Fig. 2. Prototype SOFC single cell experimental test bench.

Table 2Tubular SOFC prototype cell parameters.

Symbol Parameter Value Unit

Cell geometriesdGDL,C Cathode electrode thickness 30 lmdGDL,A Anode electrode thickness 1000 lmdel YSZ electrolyte thickness 20 lmLcell Cell length 25 mmDinner Cell tube inner diameter 7.7 mmDouter Cell tube outer diameter 9.8 mm

Cell physical propertieseC Cathode electrode porosity 0.35sC Cathode electrode tortuosity 2.84kC Cathode electrode thermal conductivity 18.2 W m�1 K�1

Cp,C Cathode electrode thermal capacity 505 J kg�1 K�1

eA Anode electrode porosity 0.4sA Anode electrode tortuosity 2.62kA Anode electrode thermal conductivity 10.3 W m�1 K�1

Cp,A Anode electrode thermal capacity 534 J kg�1 K�1

kel Electrolyte thermal conductivity 10.1 W m�1 K�1

Cp,el Electrolyte thermal capacity 538 J kg�1 K�1

Cell electrochemical propertiescC Cathode reference current density 7.3 � 107 A m�2

cA Anode reference current density 3.78 � 109 A m�2

a Electrochemical symmetry factor 0.5661CdlC Cathode double layer capacitance 0.763 mFCdlA Anode double layer capacitance 7.9 mFr0 Reference YSZ elec. conductivity 3.6 � 107 S K m�1

EC Cathode activation energy 120 kJ mol�1

EA Anode activation energy 100 kJ mol�1

EYSZ YSZ activation energy 80 kJ mol�1

Cell operating conditionsqfuel,in Inlet mixed-gas fuel flow rate 400 sccmPcathode Cathode operating pressure 0.83 atmPanode,in Anode inlet gas pressure 0.83 atm

Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185 177

composition is stabilized, different electrical current values aredrawn from the cell via electronic load and the corresponding volt-ages are recorded. At each current value change, the new currentvalue is maintained for at least 5 min to stabilize the cell beforethe voltage measurement. Between each gas composition change,the hydrogen is supplied to the cell for 15 min before beginninganother sequence.

SOFC test cool down process: once all the test sequences are fin-ished, the electronic load connected to the cell is switched-off andthe furnace is then naturally cooled. During the nature cooling ofthe furnace, hydrogen is supplied to the anode cell tube to preventcell electrode damage, until the temperature in the furnace dropsdown to the ambient temperature.

The presented experimental test bench allows the easy varia-tions of inlet mixed-gas composition. The presence of argon gas al-lows the molar fraction of H2 and H2O to vary at the same time,without changing the inlet molar flow rate and the inlet pressure.The experimental validation of the SOFC model has been con-ducted for a temperature range from 800 �C to 850 �C, a hydrogenmolar fraction range from 11.84% to 59.22% and a water vapor mo-lar fraction range from 9.87% to 59.22%.

The physical dimensions of the tubular prototype solid oxidefuel cell used for model validation are shown in Fig. 3.

In addition, the physical parameters of the prototype cell and itsoperating conditions are summarized in the Table 2. For the modelexperimental validation purpose, the axial segments number

Fig. 3. Prototype SOFC single

N = 15 is chosen based on the results from the modeling grid inde-pendency check (Table 1).

3.2. Results and discussions

The SOFC polarization curves from developed model and exper-imentation at 850 �C with different H2 molar fraction and H2O mo-lar fraction are presented in Fig. 4. The simulation results of thedeveloped SOFC model have a very good agreement compared tothe experimental data. The maximum relative error for the cellvoltage between simulation and experimentation is less than 6%.

The polarization curves at 800 �C with different H2 molar frac-tion and H2O molar fraction are shown in Fig. 5. Due to the lowertemperature, the SOFC limitation current is smaller than the previ-ous case. The model simulation results show again a good agree-ment with the experimental data. It can be noted that, the modelpredicted curve is slightly different from the experimental one atlow current values. The voltage drop at low current zone is mainly

cell physical dimensions.

0 1 2 3 4 50.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

SOFC Cell current (A)

SOFC

Cel

l vol

tage

(V)

11.84 % H2; 19.74 % H2O (Measured)

11.84 % H2; 19.74 % H2O (Model)

35.53 % H2; 19.74 % H2O (Measured)

35.53 % H2; 19.74 % H2O (Model)

59.22 % H2; 19.74 % H2O (Measured)

59.22 % H2; 19.74 % H2O (Model)

24.67 % H2; 09.87 % H2O (Measured)

24.67 % H2; 09.87 % H2O (Model)

24.67 % H2; 59.22 % H2O (Measured)

24.67 % H2; 59.22 % H2O (Model)

T = 1123.15 K (850 °C)

Fig. 4. SOFC Polarization curves at 850 �C (experimental and simulation).

0 0.5 1 1.5 2 2.5 3 3.5 40.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

SOFC Cell current (A)

SOFC

Cel

l vol

tage

(V)

35.53 % H2; 19.74 % H2O (Measured)

35.53 % H2; 19.74 % H2O (Model)

24.67 % H2; 09.87 % H2O (Measured)

24.67 % H2; 09.87 % H2O (Model)

24.67 % H2; 24.67 % H2O (Measured)

24.67 % H2; 24.67 % H2O (Model)

T = 1073.15 K (800 °C)

Fig. 5. SOFC Polarization curves at 800 �C (experimental and simulation).

0 5 10 15 20 25

0

0.1

0.2

0.3

0.4

0.5

0.6

Position in cell axis (mm)

Cur

rent

den

sity

(A/c

m2 )

Cell current = 0.5 ACell current = 1.5 ACell current = 2.5 ACell current = 3.5 A

T = 1123.15 K (850 °C)H2 = 35.53 %H2O = 19.74 %

Fig. 6. Axial steady-state current density distribution at different current (850 �C,35.53% H2, 19.74% H2O).

178 Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185

due to the activation losses in a SOFC. The voltage value differencebetween simulation and experimentation could be explained bythe under-estimation of the electrode reference exchange currentdensity at lower temperature. In a SOFC, the electrode electro-chemical reaction kinetic depends highly on the SOFC operatingtemperature.

As mentioned in the previous sections, the developed model hasN = 15 configurable axial control volumes, which allows the inves-tigation of non-uniform distribution of SOFC operating parameters.Fig. 6 presents the steady-state axial current density distributionfrom the cell inlet to the cell outlet, at different current values at850 �C, with a H2 molar fraction of 35.53% and a H2O molar fractionof 19.74%. At high current, the non-uniformity of cell current den-sity under steady-state becomes more significant. The difference ofcurrent density between the cell inlet and outlet could reach to0.05 A cm�2, which represents almost a 12% relative difference.The current density has a more uniformly distribution at low cur-rent value, due to the lower reactant gas consumption rate. It

52

05

1015

20

0

2

4

6

850

850.1

850.2

850.3

850.4

850.5

Position in cell axis (mm)

Time (s)

Elec

troly

te te

mpe

ratu

re (°

C)

850.464

850.466

850.468

850.47

850.472

850.474

850.476

850.478

850.48

850.482

850.484

I = 3.5 AT = 1123.15 K (850 °C)H2 = 59.22 %H2O = 19.74 %

Fig. 7. Temperature variation of SOFC electrolyte (850 �C, 59.22% H2, 19.74% H2O).

5

05

1015

202

0

2

4

6

850

850.1

850.2

850.3

850.4

850.5

Position in cell axis (mm)

Time (s)

Anod

e tu

be g

as te

mpe

ratu

re (°

C)

850

850.2366

850.4732

I = 3.5 AT = 1123.15 K (850 °C)H2 = 59.22 %H2O = 19.74 %

Fig. 8. Temperature variation of anode gas in the channel (850 �C, 59.22% H2, 19.74% H2O).

Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185 179

should also be noted that, at a lower H2 molar fraction, the differ-ence of current density between the cell inlet and outlet can bemore important, because it leads to a lower H2 partial pressureat electrochemical reaction zone with the same H2 consumptionrate.

The electrolyte temperature variation for a step current changefrom 0 A to 3.5 A is shown in Fig. 7 (850 �C, molar fraction of H2 of59.22% and molar fraction of H2O of 19.74%). Due to the SOFC ther-mal capacity, the cell reaches the steady-state in some seconds.The inlet part of the cell has the highest temperature, because ahigher current density value at cell inlet leads to a higher heat gen-eration rate from activation losses and ohmic loss. At the cell outletpart, the temperature is slightly lower than that of the inlet part.

In addition to the SOFC electrolyte temperature variation, theanode gas temperature variation through the anode tube is also gi-ven in Fig. 8, for the same current step change from 0 A to 3.5 A. Incontrast to the electrolyte temperature distribution, the highestgas temperature can be found at the cell outlet part, because theanode gas is heated progressively through the anode tube. Thetemperature gradient of the anode gas in the tube from the inletto the outlet is more significant, due to the convective mass flowheat transfer of the anode gas (Eq. (34)). At the cell outlet, the an-ode gas temperature reaches the cell solid electrode temperature.

In the anode gas diffusion direction (axial direction of themodel), the H2 and H2O partial pressure distributions in the tubefrom the inlet to the outlet at different current values are presentedin Fig. 9 at 800 �C, with the same H2 and H2O molar fractions equalto 24.67%. It should be noted that, the H2 and H2O partial pressuresare the same at the cell inlet in this experiment. As the H2 is con-sumed and the H2O is produced progressively through the cell dueto the electrochemical reaction, the H2 partial pressure decreases.At same time, the H2O partial pressure increases, because the pro-duced water vapor enters into the tube channel. At low current, thepartial pressure gradient in the tube is less significant than that athigh current, due to the lower gas consumption rate.

In another case, the partial pressure gradients for H2 and H2Othrough the anode channel are shown in Fig. 10 for different cur-rent values at 800 �C, with a H2 molar fraction of 35.53% and aH2O molar fraction of 19.74%. In this case, the H2 partial pressureis always higher than the H2O partial pressure in the entire cell.At higher current value, the H2 partial pressure decreases morerapidly, with a more rapid increase of the H2O partial pressure inthe cell tubular channel.

From the polarization curve at 850 �C, with a H2 molar fractionof 24.67% and a H2O molar fraction of 9.87%, the voltage losses dueto the different phenomena are presented in Fig. 11. Three major

0 5 10 15 20 25

1.7

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

x 104

Position in cell axis (mm)

Gas

pre

ssur

e (P

a)

PH2 (I = 0.5 A)

PH2O (I = 0.5 A)

PH2 (I = 1.5 A)

PH2O (I = 1.5 A)

PH2 (I = 2.5 A)

PH2O (I = 2.5 A)

T = 1073.15 K (800 °C)H2 = 24.67 %H2O = 24.67 %

Fig. 9. Axial anode gas pressures variation at different currents (800 �C, 24.67% H2,24.67% H2O).

0 5 10 15 20 25

1.6

1.8

2

2.2

2.4

2.6

2.8

3

x 104

Position in cell axis (mm)

Gas

pre

ssur

e (P

a)

PH2 (I = 0.5 A)

PH2O (I = 0.5 A)

PH2 (I = 1.5 A)

PH2O (I = 1.5 A)

PH2 (I = 2.5 A)

PH2O (I = 2.5 A)

T = 1073.15 K (800 °C)H2 = 35.53 %H2O = 19.74 %

Fig. 10. Axial anode gas pressures variation at different current (800 �C, 35.53% H2,19.74% H2O).

0 1 2 3 4 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SOFC Cell current (A)

SOFC

Vol

tage

loss

es (V

)

Cathode ActicationAnode ActivationElectrolyte ohmic

T = 1123.15 K (850 °C)H2 = 24.67 %H2O = 09.87 %

Fig. 11. SOFC voltage losses (850 �C, 24.67% H2, 9.87% H2O).

180 Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185

voltage losses can be distinguished in SOFC: cathode activationloss, anode activation loss (Eq. (5)) and electrolyte ohmic loss

(Eq. (11)). It can be seen from the figure that, the cathode activa-tion loss is the most significant voltage loss in SOFC, especially ata high current value. This is due to the slow electrochemical kineticof oxygen reduction at cathode catalyst zone. Based on Eqs. (6) and(7), the reactant partial pressures (H2, O2, H2O) have a direct im-pact on the cathode and anode exchange current values, whichare related directly to the SOFC activation losses. With a low molarfraction of H2O in this case, the anode activation loss is higher thanthe electrolyte ohmic loss in the figure. In the case of low temper-ature fuel cells such as PEMFC, the anode activation loss can usu-ally be neglected compared to the cathode one. However, in thecase of SOFC, due to the high operating temperature, the cathodeand anode activation losses have same magnitude of value. Thusthey must be both considered in a SOFC model. It should also benoted that, the electrolyte ohmic loss is only a function of temper-ature and does not depend on gas pressure at the electrolyte inter-face, as shown in Eq. (10).

4. Dynamic transient time analysis

In SOFC, two different dynamic phenomena during the opera-tion can be found: the voltage transient state due to the doublelayer capacitance of electrochemical reaction, and the temperaturetransient state due to the material thermal capacity. Those twopseudo-first order dynamic phenomena have different time con-stant ranges. In general, the electrochemical transient time is muchfaster than the thermal transient time in SOFC.

In this section, both dynamic phenomena mentioned above areinvestigated in detail. The explicit time constant expressions aregiven and discussed under different SOFC operation conditions.

4.1. Time constant of first order dynamic systems

The general form of a first order differential system can be ex-pressed as:

dyðtÞdtþ 1

syðtÞ ¼ f ðtÞ ð41Þ

where the coefficient s represents the time constant of the first or-der system.

The time constant s (in unit ‘‘second’’) is used to characterizethe dynamic response time following a step change of system in-put. A smaller time constant value leads to a faster dynamic re-sponse time, and vice versa. In general, it can be considered that,the system dynamic response reaches the steady-state after a timeof 4s–5s (e.g. system transient state time), following a step changeat t0.

Thus, by analyzing the time constant expression of a first orderdifferential system, the influencing factors on the system transientstate time, as long as the system transient time variation rangecould be known.

4.2. Transient analysis for double layer capacitances

The double layer capacitance effect is due to the presence ofcharged reactants particles at catalyst interface during the electro-chemical reaction at both cathode and anode side. Due to thesedouble layer capacitances, the SOFC voltage response of a currentstep change follows a first order dynamic, which can be character-ized by Eq. (8) at cathode side and Eq. (9) at anode side.

4.2.1. Cathode and Anode time constant expressionsFrom the cathode and anode activation losses Eqs. (8) and (9)

the expression of Eq. (41), the cathode and the anode voltage dy-namic time constant expressions can be calculated:

9.5 10 10.5 11 11.50

0.2

0.4

0.6

0.8

1

Time (ms)

SOFC

Vol

tage

(V)

Cell voltageCathode ActivationAnode Activation

T = 1123.15 K (850 °C)H2 = 11.84 %H2O = 19.74 %Step cell current = 3.5 A

Fig. 12. Voltage dynamic due to the double layer capacitances (850 �C, 11.84% H2,19.74% H2O).

9.5 10 10.5 11 11.50

0.2

0.4

0.6

0.8

1

Time (ms)

SOFC

Vol

tage

(V)

Cell voltageCathode ActivationAnode Activation

T = 1123.15 K (850 °C)H2 = 35.53 %H2O = 19.74 %Step cell current = 3.5 A

Fig. 13. Voltage dynamic due to the double layer capacitances (850 �C, 35.53% H2,19.74% H2O).

Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185 181

sC ¼Cdl;C � gact;C

ið42Þ

sA ¼Cdl;A � gact;A

ið43Þ

where the term gact represents the steady-state activation lossesfrom the Bulter–Volmer Eq. (5).

When gact is small at low SOFC current, Eq. (5) can be simplifiedto a linear equation:

gact ¼R � Tne � F

� ij0 � Ael

ð44Þ

Thus, the corresponding time constant expressions become:

sC ¼Cdl;C � T

j0;C� RAel � ne;C � F

ð45Þ

sA ¼Cdl;A � T

j0;A� RAel � ne;A � F

ð46Þ

When gact is large at high SOFC current, Eq. (5) becomes thewell-known Tafel equation:

gact ¼R � T

a � ne � Fln

ij0 � Ael

� �ð47Þ

Thus, the time constant expressions at high current are:

sC ¼Cdl;C � T

i� ln i

j0;C � Ael

!� Ra � ne;C � F

ð48Þ

sA ¼Cdl;A � T

i� ln i

j0;A � Ael

!� Ra � ne;A � F

ð49Þ

It can be concluded from Eqs. (44)–(49) that, the time constant val-ues of SOFC voltage dynamic at cathode and anode side depend mainlyon the corresponding double layer capacitance and exchange currentdensity. The dependence to the temperature is relatively smaller be-cause the temperature is expressed in Kelvin (K). At higher SOFC cur-rent, the current value influences also the voltage dynamic.

From Eq. (1), it can be seen that, the SOFC output voltage dy-namic response depends on both cathode and anode activationvoltage dynamics, which are two independent first order differen-tial systems in series. Thus, the overall SOFC output voltage tran-sient time depends only on the larger time constant valuebetween the cathode and the anode.

The time constant numeric ranges for the presented tubularSOFC prototype can be calculated from the parameter values inTable 2.

At cathode side, the time constant value is between 0.0582 msand 0.629 ms, which represents roughly a cathode dynamic tran-sient time between 0.23 ms and 2.5 ms.

At anode side, the time constant value is between 0.0325 msand 1.57 ms, which represents roughly an anode dynamic transienttime between 0.13 ms and 6.3 ms.

4.2.2. Results and discussionsThe dynamic responses of SOFC output voltage, cathode activa-

tion voltage and anode activation voltage for a step current changefrom 0 A to 3.5 A at 10 ms is presented in Fig. 12. Due to the lowanode gas partial pressures (thus low anode exchange current den-sity) and high SOFC current, the anode activation voltage transienttime is higher than that of the cathode. Thus, the output voltagetransient time follows the anode activation voltage transient time,which is about 0.84 ms.

With a higher H2 molar fraction, the voltages dynamic undersame test conditions is presented in Fig. 13. It can be seen clearlyfrom the figures that, compare to the previous case, the anodeactivation voltage transient time is smaller with a higher H2 partial

pressure (thus a higher anode exchange current density). Thus, theoverall SOFC output voltage transient time is equal to that of thecathode, which is roughly around 0.38 ms.

4.3. Transient analyze for cell thermal capacity

The SOFC temperature dynamic effect between two operatingpoints is mainly due to the thermal capacity of SOFC ceramic mate-rial. With the change of internal heat generation rate and fluidicconditions (forced convection coefficient, convective mass flowrate), the ceramic material body absorbs or releases an amountof heat. This phenomenon leads to a non-linear first order dynamicof temperature, which can be characterized by Eqs. (29)–(36).

4.3.1. Thermal time constant expressionBy applying the form of the Eq. (41), the SOFC thermal dynamic

time constant expression can be calculated:

sT ¼qEE � VEE � Cp;EE

Kfluid þ Kelecð50Þ

with:

Kfluid ¼ hfc � Ainner þkEE � Aouter

dEEþ qsp �Msp � Cp;sp ð51Þ

Kelec ¼i � DS0

2F� i2 � del

r00�

Xk¼C;A

i � Ra � ne;kF

lni

j0;kAel

! !ð52Þ

0 1 2 3 4 5 6 7850

850.4

850.8

Time (s)

Elec

troly

te te

mpe

ratu

re (°

C)

Cell current = 0.5 ACell current = 1.5 ACell current = 2.5 ACell current = 3.5 A

T = 1123.15 K (850 °C)H2 = 59.22 %H2O = 19.74 %

Fig. 14. Electrolyte temperature dynamic due to thermal capacity (850 �C, 59.22%H2, 19.74% H2O).

182 Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185

where Kfluid and Kelec are two terms that represent, respectively, thefluidic parameters influences and the electrochemical parametersinfluences on the SOFC thermal time constant.

It should be noted that, the term Kelec represents a negative nu-meric value. Thus, under the same fluidic conditions (constant Kfluid),a greater current i leads to a higher temperature transient time.

For the different partial gas pressure values, the time constant valueis estimated from 0.42 s to 1.57 s for the presented tubular SOFC, whichrepresents a temperature transient time around some seconds.

4.3.2. Results and discussionsThe temperature dynamic responses at 850 �C for different step

current values are presented in Fig. 14. It can be seen from the fig-ure that, at high current, the temperature transient time is slightlyhigher the one at low current value. The biggest temperature dy-namic transient time is estimated to about 4.6 s.

Based on the study in Sections 4.2 and 4.3, it can be concludedthat, the temperature dynamic due to the SOFC thermal capacityhas the longest transient time following a step current change. Thisthermal transient time could reach to some seconds. In contrast,the voltage dynamic responses due to the SOFC double layer capac-itance is much faster, only around some milliseconds.

5. Modeling parameters sensitivity analysis

Due to the physical complexity of a multiphysics SOFC model,some semi-empirical parameters have to be used in the model.Those parameters, such as electrode exchange current density orelectrode tortuosity, have clear physical significations in SOFC.However, they are hard or even impossible to be theoreticallydeterminated. Each semi-empirical parameter has its influencewith a different degree of sensitivity on the model output accuracy.In the case of SOFC, the most important model output parameter isthe cell output voltage.

The sensitivity study for the semi-empirical parameters in a SOFCmodel represents an important step during the model development.Such a study indicates the most and the least influencing parameterin a given set of modeling equations. Thus, it gives useful informa-tion for the choice of modeling assumptions and leads to a betterunderstanding of the interactions between physical equations.

5.1. Multi-parametric sensitivity analyze (MPSA) method

In order to investigate the sensitivity of different semi-empiri-cal parameters on the SOFC output voltage, a statistic Multi-Para-metric Sensitivity Analyze (MPSA) method has been used.

The generic MPSA method has been introduced in detail in [28]and [43]. The six main steps used by the MPSA method are:

1. Select the set of the parameters to be analyzed.2. Set the numeric variation range of each parameter.3. For each selected parameter, generate a series of 500 indepen-

dent random numbers with a uniform distribution within thedefined parameter variation range.

4. Run the SOFC model using the selected series of 500 numbers,and then calculate the corresponding objective function valueby using Eq. (53), for different SOFC current values.

5. Evaluate the relative sensitivity criteria at different SOFC currentvalues of each parameter by using Eq. (54).

6. Evaluate the sensitivity index value (overall relative sensitivitycriteria) of each parameter by using Eq. (55).

At a given SOFC current value, the objective function values arecalculated from the sum of square errors between the SOFC outputvoltage value Vcell,(i),typical obtained from the typical parameter va-lue (Table 2) and the SOFC output voltage value Vcell,(i)(k) obtainedby using the parameter value within the defined parameter varia-tion range:

fðiÞ ¼X500

k¼1

ðVcell;ðiÞ;typical � Vcell;ðiÞðkÞÞ2 ð53Þ

where f(i) is the objective function value at a current value i.The corresponding relative sensitivity criteria value /(i) which is

independent to the SOFC output voltage value can thus be calcu-lated by:

/ðiÞ ¼fðiÞ

Vcell;ðiÞ;typicalð54Þ

In order to evaluate the overall parameter sensitivity in the se-lected SOFC current range, the parameter sensitivity index value hcan be defined as the sum of the relative sensitivity criteria of dif-ferent SOFC current values:

h ¼Ximax

i¼0

/ðiÞ ð55Þ

5.2. Selected parameters and numeric ranges

In the developed SOFC model, seven semi-empirical parametersin different physical domains have been chosen for the sensitivityanalysis.

These parameters are:

1. Cathode and anode reference exchange current density, cC andcA in Eqs. (6) and (7).

2. Electrochemical symmetry factor a in Eq. (5).3. Cathode and anode gas diffusion coefficient of the diffusion lay-

ers, DO2�GDL and DH2�GDL in Eqs. (26) and (27).4. Cathode and anode diffusion layer tortuosity, sC and sA in Eq.

(27).

The typical values of those parameters in the model have beenchosen based on the reliable physical range estimation of eachparameter and experimental calibrations.

The numeric variation range for each parameter has been set to±30% of their typical values for the sensitivity analysis.

The chosen semi-empirical parameters with their typical valuesand the applied variation ranges are summarized in Table 3.

Table 3SOFC parameters sensitivity typical value and range.

Symbol Parameter Typical value Test range (%)

cC Cathode reference current density 7.3 � 107 ±30cA Anode reference current density 3.78 � 109 ±30a Electrochemical symmetry factor 0.5661 ±30DO2�GDL Cathode diffusion coefficient 3.062 � 10�2 ±30DH2�GDL Anode diffusion coefficient 47.72 � 10�2 ±30sC Cathode GDL tortuosity 2.84 � 107 ±30sA Anode GDL tortuosity 2.62 � 109 ±30

Table 4SOFC parameters sensitivity index classification.

Symbol Parameter Index value Sensitivity

a Electrochemical symmetryfactor

80.6653 Highlysensitive

cC Cathode reference currentdensity

6.5141 Sensitive

cA Anode reference current density 4.1585 SensitiveDO2�GDL Cathode diffusion coefficient 2.165 � 10�8 InsensitiveDH2�GDL Anode diffusion coefficient 8.8844 � 10�8 InsensitivesC Cathode electrode tortuosity 4.2602 � 10�7 InsensitivesA Anode electrode tortuosity 1.1738 � 10�7 Insensitive

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

SOFC cell current (A)

Rel

ativ

e se

nsiti

vity

crit

eria

Electrochemical symmetry factor alphaOverall relative sensitivity criteria value = 80.6653

Fig. 15. Sensitivity of electrochemical symmetry factor.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

0.25

SOFC cell current (A)

Rel

ativ

e se

nsiti

vity

crit

eria

Cathode reference exchange current densityOverall relative sensitivity criteria value = 6.5141

Fig. 16. Sensitivity of cathode reference exchange current density.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

SOFC cell current (A)

Rel

ativ

e se

nsiti

vity

crit

eria

Anode reference exchange current densityOverall relative sensitivity criteria value = 4.1585

Fig. 17. Sensitivity of anode reference exchange current density.

Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185 183

5.3. MPSA sensitivity analysis results and discussion

By applying the MPSA method described previously, the MPSAresults of SOFC output voltage for each parameter in the selectedcurrent range (0–4.5 A) are presented in Table 4.

It can be clearly seen from the index values that, the outputvoltage value is highly sensitive to the electrochemical symmetryfactor, a semi-empirical parameter that appears in exponentialterms in the activation loss Eq. (5). The output voltage value is alsosensitive to other electrochemical parameters, such as cathode andanode reference exchange current densities. In the fluidic domain,the electrode diffusion related parameters, such as gas diffusioncoefficient and electrode tortuosity, have very small impact onthe output voltage value in their 30% numeric variations range.

Thus, in order to have an accurate SOFC model, the electro-chemical parameters in the model should be carefully chosen.The model output voltage error between the model and theexperimentation could also be efficiently corrected by changingthe values of these parameters. Among these parameters, the elec-trochemical symmetry factor has the most significant impact onthe SOFC output voltage. However, even the diffusion relatedcoefficients of the electrode do not have a high degree of sensitivity

on the SOFC output voltage in general, it should be noted that thesefluidic parameters are important at high SOFC current. Because athigh current, the SOFC operating performance is limited by the dif-fusion related phenomena, known as ‘‘fuel cell concentration loss’’.

In addition to the index value of each parameter for the entirecurrent range, the individual relative sensitivity criteria valuesfor the three most sensitive parameters at different current are alsopresented.

The individual relative sensitivity criteria value of MPSA meth-od for electrochemical symmetry factor is illustrated in Fig. 15. Theresults show an increase of relative sensitivity of the parameterwith the increase of the cell current. Because the activation losses,especially the cathode activation loss, become more significant athigher current.

The individual relative sensitivity criteria values at differentcurrents for cathode and anode reference exchange current densityare presented in Figs. 16 and 17. It can be also seen that, the outputvoltage is more sensitive to these parameters at high current value.

As discussed previously, the influences on the output voltage fora ±30% value variation of the four fluidic parameters (e.g. DO2�GDL,DH2�GDL,sC,sA) during normal SOFC operation are not significant.However, these diffusion-related parameters could have differenceof some magnitudes in values, due to the different structures andmaterials of different SOFC electrodes. At this level, the impact ofthese parameters on the SOFC performance could become moresignificant.

184 Y. Huangfu et al. / Energy Conversion and Management 71 (2013) 172–185

6. Conclusion

In this paper, a dynamic, multiphysics, 1D model for a tubularSOFC prototype has been developed. The model covers the electro-chemical, fluidic and thermal domains. The dynamic phenomenadue to the electrochemical double layer capacitance and the cellthermal capacity have also been taken into account. The presentedmodel has been validated experimentally in eight different opera-tion conditions with different operating temperatures and gaspressures. The results show a very good agreement between thesimulations of the provided model and experimentations. Thenon-uniform distributions of the current density, gas pressuresand temperatures in SOFC at different current values have alsobeen presented and discussed.

A detailed analysis of the transient time constants for differentdynamic phenomena in SOFC has been presented at second. Basedon the physical equations, the explicit time constant expressionshave been given and discussed for the electrochemical and thermaldomains. The results show that, the transient time due to the dou-ble layer capacitances in the electrochemical domain is aroundsome milliseconds and the transient time due to the cell thermalcapacity is around some seconds for the presented SOFC prototype.

In order to investigate the influences of the different modelingparameters on the cell output voltage value, a parameter sensitiv-ity study has been realized. By using the Multi-Parametric Sensitiv-ity Analyze (MPSA) method, seven semi-empirical parametersfrom different physical domains in the model have been chosenfor analysis. The sensitivity analysis results show that, the SOFCmodel output voltage is highly sensitive to the electrochemicalsymmetry factor, which can be found in the activation loss equa-tions. The reference exchange current density values at cathodeand anode sides shows also a great impact on the SOFC output volt-age value. Thus, those parameters need to be carefully identifiedand verified during the SOFC modeling, in order to improve themodel accuracy. The results show also that, the fluidic domainparameters, such electrode tortuosity and electrode diffusion coef-ficients, have less impact on the SOFC output voltage value. How-ever, it should be noted that, the fluidic parameter values have animportant influence on the model prediction at very high current,due to the diffusion-related limitations of the reactant.

The experimentally valiated 1D SOFC model presented in thispaper gives a possibility to investigate the non-uniform distribu-tion of different parameters during SOFC operation. The modelalong with the results of the dynamic time constants analysisand the model parameter sensitivity analysis provides also a com-prehensive evaluation of the influences of different physical phe-nomena and modeling parameters to the simulation accuracy.

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