MOLTEN CARBONATE FUEL CELL MODELLING

Sergio Bittanti ∗ Silvia Canevese ∗ Antonio De Marco ∗

Gianluca Moretti ∗ Valter Prandoni ∗∗

∗ Dipartimento di Elettronica e Informazione, Politecnico diMilano, Milan, Italy

∗∗ CESI (Centro Elettrotecnico Sperimentale Italiano) S.p.A.,Milan, Italy

Abstract: Hybrid plants where a fuel cell and a gas turbine are combined have attractedthe attention of the power system community. In this paper, amodel is provided of aMolten Carbonate Fuel Cell stack and of the thermo-hydraulic equipment in which it isembedded. The model is worked out from basic physical considerations; however, it isalso simple enough for simulation and control purposes. Besides, it has been validatedagainst experimental data.Copyright©2005 IFAC.

Keywords: Fuel Cells, Models, Power Plants, Simulators

1. INTRODUCTION

Power systems based on Fuel Cells (FCs) are grad-ually gaining interest. The main reason is the lowpolluting emission in terms ofCO2, NOx, SOx, aswell as the remarkable efficiency in energy conver-sion. Modularity is also a relevant issue: generators ofdifferent size can be adapted to different users’ needs,and therefore to different markets. (Kowalenko, 2004),(Gurney, 2004), (Andrews and Weiner, 2004), (Evers,2003).

Notwithstanding the importance of these systems inthe present years, and in the years to come, the atten-tion of the control community is still relatively mod-est. The purpose of this paper is to develop a soundmathematical model for a certain class of these sys-tems, namely Molten Carbonate Fuel Cells (MCFCs),and to point out some of their main dynamical char-acteristics. This is the ground for future activity oncell control, with special reference to the project ofinnovative hybrid power plants (Prandoniet al., 2003),where these cells are coupled with gas turbines (alsofor distributed generation), in order to achieve highertotal efficiency. In the absence of operating experi-ence for such plants, in fact, a reliable model-basedstack simulator can be employed to predict the be-

haviour of relevant process variables (like temperaturedistributions, gas concentrations, overall voltage andpower) and this can help to understand relationshipsand dependencies both among them and with variablesrelated to the other devices connected to the FCs.This, in turn, is the starting point to study if operatingconstraints, tied to material limitations and requestedperformance, can be met, both in steady-state and intransient conditions, by means of suitable control ac-tions, obtained,e.g., with industrial PIDs (cmp (Lukaset al., 2000)).

2. MOLTEN CARBONATE FUEL CELLS

FCs are stratified systems, with a usually planar orcylindrical structure (in Figure 1, a typical planarstructure is represented). They work as electric gener-ators, provided that they are continuously fed with fueland oxidant: in fact, their two (porous) electrodes, theanode and the cathode, offer catalytic sites for the cellelectrochemical reactions. Roughly speaking, hydro-gen (either directly pumped into the anode or obtainedby internal hydrocarbon reforming) and oxygen (at thecathode) combine to produce not only water and heat,but also a potential difference, which enables currentflow in an external circuit.An electrolyte layer, interposed between the elec-

trodes, closes the circuit path inside the cell, allow-ing the transport of ions produced or consumed bythe reactions occurring at the electrodes. FCs can beclassified on the basis of their electrolyte; MCFCs,for instance, contain a mixture of alkali carbonates,Li2CO3 andK2CO3 mainly, which are imprisoned inan inert matrix and which, at the operating tempera-ture, around 903−943K, are actually in a molten state(EG & G Services, 2000).

Figure 1 pictures the typical structure of a MCFC.The feeding gas mixtures flow over the electrodesin perpendicular directions to each other (cross flow)and, from those bulk regions, they diffuse into theelectrodes, which act as catalysts, mainly for the re-actions generating or consuming electrons. The elec-trode layer,i.e. the porous medium, is formed in turnby sub-layers: on the one side, between electrode andgas, there is a thin porous metal plate (theporouswall), which acts as a diffusor, favouring the penetra-tion of the gas mixture into the electrode pores; theliquid electrolyte floods the other side of the porousmaterial, without reaching the gas side though, thanksto surface tension; inside the pores, the penetratedgas diffuses into the electrolyte and it is enabled toreact with its ions: this occurs just on the pores innersurface.

The gaseous components to be considered are listed inFigure 1. At the anode, the gas-shift reaction

CO+H2O ↔CO2 +H2

and the electrochemical half-reaction

H2 +CO−−

3 ↔CO2 +H2O+2e

occur, while at the cathode there is only the electro-chemical half-reaction

12

O2 +CO2 +2e ↔CO−−

3 .

Since each cell generates a voltage of about 1Vonly, many elementary cells, connected electrically inseries, are usually packed into a stack to obtain therequired voltage, typically 100V . Metal plates isolateanode and cathode gases feeding adjacent elementarycells.

The system subject of this study is a prototype of ahigh-efficiency hybrid pilot plant, under constructionat CESI; this integrates a MCFC stack, composed ofNC = 150 cells and producing 125kWe (with a 1100A current), with a micro-size gas turbine, deliveringalone about 60kWe. An intense research activity iscurrently underway, including many laboratory tests,modelling efforts and studies for the investigation ofplant control problems arising in normal and emer-gency conditions.

Here, a model of the stack and its implementation ina Matlab-Simulinkr environment are described. This

Cathodic gases (channels):

O2, CO2, N2, H2O

Anodic gases

(channels):

H2, CH4, CO,

CO2, H2O

Electrolyte

Electrodes

Diffusors

Separating plates

xchA

xchC

z

Cathodic gases (channels):

O2, CO2, N2, H2O

Anodic gases

(channels):

H2, CH4, CO,

CO2, H2O

Electrolyte

Electrodes

Diffusors

Separating plates

Cathodic gases (channels):

O2, CO2, N2, H2O

Anodic gases

(channels):

H2, CH4, CO,

CO2, H2O

Electrolyte

Electrodes

Diffusors

Separating plates

xchA

xchC

z

xchA

xchC

z

Fig. 1. The planar stratified structure of a MCFC;notice the cross-flow of anode and cathode gases

model has been connected to the models of the otherdevices operating with the stack, so as to form anoverall dynamic simulator of the MCFC sub-plant.The paper is organized as follows: in Section 3, amodel for an elementary cell, dealing with chemicalreactions, mass and energy transport and storage andelectrical phenomena, is worked out; Section 4 sumsup how the model has been implemented; then, modelvalidation against steady-state real data is summarized(Section 5); finally, some transients of interest areanalysed in Section 6.

3. MODELLING OF A MCFC

This section outlines a MCFC model which is derivedfrom first-principle considerations, with energy, massand momentum conservation equations plus suitableconstitutive equations and electrical equations. Thestack model can be obtained easily from the elemen-tary cell model by observing that the electric currentflowing through all the cells is the same, the wholestack potential difference isNC times that of the singlecell and the total anode and cathode gas flow rates are,on the average, equally subdivided among the singlecells, which they feed in parallel.

In order to avoid the typical limitations of zero-dimensional and one-dimensional models, which donot take into consideration the spatial dynamical cou-pling between anode and cathode reaction rates due togas cross flow, it is necessary to turn to at least a two-dimensional approach (Lukaset al., 2000), (Lukasetal., 1999), (Duijn and Fehribach, 1993), (Lukas andGhezel-Ayagh, 2001), (He, 1994): in fact, reactionrates strongly depend on local reactant concentrationsand on local temperatures. We therefore divide theorthogonal gas flows intoNchA anodic strips, the so-called anodic channels, and intoNchC cathodic chan-nels; channels are referred to orthogonal coordinates,xchA and xchC, as shown in Figure 1. In turn, eachanodic (cathodic, respectively) channel is subdividedinto NtA (NtC) volumes, or anodic (cathodic) nodes,so that in each anodic (cathodic) channel there areNtA = NchC (NtC = NchA) control volumes. In the sim-ulations reported later on, the assumptionNchA = 4andNchC = 8 is made. In the perpendicular direction

z to the anodic and cathodic flow planes,i.e. alongthe cell thickness, different layers have different as-sociated control volumes, for both energy and massconservation equations: this leads to the formulationof a final model that can be considered, in many re-spects, as three-dimensional (although not as far asgas velocity and pressure fields are concerned), whereeach control volume is identified by its “node-channel-layer” coordinates.

The conservation equations describing each controlvolume in the various physical layers form a set ofPartial Differential Equations complemented with al-gebraic equations.

3.1 Thermochemical model

Mass conservation in the bulkConsider each single,i− th component in the anodicand cathodic gas mixtures and thex coordinate alongwhich it moves. Denoting its mass fraction byfi(x, t),its mass conservation equation, inside the bulk, for anelementary control volume with areaA, is

ρA∂ fi

∂ t+w

∂ fi

∂x= w′

lt ( f ∗i − fi)+Hgd ( f ∗i − fi) (1)

where, in the first member, there are the usual accu-mulation and convective transport terms and, in thesecond one, the contribution given by lateral flowsbetween the channel and the porous medium.w′

lt is thetotal lateral convective gas flow rate, per unit length,at the diffusor-channel interface. On the anode side,the lateral total flow rate should, in ordinary operat-ing conditions, go from the porous medium towardsthe bulk; therefore,f ∗i = fipw (pw stands forporouswall). On the cathode side, instead, the lateral totalflow rate should always (under correct operation) bedirected from the bulk towards the porous medium;thus, f ∗i = fi.

Mass conservation in the catalystFor the i − th gaseous component inside theporousmedium (subscriptpm in the following), in the anodicand cathodic layers which are not completely filledby the electrolyte (εg will be the space filling coef-ficient, in the porous medium, by the gas mixture),there are the accumulation, diffusion and convectionterms alongz and the terms of (electro)chemical pro-duction/consumption of the component:

εgAp∂∂ t

(ρ fipm)+ εg∂∂ z

(wp fipm) = γdi f fiεgAp·

·∂∂ z

(∂ρ fipm

∂ z

)

+

(

Ri,shAsh +i′i

2F

)

PMi,

wherePMi is the component molar mass. The shiftreaction conversion rateRi,sh [kmol/m3s] can be cal-culated by multiplying by density the classical rate(Xu and Froment, 1989) function of partial pressures,

Gibbs energy and temperature. Electrochemical dy-namics, not considered here for simplicity, as observedin Section 3.3, are object of current work, based on theanalysis of ionic motion in the electrolyte and, near theelectrode pore surfaces, of the formation of potentialdifferences due to the presence of charged activatedcomplexes. This other approach, in fact, aims at de-scribing the origin of capacitive effects, which areresponsible for cell fast dynamics, developing on atime scale around 1second or less.

Boundary conditions for massFor each electrode, at the bulk-diffusor interface, a sortof radiation condition can be written:

Hgd(

fi − fipm(0, t))

+DdΩgd∂

(

ρ fipm

)

∂ z

∣

∣

∣

∣

z=0= 0,

where mass convective motion and gas mass diffusiontowards the catalyst inner layers appear.Dd is a suit-able mass diffusion coefficient in the porous medium.The boundary conditions of the layers completelyfilled by the electrolyte are null mass fluxes.

Energy conservation in the bulkWe consider, for each channel, thermal energy accu-mulation, energy convective transport alongx, workof pressure and friction forces, energy flux betweenchannel and porous medium (tied to temperature dif-ferences between the gas bulk in the channel and thediffusor), energy flux (analogous to the previous one)between channel and separating plate, energy flux dueto convective flow rates between each channel andthe two adjacent ones. The energy flux due to heatdiffusion alongx and to matter diffusion from channelto porous medium is neglected. Considering mass andmomentum conservation as well, after some manipu-lations the energy equation becomes

ρA∂e∂ t

+w∂h∂x

= Γgd(Td −T )+Γgp(Tp −T )+

+w′

lt(e∗− e),

where T (x, t), e(x, t), h(x, t) are, for the bulk gasin the channel, temperature, internal specific energyand specific enthalpy respectively;Td andTp are thediffusor temperature at the channel interface and theplate temperature;Γgd andΓgp are the heat exchangecoefficients per unit length(W/(mK)) between gasin the channel and, respectively, the diffusor and theplate; e∗ = e(Td) for anodic channels,e∗ = e forcathodic channels.

Energy conservation in the catalyst and in the elec-trolyteThe porous medium on the gas side can be roughlymodelled as a combination of two elements: the elec-trode material itself (a nickel alloy for the anode andlithiated nickel oxide, a highly conductive semicon-ductor, for the cathode) and the gas diffusing intothe pores. For those layers where the gas penetratesand where reactions can take place, we consider ther-

mal energy accumulation, energy convective trans-port by gas, thermal energy diffusive transport, (elec-tro)chemical generation, generation by Joule’s effect(due to the passing of electric current) and lateral (withrespect toz) diffusive energy fluxes. The equationsabout the porous medium layers which are adjacentto the electrolyte (where the electrolyte leaks) and theones relative to the electrolytic layer, which separatesthe electrodes, are analogous to these ones, but with-out the reaction terms (for the complete equation list,refer to (Moretti, 2003)).

Boundary conditions for energyFor each electrode, at the channel-diffusor interface,this radiation condition stands:

ΓgdAp (T −Tpm(0, t)) = −λdΩgd∂Tpm

∂ z

∣

∣

∣

∣

z=0,

whereλd is the diffusor thermal conductivity.Diffuser and electrode constitute a unique, but inho-mogeneous, porous medium: therefore, temperaturegradients are continuous in the passage from the oneto the other, so that no boundary condition is neededbetween them: only physical parameters vary.

3.2 Hydrodynamics

For each anodic and cathodic channel, we take globalmass conservation,i.e.

∂ (ρA)

∂ t+

∂w∂x

= w′

lt ,

and general momentum conservation:

∂w∂ t

+∂wu∂x

+A∂ p∂x

= τ f rdΩgd + τ f rpΩgp.

In the latter, on the left, there are the two inertialterms and the term due to pressure forces; on theright, the terms due to tangential friction on diffusorand plate (the gravity term is negligible). These equa-tions, yielding all flow rates and pressures along thechannels as functions of the anode and cathode inletand outlet pressures, are integrated over finite controlvolumes which are staggered with respect to the onesconsidered for the single component mass equationsand for energy equations.

In the porous medium layers, pressures are supposedto be equal to the pressure in the corresponding bulkcontrol volume, while flow rates can be computedthrough the total mass conservation equation: at eachporous wall, the flow rate must vanish, while acrossthe surface between the diffuser and the active layerthe flow rate is linked, of course, to electric current.At every instant and in every control volume, the lat-eral flow rates, to or from the porous medium, supplya known contribution (w′

lt) to the mass conservationequation.

exti

,N ki

,N kG

, ,nrst N kE

extGCAV

N,k strip

Fig. 2. The structure assumed for the electrical model

3.3 Electrical equations

From the electrical point of view, a single cell actsas the parallel of elementary real generators, each ofwhich is a fraction of the cell thickness from the anodeto the cathode: this is shown schematically in Figure 2,where the generic(N,k) strip (N = 1,2, . . . ,NchA, k =1,2, . . . ,NchC) is highlighted and the external circuitis assumed, for simplicity, to be a plain conductanceGext . The total current, flowing through the cell andthe external circuit, and the potential difference are,respectively,

iext =NchA,NchC

∑N=1,k=1

iN,k

VCA = Enrst,N,k −iN,k

GN,k=

iext

Gext,

whence

VCA =∑NchA,NchC

N=1,k=1 Enrst,N,kGN,k

Gext +∑NchA,NchCN=1,k=1 GN,k

,

whereEnrst,N,k is the Nernst potential associated to the(N,k) strip.In general,VCA is a function both ofiext and of externalcontrol variables which depend on the nature of theexternal circuit (think,e.g., of a stack connected to anetwork through a DC-DC converter connected to aninverter).

The Nernst potential is the open circuit reversiblevoltage between the electrodes and it is a function oflocal temperature and of reactant molar fractions inthe reaction zones just inside the liquid electrolyte: inevery strip, omitting subscriptsN andk for simplicity,

Enrst −E0 =RT2F

ln[O2]

12c [CO2]c [H2]a

[

CO−−

3

]

a[

CO−−

3

]

c [H2O]a [CO2]a,

where E0 is the standard conditions potential andwhere the neutral component concentrations refer tothe gas diffused into the liquid electrolyte. Anyway,neglecting the limits due to neutral components dif-fusion into the electrolyte and the transport ofCO−−

3

from the anode to the cathode (by assuming that[CO−−

3 ]a

[CO−−

3 ]c

= 1), one has

Enrst −E0 =RTa

2Fln

[H2]a[H2O]a [CO2]a

+

+RTc

2Fln

(

[O2]12c [CO2]c

)

,

where the (anodic and cathodic)gas concentrationsare employed, which simplifies model implementa-tion. This approximation, which is fair in station-ary conditions, is anyway usually extended to quasi-stationary conditions too (EG & G Services, 2000).It stands in the present work as well, since the maincontrol objectives that one can figure out for the wholeMCFC-microturbineplant involve relatively slow dy-namics (e.g., among operating constraints there iskeeping cell and combustor temperatures under cer-tain values), so that (electro-)chemical phenomena canbe considered as having already reached their steadystate and therefore neglected in the description: thehigh-frequency effects due to charge storage inside theFCs, in fact, become relevant only when control of theexternal electrical system, which exhibits very smalltime constants alike, is included.

4. IMPLEMENTATION OF THE GLOBALMODEL

The equations described so far draw an overall pictureof a single MCFC. Inside every(N,k) control volumein the cell, the bulk equations are integrated alongthe anodic and cathodicx coordinates, those regardingthe catalytic medium are integrated along coordinatez(cmp (Leva and Maffezzoni, 2003), (Eitelberg, 1983)):therefore, by using average variables in each volume,ODEs are obtained from PDEs. These ODEs are thentime-discretized (with sampling intervals of 0,1÷ 1second), yielding approximated equations which arefinally solved mostly with an iterative implicit method,since there are nonlinearities and in order to cope withstrong ties among some equations and with equationsubsets where slow and fast dynamics interact. Asan example of the numerical method adopted, let usconsider equation (1): first, it can be transformed into

ρN,kA∆xN,k∂ fi,N,k

∂ t+max(0,wN,k′)( fi,N,k − fi,N,k′) =

= w′

lt,N,k

(

f ∗i,N,k − fi,N,k)

+Hgd∆xN,k(

f ∗i,N,k − fi,N,k)

,

where subscriptsN,k denote the average quantities involume(N,k), subscriptsN,k′ denote quantities at theedge between volume(N,k) and volume(N,k − 1)and∆xN,k is the length of the integration interval alongthe integration directionx (k is taken as the discretespatial coordinate alongx). Then, the implicit Eulermethod is used to time-discretize the approximateequation, and the final unknowns aref ∗i,N,k, fi,N,k′ and

fi,N,k at the new time instanttn+1, while the samevariables are assumed to be known at the old instanttn. This equation is linear in the unknowns, so it can besolved directly; for nonlinear equations, it is enough toapply an iterative method, by linearizing, at each step,around the unknowns.

The numerical algorithm just described has been im-plemented in the C++ language and embedded in aMATLAB-Simulinkr environment, by means of theS-Function functionality.The cell simulator can be employed to study the dy-namics of the main physical phenomena. Moreover,it can help to plan suitable operating manoeuvres andcontrol strategies for the cell. However, these last can-not, in many cases, be conceived without an analysisof cell behaviour within its plant, whence the need tomodel also the other components of the MCFC systemconsidered here.Therefore, a thermodynamic model for each kind ofdevice in the process, based, again, on conservationprinciples and constitutive equations, has been built.In particular, each one-dimensional element is firstdivided into control volumes (nodes), along its curvi-linear abscissa; then, single component mass and en-ergy conservation equations (thermochemical equa-tions) for the fluid in each node are integrated andthe thermodynamical properties of the fluid are sub-stituted with the thus obtained average values; themomentum and global mass conservation equations(hydrodynamic equations), instead, are integrated onbranches,i.e. control volumes staggered with respectto the nodes.System topology,i.e. the corresponding net of ele-ment connections, is described by an incidence matrix,supplying, for each ordinary node, corresponding toa physical device, the list of entering and outgoingbranches, and, for each ordinary branch, the upstreamand downstream nodes; the net boundary conditions(pressure, temperature, composition) are described byfictitious terminal nodes and branches.In the overall integration technique,external nodes,whose equations and unknown hydrodynamic vari-ables actually participate in building up the globalhydrodynamic system, are distinguished frominnernodes, whose equations are solved locally in the singlemacro-branch (Dell’Oca, 1999). This way, local hy-drodynamics (local with respect to the single controlvolumes) is handled so as to maintain stability and,at same time, reduce the total hydrodynamic systemdimension, thus leading to shorter simulation times; inother words, the problem of slow dynamics interactingwith fast dynamics, which arises when the nodes ina net are numerous and many of them correspond toquite small control volumes, making the total hydro-dynamic system stiff, with consequent lack of conver-gence and/or stability, is avoided.The resolution of the equations describing the elemen-tary devices has been carried out by decoupling ther-mal equations, chemical equations and, for branches

Fig. 3. Anode temperature profile

and macro-branches, hydrodynamics from the hydro-dynamic equations for the nodes. This decouplingconcurs to reduce calculation times and it is possiblebecause transport phenomena of thermal and chemi-cal quantities occur with the fluid speed, while trans-port phenomena of hydrodynamic quantities developat sound speed (Colomboet al., 1989).

5. MODEL VALIDATION

The cell model has been validated against experimen-tal results (Scagliottiet al., 1999), about a MCFCstack composed of 14 active cells with 700cm2 usefularea each. We refer to three stationary conditions:

(1) open circuit;(2) 70A current, 642W power;(3) 70A current, 700W power.

A test at working point 2 was exploited empiricallyto identify some of the model uncertain parameters,like exchange coefficients and the internal electricalresistance, so that the implemented cell model couldlead to numerical results coherent with experimentaldata. The thus obtained parameter values were thenheld fixed in the other tests.In all the three cases, the imposed inputs were thepressures and the temperatures of the inlet anodic andcathodic gases and the related mass fractions.

As for case 1, the obtained values are shown in Table1. Figures 3 and 4 show the temperature profile of

Table 1. First case

Experimental Calculateddata data

Average cell voltage (927,4±0,9) mV 926,8 mVAnode outlet temperature ∼ 925K 928KCathode outlet temperature ∼ 925K 928K

the bulk plates at the anode and at the cathode of anelementary cell. In cases 2 and 3, respectively, theobtained results are summarized in Tables 2 and 3.

Analogous tests, carried out with other experimentaldata (Previtali and Villa, 2002), have allowed a valida-tion of the parts of the model relative to the reformer

Fig. 4. Cathode temperature profile

Table 2. Second case

Experimental Calculateddata data

H2 consumption (anode) 60% 60,4%CO2 consumption (cathode) 22% 21,7%Anode outlet temperature ∼ 945K 948KCathode outlet temperature ∼ 945K 938K

Table 3. Third case

Experimental Calculateddata data

H2 consumption (anode) 51% 51,22%CO2 consumption (cathode) ∼ 22% 21,5%Anode outlet temperature ∼ 945K 946KCathode outlet temperature ∼ 945K 937K

and the combustor supporting the MCFC stack. Forconciseness, anyway, those results are not reportedhere.

6. TRANSIENT SIMULATIONS

After validation, the implemented model was em-ployed to simulate the transient behaviour of the 150cell stack (useful area of each cell: 0,8 m2) under con-struction (Prandoniet al., 2003). In particular, startingfrom a stationary situation of open circuit at the nor-mal exercising temperature, the stack was supposed toget instantaneously connected with an external load (attime t = 50 seconds), a simple conductance, in orderto drive the produced nominal power to approximately120kW . The imposed inputs are reported in Table 4.

Table 4. Simulation inputs

Anode CathodeInlet pressure 3,498E5Pa 3,516E5PaOutlet pressure 3,479E5Pa 3,493E5PaInlet gas temperature 870K 875,89KInlet CO2 mass fraction 0,20931 0,02901Inlet H2 mass fraction 0,07031 0,0Inlet N2 mass fraction 0,0 0,72337Inlet CO mass fraction 0,14806 0,0Inlet CH4 mass fraction 0,0389 0,0Inlet H20 mass fraction 0,53341 0,07021Inlet O2 mass fraction 0,0 0,17741

0 500 1000 1500 2000 2500 3000 3500 4000870

880

890

900

910

920

930

940

sec

K

Outlet temperatures (anode channels)

3

4

2

1

Fig. 5. Outlet temperature of the anode channels ver-sus time

0 500 1000 1500 2000 2500 3000 3500 4000870

880

890

900

910

920

930

940Outlet temperatures (cathode channels)

sec

K

3 42

15

67

8

Fig. 6. Outlet temperature of the cathode channelsversus time

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7Anode outlet mass fractions

sec

CO2

H2O

COH

2CH

4

Fig. 7. Anode outlet mass fractions versus time

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

sec

Cathode outlet mass fractions

CO2 H

2O

O2

N2

Fig. 8. Cathode outlet mass fractions versus time

0 500 1000 1500 2000 2500 3000 3500 40000

20

40

60

80

100

120

140Stack potential difference

sec

Vol

t

Fig. 9. Stack voltage versus time

0 500 1000 1500 2000 2500 3000 3500 40000

5

10

15x 10

4

sec

Wat

t

Stack electric power

Fig. 10. Stack electrical power versus time

Fig. 11. Anode temperature profile

Fig. 12. Cathode temperature profile

The obtained results are shown in Figures 5, 6, 7,8, 9, 10, 11, 12, where the numbers on the curvescorrespond to the channels (from 1 to 4 for the anode,from 1 to 8 for the cathode). In particular, the outlettemperature increase with time shows that the totalelectrochemical reaction is exothermal; moreover, theanodicCO2 mass fraction increase, theH2 decreaseand the cathodicCO2 decrease correspond to the factthat the electrochemical reactions must shift to theright when the generators produce electrical power.Different orders of magnitude in raising time (i.e. thetime which is necessary for a variable to attain a newregime) appear clearly: time scales are of the order ofthousands of seconds (2 or 3hours) for temperatures,the slowest quantities, while only of tens of secondsfor mass fractions, seconds for the total voltage andpower.

7. CONCLUSIONS AND FUTUREDEVELOPMENTS

This article introduces the main aspects of a modeldeveloped for a MCFC (stack). The model describesthe main physical phenomena involved and allowsthe computation of the electric current, power, chem-ical composition, temperatures at various points, etc.;besides, it is accurate enough (since it is two-three-dimensional) to verify the constraints on process vari-ables which the control system has to satisfy for plantoperational procedures, in both normal and emergencyconditions. Of course, since the control system refersto the whole plant, this cell model, as already men-tioned, has to be integrated with the models of theother plant components (e.g. combustor and reformer).In order to increase the cell model accuracy, attentionis currently focused on reaction kinetics, so that thechemical relationship too (and not only the thermaland electrical ones) between gas mixtures and elec-trolyte can be pointed out. Such study also includesthe carbonate ion motion across the electrolyte, withits consequences on the electrical part of the model(on the Nernst equation in particular) and, in general,on system dynamics, which is so important for controlpurposes, for instance when the cell stack has to facefast and large changes in its electrical load.

8. ACKNOWLEDGEMENTS

This work has been developed in the frame of theresearch on the Italian Electrical System “Ricerca diSistema”, Ministerial Decrees of January 26, 2000 andApril 17, 2001.

Research has been supported also by the Italian Na-tional Research Project “New Techniques of Identifi-cation and Adaptive Control for Industrial Systems”and partially by CNR-IEIIT.

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