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    This paper describes a novel method to determine MEA-Resistance and Electro de Diffusion coefficient for a fuel cell (MRED-method) under in-situ conditions . It is shown that the MRED-method allows the determination of i) the mass transport free ohmic resistance of a MEA and ii) the mass-transport coefficient of the electrodes . The method is based on the galvanostatic discharge of a fuel cell without reactant supp ly from the outside. Application of the method to the cathode of a PEM fuel cell is demonstrated and the experimental results are analyzed using a theoretical model based on a simple 1-dimensional diffusion process using Fick's law. The observed independence of the O2 mass transport coefficient of gas composition and pressure indicates that diffusion in the active layer is mass transport limiting.
    公开号:CA2484670A1
    申请号:C2484670
    申请日:2004-10-08
    公开日:2005-02-08
    发明作者:Stumper Juergen；Herwig R. Haas
    申请人:Ballard Power Systems Inc；
    IPC主号:G01R31-389
    专利说明:
    [1" class="description-paragraph] ~. Introduction To meet the power density, reliability and cost requirements that will enable a wide spread use of fuel cells, cost reduction and increased power output remain key challenges. Ln order to reduce development time and -cost, a rapid and convenient means for the optimization of fuel cell desigins is needed. First and foremost this requires a sufficient understanding of the structure-property-performance relationships for fuel cells and its components. In particular the dependence of cell voltage, cell voltage degradation, etc. upon the physico-chemical- and structural properties of the components of the fuel cell needs to be. better understood. Due to the complexity of the heat- and mass transport processes occurring in fuel cells, there is typically a multitude of parameters that needs to be determined. The challenge therefore is the identification of those critical parameters, which have the highest impact on fuel cell performance. This can be done through either theoretical or empirical models, which can then be used as design tools to optimize fuel cell designs more quickly. For the development of these design tools accurate measurement tools are needed to provide reliable data for the establishment of empirical correlation, to provide model inputs or to validate model predictions.In this study we present a novel method for determination of resistance and electrode diffusion coefficient for a fuel cell membrane electrode assembly (MEA), which can be used as a measurement tool to provide valuable experimental data for the investigation of the structure-performance relationships for fuel cell electrodes. The MRED-method allows the investigation of the effect of operating conditions as well as structural properties of the electrodes on performance and thus can support the development of improved design tools for fuel cells.In general, the cell voltage losses can be kinetic, ohmic and mass transport relatedi, consequently, one of the design challenges is to optimize electrode structures in such a way that these cell voltage losses are minimized. Especially on the oxidant side, mass transport lasses associated with transport of reactants arid products to and from the active catalyst sites are a major cause for performance inefficiencies of fuel cells The most widely used method for the determination of the different voltage loss components for the catholic reaction consists of the least-squares fitting of an analytical expression comprising all the various loss components to the experimental cell voltage vs current density {E vs i) curves. Srinivasan and co-workers2 have been using the simple equation E=Err--b'log(g)-r'R (1) la to fit experimental E vs a curves up to the end of the linear region. Here, E°~ell ~ 1° , b and R denote the reversible cell voltage, exchange current density, Tafel slope and cell resistance, respectively. The typical assumption here is that the anodic contribution to the total over-potential is small compared to the catholic contribution, which is generally justified for pure HZ operation. However, the above equation contains no explicit term for the mass-transport- or diffusion over-potential and therefore does not account for deviations from the linear behavior observed at higher current densities andlor lower oxygen concentrations on the cathode. Also, the resistance R as determined from a fit of (1) to experimental data should more appropriately be called "pseudo-ohmic"3 as it depends on operating conditions such as current density, gas composition and is therefore partially determined by mass transport effects as well. Rho and coworkers compared PEM fuel cell performances for ~2/He, (~2/Ar and ~2/N2 mixtures as catholic reactants and showed that R depends on the inert component and, more strongly, on the cathode gas pressure4. In order to more accurately describe the E vs i behavior when mass transport effects are present eq {1) has been extended by including an empirical terms: E=E°Il-b~log{g)-i-R-m~exp(n-i) is This disadvantage with this approach is that it does not provide a physico-chemical interpretation of the associated fit parameters. Another alternative is the inclusion of a mass-transport related concentration over-potential based on Fick's law of diffusion6: E=Eoa --b.log(ij )-z-R-~-~E ln(t t°,~' ~) o max When fitting experimental data with an equation based on (3), however, it becomes apparent that this approach suffers from an interdependence of R and the limiting current i",~. Squadrito7 etal. have described the mass transport loss r~~ by a term such as in eq (2) multiplied with a factor a-ik introducing again parameters a,k for which there is no physico-chemical interpretation. E=Err-b°log(il )-i°R-~...F In(i 1~ i).a ~jk Consequently, the quantitative experimental determination of the mass-transport related performance loss still remains a challenge.Hence, in this study we present a novel method which allows the separation of the ohmic and mass-transport related performance losses of a fuel cell and yields the true ohmic cell resistance unaffected by mass-transport effects the effective 02- diffi~sivity of the cathode electrode
    [2" class="description-paragraph] 2. Theoretical analysis of cell voltage transients during cathode discharge During a cathode discharge experiment according to the MRED-method, the cathode compartment is separated from the gas supply by closing both inlet and outlet valves, whereas the anode side continues to be supplied with H2. Then the load is switched on (i=const_} and the resulting cell voltage transient recorded during the discharge of the fuel cell. The resulting discharge curve (see Fig. 4j is similar to those observed on batteries *.The initial voltage drop after the current has been switched on (t=0) is due to the ohmic and kinetic over-potentials, which react practically instantaneously (on a timescale larger than the charging of the double layer). However, no mass-transport losses are present at t=D as the OZ concentration at the catalyst sites is identical to the bulk concentration in the flow channels. The rapidly accelerating voltage decrease observed for times t>0, on the other hand, is determined by i) the decrease in 02 concentration in the cathode compartment and ii) the 02 concentration gradient over the electrode necessary to sustain the external current.In the following we will i) develop a solution for the 02 concentration c(0, t) at the catalyst sites by solving the corresponding differential equation with the appropriate boundary conditions and ii} show how the 02 concentration c(0, ~~ at the catalyst sites can be calculated by using the measured E~erl(t,I profiles. Using the MIZED-method it is thus possible to extract valuable information on the mass transport properties of the electrode in-situ under fuel cell operating conditions.2.1 Calculation of the ~2 concentration at the catalyst sites during catholic discharge In the following we use the schematic diagram in Fig I for the discussion of the discharge experiment. We describe the electrode as a single layer of thickness ~ ~~ith the catalyst Iocated-at z=0. To model the transport of Oz through the GDL we assume the following:i) the electrode is modelled as a single layer with a catalyst layer of zero thickness ii) a one-dimensional model for the gas diffusion layer using Fick's law iii) no production/condensation of water vapour and associated convective effects iv) the effect of liquid water in the GDL on 02 diffusion is ignored v) heat transfer effects are neglected vi) the current density i is constant during discharge Even though most fuel cells operate with air instead of oxygen on the cathode, we will compare our model predictions with experimental data obtained for pure 02, the intent is to include 02/inert gas mixtures in the future.VJe denote by c(z, t) the 02 concentration in the GDL and by c~~,(t) the 02 concentration in the flowfield channels. The differential equation for c(z, t) is then C~t C(Z't) D ~ (~ZZ C(Z't) (3) where D is the effective 02 diffusion coefficient of the cathode electrode. The boundary conditions are as follows:c(zz~) = co 4-F~D-~ZC(O,t)=i fort>0 ( ) where F is Faraday's constant. The boundary condition at the GDLlchannel interface can be written as c(-~~ t ) = c'n (t) where c~h(t) fulfils the following PDEa ~ (t) _ '~~D~ a ~(Z~t) (~) at 'h vCh aZz=Lwith the initial condition c~ (~) = Co (~) The solution of equations (3-7) is explained in more detail in the appendix A. Fig 3 shows a plot of c(0, t) and c~h(t) for a range of diffusion coefFcients and i= 1 A/cm2.Fig. 3 shows that after a short initial relaxation time on the order of t,. ~ LZIDe~the OZconcentration at the catalyst sites decreases with the same rate as the OZ concentration in the channel. We also note that the concentration difference dc= c~h(t)-c(0, t) is time independent and a function of Degonly. Using Fick's Iaw and assuming a linear concentration gradient over the GDL we can write be = c'h (t) - c(~~ t) = 1- ~ ($) 4'F°D~~-Therefore, if it do can be derived from experimental data then eq (7) allows the determination of the effective 02 diffusion coefficient De~of the electrode. Whereas c'h(t) can be measured simply using a pressure sensor, the calculation of c(D, t) from the cell voltage transient during discharge is explained in the next section.2.2 Calcuia~on of c(~,t) from ~C~"(t~ The cell voltage can be expressed as cell (t) _ ~aa(t)-~'(t)-~a -1'Rwhere E°'etr is the reversible cell voltage, r~ the over-potential and the sub scripts a,c denote the anode and cathode, respectively. We further assume that the over-potential for the anode is constant for t>0 as H2 is continuously being supplied during the discharge, i.e. the cell voltage decrease is interpreted exclusively as change in over-potential on the cathode side. The cell voltage decay can be expressed in most general form as follows: Ecell (~) Ecell (t) '- ~c0e4 (~) - ~' cell (t) + ~c (~) ~c (~) 1 using ~ '(t) _ (Eo -Ec )(t) = E~(t)-~c (t) (11) it follows cell (O) cell (t) - Ec (~) ~c (t) -=fE'(o)-~°(~)-(E'(t)-~°(t))~+E~(~)-EQ(t) (12) The first term in curly brackets can now be calculated using the Butler-Volmer equation: E'(t)-E°(t)=-b~ ~log(io ~C'~(~)) C ( 9 ) and the second term using the Nernst equation ref F' ~ (t) - ~' ~ ref -'t- ~ ~ T ln( C~~ f ) ' ( CH+ 2 ' CHZG ( ~ ~l ) . ~'' C CN+ ref GHZOto yield s Ecell (~) - E' cau (t) - -~c ~ I~g( Ip (t) ~ Cch (~} . C(~, t) ) + ~ ' ~ ~{ C(~,~} a ( Cly+ (O) } 2 . CHZp (~) tc (~~ Cch (t) C(~o) 2 ' ~' C(~, t) CH. (t} CHZO (~) (15) if we now further assume that c~f and cHlo are time independent during the transient, we notice that the voltage decay depends only on the changes in O~ concentration both at the catalyst sites c(0, t) and in the ~Iow channels c~h(t). if we take the exchange current on the cathode r'° to be proportional to c~h(t) (which equals c(O,t) in the limit of law current} g l ~ (t} = k~ ° c~h (t) { I 6}and note that at equilibrium in eq. (14} c(0,t)=cC~(t) then (I S) simplifies to E~erl f0) - E~eu (t) = b~ - IogtC~~' j ) + ~.~F In( C h ( ) ) (17}Equation {16) does not contain any ohmic contribution as the ohmic over-potential remains constant during the transient_ Thus the change in cell voltage over time is due to the change of the ratio c(0, 0)lc(0, t) at the catalyst sites corrected with respect to the change in oxygen concentration in the flowfield channels which act as 02 reservoir. The cell voltage transient during cathode discharge can therefore be considered as an electro-titration of 02 at the cathode catalyst sites9 p250.With (17} we can now calculate the local 02 concentration c(0,t) at the catalyst sites as a function of time using the fact that without current (t=0) the 02 concentration at the catalyst sites equals the bulk concentration in the cathode compartment. With ~(~70) = C~ (o) _ ~p (I s}follows E«afp)- ~uLt) ~F 1~ cGc°r ) hf ) ) c(0, t} = c~ .10 (I 9) Determination of electrode diffusivity During the discharge of the fuel cell a 02 concentration gradient will develop over the Gas diffusion layer (GDL} which is su~cient to sustain the current drawn by the external Ioad. Eq {8) Yields _ 1~1, De$ 4°F~~c {t) and Deb-cari be determined from the sloe of a plot of do against i. do can be calculated using (19) and the 02 partial pressure p ~(t) on the cathode side: dc(t} = c~n (t) ' c~0~ t) _ ~2 (~.,) - C{0~ t) (21 ) Z. Experimenfa! The experimental results were obtained using a single BallardTM Mk 9 membrane electrode assembly (MEA} with NafionTM I 12 membrane. The data acquisition system consisted of a PXI-1002 chassis with an embedded processor coupled to a SCXI-I 001 chassis via a PXI-6052E I6bit data acquistion card from National Instruments Corp.connected to a Personal Computer.The test cell was operated on a custom designed test station allowing accurate control and monitoring of all operating parameters. Steady state polarization curves were obtained under constant fuel- and oxidant stoichiometry with respect to the total cell current. For a transient polarization, the cathode inlet and both outlet valves (see Fig I ) were closed.The anode inlet valve was left open in order to keep the I-I2 pressure constant during discharge. After each discharge the cathode inlet and outlet valves were opened for ~ 1 s to recharge the cell for the next transient experiment. Cell temperatures are given by the coolant inlet- and outlet temperatures, respectively.
    [3" class="description-paragraph] 3. Results and Discussion Fig. 4 shows a typical cell voltage transient for H2/02 operation at i=1 A/crn2 together with the load current. At t=0 the current is switched on and the the cell voltage drops to its initial value E~Ir(~). The current stays constant until ~~1~ (t) drops to about 0.4 V, a cutoff point which was determined by the characteristics of the electronic Load used for the experiment. Fig 5 shows a series of cell discharge voltage profiles obtained at a range of different current densities. When displayed against charge instead of time it can be seen that all transients correspond to the same capacity Q~~ co'4'~'ifor of about 1500 C {= 0.41 Ah} and by using (22) the total cathode free volume Y~ot can be estimated to be about 38.8 cm3. This is only slightly larger than the channel volume of Y~ X35 cm3 measured with liquid water leading to the conclusion that most of the product water generated during the cathode discharge is in the liquid state. Using the experimental value for Q the amount of product water per discharge can be estimated to about O.I35 cm3 which is negligible compared to the total cathode volume Ytor-From graphs such as Fig. S the initial or "transient" cell voltage E~II(0, i) can be extracted and plotted against current density i {transient polarization curve}, which is shown in Fig b together with a traditional steady state polarization curve for comparison. The transient polarization curve lies above the steady state polarization by about 40 mV. This indicates the presence of mass transport losses under steady state conditions. The results of a fit of equation {I) to the experimental data is shown in Table 1. The fact that the resistance is larger for the steady state case, is supporting evidence for the argurrient that there are significant mass-transport cantributions to the steady state resistance. possible causes are most likely O2 concentration gradients along the flowchannel and through the GDLwhich are not present in the transient case. These concentration gradients in turn lead to inhomogenous current density distributions over the active area and consequently hydration gradients in the membrane. Furthermore, these hydration gradients are likely dependent on the overall water production rate i.e. the total current and hence could lead to a distortion of the steady state polarization curve. The higher Tafel slope for the transient case indicates that these distortions also affect the kinetic region of the polarization curve.Table 1: Results of a least squares fit of eel (1) to the transient and steady state polarization in la'ig 6.The errors for the fit parameters correspond to a doubling of the x2 error (while atl other fit parameters are held constant, s is the maean relative error of the fit.Fig. 7 shows a typical result for the time dependence of the 02 concentrations in the flowchannels together with the O2 concentration the catalyst sites calculated using equations (19),(2I). As expected from eq (6) the O~, concentration in the flow channels decreases linearly with time as long as the current i is constant. Also, flee concentration at the catalyst sites c(~,t) shows the behaviour suggested by eq (8) except at short times (< 0.5 s) where relaxation effects 1 ~ play a dominant role. The observed time-independence of dc(t) shows that a simple I-dimensional Fickian diffusion model with a time-independent 02 diffusion coefficient provides a satisfactory explanation of the observed voltage transients during cathode discharge. Ivloreove:r, this also implies that the effective 02 diffusion coefficient is pressure independent. The total cathode volume htoi 42.1 cm3 used for the model calculations is slightly larger than the value determined from eq (22), the difference indicating that there probably remains some residual 02 after completion of the transient.For the experimental determination of Deb according to eq (8) the discharge experiment is repeated for different currents and the resulting do plotted against i. Fig. 8 shows that do is indeed proportional to the current density. A.s c(0, t) is very sensitive to the the value of ECelr(0), the initial cell voltage after switching on the current, a non-linear fit routine was used to extract those values for ~~r'(0) that mimimized the error between the calculated c(D,t) and a straight line parallel to c;~~,(t). In aII cases the fitted L'~e'j (0) was within ~ 5 mVof the experimental value.The experimental value for the mass transfer coeff dent of De~L= 2.44 10-3 m/s determined from the linear fit is close to tI-~e value of De~L = 2.8 1 d-3 m /s that was determined independently using a non-linear four-parameter-f t to experimental helox and air polarization curves obtained with a similar setup and fuel cellr 1. Using L= 2.5 I0~m these experimental values correspond to Deg= 6.1 107 m2/s and De~-= 7.2 i0-~ m~/s , respectively; which are more than an order of magnitude smaller than the binary diffusion coe~cients DjJ for 02 in H2O vapour, N2 or He (see 'Table 2). This difference could be due to i) presence of liquid water in the GDL reducing porosity or ii} presence of a secondary layer limiting the 02 diffusivity of the electrode.Case i): The presence of a porous medium generally inhibits gas diffusion which can be taken into consideration using a Druggeman-type correction Des. - ~_ . Da= (23) where s;i are the porosity and tortuosity of the porous material, respectively and D~2 the free space OZ diffusion coefficient. Using a typical value for the GDL tortuosity of i 1.513 it follows that a diffusivity reduction by a factor of 10 would mean a porosity of EU.21, compared to a valhe of U.7-~.8 for the porosity of a dry GDL. In order to explain the low experimental value for Des-(snore than one order of magnitude difference) we therefore have to assume a liquid water saturation of >75% for the GDL, which, seems rather high.Case ii):The fact that the mass transfer coefficient is essentially independent of the gas composition could also indicate that another transport process such as diffusion through the active {catalyst) Iayer (in series with diffusion though the GDL) determines the 02 flux to the catalyst sites. Typically, electrodes are fabricated using a highly porous substrate such as carbon fiber paper which is coated. with one or more (active) layers containing carbon black, polytetrafluoroethylene {PTFE), a proton conductor (Nafion) and catalyst particles 14.Table 2: Binary diffusion coefficients for gas mixtures calculated using the p'uller, Schettler, Giddings relationshipl2 for p=300 kPa, T= 70 °C.Table 3: ~Z diffusion coefficients for Nafon membranes and the active Layer reported in the Literature together with the diffusion lengths Leap calculated using the experimental result for DPI.'T=80°Cb T= 70 °C Using the 02 diffusion coefficents in the membrane or the active layer reported in the literature (see Table 3) and the experimental value for Des-lL, the diffusion length Lei in the active layer can be estimated. Table 3 shows that L~ ranges from 0.02 to 8 pm.This is appears reasonable considering the physical thickness of the active layer (typically in the order of several 10 pm) which should constitute the upper boundary for the diffusion length. Consequently, taking into account the range of values reported in Table 3 and the structural properties of the electrode the assumption of a transport-limiting step in the active layer seems the more plausible explanation.The bi-layer model for the electrode also explains the observed independence of Deb on pressure. Whereas in the substrate layer the transport of OZ is governed by continuum diffusion (and/or viscous flow) where molecule-molecule collisions dominate over molecule-wall conditions, 02 transport through the active layer most likely is dominated by Knudsen diffusion due to the much smaller pore size. The binary diffusion coefficients D3 (see Table 2) describing continuum diffusion are inversely proportional to the total gas pressure which consequently would lead to a pressure (time) dependence of dc.Knudsen diffusion is dominated by interactions with the pore walls and the corresponding Knudsen flow parameter D;K is independent of total gas pressure2o.
    [4" class="description-paragraph] 4. Conclusions A novel in-situ method has been developed to determine MEA ~tesistance and 02 Electrode Diffusivity (MRED-method) of a fuel cell electrode under operating conditions. The MRED method provides -i) information on both the kinetic and resistive cell parameters without interference by mass transport effects (by means of the transient polarization) and ii) additionally allows the determination of the effective reactant diffusion coefficient Deff of the electrodes. , The MRED method was applied to the cathode electrode of a PEM fuel cell to determine the pure ohmic resistance of the MEA and the effective diffusion coefficient for 02 (Den{O2)) for the cathode electrode. A comparison of the results for pure 02, O2/N2 and O2IHe mixtures showed that De~02) is i) essentially independent of reactant gas composition and ii) independent of total gas pressure indicating the presence of an active layer in the electrodes where Knudsen- diffusion most probably is mass transport limiting. The calculated diffusion length in the active layer covers a relatively wide range from 0.02 to 8 IZm due the Large spread in literature values for the active layer 02 diffusion coefficient.An analytical solution was developed far the cathode discharge of a fuel cell using an iterative procedure. Assuming a simple 1-dimensional Fickian diffusion process with time independent diffusion coe~cient, good agreement was found between model and experiment.While experimental results reported in this study only apply to the cathode of a PEM fuel cell, the MRED method can also be used on the anode side to provide information on the H2 diffusion properties of the anode. Furthermore, the method appears to be applicable to any type of fuel cell that can tolerate a pressure differential across the gas separator. Io Appendix AIn this section we describe the solution of equations {2)-(6). z is the axis across the GDLperpendicular to the membrane, z E [0, L] , where L is the width of the GDL. The interface between catalyst and CrDL is located at z=D and at z=L the interface Between the channel and the GDL. c(z, t) and cue, (t) denote the concentration of 02 in the GDLand flow channel respectively. A.1 Non-dimensional formulation It is convenient to introduce the non-dimensional variables x = x I L, ~ = t -D I LZ . Equations (3)-(7) become: a c{z't ) = a z c(Z't ) (AI) C{Z,O) - CO4. L.D.~c(O,t)=i (A3) c(l,t) = c~h (t) (A4) where the equations for c~ {t) a~ ~~h (t ) - vch '~ ~ ~ C{I' t ) {AS) c~h (0) = co In what follows we will write z, t instead of z, t . A.2 Discretization Since all the oxygen initially present in the flow channel will be consumed, the concentration will vanish at a finite time, say r . We take small time intervals jt j , tl+t ) 1 >_ 1 with t, = 0 and constant increments t'+~ - t, = At, which we take as small as ~ 10~ to make sure results are accurate within S% .The process to solve far c(z,t) will be iterative in 1, the time-step interval. For each l, the solution c' (z, t) involves an infinite sum in cosines; for our accuracy purposes of 5%, it will be enough to take the first 100 components (see eq. A14).In each interval the concentration of oxygen in the channel, c~, {t) , is assumed to be , constant. After obtaining a solution c' {z, t) for z E [0,1] and t E jtt, tl+~ ) . At t = tl+I , the concentration is updated using equation {AS). Taking the limit dt --~ 0 , keeping z fixed, provides the analytic solution c(z, t) {see Pig. 9). Discretizing equations (Al)-{AS): t c'(z't) ~ c~2z c'(z't) z in [0,1], t ~ [tE,p+r) (A6) C' (Z~ tl ) -- ~' '(Z~ tl ) (A7) r3 ~, {~~ t) _ a (Ag) r~z (A9) cr (I,t) = e' where the values for c '-' (z, t) is calculated from the previous step, with c° (z, tl ) = co , a 4 1F D , and e; = c~ (t' ) = e,_i + at ~ y L - ~Z c'-' (1, t, ) , for 1 >_ 2 , with eo = e, = co .Equation {A7) is a continuity condition: the initial value for c' , at t = t, , is given by c '-' at the end of its time interval.We set the boundary conditions at z=0 and z=1 according to the change of variables c'(z,t)=Wl(z,t)+a~z-~-e'-a and solve:z W'(z,t)=~ zW'(z,t) (A10) W'(0,t)=W'(I,t)=0 (All) W '(z~ tr ) = W ~ ~ (z~ t~ ) - (er - e'-~ ) (AI2) In particular, W'(z,U)=b-a-z-e, +a.By separation of variables, set W (z, t) = Z(z) ~ T (t) , and use (A I O) to obtain Z(z) ~ T' (t) = T {t) - Z" (z) , thus Z. ~ (z) _ T. (t) . _~,2 Z(z) T(t) so, Z(z) = A cos(~, ~ z) + B sin(, - z) , T (t) = exp( -RZ ~ t) , and the general solution to (A10) is W 1 (z, t) _ ~ {A,', ~ cos(~,n ~ z) + B,'~ ~ sin(,;, ~ z)) ~ exp(-(~,;, ) z ~ t) n~o For our accuracy purposes, it suffices to take the first 100 components. For each 1, the boundary condition {A 1 I ) gives first B,', = 0 for ail h, and then cos( ~.;, ) = 0 , or ~,;, _ ~,n = (2n + I) ~ (AI 3) {that is, ~.n does not depend on ~ the time interval}. ~A,d, $n9o Can be calculated from {AI2):'~Z~ tf ~ - ~ An - COS( ~.n ~ Z~ = W 1 ~ ~Z, tl ~ el el-i n=0 by multiplying both sides by cos(~"2 - z) and integrating between 0 and l; using the orthogonality of the vectors ~cos(~."z - z)~m=i alI these integrals are zero except when n=m, in which case:~n = ~ (~jl i (Z~ ~j ) (ef el-1 )) ( n ) - cos 7~ ' z ' dz The values for ~A,', ~ ~ are then calculated by iteratian on 1. A.3 Iteration We proceed as follows: for I =1 we obtain {A,', ~~9 o as ~n =~(b-a-z-ei+a}-cos(~.n'z)'dz=(b-ei)-(~)n +a-~.n n where ~.n is as in (A13)- Thus, Ci(Z,t}=~An 'COS(/~,n -Z)°eXp(-/~,n -t}+2i -a+a-Zn=0 Assume ~An-' ~n90 1S known. Then: An = ~ (W 1 1 {Z' tj ) - (e' _ el-i )) ( n ) -cos ~, -z -dz= _ ~ {~ Am i - COS(/~,m Z) ' e~( ~'m ~ (~! tl-i )~) - COS(/~.n - Z) - dZ - ~ {el - G'l_1 ) - COS(/~,n - Z) -dZm=1 Using the orthogonality of the vectors ~cos(~,m - z)~m=~ , the first integral is zero except when n=m, therefore r ~° _ ~ ~ An-' - exp(~,,z, - (t' - tl-' )) ' ~ - ~ (er - ca-r } ' cos(~,n - z) - dz =_ In _ ~ An ' - exp(~,n - (tl - tl-i )) - (ef - el-i ) ° ( ~ ) n Thus, 99 _ C' (z, t) _ ~ A; - cos(~.n ' z) ' exp(-~," ' (t - t' )} + e' - a + a ' z n=0 The error made in the Fourier series for c' {z, t) when taking only 100 components, El , can be estimated by: E' <_, E' +2S'~E' -a~2S+l~l {A14) where S - ~ I < ~ I < ~ ~ I 0-22 n=soo a.n -exp ~,ns n,=31S m - exh m ~ Ei = 2. I~-s . °° ~,,n z < 2 .I~-s '° m 2 ~ I~-2i exp ~,n s ~ ~5 exp m s Thus, for D = 6. i-10-' m2/s, a time step interval of dt=5~ I O~, and a channel OZconsumption time of 5 seconds, E' <_ E' +2S~E~ -a~2S+I~' <_IO-z' +IO-z3.exp~l~ZR~<Ip-zo Symbols and notation E, E~ {t) cell voltage V Eo= E~u + log fit parameterV(i) E~l (t) reversible cell voltage V i current density Al m2 i (t} exchange current density A/ m2 R cell resistance S2-m2 ideal gas constant 8.314 J/K~mol b Tafel slope V/decade r~ (t)over-potential V c(z,t)oxygen concentration in the mol/m3 GDL cue, (t) oxygen concentration in the mol/m3 channel c =c~, (0)initial oxygen concentration mol/m3 =c(O,t) D= Des. effective diffusion coefficientm2/s D;,~ binary diffusion coefficient m2/s Dofree space O2 diffusion coefficientm2/s z total cathode free volumem3 T ;~ Tl~, channel free volume m3 A cell active area m2 F Faraday's constant 9.648~I04 C/moI T temperature K L thickness of the GDL m Q cell capacityC c'(z,t) concentration, of Oz in the GDL for t ~ [t' ,t'+, ) (see appendix)mol/m3 e'value of the concentration of OZ at the channel for t EmoI/m3 [tt,t'+,) A,', n~'-Fourier coefficient for moUm3 c'(z,t) n~'-Eigenvalue m i Sub scripts a anode c cathode References I A. Weber, R. Darling, J. Meyers, and J. Newman, iri Handbook of'Fuel Cells; Col.1 Fundamentals and survey ofsystems, edited by W. Vielstich, A. Lamm, and H. Gasteiger (Whey, Chichester, West Sussex, P019 BSQ, 2003), p. 47-69.2 S. Srinivasan, E. A. Ticianelli, C. R. Derouin, and A. Redondo, J. Power Sources 22 359-375 (i988).3 R. Mosdale and S. Srinivasan, Electrvchim. Acta 4U 4I3-421 (1995}.4 Y. W. Rho, 4. A. Veiev, and S. Srinivasan, J. Electrochem. Soc. 14I 2084 (I994). S J. Kim, S. M. Lee, S. Srinivasan, and C. E. Chamberlin, Journal of the Electrochemical Society 142 2670-2674 (1995).6 M. Eisenberg, Electrochim. Acta 6 93-1 I2 (1962). G. Squadrito, G. Maggio, E. Passalacqua, F. Lufrano, and A. Pani, Journal of Applied Electrochemistry 29 1449-1455 {1999).A. Parthasarathy, S. Srinivasan, A. J. Appleby, and C. R. Martin, J. Electrochem. Soc. 139 2856 (1992}. A. J. Bard and L. R. Faulkner, Electrochemical Methods {Wiley, 1980}. K_ Promislow and J. M. Stockie, SIAM J. of Appl. Math. 62 180 (2001).1 I P. Berg, K. Promislow, J. St-Pierre, J. Stumper, and B. Wenon, subm. to J. Electrochem. Soc (2003).I2 R. H. Perry, J'erry's chemical engineer's handbook (McGraw-Hill, N. Y., 1984}.13 S. Um, C.-Y. Wang, and K. S. Chen, J. Electrochem. Soc.147 4485 (2000). I4 S. A. Campbell, J. Stumper, D. P. Wilkinson, and M. 't. Davis, US Patent #6060190 {2000). F. Jaouen, G. Lindbergh, and G. Sundholm, Journal of the Electrochemical Society 149 A437-A447 (2002).I6 y, W, Rho, S. Srinivasan, and Y. T. Ko, J. Electrochem. Soc. 1412089 {I994).I7 A. Parthasarathy, G. R. Martin, and S. Srinivasan, J. Electrochem. Soc. 138 (199I)_ I 8 A. Parthasarathy, S. Srinivasan, and A. J. Appleby, J. Electrochem. Soc. (1992).I9 P. D. Beanie, V. I. Basura, and S. Holdcroft, J. Electroanal. Chem. 468 I80 (1999)_ E. A. Mason and A. P. Malinauskas, Gas transport in porous media: The dusty Gas model (Elsevier, New York, 1983).rs
    权利要求:
    Claims THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:
    1. A method for in-situ determination of membrane electrode assembly (MEA) resistance and electrode diffusivity of a fuel cell, comprising determining the mass transport free ohmic resistance of the MEA, and the mass-transport coefficient of the cathode and/or anode electrode.
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    同族专利:
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    引用文献:
    公开号 | 申请日 | 公开日 | 申请人 | 专利标题
    CN109946609A|2017-12-15|2019-06-28|中国科学院大连化学物理研究所|A kind of electric conductivity test device, test method and its application in a fuel cell|
    法律状态:
    2007-01-12| FZDE| Dead|
    优先权:
    申请号 | 申请日 | 专利标题
    US51450203P| true| 2003-10-23|2003-10-23||
    US60/514,502||2003-10-23||
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