Polymer electrolyte fuel cell model

2334 J. Electrochem. Soc., Vol. 138, No. 8, August 1991 9 The Electrochemical Society, Inc. 5. Un i o n Carbide Corporation, Carbon Products Division (now AMOCO Performance Products), Technical Information Bulletins 465-223,465-225, 465-246. 6. P. Wagner, J. Am. Ceramic Soc., 55, 214 (1971). 7. O. M. Baycura, IEEE Trans. Indust., 5, 208 (1968). 8. K. Kinoshita and S. C. Leach, This Journal, 129, 1993 (1982). 9. "Chemical Engineers Handbook," 5th ed., R. H. Perry and C. H. Chilton, McGraw-Hill, N ew York (1973). 10. W. L. I n g m an so n et aI., TAPPI, 42, 840 (1959). Polymer Electrolyte Fuel Cell Model T. E. Springer,* T. A. Zawodzinski,* and S. Gottesfeld* Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ABSTRACT We present here an isothermal, one-dimensional, steady-state model for a complete polymer electrolyte fuel cell (PEFC) with a 117 Nation | membrane. In this model we employ water diffusion coefficients electro-osmotic drag coefficients, water sorption isotherms, and m e m b r a n e conductivities, all measured in our laboratory as functions of m e m b r a n e water content. The m o d e l pre.dicts a net-water-per-proton flux ratio of 0.2 H20/H § under typical operating conditions, which is m u c h less than the measured electro-osmotic drag coefficient for a fully hydrated membrane. It also predicts an increase in m e m b r a n e resistance with increased current density and demonstrates the great advantage of a thinner membrane in alleviating this resistance problem. Both of these predictions were verified experimentally under certain conditions. Fuel cells employing hydrated Nation or other hydrated perfluorinated ionomeric materials as the electrolyte are promising candidates for electric vehicle applications (1). The polymer electrolyte provides room temperature startup, elimination of m a n y corrosion problems, and the potential for low resistance losses. Resistive losses within the fuel cell result, in part, from the decrease of m e m b r a n e protonic conductivity following partial dehydration of the membrane. On the other hand, cathode flooding problems are caused w h e n too m u c h water is in the system. Clearly, water m a n a g e m e n t within the fuel cell involves walking a tightrope between the two extremes. Spatial variations of water content within the polymeric electrolyte of a current~carrying fuel cell result from the electro-osmotic dragging of water with proton transport from anode to cathode, the production of water by the oxygen reduction reaction at the cathode, humidification conditions of the inlet gas streams, and "back-diffusion" of water from cathode to anode, which lessens the concentration gradient. The water distribution within a polymer electrolyte fuel cell (PEFC) has been modeled at various levels of sophistication by several groups. Verbrugge and Hill (2-4) have carried out extensive modeling of transport properties in perfluorosulfonate ionomers based on dilute solution theory. Fales et al. (5) reported an isothermal water map based on hydraulic permeability and electro-osmotic drag data. Though the model was relatively simple, some broad conclusions concerning m e m b r a n e humidification conditions were reached. Fuller and N e w m a n (6) applied concentrated solution theory and employed literature data on transport properties to produce a general description of water transport in fuel cell membranes. The last contribution emphasizes water distribution within the membrane. Boundary values were set in these cases rather arbitrarily. A different approach was taken by Bernardi (7). She considered transport through the gas diffusion electrodes, assuming the m e m b r a n e to be uniformly hydrated, corresponding to an "ultrathin m e m b r a n e " case. We present an isothermal, one-dimensional, steady-state model for water transport through a complete PEFC. The model includes transport through the porous electrodes, based on calculated diffusivities corrected for porosity, and transport through the m e m b r a n e electrolyte, based on experimentally determined transport parameters. In this model, we employ water-diffusion coefficients, electroosmotic drag coefficients, water sorption isotherms, and m e m b r a n e conductivities, all of which are measured in our laboratory as functions of m e m b r a n e water content. We d e e m e d it highly desirable to use in our model a complete * Electrochemical Society Active Member. set of experimental data generated under conditions in which the m e m b r a n e is routinely handled (for example, pretreated) and tested for P E F C work as consistently as possible. Furthermore, we explicitly measured parameters over the fullest possible range of water contents to enable us to include the effects of the variation of water content as a function of position within the m e m b r a n e during any current flow. General Aspects of the Model The distribution of water in a P E F C at steady state (constant current and fluxes) is calculated in this model by considering water flow through the following five regions of unit cross-sectional area within the fuel cell, as schematically illustrated in Fig. 1: the two inlet channels, the two gas diffusion electrodes, and the Nation membrane. Not only does water enter the fuel cell assembly as a component of the humidified fuel and oxidant streams, but water is also generated at the cathode by the oxygen reduction reaction. We entered into the model the cell current, cel] and humidifier temperatures, anode and cathode pressures, inlet gas flow rates, and oxygen/nitrogen ratio in the cathode gas feed. The cell is considered isothermal between the inlet channels. The water flow into the inlet channels is set by the temperatures of the external gas humidifiers and the cell. Water vapor is equilibrated at the saturation pressures set by the humidifier temperatures in the inlet flow streams. Entrained droplets may be considered to form and be transported with the flow stream if the cell temperature is less than the humidifier temperature. These droplets will re-evaporate in the inlet flow channels if the condition of the assumed thoroughly m i x ed fluid there become less than saturation. U n d er these isothermal, thoroughly m i x e d or m e a n conditions, the model can use entrained droplets as a transport m e c h a n i s m for water without having to specify the details. We will discuss elsewhere a qualitative treatment of electrode "flooding" (large excess of liquid water) when it occurs. Water vapor leaves the two inlet flow channels at the mean role fraction of the respective channels. Reactant gases split their flow between the gas-diffusion electrodes and the channel exits, as determined by the current and the respective stoichiometry. Binary diffusion of hydrogen and water vapor occurs through the anode, and ternary diffusion of oxygen, nitrogen, and water vapor occurs through the cathode. The flow of gas streams through the gas diffusion electrode is modeled in a m a n n e r similar to that described by Bernardi (7). Interdiffusion of gases through the porous electrode is calculated from tabulated data using the Stefan Maxwell equations with a B r u g g e m a n n correction Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. Electrochem. Soc., Vol. 138, No. 8, A u g u s t 1991 9 The Electrochemical Society, Inc. H20, 0 2 , N2 I H.O, H. 2335 sat = PA/PA; XwA 1 = psat/p C C [2] Xwc F r o m flow balance, t h e a n o d e inlet w a t e r flow less t h e e x i t w a t e r flow is t h e flow c r o s s i n g interface 1 I VHXwhI NIwA -- NLA = NwA;1 - XawA (VH -- 1)XwlI 1 - X~l -- aI [3] w h e r e VH, t h e s t o i c h i o m e t r i c coefficient, is the ratio: N~2/ NH2,~. On t h e c a t h o d e side t h e r e are t h r e e species. T h e inlet dry gas m o l e fraction of oxygen, XON, is k n o w n . T h e inlet 02 m o l e fraction of t h e s a t u r a t e d gas is x~ = (1 - X wI c ) X o N . T h e t h r e e species inlet flow rates a n d e x i t flow rates exp r e s s e d p r o p o r t i o n a l to I are I Niwc - N w c - N w c + (1 + a)l; N L L H20, O2, Nz CATHODE EXIT HzO, Hz ANODE EXIT _ voI(1 - Xor~) [4] 2XoN X~cVoI voI -- _-Z-7~ -; N~ = ~ - ; N ~ 2(1 - Xwc)XoN I (vo 2 _ i)!;N~ pol(l - XoN) [5] 2XoN _ T h e total c a t h o d e e x h a u s t flow, o b t a i n e d by s u m m i n g t h e t h r e e exit flows in Eq. [5], are N~ot~,c - 2(1 - X~c)XoN ~ ~ + I [0] Fig. 1. Schematicdiagram of fuel cell model. a p p l i e d to t a k e a c c o u n t of t h e e l e c t r o d e porosity. The effect of t h e p r e s e n c e of w a t e r droplets is m o r e difficult to treat rigorously. I n this initial a t t e m p t at m o d e l i n g w a t e r t r a n s p o r t in t h e fuel cell, w e deal w i t h t h e droplet p r o b l e m only q u a l i t a t i v e l y (see below). E q u i l i b r i u m of w a t e r activity is a s s u m e d at the elect r o d e / m e m b r a n e interface and d e t e r m i n e s t h e local a m o u n t of w a t e r at t h e m e m b r a n e surface. A n e x p e r i m e n tally d e t e r m i n e d i s o t h e r m for w a t e r s o r p t i o n into t h e m e m b r a n e is u s e d to c o n v e r t f r o m w a t e r v a p o r activity to w a t e r c o n t e n t in t h e m e m b r a n e at the interface. The n e x t flux of w a t e r across t h e m e m b r a n e is calculated u s i n g t h e e l e c t r o - o s m o t i c drag and diffusion coefficients of w a t e r as f u n c t i o n s of m e m b r a n e w a t e r content. W h e n t h e e q u a t i o n s that d e t e r m i n e t h e w a t e r balance are solved, t h e m e m b r a n e resistance and the O2 concentration at t h e catalyst interface are k n o w n . The c a t h o d e overp o t e n t i a l and a V-I c u r v e for t h e fuel cell can t h e n be determ i n e d . A n o d e catalytic activity is c o n s i d e r e d so h i g h that a n o d e o v e r p o t e n t i a l is neglected. T h e o n s e t of c a t h o d e flooding will p r e c e d e any a n o d e flooding. On t h e o t h e r hand, m e m b r a n e d r y i n g n e a r the a n o d e c o u l d cause an ext r e m e i n c r e a s e in t h e overall resistance. The active catalyst layer in t h e c a t h o d e is a s s u m e d to exist only as a t h i n p l a n e at t h e c a t h o d e / m e m b r a n e interface. T h e rest of t h e c a t h o d e serves o n l y as a gas diffusion region. Mathematical Model Continuity equations in inlet flow channeL--Figure 1 s h o w s t h e five regions and t h e four interfaces of the model. W h e n t h e c o n s t a n t cell c u r r e n t is J (A/cm2), t h e m o l a r flux of H2 t h r o u g h t h e a n o d e is I = J/2F moYcm2-s. F l o w s of H2 a n d w a t e r are c o n s i d e r e d to be p o s i t i v e w h e n m o v i n g to the left t o w a r d i n c r e a s i n g interface n u m b e r s . T h e O~ flow is positive in t h e o p p o s i t e direction. U n d e r steady-state c o n d i t i o n s t h e N2 flow will always be zero. F l u x e s and flow rates, d e s i g n a t e d N, are n u m e r i c a l l y identical b e t w e e n interfaces 1 and 4, b e c a u s e t h e cross-sectional area is unity. I f c~is t h e ratio of t h e w a t e r flux c r o s s i n g interface 1 (or 2) to t h e m o l a r flux I, a n d NwA and Nwc are the a n o d e and c a t h o d e w a t e r flow rates (or fluxes) t h r o u g h interfaces 1 and 4, respectively, t h e n t h e fluxes are all related by I = J/2F = NHa,1 : 2No2.4 = NwA/Ot = Nwc/(1 + a) [1] T h e fluid c o m p o s i t i o n in t h e a n o d e a n d c a t h o d e flo~w channels is a s s u m e d to be uniform. T h e a n o d e a n d c a t h o d e feed s t r e a m s are e a c h a s s u m e d to b e s a t u r a t e d w i t h w a t e r v a p o r at the h u m i d i f i e r t e m p e r a t u r e s , as The m o l e fraction of w a t e r and o x y g e n at interface 4 and t h r o u g h o u t the c a t h o d e flow c h a n n e l is t h e ratio of the w a t e r or o x y g e n flow to t h e total f l o w t h r o u g h interface 4. Thus X w 4 -- I XwcVo + 2(1 + ~)(1 - XwC)XON I Vo + (2a + 1)(1 - X~c)XoN ' (Vo -- 1)(1 -- XIwc)XoN XO4 -- VO + (2a + 1)(1 -- X~c)Xo~ [7] Similarly, t h r o u g h o u t the a n o d e flow c h a n n e l f r o m Eq. [3] w e get, for t h e m o l e fraction of w a t e r at interface 1 i P H X w A -- Xw, - ~(1 i __ XwA ) ~wA - ~ ( 1 - X~wA) + , ~ - 1 [8] T h e c o n t i n u i t y c o n d i t i o n s h a v e g i v e n us Xwl, xw4, and Xo4 as the i n p u t p a r a m e t e r s and t h e (as yet u n k n o w n ) ratio ~ of w a t e r flux passing t h r o u g h interface 1 to flux I. N e x t w e d e v e l o p t h e e q u a t i o n s for w a t e r t r a n s p o r t t h r o u g h t h e two e l e c t r o d e s and the m e m b r a n e . T h e region b e t w e e n interfaces 1 and 2 has b i n a r y diffusion of H~ and w a t e r vapor; the r e g i o n b e t w e e n interfaces 2 and 3 has w a t e r drag of p r o t o n i c c u r r e n t and w a t e r diffusion; and the region bet w e e n interfaces 3 and 4 has t e r n a r y gas-phase diffusion. Gas diffusion in the electrodes.--We a s s u m e that t h e a n o d e and c a t h o d e gas m i x t u r e s w i t h w a t e r v a p o r act as ideal gases. L e t z i n c r e a s e in t h e direction of r e a c t a n t flow, that is, f r o m interface 1 to 2 and f r o m interface 4 to 3. T h e S t e f a n - M a x w e l l e q u a t i o n for m u l t i c o m p o n e n t diffusion defines the g r a d i e n t in m o l e fraction of t h e c o m p o n e n t s dxi x~N~ - xjNi dz - R T ~ PD~j [9] T h e b i n a r y diffusion coefficients w e r e calculated for m a s s diffusivity f r o m t h e S l a t t e r y and B i r d (8) e s t i m a t i o n at l o w pressures. T h e pressure-diffusivity p r o d u c t (atm-cm2/s) is e s t i m a t e d f r o m critical t e m p e r a t u r e , critical pressure, and m a s s of t h e r e s p e c t i v e c o m p o n e n t s A and B p. DAB=a( T ~b(pcAP~B)l/3 (TcATcB)5/12 + Ms/ ~3/2 [10] Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. E l e c t r o c h e m . 2336 Soc., Vol. 138, No. 8, August 1991 9 T h e w h e r e a = 0.0002745, b = 1.832 for H2, O2, a n d N2, a n d a = 0.000364 a n d b = 2.334 for w a t e r v a p o r . T h e e~/2t e r m is a B r u g g e m a n n c o r r e c t i o n of t h e d i f f u s i o n coefficients to a c c o u n t for t h e p o r o s i t y o f t h e e l e c t r o d e (9). This model assumes only concentration gradients, not t o t a l p r e s s u r e g r a d i e n t s , a c r o s s t h e e l e c t r o d e s . (A p r e s s u r e d r o p c a n o c c u r a c r o s s t h e m e m b r a n e if t h e a n o d e a n d c a t h o d e p r e s s u r e s a r e different). F r o m Eq. [9] w e o b t a i n t h e foll o w i n g d i f f e r e n t i a l e q u a t i o n s for x~A a c r o s s t h e a n o d e a n d for X~c a n d Xoc a c r o s s t h e c a t h o d e dXwA dz RTI --[X~A(1 + a) -- a] P ADwn [11] E l e c t r o c h e m i c a l Society, Inc. 14 12. ,~ 10. ~. 8. ~t 6. [] r162 4. 2. O. dxo RTI[xo(l+cO+O.5xwc - -- dz' dx~c dz' - Pc Dwo t- 1 - X~c - Xo] 0.0 [12] -~ON o.2 o.4 o.6 o.8 1.0 Water Vapor Activity (Pw / P,a, ) Fig. 2. Measured membrane water content vs. water activity for Nation 117 at 30~ and according to Eq. [16]. RTI [(1 - Xwc - Xo)(1 + ~) Pc L (SO~H +) sites. A fit of t h e e x p e r i m e n t a l r e l a t i o n s h i p of X vs. w a t e r v a p o r activity, a, u s e d i n t h e m o d e l is D~N + 0.5X~c + xo(1 + ~)7 -J DoN [13] E q u a t i o n [11] c a n b e i n t e g r a t e d a c r o s s t h e a n o d e t h i c k n e s s , tA, f r o m t h e initial c o n d i t i o n of Xwh = Xw~ at z = 0 at interface 1, w h i c h w e k n o w f r o m Eq. [8], to o b t a i n Xw2 at z = tA at i n t e r f a c e 2 Xw2 = x~, - : ~ o e x p ~P---~w / + - l + a [14] Similarly, Eq. [12] a n d [13] c a n b e s i m u l t a n e o u s l y integ r a t e d f r o m t h e k n o w n initial c o n d i t i o n s of Eq. [7] at interface 4 to o b t a i n Xo3 a n d Xw3 at i n t e r f a c e 3. A l t h o u g h t h e s e e q u a t i o n s c a n b e i n t e g r a t e d analytically, w e c h o s e to integ r a t e t h e m n u m e r i c a l l y . We n o w h a v e t h e w a t e r m o l e fract i o n i n t h e g a s m i x t u r e i n c o n t a c t w i t h e a c h side o f t h e m e m b r a n e i n t e r m s o f a a n d t h e i n p u t p a r a m e t e r s a n d gas phase (electrode) transport parameters. I n t h i s m o d e l w e h a v e n o t c o n s i d e r e d t h e p o s s i b i l i t y of large e x c e s s e s of l i q u i d w a t e r i n t h e flow c h a n n e l s or within the electrode. In solving the continuity equations we did include, however, marginal situations in which a s l i g h t v o l u m e o f l i q u i d w a t e r f o r m s w i t h i n t h e c h a n n e l s of the electrodes. For the continuity equations, no distinction w a s m a d e b e t w e e n w a t e r gas a n d w a t e r l i q u i d m o l e c u l e s . I f Pw e x c e e d e d Psat, t h e n t h e e x c e s s w a t e r w a s c o n s i d e r e d i n t h e c o n t i n u i t y e q u a t i o n s as if it w e r e finely d i s p e r s e d d r o p l e t s ( w i t h zero v o l u m e ) , h a v i n g t r a n s p o r t p r o p e r t i e s i d e n t i c a l to t h o s e o f w a t e r v a p o r . O n l y w h e r e t h e p r e s s u r e of t h e r e a c t i n g gas ( o x y g e n ) h a d to b e c o n s i d e r e d for t h e c a t h o d e k i n e t i c s w a s t h e f r a c t i o n o f l i q u i d water, x~,q, subt r a c t e d a n d t h e r e a l m o l e f r a c t i o n for g a s p h a s e a l o n e w a s employed. Thus, the partial pressure of oxygen at the catho d e is PcxoJ(1 - xnq). Water transport in membrane.~Memhrane w a t e r cont e n t at t h e i n t e r f a c e is d e t e r m i n e d b y t h e a c t i v i t y o f w a t e r v a p o r at t h e e l e c t r o d e / m e m b r a n e interface, a s s u m i n g e q u i l i b r i u m . T h e a c t i v i t y i n t h e v a p o r p h a s e is xwP/Psat. T h e s a t u r a t i o n p r e s s u r e o f w a t e r , P~t, u s e d h e r e a n d previo u s l y i n Eq. [2] w a s fitted to t h e f o l l o w i n g e m p i r i c a l exp r e s s i o n a c c o r d i n g to t a b u l a t e d v a l u e s (10) logl0P~t = - 2 . 1 7 9 4 + 0.02953T - 9.1837 910-ST 2 + 1.4454 910-TT 3 [15] Membrane water content was measured by Zawodzinski et at., (11) as a f u n c t i o n of w a t e r a c t i v i t y for t h e N a t i o n 117 membrane by weighing membranes equilibrated above a q u e o u s s o l u t i o n s of v a r i o u s l i t h i u m c h l o r i d e c o n c e n t r a tions. F i g u r e 2 s h o w s t h e r e s u l t i n g " i s o p i e s t i c c u r v e " m e a s u r e d at 30~ W a t e r c o n t e n t , X, is g i v e n as t h e ratio of t h e n u m b e r o f w a t e r m o l e c u l e s to t h e n u m b e r of c h a r g e X(~0c) = 0.043 + 17.81a - 39.85a z + 36.0a 3 for 0 < a -<- 1 [16] The measured value of h in equilibrium with saturated w a t e r v a p o r a t 30~ is 14 H20 p e r c h a r g e d site (see Fig. 2). I n t h e a b s e n c e , a t t h e m o m e n t , of i s o p i e s t i c d a t a at 80~ w e a r e a s s u m i n g t h a t t h e 30~ d a t a a p p l i e s to m e m b r a n e e q u i l i b r i u m w i t h w a t e r v a p o r at 80~ O u r m e a s u r e m e n t s also s h o w t h a t w h e n t h e m e m b r a n e , f o l l o w i n g p a r t i a l drying, is i m m e r s e d i n l i q u i d water, X c a n go as h i g h as 22 w h e n at Water b o i l i n g t e m p e r a t u r e a n d to 16.8 w h e n at 80~ T h i s a p p a r e n t a n o m a l y , i n w h i c h w a t e r v a p o r a n d liqu i d w a t e r (in e q u i l i b r i u m w i t h e a c h o t h e r ) e q u i l i b r a t e sepa r a t e l y to d i f f e r e n t m e m b r a n e w a t e r c o n t e n t s , is of t h e o r e t i c a l a n d p r a c t i c a l i n t e r e s t b u t will n o t b e f u r t h e r d i s c u s s e d here. F o r m o d e l i n g p u r p o s e s , w e a l l o w e d X to inc r e a s e l i n e a r l y f r o m 14 to 16.8 as t h e m o l e f r a c t i o n of w a t e r e x c e e d e d s a t u r a t i o n a n d i n c r e a s e d f r o m xw.s~t to 3Xw,~t, t h a t is, for <-- Psat ~< 3, h = 14 + . \ - ~ a t -- 1 ) [17] We m e a s u r e d , i n N a t i o n 117, t h e n u m b e r o f w a t e r m o l e c u l e s d r a g g e d p e r H § i o n m o v e d b y e l e c t r i c field t h r o u g h t h e m e m b r a n e a n d f o u n d it to b e 2.5 _+ 0.2 for a fully hyd r a t e d m e m b r a n e i n e q u i l i b r i u m w i t h l i q u i d w a t e r at 30 or 50~ (12) (the m e m b r a n e i n t h e s e last e x p e r i m e n t s h a d b e e n p r e t r e a t e d i n b o i l i n g water). B a s e d o n w o r k of LaC o n t i et al. (13), w e a s s u m e t h a t t h e w a t e r d r a g coefficient is l i n e a r l y p r o p o r t i o n a l to w a t e r c o n t e n t . We h a v e r e c e n t l y measured a value of only -0.9 water molecules dragged per H § in partially hydrated Nation with a water content X of 11, w h i c h w o u l d i n d i c a t e t h a t t h i s p r o p o r t i o n a l i t y ass u m p t i o n will n o t u n d e r e s t i m a t e w a t e r d r a g at l o w m e m b r a n e w a t e r c o n t e n t . T h u s , for t h e " c o m p l e x " H+(H20)~ mig r a t i n g t h r o u g h t h e m e m b r a n e a t 80~ w e will d e s i g n a t e t h e n u m b e r of w a t e r m o l e c u l e s p e r p r o t o n as ndrag, a n d then ~7~drag 2.5k/22 a n d Nw,drag = ndrag (21) = " [18] (The H § flux (2I) is t w i c e t h e H2 flux, I.) U s i n g t h e p u l s e d - f i e l d g r a d i e n t s p i n - e c h o n u c l e a r magn e t i c r e s o n a n c e (NMR) t e c h n i q u e , Z a w o d z i n s k i et al. (11) also m e a s u r e d t h e i n t r a d i f f u s i o n c o e f f i c i e n t of w a t e r in N a t i o n 117 m e m b r a n e s e q u i l i b r a t e d o v e r a q u e o u s solut i o n s o f LiC1 to s e t t h e w a t e r a c t i v i t y i n c o n t a c t w i t h t h e m e m b r a n e . We a s s u m e h e r e t h a t t h i s i n t r a d i f f u s i o n coefficient, D', r e l a t e s a w a t e r d i f f u s i o n flux to t h e g r a d i e n t i n c h e m i c a l p o t e n t i a l or t h e g r a d i e n t i n t h e l o g a r i t h m of activity. I n t u r n , t h e g r a d i e n t , w i t h r e s p e c t to t h e l a b o r a t o r y c o o r d i n a t e z' o f w a t e r c o n t e n t h m a y b e u s e d D'cw V~ RT Nw,dif = - - - D'cw d(ln a) dk -V(RT In a) = - D ' C w - RT dk dz' [19] Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. Electrochem. Soc., Vol. 138, No. 8, August 1991 9 The Electrochemical Society, Inc. In o r d e r to e l i m i n a t e t r a c k i n g t h e m e m b r a n e swelling in t h e m o d e l , w e c o n v e r t to t h e e q u i v a l e n t w a t e r concentration in a dry m e m b r a n e , kpa~y/Mm, and to a n e w coordinate, z, fixed to t h e d r y m e m b r a n e . To a c c o u n t for m e m b r a n e swelling, dry m e m b r a n e t h i c k n e s s d i m e n s i o n s are exp a n d e d by t h e factor (1 + sk). We define t h e c o r r e c t e d diffusion coefficient Dk, w h i c h relates flux to g r a d i e n t w i t h r e s p e c t to t h e dry m e m b r a n e c o o r d i n a t e s in w a t e r c o n t e n t >, and w h i c h is a f u n c t i o n of }, d(ln a) dk Nw,dif = -D'cw-- dk 6 . . . . 2337 i . . . . D, ~ • s ~ . . . . ,s ~ / / Z e dz' r~ = _D,(kpd~y/Mmk d a ) dk \(l+s>,) ~a~ d~= pdry dk MmDX~zz [20] B y differentiating the fit (at 30~ o f k vs. a, f r o m Eq. [16], to s u b s t i t u t e for da/dX in Eq. [20], t h e t r a n s f o r m e d diffusion coefficient for a c o o r d i n a t e fixed to t h e d r y m e m b r a n e is . 1 k ) D' D~.a0c = (1 + sk) ~ a(17.81 - 79.70a + 108a ~) [21] F r o m t h e m e a s u r e d t h i c k n e s s of dry and fully h y d r a t e d Nation 117 m e m b r a n e s , w e d e t e r m i n e d s to h a v e a v a l u e of 0.0126. G i v e n k, Eq. [16] is s o l v e d i m p l i c i t l y to d e t e r m i n e a and is d i f f e r e n t i a t e d to get da/dX for use in Eq. [21]. Figure 3 is a plot of D', D~, and their ratio as a f u n c t i o n of Nation 117 w a t e r c o n t e n t ~. T h e p e a k in the correction factor and in D~ around k = 3 (Fig. 3) is very sensitive to the exact form of the strong variation in water activity in this region (see Fig. 2). In the model, Dx was represented by a cubic polynomial for k > 4 and by tabular interpolation for k -< 4. Equation [22] is the fit for k > 4 of D~ vs. ~ at 30~ corrected, by an activation energy term, to the other temperatures we used _ 1 D~>4 = 10-6 e x p I 2 4 1 6 ( Z o , , i i 0 i 5 i i i i r 10 . . . . 5 ~, (H~O/$O;) Fig. 3. Measured intradiffusion coefficient, D', at 30~ the corrected diffusion coefficient, Dx, and the correction factor, DJD', calculated from measured activity and swelling, plotted against water content per charge site for Nation 117. f o u n d and a s s u m e d to a p p l y at all v a l u e s of ~. The following fit of c o n d u c t i v i t y in (~-cm)-' as a f u n c t i o n o f k was obtained ~30 = 0.005139k - 0.00326 for h > 1 or(Tee,) = e x p 1268 3 3 273 + Tcel] %0 [25] B e l o w one w a t e r per c h a r g e site, the m e m b r a n e c o n d u c t i v ity was a s s u m e d constant. T h e m e m b r a n e resistance in ~ - c m 2 is o b t a i n e d u s i n g Eq. [24] and [25] by integration o v e r the m e m b r a n e thickness, tin, as 273 + Tcell) ] Rm = (2.563-0.33>, + 0.0264k ~ - 0.000671X a) [22] T h e activation e n e r g y of Eq. [22] was b a s e d on measu r e m e n t s of Yeo and E i s e n b e r g (14) on w a t e r diffusion in acid f r o m Nation of 1155 e q u i v a l e n t weight. T h e n e t w a t e r flux t h r o u g h t h e m e m b r a n e is N w = ~ I -- , X Pdry dk ndrag" (2I)" 22 -- M~ D~ d-z [23] m ~(k) dz f~ [26] Cathode overpotential.--To g e n e r a t e the c o m p l e t e cell polarization curve, w e a s s u m e a c a t h o d i c o v e r p o t e n t i a l loss in a d d i t i o n to t h e o h m i c loss in t h e m e m b r a n e . To calculate this c a t h o d e loss w e u s e d t h e s i m p l e Tafel expression for J vs. ~ g i v e n in Eq. [27]. T h u s J=JoPc Xo3 Fo.sF ] exp L-~-~J [27] 1 -- Xli q and h e n c e dk - ~ZZ- [ 2ndrag . ~ _ ~] ~ IM~ pdr~(~) [24] E q u a t i o n [24] can be i n t e g r a t e d n u m e r i c a l l y f r o m interface 2, w h e r e ~2 is d e t e r m i n e d by x~2 and Eq. [14]. This allows us to calculate k3 and x'3. Iterative solution.--The k e y to solving the w a t e r transport t h r o u g h t h e P E F C occurs at this point, w h e r e w e m u s t a p p l y in iteration to d e t e r m i n e t h e p r o p e r v a l u e of u. We d i s t i n g u i s h x~3, w h i c h is o b t a i n e d f o l l o w i n g t h e integration of Eq. [24], f r o m x~3, w h i c h is o b t a i n e d following i n t e g r a t i o n f r o m t h e c a t h o d e e n d u s i n g Eq. [7] and [13]. T h e s e two v a l u e s will be e q u a l w h e n t h e p r o p e r v a l u e of is used. This iteration i n c l u d e s Eq. [1]-[24], w h i c h w e r e disc u s s e d in t h e t h r e e p r e v i o u s sections and w h i c h are all f u n c t i o n s of ~. M e m b r a n e resistance and voltage d r o p . - - W h e n t h e w a t e r profile in t h e m e m b r a n e has b e e n d e t e r m i n e d , w e can d e t e r m i n e t h e m e m b r a n e r e s i s t a n c e and t h e potential d r o p across it. T h e c o n d u c t i v i t y of N a t i o n 117 was measu r e d a b o v e w a t e r for a r a n g e of w a t e r activities at 30~ T h e c o n d u c t i v i t y was also m e a s u r e d at 30 a n d 80~ for a fully h y d r a t e d m e m b r a n e f r o m w h i c h an a c t i v a t i o n e n e r g y was w h e r e ~ = Voc - Vceii - J " Rm.... T h e e x c h a n g e c u r r e n t density Jo is r e f e r e n c e d to p u r e o x y g e n at 1 arm at the o p e n cell potential Voc. As m e n t i o n e d earlier, t h e m o l e fraction of 02 is c o r r e c t e d to a gas-phase fraction. F o r the sake of simplicity, w e do n o t c o n s i d e r in Eq. [27] t h e rather i m p o r t a n t effects of e x c e s s l i q u i d w a t e r on o x y g e n mass t r a n s p o r t rates w i t h i n t h e cathode. B e c a u s e w e initially specify t h e c u r r e n t d e n s i t y to get t h e w a t e r and o x y g e n c o n c e n t r a t i o n s and m e m b r a n e resistance R . . . . Eq. [27] is solved i m p l i c i t l y to d e t e r m i n e t h e v a l u e of Vceu (only p o s i t i v e voltage allowed) corres p o n d i n g to t h e specified c u r r e n t density. Results and Discussion T h e m o d e l has 16 i n p u t p a r a m e t e r s plus specific isopiestic, diffusion, and e l e c t r o - o s m o t i c drag data that can v a r y for different m e m b r a n e materials and m a n y o u t p u t variables t h a t c o u l d be e x a m i n e d . We only f o c u s e d on a few of t h e s e p a r a m e t e r s a n d variables to illustrate the effects of w a t e r t r a n s p o r t on t h e fuel cell system, as p r e d i c t e d by the a s s u m p t i o n s of the m o d e l , and to o b s e r v e w h i c h aspects of t h e m o d e l s u p p o r t and w h i c h fail to s u p p o r t certain experi m e n t a l results. While d e v e l o p i n g t h e e q u a t i o n s for the model, it was c o n v e n i e n t to i n t r o d u c e c~, t h e ratio of net w a t e r flux to w a t e r flux p r o d u c e d at t h e cathode. To evaluate t h e m o d e l results, w e will n o w use ~, t h e net w a t e r flux Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. Electrochem. Soc., Vol. 2338 Table I. The base-case input parameters and some computed variables for a cell with Nation 117 membrane. Input parameters Computed variables J = 0.5 A/cm 2 j = 0.01 A]cm2 v. = 4 vo = 6 PA = 3 atm Pc = 3 atm tA = 0.0365 cm tc = 0.0365 cm t ~ = 0.0175 cm Tr = 80~ T~t~ = T~tc = 80~ Vo~ = 1.1 V X~N = 0.21 (air) w~ X~c= 0.1558 Vcel] = 0.668 V Rm~r, = 0.285 12-cmz ct = 0.4 ~ = 0.2 H20 flux/H + flux xwi = 0.1015 x~a = 0.1013 x~a = 0.2327 x ~ = 0.2264 Xo~ = 0.1329 Xo4 = 0.1403 Rel Hum~ = 0.6515 Rel Hum4 = 1.4535 X]iq4 = 0.077 p e r p r o t o n , w h i c h is s i m p l y 0.5~. T h e set of m o d e l p a r a m e t e r s u s e d for o u r b a s e c a s e is l i s t e d i n T a b l e I a l o n g w i t h s o m e of t h e c o m p u t e d v a r i a b l e s for t h i s case. W h e n curr e n t d e n s i t y is v a r i e d , vH a n d Vo a r e s p e c i f i e d h e r e for t h e m a x i m u m c u r r e n t d e n s i t y of 2 A / c m 2 a n d i n l e t flow r a t e s are h e l d c o n s t a n t at t h i s v a l u e (i.e., v = ~m~x" J ~ J J ) . T h u s , i n T a b l e I, a l t h o u g h vH is 4 for a c u r r e n t d e n s i t y o f 0.5 A / c m 2, vH will b e 1 w h e n t h e c u r r e n t d e n s i t y is inc r e a s e d to 2 A / c m 2 at t h e s a m e h y d r o g e n flow rate. F o r t h e b a s e case, t h e a n o d e a n d c a t h o d e gas s t r e a m s b o t h e n t e r at s a t u r a t e d c o n d i t i o n s . T h e r e l a t i v e h u m i d i t y d r o p s in t h e a n o d e flow c h a n n e l ( x ~ < X~A) b u t f o r m s add i t i o n a l l i q u i d w a t e r i n t h e c a t h o d e c h a n n e l (x~4 > X~c). F i g u r e 4 s h o w s t h e c o m p u t e d w a t e r profile as a f u n c t i o n of t h i c k n e s s f r a c t i o n m e a s u r e d f r o m t h e c a t h o d e o n t h e left a c r o s s t h e N a t i o n 117 m e m b r a n e for f o u r c u r r e n t densities. T h e J = 0.5 A / c m 2 curve, c o r r e s p o n d i n g to t h e v a l u e s of T a b l e I, h a s a n e t w a t e r flux ratio [3 o f 0.2 w a t e r m o l e c u l e s per H § ion transported through the membrane. The net w a t e r flux m o v e s f r o m r i g h t to left or i n t h e d i r e c t i o n of increasing water content. At the higher current density there is m o r e n e t w a t e r flux e v e n t h o u g h t h e r a t i o of n e t w a t e r flux p e r p r o t o n is less; t h u s , at t h e c a t h o d e , t h e v a l u e s o f v a r y f r o m 14.1 t o 15 as t h e c u r r e n t d e n s i t y i n c r e a s e s f r o m 0.1 to 0.8 A / c m 2. S o m e e x c e s s l i q u i d is p r e s e n t at t h e c a t h ode, so Eq. [17] d e s c r i b e s t h e d e t e r m i n a t i o n o f ;~ at t h e membrane surface in the presence of excess liquid water at i n t e r f a c e 3. T h e m a g n i t u d e o f t h e s l o p e o f t h e w a t e r profiles d r o p s f r o m c a t h o d e to a n o d e , chiefly b e c a u s e t h e w a t e r d r a g flux is p r o p o r t i o n a l to w a t e r c o n t e n t and, t h u s , less d i f f u s i o n flux is r e q u i r e d to offset it at s t e a d y s t a t e nearer the anode. An examination of the water and oxygen mole fractions i n T a b l e I gives s o m e i n s i g h t i n t o gas d i f f u s i o n a n d t h e 138, No. 8, A u g u s t 1991 transport of water in the electrodes. Fully saturated water v a p o r e n t e r s b o t h a n o d e a n d c a t h o d e flow c h a n n e l s at 3 a t m p r e s s u r e , 80~ c o r r e s p o n d i n g to a w a t e r m o l e fract i o n o f 0.1558. T h e m e m b r a n e w a t e r flux l e a v i n g t h e a n o d e flow c h a n n e l a n d e n t e r i n g t h e m e m b r a n e d r o p s t h e rem a i n i n g w a t e r v a p o r to a m o l e fraction, x ~ o f 0.1015. T h i s s a m e m e m b r a n e w a t e r flux, p l u s t h e w a t e r p r o d u c e d at t h e c a t h o d e , a d d to t h e m o l e f r a c t i o n o f w a t e r i n t h e c a t h o d e a n d t h e c a t h o d e flow c h a n n e l . T h r o u g h t h e g a s - d i f f u s i n g anode, the mole fraction of water vapor drops very slightly to Xw2 of 0.1013. T h e s e s l i g h t v a r i a t i o n s i n a n o d e gas comp o s i t i o n b e t w e e n i n t e r f a c e s 1 a n d 2 (across t h e a n o d e ) m e a n t h a t i n t e r p l a y b e t w e e n w a t e r a n d h y d r o g e n gas t r a n s p o r t a c r o s s t h e a n o d e are n o t s i g n i f i c a n t for overall cell p e r f o r m a n c e . I n fact, c o n s i d e r i n g i n t e r f a c e 2 e q u i v a l e n t to i n t e r f a c e 1 w o u l d b e of little c o n s e q u e n c e . U n l i k e t h e c a s e of t h e a n o d e , a c r o s s t h e c a t h o d e t h e w a t e r v a p o r flux d i f f u s e s c o u n t e r to t h e o x y g e n flux, a n d t h e c h a n g e in m o l e f r a c t i o n a c r o s s t h e c a t h o d e is n o t so slight. B e c a u s e t h e initial c a t h o d e i n l e t s t r e a m w a s a l r e a d y s a t u r a t e d w i t h w a t e r , e x c e s s l i q u i d is p r e s e n t a n d is a s s u m e d to b e t h o r o u g h l y d i s p e r s e d i n t h e c a t h o d e a n d t h e c a t h o d e flow c h a n n e l w h e r e t h e m o l e f r a c t i o n o f l i q u i d water, x]~q,4, is 0.077. C o m p a r i s o n of Xo3 to Xo4 ( T a b l e I) s h o w s t h a t t h e 02 p r e s s u r e h a s d r o p p e d (at 0.5 A / c m 2) o n l y b y a b o u t 5% across the cathode. This means that the limiting current s e t b y o x y g e n d i f f u s i o n t h r o u g h t h e c a t h o d e is a b o u t 10 A / c m 2. U n d e r d i f f e r e n t o p e r a t i n g c o n d i t i o n s , t h e m o d e l predicts good performance with no significant liquid water p r e s e n t . F o r e x a m p l e , r e f e r r i n g to t h e p r e v i o u s b a s e case, if t h e a n o d e h u m i d i f i e r t e m p e r a t u r e is r a i s e d to 105~ w h i l e t h e c a t h o d e ' s is l o w e r e d to 20~ (cell t e m p e r a t u r e 80~ a n d t h e c a t h o d e s t o i c h i o m e t r y is r a i s e d f r o m 6 to 8.4, t h e n t h e r e l a t i v e h u m i d i t y is 0.914 in t h e a n o d e flow c h a n n e l a n d is 0.930 i n t h e c a t h o d e flow c h a n n e l . A v e r y s m a l l a m o u n t of l i q u i d is p r e s e n t at i n t e r f a c e 3 (X~q,3 = 0.0035), b u t t h i s e v a p o r a t e s d u r i n g t h e d i f f u s i o n p a t h to i n t e r f a c e 4. E x a m p l e s are c i t e d in w h i c h t h e c a t h o d e h u m i d i f i e r is at 80~ b e c a u s e , i n t h e l a b o r a t o r y , less h u m i d i f i c a t i o n to t h e c a t h o d e g a v e p o o r cell p e r f o r m a n c e , c o n t r a r y to s u c h m o d e l p r e d i c t i o n s . We b e l i e v e t h i s is b e c a u s e o f t h e p a r t i a l d r y i n g of t h e m e m b r a n e t h a t a p p a r e n t l y t a k e s p l a c e bec a u s e of t h e e x c e s s i v e e v a p o r a t i v e loss t h r o u g h t h e g r a p h ite c o l l e c t o r p l a t e s h e l d at 80~ W h e n t h i s " a d d i t i o n a l " c a t h o d e w a t e r r e m o v a l p a t h b e c o m e s u n d e r s t o o d , it will be included in future models. F i g u r e s 5-9 d i s p l a y t h r e e v a r i a b l e s (cell voltage, V~en, m e m b r a n e r e s i s t a n c e , Rm~m, a n d s t e a d y - s t a t e w a t e r - p e r 1.0 / 14 0.6 12 0.4 ~ o~'~ ~,.~ ~ _-8ooc ~ 0.2 N 8 o 6 ' 0.2 0.5 0.8 9 * 0.o x tm,m = 1 7 ~ ~tm ",,\', X ~. 0.5 x 0 ~ . c m z) "''~'.... ~" 0.4 2 ' 0.8 16 U'J 9 The Electrochemical Society, Inc. o'.= q. T=t~ = 80~ Tutc = 80oC o'., 0'., ;., ,.o THICKNESS FRACTION Fig. 4. Computed water profiles in Notion ] ] 7 for the base case for four current densities holding gas flow rates constant, VH = ] at ] A]cm 2. The cathode is on the left, and the net water flux moves from right to left, ..... ... 0.3 0 0.2 :~ 0.1 ........... "OH = 2. ". . . . . . . . ....... 0.0 :::::::::::::::::::::::: ~"---.:" .... t). =4 I 0 1 CURRENT 2 DENSITY A / cm 2 Fig. 5. Cell voltage, membrane resistance, and H20/H + flux ratio v s . current density for different anode stoichiometry with anode humidifier at 80~ Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. Electrochem. Soc., Vol. 138, No. 8, August 1991 9 The Electrochemical Society, Inc. 1.0 2339 1.0 ~ . 0.8 (V) ~ ,, 0.4 Ir ~ ~ , / - 0.2 ...~--'~,.:':_ ~ x - - ~ u ::::.:::: \ ...... .--:2 ............. =1 r- i o : 0 proton flux ratio, [3) as a function of current density from 0 to 2 A/cm 2. Except where indicated, all parameters correspond to the base case of Table L Figure 5 shows the effect on the base case of varying the anode stoichiometry. The only effect of variations in VHiS through the role of H2 as a carrier for water vapor in the inlet gas stream because the model assumes that overpotential occurs only at the cathode/membrane interface. Note that the steady-state water flux per H § ratio decreases with increasing current density for a fixed H2 flow, and it increases with increased He stoichiometry when more water vapor is supplied at the anode. A higher water flux results in lower m e m b r a n e resistance and higher cell voltage at a given current density. Figure 6 shows the effect of raising the anode humidifier temperature above the cell temperature. The m e m b r a n e at low currents maintains a water content near 14 water molecules per m e m b r a n e charge site throughout its thickness, 0.8 \ i 0.6 // ~ tmem = 175 g m / .... x~:,~,, / ,.,%+, 175gin 1 2 C U R R E N T DENSITY A / c m ~ 2 Fig. 6. Cell voltage, membrane resistance, and H20/H + flux ratio vs. current density for different anode stoichiometry with anode humidifier at 105~ 1.0 ...... t.... 0.0 CURRENT DENSITY A / cm ~ \ ) -. -~. "-..~ o 0.1 ff ' 1 . ", - ' , _ ~ .... t 501xm ............ t . . . . 1001xm ......... t . . . . 140gin , .... ~ _ o.2 1 i 9 7 " / ,,\ 0.6 ~ " C , ~ , ~ .. _ 0.2 / T.,=80~ tmem = 175 ~tm 0.6 / .,,.--'l T~tA = 1 0 5 ~ Fig. 8. V~u, R~u, and H~O/H + flux ratio [3 vs. membrane thicknesses at T,,~ = 80~ J for four Nation 117 and the resistance approaches 0.13 ~-cm 2. The addition of more water vapor by raising the humidifier temperature has improved cell performance by lowering the m e m b r a n e resistance considerably. This is clearly seen by comparing the cell resistance and cell polarization curves shown in Fig. 5 and 6. The initial high H20/H + flux ratio [3 of about 2.3 at 0.1 A/cm ~ occurs because the 105-80~ difference in anode-to-cathode humidifier temperatures provides a gradient in m e m b r a n e water content. (Remember that we allowed (in Eq. [17]) >, to increase with excess liquid water at the interface.) At higher current and lower vH, the flux ratio drops. At VH= 1, we cannot transport much water vapor to the anode, which causes h to drop in the membrane. This drops the electro-osmotic drag but also raises the resistance. Figure 7 shows the effect of holding VHat i (defined for 2 A/cm ~) and varying the temperature of the anode humidifier, T~tA. Here the resistance increase with current is much higher than in Fig. 5, because less water is available from the anode feed stream for the membrane. The flux ratio [3 increases as T~atAincreases, principally because the 1.0 i / / / .,.,.- ,,,, ,o.oS ooO,S ~ u (V) D. = 1 t 0.8 , ~.o 0.6 0.4 0.4 ,...... ,, 0.2 R~..-~O'cmZ> t , ~',Jj",.. N \ . -,t T~tA 80oc ............ 88oc .......... 96 C . . . . . . 105oc - - \ x 9 , ...9 ' . . " , , , -u o ............ \ -.,. ) "~ ' ~ ' ~ ' -- ' " ' ~" " ~ I "--= ? ~ ,=~ , ' ~ . ' ~ .=--" .=--= -=--=-=-'- e--" r 1 CURRENT DENSITY A / em 2 ..~ k "~ x 01~" ~ 50p, m tmem " ~ ~ ............ t = . .......... t . . 100gm 140p.m 175~tm -Ir 0 0 1 2 CURRENT DENSITY A / cm 2 Fig. 7. Cell voltage, membrane resistance, and H20/H + flux ratio vs. current density for different anode humidifier temperatures with anode stoichiometry of 1 at full current density. Fig. 9. Vcell , Re,,, and H20/H + flux ratio [3 vs. membrane thicknesses at T~tA = 105~ J for four Notion 117 Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. Electrochem. Soc., Vol. 138, No. 8, August 1991 9 The Electrochemical Society, Inc. 2340 drag coefficient is proportional to k. The inflections observed near ~ = 1.59 occur because the slope of the k v s . water activity curve changes w h e n the water vapor at the interface reaches saturation. (The value of n d ~ is 1.59 at k = 14.) In general, the value of ~ is less than ndr~g because backdiffusion ameliorates the effect of the electro-osmotic drag. Figure 8 shows the significant advantage of thinner membranes. At T~tA = 80~ four m e m b r a n e thicknesses are compared. With the thin 50 ~m membrane, not only is the resistance less than 0.05 t2-cm 2, but the flux ratio ~ is very low compared with the drag coefficient. Thus, the need to continuousl~ replenish the anode with water from the cathode in real stack operation becomes m u c h smaller. The backdiffusion of water is now able to supply the anode need. Figure 9 shows the effect of the same m e m b r a n e thickness variation but at an anode humidifier temperature of 105~ Now the steady-state gradient of m e m b r a n e water content keeps the resistance lower for the thicker membranes, but it also causes more water to m o v e to the cathode. We performed several measurements to verify the predictions of the water transport aspects of this model. In the first m e a s u r e m e n t we collected water from the outlet gas streams of a 50 c m 2 cell with a 117 Nation membrane, using Prototech electrodes, and compared it with the inlet streams. We found at 500 m A /c m 2 that 0.2 H20 was transported per H + at 80~ The second type of m e a s u r e m e n t was that of high-frequency resistance ("ohmic resistance") of a 1 cm 2 Nation 117 cell operating on air and hydrogen at 80~ cell and humidifier temperatures, and at 3 atm anode pressure and 5 atm cathode pressure. The anode-cathode high-frequency resistance is measured at 5 kHz, where concentration polarization and faradaic effects are eliminated, leaving, presumably, only the m e m b r a n e resistance and any contact resistance. Figure 10 shows that both the experiment and the model indicate an increase in high-frequency resistance (RHF) with cell current. When the anode and cathode humidifier temperatures were raised to 105~ the high-frequency resistance actually decreased slightly with current density. We do not yet understand the observed d e p e n d e n c e of RHF on cell current when the cell is well humidified (Tsat = 105~ S o m e general c o m m e n t s about the approach described in this manuscript are in order. We would like to stress again the strong reliance, in this modeling effort, on membrane parameters that have been measured in our laboratory. We believe that in the effort to m o d e l a complex syst e m such as this, where the results depend on so many model parameters, the results of the code are of little value unless they rely to the m a x i m u m extent possible on experimentally derived parameters. In this context, let us briefly 0.5t r 0.4 80~ ANODE STOICH = 3 MEASURED HiGH @2 AJcm FREQUENCY C~ o,3 Z ~p, 0.2. oo PLUS 0.065 CONTACT RESISTANCE LU rr 0.1. 0.0 o.o 0:2 0'.4 o:s o.s CURRENT DENSITY A/cm 2 Fig. 10. Measured high-frequency resistance of a fuel cell with Nation i 1 7 compared with model membrane resistance plus 0.065 ~-cm 2 resistance. The cathode pressure was raised to 5 atm (3 atm on anode) to squeeze the membrane and electrodes. review the experiments performed, to highlight several aspects. The isopiestic t ech n i q u e has been shown here and elsewhere (11) to provide a good basis for establishing a range of well-defined water contents in the ionomeric membrane, allowing us to derive the dependence of membrane protonic conductivity and water diffusion coefficient on water content. Water (intra)diffusion coefficients have been derived over a range of m e m b r a n e water contents using the 1H-pulsed gradient spin-echo (PGSE) NMR t ech n i q u e (11). The ~H intradiffusion coefficient was found to vary across a full order of magnitude as the membrane water content varies between 2 and 14 water molecules per sulfonate group. F r o m arguments of H~O/H + population ratio in the m e m b r a n e at high water contents and from similar apparent values of D derived from NMR and from protonic conductivity at small m e m b r a n e water contents, we conclude that the NMR measured D of 1H is very close to D~20 in the range of water contents probe (11). We do intend, however, to further test the full accuracy of this conclusion in some future 170 NMR experiments. Another subtle aspect needs to be mentioned in the context of applying an intradiffusion coefficient, (as derived from NMR or from a radioisotope labeling experiment), as an interdiffusion coefficient, that is, using the intradiffusion coefficient in a Fick's first law formalism (see Eq. [t9]). In a strict analysis the intradiffusion coefficient (NMR) and the interdiffusion coefficient (Eq. [19]) are not necessarily identical (15). However, to complete this stage of work on the P E F C model, we felt that the proton diffusion coefficient derived by NMR as a function of m e m b r a n e water content can yield, upon correction for activity coefficient, a satisfactory approximation for the interdiffusion coefficient of water in the membrane. The values we obtained for the intradiffusion coefficient of water are reasonable; they fall between one order of magnitude smaller (for a fully hydrated membrane) and two orders of magnitude smaller (for a m e m b r a n e with k = 2) than the intradiffusion coefficient of water in liquid water, and they are in general agreement with other reported values of D(intra) at high (2-4) and very low (16) water contents. In fact, when the water activity gradient is used in a Ficks first law expression (Eq. [19]), the effect D becomes almost i n d ep en d ent of water content (Fig. 3), at a level of 1.2 - 10 -6 cm2/s. This, again, is a very reasonable result for the diffusion coefficient of water in a water-absorbing ionomeric m e m b r a n e like Nation. It is similar in magnitude to that reported by Yeo and Eisenberg (14) for measurements under interdiffusion conditions. The reasonable order of magnitude of Dwat~r evaluated here is stressed because, in other previous contributions, m u c h higher values of Dwater w e r e mentioned, In fact, Dwater higher than the diffusion coefficient of water in liquids at the same temperature was invoked (6). Such large values of Dwater w e r e required to explain experimental observations, because it was assumed (4, 6) that the electro-osmotic drag is not lowered as the membrane water content becomes smaller. Our results show clearly that experimental observations can be fully accounted for with the apparently very conservative IeveIs of Dw,ter in the membrane. These observations are based on the NMR m e a s u r e m e n t s and their interpretation. The capability to account for observed cell performance with such conservative water diffusion coefficients is, in turn, based on the lowering of the electro-osmotic water drag in the m e m b r a n e with the lowering in m e m b r a n e water content (Eq. [18]). It is the interplay between the diffusion coefficient of water and the degree of water drag, including the variations of these two parameters with m e m b r a n e water content, which, together, determine the water profile in the m e m b r a n e under given humidification conditions and cell current density. In the context of the latter point, we would like to stress the difference between the electro-osmotic drag, designated by nd~,g in this paper, and the steady-state flux ratio H20/H + designated by ~ because these two parameters have sometimes been intermixed in previous discussions of this system. It should be clear that nd~g is what one measures for a m e m b r a n e fully i m m e r s e d in liquid water, when the electro-osmotic drag is recorded at unit activity Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp J. Electrochem. Soc., Vol. 138, No. 8, August 1991 9 The Electrochemical Society, Inc. of water throughout the system. On the other hand, in a fuel cell operating at steady state at a certain current density, there would be usually a significant gradient of water content (see Fig. 4), and, hence, the measured flux of water from anode compartment to cathode compartment per proton passed through the m e m b r a n e ~ would be significantly smaller than ndrag(see Eq. [23]). This distinction between 13 and ndrag has significant practical implications in the context of the extent o f " b a c k - p u m p i n g " of water from cathode to anode, which would be required during steadystate cell operation. It should be clear that the level of water recirculation required is given by ~, not by nd~g, and thus is significantly lower than would be suggested by naive inspection of only nd~ag. Finally, we should bring up two discrepancies between the real PEFC system with which we have experimented and the validity domain of this model. One noticeable discrepancy is that the model does not predict the need to humidify the cathode feed stream continuously at any appreciable current density. I n practice,however, we found that in cells based on the Nation 117 membrane, the highest performance is obtained only with well-humidified cathode feed streams. We believe that this is because evaporative water losses from the cell are substantial, in practice, at least in our test cell configurations. Unless supplied with an aerosol of water (T,,t = 105~ the m e m b r a n e loses water at an excessive rate, which cannot be compensated by water production at the cathode. This evaporative loss is not considered in our model at present, and, therefore, the model does not predict the need for extensive cathode humidification. On the other hand, as stressed before, the model does not consider the effect of substantial excess of liquid water in the cathode. One important effect expected is an additional thin film diffusion barrier for reactant gas transport (17). Such an excess of liquid water may be essential, however, to achieve the highest degree of membrane hydration, and, thus, the highest possible protonic conductivity. How to compromise between these two conflicting effects of excess liquid water at the cathode is a key to the successful operation of a PEFC. Conclusions We developed a simple, one-dimensional, isothermal model of a complete polymer electrolyte fuel cell that has provided useful insight into the cell's water transport mechanisms and their effect on the cell's performance. We applied equilibrium conditions between m e m b r a n e water and electrode water vapor at the membrane/electrode interfaces and considered the electro-osmotic and diffusion driving forces for water in the m e m b r a n e and diffusion for water vapor and reactant gases in the electrodes to obtain material balances throughout the cell. The model used data that we measured for 117 Nation membranes, including water content vs. water activity and water diffusion coefficient, protonic conductivity, and electro-osmotic water drag as a function of m e m b r a n e water content. The model was designed for water in the vapor state in the electrodes, but it could accommodate some excess liquid water assumed to be very finely dispersed. Experimental verification of some of the model predictions was successful when the measured cell was operated u n d e r conditions in which excess liquid was not present in the cathode. Measured net steady-state water flux per H § in a 50 cm 2 cell of 0.2 H20/H + agrees with model predictions at 80~ cell and humidifier temperatures. Likewise, the measured high-frequency resistance of a 1 cm 2 cell at an 80~ humidifier temperature increased with current density, as the model predicted. This effect was not observed when a 105~ humidifier temperature was used. An important conclusion is that the net water per H § flux ratio in a PEFC can be as little as one-tenth of the electroosmotic drag coefficient measured for a fully hydrated membrane, thus reducing the problem of water managem e n t for PEFC stacks. Acknowledgments We wish to acknowledge our m a n y fruitful discussions with Mr. Thomas Fuller, who contributed to the developm e n t of the work described herein. This work was sup- 2341 ported by the U.S. Department of Energy, Office of Conservation and Renewable Energy. Manuscript submitted Dec. 10, 1990; revised manuscript received Feb. 11, 1991. Los Alamos National Laboratory assisted in meeting the publication costs of this article. LIST OF SYMBOLS Subscripts 1 anode p l e n u m to anode interface 2 anode electrode to m e m b r a n e interface 3 m e m b r a n e to cathode interface 4 cathode to cathode p l e n u m interface A anode region, or component A B component B C cathode region H hydrogen, H2 N nitrogen, N2 O oxygen, O2 mem membrane sat saturation w water Superscripts I initial value entering inlet flow channels L final value leaving inlet flow channels Parameters and variables D diffusion coefficient, cm2/s F Faraday constant, 96,484 C/mol I water molar flux produced at cathode J/2F, mol/ cm 2 s J Mm Mw ndrag N p~ P R Rcell t T Tc V x z p current density, A/cm 2 equivalent weight of m e m b r a n e molecular weight of water electro-osmotic drag coefficient molar flux, mol/cm 2 S critical pressure, atm pressure, atm molar gas constant m e m b r a n e resistance, tl c n ~ 2 thickness, cm temperature, ~ critical temperature, K cell potential, V mole fraction distance variable, cm ratio of net H~O flux in m e m b r a n e to H20 flux product at cathode ratio of net H20 flux in m e m b r a n e to H § flux in membrane water content or local ratio H~O/SO~ in the membrane density, g/cm 3 stoichiometric coefficient REFERENCES 1. R. A. Lemons, J. Power Sources, 29, 251 (1990). 2. M. Verbrugge and R. Hill, J. Phys. Chem., 92, 6778 (1988). 3. M. Verbrugge, This Journal, 136, 417 (1989). 4. R. Hill and M. Verbrugge, ibid., 137, 886 (1990). 5. J. L. Fales, N. E. Vanderborgh, and P. Stroeve, in "Diaphragms, Separators, and Ion-Exchange Membranes," J.W. van Zee, R. E. White, K. Kinoshita, and H.S. Burney, Editors, PV86-13, p. 179, The Electrochemical Society Softbound Proceedings Series, Pennington, NJ (1986). 6. T. Fuller and J. Newman, in "Fuel Cells," R. E. White and A. J. Appleby, Editors, PV 89-14, p. 25, The Electrochemical Society Softbound Proceedings Series, Pennington, NJ (1984). 7. D. Bernardi, ibid., p. 51. 8. J. C. Slattery and R. B. Bird, AtChE J., 4, 137 (1958). 9. R. E. De La Rue and C. W. Tobias, This Journal, 106, 827 (1959). 10. "Handbook of Chemistry and Physics," 62nd ed., CRC Press, Boca Raton, FL (1981). 11. T. Zawodzinski, M. Neeman, L. Sillerud, and S. Gottesfeld, J. Phys. Chem., Submitted. 12. C. Derouin, J. Pafford, S. Radzinski, T. Springer, and S. Gottesfeld, Abstract 627, p. 903, The Electrochemical Society Extended Abstracts, Vol. 89-1, Los Angeles, CA, May 7-12, 1989. Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp 2342 J. Electrochem. Soc., Vol. 138, No. 8, August 1991 9 The Electrochemical Society, Inc. 13. A. B. LaConti, A . R . Fragala, and J . R . Boyack, in "Electrode Materials and Processes for Energy Conversion and Storage," J. D. E. McIntyre, S. Srinivasan, and F. G. Will, Editors, PV 77-6, p. 354, The Electrochemical Society Softbound Proceedings Series, Pennington, N J (1977). 14. S. C. Yeo and A. Eisenberg, J. Appl. Polym. Sci., 21, 875 (1977). 15. J. Crank, "The Mathematics of Diffusion," 2nd ed., Clarendon, Oxford (1975). 16. T. Nguyen, J. C. Hedstrom, and N. Vanderborgh, in "Fuel Cells," R. White and A. Appleby, Editors, PV 89-14, p. 39, The Electrochemical Society, Softbound Proceedings Series, Pennington, N J (1989). 17. T.E. Springer and I. D. Raistrick, This Journal, 136, 1594 (1989). Measurements of Double-Layer Capacitance at the Preoxidized Ni/Fused Na2S04 Interface Yiing-Mei Wu*" Department of Materials Science and Engineering, The Ohio State University, Columbus, Ohio 43210 ABSTRACT The double-layer capacitance in a fused Na2SO4 film on a preoxidized Ni electrode at 1200 K was measured by electrochemical i m p ed an c e spectroscopy. The capacitance-potential curve has a m a x i m u m at a potential of -0.6 V vs. Ag/Ag2SO4, silica reference electrode, resulting from the additional contribution of a faradaic reaction. Most of the studies of double-layer properties have been conducted on metal electrodes in aqueous solutions (1-5). Only a few researchers investigated the capacitance (C)potential (E) relation in molten salt electrolytes. Graves and I n m a n (6) have reviewed the case for a liquid lead electrode in KC1-LiC1 melts. The capacitance is thought to comprise both the double layer and faradalc contributions. The shape of the C-E curve is parabolic with a m i n i m u m at the Epzc (potential of zero charge). More recently, Painter et al. (7) used a model of charged hard sphere fluid in contact with a charged hard electrode wall to explain the features of the capacitance for the metaYmolten salt (alkali halides) interface. Although the magnitude of the calculated capacitance is comparable with experimental values, the temperature d e p e n d e n c e of capacitance at Epzc cannot be reproduced by the hard sphere model. This probably arises from "relaxation" of the structure, especially near the metal/molten salt interface, where the interactions between metal-ion and ion-ion are strongest. In the current study, the capacitance in a fused Na2SO 4 film on a preoxidized Ni electrode was measured by electrochemical i m p ed an c e spectroscopy. With an appropriate equivalent circuit, obtaining capacitance values from impedance m e a s u r e m e n t s is straightforward and accurate, as compared to other pulse m e th o d s (8). This system was chosen because of its relevance to the hot corrosion degradation of metals or alloys in gas turbines and other fuel combustion systems. Furthermore, as the d e v e l o p m e n t of the molten salt fuel cells and batteries advances, the need for information on the structure of the electrode/molten salt interface has b e c o m e rather urgent. Double-layer capacitance, among other surface properties, is an important and experimentally accessible parameter in that respect. tacted the nickel WE to act as the second RE. The combination of the zirconia and silica reference electrodes provided measures for both the basicity and oxygen pressure at the WE. The Ni WE was preoxidized in pure oxygen at 1200 K for 2 h, resulting in an oxide thickness of 4.3 ~m. After this preoxidation, Ni WE arrangement was immersed briefly into fused Na2SO4 contained in a silica crucible, and the gas atmosphere was changed to a catalyzed 0.1% SO2-O2 gas mixture. U p o n withdrawal of the sample from the melt, a thin fused salt film covered the electrodes. The thickness of t h e fused salt film between the WE and CE was about 2 mm. The previous study (9) had shown that the WE was passive u n d er these conditions, i.e., the salt film did not directly contact the Ni metal substrate. Electrochemical i m p ed an ce m e a s u r e m e n t s were performed with a Princeton Applied Research (PAR) 5208 two-phase lock-in analyzer and a P A R 273 potentiostat interfaced through an I E E E 488 bus to an IBM PS-2 computer. This system generated a sinusoidal voltage which was applied to the WE with a m a x i m u m amplitude less than 10 mV for frequencies ranging from 10 -3 to 105 Hz. A fast Fourier transform (FFT) t ech n i q u e was employed for frequencies from 10 -3 to 10 Hz to increase m easu r eme nt speed and lower the degree of perturbation to the cell. Through use of the potentiostat, polarization of the WE with respect to the silica RE was performed and impedance m e a s u r e m e n t s were made after the polarization current was stabilized (change of less than 10% per 10 min). I m p e d a n c e data were s t o r e d and plotted by the computer. Potentiodynamic polarization was conducted with the same potentiostat in a separate experiment. The scan rate was 1 mV/s. Experimental Procedure Results and Discussion Thin film i m p ed an c e m e a s u r e m e n t at 1200 K were carried out using the apparatus and procedures reported earlier (9). The working electrode (WE) was a 99.9975% pure preoxidized Ni wire w o u n d around the outer surface of an yttria-stabilized zirconia reference electrode (RE). A platin u m foil counterelectrode (CE) encircled the zirconia RE without touching the preoxidized nickel WE. The space between the CE and WE was made small (about 2 mm) so that a thin layer of used salt could be held there during an experiment. A fused silica reference electrode also con* Electrochemical Society Student Member. Present address: Mobil Research and Development Company, Paulsboro, New Jersey 08066-0486. Figure 1 shows the typical i m p ed an ce spectrum at the open-circuit potential and the corresponding equivalent circuit, which were reported earlier (9). In Fig. 1, the lowfrequency Warburg i m p ed an ce corresponds to the diffusion of oxidant, $2072-, in the thin salt film Wo = ~or - j), j = ~ [1] where % is the Warburg coefficient, as shown in the equivalent circuit in Fig. 1, co = 2~rf and f is the frequency. The two semicircles at m e d i u m and high frequencies were attributed to the i m p ed an ce for the charge transfer at the oxide-salt interface, Ro/s, and the resistance to the m o v e m e n t Downloaded 20 Nov 2009 to 140.112.2.121. Redistribution subject to ECS license or copyright; see http://www.ecsdl.org/terms_use.jsp