Stability Analysis for Liquid Water Accumulation in Low Temperature Fuel Cells

Transcription

1 Proceedings of the 47th IEEE Conference on Decision and Control Cancun Mexico Dec TuB7.5 Stability Analysis for Liquid Water Accumulation in Low Temperature Fuel Cells B.A. McCain and A.G. Stefanopoulou and I.V. Kolmanosky Abstract In this paper we analyze the stability and twophase dynamics of the equilibrium water distributions inside the porous media of a polymer electrolyte membrane PEM fuel cell gas diffusion layer GDL whi is described with first and second-order parabolic partial differential equations PDE. A Lyapuno function-based stability criterion is implemented for the liquid mass continuity PDE while physics and mathbased arguments are used to demonstrate a key range of liquid distribution that cannot attain equilibrium. The quantity of water within the fuel cell directly affects performance efficiency and durability. High membrane humidity is desirable for proton conductiity yet excess liquid water in the anode has been experimentally shown to be a cause of output oltage degradation. We identify the unstable states representing unbounded growth of liquid water and show that stabilization of annel water mass is sufficient for a stable liquid distribution. I. INTRODUCTION AND MOTIVATION A compact model of the multi-component oxygen hydrogen water two-phase apor and liquid water spatiallydistributed and dynamic behaior across the gas diffusion layer Fig. 1 is inestigated. The time-arying constituent distributions in the GDL of ea electrode are described by three second-order parabolic PDEs for reactant oxygen in the cathode and hydrogen in the anode concentration water apor concentration and liquid water olume. The electroemical reactions on and the mass transport through the catalyst-coered membrane couple the anode and cathode behaiors and together with the annel conditions proide the time-arying boundary alues for these PDEs. The water liquid and apor PDEs are coupled through the eaporation/condensation rate. Further the liquid water becomes a nonlinearly distributed parameter that inhibits reactant gas and water apor diffusion. Liquid water occupies pore space in the GDL impedes the diffusion of reactant flow towards the membrane and reduces the fuel cell actie area [1] causing performance degradation. We focus on the stability of the anode GDL water distribution and associated anode annel water mass for two reasons. First mobile applications of fuel cells spend a considerable time under low-to-medium load conditions where net water transport is from cathode to anode through the polymer electrolyte membrane. Second remoal of accumulated liquid water is necessary to regain performance whi is typically accomplished by surging an inlet flow B.A. McCain bmccain@umi.edu A.G. Stefanopoulou annastef@umi.edu and I.V. Kolmanosky ilya@umi.edu are with the Fuel Cell Control Laboratory Me. Engr. Dept. Uni. of Miigan. Funding is proided from NSF-GOALI CMS Oxygen Ca Hydrogen de Catalyst Layer Membrane Channel x= x=l Gas Diffusion Layer Channel Liquid Water Fig. 1. Conceptual sematic showing accumulation of liquid water in the GDL and subsequent flow to the annel where reactant-blocking film is formed e.g. anode supply resulting in efficiency loss. Allowing liquid water to accumulate is undesirable for performance efficiency and membrane durability reasons hence estimation and control of the liquid water distribution is critical for effectie fuel cell management. The phenomenon of anode GDL flooding-induced oltage degradation loss is addressed in this work though the methodology presented is applicable to the cathode GDL. In fact in order to understand and model the anode GDL water accumulation and transport the cathode water apor distribution must also be determined because the membrane transport model requires cathode water apor concentration. Therefore while the focus is on the anode GDL and annel the cathode water apor distribution and annel balance are included supplementally to complete the annel-to-annel model. Finally since the cathode liquid accumulation does not affect the anode water distributions it is not modeled and all discussions of liquid refer to that of the anode side. A. Obseration of System Instability We know from experiment and simulation that the fuel cell will experience unbounded liquid water growth under many normal operating conditions flow regulated anode inlet high utilization etc. Figure 2 demonstrates the oltage degradation phenomena obsered experimentally during dead-ended anode operation utilization = 1% between purges; namely that anode flooding is related to oltage degradation. It turns out that the apparent instability is directly related to the annel water mass state. Specifically we deelop the following result: /8/$ IEEE 859

2 47th IEEE CDC Cancun Mexico Dec TuB Cell Voltage Experiment Model Prediction W l x t is the liquid water mass flow kg/s at time t at a cross-section of the GDL located at x x L; Sx t is the reduced water saturation m 3 /m 3 at time t at a cross-section of the GDL located at x x L V mg 2 A annel water mass Stack Current Purge Purge Time sec Fig. 2. Experimental results support the hypothesis that anode flooding causes oltage degradation that is recoered through anode annel purging. Claim: Gien the assumptions partial differential equations and boundary conditions BC described in Sec. II and the analytic solutions from Sec. III the anode liquid water distribution is stable if the annel liquid water is stabilized. Sket of the Proof: The GDL liquid water is locally exponentially stable for fixed annel liquid BC see Sec. IV. Therefore stabilization of the annel water mass state results in oerall water system stability. Thus if we iew the annel water mass as a controlled output we essentially establish the stability of the resulting infinite-dimensional zero dynamics within the GDL as this output is held constant. We proceed with analysis to justify this result. II. FUEL CELL MODEL Due to space limitations only a summary of the first principles model will be proided here. The details of the model are identical to those presented in [2]. We proceed with a one-dimensional treatment focusing on the anode using the following assumptions. The model is spatially isothermal. Gas conectie flow in the GDL is neglected. Mass transport is in 1D normal to the membrane. Water transport out of the anode annel is assumed to be in apor form. We designate x as the spatial ariable with x= corresponding to the membrane location and x=l corresponding to the anode annel location and we let t denote time. The state ariables are as follows: c H2 x t is the concentration mol/m 3 at time t at a cross-section of the GDL located at x x L; c an x t is the concentration of water apor at time t at a cross-section of GDL located at x x L; The water saturation sx t is the fraction of liquid water olume V L to the total pore olume V p s = VL V p. s is thus a concentration-like ariable for the liquid water at time t at a cross-section of GDL located at x x L. The following intermediate ariables are useful: Sx t = sx t s im 1 s im 1 where it should be noted that S = for s < s im. The immobile water saturation s im works as stiction i.e. there is no liquid flow unless the water saturation s exceeds s im. Finally the eaporation rate r is defined as { γ c r c an = sat c an for s > min { γc sat c an } for s = where γ is the olumetric condensation coefficient and c sat is the apor saturation concentration. Note that eaporation can only occur if there is liquid water s > in the GDL. Conseration of gas consituents and the liquid mass are employed to determine the three second-order parabolic PDEs that goern the anode reactant water apor and water liquid c H2 = c H2 H t 2 2 c an t = t = b 1S b 2 S c an + r c an 3 M r c an 4 where Dy sim is the diffusiity of y inside the GDL porous medium M is the apor molar mass and is the density of the liquid water. A. Boundary Conditions The boundaries for the GDL distributions are at the interfaces with the membrane x= and the annel x=l. The boundary condition for the reactants at the membrane- GDL interface is a function of the current drawn. Water apor concentration on one side of the membrane influences the water apor concentration on the other side thus there are four BC needed to simultaneously sole both the anode and cathode water apor concentration second-order PDE. These BC are influenced by inlet flow stoiiometries temperatures and relatie humidities as well as the disturbance stack current and the controllable outlet ale on the anode outlet. For c H2 x t mixed Neumann-Dirilet type BC are imposed. The annel boundary condition is c H2 x=l = c = p / RT 5 where R is the uniersal gas constant T is the temperature and the partial pressure in the anode annel p depends on the control input ū as discussed in Sec. II-B. The membrane mb boundary condition is c H2 1 = it x= x= 2F = Nrct 6 x= 86

3 47th IEEE CDC Cancun Mexico Dec TuB7.5 where the rct indicates the reaction of at the anode catalyst whi depends on the current density i A/m 2 and F is Faraday s constant. For c an x t similar mixed BC are imposed: c an x=l = c an = p an/ RT 7 where p an is the apor partial pressure in the annel. The slope of the water apor distribution at the membrane is determined by the water apor flux across the membrane c an = Nmb 8 x= D sim where the membrane water molar flux N mb is goerned by electro-osmotic drag and back diffusion. For the liquid water PDE mixed BC are again imposed. Specifically since water passing into the GDL is in apor form due to the assumed presence of a micro-porous layer S =. 9 x= The liquid water boundary condition at the GDL-annel interface is an issue of ongoing resear. A common assumption is that the hydrophobic nature of the GDL porous media essentially pushes the liquid water into the annel formed of hydrophillic material with zero resistance [3]. On the other hand Wang and his colleagues [4][5] suggest that the presence of liquid water on the GDL-annel interface constitutes a resistance to flow into the annel. There does not yet exist a physics-based method to relate the amount of liquid water in the annel to resistance to flow from the GDL. In [6] arious alues of sl t were selected without reason and kept fixed for simulation and in [5] a constant capillary back-pressure was assumed 7.7kPa without explanation for the oice. Here we consider a general form of the liquid boundary condition at the GDL-annel interface: SL t = gm l t 1 where m l t is the mass of liquid water in the annel and g is an unknown function of that liquid. A form for g is not proposed and for the purposes of the claims made herein it is not necessary to know the function. It is assumed howeer that g is monotonically increasing with m l and has an exponential bound of the form gm l t a e bm l t 11 where a and b are positie constants. Thus 1 represents leels of liquid accumulation in the annel su that the GDL liquid boundary condition aries with the quantity present. B. Anode Channel Equations For the anode the goerning equations for and water: dm /dt = W in W out + W GDL H2 dm w /dt = W GDL w W out W in. 12 In the general case the anode will hae a humidified inlet stream with the relatie humidity RH in and the inlet flow rate W in prescribed thus where P in W in = P in M PH in WH in 2 M 2 13 H2 = RH in P sat and P in is dry inlet gas pressure. The and water apor partial pressures whi represent the annel side GDL BC are calculated from: p = m RT M H2 Van p { = min m RT M V p sat p = p + p. } 14 The anode exit flow rate to the ambient amb is modeled as a linearly proportional nozzle equation W out = ū k out p p amb 15 where k out is an experimentally determined nozzle orifice constant and ū is a controllable ale flow ū 1 to remoe accumulated annel water and unfortunately W out W out = m H2 m W out = W out W out 16 where m = m + p V M /RT. The and water mass flow rate from the GDL to the annel are calculated using: W GDL = εa fc M H2 c H2 Ww GDL S b2 = εa fc b 1 S + M D sim III. ANALYTIC SOLUTIONS MODEL x=l c. x=l 17 Analysis of the state equations 2-4 allowed us in [2] to employ a steady-state analytic solution for the gaseous constituents inside the GDL combined with the numeric implementation of the liquid water state. We call these set of equations as the semi-analytic solution model and will be summarized here along with analytic solutions for the steady-state liquid water distribution because it will facilitate the deriation of equilibria distributions and their stability analysis. A. The Water Vapor Solution From [2] the steady-state solutions implemented for the cathode and anode GDL water apor distributions are c an x t = α 1 e βx + α 2 e βx + c sat 18 c ca x t = ν 1 e βx + ν 2 e βx + c sat 19 where β = γ/. 2 The α i and ν i are functions of the membrane water apor transport and the annel conditions [2]. 861

4 47th IEEE CDC Cancun Mexico Dec TuB7.5 B. Liquid Water Goerning Equation An alternate form of the liquid water distribution in the porous medium is obtained by replacing the c an x t coupling term in 4 by its steady-state solution 18 since it has been shown that the time constant of the water apor is orders of magnitude faster than that of the liquid water t = b 1 S b2 S for s s im and for < s < s im : + M γ α 1 e βx + α 2 e βx 21 t = M γ α 1 e βx + α 2 e βx. 22 For the sx t s im case 21 can be integrated twice to obtain the steady-state solution [ S ss x = β z βα 1 α 2 x L + c an c sat 23 α 1 e βx + α 2 e βx + ] 1 b SL tb β z where β and α i are as defined in the c an x t solution preiously and β z = M γb 2 + 1/ β 2 b 1. For the < sx t < s im case the time-arying solution can be found easily by integrating 22 with respect to time sx t= M γα 1 e βx + α 2 e βx t+s x 24 where s x is the water saturation distribution at t = t. The goerning behaior for < sx t < s im is su that the water liquid will grow to exceed s im and go back in the 21 case or will decrease with time and cause all the water liquid to be eaporated s = as discussed in detail in later section. Figure 3 demonstrates that the steady-state solutions to the water PDEs lead to an exponential form for the apor and a fractional power polynomial for the liquid. These shapes are highly dependent upon the oice of BC as eidenced by results from [7] where numeric results show mu higher water saturations s>.8 at the membrane and zero liquid water at the annel due to the assumptions of liquid water transport across the membrane and s= at the GDL-annel interface. Recent results [8] support the membrane boundary condition 9 used here. The dash-dot line in Fig. 3 demonstrates that GDL liquid water in excess of the immobile saturation i.e. flooding occurs when the water apor concentration reaes its maximum alue of c sat in the annel. Under these conditions the liquid water will grow unbounded in the annel instability. The solid line represents a lower annel water apor concentration and we obsere a annel condition c = c below whi the GDL liquid water begins to recede into the GDL. The dotted line depicts the steady-state distributions if c < c where the liquid water front has receded into the GDL stable with sl =. Concentration mol/m 3 Liquid Fraction m 3 /m c sat c mb s flooding s balanced sl < s im c flooding c balanced c L < c * c * mb Position mm Fig. 3. Anode distribution of liquid water ratio for arying annel water apor concentrations. IV. STABILITY ANALYSIS We need to show the stability of the liquid water distribution when s > s im for x L whi will cause a nonzero liquid flow to the annel. We show that if the liquid water in the annel feeds the boundary condition of the liquid water distribution is kept stable then the liquid water GDL distribution is also stable. Linearization of the GDL liquid PDE 21 for s > s im is accomplished by substitution of a perturbation from an equilibrium distribution Sx t = S o x + S t = b 1 b S ox + S b f ox 25 where f o x = Mγ α 1 e βx +α 2 e βx. A linear approximation for polynomial expansion is S ox+ S b 2+1 S ox b 2+1 +b 2 + 1S ox b 2 S. 26 Inserting 26 into 25 it can be seen that the initial steadystate solution is embedded within Since we get t = ṡ o = { }} { b 1 2 b S ox b2+1 + f o x b 1 2 S o x b2 S. S = s 1 s im 28 b 1 2 ṡ = 1 s im 2 So b2 s Defining w S b2 o s results in 29 w t = K 2 S b2 o w xx = ψxw xx 3 where K 2 =b 1 /1 s im and ψx K 2 S b2 o. 862

5 47th IEEE CDC Cancun Mexico Dec TuB7.5 The BC of the new PDE are found by substituting the original BC into the transformation. At x= w x t = b 2 S b2 1 o S o s s + Sb2 o = 31 because both the initial steady-state and perturbation distributions must satisfy the zero-slope requirement at x=. For the x=l Dirlet BC wl t = S b2 o L sl = 32 because a fixed x=l BC whi is a requirement for steadystate conditions requires sl remain zero. Thus for any constant boundary condition at x=l the BC of the transformed system will be zero. Modifying the steps described in [9] to fit our application a Lyapuno stability analysis is applied to the plant of the transformed system 3. L=1 is taken for conenience since a coordinate transformation z = x L could be implemented without loss of generality. Using the candidate Lyapuno function V w w x = 1 2 w 2 x t dx + 1 ψx 2 w 2 xx tdx 33 where it should be noted that ψx oer the range x [ 1]. Since ψ1 = K 2 So b2 1 may be zero for confirmation we eck w 2 1 t ψ1 = Sb2 o 1 s12 = Sb2 o 1 s2 1 = 34 K 2 So b2 1 K 2 and thus V is defined oer the entire range. The time deriatie of the candidate Lyapuno function wwt V = ψx 1 w 2 ψ 1 tx dx + w xw txdx 35 2 ψx 2 where ψ t x = since ψx is not a function of time. Substituting the transformed PDE w t = ψxw xx and integrating by parts V = 1 V = ww x ww xx dx + 1 wxdx 2 + w x w t From 31 and 32 and w t 1 = V = w 2 xdx w x w tx dx 36 ψw 2 xxdx. 37 ψw 2 xxdx. 38 Since V we hae stability in the sense of Lyapuno. To show exponential stability recognize that < ψ min ψ and apply Poincare s inequality to show and w 2 x dx γψ min ψw 2 xx dx γψ min w 2 dx 39 ψ wx 2 dx 4 where γ = 1 4. Then V γψ min whi becomes w 2 ψ dx + wxdx 2 41 V 2γψ min V 42 indicating that V exponentially as t. To show pointwise conergence use Agmon s inequality with w1 t = max wx x [1] t 2 2 w t w x t 43 and the inequality 2 a b a 2 + b 2 to find max wx x [1] t 2 w t 2 + w x t 2 44 where denotes the L2-norm. Since ψx > for all x and V wx t and w x x t as t implying max wx t as t. 45 x [1] Pointwise conergence is thus shown and the liquid water distribution within the GDL is exponentially stable. V. ANALYSIS OF EQUILIBRIA The form of 22 suggests that an equilibrium condition for < sx t < s im does not exist because the α i are not functions of s. It is also shown here that the concept of immobile saturation introduces a range of s where equilibrium does not exist. A. Claim: No equilibrium solution s exists for whi < sx < s im at some x We claim that there exists no equilibrium condition su that an sx within the GDL is non-zero but less than the immobile saturation. To see this we take 4 and substitute the eaporation term x t t = M γ c sat c x. 46 First considering the cases where c ss x c sat the RHS of the 46 dynamic equation is not a function of sx t and therefore no condition exists that will satisfy / t =. Thus while <sx t<s im sx t will either grow positie / t i.e. c x>c sat until it exceeds s im causing S to be re-introduced as an opposing factor for the condensation or decrease under eaporation negatie / t i.e. c x<c sat whi will cease only when sx t=. It remains to show c ss x c sat when < sx < s im. B. Claim: c ss x c sat for < sx < s im We claim that anywhere in the GDL that the liquid saturation steady-state is less than the immobile saturation the water apor pressure will not mat the saturation pressure. To demonstrate this we proide a continuity of solution sket of proof. This claim can be proen by contradiction. Specifically if there exists x su that c ss x = c sat while < s x < s im it can be first shown by continuity that < sx < s im 863

6 47th IEEE CDC Cancun Mexico Dec TuB S mb Increasing SLt position [m] 4 5 x 1 4 Fig. 4. Example of anode GDL liquid distributions where Wl GDL as the product of S b 2L and ds dx x=l reaes equilibrium when SL increases by mating W mb with decreased ds dx x=l. for all x in an open interal U containing x. From 18 and 22 it then follows that c ss x c sat =α 1 e βx +α 2 e βx = e 2βx = α 1 α 2 47 for x U. Since β the last relation cannot hold for all x U and we rea a contradiction whi implies the truth of Claims A and B. VI. BEHAVIOR OF THE CHANNEL WATER MASS In the preious section we showed that the liquid water distribution within the GDL is stable if the annel water mass has been stabilized. Within the literature constant alues of SL t suggested include SL t= [3]; SL t= [6]; and SL t=.153 [5]. Through control of the anode annel outlet ale described in Sec. II- B using feedback of oltage and relatie humidity it is possible to stabilize the annel liquid mass [1]. This stabilization can take the form of remoal of the annel liquid water m l = or maintenance of annel liquid mass below a predetermined threshold. In either case the method of Sec. IV can be used to demonstrate stability of the GDL liquid water distribution for typical annel liquid BC found in the literature. Figure 4 demonstrates the balancing role played by the liquid water transport out of the GDL. As can be seen from 4 the mass flow rate of liquid water is a function of the olume of liquid present SL and the slope of the distribution at the GDL-annel interface ds dx x=l. As the x = L BC grows the magnitude of the slope decreases with the final alue determined by the GDL-annel flow required to balance the liquid mass in the GDL. Thus for eery constant annel BC the distribution reaes equilibrium and as SL gets larger the equilibrium distribution flattens to nearly constant. VII. CONCLUSIONS It has been demonstrated that for a PEM fuel cell model with a GDL liquid water boundary condition coupled to the amount of liquid water in the annel system liquid water stability can be attained by stabilization of the annel water mass. This is the same result as for the commonly assumed case where the annel liquid has no influence on the GDL liquid. Lyapuno stability analysis is used to show that the liquid water distribution within the anode GDL is locally exponentially stable does not experience unbounded growth if the anode annel liquid water mass is bounded. This proides confidence that the stability results are robust to ariation in annel BC as long as those conditions can be stabilized. Analysis of the GDL liquid water dynamics indicates that the inclusion of an immobile saturation creates a range for whi the water saturation s will not rea equilibrium <s<s im. This result influences stability and control as this unstable region acts as a buffer delay to obserability of the GDL liquid ia the annel water state. While sl t < s im liquid flow from GDL to annel is zero and therefore the effect of anges to annel water concentration on GDL liquid distribution cannot be obsered using measurable outputs cell oltage outlet humidity. Feedback control using these outputs must consider this non-equilibrium phenomenon. REFERENCES [1] D. A. McKay J. B. Siegel W. T. Ott and A. G. Stefanopoulou Parameterization and prediction of temporal fuel cell oltage behaior during flooding and drying conditions J. Power Sources ol. 178 no. 1 pp [2] B. A. McCain A. G. Stefanopoulou and I. V. Kolmanosky A multi-component spatially-distributed model of two-phase flow for estimation and control of fuel cell water dynamics. New Orleans LA USA: 46th IEEE CDC CDC Dec 27. [3] J. H. Nam and M. Kaiany Effectie diffusiity and water-saturation distribution in single- and two-layer PEMFC diffusion medium Int. J Heat Mass Transfer ol. 46 pp [4] F. Y. Zhang X. G. Yang and C.-Y. Wang Liquid water remoal from a polymer electrolyte fuel cell J. Electroemical Society ol. 153 no. 2 pp. A225 A [5] P. Sinha and C.-Y. Wang Pore-network modeling of liquid water transport in gas diffusion layer of a polymer electrolyte fuel cell Electroimica Acta ol. 52 pp [6] M. Gan and L.-D. Chen Analytic solution for two-phase flow in PEMFC gas diffusion layer. Irine CA USA: ASME Fuel Cell 26 ASMEFC June 26. [7] D. Natarajan and T. V. Nguyen A two-dimensional two-phase multicomponent transient model for the cathode of a proton exange membrane fuel cell using conentional gas distributors J. Electroemical Society ol. 148 no. 12 pp. A1324 A [8] J. P. Owejan J. E. Owejan T. W. Tighe W. Gu and M. Mathias Inestigation of fundamental transport meanism of product water from cathode catalyst layer in PEMFCs. San Diego CA USA: 5th Joint ASME/JSME Fluids Engineering Conference July 27. [9] M. Krstic and A. Smyshlyae Boundary control of PDEs: A short course on backstepping designs SIAM to appear May 28. [1] B. A. McCain Modeling and analysis for control of reactant and water distributions in fuel cells Ph.D. dissertation Uni. of MI