Chemical Engineering Science 58 (2003 1977 1988 www.elsevier.com/locate/ces Moeling an analysis of hyrogen permeation in mixe proton electronic conuctors Lin Li a;b;1, Enrique Iglesia a;b; a Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720-1462, USA b Division of Materials Sciences, E.O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA Receive 22 August 2002; receive in revise form 20 January 2003; accepte 27 January 2003 Abstract A rigorous moel for hyrogen permeation through ense mixe conuctors was erive using the formalism of non-equilibrium thermoynamics for various operating moes an process conitions. The concentrations of charge carriers were rigorously inclue in this moel through efect equilibria with the chemical environment at each membrane surface an through balance equations an a virtual pressure formalism within the membrane. Hyrogen permeation rates through proton electron hole mixe conuctors were simulate using this framework uner open-circuit, short-circuite, an applie potential operating moes. The sensitivity of H 2 permeation rates to the reuction oxiation potentials at each sie of the membrane an to the membrane properties (e.g. electron/hole iusivity, oxygen bining energy was examine in terms of the mobility an concentration of each charge carrier in orer to ientify rate-limiting steps for H 2 transport. These simulations showe that electronic transport controls H 2 permeation rates in proton electron hole mixe conuctors typically use for H 2 permeation, especially when hyrogen chemical potentials are signicantly ierent in the two sies of the membrane. These electronic conuction limitations arise from a region of very low electronic conuctivity within the membrane, cause by a shift in the preominant charge carriers from electron to holes with ecreasing hyrogen chemical potential. Uner these asymmetrical conitions, H 2 permeation rates increase more markely when an external electron-conucting path is introuce than at lower chemical potential graients. Such interplay between rate-controlling variables leas to complex eects of H 2 chemical potential graients on permeation rates. The eects of intrinsic membrane properties on H 2 permeation were examine by systematic changes in the efect equilibrium constants. A ecrease in oxygen bining energy, manifeste in a stronger tenency for reuction of the oxie membrane material, leas to higher electron concentrations an to higher rates for open-circuit operation, uring which electron conuction limits H 2 transport rates.? 2003 Elsevier Science Lt. All rights reserve. Keywors: Moel; Simulation; Mass transfer; Membrane; Proton conuction 1. Introuction Metal oxies with mixe ionic electronic (ambipolar conuctivity have attracte consierable attention as ense membranes useful in separations, chemical reactors, an fuel cells. Most stuies of ionic transport through ense oxies have focuse on oxygen-permeable materials (Bouwmeester & Burggraaf, 1996, but high-temperature proton conuctors have been known for some time (Iwahara, 1996; Nowick & Du, 1996; Kreuer, 1997, 1999, 2000; Norby, 1999. The Corresponing author. Department of Chemical Engineering, University of California at Berkeley, Berkeley, CA 94720-1462, USA. Tel.: +1-510-642-9673; fax: +1-510-642-4778. E-mail aress: iglesia@cchem.berkeley.eu (E. Iglesia. 1 Current aress: UOP LLC, 25 East Algonquin R., Des Plaines, IL 60017, USA. efect chemistry an the proton conuctivity of these materials uner electrochemical conitions has been stuie, but few reports have aresse the ability of these materials to permeate hyrogen (Balachanran et al., 2001; Guan, Dorris, Balachanran, & Liu, 1998; Hamakawa, Hibino, & Iwahara, 2002; Li & Iglesia, 2001; Qi & Lin, 2000. As a result, kinetic an mechanistic escriptions of hyrogen permeation behavior through proton electronic mixe conuctors remain incomplete. Some theoretical guiance is available from treatments of oxygen transport (Bouwmeester & Burggraaf, 1996, but the conitions an the relevant transport mechanisms ier signicantly in these two systems. For example, both sies of oxygen transport membranes may be in contact with oxiizing streams. Uner such conitions, the require electronic conuction is carrie out by holes (p-type conuction. In 0009-2509/03/$ - see front matter? 2003 Elsevier Science Lt. All rights reserve. oi:10.1016/s0009-2509(0300057-5
1978 L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 contrast, H 2 permeation involves at least one sie expose to a reucing environment. This leas to very low hole conuctivities, because hole concentrations in acceptor-ope perovskites ecrease with ecreasing oxygen thermoynamic activity. These reucing or asymmetric (reucing in one sie an oxiizing in the other conitions lea to changes in rate-etermining charge carriers as a function of position within the membrane thickness. The characterization an preiction of the relevant rate processes require rigorous transport moels of greater complexity than those neee for oxygen transport systems. Recently, Norby an Larring (2000 reporte a hyrogen permeation moel for proton conuctors. Their moel escribes the behavior of such systems for open-circuit conitions, in which no net current ows across the membrane an the electrical potential graient equals the net chemical potential graients for all charge carriers. Hyrogen ux equations were erive by assuming electron transference numbers of unity (conuctivities much greater for electrons than for protons an then expressing the ux as a function of the proton conuctivity an the hyrogen partial pressure. The earth of permeation ata remains a signicant hurle to the valiation an improvement of available permeation moels. Simulations base on the Norby an Larring (2000 moel have not been compare with permeation ata; also, this moel is useful only for materials with high electronic conuctivity, for which the hyrogen ux is controlle only by proton transport. Yet, H 2 permeation ata inicate that electronic conuction is a kinetically relevant step in the overall transport process (Balachanran et al., 2001; Guan et al., 1998; Li & Iglesia, 2001; Qi & Lin, 2000. Proton conuction through ense membranes involves proton migration via hopping of hyrogen atoms among lattice oxygen atoms within the structure of perovskite materials, without concurrent motion of oxygen anions. Inee, immobile oxygens, generally associate with electronic insulators, are essential in orer to avoi oxygen migration an membrane over-reuction uring hyrogen permeation (Bouwmeester & Burggraaf, 1996; Tuller, 1981; Smyth, 1992. Also, hyrogen separation membranes must be operate at relatively low temperatures ( 1000 K in orer to prevent oxygen permeation an membrane reuction. Electronic conuctivities show higher activation energies than the corresponing conuctivities for protons (Tuller, 1981; Smyth, 1992; as a result, electron transport rates are typically lower than proton transfer rates, leaing to electron transference numbers signicantly less than unity. In all previous mass transfer moels of mixe conuctors (Norby & Larring, 2000; Guan et al., 1998; Qi & Lin, 2000, hyrogen permeation has been escribe in terms of proton an electronic conuctivities that are taken either as constant across the membrane cross-section or as empirical functions of H 2 partial pressure. In reality, these conuctivities, as ene, epen not only on the transport properties of the soli (i.e., charge carrier mobility, but also on the concentrations of each charge carrier. These concentrations, in turn, epen on the chemical composition of the contacting gas phase an on the position across the membrane thickness, because of the prevalent chemical potential graients responsible for net mass transport. As a result, conuctivities have retaine a qualitative character in all previous attempts at rigorous escriptions of net transport rates, because they o not reect merely the intrinsic properties of the materials, but epen also on conitions, the require etails of which are not always reporte or even measure in parallel experiments. Recently, Huggins (2001 iscusse charge carrier transport insie soli electrolytes by using efect equilibrium iagrams, but his analysis focuse on electrical properties of soli electrolyte material, an not on the rate of proton transfer processes. We propose here a comprehensive moel esigne to escribe hyrogen permeation behavior in proton electron mixe conuctor membranes. Such moels, when accurate an rigorous, provie essential guiance in the esign an unerstaning of mixe conucting materials. The moel propose uses the formalism of non-equilibrium thermoynamics in orer to provie rigorous an accurate escriptions of transport processes as a function of chemical environment an operating moes. The ientity of the limiting charge carriers is specically probe with the objective of guiing the search for materials that overcome transport bottlenecks in currently available conuctors. 2. Mass transfer moel for proton electronic mixe conuctor membranes In mixe conuctors, chemical an electrical potential graients co-exist an they act as the riving force for mass transport. Along the imension of net transport (x = membrane thickness, mass transfer rates per cross-sectional area (uxes for component k are given by (Bouwmeester & Burggraaf, 1996 J k = c kd k ( k x z kf x : (1 In Eq. (1, the rst an secon terms represent transport riven by chemical an electrical potential graients, respectively. If we choose component 1 as the basis component, the electrical potential graient = x can be expresse as a function of the chemical potentials of all other charge carriers. The ux for species of 1 then becomes (Appenix I J 1 = c 1D 1 + c 1D 1 I F n n z2 k c kd k c k D k z k c 1 D 1 1 x k x ; (2 where n = zkc 2 k D k : (3 k=1
L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 1979 Eq. (2 escribes uxes for each charge carrier as a function of both their chemical potential graient an those for all other species. This gives rise to a signicant practical hurle, because it is icult to measure the chemical potential of each charge carrier within the membrane material, as require for the solution of Eq. (2. Thus, we must escribe these potentials as a function of the concentrations an chemical potentials of the gas-phase species (e.g., H 2 ; H 2 O; O 2 in contact with each sie of the membrane. If both sies are brought in contact with the same gas mixture, the concentrations of all charge carriers will ultimately reach equilibrium with this gas phase, an their concentrations an chemical potentials at each point will epen only on the thermoynamics of the gas soli reactions responsible for these equilibria. In proton electron hole mixe conuctors, charge carriers inclue protons (OH., electrons (e, electron holes (h. an vacancies (VO For typical proton conuctors, such as SrCe 0:95 Yb 0:05 O 3, the gas soli reactions can be escribe by the following stoichiometric equations (Iwahara, 1996; Schober, Schilling, & Wenzl, 1996: K 1=2O 2 +V 1 O =O O +2h. ; (4 H 2 +2O O H 2 O+V O +O O K 2 = 2OH. +2e ; (5 K 3 = 2OH. : (6 These three reactions, however, are not inepenent, because the H 2 O; H 2 an O 2 partial pressures must satisfy the thermoynamics for the reaction H 2 + 1 2 O K w 2 =H2 O: (7 At each point within the membrane material, the charge carrier concentrations can be consiere to be in quasi-equilibrium with a set of virtual partial pressures of H 2 ; O 2, an H 2 O( j. These virtual partial pressures act as surrogates for the chemical potentials of these respective species at each point within the soli (Bouart & Djega-Mariaassou, 1984. Therefore, the ux equations can be expresse as functions of graients in the virtual pressures of hyrogen an water (Appenix A: j OH = OH i OH 2 j V = 2 V V ln( w +4 V V i + V V ( h h + e e ln( H 2 ( h h + e e ln( H 2 ; (8 ( h h + e e + OH ln( w : (9 Eqs. (8 an (9 are general ux equations that escribe mass transport in mixe conuctor membranes. These ux equations can be simplie to obtain expressions for specic membrane compositions, chemical environments, an operating conitions, as we escribe in etail below. 2.1. Open-circuit conitions For open-circuit conitions (I = 0, the uxes of protons an vacancies are given by ( j OH open = OH ( h h + e e ln( H 2 2 ln( w +4 V V ; (10 ( j V open = V V ( h h + e e + OH ln( w ( h h + e e ln( H 2 : (11 At steay state, j OH an j V must be constant at every point within the membrane an the above equations can be re-arrange to give (Appenix A (ln H2 (ln w = 4j V ( h h + e e 2j OH h h + e e + OH ; (12 OH h h + e e = j V V V 2j OH OH : (13 With appropriate bounary conitions, Eqs. (12 an (13 can be integrate to obtain j V an j OH. The charge carrier concentrations involve in these two equations can be obtaine from the interactions between gas-phase molecules an the membrane surface, as escribe in Section 3. 2.2. Short-circuite conitions For short-circuite conitions, potential ierences across the membrane become negligible because a connecting external conuctor provies a path of negligible resistance for electron transport (i.e. =x = 0. Then, we obtain from Eq. (1 (J OH close = c ( OHD OH OH ; (14 (J V close = c V D V ( V x x : (15 By analogy with the open-circuit case, uxes can be expresse as functions of the H 2 an H 2 O virtual partial pressure graients within the membrane an then re-arrange to give (ln H2 (ln w = 2j OH OH ; (16 = j V V V 2j OH OH : (17 Eqs. (16 an (17 can then be integrate in orer to obtain the uxes, after applying appropriate bounary conitions
1980 L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 an introucing a relation between charge carrier concentrations (c V an c H an gas-phase concentrations (p H2 ;p w. 2.3. Electrochemical pumping The membrane system can also be operate as an electrochemical pump by applying a irect voltage across the two sies. In this case, Eqs. (8 an (9 are require in orer to escribe the ux. Comparing Eqs. (8 an (9 with Eqs. (10 an (11 an ening the transference numbers for protons (t OH, vacancies (t V, an electron/holes (t el as t OH = c OHD OH t V = c V D V = OH ; (18 = V V ; (19 t el = c ed e + c h D h = e e + h h ; (20 the uxes across the membrane become j OH = t OH i +(j OH open (21 j V = t V i +(j V open (22 In Eqs. (21 an (22, the rst an secon terms reect mass transport riven by electrical potential graients impose by an external voltage an by chemical potential graients impose by ierent gas compositions in the two sies of the membrane, respectively. When proton transference numbers are inepenent of applie voltage, the ux epens linearly on the current i with a slope equal to t OH an a y-intercept given by the open-circuit hyrogen ux at a given chemical potential graient across the membrane. In many cases, the proton transference number epens on the applie voltage, because membrane materials are semiconuctors, an the relationship between the ux j OH an the current ensity i becomes non-linear (Iwahara, 1999; Li & Iglesia, 2001. Fig. 1 shows the expecte epenence of the hyrogen ux on the current across the membrane for ierent operating moes. The soli an ashe lines in Fig. 1 represent ux behavior for membranes with proton transference numbers of one an less than one, respectively. For open-circuit conitions, the proton ux for membranes with proton transference numbers near unity woul be zero, corresponing to point O in Fig. 1. For proton transference numbers below unity, proton transfer will occur even uner open-circuit conitions (point A in Fig. 1, but the total net current across the membrane is zero, because currents carrie by protons an electrons are equal but opposite in irection. Uner short-circuite conition, all electrons are transporte via the external circuit, because its resistance is much smaller than that of the membrane. In this case, the net current across the membrane is equal to the proton current, an hyrogen permeation rates become inepenent of the proton transference number; the two lines which represent the ux for membranes with t OH = 1 an 1 cross at point B in Fig. 1. (J short-circuite (J open J H2 A O I OH=-I e B (I short-circuite I/zF Fig. 1. Hyrogen ux uner various operation moes. The soli an ashe lines represent the ux of membranes with t OH = 1 an 1, respectively. An external voltage can lea to higher proton uxes. Eq. (21 shows that proton uxes uner electrochemical pumping conitions are linear functions of the current when t OH is inepenent of applie voltage an the proton uxes can become signicantly greater than that uner short-circuite conitions. A combination of hyrogen permeation rate measurements in these various operating moes can be use to etermine the ientity of the rate-controlling charge carriers an their respective electronic transference numbers. For example, comparisons of uxes uner open-circuit an short-circuite conitions give the average proton transference number t OH (the slope of line AB in Fig. 1. 2.4. The relationship between uxes of charge carriers (J OH ;J V an of molecules (J H2, J w an J O2 In general, the motion of vacancies an protons leas to transport of H 2 ; O 2 an H 2 O across the membrane (the other two charge carriers, electrons an holes, o not irectly contribute to the transport of molecular or atomic species. The specic origin of the proton cannot be rigorously establishe (from H 2 or H 2 O because of the H 2 H 2 O equilibrium. Vacancies can react with either O 2 or H 2 O. However, the mass balance requires that J OH =2J H2 +2J w ; (23 J V = 2 J O2 J w : (24 The irection an magnitue of J H2 ;J W an J O2 epen on: (1 the chemical environment in each sie of the membrane (e.g. H 2 ; H 2 O an O 2 pressures; (2 the membrane properties (the thermoynamics for reactions (4 (7; an (3 the mobility of charge carriers (D V ;D OH within the membrane material.
For selective hyrogen permeation, the transport rates of vacancies must be much lower than that of protons. Then, J V 0 an J w 0, an J H2 =1=2J OH : (25 L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 1981 3. Charge carrier concentrations in mixe conuctors For mixe conuctors, the gas-phase composition in- uences charge carrier concentrations an their graients across the thickness of the membrane material. In orer to rigorously account for these changes as a function of gas-phase environment an location within the membrane, we must relate the concentration of each charge carrier to one another an to the external environment at each membrane surface. Simulations of hyrogen permeation using the approach escribe above require that we estimate charge carrier concentrations as a function of H 2 O an H 2 partial pressures. For mixe conuctors, these concentrations are obtaine by consiering the soli gas reactions in Eqs. (4 (7. In aition, electronic equilibrium requires that: 0 =h. Ke +e; (26 an electrical neutrality imposes the aitional constraint 2c V + c OH + c h = c opant + c e ; (27 where c opant is the opant concentration in the membrane. In SrCe 0:95 Yb 0:05 O 3 membrane, Yb acts as the opant use in orer to increase the electronic conuction of these materials. The equilibrium between gas phase an membrane shoul be escribe using chemical potential, which poses a signicant challenge. However, Kreuer (1999 propose that interactions with gas-phase species can be escribe by assuming ieal behavior for all species involve in efect formation for cubic an istorte cubic perovskites. Therefore, we treate gas phase membrane interaction as an ieal system, an we can then solve the concentrations of four charge carriers for a given set of H 2 an H 2 O partial pressure by combining the equilibrium of Eqs. (4 (7 with the electrical neutrality conition (Eq. (26 (Appenix B. The charge carrier concentrations epen on the equilibrium constants for reactions (4 (7 an (26 (K 1 ; K 2, K w an K e for a given membrane composition (c opant = constant an temperature; they can be expresse in this manner as functions of p H2 an p w. Fig. 2 shows the concentrations of these four charge carriers with changes in p H2 an p w for SrCe 0:95 Yb 0:05 O 3 at 1000 K. The equilibrium constants, K 1 ;K 2 ;K w an K e reporte by Schober, Frierich, an Conon (1995 were use in these calculations. Water partial pressures inuence proton concentrations more strongly than H 2 partial pressures, an the eect of the latter becomes more pronounce at high H 2 pressures (p H2 100 Pa. In contrast, p H2 strongly inuences electron an hole concentrations over the entire range of Fig. 2. Charge carrier concentration as functions of hyrogen an water partial pressure (SrCe 0:95 Yb 0:05 O 3 ; 1000 K: (a proton concentration; (b electron concentration; (c hole concentration; an ( vacancy concentration. partial pressures. Electronic conuction is provie by electrons for reucing environments (high p H2 an by holes for less reucing conitions (low p H2. Therefore, the chemical composition of the gas streams inuences not only the mass transfer riving force but also the charge carrier concentrations within the membrane, an hence the conuctivities of charge carriers, as ene. Conuctivities are also inuence by the equilibrium constants, the values of which epen on the membrane material. For example, a higher value of K 2 inicates a higher membrane anity for hyrogen, while higher K 1 values reect higher membrane anities for oxygen. By
1982 L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 0.07 0.06 0.05 0.04 Close circuit 10 2 j OH 0.03 10 1 0.02-1 10 0.01 γ e =γ h =10-2 0.00 0 10 20 30 40 50 ln (p H2,1 /p H2,2 Fig. 3. Eect of electron/hole iusivity on proton ux through SrCe 0:95 Yb 0:05 O 3 membrane; p H2 ;1 =0:1 MPa;p w =10 6 MPa; T = 950 K. Fig. 2. continue. incorporating the epenence of charge carrier concentrations on H 2 an H 2 O partial pressures into the moel equations erive in the previous section, hyrogen permeation rates can be preicte for mixe conucting membranes. The resulting simulations account rigorously for the eects of operating conitions on both the mass transfer riving force an the concentrations of charge carriers. 4. Simulation of hyrogen permeation behavior For mixe conuctors, the proton ux across the membrane is etermine by the concentration an iusivity of protons an by the electron transfer ability of the material, which is usually expresse as its electronic conuctivity. As we will see later, the electronic conuctivity is a function of position in the membrane because it epens on the electron concentration, which also varies with position. The eective electron-conucting ability of the membrane, which can be ene as an average electronic conuctivity, epens on: (i the concentrations of electrons an holes in the membrane an (ii the iusivities of electrons an holes. The concentrations of electrons an holes are etermine by the properties of the materials (e.g., reucibility an by the operating conitions. In this section, we examine the eects of membrane reucibility an iusivity on proton uxes. Fig. 3 shows the eects of electron/hole iusivity on proton ux uner open-circuit an short-circuite conitions; the latter correspons to very high electron/hole iusivities (provie by the external circuit. For a given mass transfer riving force (ln(p H2;1=p H2;2, the proton ux (j OH increases with increasing of electronic mobility, because charge neutrality requires the concurrent motion of protons an electron/holes. When e an h are very high ( 10 2, aitional increases in electronic mobility o not inuence H 2 permeation rates, because uxes become limite by proton mobility. The relationship between the proton ux (j OH an the chemical potential riving force for mass transfer {ln(p H2;1=p H2;2} epens on the electronic mobility of the membrane. In general, at low values of ln(p H2;1=p H2;2( 5, both sies of the membrane are expose to reucing mixtures (symmetric operating moe an electronic conuction occurs preominately via electron transport. Higher ln(p H2;1=p H2;2 values will lea to higher proton transport rates, because average electronic transference numbers will be largely inepenent of ln(p H2;1=p H2;2. At suciently high ln(p H2;1=p H2;2 values, however, the
L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 1983 1.0 γ e =γ h =10 1 γ e =γ h =10 2 1.0 0.8 0.8 K 1 =5x10-7 0.6 t el 0.6 γ e =γ h =10 2 t el 0.4 K 1 =5x10-6 0.4 0.2 K 1 =5x10-5 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 4. Eect of electron/hole iusivity on electronic transference number istribution insie membrane; p H2 ;1 =0:1 MPa;p w =10 6 MPa; T = 950 K; ln(p H2 ;1=p H2 ;2 = 45. high oxygen activity in the permeate sie leas to very low electron concentrations an to low average electronic transference numbers. As a result, the higher mass transfer riving force impose by these high ln(p H2;1=p H2;2 values will be partially oset by a concurrent ecrease in the electronic transference number. Ultimately, high values of ln(p H2;1=p H2;2 lea to an increase in the electronic transference number an to higher uxes, because holes become the preominant charge carriers, at least in the membrane regions near the permeate sie. The eects of electron/hole iusivity can also be examine by calculating electronic transference numbers (Eq. (20 as a function of position within the membrane. Fig. 4 shows that t el epens strongly on position within the membrane for asymmetrical conitions {ln(p H2;1=p H2;2=45}.At these conitions, the membrane regions near the process sie interact with a highly reucing environment an electrons are the preominant electronic carriers. As increases, the virtual H 2 partial pressure ecreases an the lower electron concentrations lea to lower electronic transference numbers. After reaching a minimum value, electronic transference numbers increase with increasing because the permeate sie is expose to oxiizing conitions, which lea to higher hole concentrations an more eective electronic conuction. This conclusion is consistent with electronic conuctivity measurements reporte by Hamakawa, Hibino, an Iwahara (1994. Uner asymmetrical conitions, there is a transition region within which t el is very low an electron conuction etermines the electronic conuctivity an the hyrogen permeation rates. As the electron/hole mobility increases, the minimum t el increases an the low t el tran- ξ 0.0 0.0 0.2 0.4 0.6 0.8 1.0 ξ Fig. 5. Eect of reucibility (K 1 on electronic transference number istribution insie membrane; p H2 ;1 =0:1 MPa, p w = 10 6 MPa; T = 950 K; ln(p H2 ;1=p H2 ;2=45; e = h =10 2 ; V =10 5. sition region becomes narrower; as a result, the average t el for the membrane increases. Symmetrical conitions lea to reucing environments throughout the membrane thickness an the t el prole within the membrane is given by the left region in Fig. 4. Within this region, t el ecreases with increasing because low H 2 virtual pressures lea to low concentrations of electrons; no increase in the electronic transference number is observe because the H 2 virtual pressure is never suciently low for holes to become the preominant charge carriers. The average t el is higher than uner asymmetrical conitions an the eects of varying the electronic conuctivity or of closing an external circuit on proton uxes are stronger for asymmetrical (ln(p H2;1=p H2;2 40 than for symmetrical (ln(p H2;1=p H2;2 4 conitions. Our simulations also show that the eects of the chemical nature of the permeate stream epen on the membrane reucibility an on the mobility of charge carriers. When electronic mobility is low, hyrogen permeation rates are controlle by electronic conuctivity. In this case, a shift from reucing to oxiizing conitions in the permeate sie (symmetrical to asymmetrical moes leas to small eects in permeation rates. When the electronic conuctivity is high, either because of the properties of the membrane properties or because of an external circuit, more oxiizing environments in the permeate sie lea to larger permeation rates, because the corresponing increase in permeation riving force is not oset by a concurrent ecrease in electronic conuctivity as a result of changes in the permeate stream. The reucibility of the membrane material (given by K 1 in our moel also inuences hyrogen permeation rates. Higher values of K 1 reect stronger oxygen bining an less reucible membrane materials. Fig. 5 illustrates the eect of varying K 1 on the electronic transference number t el.ask 1 increases, electronic transference numbers within the more
1984 L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 0.20 0.008 0.18 0.16 j OH 0.14 0.12 0.10 K 1 =5x10-5 0.006 K 1 =5x10-6 K 1 =5x10-7 0.08 0.06 0.04 0.02 K 1 =5x10-6 K 1 =5x10-7 j OH 0.004 0.002 K 1 =5x10-5 0.00 0 10 20 30 40 50 ln (p H2,1 /p H2,2 Fig. 6. Eect of reucibility on proton ux uner short-circuite conitions; p H2 ;1 =0:1 MPa; p w =10 6 MPa; T = 950 K. reucing regions of the membrane (near the process sie ecrease because of a ecrease in electron concentrations. The location of the minimum electronic transference number moves towars the process sie, suggesting that the region within which electrons are the ominant charge carriers becomes thinner. For the range of K 1 values we examine, the average t el within the membrane ecreases with ecreasing of membrane reucibility. However, further increase of K 1 may lea to higher average t el, because holes ultimately become the preominant charge carriers within a large fraction of the membrane thickness. These nings suggest that for typical mixe conuctors, such as SrCe 0:95 Yb 0:05 O 3, hyrogen permeation rates are largely controlle by electron conuction. Higher electronic conuctivities will require higher concentrations or iusivities of electrons or holes. In oxygen transport membranes, holes are the preominant charge carriers throughout the membrane thickness. In H 2 transport processes, at least one membrane surface is in contact with a strongly reucing environment; therefore, hole conuction cannot preominate throughout the entire membrane thickness. One potential solution is to increase electron concentration by selecting more reucible membrane materials. Reuction of membrane materials uring operation, however, can lea to changes in the perovskite structure an to loss of mechanical integrity; thus, any such improvements in proton transport by increasing membrane reucibility is likely to be limite by these practical consierations. In aition to its eects on electron/hole concentrations an t el, the reucibility of the membrane also inuences proton concentrations an hyrogen permeation rates. Figs. 6 an 7 show the eect of K 1 on proton ux uner short-circuite an open-circuit operating moes, respectively. Uner short-circuite conitions, j OH increases with increasing of K 1, an the sensitivity j OH to the riving force term ln(p H2;1=p H2;2 increases, inicating that a given mass 0.000 0 10 20 30 40 50 ln (p H2,1 /p H2,2 Fig. 7. Eect of reucibility on proton ux uner open-circuit conitions; p H2 ;1 =0:1 MPa;p w =10 6 MPa; T=950 K; e = h =10 2 ; V =10 5. transfer riving force becomes more eective for H 2 transport as K 1 increases, because of the higher prevalent proton concentrations. In open-circuit moe, hyrogen permeation rates are largely controlle by electronic conuction. Therefore, higher values of K 1 lea to lower proton uxes, as a result of the lower average electron concentrations prevalent as the membrane material becomes more icult to reuce. 5. Usefulness an limitations of hyrogen transport moel The rigorous moel presente here escribes the transport properties of mixe conuctors using the iusivities an concentrations of each charge carrier as the funamental relevant parameters responsible for net macroscopic motion. In oing so, it avois the use of lumpe phenomenological parameters, such as conuctivities, which are appropriate for ieal electrical conuctors, but not for the transport of chemical species through mixe conuctors. The approach presente here also inclues rigorous equilibria between the chemical environments in the process an permeate sie an the efect species present at each membrane surface. As a result, the simulations implicitly contain the eects of these chemical environments on both the riving force for mass transfer an the concentrations of charge carriers. One remaining issue, not currently treate in this moel, is the potential eect of the chemical environment in the process an permeate sie on the intrinsic mobility (iffusivity of each charge carrier. Such eects may be important an icult to measure, an they cannot be rule out. We suggest, base on heuristic arguments an previous ata, that they are not critical in etermining transport rates. First, the concentrations of efects are small within typical membrane materials; therefore, non-iealities
L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 1985 arising from efect efect interactions are unlikely to arise. Also, Yajima an Iwahara (1992a, b have foun that measure proton mobilities o not epen on proton concentrations for SrCe 0:95 Yb 0:05 O 3. When require, such eects must be escribe in terms of the atomic-level bon-making an bon-breaking mechanisms responsible for migration of hyrogen an oxygen species within such solis. Recent examples show the emerging feasibility of such funamental approaches for all charge carriers in mixe conuctors (Kreuer, 1996, 1997, 1999, 2000, but etaile knowlege of charge carrier transport within these solis remains very limite. External bounary layer or surface asorption resistances can easily be incorporate into the hyrogen permeation moel escribe here. Both of these eects are occasionally important for oxygen transport membranes (Bouwmeester & Burggraaf, 1996. Bounary layer eects have been neglecte here for H 2 transport membranes because H 2 molecules iuse much more rapily than O 2 molecules through a gas-phase bounary layer, while membrane permeation rates in practical applications are generally similar for both molecules. Dissociative chemisorption of H 2 is expecte to occur more rapily than for O 2, because of the smaller bon energies in H 2. Also, H 2 transport membranes must be operate below 1000 K in orer to avoi oxygen transport an membrane over-reuction; as a result proton uxes ten to be lower than for O 2 transport membranes, which operate at much higher temperatures ( 1300 K. Finally, our recent stuies have shown that even for very thin SrCe 0:95 Yb 0:05 O 3 ( 10 m lms, the proton ux remains proportional to the inverse of the lm thickness (Hamakawa, Li, Li, & Iglesia, 2002; thus, surface processes remain kinetically irrelevant even for very high H 2 permeation rates. In some sense, this reects the lower permeation rates for H 2 in proton electronic conuctors at their typical temperatures (1 bar H 2 ; 950 K; 5 3 Scm 1 for SrCe 0:95 Yb 0:05 O 3 ; Iwahara, 1996 than for O 2 in oxie-electronic conuctors (0:2 bar O 2 ; 1273 K; 0:5 Scm 1 ; Bouwmeester & Burggraaf, 1996. Thus, it appears that the inclusion of external transport an asorption kinetics is not require for currently available proton mixe conuctor membranes. As H 2 permeation rates continue to increase with avances in the hyrogen permeability of these materials, both external transport an H 2 chemisorption kinetics must be rigorously escribe in orer to replace the equilibrium between the surface an the bulk ui assume within the current moel. 6. Conclusions A mass transfer moel for hyrogen permeation through proton electronic mixe conuctors was evelope by using the formalism of non-equilibrium thermoynamics an escribing the concentrations of all charge carriers through equilibrium with the contacting gas phase at surfaces an through a virtual pressure formalism within the membrane material. This moel was use to escribe H 2 permeation through ense SrCe 0:95 Yb 0:05 O 3 membranes. These simulations conrme the rate-controlling role of electronic transport in H 2 permeation processes, especially when the membrane is operate at very ierent hyrogen chemical potentials in the process an the permeate sies. Uner such asymmetric operating conitions, an internal region with very low electronic transference number arises from a transition from electronic to hole conuction as the hyrogen chemical potential within the membrane ecreases from high values near the process sie to much lower values near the permeate sie. As a result, when electronic mobility is high, either because of the nature of the membrane material or because of the presence of an external conucting circuit, large H 2 chemical potential graients across the membrane exploit the full riving force without the averse eects of the lower electronic transport rates associate with such chemical potential graients. When electronic transport limits H 2 permeation rates, the full eects of these large chemical potential graients cannot be exploite, because of their negative eects on electron transport rates. Improvements in electronic (an H 2 transport rates require higher electron/hole iusivities or more reucible membrane materials, which increase electron concentrations for a given chemical environment. Simulation results illustrate the nature an the unerlying mechanism for the observe eects of the chemical environment in each sie of the membrane on H 2 permeation rates an how such eects epen on membrane properties, such as reucibility an charge carrier mobility. Heuristic arguments an experimental results are use to neglect the eects of external mass transport an of H 2 issociation rates on the rate of H 2 permeation through typical mixe proton electronic conuctors. These external transport an asorption eects can be reaily incorporate into the current framework when the experimental situation requires it. Notation c h C T D k F i I I k j k J k K i L p k concentration of charge carrier k, mol m 3 molar concentration of membrane (given by membrane ensity ivie by molecular weight, mol m 3 iusivity of charge carrier k; m 2 s 1 Faraay constant C mol 1 Dimensionless current ensity total current ensity, A m 2 current ensity carrie by charge k; Am 2 imensionless ux of component k, ene by Eqs. (A.19 an (A.20 ux of component k; mol m 2 s 1 equilibrium constant, ene by Eqs. (B.1 (B.4 membrane thickness, m partial pressure of component k, Pa
1986 L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 t k x z k transference number of charge carrier k istance along membrane thickness, m charge on species k Greek letters k imensionless mobility of charge carrier k, ene by Eqs. (A.22 (A.24 k imensionless concentration of charge carrier k, e- ne by Eqs. (A.25 (A.28 k electrochemical potential of species k; J mol 1 ene by Eq. (A.30 ene by Eq. (3 k chemical potential of species k; J mol 1 imensionless istance, = x=l k virtual partial pressure of component k, Pa electrical potential, V Subscripts h holes e electrons OH proton efect V vacancy w water an the total current across the membrane is given by n n I = I k = I 1 + z k FJ k : (A.4 k=1 Substituting the expression for J k into Eq. (1, we obtain n n c k D k z k k n z k J k = x c k D k zk 2 F x ; (A.5 an rearranging provies an expression for the electrical potential graient: n F x = z kj k + n n c k D k z 2 k c k D k z 2 k k x : (A.6 Substituting Eqs. (A.3, (A.4, an (A.6 into Eq. (1 we obtain Eq. (2. From Eqs. (4 (7, H2 =2 OH +2 e ; (A.7 1=2 O2 + V =2 h ; (A.8 w + V =2 OH ; (A.9 Acknowlegements This work was supporte by the Division of Fossil Energy of the U.S. Department of Energy (Contract No. DE-AC03-76SF00098 uner the technical supervision of Dr. Daniel Driscoll. Appenix A. Derivation of ux equations uner various conitions In mixe conuctors, both chemical an electrical potential graients can exist an act as the riving force for mass transport. In one imension (x=membrane thickness imension, the mass transfer rate per cross-sectional area (ux of component k is given by (Bouwmeester & Burggraaf, 1996 J k = c kd k k x ; (A.1 where k, the electrochemical potential of component k, contains both the chemical ( k an the electrical potentials ( riving the transport of molecules across the membrane: k = k + z k F: (A.2 Substituting Eq. (A.2 into Eq. (A.1, we obtain Eq. (1. The charge current carrie by component k is I k = z k FJ k ; (A.3 H2 +1=2 O2 = w : (A.10 Base on Eqs. (A.7 (A.10, the ux equations can be expresse as functions of graients of chemical potential of hyrogen an water: J OH = c OHD OH J V = 2c V D V I F c OHD OH (c h D h + c e D e 1 H2 2 x +2c w V D V ; (A.11 x I F + c V D V (c h D h + c e D e + c OH D OH w x (c hd h + c e D e H 2 x : (A.12 The chemical potential of H 2 an H 2 O can be calculate in terms of their respective virtual pressures: H2(g = 0 H 2 + ln( H2 ; w(g = 0 w + ln( w ; H2 x = 1 H2 H2 x ; w x = 1 w w x : (A.13 (A.14 (A.15 (A.16
L. Li, E. Iglesia / Chemical Engineering Science 58 (2003 1977 1988 1987 Then the ux equations can be expresse as functions of hyrogen an water partial pressure graients: J OH = c OHD OH I F c OHD OH (c h D h + c e D e 2 1 H2 J V = 2c V D V H2 x +4c 1 w V D V w x I F + c V D V w x (c hd h + c e D e 1 H2 ; (A.17 (c h D h + c e D e + c OH D OH 1 H2 x w : (A.18 Eqs. (A.17 an (A.18 can be nonimensionlize to Eqs. (8 an (9 by ening j H = j V = i = J OHL C T D OH ; J V L C T D OH ; IL C T D OH F ; h = D h =D OH ; e = D e =D OH ; V = D V =D OH ; h = c h =C T ; e = c e =C T ; V = c V =C T ; (A.19 (A.20 (A.21 (A.22 (A.23 (A.24 (A.25 (A.26 (A.27 The above equations can be non-imensionlize as ( j OH close = OH 2 ( j V close = V V ln( w ln( H2 ; (A.33 ln( H2 V V : (A.34 Eqs. (A.33 an (A.34 can be rearrange to give Eqs. (16 an (17. Appenix B. Calculation of charge carrier concentrations The corresponing equilibrium expressions for reactions (4 (7 an (26 are given by K 1 = ch=c 2 V p 1=2 O 2 ; (B.1 K 2 = c 2 OHc 2 e=p H2 ; K w = p w =p H2 p 1=2 O 2 ; K e = c h c e : (B.2 (B.3 (B.4 Base on Eqs. (B.1 (B.4, a non-linear equation is obtaine for the proton concentration within the membrane: c 3 OH + ac 2 OH + bc OH + c =0; where a = K 1K 2 p H2 p 1=2 O 2 2Ke 2 (B.5 + K 1p 1=2 O 2 K2 p H2 2K e ; (B.6 K 1 K 2 p H2 p 1=2 O b = c 2 opant 2Ke 2 ; (B.7 c = K 1p 1=2 O 2 (K 2 p H2 3=2 : (B.8 Then, 2K 2 e OH = c OH =C T ; = x=l; (A.28 (A.29 c e = c h = K2 p H2 c OH ; (B.9 K ec OH K2 p H2 ; (B.10 = =(C T D OH = h h + e e + OH +4 V V ; (A.30 where L the membrane thickness an C T is the mole concentration of the membrane. For short-circuite conitions, using Eqs. (A.13 (A.16, Eqs. (14 an (15 can be expresse as (J OH close = c OHD OH 2 H2 (J V close = c V D V w H2 x ; w x c V D V H2 H2 x : (A.31 (A.32 c V = K 2 e c 2 OH K 2 p H2 K 1 p 1=2 O 2 : (B.11 References Balachanran, U., Lee, T. H., Zhang, G., Dorris, S. E., Rothenberger, K. S., Howar, B. H., Morreale, B., Cugini, A. V., Siriwarane, R. V., Poston Jr., J. A., & Fisher, E. P. (2001. Development of ense ceramic membranes for hyrogen separation. Stuies in Surface Science an Catalysis, 136, 465 470. Bouart, M., & Djega-Mariaassou, G. (1984. Kinetics of heterogeneous catalytic reactions. Princeton, NJ: Princeton University Press.
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