Thermal-Electrochemical Modeling of Battery Systems

Transcription

1 90 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Thermal-Electrochemical Modeling of Battery Sytem W. B. Gu* and C. Y. Wang**,z GATE Center for Advanced Energy Storage, Department of Mechanical and Nuclear Engineering, The Pennylvania State Univerity, Univerity Par, Pennylvania 680, USA A general form of the thermal energy equation for a battery ytem i derived baed on firt principle uing the volume-averaging technique. A thermal-electrochemical coupled modeling approach i preented to imultaneouly predict battery electrochemical and thermal behavior. Thi approach couple the thermal energy equation with the previou multiphae micro-macrocopic electrochemical model via the heat generation and temperature-dependent phyicochemical propertie. The thermal-electrochemical model i multidimenional and capable of predicting the average cell temperature a well a the temperature ditribution inide a cell. Numerical imulation are performed on a Ni-MH battery to demontrate the ignificance of thermal-electrochemical coupling and to invetigate the effect of thermal environment on battery electrochemical and thermal behavior under variou charging condition. 000 The Electrochemical Society. S (00) All right reerved. Manucript ubmitted February 3, 000; revied manucript received May 8, 000. A a follow-up of previou wor,, the preent wor i intended to develop a thermal and electrochemical coupled model capable of predicting the patial ditribution and temporal evolution of temperature inide a battery. It i nown that temperature variation inide a battery may greatly affect it performance, life, and reliability. Battery phyicochemical propertie are generally trong function of temperature. For example, the equilibrium preure of hydrogen aborption-deorption, which ignificantly affect the open-circuit potential of the metal hydride electrode and hence the performance of nicel metal hydride batterie, i trongly dependent on temperature. 3 Capacity loe occur at low temperature due to high internal reitance and at high temperature due to rapid elf-dicharge. 4 Therefore, a proper operating temperature range i eential for a battery to achieve optimal performance. In order to prolong the battery cycle life, balanced utilization of active material i deired, which require a highly uniform temperature profile inide the battery to avoid localized degradation. More important, the battery temperature may increae ignificantly due to the elf-accelerating characteritic of exothermic ide reaction uch a oxygen reaction in aqueou batterie, eventually cauing thermal runaway. 5-8 An optimal operating range and a high uniformity in the internal temperature ditribution contitute two thermal requirement for a battery to operate afely. Thee two are particularly important for advanced electric-vehicle batterie becaue of their high energy and power denitie, large ize, and high charge and dicharge rate. Although experimental teting and microcalorimetric meaurement 9- are neceary to obtain battery thermal data for deign and optimization, a mathematical model baed on firt principle i capable of providing valuable internal information to help optimize the battery ytem in a cot-effective manner. In general, a battery thermal model i formulated baed on the thermal energy balance over a repreentative elementary volume (REV) in a battery. The differential equation that decribe the temperature ditribution in the battery tae the following conervation form,3 cp vt T q accumulation convection conduction heat generation where T i the temperature, v i velocity vector of the electrolyte, q i the volumetric heat generation rate, and, c p, and are the volume-averaged denity, pecific heat, and heat conductivity of the REV, repectively. The thermophyical propertie can be aniotropic due to the inhomogeneity of battery component. ** Electrochemical Society Student Member. ** Electrochemical Society Active Member. * z cxw3@pu.edu [] In batterie with flowing electrolyte, the convection term play an important role. However, it can be neglected in mot tationary batterie, and then Eq. i reduced to the tranient heat conduction equation. Subect to proper boundary condition, Eq. and it variou implified form have been olved to obtain the temperature ditribution in lead-acid, 4-6 nicel-hydrogen, 7 lithium-polymer, 8- and lithium-ion 8, battery module, with a ingle cell a the minimum REV. When the lumped-parameter approach i applicable (cf. Eq. 40) or only the average cell temperature i deired, the conduction term can be further diminihed by integrating boundary condition into the equation. The reulting time-dependent ordinary differential equation (i.e., Eq. 4) ha been widely ued in lead-acid, 3-5 nicel-hydrogen, 6,7 lithium-polymer, 8,9 and lithium-ion 30 battery model. For a thermal model a battery can be thermally and electrochemically coupled or decoupled, depending on how the heat generation term i treated. During battery operation, the heat generation rate depend not only on the cell temperature but alo on charge or dicharge regimen. A fully coupled model ue newly produced current and potential information from the model to calculate the heat generation rate and hence temperature ditribution, which in turn determine the current and potential,,5,7 wherea a decoupled model may employ empirical equation (e.g., the Shepherd equation) decribing experimental battery charge/dicharge curve of different rate at contant temperature. 8-0, The decoupled model i much impler, but accurate only when battery performance i inenitive to temperature. The complexity of the coupled model can be ignificantly reduced by the partially coupled approach propoed by Pal and Newman; 7 that i, etimating the heat generation rate during noniothermal dicharge from that obtained at contant temperature from an iothermal cell model. In other word, heat-generation rate are approximated to be independent of dicharge hitory. The heat generation rate depend on the thermodynamic propertie of the reaction proceeding in a cell, the potential-current characteritic of the cell, and the rate of charge and dicharge. By utilizing the firt law of thermodynamic for an iobaric battery ytem, Bernardi et al. 3 gave a general energy balance equation for a cell in which the rate of heat generation wa given by q I U T U IV enthalpy-of-mixing term T av av phae change term [] where I i the volumetric partial reaction current reulting from electrode reaction, U av i the correponding open-circuit potential (OCP) with upercript av referring to the value evaluated at the average compoition, I the total current in the unit of A/cm 3, and V

2 Journal of The Electrochemical Society, 47 (8) 90-9 (000) 9 S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. the cell potential. The firt term on the right ide (RS) of Eq. repreent the enthalpy of charge-tranfer reaction. The econd term tand for the electrical wor done by the battery. The third term or the enthalpy-of-mixing term repreent the heat effect aociated with concentration gradient developed in the cell. The lat term or the phae-change term tand for the heat effect due to phae tranformation. In arriving at Eq., the energy balance wa performed over the entire cell with the aumption of uniform cell temperature. When a cell i thin and the end effect are negligible, the uniformity in temperature i a good approximation. Apparently, when multiple electrode reaction occur imultaneouly, the partial current of each reaction mut be nown in order to calculate the heat generation rate uing Eq.. For a battery ytem that involve an inertion reaction, uch a lithium-baed and nicel-baed batterie, the OCP i a trong function of the local tate of charge (SOC), which i often controlled by olid-tate pecie diffuion. During operation at high rate, the pecie concentration ditribution in a cell could be highly nonuniform and reult in nonuniform electrochemical reaction rate. Neglecting enthalpy-of-mixing and phae-change term, Rao and Newman 3 mot recently preented a general energy balance equation for inertion battery ytem, in which the rate of heat generation i written a q a i U T U dv IV n c where a i the pecific urface area active for electrode reaction, in the tranfer current denity due to reaction, and U the local OCP of reaction. A imilar expreion of the heat generation rate wa recently preented by De Vidt et al. 7 for a nicel-hydrogen cell, with the preure wor additionally taen into account. Unlie Eq., Eq. 3 relate the heat generation to the local electrochemical reaction rate and the local OCP and thu i capable of calculating heat generation rate when cell undergo relaxation under dynamic condition. 3 It i expected that thermal runaway i firt triggered by the hottet pot in a cell. There i thu a need to predict the temperature ditribution within a cell in order to capture the thermal runaway proce. Baed on overall heat balance of a cell, both Eq. and 3 become inadequate when the temperature profile inide a cell i deired. In the next ection, a thermal energy equation capable of decribing the internal temperature ditribution of the cell i developed baed on firt principle uing the volume-averaging approach. A fully coupled thermal and electrochemical model i then developed by coupling the thermal equation with the previou multiphae tranport and electrochemical model. Simplification to variou calculation of the heat generation rate are dicued. The full capability of the model to predict temperature ditribution inide cell ha been demontrated elewhere. 49 In thi paper, numerical imulation baed on a lumped thermal model are carried out to examine the ignificance of thermal and electrochemical coupling and to invetigate the effect of thermal environment on the electrochemical and thermal behavior of a Ni- MH (metal hydride) cell under variou charging mode. Model Development Conider an electrochemical ytem coniting of porou electrode, an electrolyte, and a ga phae. The electrolyte can be either liquid or olid. The ga phae i preent in batterie due to ga generation accompanying the primary reaction. General thermal energy equation. For a multicomponent ytem uch a electrolytic olution, a general differential equation of thermal energy balance ha been deduced baed on firt principle. 33,34 With the aumption of negligible heat effect due to vicou diipation and preure wor, no body force, and no homogeneou chemical reaction, thi general energy equation i reduced to cp vt q H J ˆ pecie [3] [4] where and c p are the denity and pecific heat, repectively, T i the temperature, v i the velocity vector, q i the heat flux, J i the molar flux of a pecie due to diffuion and migration, and Ĥ i the pecie partial molar enthalpy, with ubcript denoting in phae. The econd term on the RS of Eq. 4 thu repreent thermal tranport due to pecie diffuion and migration, with the ummation carried out over all pecie in phae. In general, the heat flux q include conductive flux (or Fourier flux), flux caued by interdiffuion of variou pecie, and the Dufour energy flux (or diffuion-thermo effect). Becaue Dufour energy flux i uually negligible, 33,34 the heat flux q can be expreed a q T H ˆ J [5] pecie where i the thermal conductivity of phae. Applying Eq. 5 and the continuity equation for phae v 0 [6] Equation 4 become cp ( v T ) ( T ) J Hˆ [7] pecie Ue the thermodynamic relationhip 34 Ĥ T [8] T along with the electrochemical potential defined by 34 o a RT ln zf [9] a,ref where o and a are the tandard chemical potential and the activity of a pecie in phae, repectively, and a,ref i the pecie activity at a reference tate. One ha a ln H ˆ a,ref R T zf T T [0] Equation 0 hold true with no additional aumption. The firt term on the RS of Eq. 0 i cloely related to the enthalpy-of-mixing and ha been generally neglected in practice. 7,30,3 Further ignoring the temperature dependence of phae potential for implicity, Eq. 0 i then implified to Ĥ zf Subtituting Eq. into Eq. 7 yield cp ( vt ) T zfj pecie [] [] Noting that the current through phae reult from diffuion and migration of ionic pecie in the phae under the aumption of electroneutrality, i.e. Equation can be rewritten a i pecie zfj cp ( vt ) T i [3] [4]

3 9 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. The econd term on the RS of Eq. 4 repreent the converion of electrical energy to thermal energy (i.e., Joule heating) and i an important difference between electrical and nonelectrical ytem. Electrochemical reaction occur at the interface between the electrode and the electrolyte. Heat balance over the interface reult in 35,36 e Ten e T n in i n [5] where n repreent the normal unit vector pointing outward from a phae, with ubcript e and denoting the phae of electrolyte and the phae of olid active material, repectively, and i n i the local tranfer current denity due to the electrode reaction. The RS of Eq. 5 tand for the heat generated at the electrode/electrolyte interface and i divided into two part. The firt term i the irreverible reaction heat due to the electrochemical reaction reitance at the interface, imilar to Joule heating. It i proportional to the urface overpotential of the electrode reaction and i alway poitive. The econd term i the reverible part of the reaction heat mainly due to the entropy change of the electrode reaction. It i called Peltier heat and change ign with changing current direction. The Peltier coefficient can be determined experimentally. 37 It i worth mentioning that in arriving at Eq. 5, the heat effect aociated with nonelectrochemical procee, uch a water condenation and evaporation occurring in a nicel-hydrogen cell, 6 hydride formation in the MH electrode, and active material decompoition in lithium-ion cell, 30 ha not been conidered. More generally, Eq. 5 can be rewritten a Tn m Tmn m ain( ) ( h hm ) m [6] am for the interface at which phae tranformation a well a multiple electrochemical reaction tae place. Here m repreent the phae tranformation rate at the -m interface from phae m to phae, a i the pecific urface area active for electrode reaction, a m A m /V o i the pecific urface area of the -m interface within the averaging volume V o, and h i the enthalpy with ubcript and m referring to phae and m, repectively. On the RS of Eq. 6, the econd term account for the heat effect due to the phae tranformation taing place at the interface, wherea the firt term include the heat effect due to the electrochemical reaction. For interface at which no electrochemical reaction occur, uch a ga-involved interface, the firt term on the RS imply vanihe. Let V o be the volume of an REV coniting of V ( e,, and g for electrolyte, olid active material, and ga phae, repectively) and follow the procedure decribed in Ref.. Volume-averaging of Eq. 4 over the REV yield < T > cp T < > < v > with and [ < > ] eff a, T < > < > Qm d Qm i Q Joule m Qm d TndA Vo A m Qm cpt ( w v ) nda o A m [7] [8] [9] J Q oule < i > < > nda Vo A m dv [0] o V ( i < i > ) ( < > ) where eff i the effective thermal conductivity of phae and a, i the diperion coefficient in phae. While eff include the effect of tortuoity and may follow the Bruggeman correction (i.e., eff.5 ), a, repreent the effect of hydrodynamic diperion that reult from variation of the microcopic velocity and temperature and vanihe in the abence of fluid motion. The econd term on the RS of Eq. 7 i the um of interfacial heattranfer effect. Qm d repreent the interfacial heat-tranfer rate due to conduction, wherea Q m tand for the thermal effect due to the interface movement at a velocity of w. In view of the mean value for integral, Q m can be modeled a the product of the average interfacial temperature by the phae tranformation rate at the interface, i.e. [] where T m i the area-averaged temperature at the -m interface. The lat term on the RS of Eq. 7, Q Joule, arie from volume-averaging the Joule heating term in Eq. 4. Apparently, it would vanih when electrical equilibrium hold true in a phae, i.e., < >. However, electrical nonequilibrium i expected if the phae conductivity i low and/or the applied current denity i high. The conductivity of emiconductor-lie active material (e.g., NiOOH) can be a low a 0 5 S/cm. Such a low electronic conductivity may caue a ignificant microcopic ohmic drop acro the active material layer coated on a ubtrate. In thi cae, it can be hown that the firt term on the RS of Eq. 0 i till negligible (ee the Appendix). However, the magnitude of the econd term cannot be eaily etimated and, thu, it i left unmodeled in the preent wor. Phyically, thi term arie from fluctuation in the profile of microcopic current and potential. Quantification of thi term will be attempted in future wor. Summation of Eq. 7 over all phae involved in the REV (i.e., the electrolyte phae, the olid active material phae, and the ga phae) and ue of Eq. 8 yield Applying Eq. 6, Eq. become < T > c p < T > < v > T da c T n p mm m m o A m < T > c p < T > < v > Qm cptmm { [ ]} < > eff a, T < > < > i < T > a i eff n Π [ ] ( h hm ) m ( cp cpm) Tmm m m < > < > i [] [3]

4 Journal of The Electrochemical Society, 47 (8) 90-9 (000) 93 S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Auming that local thermal equilibrium exit in the ytem under conideration, i.e. < T > < Tm > m Tm Tm T [4] and dropping the volume-averaged ymbol for convenience, Eq. 3 become where and the heat-ource term, q, i given by cpt vt T q cp cp v cp < v > eff a, [5] [6] [7] [8] q ain h* m m m < > < > ( i ) [9] e& with h* (h h m ) (c p c pm )T [30] The volume-averaged current denity through phae tae the form of eff e eff < i > < e > e i D ln < c > i or [3] for the phae of a concentrated binary electrolyte, and eff < i > < > [3] for the phae of a olid active material, where i the ionic conductivity of the electrolyte, D i the diffuional conductivity, and i the olid electronic conductivity, with upercript eff indicating the effect of poroity and tortuoity included. Note that there i no current flowing through the gaeou phae. Equation 9 how that the thermal effect are due to electrochemical reaction, phae tranformation, and ohmic Joule heating in both the electrolyte and olid active material phae. Becaue the reaction heat i expreed in term of the local tranfer current denity, the heat effect reulting from a highly nonuniform ditribution in the reaction rate can be aeed. Equation 5 enable one to determine temperature ditribution inide a cell rather than to obtain only an average temperature of the cell. In order to calculate the heat generation rate uing Eq. 9, the Peltier coefficient,, mut be nown. An expreion for Peltier coefficient ha been derived by Newman 35 baed on the general multicomponent tranport equation and electrode reaction. With the Dufour energy flux neglected, it i reduced to T S [33] nf where S i the entropy change of electrode reaction. Entropy change of a number of electrode reaction were calculated by Lampinen and Fomino 38 and by Xu et al. 39 Alternatively, uing the following thermodynamic relationhip between the OCP U and the entropy change S 40 S n F U the Peltier coefficient can be rewritten a T U Subtituting Eq. 35 and e e U into Eq. 9 give q a i U T U n ain e e [34] [35] h* m i [36] m m e& Equation 5, along with the expreion for the heat generation rate, Eq. 36, contitute a general thermal model that decribe the temperature field inide a cell. Thermal and electrochemical coupling. Temperature-dependent phyicochemical propertie, uch a the diffuion coefficient and ionic conductivity of the electrolyte, are needed to couple the thermal model with the multiphae ma-tranport and electrochemical inetic model. More pecifically, the dependence of the phyicochemical propertie on the temperature can be decribed by Arrheniu equation,5,36 act, ref exp E [37] R T T ref where i a general variable repreenting the diffuion coefficient of a pecie, conductivity of the electrolyte, exchange current denity of an electrode reaction, etc., with ubcript ref denoting the value at a reference temperature. E act, i the activation energy of the evolution proce of. It magnitude determine the relative enitivity of the cell to temperature. The greater it activation energy, the more enitive i the parameter to temperature. The ionic conductivity and electrolyte diffuion coefficient uually are trong function of the temperature; uch data are available for Ni-MH 4 and Li-baed 7 batterie. In addition, the OCP of electrode reaction, U, i uually approximated a a linear function of temperature U U U,ref ( T Tref ) [38] The heat generation rate due to electrochemical reaction and Joule heating are calculated locally via a detailed electrochemical model, and ubequently are ued in the energy conervation equation to calculate the temperature evolution. Thi temperature information i, in turn, fed bac to update the electrochemical calculation through temperature-dependent phyicochemical propertie. Figure how a chematic diagram of the preent coupled modeling approach. Lumped thermal model. For mot battery ytem, the convection term in Eq. 5 can generally be neglected, and a tranient heat conduction equation i ufficient to decribe thermal phenomena in batterie, i.e. ( cpt) ( T) q [39] Here, the thermal conductivity i highly aniotropic and microtructure-dependent becaue a battery cell i compoed of a variety of complex material. For ingle cell of mall thicne, one may have hl Bi << < > < > [40]

5 94 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Figure. Diagram of the thermal-electrochemical coupled modeling approach. where h i the convective heat-tranfer coefficient, and L and are the thicne and the thermal conductivity of the cell, repectively. Under the condition given by Eq. 40, the aumption of a uniform temperature acro the cell i valid and hence a lump-parameter model of energy balance can be applied. Equation 39 i thu reduced to d( cpt) Q qd [4] dt where the heat removal rate per unit volume from the cell to the urrounding, Q, can be expreed uing an equivalent convective heat tranfer coefficient, h, a follow ha T T Q c ( a ) [4] where V c i the cell volume, A c the urface area through which heat i removed from the cell, and T a the temperature of urrounding. <q> i the volume-averaged heat generation rate in the form of a i U T U n V T dv ain e e dv h m dv c V ( * ) c c m m dv [43] ( < i > < > ) e& We rewrite the econd integral of Eq. 43 to obtain V c e ain e dv e e ( < > ) ( < > ) a i U T U n dv ( < > < > ) ain e V c [ ] e dv ( h* m) dv < i > < > dv V V c m m c e& [44] The third term on the RS of Eq. 44 reult from the electrical nonequilibrium that exit in the olid active material and electrolyte phae. Auming that the heat effect due to the electrical nonequilibrium i negligible, Eq. 44 then become a i U T U n V T dv e a i dv n e ( < > < > ) h dv dv ( * m) c V ( < i > < > ) c m m c e& [45] We aume one-dimenional electrochemical procee acro the cell thicne, which i generally the cae becaue of the high electronic conductivity of the current collector bound to the electrode. Conervation of charge in both olid and electrolyte phae require < ie > < i > ain [46] Applying Eq. 46 to the econd integral in Eq. 45 and integrating it by part yield e nd integral in Eq. 45 < ie >< e > dv < i >< > dv IV dv [47] ( < i > < > ) e& where I ia/v c i/l i the total volumetric current denity (A/cm 3 ) applied to the cell, with A and L denoting the proected electrode area and the cell width, repectively. To obtain Eq. 47, the following boundary condition were ued < i e > 0 at x 0 and L [48] < i > i at x 0 and L [49] with the definition of cell potential V < > xl < > x0 [50] Subtituting Eq. 47 into Eq. 45, one ha a i U T U n dv IV V c h m dv [5] ( * ) m m When the heat effect due to phae tranformation i ignored, Eq. 5 i implified to Eq. 3 preented by Rao and Newman 3 for inertion battery ytem. Two condition can be applied to further implify Eq. 5: (i) the OCP term, a hown in the parenthei in Eq. 5, i contant; and (ii) the electrode reaction rate are patially uniform. In the cae of contant OCP, Eq. 5 can be reduced to the following by taing the OCP term out of the integrand where < q > I U U IV I aindv c [5] [53]

6 Journal of The Electrochemical Society, 47 (8) 90-9 (000) 95 S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. In the cae of uniform reaction rate, the heat generation rate can be rewritten a I U T U dv IV T I U T U IV av [54] Equation 5 or 54 i identical to Eq. preented by Bernardi et al. 3 with neglect of enthalpy-of-mixing and phae-change term. If only one overall cell reaction (i.e., one pair of electrode reaction) mut be conidered, Eq. 5 or 54 can be rewritten a I U T U IV I U V T U T i [55] L U V T U where U i the open-circuit cell potential determined by the difference between poitive and negative electrode OCP. Equation 55 ha been widely ued in lead-acid, 3,4 lithium-polymer, 7,8-0,8,9 and lithium-ion battery thermal model. It i clearly hown by the implification made that Eq. 55 i accurate only for the overall thermal balance of a cell with no ignificant heat effect due to phae change, no concentration gradient preent in the cell, contant OCP or uniform reaction rate, and a ingle overall reaction contributing to reaction heat. Rao and Newman 3 have illutrated that ignificant error may occur when Eq. 55 i ued to calculate the heat generation rate intead of Eq. 3. The extent of error depend greatly on the temporal behavior of the electrode OCP. Equation 55 virtually fail for a dynamic dicharge in which cell relaxation i involved. 3 Application to Ni-MH Cell A thermal and electrochemical coupled model reult by combining the thermal equation derived previouly with the micro-macrocopic multiphae tranport and electrochemical model previouly developed for Ni-MH cell. The model not only account for the microcopic diffuion of proton and hydrogen in olid active material, but alo incorporate oxygen reaction and tranport through both electrolyte and ga phae. Model detail have been preented in Ref. and hence are not repeated here. Equation 4, i.e., the lumpedparameter thermal equation, i employed in the following imulation a a firt tep. Applying Eq. 5 to a Ni-MH cell, one ha a i U T U n dv IV V c [56] where h* hyd i the enthalpy of metal hydride formation. Equation 56 i ued to calculate the volume-averaged heat generation rate of the Ni-MH cell. The model equation are ummarized in Table I, ubect to the following initial and boundary condition. Initial condition. Specie concentration are uniform at time 0, i.e. OH OH O O O O H H hhyd * a3in3 dv c co, ce ce,o, cg cg,o, c co, and T To [57] V c Table I. Summary of model equation for a Ni-MH battery. Specie concentration in liquid phae Specie concentration in ga phae Specie concentration in olid phae Liquid phae potential Solid phae potential Cell temperature OH o ( ec ) OH OH t OH Deff c F O ( ec e ) O O O O D e,eff ce eg t 4F J O cg Vg ( c H ) H F H D c H H H e c le aef eff eff e D OH OH ln c 0 a V eff OH b b 0 Rb OH e Re ae O Jeg dv d( cpt) ha ( T T ) c a dt a i V U T U T dv IV n * ( hyd 3 n3) V c h a i dv

7 96 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Boundary condition. There i no flux of pecie and all current L goe through the olid, therefore < > < > a i U T U P e n e dx L 0 OH O c c e 0, 0, and e 0 x x x and 3 [6] at the electrode/current collector interface [58] The boundary condition for olid-phae potential are dependent on the operation mode. At the poitive electrode/current collector interface (x L) 0 (reference potential) [59] At the negative electrode/current collector interface (x 0) V for floating charge [60] eff i for galvanic charge x where V and i are applied voltage and current denity, repectively. Cell in the previou wor i ued to invetigate the electrochemical and thermal behavior of a Ni-MH cell under variou operation mode and thermal condition. It cell-pecific parameter have been given therein. The operation mode include contant current charging at the C rate (i.e., i 35.7 ma/cm ) and float charging at a contant cell voltage of.5 V. Four thermal condition are conidered in term of an equivalent heat-tranfer coefficient impoed on the cooling urface of the cell. Wherea h 0 and correpond to adiabatic and iothermal condition, repectively, h 5 and 5 W/m K refer to typical value of air-free convection and forced convection (e.g., via a cooling fan). In other word, active thermal management i neceary in order to achieve a convective heat-tranfer coefficient of 5 W/m K. Table II lit the value of parameter ued in the thermal modeling of the Ni-MH battery. Other value of parameter needed in the imulation can be found in Ref.. The partial heat generation rate to be dicued in the next ection are defined a L < > < > a i U T U O e O n e O L dx 0 and 4 [6] H q h* hyda3in3 dx [63] L MH 0 and L J < > < > ( i ) dx [64] L 0 e& where the upercript P, O, H, and J denote the partial heat generation rate due to the primary reaction and 3, oxygen reaction and 4, hydride formation, and Joule heating, repectively. L MH i the thicne of the metal hydride electrode. The partial current preented in the Reult and Dicuion ection are defined a and LMH < > i P LL Ni L L io aindx LL Ni aindx [65] [66] where i p and i O repreent the partial current due to the primary reaction and oxygen generation reaction in the nicel electrode, repectively. L Ni i the thicne of the nicel electrode. Table II. Value of parameter ued in the thermal modeling of a Ni-MH battery. Symbol Value Unit Decription e.5 g/cm 3 Denity of KOH electrolyte 4 Ni 3.55 g/cm 3 Denity of nicel electrode 4 MH 7.49 g/cm 3 Denity of MH electrode 4 ep 0.9 g/cm 3 Denity of polyamide eparator a c p,e 3. J/g K Specific heat of KOH electrolyte 4 c p,ni 0.88 J/g K Specific heat of nicel electrode 4 c p,mh 0.35 J/g K Specific heat of MH electrode 43 c p,ep.9 J/g K Specific heat of polyamide eparator 4 (du/dt) ref.5 mv/k Temperature coefficient of reference electrode OCP 44 (du/dt) Ni.35 mv/k Temperature coefficient of nicel electrode OCP 44 (du/dt) MH mv/k Temperature coefficient of MH electrode OCP 44 b (du/dt) O.68 mv/k Temperature coefficient of oxygen reaction OCP 44 h* hyd 30.4 J/mol H Enthalpy of metal hydride (LaNi 5 H 6 ) formation 4,43 E act,ni 0. J/mol Activation energy of nicel electrode reaction 45 E act,mh 30. J/mol Activation energy of MH electrode reaction 4 E act,o 50. J/mol Activation energy of oxygen reaction 45 E act,d OH 4. J/mol Activation energy of electrolyte diffuion 46 E act,d O 4. J/mol Activation energy of oxygen diffuion in KOH electrolyte 47 E act,d H 9.6 J/mol Activation energy of proton diffuion in nicel active material 48 E act,d H 0. J/mol Activation energy of atomic hydrogen diffuion in MH particle a E act, 3 J/mol Activation energy of electrolyte conductivity 4 c a Etimated value. b Tae the data of H O(l)/H (g), OH electrode. c Evaluated uing the data in Fig. 5 of Ref. 4.

8 Journal of The Electrochemical Society, 47 (8) 90-9 (000) 97 S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Reult and Dicuion Significance of thermal-electrochemical coupling. The need for a thermal-electrochemical coupled model can be demontrated clearly by comparion of predicted cell potential, preure, and temperature uing the coupled and decoupled model, repectively. The decoupled model aume that the electrochemical ubmodel i eentially temperature-independent. However, becaue temperature i included in the Butler-Volmer equation, the decoupled model reult are till dependent on the thermal hitory and thu on the heat-tranfer coefficient. Figure diplay comparion of predicted cell potential curve during C charging. An apparent dicrepancy between the coupled and decoupled model prediction begin from 0% charge input and become ignificant when approaching the overcharge region. The dicrepancy decreae with increaing heat-tranfer coefficient, becaue a higher heat-tranfer coefficient yield a larger heat diipation rate and hence a maller temperature rie during battery charging. A expected, the decoupled model i applicable only when the variation in cell temperature i ufficiently mall. Unlie the cell potential, the cell temperature and preure are inenitive to the thermal-electrochemical coupling during contantcurrent charging, a hown in Fig. 3. It alo how that the cell temperature decreae ignificantly with increaing heat-tranfer coefficient, while the reduction in cell preure i very mall. Figure 4 plot the current denity applied to the cell during float charging at a voltage of.5 V. A ignificant dicrepancy i oberved between the coupled and decoupled model prediction when the heat diipation i poor (correponding to a mall heat-tranfer coefficient). The dicrepancy become maller when a higher heat-tranfer coefficient i applied. Thi, again, indicate that the decoupled model i valid only when the variation in the cell temperature i inignificant. Figure 5 how the comparion of predicted cell preure and temperature during the float charging. The decoupled model ignificantly underpredict the cell preure and temperature under poorer heat diipation condition. A thermal-electrochemical coupled model i neceary in order to accurately predict preure buildup in a Ni-MH battery and enure it afe operation. In view of the inaccuracy of the decoupled approach, the coupled model i ued to perform all following numerical tudie. Thermal effect during contant-current charging. Figure 6 how electrochemical and thermal behavior of the Ni-MH cell during C charging. A the heat-tranfer coefficient decreae, the cell potential increae more lowly and the potential pea become more pronounced. The cell temperature increae with time in all Figure 3. Comparion of predicted cell temperature and preure evolution between the coupled and decoupled thermal-electrochemical model during C charging. cae, indicating that the cell i exothermic when charged at C. A larger heat-tranfer coefficient correpond to a larger rate of heat diipation, reulting in a maller temperature rie. When the heattranfer coefficient i larger than 5 W/m K, the cell temperature rie i le than 5C when the charge input i le than 90% of the nominal cell capacity. After that tage, the cell temperature increae dra- Figure. Comparion of predicted potential curve between the coupled and decoupled thermal-electrochemical model during C charging. Figure 4. Comparion of predicted current denity variation between the coupled and decoupled thermal-electrochemical model during.5 V charging.

9 98 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Figure 5. Comparion of predicted cell temperature and preure evolution between the coupled and decoupled thermal-electrochemical model during.5 V float charging. Figure 6. Cell potential and temperature evolution during C charging under variou thermal condition. matically up to 64C at 0% charge input when the heat-tranfer coefficient equal 5 W/m K. A the heat-tranfer coefficient increae, the final cell temperature decreae. A le than 0C increae from the original temperature i oberved when the heat-tranfer coefficient i a large a 5 W/m K. The cell temperature exceed the afety limit for an aqueou cell (80C) when the cell i charged in an adiabatic condition, howing the need for thermal management for Ni-MH batterie. The overall trend in the cell temperature during C charging can be explained uing Fig. 7, which plot the total and partial heat generation rate of the cell v. charge input. The total heat generation rate increae lowly before 90% charge input, ump quicly thereafter, and finally reache a teady tate, cloely matching the trend in temperature variation. Surpriingly, the thermal environmental condition ha little effect on the heat generation rate. Becaue the overall reaction current i fixed when a battery i charged at a contant current, light variation in the total heat generation rate reult only from the difference in the ratio of primary to econdary reaction rate under different cooling condition. The primary reaction, oxygen reaction, MH formation, and Joule heating contribute to the total heat generation rate, a decribed by Eq. 9. Figure 7 alo diplay thee contribution of heat generation in the Ni-MH cell for h 5 W/m K. Initially, the heat effect due to primary reaction offet that due to MH formation, while the oxygen reaction are inignificant and the Joule heating negligible. With time, the heat aborbed by the primary reaction decreae becaue of the reduction in their enthalpy potential (i.e., U T U / ) 3,40 and the heat generated from the metal hydride formation remain contant. A a reult, the total heat generation rate gradually increae. When the cell i being charged upon a condition at which the oxygen reaction become ignificant, the total heat generation rate increae dramatically becaue the enthalpy potential of oxygen reaction are comparatively large and the heat aborbed by the primary reaction i negligibly mall. When the cell i being overcharged at a rate a large a C, almot all the current applied to the cell i ued to generate oxygen at the poitive electrode and only a portion of oxygen can be reduced at the negative electrode. In other word, the primary reaction at the MH electrode till account for a large portion of applied current due to the preence of deignated overcharge reerve, a indicated in Fig. 7 by the heat effect due to MH formation. The net heat generation rate i large, but teady tate i reached becaue of a contant reaction rate. Figure 8 how the variation in reaction current during C charging. Initially, the oxygen reaction i negligible, and all the current applied to the cell i ued to convert the active material from the dicharged to charged tate. The oxygen reaction become ignificant when the charge input exceed 90% of nominal cell capacity. It i obviou that the oxygen reaction current increae and the primary reaction current decreae, with the total current remaining the ame. The charge acceptance, defined a the ratio of primary reaction current to the total current, then exactly follow the curve of primary reaction current, with a maximum value of 00%. Apparently, the heat diipation rate affect the charge acceptance only after 00% of cell nominal capacity i applied. The wore the heat diipation, the maller the charge acceptance. Thermal effect during float charging. In addition to the contant-current charging mode, float charging at a contant voltage i alo frequently applied to Ni-MH batterie. Figure 9 how the reaction current when a contant voltage of.5 V i applied to the Ni-

10 Journal of The Electrochemical Society, 47 (8) 90-9 (000) 99 S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Figure 7. Total and partial heat generation rate during C charging under variou thermal condition. The partial heat generation rate are defined by Eq MH cell. The current due to oxygen reaction i negligible when the charge input i le than 60% of nominal cell capacity in all cae. A poorer cooling condition caue an earlier occurrence of oxygen reaction reulting from the higher cell temperature. The current due to primary reaction decreae very quicly during the firt 0% of Figure 8. Primary and oxygen reaction current at nicel poitive electrode during C charging under variou thermal condition. The partial current are defined by Eq. 65 and 66. Figure 9. Reaction current during.5 V float charging under variou thermal condition. nominal cell capacity becaue the urface overpotential, the driving force for the electrochemical reaction, drop quicly due to the continual increae of the electrode OCP during charging. The primary reaction current i trongly affected by cooling condition. The higher the heat diipation rate, the lower the cell temperature, and hence the lower reaction current. The reaction current increae even after the initial quic drop, indicating that the reaction are facile at high temperature. The total reaction current from both primary and oxygen reaction i expected to have a minimum value when both the primary and oxygen reaction current are mall. A lower teady total current can be obtained at a higher heat diipation rate. Figure 0 how the cell temperature and preure variation during.5 V float charging. The cell temperature i trongly affected by the cooling condition. While the cell temperature continue to increae under poor cooling condition, it decreae continuouly after an initial quic rie. The larger the heat-tranfer coefficient, the earlier the decreae occur. Thi decreae in cell temperature i arreted when the cell i near it fully charged tate and oxygen reaction become dominant. The cell temperature then increae dratically, the magnitude of which depend on the cooling condition. While the cell temperature tend to approach a contant value at high cooling rate, the cell virtually underegoe thermal runaway at low cooling rate. The oxygen reaction contribute to the cell preure buildup. The earlier the oxygen reaction tae place, the more the cell preure build up. While the cell preure can be maintained at a low level when the heat diipation rate i high and hence the cell temperature i low, it build up o quicly under poor cooling condition that afety become a concern.

11 90 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. Figure 0. Cell temperature and preure evolution during.5 V float charging under variou thermal condition. A found earlier, the heat generation rate of the cell i nearly independent of thermal condition during contant-current charging. Thi i not the cae during float charging. Figure diplay the heat generation rate of the Ni-MH cell during.5 V float charging. The total heat generation rate continue to decreae before the oxygen reaction, which reult in a dramatic increae in the total heat generation rate, become ignificant. It decreae more quicly at a larger heat diipation rate. The mallet heat generation rate occur when the heat diipation rate i infinitely large and the cell i accordingly iothermal. When the cell i adiabatic, the heat generation rate i nearly contant after an initial quic drop. Figure alo how the partial heat generation rate that contribute to the total rate for h 5 W/m K. The heat effect due to the hydride formation i dominant when the charge input i below 80% of nominal cell capacity, wherea the heat effect due to the oxygen reaction i dominant when the charge input i larger than 95% of nominal cell capacity. The heat effect due to Joule heating experience a minimum cloe to 00%, indicating that total current due to both primary and oxygen reaction ha a minimum value there. The heat effect due to MH formation decreae gradually with time and reache a minimum at a charge input of 0% of nominal cell capacity, meaning that the primary reaction till proceed at the MH electrode and i facile due to the increae in the cell temperature. A a reult, the heat effect due to the primary reaction at both nicel and MH electrode increae with increaing oxygen reaction. Figure how the effect of thermal condition on the charge acceptance of the Ni-MH cell during.5 V float charging. The charge acceptance reflect the efficiency of the charging proce. Figure. Total and partial heat generation rate during.5 V float charging under variou thermal condition. The difference in the charge acceptance i indicernible for all cae when the charge input i le than 60% of nominal cell capacity. However, it drop quicly a oon a the oxygen reaction become ignificant. When the charge input reache 0% of nominal cell Figure. Charge acceptance during.5 V float charging under variou thermal condition.

12 Journal of The Electrochemical Society, 47 (8) 90-9 (000) 9 S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. capacity, the charge acceptance virtually diappear. The oxygen reaction i much more facile at a higher cell temperature that reult from a poorer cooling condition. Accordingly, the charge acceptance drop earlier when the heat-tranfer coefficient i maller. Thi behavior i different from that diplayed in Fig. 8 during C charging. Concluion A general thermal energy equation ha been derived uing the volume-averaging technique, along with a local heat generation rate reulting from electrochemical reaction, phae tranformation, and Joule heating. The thermal model i fully coupled to the micro-macrocopic electrochemical model previouly developed via temperaturedependent phyicochemical propertie. The thermal-electrochemical coupled model i multidimenional and capable of predicting the temperature ditribution inide a cell in addition to the average cell temperature, thu providing a cot-effective tool to accurately predict the cell electrochemical and thermal behavior, and mot important, to identify the mechanim reponible for thermal runaway. Numerical imulation were performed to illutrate the ignificance of thermal and electrochemical coupling. The electrochemical and thermal behavior of a Ni-MH battery were then explored numerically uing the fully coupled thermal-electrochemical model. Variou operation mode and thermal condition were examined. The cell temperature rie i ignificant when the cell i charged at high rate and under poor cooling condition, primarily due to the oxygen reaction occurring near the end of full charge. The cell thermal behavior during contant-current charging wa found to differ ignificantly from that during float charging. Wor i underway to examine the predictablity of the preent model for hot pot within a cell. The thermal effect on the active material degradation and hence battery cycle life will be incorporated in future wor. Acnowledgment Thi wor wa upported, in part, by the Defene Advanced Reearch Proect Agency (DARPA), Tactical Technology Office, Electric and Hybrid Vehicle Technology Program, under cooperative agreement no. MDA and The Pennylvania State univerity aited in meeting the publication cot of thi article. Appendix An Order-of-Magnitude Etimate of the Firt Term in Eq. 0 The firt term in Eq. 0, Q Joule, repreent the Joule heating rate due to fluctuation at the microcopic level. It can be rewritten a < i t term in Eq. 0 > am m [A-] ( < > ) Note that the microcopic ohmic drop can be etimated from the tranfer current denity normal to the electrode/electrolyte interface a follow m < > r i n [A-] where r i the microcopic length cale (or the particle radiu). Subtituting Eq. A- into A- reult in eff < i t term in Eq. 0 > r a e i < > n e n r a i [A-3] where ue ha been made of the volume-averaged Ohm law, Eq. 3. Furthermore, it follow from the conervation of charge over an entire electrode that i ae i < > n [A-4] Le where L e i the macrocopic thicne of the electrode. Subtituting Eq. A-4 into A-3 and comparing thi term to the macrocopic Joule heating rate yield eff t term in Eq. 0 r r < i > < > Le Le << [A-5] ince the ratio of micro to macro length cale i much maller than unity. Lit of Symbol A proected electrode area, cm A c urface area through which heat i removed from the cell, cm a activity of a pecie a pecific urface area active for electrode reaction, cm /cm 3 c i volume-averaged concentration of pecie i over a phae, mol/cm 3 c p pecific heat, J/g K D i diffuion coefficient of pecie i, cm / E act activation energy, J/mol F Faraday contant, 96,487 C/mol Ĥ partial molar enthalpy of a pecie, J/mol h enthalpy of pecie participating in the phae tranformation, J/mol h equivalent heat-tranfer coefficient, W/cm K I volumetric current, A/cm 3 i current denity vector, A/cm i applied current denity, A/cm in tranfer current denity of reaction, A/cm J molar flux of a pecie, mol/cm J O eg ma-tranfer rate of oxygen acro ga/electrolyte interface, mol/cm 3 i reaction current reulting in production or conumption of pecie i, A/cm 3 L cell width, cm L e electrode thicne, cm l e microcopic diffuion length of pecie in a olid phae, cm n number of electron tranferred in reaction Q volumetric heat removal rate from the cell, J/cm 3 Q Joule microcopic Joule heating term due to electrical nonequilibrium, J/cm 3 Q d m interfacial heat-tranfer rate due to conduction, J/cm 3 Qm thermal effect due to the interface movement, J/cm3 q heat flux, J/cm q volumetric heat generation rate, J/cm 3 R univeral ga contant, J/mol K R o applied load, R b, area-pecific electrical reitance acro the olid/ubtrate interface, R e cm Ŝ partial molar entropy, J/mol K toichiometric coefficient of a pecie involved in reaction T abolute temperature of the cell ytem, K t time, t o tranference number of OH with repect to the olvent velocity U open-circuit potential of electrode reaction, V V applied voltage, V V c cell volume, cm 3 V g volume occupied by the ga phae, cm 3 v velocity vector, cm/ w interface velocity vector, cm/ x coordinate along the cell width, cm z charge number of an ionic pecie Gree volume fraction of a phae urface overpotential of electrode reaction, V the Peltier coefficient, V phae tranformation rate, mol/cm conductivity of an electrolyte, S/cm D diffuional conductivity of a pecie, A/cm denity, g/cm 3 conductivity of olid active material, S/cm thermal conductivity, W/cm K electrochemical potential of a pecie, J/mol o tandard chemical potential of a pecie, J/mol h* enthalpy change due to phae tranformation, J/mol potential in a phae, V urface potential in a phae, V Subcript b ubtrate e electrolyte phae eff effective eg electrolyte/ga interface g ga phae hyd hydride formation

13 9 Journal of The Electrochemical Society, 47 (8) 90-9 (000) S (00)0-05- CCC: $7.00 The Electrochemical Society, Inc. m interface between phae and m MH metal hydride active material Ni nicel active material ref with repect to a reference tate olid phae b olid/ubtrate interface e olid/electrolyte interface ep eparator o initial value Supercript av average eff effective H pecie hydrogen or proton H O olvent water OH pecie OH overbar, average over an interface Reference. C. Y. Wang, W. B. Gu, and B. Y. Liaw, J. Electrochem. Soc., 45, 3407 (998).. W. B. Gu, C. Y. Wang, S. L. Li, B. Y. Liaw, and M. M. Geng, Electrochim. Acta, 44, 455 (999). 3. T. F. Fuller and J. Newman, in Modern Apect of Electrochemitry, Vol. 7, R. E. White, J. Bocri, and B. Conway, Editor, p. 359, Plenum Pre, New Yor (995). 4. G. Zheng, B. N. Popov, and R. E. White, J. Appl. Electrochem., 7, 38 (997). 5. D. Berndt and E. Meiner, INTELEC 90 Proceeding of th International Telecommunication Energy Conference, p. 48 (990). 6. D. Pavlov, J. Power Source, 64, 3 (997). 7. C. R. Pal and J. Newman, J. Electrochem. Soc., 4, 374, 38 (995). 8. Y. Chen and J. W. Evan, J. Electrochem. Soc., 43, 708 (996). 9. F. Joubert, J. Bouet, and J.-F. Fauvarque, in Batterie for Portable Application and Electric Vehicle, C. F. Holme and A. R. Landgrebe, Editor, PV 97-8, p. 837, The Electrochemical Society Proceeding Serie, Pennington, NJ (997). 0. M. S. Wu, Y. Y. Wang, and C. C. Wan, J. Power Source, 74, 0 (998).. Z. L. Zhang, M. H. Zhong, F. M. Liu, F. P. Zhong, and F. Wu, J. Power Source, 70, 76 (998).. J. Lee, in Electrochemical and Thermal Modeling of Battery, Fuel cell, and Photoenergy Converion Sytem, J. R. Selman and H. C. Maru, Editor, PV 86-, p. 06, The Electrochemical Society Proceeding Serie, Pennington, NJ (986). 3. J. Newman and W. Tiedemann, J. Electrochem. Soc., 4, 054 (995). 4. K. W. Choi and N. P. Yao, J. Electrochem. Soc., 6, 3 (979). 5. J. Lee, K. W. Choi, N. P. Yao, and C. C. Chritianon, J. Electrochem. Soc., 33, 86 (986). 6. H. Huang and T. V. Nguyen, J. Electrochem. Soc., 44, 06 (997). 7. J. Kim, T. V. Nguyen, and R. E. White, J. Electrochem. Soc., 39, 78 (99). 8. Y. Chen and J. W. Evan, J. Electrochem. Soc., 40, 833 (993). 9. Y. Chen and J. W. Evan, J. Electrochem. Soc., 4, 947 (994). 0. Y. Chen and J. W. Evan, Electrochim. Acta, 39, 57 (994).. M. K. Verbrugge AIChE J., 4, 550 (995).. Y. Chen and J. W. Evan, INTELEC 96 Proceeding of the 8th International Telecommunication Energy Conference, p. 465 (996). 3. W. Tiedemann and J. Newman, in Battery Deign and Optimization, S. Gro, Editor, PV 79-, p. 3, The Electrochemical Society Proceeding Serie, Pennington, NJ (979). 4. W. Tiedemann and J. Newman, in Advance in Lead-Acid Batterie, K. R. Bulloc and D. Pavlov, Editor, PV 84-4, p. 336, The Electrochemical Society Proceeding Serie, Pennington, NJ (984). 5. H. Huang and T. V. Nguyen, J. Electrochem. Soc., 44, 40 (997). 6. J. Kim, T. V. Nguyen, and R. E. White, J. Electrochem. Soc., 4, 333 (994). 7. P. De Vidt, J. Delgado, B. Wu, D. See, K. Koanovicb, and R. E. White, J. Electrochem. Soc., 45, 3874 (998). 8. C. R. Pal, M. Doyle, T. F. Fuller, and J. Newman, in Dougla N. Bennion Memorial Sympoium, R. E. White and J. Newman, Editor, PV 94-, p. 48, The Electrochemical Society Proceeding Serie, Pennington, NJ (994). 9. M. Doyle and J. Newman, Electrochim. Acta, 40, 9 (995). 30. G. G. Botte, B. A. Johnon, and R. E. White, J. Electrochem. Soc., 46, 94 (999). 3. D. Bernardi, E. Pawliowi, and J. Newman, J. Electrochem. Soc., 3, 5 (985). 3. L. Rao and J. Newman, J. Electrochem. Soc., 44, 697 (997). 33. R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Tranport Phenomena, John Wiley & Son, New Yor (960). 34. J. Newman, Electrochemical Sytem, Prentice-Hall, Englewood Cliff, NJ (99). 35. J. Newman, Ind. Eng. Chem. Re., 34, 308 (995). 36. Y. V. Kuzminii, L. I. Nyova, and A. A. Andriio, J. Power Source, 46, 9 (993). 37. Y. V. Kuzminii and A. V. Gorodyii, J. Electroanal. Chem., 5, (988). 38. M. J. Lampinen and M. Fomino, J. Electrochem. Soc., 40, 3537 (993). 39. Q. Xu, S. Keltrup, and B. Hafold, Electrochim. Acta, 43, 597 (998). 40. H. F. Gibbard, J. Electrochem. Soc., 5, 353 (978). 4. D. Berndt, Maintenance-Free Batterie Lead-acid, Nicel/Cadmium, Nicel/ Hydride, John Wiley & Son, New Yor (993). 4. B. Paxton and J. Newman, J. Electrochem. Soc., 44, 388 (997). 43. D. Ohlendorf and H. E. Flotow, J. Le-Common. Met., 73, 5 (980). 44. S. G. Bratch, J. Phy. Chem. Ref. Data, 8, (989). 45. B. V. Ratnaumar, P. Timmerman, C. Sanchez, S. Di Stefano, and G. Halpert, J. Electrochem. Soc., 43, 803 (996). 46. Y. Oinaa, J. Electrochem. Soc., 7, 89 (970). 47. R. E. Davi, G. L. Horvath, and C. W. Tobia, Electrochim. Acta,, 87 (967). 48. D. M. MacArthur, J. Electrochem. Soc., 7, 79 (970). 49. W. B. Gu and C. Y. Wang, in Lithium Batterie, R. A. Marh, Z. Ogumi, J. Praah, and S. Surampudi, Editor, PV 99-5, p. 748, The Electrochemical Society Proceeding Serie, Pennington, NJ (999).