A00 Journal of The Electrochemical Society, 15 10 A00-A008 005 0013-651/005/1510/A00/7/$7.00 The Electrochemical Society, Inc. Efficient Macro-Micro Scale Couple Moeling of Batteries Venkat. Subramanian,*,z Vinten D. Diwakar,** an Deepak Tapriyal Department of Chemical Engineering, Tennessee Technological University, Cookeville, Tennessee 38505, USA In this paper, efficient approximate solutions are evelope for microscale iffusion insie porous electroes. Approximate solutions evelope for the microscale iffusion are then couple with governing equations for the macroscale to preict the electrochemical behavior of a lithium-ion cell sanwich. Approximate solutions evelope facilitate the numerical simulation of batteries by reucing the number of ifferential algebraic equations resulting from the iscretization of governing equations. 005 The Electrochemical Society. DOI: 10.119/1.037 All rights reserve. Manuscript submitte August 19, 00; revise manuscript receive April 6, 005. Available electronically August, 005. Electrochemical moels that preict the performance of batteries accurately are usually complex because of the nonlinear coupling of the epenent variables in the governing equations an nonconstant kinetic an transport parameters. 1- These moels have been use by various researchers to optimize the cell esign an to stuy the effect of system parameters an thermal behavior. These moels for porous electroes involve two scales: electrolyte transport, electrolyte conuction, an soli-phase conuction in the macroscale x an soli-state iffusion in the microscale insie the particle r. Battery moels typically solve electrolyte concentration an electrolyte potential in the separator; an electrolyte concentration, electrolyte potential, soli-state potential, an soli-state concentration in the porous electroe. 1- Even when one-imensional transport in the macroscale x is consiere, these moels involve two couple nonlinear partial ifferential equations PDEs in x,t in the separator an three couple nonlinear PDEs in x,t in the porous electroe. 1- In aition, soli-state iffusion shoul be solve in the pseuoimension r,t in the electroe. For preicting the thermal behavior, one has to a an aitional equation for temperature. The finite-ifference technique,3 is the most common technique use for simulation of batteries, though collocation, finite element, an other techniques have also been use by some researchers. 3 If one has to simulate these battery moels rigorously, it woul involve iscretization of spatial erivatives in the macroscale an also in the microscale. For example, if one were to use 100 noe points in the macroscale x an 0 noe points in the microscale r, one has to solve 100 0 = 8000 ifferential algebraic equations resulting from the iscretization of the governing equations. Duhamel s superposition theorem was use by Doyle et al. 3 to eliminate the pseuoimension r. This involves an infinite series an has to be taken care of while integrating macroscale equations numerically in time. This means that one cannot use existing avance numerical solvers irectly for the integration of macroscale equations in time.,3 In aition, this metho cannot be use for concentration-epenent iffusion coefficients. Wang an co-workers 3, use volume average equations an a parabolic profile approximation for soli-phase concentration. This helpe reuce the soli-phase partial ifferential equation to two ifferential algebraic equations. Their work for Ni C an Ni MH systems inclue the aition of an empirical term to take care of the iscrepancy in the short-time solution. Along similar lines, Srinivasan et al. 5 use a volume average approach for the analysis of thermal behavior of Li-ion cells. However, they i not enumerate when their moels fail. The two-parameter moel introuce in this paper yiels results similar to the parabolic profile moel escribe by Wang an co-workers. It is also note that the approach evelope in this paper can be use to increase the accuracy of the * Electrochemical Society Active Member. ** Electrochemical Society Stuent Member. z E-mail: vsubramanian@tntech.eu moel by aing parameters, instea of using empirical terms. The performance of the new moel is foun to be vali even at short times. In this paper, efficient approximations are evelope for the microscale iffusion, which reuce the microscale iffusion 1 PDE to two or three ifferential algebraic equations. These approximations are evelope by assuming that the soli-state concentration insie the spherical particle can be expresse as a polynomial in the spatial irection. 5 Subramanian et al. 5 evelope approximate solutions for soli-phase iffusion base on polynomial profile approximations for constant pore wall flux at the surface of the particle. However, these moels cannot be use for battery moeling irectly because the pore wall flux at the surface of the particle changes both as a function of time an istance across the porous electroe. In this paper, approximations are evelope for the microscale iffusion for time-epenent pore wall flux. These approximations are then teste with the exact numerical solution of particle iffusion for various efine functions in time for the pore wall flux. Next, these approximations are use with the macroscale moel to preict the electrochemical behavior of an Li-ion cell sanwich. The approximations evelope reuce the computation time for simulation without compromising accuracy. Approximate Solution for Microscale Diffusion Transient iffusion in a spherical electroe particle having an initial concentration c 0 is given by c t D 1 S r rr c Initial an bounary conitions are r =0 c = c 0 at t = 0 an for 0 r fully charge state c s r 1 =0 atr = 0 an for t 0 3 c r = j n at r = an for t 0 where j n is the pore wall flux at the surface of the particle an is the raius of the particle. The pore wall flux is a function of both the istance across the electroe x an time t. Next, we evelop approximate solution for Eq. 1 base on polynomial profile approximations evelope earlier for constant pore wall flux at the surface. 5 Two-parameter moel. The concentration profile insie the particle is assume to be a parabola in r 5- cr,t = at + bt r Substituting Eq. 5 in Eq. 1, we obtain 5
at + r bt 6D Sbt 6 t t The bounary conition at r = 0 is automatically satisfie. The bounary conition at r = Eq. becomes D S bt = j n 7 For battery moeling, we are mainly intereste in the average concentration for state of charge an surface concentration for electrochemical behavior. Hence, constants at an bt are expresse in terms of the volume-average concentration c t an surface concentration c S t. The volume-average concentration is given by 3 r cr,t c t r =r=0 P Applying Eq. 5 in Eq. 8, we get c t = at + 3 5 bt Surface concentration c S is obtaine by substituting r = in Eq. 5 c S t = at + bt Equations 9 an 10 are solve to obtain Journal of The Electrochemical Society, 15 10 A00-A008 005 8 9 10 cr,t = at + bt r + p t r p Substituting Eq. 17 in Eq. 1 gives A003 17 at + r bt + r t D S t t t 3bt +10 r p t P =0 18 The bounary conition at r = 0 is automatically satisfie again. The bounary conition at r = now becomes D S bt + D St = j n 19 Next, the constants at, bt, an t are solve in terms volumeaverage concentration c t, surface concentration c S t, an volume-average concentration flux q t. The volume-average concentration flux is a physically meaningful term, which efines the average change of concentration with respect to the position in the system. 6 Volume-average soli-phase concentration can be evaluate using Eq. 8 an is foun to be c t = 3 7 t + 3 bt + at 5 0 Surface concentration is obtaine by evaluating cr,t at the surface at = 3 c St + 5 c t bt = 5 c t + 5 c St 11 1 The concentration profile given by Eq. 5 is now purely in terms of the volume-average concentration c t an the surface concentration c S t cr,t = 3 c St + 5 c t + 5 c t + 5 c St r 13 We now nee two equations to evaluate the average concentration c t an the surface concentration c S t. The volume-average concentration can be evaluate by volume averaging the entire governing Eq. 1 3 r c t D 1 S r rr c 1 x=0 r r =0 Substituting Eq. 1 in Eq. 13 an evaluating, we have t c t +3j n =0 15 The secon equation require to evaluate for c S t is obtaine by evaluating the bounary conition at r =. Evaluating Eq. 5 using Eq. 1, we get c S t c t = j n 16 5 It is clearly seen that Eq. 1 an 15 are purely in terms of the average concentration an surface concentration. It must be note here that Eq. 15 an 16 are vali even if the pore wall flux j n is a function of time. Higher-orer polynomial profile moel. As shown in the previous publication, two-parameter moels may not be vali at higher ischarge rates. 5,6 In this section, an efficient three-parameter moel is now evelope. 6 The concentration profile is taken to be c S t = at + bt + t 1 The volume-average concentration flux term is evaluate using the following expression cr,t 3 r r r q t =r=0 Substituting Eq. 17 in Eq., the average concentration flux is foun to be q t = t + 3 bt 3 Equations 0, 1, an 3 can be solve for constants at, bt, an t in terms of the average concentration c t, the surface concentration c S t, an the average concentration flux q t to obtain at = 39 c St 3q t 35 c t bt = 35c S t +10q t +35c t 5 t = 105 c St 7q t 105 c t 6 The concentration profile given by Eq. 17 is now purely in terms of the volume-average concentration c t, the volume-average concentration flux q t, an the surface concentration c S t cr,t = 39 c St 3q t 35 c t + 35c St +10q t +35c t r 105 + c St 7q t 105 c t r 7 We now nee three equations to solve for the average concentration, the surface concentration, an the average flux. The equation for the volume-average concentration is obtaine by volume averaging the entire governing Eq. 1 as given by Eq. 1
A00 Journal of The Electrochemical Society, 15 10 A00-A008 005 t c t +3j n =0 8 The secon equation for the volume-average flux is obtaine by volume averaging the ifferential of the governing equation c 3 r x=0 t D 1 S r rr c P r Equation 9 yiels t q t +30D S r r =0 q t + 5 j n 9 30 The thir equation can be obtaine by evaluating the bounary conition at r = given by Eq. 19 35 D S c s c t 8 q t = j n 31 It is clearly seen that Eq. 8, 30, an 31 are purely in terms of the average concentration, surface concentration an average concentration flux. It must be note here that the pore wall flux j n can be a function of time t. Note that the approximate equations are evelope in imensional form so that they can be couple with the governing equations for the macroscale moels of batteries. However, to test the accuracy of the approximate moels evelope, it is convenient to convert the equations to imensionless form as illustrate in the next section. Dimensionless Analysis Equation 1 is converte to imensionless form using the following imensionless variables C = c, z = r, an = D St 3 c 0 The governing equation in imensionless form is C = 1 z zz z C 33 The bounary an initial conitions in imensionless form are C =1at = 0 an for 0 z 1 3 C =0atz = 0 an for 0 z 35 C = at z = 1 an for 0 z 36 where = j n / c 0 is the imensionless pore wall flux. Two-parameter moel. The parabolic profile approximation takes the form Cz, = a + bz 37 Following the proceure escribe earlier for imensional equations, the governing equations for volume-average imensionless concentration an surface concentration are obtaine as C +3 =0 38 C S C = 39 5 Next, to verify the accuracy of the approximations evelope, ifferent test functions are trie for the imensionless pore wall flux. Case (i): Linear function =. Substituting = in Eq. 38 an 39 an solving for the average concentration an the surface concentration in terms of the inepenent variables time an position in the system z, weget C =1 3 0 C S =1 5 3 1 A close form expression is obtaine for the concentration as a function of imensionless position in the system z an imensionless time Cz, =1+ z + 3 10 3 Case (ii): Oscillatory function = sin. Substituting = sin in Eq. 38 an 39 an solving for the average concentration an the surface concentration, we get C = +3cos 3 C S = sin + 3 cos 5 The concentration profile is obtaine as Cz, = sin z + 3 sin + 3 cos 5 10 Case (iii): Exponential function = exp. Substituting = exp in Eq. 38 an 39 an solving for the average concentration an the surface concentration, we get C = +3exp 1 exp C S + 5 The concentration profile is obtaine as 33 exp exp z Cz, = + 10 6 7 8 Three-parameter moel. The polynomial profile is taken to be Cz, = a + bz + z 9 Following the proceure escribe earlier for imensional equations, the governing equations for volume-average imensionless concentration, volume-average imensionless flux, an surface concentration are obtaine as C +3 =0 50 t Q +30Q + 5 51 t 35c s c t 8Q = 5 Next, ifferent test functions are trie for imensionless pore wall flux. Only the final equations are reporte. Case (i): Linear function =. C = 3 +1 Q = 3 1 0 + 1 exp 30 0 5 53
Journal of The Electrochemical Society, 15 10 A00-A008 005 A005 Figure 1. Comparison of approximate moels evelope for =. Figure 3. Comparison of approximate moels evelope for =10. C S = 5 + 17 175 + 1 175 exp 30 + 3 55 Cz, = 17 100 + z z + 0 10 3 + 3 + 1 0 z 7 100 1 0 zexp 30 56 Case (ii): Oscillatory function = sin. C = 3cos + 57 Q = 675 5 5 sin cos + exp 30 58 901 180 180 C S =+ 6301 36 18957 sin + exp 30 31535 6307 6307 cos 59 Cz, =+ 9 3 901 z + 360 z 1610sin 37761 + 5 180 z 37599 161 5 901 zcos + 5 901 z 3 161 5 180 zexp 30 60 Case (iii): Exponential function = exp. C = 3exp + 61 Q = 5 5 exp exp 30 6 58 58 C S = 36 836 exp 30 exp 63 03 1015 Figure. Comparison of approximate moels evelope for =5. Figure. Comparison of approximate moels evelope for = sin.
A006 Journal of The Electrochemical Society, 15 10 A00-A008 005 Figure 6. Lithium-ion cell sanwich, consisting of lithium-foil, separator, an porous electroe. Figure 5. Comparison of approximate moels evelope for =exp. Cz, =+ 16 9 z 3 116 z 1379 060 exp + 3 06 5 9 z + 5 58 zexp 30 6 Accuracy of approximate moels. Electrochemical behavior is governe by the surface concentration. Hence, the accuracy of the approximate moels evelope is verifie by plotting the imensionless surface concentration. To compare the accuracy, a rigorous numerical solution is obtaine by solving Eq. 33 subject to bounary conitions/initial conitions given by Eq. 3-36 using numerical metho of lines. 17,18 The exact numerical solution is obtaine by applying finite ifferences in the spatial irection an by integrating the resulting system of couple orinary ifferential equations numerically in time. Twenty noe points were foun to be sufficient for the purpose of this paper. In Fig. 1, imensionless surface concentration is plotte as a function of imensionless time for a linear function for the imensionless pore wall flux, =. From Fig. 1, we conclue that both two- an three-parameter moels fit exactly with the exact numerical solution for the linear function of time. In Fig., imensionless surface concentration is plotte as a function of imensionless time for a linear function for the imensionless pore wall flux, =5. From Fig., we conclue that the three-parameter moels fit better than the two-parameter moel with the numerical exact moel for the linear function of time 5. The increase in magnitue causes the two-parameter moel to show a small eviation. In Fig. 3, imensionless surface concentration is plotte as a function of imensionless time for a linear function for the imensionless pore wall flux, =10. From Fig. 3, we conclue that the three-parameter moels fit better than the two-parameter moel with the numerical exact moel for the linear function of time 10. The increase in magnitue causes the two-parameter moel to pre- Table I. Summary of moel equations an bounary equations in the macroscale. Variables Equations Bounary conition Separator At x = 0 c c 0.5 = D c t S x D c x = i app1 t + 1.5 S F x x T1 t+ F Composite cathoe 1 1 x x x T1 t+ F c j n = Kc max c c exp af 1 U T U = ln c x =0 =0 At x = L s + L c = afj n 1 k x = i app ln c x c p t = D 1.5 c P x + aj n1 t + exp 1 cf 1 U T.707 36.19 c s + 10.813 c s + 19.91 c s 3 + 111.818 c s 35.705 c s 5 1 7.598 c s + 1.779 c s 30.959 c s 3 + 3.63 c s + 7.87 c s 5 = afj n eff x =0 c x =0
Journal of The Electrochemical Society, 15 10 A00-A008 005 A007 Table II. Values for various parameters for LiCoO. Parameters Value Parameters Value 1.0 10 13 T 100 C D 7.5 10 10 L s 3 m i app 60 A/m L c 9 m a, c 0.5 c 0 1000 mol/m 3 5155 mol/m 3 s 0.7 K.5 10 6 m /mol s p 0.39 c max 5155 100 S/m Figure 7. Potential obtaine by using an approximate moel for the soli phase concentration is compare with potential obtaine by using 0 noes in the particles. Soli line represents the noe moel rigorous solution an soli ots represent the approximate moel. Figure 8. Electrolyte concentration insie the cell sanwich preicte using an approximate moel for the soli phase concentration is compare with electrolyte concentration obtaine using 0 noes in the particles. Soli line represents the noe moel rigorous solution an soli ots represent the approximate moel. ict erroneous results. Further increase in the magnitue causes the two-parameter moel to eviate further from the exact numerical moel. In Fig., imensionless surface concentration is plotte as a function of imensionless time for a linear function for the imensionless pore wall flux, = sin. From Fig., we can infer that both two- an three-parameter moels fit exactly with the numerical exact moel for a sinusoial function in time. In Fig. 5, imensionless surface concentration is plotte as a function of imensionless time for a linear function for the imensionless pore wall flux, = exp. We observe that the twoparameter moel fails at very low values of time. The threeparameter moel shows goo fit to the exact numerical moel at all the values of state of charge. It is interesting to note, however, that the two-parameter moel shows goo fit to the exact numerical moel at high values of time. From Fig. 1-5 we can conclue that three-parameter moels can be use safely without compromising the accuracy. Also, twoparameter moels can be use as long as 1. Also, even for higher values of, two-parameter moels can be use at long times. In aition, two parameters work if starts from zero. This observation is useful for battery moeling because pore wall flux is zero before the beginning of ischarge. After the ischarge begins, the pore-wall flux insie the porous electroe increases with time. The approximate moels evelope for the microscale iffusion are couple with the macroscale equations to preict the electrochemical behavior of an Li-ion cell sanwich in the next section. Electrochemical Behavior of an Li-Ion Cell Sanwich In the previous sections, approximate moels were evelope for microscale iffusion of ions insie a spherical particle. The approximate moels evelope convert one PDE to two or three ifferential algebraic equations. The approximate moels evelope were then valiate by comparing with the exact numerical solution. If one has to solve just one PDE insie the particle, then one can get a closeform solution or a numerical solution quite easily. 7-30 The approximate moels evelope, though accurate, are not neee if one has to solve just a single PDE. However, for battery moeling, a single PDE for the microscale iffusion is couple with other PDEs in the macroscale.,3 Next, we show the utility of the approximate moels evelope for battery moeling. The geometry moele is shown in Fig. 6. The Li-ion cell sanwich moele consists of lithium foil, a separator, an a porous electroe. The governing equations an bounary conitions for the in the macroscale are liste in Table I. The typical values of the constants are given in Table II. There are two PDEs in the separator an three PDEs in the porous electroe.,3 In aition, for the porous electroe, the soli-phase iffusion is solve insie the particle in the pseuoirection r microscale Eq. 1-. For simulation purposes, the governing equations in the macroscale are iscretize in the x irection using 100 noe points in the separator an 100 noe points in the porous electroe. For a rigorous numerical solution, if 0 noe points are use insie the particle in the microscale, then we have 100 0 = 8000 ifferential algebraic equations in the porous electroe.,3 When our approximate moel Eq. 15 an 16 is use for the microscale iffusion, we have 100 = 800 ifferential algebraic equations in the porous electroe. Both the rigorous numerical solution with rigorous solution of particle iffusion an the numerical solution with approximate moels for the particle iffusion are compare in Fig. 7. We observe that the approximate moel for the particle iffusion matches exactly with the rigorous simulation. In aition, we observe that the approximate moel evelope in this paper preicts the electrolyte concentration an soli-phase concentration insie the porous electroe accurately, as shown in Fig. 8 an 9. igorous solution to preict one ischarge curve with numerical iscretization of micro- an macroscale moels takes aroun min in a.6 GHz processor. Numerical solution with the approximate moel for the microscale evelope takes only 10 s to preict one ischarge curve. Discharge curves are compare for applie current ensity 60, 90, an 10 A/m. Note that higher-orer polynomial moels Eq. 8, 30, an 31 preict the same behavior. The twoparameter moel is foun to be sufficient for this system. However, for porous electroes with high values of pore wall flux or very low soli-phase iffusion coefficients, one might nee three-parameter moels. Also, because it is expensive to get a rigorous numerical solution, the best option is to simulate the behavior using both twoparameter an three-parameter moels for the highest possible ap-
A008 Journal of The Electrochemical Society, 15 10 A00-A008 005 NO 000-03-C-01. The authors woul like to acknowlege the Center for Electric Power, Tennessee Technological University for the grauate research assistantship provie to V.D.D. an D.T. Tennessee Technological University assiste in meeting the publication costs of this article. List of Symbols Figure 9. Soli-phase surface concentration at the particle/electrolyte interface insie the cell sanwich preicte using an approximate moel for the soli phase concentration is compare with soli-phase surface concentration obtaine using 0 noes in the particles. Soli line represents the noe moel rigorous solution an soli ots represent the approximate moel. plie current an see if they match. If they match, then one can safely use the two-parameter moels even for pulse currents, voltammetry, an other complicate bounary conitions, geometry, an systems. In this paper, approximations were evelope an implemente only for microscale iffusion in spherical particles. However, the same methoology has been use by the authors for cylinrical an rectangular particles with the same effect. The simplifie equations are very similar to the simplifie equations for the spherical particle Eq. 15, 16, 9, 31, an 3 an are available upon request from the corresponing author. Currently, we are working on eveloping optimize twoparameter an three-parameter moels for various electrochemical systems for both macroscale an microscale phenomena. Conclusions In this paper, approximate moels were evelope for soliphase iffusion in the microscale. The approximate moels evelope were teste for arbitrary functions of pore wall flux. In aition, the approximate moels evelope were then use to preict the ischarge curves of a lithium-ion cell sanwich consisting of a lithium foil, separator, an porous electroe. The approximate moels evelope save computation time by more than 80% without compromising accuracy. In this paper, the utility of the approximate moels evelope was shown for one porous electroe only. The same concept can be irectly extene to preict the electrochemical behavior of lithiumion batteries with two porous intercalation electroes. In aition, similar approximate moels have been foun to be of use in preicting the electrochemical behavior of PEM fuel cells an will be publishe later. Note that the approximate moels evelope Eq. 15, 16, 9, 31, an 3 are vali even if the soli-state iffusion coefficient is a function of soli-state concentration. This will be iscusse in a later publication. The approximate moels evelope in this paper are foun to be useful in preicting the impeance response of Li-ion batteries also an will be communicate later. Acknowlegments The authors are grateful for the financial support of the project by National econnaissance Organization NO uner contract no. at,bt,t c c c s C r t x D a j n t + i app L c,l s F n cs av Greek p s 1 time-epenent constants in the polynomial approximations concentration of electrolyte mol/m 3 soli-phase concentration mol/m 3 soli-phase concentration at the surface of the particle mol/m 3 imensionless soli-phase concentration istance from the center of the particle m, microscale time s istance m, macroscale iffusion coefficient of the electrolyte m /s iffusion coefficient of the electrolyte in the soli particles m /s interfacial area m 1 pore wall flux of Li ions mol/m /s transfer number current ensity A/m length of the electroe m, length of the separator m Faraay s law constant 96,87 C/mol number of electrons transferre n = 1 for the simulation average concentration in the soli particle porosity of the electroe porosity of the separator soli phase potential solution phase potential electrolyte conuctivity S/m ionic conuctivity S/m eferences 1. 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