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Computational Fluid Dynamics Modeling of a Lithium/Thionyl Chloride Battery with Electrolyte Flow

W.B. Gu and C.Y. Wang GATE Center of Excellence for Advanced Energy Storage Department of Mechanical Engineering Pennsylvania State University University Park, PA 16802 Email: cxw31@psu.edu John Weidner Department of Chemical Engineering University of South Carolina Columbia, SC 29208 Rudolph G. Jungst Lithium Battery Research and Development Department 1521 Sandia National Laboratories P.O. Box 5800, MS 0613 Albuquerque, NM 87185

Submitted to Journal of Electrochemical Society March, 1999

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Abstract A two-dimensional model is developed to simulate discharge of a primary lithium/thionyl chloride battery. The model accounts for not only transport of species and charge but also the electrode porosity variations and electrolyte flow induced by the volume reduction upon electrochemical reactions. Numerical simulations are performed using a finite volume method of computational fluid dynamics. The predicted discharge curves for various temperatures are compared to the experimental data with excellent agreement. Moreover, the simulated results in conjunction with computer visualization and animation techniques reveal that the cell performance in the parametric range of interest is limited by pore plugging on the front side of the cathode as a result of LiCl precipitation. The detailed two-dimensional flow simulation also shows that the electrolyte feed to the cell is predominantly from the separator to the cathode during most part of discharge, especially for higher cell temperatures. This finding explains why an earlier one-dimensional model of Jain et al. (1998) accounting for unidirectional electrolyte flow is successful in predicting the cell discharge performance.

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Introduction In recent years there has been increased use of battery modeling and simulation in the exploration and development of advanced batteries for electric vehicles and consumer electronic applications. Unique advantages of coupling first-principle modeling with experimentation include: ? Providing a more complete understanding of the fundamental chemical and transport processes occurring inside a battery, and hence helping identify key mechanisms and parameters governing battery performance. ? Permitting what-if parametric studies to establish guidelines for more rational and therefore less costly experimental development efforts, particularly when destructive experiments are involved, such as abuse, thermal runaway and battery cycle life tests. ? Accelerating battery development cycles by allowing engineers to evaluate many design alternatives prior to constructing a prototype cell and establishing a rational basis for design optimization and innovation. ? Facilitating integration and interfacing of batteries with application devices like vehicles as well as the development of battery infrastructure such as chargers and thermal management systems by means of high-fidelity battery simulators rather than physical battery modules and packs. Much prior work has demonstrated the application of computer models to the first task with important examples of lead-acid,1-3 nickel-metal hydride,4-8 and lithium-based batteries.9-12 Most recently, an Internet-based battery simulation environment was created to provide a convenient tool to undertake the last three tasks mentioned above.13 This online simulation system allows users to submit input files, execute simulations, and receive results via the Internet in an encrypted format anywhere, anytime. The system has fully interactive pre- and post-processing interfaces and is particularly useful for battery designers and application engineers to collaborate via the Internet in real-time. The present paper is a continuation of the recent series of work to explore computational fluid dynamics (CFD) techniques in conjunction with experimentation for fundamental battery research. The application of interest in this work is a primary lithium/thionyl chloride battery. This battery has many desirable characteristics as a power source such as high energy and power densities, high operating cell voltage, excellence voltage stability over 95% of the discharge, and a large operating temperature range. As such, there has been a number of modeling studies14-18 in the literature. Earlier modeling efforts by Szpak et al.14 and Cho15 focused on the battery’s high-rate discharge and the ensuing thermal behaviors, whereas a most recent investigation of Jain et al.17-18 scrutinized fundamental electrochemical and transport phenomena at low to moderate discharge rates. An interesting phenomenon found to govern battery discharge in the latter regime is the electrolyte flow from the electrolyte header into the cell to replenish voids created by the volume reduction from reactants to products upon electrochemical reactions. This critical phenomenon cannot be addressed by almost all previous battery models that are one-dimensional in nature. Jain and Weidner17 showed that substantial modifications to a one-dimensional model are necessary in order to account properly for the reaction-shrinkage induced electrolyte flow. While the modified one-dimensional model of Jain et al.18 is simple and efficient in predicting the discharge of a Li/SOCl2

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battery, a multi-dimensional model fully accounting for the electrolyte flow without making ad hoc approximations is deemed valuable to gain a more fundamental understanding of the processes occurring in this type of batteries. The present paper describes such a model and a finite-volume method of computational fluid dynamics (CFD) to simulate a Li/SOCl2 battery in the operating regime of low to intermediate discharge rates. Computational fluid dynamics (CFD) is a numerical tool to analyze and optimize fluid flow, mass and thermal transport, and related phenomena (e.g. chemical reactions) that may simultaneously take place in a complex system. The broad scope, power, convenience, user interface, and pre- and post-processing capabilities developed for CFD over the last decade has made the technique very attractive. Over the past few years we have demonstrated its adaptation to solve electrochemical problems, in particular, battery problems that may simultaneously include electrolyte flow, charge transfer, species transport via diffusion, migration and convection, nonlinear charge transfer kinetics, and electrode structural changes.3,7,8 In the present work, we shall further demonstrate that the CFD technique in conjunction with visualization and animation tools can provide a viable means to fundamentally understand and diagnose advanced battery systems. Numerical Model Description of a Li/SOCl2 System A schematic of a lithium/thionyl chloride (Li/SOCl2) cell is shown in Figure 1. The system is identical to the one studied most recently by Jain et al.17-18 Basically the cell consists of a lithium foil anode, a LiCl film, a separator (glass matting), and a porous carbon cathode with the electrolyte being lithium tetrachloraluminate (LiAlCl4) dissolved in thionyl chloride (SOCl2). The passive lithium chloride (LiCl) film forms on the lithium foil due to lithium corrosion; namely, 4Li + 2SOCl2 → 4LiCl(s) + SO2 + S (1) The LiCl film can further be divided into two sub-regions: a relatively thin, compact film adjacent to the Li anode, known as the solid-electrolyte interface (SEI), and a much thicker, porous LiCl film formed next to the SEI, known as the secondary porous layer. No reaction occurs in the secondary layer and its matrix is not conducting; thus, it behaves much like the separator. During discharge, the following electrochemical reactions take place: Anode Cathode Li (s) → Li+ + e4Li + 4e + 2SOCl2 → 4LiCl(s) + SO2 + S

+ -

(2) (3)

That is, the lithium is oxidized at the anode and SOCl2 is reduced at the cathode, followed by precipitation of LiCl. Significant volume reduction occurs during reaction (3) due to the much larger molar volume of SOCl2 than LiCl precipitate, thereby resulting in electrolyte flow from the top header into the cell. See Fig.1. Assumptions The following assumptions are invoked in this work; they have been justified by Jain et al.18 for cell discharge at low to intermediate rates.

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1. 2. 3. 4.

LiCl precipitation occurs completely and instantaneously at the cathode; The cathode thickness remains constant; The porous LiCl film on the anode has a constant thickness and porosity; The solid-electrolyte interface (SEI) is treated as the Li anode surface and its effect on the cell performance is accounted for via the electrode kinetics; 5. The electrolyte is treated as a binary solution with constant properties. The effects of products S and SO2 are implicitly included in the properties of the solvent; 6. The cell remains isothermal; and 7. Effects of double-layer charging and self-discharge are ignored. Governing equations Based on the above-stated assumptions, a two-dimensional model for Li/SOCl2 batteries can be derived from the general modeling framework previously developed by Wang et al.6 Specifically, the present model consists of equations governing conservation of species, charge, mass and momentum in electrolyte and electrode matrix phases, respectively. Conservation of species.? Applying the general species conservation equation in Table I of Wang et al.6 to the species of interest in the Li/SOCl2 system, one obtains

o ?ε e c 1 ? t+ j + ? ? (vc ) = ? ? D eff ?c + F ?t

(

)

(4)

for Li+, and

?ε e c o 1 + ? ? (vco ) = ? ? D eff ?c o + jo ?t 2F

(

)

(5)

for SOCl2. Here ? =

? v ? v i+ j is for the 2-dimensional Cartesian coordinate system, c ?x ?y the concentration of a species in the electrolyte and j the volumetric reaction current resulting in production or consumption of the species. Subscript o denotes the solvent SOCl2, εe the volume fraction of the liquid electrolyte, v the superficial electrolyte o velocity, Deff the effective diffusivity of the electrolyte, and t + the transference number of lithium ion with respect to the volume-averaged velocity. The reaction currents are given by

_ ? a ? 1 i n1 at anode j=? _ ? a ? 2 i n 2 at cathode

(6)

and

? 0 ? jo = ? _ ? a i n2 ? 2 at anode at cathode

(7)

where the transfer current densities are expressed by the Butler-Volmer equation as

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_ ? ?α F ? ? c i n1 = io1,ref ?exp? a1 η1 ? ? ? ? ? ? ? RT ? c ref ?

? α F ?? ? exp? ? c1 η1 ? ? ? ? ? RT ?? ? ? ? ? ? ?

2

(8)

and i n 2 = io 2,ref

_

? α F ? ? c ?exp? ? a2 η2 ? ? ? ? ? RT ? ? ? c ref ?

? co ? ?c , ? o ref

? α F ?? ? exp? ? c 2 η 2 ?? ? ? ? RT ?? ? ?

(9)

In the above, the surface overpotential that drives the electrochemical reaction at the electrode/electrolyte interface is defined as

η j = φ s ? φ e ? U j ,ref

j = 1, 2

(10)

where φ is the electric potential, with subscript s and e denoting solid and liquid phases, respectively. The last term, Uj,ref, is the open-circuit potential of electrode reaction j measured using a reference electrode under the reference conditions. In Eqs. 6 and 7, a is the electrochemically active surface area per unit volume, with subscripts 1 and 2 denoting anode and cathode, respectively. While a1 is assumed to be constant, a2 varies during discharge due to LiCl precipitation via the following correlation19 ? ? ε o ? ε ?ζ ? o e e ? a = a ?1 ? ? (11) o ? ? ? ε ? ? e ? ? ? ? where superscript o represents initial values. The exponent ζ is an empirical parameter used to describe the morphology of the electroactive surface. The effective diffusivity of species in the electrolyte accounting for the effects of porosity and tortuosity is evaluated via the Bruggman correction

1.5 D eff = ε e D (12) where D is the diffusion coefficient of the electrolyte. Conservation of charge.? Applying the potential equations in Table I of Wang et al.,6 one has eff ? ? κ eff ?φ e + ? ? κ D ? ln c + j = 0 (13)

(

)

(

)

for the liquid phase potential and

? ? σ eff ?φ s ? j = 0

(

)

(14)

for the solid phase potential. Here κeff is the effective ionic conductivity of the electrolyte, and σeff is the effective electronic conductivity of the solid matrix, both of which can be evaluated by a similar expression to Eq. 12. The effective diffusional ionic eff conductivity, κ D , is given by

eff = κD

2 RTκ eff F

?0 c ? ?t + ? 1 + ? 2c o ? ?

(15)

for the reactions described by Eqs. (2) and (3).

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Conservation of mass.? Mass balance for the solid phase yields

j ∧ ?ε s = ? o V LiCl ?t F

(16)

∧

where εs is the volume fraction of the solid phase and V LiCl is the partial molar volume of species LiCl. Equation 16 implies that the volume fraction of the solid phase at the cathode increases due to the precipitation of LiCl film upon reaction (here jo<0), while it remains constant within the anode. For the binary electrolyte it follows that

c V + co V o = 1

∧ ∧ ∧ ∧

∧ ∧

(17)

where V and V o are the partial molar volumes of the electrolyte salt (LiAlCl4) and the solvent (SOCl2), respectively. Adding Eq. 4 × V to Eq. 5 × V o and making use of Eq. 17 yield o ∧ ∧ ?ε e 1 ? t+ 1 +??v = jV + jo V o (18) ?t F 2F Two scenarios may result from the reaction-induced electrolyte volume reduction during discharge: (1) there is excess electrolyte available in the header located above the cell and the cell is fully filled with the electrolyte all the time; and (2) there is no excess electrolyte present, causing part of the electrode to dry out and become electrochemically inactive with time. In Case 1 the following holds true throughout the cell:

εe + εs =1

Summing Eq. 18 and Eq. 16 and recognizing Eq.(19) thus result in

o ∧ ∧ 1? t+ 1 ?∧ ? ??v = jV + j o ?V o ? 2 V LiCl ? F 2F ? ?

(19)

(20)

Equation (20) represents the continuity equation of the liquid electrolyte and, when coupled with the electrolyte momentum equation, can be used to solve for the electrolyte flow field. By integrating Eq. 20 over the whole cell and noting the fact that the reaction current j cancels with each other in the anode and cathode, one obtains

HLc < ∫x=0 (v )y= H dx =

L

∧ jo > ? ∧ ? ? V o ? 2V LiCl ? 2F ? ?

(21)

where vy=H is the electrolyte feeding velocity at the top surface of the cell, H is the electrode height, Lc is the thickness of the cathode, and <jo> is the volume-averaged reaction current within the cathode. Equation 21 indicates that the total amount of the electrolyte sucked into the cell from the top header is directly proportional to the volume reduction of active materials inside the cell. In Case 2 there is no electrolyte replenishing the cell and thus the cell partly dries out as the discharge proceeds. The electrolyte velocity, vy=H, at the free surface of electrolyte is then equal to the moving velocity of the receding interface, thus yielding

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∧ 1 d < H > Lc < jo > ? ∧ ? = ?V o ? 2V LiCl ? <H > dt L 2F ? ?

(22)

where H is the local electrolyte height and the symbol <> denotes its average across the cell. Integrating Eq. 22 with time yields

∧ ? L ?∧ ? ? t < H >= H o exp ? c ? V o ? 2V LiCl ? ∫ < jo > dt ? ?o ? 2 FL ? ?

(23)

where <H> is the average electrolyte height in the cell and H o is the initial height. No electrochemical reactions take place where y>H due to the absence of electrolyte. Because Eq. 19 is valid for y≤H, Eq. 20 still holds true in this region of y≤H. Equation (23) allows to calculate the change in the domain geometry as the electrolyte recedes and hence enables re-gridding of the mesh in numerical simulations. A moving-boundary numerical technique is warranted to simulate Case 2. Conservation of momentum.? For creeping flow of the electrolyte through a porous electrode, the momentum equation in Table I of Wang et al.6 reduces to the familiar Darcy’s law; i.e. K v = ? ?p (24)

?

where p is the pressure, K the permeability of the porous medium, and ? the dynamic viscosity of the electrolyte. Substituting Eq. 24 into Eq. 20 results in

o ∧ ∧ ?K ? 1 ? t+ 1 ?∧ ? ? ??? ? p + j V + j V ? 2 V o LiCl ? = 0 ? o ?? ? F 2F ? ? ? ?

(25)

This equation is solved for the pressure field in the electrolyte, with the permeability evaluated by the Kozeny-Carman relation, K=

ε 3d 2 2 180( 1? ε )

(26)

where d is the diameter of particles that make up the porous electrode. Once the pressure field is determined, the electrolyte flow field can be calculated from Eq.(24). Equations 4, 13, 14, 16, and 25 form a complete set of governing equations for five unknowns: c, φe, φs, εs, and p. Their corresponding initial and boundary conditions are as follows: Initial conditions c = c o , and ε s = 1 ? ε eo at t = 0 (27) Boundary conditions ?c ? c = c o for Case 1 and = 0 for Case 2 at y = H ? ?y ? Concentration: (28) ? ? c ? = 0 at other boundaries ? ? ?n

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Pressure:

? p = 0 at y = H ? ? ?p = 0 at other boundaries ? ? ?n

(29)

? ?φ e ? ?n = 0 at all boundaries ? ?φ ? s = 0 at y = 0 and H ? ?y ? φ s = 0 at x = 0 ? Potential: (30) ? ? ? ? φ s = V for potentiostatic discharge ? ? I ? eff ?φ s ? ?σ = for galvanic discharge ? at x = L ?x A ? ? φs ? eff ?φ s for constant load discharge? = ? ?σ ? ?x Ro A ? ? where V, I and Ro are the applied cell voltage, current, and load, respectively, and A is the projected effective electrode area. Numerical Procedures The conservation equations, Eqs.4, 13, 14, 16, and 25, are discretized using a finite volume method20 and solved using a general-purpose computational fluid dynamic (CFD) code. Details have been given in the previous work.3,21 It should be mentioned that stringent numerical tests were performed to ensure that the solutions were independent of the grid size and time step. It was found that a 47×32 mesh provides sufficient spatial resolution and the time step normally ranges from 600-1800 seconds, except for the very beginning and the end of discharge where smaller time steps were required. The equations were solved as a simultaneous set, and the solution was considered to be convergent when the relative error in each field between two consecutive iterations was -5 less than 10 . A typical transient simulation for two-dimensional problems with electrolyte flow required about 30 minutes of CPU time on an SGI Octane workstation or a 400MHz PC. Results and Discussion Experimental Verification To validate the numerical model developed in the preceding section, comparisons were made to the experimental data most recently reported by Jain et al.18 for a D-sized, spirally wound Li/SOCl2 cell over a range of temperatures. The geometrical parameters of the cell along with the electrochemical and transport properties, all taken from Jain et al.,18 are summarized in Table I and Appendix A, respectively. Figure 2 compares the predicted cell potential curves with the experimental results for constant load discharge at 50 ?. Good agreement can be seen for all three operating temperatures, indicating that the present two-dimensional model can accurately predict

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the cell performance. Consistent with the previous work of Jain et al.,18 the end of discharge marked by a drastic drop in the cell potential as seen from Fig.2 for all three temperatures is due to the front of the cathode clogged with LiCl precipitate. It is also interesting to note that the modified one-dimensional model of Jain et al.17-18 accounting for a unidirectional electrolyte flow across the cell produced almost the same kind of comparisons with the experimental data. This fact prompts us to show that the present two-dimensional model is reduced identically to the model of Jain et al.17-18 under onedimensional assumptions. Appendix B describes this derivation in detail and concludes that the two key equations in Jain and Weidner’s one-dimensional model17 are indeed recovered. Moreover, the success of the one-dimensional model to capture the cell behavior implies that the unidirectional electrolyte flow must be a good assumption in the three cases examined in Figure 2. This flow pattern is not apparent from the first glimpse but can be corroborated by the present two-dimensional flow simulation, as to be shown in the following. Visualizing Electrolyte Flow A unique advantage of detailed two-dimensional computer simulation is the ability to visualize the electrolyte flow inside a battery, thus providing much insight into the battery internal operation during discharge. This capability, while highly desired for fundamental battery research, is difficult if not impossible to realize by pure experimentation. The electrolyte flow fields simulated by the present two-dimensional model are displayed in Figs. 3 and 4 at four representative instants of time during battery discharge at –18oC and –55oC, respectively. These plots are in the form of streamlines, i.e. lines everywhere tangent to the velocity vector at a given instant. Different colors in the plots represent the magnitude of the velocity component in the vertical direction with the scale illustrated by the color bar on the side. It is clearly shown from Fig. 3 that the electrolyte flow induced by volume reduction due to electrochemical reaction is in the horizontal direction from the separator into the cathode for most part of discharge. This is because the separator is highly porous and remains to be during discharge. Also, due to the large aspect ratio of cell height to the cathode width the horizontal flow is preferred over the vertical flow from the top of the cell to the bottom. Therefore, the separator essentially serves as a reservoir to feed excess electrolyte into the cathode. This flow pattern persists until the front side of the cathode becomes nearly plugged thus leading to blockage of electrolyte flow from the separator. At this moment, the electrolyte needed to fill voids created by the electrochemical reaction in the cathode must come from the top permeable surface. The transition in the flow patterns is illustrated in Fig. 3 at t=127 hrs. As the end of discharge approaches (i.e. at t=135 hrs), the front side of the cathode is completely plugged so that the electrolyte flows in the separator and cathode are entirely separated. Within the separator the electrolyte volume begins to expand due to the continual Li+ production from the anode, causing over-flow of the electrolyte towards the top of the cell although the magnitude of its velocity is small. By contrast, the electrolyte is sucked into the cathode from the upper-right corner and flows towards the reaction sites at the separator-cathode interface where volume reduction continues to occur due to the cathode reaction. The electrolyte flow fields during discharge at –55oC as displayed in Fig. 4 exhibit the same features as in the case of –18oC, except that the flow becomes two-dimensional

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at earlier stages of discharge (see the image at the upper-right corner of Fig. 4). This is because the electrochemical reaction inside the cathode is more non-uniform at a lower cell temperature and hence more non-uniform electrolyte feed from the top header. Electrolyte Concentration, Reaction Rate and Electrode Porosity Distributions Figures 5 and 6 show the predicted electrolyte concentration profiles at the half height of the cell and how they evolve with time in the cases of –18oC and –55oC, respectively. The concentration increases from its initial value in the separator owing to the Li+ production from the anode, whereas the electrolyte is consumed in the cathode. As discharge proceeds, LiCl precipitate begins to fill the pores of the cathode and the concentration gradient becomes greater at the separator-cathode interface due to the smaller porosity of this region which reduces significantly the effective diffusion coefficient. The electrolyte transport thus becomes more difficult at later times. In the discharge case of –18oC (i.e. Fig. 5), the separator-cathode interface is completely shut off at t=135 hrs and the electrolyte becomes nearly depleted within the cathode, leading to the end of discharge. The end of discharge in the case of –55oC occurred around 58 hours (Fig. 6). The reaction rate profiles within the cathode, also at the half height of the cell, are displayed in Figs. 7 and 8 at various times for –18oC and –55oC, respectively. At early stages of discharge, the reaction rate is maximal at the front and minimal at the back of the cathode, as typically found in most batteries. The profile reaches a pseudo-steady state around 80-100 hours in the case of –18oC discharge (i.e. Fig.7). Thereafter, the reaction rate along the separator-cathode interface decreases because the local surface area available for reaction is reduced due to LiCl precipitation. Note that the volumetric reaction rate depicted in Figs. 7 and 8 is a product of the transfer current density and the electrochemically active surface area per unit volume. The reduction in the reaction rate as the pores at the front of the cathode become more plugged leads to a valley in the profile in the vicinity of the cathode front side, as can be seen from both Figs. 7 and 8. Figure 9 shows the final porosity profiles at the half height of the cell when the cell is fully discharged at –55oC, -18oC, and 25oC, respectively. It is clearly seen that the cell lifetime at all three temperatures is limited by pore plugging at the front of the cathode where the porosity goes to zero. Furthermore, the final porosity averaged across the cathode, denoting the amount of Li precipitate, is a measure of the cell capacity. The higher the cell temperature, the more uniform the electrode porosity is before the front side of the cathode becomes clogged and hence the higher capacity. In contrast, at a lower temperature (e.g. -55oC) only the very front portion of the cathode is utilized and becomes plugged, causing a quicker end of discharge. Much more simulation results, particularly the two-dimensional contours of electrolyte concentration, reaction rate and electrode porosity, are available from the web site at: http://mtrl1.me.psu.edu/mtrl/ResProj8.html. These graphs are best illustrated in color and thus are not contained in this communication. Furthermore, these twodimensional plots have been animated to visualize the entire discharge process and hence provide a great tool to identify the battery limiting mechanisms. These animated movies are also accessible for viewing from the web site.

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Conclusions A two-dimensional model of a Li/SOCl2 battery has been developed based on the multiphase modeling framework of Wang et al.6 The present model includes, among other features, the electrolyte flow induced by the volume reduction upon electrochemical reactions. The model predictions were compared to the experimental data with good agreement over a range of cell temperatures. More significantly, the two-dimensional model allowed for numerical visualization of the electrolyte flow occurring inside the battery for the first time. This capability in conjunction with the computer animation technique yielded much insight into the battery internal operation and provided convincing evidence to justify for the one-dimensional assumptions made in the earlier model of Jain et al.17-18 Future work includes the extension of this model to high rate discharge where thermal effects become significant and must be considered along with charge transfer kinetics, mass and charge conservation, electrode structural variations and electrolyte flow. The extension to couple the present formulation with a thermal model is straightforward within the present computational fluid dynamics (CFD) framework. Such an extended model will be extremely useful to address the cell abuse and thermal runaway issues. Acknowledgments C.Y. Wang acknowledges the partial support of this work by a NSF CAREER Award, grant no. CTS-9733662.

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Table I. Values of parameters used in the simulations18 Symbol Geometric A ao H Lf Ls Lc εf εs εο ζ Chemical co Value 180 1000 4.445 0.001 0.023 0.085 0.95 0.95 0.835 0.05 0.001 77.97 20.5 72.63 0.7 Eq. A-1 1×10-5 1×10-2 Eq. A-2 45.5 Eq. A-3 Eq. A-4 0.8, 0.2 1.7, 0.3 -E 0 Eq. A-5 2.28×10-4 Unit cm2 cm2/cm3 cm cm cm cm Description Projected electrode area Initial specific interfacial area of cathode Electrode height Thickness of LiCl film at Li anode Thickness of separator Thickness of carbon cathode Porosity of LiCl film at Li anode Porosity of separator Initial porosity of carbon cathode Morphology correction factor Initial concentration of electrolyte Partial molar volume of LiAlCl4 salt Partial molar volume of SOCl2 solvent Partial molar volume of LiCl solid Transference number of lithium ion Molecular diffusivity of electrolyte Permeability of porous separator Dynamic viscosity of electrolyte Ionic conductivity of electrolyte Electronic conductivity of solid matrix Exchange current density of anode reaction Exchange current density of cathode reaction Transfer coefficients of anode reaction Transfer coefficient of cathode reaction Open-circuit potential of anode reaction Open-circuit potential of cathode reaction Cell open-circuit voltage Temperature dependence of E

mol/cm3 cm3/mol cm3/mol cm3/mol

V

Vo

∧

∧

V LiCl Transport

o t+ D Κ ? κ σ Kinetic io1,ref io2,ref αa1, αc1 αa2, αc2 U1,ref U2,ref E dE dT

∧

cm /s cm2 g/cm?s S/cm S/cm A/cm2 A/cm2

2

V V V V/K

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Appendix A: Property Correlations18 1. Diffusion coefficient of LiAlCl4-SOCl2 electrolyte

? 2.315 × 10 4 2.395 × 10 6 ? ? + D = 1.726 × 1016 exp? ?? ? T T2 ? ?

(A-1)

2. Ionic conductivity of LiAlCl4-SOCl2 electrolyte

? ? 4.88 × 10 5 c ? 71.73 ? 5 2 9 . 79 c exp 2039 . 09 c 2 . 5055 10 c exp ? × ? ? for c < 1.8 M ?? T ? ? ? ? ? 4.88 × 10 5 c ? 71.73 ? ? κ = ?1.6 × 10 ? 2 exp 1.63 × 10 3 c exp ?? ? for 1.8 M ≤ c < 2.0 M T ? ? ? ? 5 ? ? ? 2.11 × 10 ?2 ? 2.53c exp 1.63 × 10 3 c exp ?? 4.88 × 10 c ? 71.73 ? for c > 2.0 M ? T ? ? ? (A-2) 3. Exchange current density for the anode reaction

(

)

(

)

(

)

(

)

(

) (

)

(

)

? 4641 ? io1, ref = 1.157 × 103 exp? ? ? T ? ?

4. Exchange current density for the cathode reaction

(A-3)

? 5500 ? io 2, ref = 2.5 × 103 exp? ? ? T ? ?

5. Open-circuit voltage of Li/SOCl2 cell

E = 3.723 + T dE dT

(A-4)

(A-5)

15

Appendix B: Reduction to Jain and Weidner’s One-Dimensional Model17-18 Equation 4 in the present two-dimensional model can be reduced to Jain and Weidner’s model17-18 under two assumptions: (1) the cell is one-dimensional so that all variables are uniform along the electrode height, and (2) the electrolyte flow within the cell is caused only by the net volume reduction of the electrolyte due to the electrochemical reaction expressed by Eq. 2. Integrating Eq. 4 over the electrode height results in

∫

H

0

o H ? ? ? ? ?(ε e c ) ? (v x c ) ? (v y c )? ? ? eff ?c ? 1 ? t + eff ?c ? ? ? = + + dy D D + + ? ? ? ? ∫0 ? ?y ? ?x ? ?y ? ?y ? F ?x ? ? ? ?t ? ?x ?

? j ?dy ?

(B-1)

Under Assumption 1, Eq. (B-1) is then simplified to

o N y =H ?(ε e c ) ?(v x c ) ? ? eff ?c ? 1 ? t + j? + = ?D ?+ F H ?t ?x ?x ? ?x ?

(B-2)

where

? ? eff ?c N y= H = ? ? ? D ?y + v y c ? ? ? ? y =H

(B-3)

representing the net species flux from the top permeable surface. To arrive at Eq. (B-2), use has been made of the no-flux boundary condition at the cell bottom surface. In Case 1 where the excess electrolyte with initial concentration co enters the electrode, the molar flux of the electrolyte at y = H can be expressed, according to Assumption 2, as

N y= H = c o

∧ jo ? ∧ ?V o ? 2 V 2F ? LiCl

? ?H ?

(B-4)

Noting that

jo = j =

?i e ?x

(B-5)

Eq. (B-4) thus becomes

N y= H H

∧ j co ? ∧ ? = ?V o ? 2V LiCl ? F 2 ? ?

(B-6)

In Case 2 where no excess electrolyte is present at the top and the free surface of the electrolyte recedes due to the electrolyte volume reduction, Eq. (B-3) is reduced to

N y = H = (v y c )y = H

(B-7)

where H is the moving electrolyte height, the velocity of which varies in accordance with the amount of volume reduction occurring inside the cell; namely, vy

εe

=

∧ j ?∧ ? ?V o ? 2V LiCl ? H 2F ? ?

(B-8)

Combining Eq. (B-7) with Eq. (B-8) yields

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∧ j ε ec ? ∧ ? ?V o ? 2V LiCl ? F 2 ? ?

N y =H H

=

(B-9)

Substituting Eq. (B-6) or Eq. (B-9) into Eq. (B-2) and making proper arrangements result in o ∧ ? j ? co ? ∧ ie t + ?ε e ?c ? ? ?? eff ?c ? (B-10) εe D cv 1 V 2 V +c =? ? ? + + + ? ? o LiCl ? ? ? x? ? ? F 2 ? ?t ?x ?x ? ?x ?? ? F? for Case 1 where excess electrolyte is available above the cell, or

o ∧ ? j ? εec ? ∧ ie t + ?ε ?c ? ? ?? eff ?c ? D cv 1 V 2 V +c e =? ? ? + + + ? ? o LiCl ?? ? x? ? ? F 2 ? ?t ?x ?x ? ?x ?? ? F?

εe

(B-11)

for Case 2 where there is no excess electrolyte in the battery. Equations (B-10) and (B-11) are identical to the one-dimensional species conservation equations developed by Jain and Weidner.17-18 In the parametric range examined in the present paper, the one-dimensional assumption of horizontal electrolyte flow appears reasonable as verified by the two-dimensional flow field presented in this paper. However, caution should be exercised that the one-dimensional model may not hold true in other circumstances. Nomenclature A aj c D d F H I i nj ioj j K L Lc p R Ro T t o t+ Uj projected effective electrode area, cm2 specific interfacial area active for reaction j, cm2/cm3 concentration of a species, mol/cm3 diffusion coefficient of the electrolyte, cm2/s diameter of particles that make up the porous electrode, cm Faraday constant, 96487 C/mol electrolyte height, cm applied current, A transfer current density of reaction j at the electrode/electrolyte interface, A/cm2 exchange current density of reaction j, A/cm2 volumetric reaction current resulting in production or consumption of a species, A/cm3 permeability of the porous medium, cm2 cell width, cm thickness of the cathode, cm electrolyte pressure, g/cm2 universal gas constant, 8.3143 J/mol?K applied load, ? absolute temperature of the cell, K time, s transference number of lithium ion with respect to the volume-averaged velocity open-circuit potential of electrode reaction j, V

17

v v

volume-averaged velocity vector, cm/s velocity component in y-direction, cm/s partial molar volume of a species, cm3/mol x-coordinate in Cartesian coordinate system, cm y-coordinate in Cartesian coordinate system, cm anodic and cathodic transfer coefficients of reaction j porosity of a porous medium or volume fraction of a phase morphology correction index in Eq. 11 surface overpotential of reaction j, V ionic conductivity of the electrolyte, S/cm diffusional conductivity, A/cm dynamic viscosity of the electrolyte, g?cm/s electronic conductivity of the solid matrix, S/cm phase potential, V

V x y

Greek αaj, αcj ε ζ ηj κ κD ? σ φ

∧

Subscript e electrolyte phase o solvent ref at reference conditions s solid phase Superscript o initial value eff effective References 1. D.M. Bernardi and M.K. Carpenter, A mathematical model of the oxygenrecombination lead-acid cell, J. Electrochem. Soc., 142, 2631 (1995). 2. J. Newman and W. Tiedemann, Simulation of recombinant lead-acid batteries, J. Electrochem. Soc., 144, 3081 (1997). 3. W.B. Gu, C.Y. Wang and B.Y. Liaw, Numerical modeling of coupled electrochemical and transport processes in lead-acid batteries, J. Electrochem Soc, 144, 2061 (1997). 4. P. De Vidts, J. Delgado and R.E. White, Mathematical modeling for the discharge of a metal hydride electrode, J. Electrochem. Soc., 142, 4013 (1995). 5. B. Paxton and J. Newman, Modeling of nickel/metal hydride batteries, J. Electrochem. Soc., 144, 3831 (1997). 6. C.Y. Wang, W.B. Gu and B.Y. Liaw, Micro-macroscopic modeling of batteries and fuel cells. Part 1. model development, J. Electrochem. Soc., 145, 3407 (1998).

18

7. W.B. Gu, C.Y. Wang and B.Y. Liaw, Micro-macroscopic modeling of batteries and fuel cells. Part 2. application to nickel-cadmium and nickel-metal hydride cells, J. Electrochem. Soc., 145, 3418 (1998). 8. W.B. Gu, C.Y. Wang, S. Li, M. Geng and B.Y. Liaw, Modeling discharge and charge characteristics of nickel-metal hydride batteries, Electrochimica Acta, accepted for publication, 1999. 9. M. Doyle, T.F. Fuller and J. Newman, Modeling of galvanostatic charge and discharge of the lithium/polyer/insertion cell, J. Electrochem. Soc., 140, 1533 (1993). 10. T.F. Fuller, M. Doyle and J. Newman, Simulation and optimization of the dual lithium ion insertion cell, J. Electrochem. Soc., 141, 10 (1994). 11. M. Doyle, J. Newman, A.S. Gozdz, C.N. Schmutz, and J.-M. Tarascon, Comparison of modeling predictions with experimental data from plastic lithium ion cells, J. Electrochem. Soc., 143, 1903 (1996). 12. P. Arora, R.E. White and M. Doyle, Capacity fade mechanism and side reactions in lithium-ion batteries, J. Electrochem. Soc., 145, 3647 (1998). 13. X.O. Chen, W.B. Gu and C.Y. Wang, Internet-based modeling and simulation of advanced battery systems, Abstract No.53, presented at 194th Electrochemical Society Mtg, Boston, Nov. 1998. 14. A. Szpak, C.J. Gabriel and J.R. Driscoll, Catastrophic thermal runaway in lithium batteries, Electrochimica Acta, 32, 239 (1987). 15. Y.I. Cho, Thermal modeling of high rate Li-SOCl2 primary cylindrical cells, J. Electrochem. Soc., 134, 771 (1987). 16. T.I. Evans, T.V. Nguyen and R.E. White, A mathematical model of a lithium/thionyl chloride primary cell, J of Electrochem. Soc., 136, 328 (1989). 17. M. Jain and J.W. Weidner, Material balance modificiation in one-dimensional modeling of porous electrodes, J of Electrochem. Soc., submitted for publication, 1998. 18. M. Jain, G. Nagasubramanian, R.G. Jungst and J.W. Weidner, Mathematical model of a lithium/thionyl chloride battery, J of Electrochem. Soc., submitted for publication, 1998. 19. J.S. Dunning, D.N. Bennion and J. Newman, J. Electrochem. Soc., 120, 906 (1973). 20. S.V. Patankar, Numerical Heat Transfer and Fluid Flow, Taylor & Francis (1980). 21. W.B. Gu, C.Y. Wang and B.Y. Liaw, “The use of computer simulation in the evaluation of electric vehicle batteries,” J. Power Sources, 75/1, 154 (1998).

19

List of Figures Figure 1 Schematic of a Li/SOCl2 cell. Figure 2 Comparison of experimental and predicted discharge curves for a 50 ? load at 55, -18, and 25oC. The symbols represent the experimental data, while the solid lines are the predicted results. Figure 3 Electrolyte flow within the cell ? streamline plot. The cell is discharged at – 18oC with a 50 ? load and the end of discharge is 135.2 hours. Figure 4 Electrolyte flow within the cell ? streamline plot. The cell is discharged at – 55oC with a 50 ? load and the end of discharge is 58.2 hours. Figure 5 Electrolyte concentration profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –18oC. The end of discharge is 135.2 hours. Figure 6 Electrolyte concentration profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –55oC. The end of discharge is 58.2 hours. Figure 7 Reaction current profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –18oC. The end of discharge is 135.2 hours. Figure 8 Reaction current profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –55oC. The end of discharge is 58.2 hours. Figure 9 Porosity profiles along the half height of the electrode at the end of discharge with various temperatures.

20

–

Current Collectors

+

Vapor Excess Electrolyte (SOCl2+LiAlCl4)

Lithium Anode

LiCl Film

Separator

Porous Carbon Cathode

H

y x

Lf Ls Lc

Figure 1. Schematic of a Li/SOCl2 cell.

21

4.0

3.6

25 oC -18 C

o

Cell potential (V)

3.2

-55 oC

2.8

2.4

2.0

0

2

4

6

8

10

12

14

16

18

Capacity (Ah)

Figure 2. Comparison of experimental and predicted discharge curves for a 50 ? load at –55, -18, and 25oC. The symbols represent the experimental data, while the solid lines are the predicted results.

22

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

t = 0.1 h

t = 50 h

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

t = 127 h

t = 135 h

Figure 3. Electrolyte flow within the cell ? streamline plot. The cell is discharged at –18 oC with a 50 ? load and the end of discharge is 135.2 hours. The color bars on the side denote the vertical velocity component in cm/s.

23

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

t = 0.1 h

t = 25 h

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

(-) anode

cathode (+)

-1.0E-09 -5.3E-07 -1.1E-06 -1.6E-06 -2.1E-06 -2.6E-06 -3.2E-06 -3.7E-06 -4.2E-06 -4.7E-06 -5.3E-06 -5.8E-06 -6.3E-06 -6.8E-06 -7.4E-06 -7.9E-06 -8.4E-06 -8.9E-06 -9.5E-06 -1.0E-05

t = 50 h

t = 58 h

Figure 4. Electrolyte flow within the cell ? streamline plot. The cell is discharged at –55 oC with a 50 ? load and the end of discharge is 58.2 hours. The color bars on the side denote the vertical velocity component in cm/s.

24

8

Electrolyte concentration (M)

0.1 h 6 10 h 50 h 100 h 135 h 4

2

0

0

0.02

0.04

0.06

0.08

0.1

Cell width (cm)

Figure 5. Electrolyte concentration profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –18oC. The end of discharge is 135.2 hours.

25

8

Electrolyte concentration (M)

0.1 h 6 5h 25 h 50 h 58 h 4

2

0

0

0.02

0.04

0.06

0.08

0.1

Cell width (cm)

Figure 6. Electrolyte concentration profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –55 oC. The end of discharge is 58.2 hours.

26

-0.01

-0.01

cathode

-0.008 -0.008

cathode

Reaction current (A/cm )

100 h 120 h 125 h 127 h 130 h

3

Reaction current (A/cm )

3

-0.006

-0.006

0.1 h

-0.004

-0.004

20 h 40 h

-0.002

60 h

80 h

-0.002

135 h

0

0

0.02

0.04

0.06

0.08

0.1

0

0

0.02

0.04

0.06

0.08

0.1

Cell width (cm)

Cell width (cm)

Figure 7. Reaction current profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –18 oC. The end of discharge is 135.2 hours.

27

-0.025

-0.025

30 h 20 h

-0.02

cathode

-0.02

40 h 50 h 54 h

-0.015

cathode

Reaction current (A/cm )

Reaction current (A/cm )

3

3

-0.015

10 h

-0.01

55 h 56 h

-0.01

-0.005

0.1 h

-0.005

58 h

0

0

0.02

0.04

0.06

0.08

0.1

0

0

0.02

0.04

0.06

0.08

0.1

Cell width (cm)

Cell width (cm)

Figure 8. Reaction current profiles at the half height of the electrode when the cell is discharged with a 50 ? load at –55oC. The end of discharge is 58.2 hours.

1

0.8

-55 C

o

Porosity

0.6

- 18 oC

0.4

0.2

25 C

o

0

0

0.02

0.04

0.06

0.08

0.1

Cell width (cm)

Figure 9. Porosity profiles along the half height of the electrode at the end of discharge with various temperatures.

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