Analytical solution for electrolyte concentration distribution in lithium-ion batteries

Abstract

In this article, the method of separation of variables (SOV), as illustrated by Subramanian and White (J Power Sources 96:385, 2001), is applied to determine the concentration variations at any point within a three region simplified lithium-ion cell sandwich, undergoing constant current discharge. The primary objective is to obtain an analytical solution that accounts for transient diffusion inside the cell sandwich. The present work involves the application of the SOV method to each region (cathode, separator, and anode) of the lithium-ion cell. This approach can be used as the basis for developing analytical solutions for battery models of greater complexity. This is illustrated here for a case in which non-linear diffusion is considered, but will be extended to full-order nonlinear pseudo-2D models in later work. The analytical expressions are derived in terms of the relevant system parameters. The system considered for this study has LiCoO₂–LiC₆ battery chemistry.

Appendix

The governing equations are given in dimensional form as:

$$ \varepsilon_{\text{p}} \frac{{\partial c_{1} }}{\partial t} = D_{{{\text{eff}},{\text{p}}}} \frac{{\partial^{2} c_{1} }}{{\partial x^{2} }} + a_{\text{p}} \left( {1 - t_{ + } } \right)j_{\text{p}} \quad 0 \le x \le l_{\text{p}} \, $$

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$$ \varepsilon_{\text{s}} \frac{{\partial c_{2} }}{\partial t} = D_{\text{eff,s}} \frac{{\partial^{2} c_{2} }}{{\partial x^{2} }}\quad l_{\text{p}} \le x \le l_{\text{s}} { + }l_{\text{p}} \, $$

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$$ \varepsilon_{\text{n}} \frac{{\partial c_{3} }}{\partial t} = D_{{{\text{eff}},{\text{n}}}} \frac{{\partial^{2} c_{3} }}{{\partial x^{2} }} + a_{n} \left( {1 - t_{ + } } \right)j_{\text{n}} \quad l_{\text{p}} + l_{\text{s}} \le x \le l_{\text{n}} + l_{\text{p}} { + }l_{\text{s}} $$

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With BCs determined from continuity of concentration and mass flux given as:

$$ - D_{{{\text{eff}},{\text{p}}}} \frac{{\partial c_{1} }}{\partial x} = 0\quad {\text{at}}\;x = 0,\quad - D_{{{\text{eff}},{\text{n}}}} \frac{{\partial c_{3} }}{\partial x} = 0\quad {\text{at}}\;x{ = }l_{\text{p}} + l_{\text{s}} + l_{\text{n}} $$

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$$ - D_{{{\text{eff}},{\text{p}}}} \frac{{\partial c_{1} }}{\partial x} = - D_{{{\text{eff}},{\text{s}}}} \frac{{\partial c_{2} }}{\partial x}\quad{\text{at}}\;x = l_{\text{p}} ,\quad - D_{{{\text{eff}},{\text{s}}}} \frac{{\partial c_{2} }}{\partial x} = - D_{{{\text{eff}},{\text{n}}}} \frac{{\partial c_{3} }}{\partial x}\quad {\text{at}}\;x = l_{\text{p}} + l_{\text{s}} $$

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$$ c_{1} \left( {x,t} \right) = c_{2} \left( {x,t} \right)\quad{\text{at}}\;x = l_{\text{p}} ,\quad c_{2} \left( {x,t} \right) = c_{3} \left( {x,t} \right)\quad {\text{at}}\;x = l_{\text{p}} { + }l_{\text{s}} $$

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With IC is given as:

$$ c_{1} \left( {x,0} \right) = c_{2} \left( {x,0} \right) = c_{3} \left( {x,0} \right) = c_{0} \left( x \right) $$

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For this study, the pore wall flux in the electrode is assumed to be constant across space and time. In order to maintain a charge balance, the ionic fluxes in the positive and negative electrodes are not necessarily identical in magnitude, but scaled according to the electrode thickness and specific surface area. This leads to the following relations for the pore wall flux:

$$ j_{\text{p}} = - \frac{{i_{\text{app}} }}{{a_{\text{p}} Fl_{\text{p}} }} \, $$

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$$ j_{\text{n}} = + \frac{{i_{\text{app}} }}{{a_{n} Fl_{\text{n}} }} \, $$

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The diffusion coefficients at the respective electrodes are expressed in terms of the Bruggeman coefficient and porosity within each electrode.

$$ D_{{{\text{eff}},i}} = D\varepsilon_{i}^{\text{Brugg}} \quad {\text{for}}\;i = {\text{p}},{\text{s}},{\text{n}} $$

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The following dimensionless variables are used in order to transform the above equations to dimensionless form and to ensure that the solution is solved in terms of the system parameters ($l_{\text{p}} ,l_{\text{s}} ,l_{\text{n}} ,p,q,\varepsilon_{\text{p}} ,\varepsilon_{\text{s}} ,\varepsilon_{\text{n}}$):

$$ C_{i} = \left( {\frac{{c_{i} }}{{c_{0} }} - 1} \right) /J{\text{p}}\quad {\text{for}}\;i = 1, 2, 3 $$

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$$ p = \frac{{l_{\text{p}} }}{L},\quad q = \frac{{l_{\text{p}} + l_{\text{s}} }}{L} \, $$

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$$ \tau = \frac{Dt}{{L^{2} }} \, $$

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$$ X = \frac{x}{L} $$

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$$ J_{i} = \frac{{i_{\text{app}} \left( {1 - t_{ + } } \right)L^{2} }}{{\varepsilon_{i} Dc_{0} l_{i} F}}\quad {\text{for}}\;i = {\text{p}},{\text{n}} $$

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$$ K = \frac{{J_{\text{n}} }}{{J_{\text{p}} }} $$

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where

$$ L = l_{\text{p}} + l_{\text{s}} + l_{\text{n}} \, $$

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