Electrolyte Transport Equations
A modified version of Ohm’s law describes the charge transport in the electrolyte. In the reservoir, the current density expression is defined as
where σl denotes the electrolyte conductivity, R the molar gas constant, T the temperature, F Faraday’s constant, and t+ the transport number.
In the porous domains (the separator and the porous electrodes), it is defined as
Current conservation is described by . Here Ql denotes the sum of all current contributions.
The dissociated salt ions can be transported due to convection, migration and diffusion. The molar flux vector, Nl, (SI unit: mol/m2 s), is written as:
Where D (SI unit: mol/(s /m2)) is the binary diffusion coefficient into which the migration effects are incorporated, and u (SI unit: m/s) is the volume averaged velocity.
In the reservoir, the material balance equation is
In the porous electrodes the electrochemical reactions give rise to sources in the material balance equation, resulting in
where Rl is the a source term (SI unit: mol/s /m3) resulting from the electrochemical reactions. This source term is according to the equation below:
where nm denotes the number of electrons involved in reaction (dimensionless), the partial molar volume of the solvent (H2O) (SI unit: m3/mol), the partial molar volume of the electrolyte (SI unit: m3/mol),
the stoichiometric coefficients for the proton,
the stoichiometric coefficient for the bisulfate ion, and
the stoichiometric coefficient for water in reaction m.
In the separator, the corresponding transport equation is used for the electrolyte, but where the source term, Rl, is zero.
For a boundary, the flux of electrolyte species due to the electrochemical reactions is calculated according to
As the solid material in the electrodes react, the porosity changes due to volume changes. This is described by the equation below:
where denotes the molar volumes (SI unit: m3/mol) for the solid materials in the electrodes,
the stoichiometric coefficient of lead,
the stoichiometric coefficient of lead oxide, and
the stoichiometric coefficient for lead sulfate in reaction m. For a species with a known density, ρi (SI unit: kg/m3), and molar mass Mi it can be calculated as
In the separator the porosity is constant.
For results and analysis purposes the following state-of-charge expression, soc, for the electrodes, is also defined
Assuming that the main Pb and PbO2 reactions are the main contributions to the currents in each electrode, the average superficial velocity in each electrode compartment can be calculated as
and
Because the reservoir is normally adjacent to the positive PbO2 electrode,
is usually a good approximation in the reservoir and
can be used in the separator.
These assumptions result in a noncontinuous velocity, and hence a noncontinuous convection contribution to the electrolyte transport, at the interface between the separator and reservoir domains. The physical background to this is that as water is either consumed or created at the positive electrode during charge or discharge, the liquid level in the reservoir either rises or falls. However, the combination of low velocities and moderate concentration gradients at the separator/reservoir interface makes these velocity approximations acceptable in most cases.