  We present porous electrode theory for capacitive deionization (CDI) with
electrodes containing nanoparticles that consist of a redox-active
intercalation material. A geometry of a desalination cell is considered which
consists of two porous electrodes, two flow channels and an anion-exchange
membrane, and we use Nernst-Planck theory to describe ion transport in the
aqueous phase in all these layers. A single-salt solution is considered, with
unequal diffusion coefficients for anions and cations. Similar to previous
models for CDI and electrodialysis, we solve the dynamic two-dimensional
equations by assuming that flow of water, and thus the advection of ions, is
zero in the electrode, and in the flow channel only occurs in the direction
along the electrode and membrane. In all layers, diffusion and migration are
only considered in the direction perpendicular to the flow of water. Electronic
as well as ionic transport limitations within the nanoparticles are neglected,
and instead the Frumkin isotherm (or regular solution model) is used to
describe local chemical equilibrium of cations between the nanoparticles and
the adjacent electrolyte, as a function of the electrode potential. Our model
describes the dynamics of key parameters of the CDI process with intercalation
electrodes, such as effluent salt concentration, the distribution of
intercalated ions, cell voltage, and energy consumption.
