  Electrochemical desalination devices that use redox-active cation
intercalation electrodes show promise for desalination of salt-rich water
resources with high water recovery and low energy consumption. While previous
modeling and experiments used ion-exchange membranes to maximize charge
efficiency, here a membrane-free alternative is evaluated to reduce capital
cost by using a porous diaphragm to separate Na$_{1+x}$NiFe(CN)$_6$ electrodes.
Two-dimensional porous-electrode modeling shows that, while charge efficiency
losses are inherent to a diaphragm-based architecture, charge efficiency values
approaching the anion transference number (61$\%$ for NaCl) are achievable for
diaphragms with sufficiently low salt conductance. Closed-form equations are
thereby derived that relate charge efficiency to the non-dimensional P\`{e}clet
and Damk\"ohler numbers that enable the selection of current and flow velocity
to produce a desired degree of desalination. Simulations using these conditions
are used to quantify the tradeoffs between energy consumption and salt removal
rate for diaphragm-based cells operated at a range of currents. The simulated
distributions of reactions are shown to result from the local salt
concentration variations within electrodes using diffusion-potential theory. We
also simulate the cycling dynamics of various flow configurations and show that
flow-through electrodes exceed the degree-of-desalination compared with flow-by
and flow-behind configurations due to solution stagnation within electrodes.
