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Electrochemical desalination devices that use redox-active cationintercalation electrodes show promise for desalination of salt-rich waterresources with high water recovery and low energy consumption. While previousmodeling and experiments used ion-exchange membranes to maximize chargeefficiency, here a membrane-free alternative is evaluated to reduce capitalcost by using a porous diaphragm to separate Na$_{1+x}$NiFe(CN)$_6$ electrodes.Two-dimensional porous-electrode modeling shows that, while charge efficiencylosses are inherent to a diaphragm-based architecture, charge efficiency valuesapproaching the anion transference number (61$\%$ for NaCl) are achievable fordiaphragms with sufficiently low salt conductance. Closed-form equations arethereby derived that relate charge efficiency to the non-dimensional P\`{e}cletand Damk\"ohler numbers that enable the selection of current and flow velocityto produce a desired degree of desalination. Simulations using these conditionsare used to quantify the tradeoffs between energy consumption and salt removalrate for diaphragm-based cells operated at a range of currents. The simulateddistributions of reactions are shown to result from the local saltconcentration variations within electrodes using diffusion-potential theory. Wealso simulate the cycling dynamics of various flow configurations and show thatflow-through electrodes exceed the degree-of-desalination compared with flow-byand flow-behind configurations due to solution stagnation within electrodes.

Quantifying the Trade-offs between Energy Consumption and Salt Removal Rate in Membrane-free Cation Intercalation Desalination