EDI apparatus for demineralizing a liquid flow is assembled in a housing having a cylindrical shape, and includes two metal electrodes, and one or more leafs, each leaf comprising a pair of selectively ion-permeable membranes arranged parallel to each other and spaced apart by spacing elements that allow liquid to flow in the interstitial space between membranes, thus forming an arrangement of dilute and concentrate cells in a desired flow configuration. Spacing elements between membranes, as well as between leaves, can be formed of inert polymer material, ion exchange beads, ion exchange fibers, a combination of two or more these elements, or a porous media incorporating one or more of such elements as an intrinsic part. An inner or central electrode and an outer or perimeter electrode establish a generally uniform and radially-oriented electrical or ionic current between the inner and the outer electrodes, across the helical flow spaces defined by the membrane/spacer windings. One or both electrodes may include a pocket, and the adjacent flow cells lie parallel to the electrode and free of shadowing and field inhomogeneity around a full circumference of the electrode. Flow paths within the helical cells are defined by barrier seals, which may form a path-lengthening maze, while unfilled cell regions may disperse or collect flow within a cell and define pressure gradients promote directional flows. Impermeable barriers between membranes further prevent the feed and concentrate flows from mixing. In various embodiments, seals along or between portions of the flow path may define a multi-stage device, may define separate feed and/or concentrate flows for different stages, and/or may direct the feed and concentrate flows along preferred directions which may be co-current, counter-current or cross-current with respect to each other within the apparatus.