Electrochemical deposition (ECD) from unsupported electrolyte in two-dimensional radial cells provides a model system for the study of branched aggregates which also arise in other depositional growth systems. It is shown that ECD aggregates grow within a concentration boundary layer that forms a smooth envelope around the branch tips. A material balance around this front allows calculation of the local driving force for growth and reveals that a limiting current can be attained in stationary solution for sufficiently high rates of growth at a well-defined limiting velocity. This model of the concentration boundary layer is applied to the problems of velocity and morphology selection.