The second-order reaction by which a species is transferred from solution to a fixed site in a porous solid is considered. A web of the porous solid moves under a reservoir of the solution, which saturates as liquid percolates downward. The saturated web then passes through a zone in which there is no downward flow and on to a pair of rollers where an arbitrary fraction of the solution is physically expressed and the reaction continues in time. The salient behavior of this class of reactors is described in the hodograph plane. Accommodating optimized designs are detailed under constraints which arise naturally. Comparison is made with experiment for the application to chelation of sodium pyrithione to zinc sites in cellulose sponge.