Flow equations for salt and water in double-layer membranes are suggested and salt rejection as a function of transport coefficients is derived. Salt rejection at given volume flow is completely determined by the reflection coefficient and salt permeability of the dense layer.
Salt rejection was measured as a function of flow rate for modified cellulose-acetate membranes heat-treated at different temperatures. The maximal salt rejection observed with these membranes ranged from 69-99 ~. Salt-rejection curves were calculated from the independently measured transport-coefficients. Good agreement between the calculated and observed curves was found.
This approach makes it possible to separate the influence of flow rate on salt rejection from eventual pressure-induced changes in membrane structure.
In the preceding article (ref.
(1) denoted in the following by (I)), salt rejection during hyperfiltration was derived as a function of two membrane parameters --t h e permeability and reflection coefficients-and the rate of volume flow. Only homogeneous membranes were considered there. The modified cellulose-acetate membranes developed by Loeb (2) consist of two layers: a very thin and dense layer and a porous support (3).
In the following, equations for water and salt flow through double-layer membranes are derived. These expressions are compared to hyperfiltration through double layer membranes of various selectivities. Notations are identical with those in (I). FLOW EQUATIONS IN TWO-LAYER MEMBRANES Calculation of salt rejection in two-layer membranes starts from the local flow equation given in (I) dc J~ = -P ~ + J~(1 --a)c (1) This equation has now to be integrated in two parts, assuming that the transport coefficients are constant in each of the layers separately. In the stationary state, flows are equal in both layers. Denoting by subscript a layer a and by b layer b, one obtains by integrating across each of the layers two equations for two variables c ~ and c" c " -ci(1 -aa) J~(1 -~ro)Ax. = In c" (2) /~o -c'(1 -go) 327