## Mass Transport Through Porous Membranes

The pore model is based on the assumption that the membrane pores are much smaller than the diameter of the retained particles but are permeable to water. In contrast to the solution-diffusion model, water does not diffuse through the pores; rather it flows under the influence of pressure and frictional forces. The pore system is ideally straight and parallel; and each pore has the same circular size. As a result of the small capillary diameter, laminar flow conditions are given and Hagen-Poiseuille's law is valid. For pores with diameter d and length L, one can write in dimensionless notation:

Re with Reynolds number: wpod

with wpo as the flow rate in a capillary pore.

In reality, the length of pores L is greater than the membrane thickness Az. Therefore, a mean tortousity p is defined:

The flow rate of permeate —p is now given by Hagen-Poiseuille's law (using Eqs. 12.30 to 12.33) with porosity e:

Ap d2 e

Carman and Kozeny's pore model (Carman 1956) assumes a pore system formed of equally sized spheres in a packed bed. With permeate flux:

Ap e3

the hydraulic diameter dh for packed beds is introduced (Rautenbach and Albrecht 1981):

(1-e) av where av is the volume-specific surface area.

Equation (12.35) is known as the Carman-Kozeny equation. For sphere-packed beds, the tortousity is given as p=25/12 (Rautenbach and Albrecht 1989). With the membrane constant A*:

2ç (1-e)2 a;pAz the equation shows the linear dependence of permeate flux on the driving force, i.e. the pressure gradient across the membrane:

Comparing the diffusive water flux according to Eq. (12.16), A* corresponds to the product of the density and the membrane constant A used in the solution-diffusion model.

The linear behavior of Jp in relation to Ap depends only on the membrane constant A* of a given membrane. A* must be determined experimentally because it depends on viscosity, which is a function of temperature. Figure 12.4 shows the membrane-controlled flux.

This model fits micro- and ultrafiltration processes, but in practice a gel layer at the surface of the membrane and a depositing of particles also affects mass transport (see Section 12.3).

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