## Mass Transport Through Nonporous Membranes

The solution-diffusion model describes reverse osmosis in ideal membranes as well as nanofiltration processes in non-porous membranes.

The transport of a component through a non-porous membrane is only possible by dissolution and diffusion. In this idealized model of mass transport, the membrane is treated like a liquid. Figure 12.3 shows the concentration and pressure gradients for an asymmetric membrane, where the active layer is responsible for separation and the supporting layer for mechanical stability.

The specific mass transfer rate of diffusion is described by Fick's first law:

where c' is the concentration of the dissolved component and D is the diffusion coefficient.

The diffusion coefficient D is idealized to be independent of concentration. That means D is constant and not affected by location in the membrane or by concentration. For one-dimensional diffusion, the diffusive flux follows as:

This is a solution for the steady-state mass balance for diffusion across two planes (e.g. through a thin plate membrane at the points z and z + dz):

dz and considering Eq. (12.7):

dz dz dz2

According to Henry's law, we obtain for ideal systems with linear sorption characteristics with respect to the dissolved impurity concentration in the membrane c':

with:

The membrane constant for the dissolved impurity B is independent of pressure but is a function of temperature because H and D vary with temperature.

The diffusive water flux follows in analogy to (12.8):

cow c^w

Because of the water surplus, the absorption equilibrium can be described via (partial) pressure:

and the water flux follows taken together with Eq. (12.13):

Dw Hw

Usually, the water flux is written as a function of the pressure difference:

with:

pAz as the membrane constant A and p as the density of water. Note that, in contrast to the dimensionless H (see Eq. 5.6), HW has the dimension g (m3 bar)-1 (Eq. 12.14).

The membrane constant A for the solvent (here water) is a function of temperature as well as a function of membrane properties [DW(T), HW(T)] and the thickness of the active layer Az.

Equation (12.16) is valid if the osmotic pressure n is negligible. Otherwise the diffusive flux is:

with:

Vi it follows: RT

where ai is the solvent activity, Yi is the activity coefficient, xi is the mole fraction, xj is the mole fraction of the solute, Vi is the molar volume of the solvent and R is the gas constant.

For ideal mixtures or for very low solute concentrations (Xj<<1) the activity coefficient Yi is unity (Mulder 2000) and the activity becomes:

Thus, using the equation for the state of ideal gases for diluted solutions the molar concentration follows according to the amount of moles Nj and Nz:

At least the osmotic pressure difference between membranes follows:

respectively for dissociating compounds, including the osmotic coefficient p:

where c0 is the feed molar concentration of the solute, c2 is the permeate molar concentration of the solute and p is the osmotic coefficient for the change in numbers of moles by dissociation, depending on the degree and stoichiometrics of the dissociation reaction (Rautenbach and Albrecht 1981).

From Eqs. (12.23) and (12.24) it follows that An is high for high values of c0 and for low values of c2. Considering concentration polarization, the solute concentration at the membrane surface is c3 >c0 (Section 12.3).

The membrane constants A and B depend on temperature according to exponential functions (Rautenbach 1997) because of its influence on D and H (B in Eq. 12.12) and on DW and HW [A in Eq. (12.17)]. Yet the temperature dependency of B is often neglected for practical purpose. A and B have to be determined experimentally. They characterize the permeability of the membrane; the quotient A/B is a degree for the selectivity of the system. For a low A/B the selectivity is high.

Often the designer's aim is to calculate:

• The flow rate of the water through membrane wp.

• The permeate concentration c2.

With the flow rate of permeate wp (Fig. 12.3), we obtain the mass flux JD for the solute with c2:

and for water:

From Eqs. (12.18) and (12.26), the permeate flow rate can be calculated:

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