Transport through membranes. A unified approach

We shall try to cover all the membrane processes within one model at the end of fliis chapter, in order to relate the various membrane processes with each other in terms of driving forces, fluxes and basic separation principles. To do so, the starting point must be a simple model, such as a generalised Pick equation [41 or a generalised Stefan-MaxweU equation [42]. In order to describe transport through a porous membrane or through a nonporous membrane, two contributions must be taken into account, the diffusional flow (v) and the convective flow (u). The flux of component i through a membrane can be described as the product of velocity and concentration, i.e. 

The contribution of convective flow is the main term in any description of transport through porous membranes. In nonporous membranes, however, the convective flow term can be neglected and only diffusional flow contributes to transport.It can be shown by simple calculations that only convective flow contributes to transport in the case of porous membranes (microfiltration). Thus, for a membrane with a thickness of 100 pm, an average pore diameter of 0.1 pm, a tortuosity C of 1 (capillar) membrane) and a porosity e of 0.6, water flow at 1 bar pressure difference can be calculated from the Poisseuille equation (convective flow), i.e. 

The driving force for diffusion is the difference in chemical potential, and both the concentration (activity) and the pressure contribute to this driving force. However it can be assumed that the concentration (or activity) on either side of the membrane is equal in microfiltrationand hence the pressure difference must be the only driving force. Indeed, diffusive water flow as a result of this driving force is very smaU, as can be demonstrated as follows. The chemical potential difference can be written as  

Considering only the extreme cases, it can be stated that transport in porous membranes occurs by convection and in nonporous ammbranesby diffusion. Howevei in going from porous to nonporous membranes, an intermediate region exists where both 

The last part of.Ais chapter will be devoted to a comparison of meiribr c processes v where transport occurs through nonporous membranes. A solution-diffusion model will be used where each component dissolves into the membrane and diffuses through the membrane independently [41]. A similar approach was recently followed by Wijmans[43]. As a result, simple equations will be obtained for the component fluxes involved in the various processes which allows to compare the processes in terms of transport parameters. 

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