Turbulent boundary/flow, membranes

The thermodynamic approach does not make explicit the effects of concentration at the membrane. A good deal of the analysis of concentration polarisation given for ultrafiltration also applies to reverse osmosis. The control of the boundary layer is just as important. The main effects of concentration polarisation in this case are, however, a reduced value of solvent permeation rate as a result of an increased osmotic pressure at the membrane surface given in equation 8.37, and a decrease in solute rejection given in equation 8.38. In many applications it is usual to pretreat feeds in order to remove colloidal material before reverse osmosis. The components which must then be retained by reverse osmosis have higher diffusion coefficients than those encountered in ultrafiltration. Hence, the polarisation modulus given in equation 8.14 is lower, and the concentration of solutes at the membrane seldom results in the formation of a gel. For the case of turbulent flow the Dittus-Boelter correlation may be used, as was the case for ultrafiltration giving a polarisation modulus of ... 

Equation (4) states that the linear deposition rate vj is a diffusion controlled boundary layer effect. The quantity Ac is the difference in foulant concentration between the film and that in the bulk flow and c is an appropriate average concentration across the diffusion layer. The last term approximately characterizes the "concentration polarization" effect for a developing concentration boundary layer in either a laminar or turbulent pipe or channel flow. Here, Vq is the permeate flux through the unfouled membrane, 6 the foulant concentration boundary layer thickness and D the diffusion coefficient. 

The positive effect of velocity on the permeate flux is a result of enhanced hydrodynamic effects at the membrane surface, since high velocities lead to high shear and turbulent flow, which results in the formation of vortices and eddies that minimize the concentration polarization effects and the development of a fouling layer. The bigger the thickness of this layer, the higher its flow resistance and the smaller the permeate flux through the membrane becomes. Under turbulent flow conditions, shear effects induce hydrodynamic diffusion of the particles from the boundary layer back into the bulk, with a positive effect on the permeate flux. 

Reverse osmosis is a cross-flow process and, as in any dynamic hydraulic process, the fluid adjacent to the membrane moves slower than the main stream. While the main stream flow may be turbulent, the layer next to the membrane surface is laminar. This thin, laminar flow film is called the boundary layer. When water permeates through the membrane, nearly all of the salt remains behind in the boundary layer next to the membrane. The salt must then diffuse across the boundary layer and back into the bulk stream. This results in a boundary layer with a salt concentration which is more concentrated than the bulk stream. The effect has been termed concentration polarization, and it is defined by the following equation ... 

Concentration polarization cannot be eliminated, but it can be minimized by decreasing boundary layer thickness. This is done by increasing the flow rate across the membrane surface or introducing turbulence promoters into the feed/reject stream. In order to achieve optimum performance, most membrane manufacturers will recommend a minimum feed rate to or from their elements and a maximum recovery in order to minimize the effects of concentration polarization. 

Figures 6.2 and 6.3 indicate the order of magnitude of concentration polarization for laminar and turbulent flows through tubular membranes. The diagrams illustrate the dependence of the concentration boundary layer on flow conditions along the membrane (Re) and on the permeation flux (Pew).

The contribution of concentration polarization to overall dialytic mass transfer resistance suggests that dialyzers should be operated with flows in the turbulent flow region to minimize boundary-layer formation. While this is typically the case on the dialysate side of the membrane, dialyzers usually are operated with laminar flow on the feed side. In hemodialysis, this is partly to avoid undue mechanical stress on the blood cells which may result in their destruction. However, a consideration of fluid dynamics dictates that feed-side flow be laminar in nearly all applications. 

Preparatory work for the steps in the scaling up of the membrane reactors has been presented in the previous sections. Now, to maintain the similarity of the membrane reactors between the laboratory and pilot plant, dimensional analysis with a number of dimensionless numbers is introduced in the scaling-up process. Traditionally, the scaling-up of hydrodynamic systems is performed with the aid of dimensionless parameters, which must be kept equal at all scales to be hydrodynamically similar. Dimensional analysis allows one to reduce the number of variables that have to be taken into accoimt for mass transfer determination. For mass transfer under forced convection, there are at least three dimensionless groups the Sherwood number, Sh, which contains the mass transfer coefficient the Reynolds number. Re, which contains the flow velocity and defines the flow condition (laminar/turbulent) and the Schmidt number, Sc, which characterizes the diffusive and viscous properties of the respective fluid and describes the relative extension of the fluid-dynamic and concentration boundary layer. The dependence of Sh on Re, Sc, the characteristic length, Dq/L, and D /L can be described in the form of the power series as shown in Eqn (14.38), in which Dc/a is the gap between cathode and anode Dw/C is gap between reactor wall and cathode, and L is the length of the electrode (Pak. Chung, Ju, 2001) ... 

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