Channel flow with soluble walls

Channel Flow with Soluble or Rapidly Reacting Walls... 

Channel Flow with Soluble or Rapidly Reacting Walls 67 with the function satisfying the recursion relation... 

For specificity let us first consider the soluble wall problem sketched in Fig. 4.2.3. Now in a channel (or pipe) flow u = 0, and near the surface the streamwise velocity component is given by u (lylh). This behavior of the velocity profile is the same as for a fully developed thin liquid film on a vertical wall, falling under gravity with a free surface at atmospheric pressure. The velocity profile is parabolic with the fall velocity and has a maximum at the free surface equal to... 

For the fully developed velocity profile the mathematical problem again reduces to the solution of the steady form of the convective diffusion equation for the solute concentration. In contrast to the diffusion equation treated in the channel flow problem with soluble or rapidly reacting walls, it is necessary to include here the lateral convection term to account for the product removal through the membrane walls, putting... 

Since v D for an infinite diluted solution, the entrance concentration region has the length Ly Ly. Fig. 6.3 shows the case when dissolved substance gets into the flow from the soluble walls, with the boundary conditions C = 0 at the entrance and C = Cw = Csat at the wall for x > 0. Dissolved substance diffuses from the wall to the channel axis. 

FEE is an alternative separation technique to SEC that crosses the borders from soluble region into particle region. The separation takes place in a thin channel between two parallel plates where the sample flows in the axial direction. The flow profile in the channel is parabolic with the maximum velocity in the middle. External field (e.g., flow, thermal) applied in the perpendicular direction forces molecules to different distances from the channel wall, i.e., to different velocity layers. Thus, the elution time increases with decreasing distance of particles from the channel wall. 

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