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Fig. 6. Variations of local transmembrane pressure (a) and dimensionless cumulative flow rate (b) as a function of radial position. Other parameters are at their ‘standard’ values.

Figure 6 Variations of local transmembrane pressure (a) and dimensionless cumulative flow rate (b) as a function of radial position. Other parameters are at their ‘standard’ values.

Related Figures (18)

Fig. 2. Closer view of the rotating membrane disk.
Fig. 2. Closer view of the rotating membrane disk.
where p is the fluid density, v the Eulerian fluid velocity vector, p the pressure, 4 the dynamic fluid viscosity and f the body force vector. The fluid flow equations (Eqs. (1) and (2)) is then solved only at nodal points on the FEM mesh. Inside each mesh element the velocity and pressure vectors are extrapolated.
where p is the fluid density, v the Eulerian fluid velocity vector, p the pressure, 4 the dynamic fluid viscosity and f the body force vector. The fluid flow equations (Eqs. (1) and (2)) is then solved only at nodal points on the FEM mesh. Inside each mesh element the velocity and pressure vectors are extrapolated.
Summary of parameter ranges Table 1
Summary of parameter ranges Table 1
Fig. 4. Standard dimensions and parameters.
Fig. 4. Standard dimensions and parameters.
Fig. 5. Variations of feed (a) and permeate (b) side pressure as a function of radial position. Other parameters are at their ‘standard’ values.
Fig. 5. Variations of feed (a) and permeate (b) side pressure as a function of radial position. Other parameters are at their ‘standard’ values.
Fig. 7. Variations of the global transmembrane pressure as a function of rotational speed. Other parameters are at their ‘standard’ values (i.e. velocity through the membrane) exhibits the same variation as local transmembrane pressure (Fig. 8(b)) since they are linked by the permeability model (Eq. (3)) which, under simplifying assumptions, reduces to Darcy’s law. Thus, the permeation velocity is higher at radial positions near the shaft where local transmembrane pressure is greater and smaller or negative for more distant radial positions. Irregularities at extreme values of radial distance (near the shaft and the membrane tip) are numerical artifacts that arise from the no-slip condition imposed to the washer and retaining seal. Axial velocity was set to zero at R’ =0 and | which induces the nearest node velocity to be higher or smaller at the tip to keep the permeate flux in agreement with the local transmembrane pressure (Fig. 6(a)). For no rotation, the radial velocity remains equal to zero and no slip occurs at the membrane surface (Fig. 8(a)). As the disk rotation increases, the fluid starts to slip. Although the slip velocity increases with both radial position and speed of rotation, it remains small in comparison with the axial velocity. tip along the membrane surface (positive velocity for the highest dimensionless axial position) and returns radially to the shaft along the end-wall surface (negative velocity for the smallest dimensionless axial position). By comparison with this overall circulation, the bulk fluid moves very little.
Fig. 7. Variations of the global transmembrane pressure as a function of rotational speed. Other parameters are at their ‘standard’ values (i.e. velocity through the membrane) exhibits the same variation as local transmembrane pressure (Fig. 8(b)) since they are linked by the permeability model (Eq. (3)) which, under simplifying assumptions, reduces to Darcy’s law. Thus, the permeation velocity is higher at radial positions near the shaft where local transmembrane pressure is greater and smaller or negative for more distant radial positions. Irregularities at extreme values of radial distance (near the shaft and the membrane tip) are numerical artifacts that arise from the no-slip condition imposed to the washer and retaining seal. Axial velocity was set to zero at R’ =0 and | which induces the nearest node velocity to be higher or smaller at the tip to keep the permeate flux in agreement with the local transmembrane pressure (Fig. 6(a)). For no rotation, the radial velocity remains equal to zero and no slip occurs at the membrane surface (Fig. 8(a)). As the disk rotation increases, the fluid starts to slip. Although the slip velocity increases with both radial position and speed of rotation, it remains small in comparison with the axial velocity. tip along the membrane surface (positive velocity for the highest dimensionless axial position) and returns radially to the shaft along the end-wall surface (negative velocity for the smallest dimensionless axial position). By comparison with this overall circulation, the bulk fluid moves very little.
Fig. 8. Variations of the radial (a), axial (b) and azimuthal (c) velocity as a function of radial position (z = 7.14 cm). Other parameters are at their ‘standar values.
Fig. 8. Variations of the radial (a), axial (b) and azimuthal (c) velocity as a function of radial position (z = 7.14 cm). Other parameters are at their ‘standar values.
Fig. 10. Streamline contour plot and the subsequent values of the stream function for a rotational speed of 0 rpm. Other parameters are at their ‘standard’ values.
Fig. 10. Streamline contour plot and the subsequent values of the stream function for a rotational speed of 0 rpm. Other parameters are at their ‘standard’ values.
Fig. 11. Streamline contour plot and the subsequent values of the stream function for a rotational speed of 1500 rpm. Other parameters are at their ‘standard’ values.
Fig. 11. Streamline contour plot and the subsequent values of the stream function for a rotational speed of 1500 rpm. Other parameters are at their ‘standard’ values.
Fig. 12. Effect of permeate flux (J) on local transmembrane pressure and dimensionless cumulative flow rate (a) and on the global transmembrane pressure (b). Other parameters are at their ‘standard’ values.
Fig. 12. Effect of permeate flux (J) on local transmembrane pressure and dimensionless cumulative flow rate (a) and on the global transmembrane pressure (b). Other parameters are at their ‘standard’ values.
Fig. 13. Effect of the distance between disk tip and vessel (Gap1) on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 13. Effect of the distance between disk tip and vessel (Gap1) on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 14. Effect of washer length on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 14. Effect of washer length on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 15. Effect of membrane permeability (Lp mp) on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 15. Effect of membrane permeability (Lp mp) on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 16. Effect of membrane permeability (Lp mp) on the optimal radial position of membrane tip for a rotational speed of 1500 rpm. Other parameters are at their ‘standard’ values.
Fig. 16. Effect of membrane permeability (Lp mp) on the optimal radial position of membrane tip for a rotational speed of 1500 rpm. Other parameters are at their ‘standard’ values.
Fig. 17. Effect of support permeability (Lp supp) on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 17. Effect of support permeability (Lp supp) on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 18. Effect of support thickness on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 18. Effect of support thickness on local transmembrane pressure and dimensionless cumulative flow rate. Other parameters are at their ‘standard’ values.
Fig. 19. Effect of the clearance between the membrane disk and end-wall (Gap2) on local transmembrane pressure and dimensionless cumulative flow. Other parameters are at their ‘standard’ values.
Fig. 19. Effect of the clearance between the membrane disk and end-wall (Gap2) on local transmembrane pressure and dimensionless cumulative flow. Other parameters are at their ‘standard’ values.
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