Velocity profile, dimensionless radial

The value AP can change in the axial direction in the hollow fiber (AP is the pressure drop in the membrane matrix due to the momentum transfer, the velocity through the membrane is u0 , where e is the membrane porosity). Kelsey etal. [11] have solved the equation system in all three cases, namely for closed-shell operation, partial ultrafiltration and complete ultrafiltration and have plotted the dimensionless axial and radial velocities as well as the flow streamlines. Typical axial and radial velocity profiles are shown in the hollow-fiber membrane bioreactor at several axial positions in Figure 14.8 plotted by Kelsey etal. [ 11]. This figure illustrates clearly the change of the relative values of both the axial and the radial velocity [V=vL/(u0Ro), U=u/u0 where uc is the inlet centerline axial velocity]. 

The dimensionless radial velocity profile, expressed by a(p) is obtained by solving the differential equation... 

The absolute values of the dimensionless velocity vary between 1 and 0. The minus sign in the figure indicates that the velocities are in opposite directions. Figure 1.8 also shows the variation of pressure along the radial direction. The velocity profiles in the x direction shown in this figure are different from those based on the theoretical model in Fig. 1.5. This is because the experimental profiles in the jet are affected by the drag forces of the stagnant atmosphere. 

In the previous section, the importance of the uniformity of the radial flow profile was established. In the present section, the fluid mechanical equations for all four flow configurations in Figure 1 are derived and solved for comparison. The development of equations closely follows the approach of Genkin et al. (1,2). Here we extend their work to include both radial and axial flow in the catalyst bed. Following our derivation in reference (16), the dimensionless equations for the axial velocity in the center-pipe for all four configurations are (the primes denote derivative with respect to the dimensionless axial coordinate). 

The Reynolds number ensures turbulent flow and with it effective radial mixing. The axial aspect ratio ensures that axial dispersion is minimal. The radial aspect ratio ensures that channeling does not occur. Chan-neling refers to the situation in which the fluid close to the reactor walls travels faster than the fluid at the center of the tube. When these three dimensionless conditions are satisfied, one can usually model the reactor as a PFR. The velocity profiles are complex, however, and broad generalizations should be used with caution [28]. 

In terms of Eq. (7.12), the variations of the radial profiles of the dimensionless tangential velocity V with Rew are plotted as shown in Fig. 7.6. It is seen that the dimensionless tangential velocities reach a maximum at r < 0.5 for values of Rew varying from 10 to 30. 

Radial profiles of the dimensionless axial velocity. Effect of tube-to-particle diameter ratio at Z =. z Z, = 10 for Rep = 175. From Papageorgiou and Froment [1995]. 

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