Membranes flow profile

The images obtained from the SPAN module (Figure 4.6.3(c and d)) show completely different characteristics compared with those from the SMC module. Noticeable features are the dense and evenly packed capillary membranes and the lower quality and inhomogeneities in the images. As already discussed for the ID profile (Figure 4.6.2(b)), both intra- and inter-membrane flow seems unhindered... 

Convective Flow Profile in a Hollow-Fiber Membrane 14.4.4.1 Without Cake and Polarization Layers... 

The basic hydrodynamic equations are the Navier-Stokes equations [51]. These equations are listed in their general form in Appendix C. The combination of these equations, for example, with Darcy s law, the fluid flow in crossflow filtration in tubular or capillary membranes can be described [52]. In most cases of enzyme or microbial membrane reactors where enzymes are immobilized within the membrane matrix or in a thin layer at the matrix/shell interface or the live cells are inoculated into the shell, a cake layer is not formed on the membrane surface. The concentration-polarization layer can exist but this layer does not alter the value of the convective velocity. Several studies have modeled the convective-flow profiles in a hollow-fiber and/or flat-sheet membranes [11, 35, 44, 53-56]. Bruining [44] gives a general description of flows and pressures for enzyme membrane reactor. Three main modes... 

Kelsey LJ, Pillarella MR, Zydney AL (1990) Theoretical analysis of convective flow profiles in a hollow-fiber membrane bioreactor. Chem Eng Sci 45 3211-3220... 

The separation channel in asymmetrical flow FFF (AF4) is approximately 30 cm long, 2 cm wide, and between 100 and 500 pm thick. A carrier flow which forms a laminar flow profile streams through the channel. In contrast to the other FFF methods, there is no external force, but the carrier flow is split into two partial flows inside the channel. One partial flow is led to the channel outlet and, afterward, to the detection systems. The other partial flow, called the cross-flow, is pumped out of the channel through the bottom of the channel. In the AF4, the bottom of the separation channel is limited through a special membrane and the top is made of an impermeable plate (glass, stainless steel, etc.). The separation force, therefore, is generated internally, directly inside the channel, and not by an externally applied force. 

The characteristic feature of flow FFF is the superimposition of a second stream of liquid perpendicular to the axis of separation. This cross-flow drives the injected sample plug toward a semipermeable membrane that acts as the accumulation wall. The cross-flow liquid permeates across the membrane and exits the channel, whereas the sample is retained inside the channel in the vicinity of the membrane surface. Sample displacement by the cross-flow is countered by diffusion away from the membrane wall. At equilibrium, the net flux is zero and sample clouds of various thicknesses are formed for different sample species. As with other FFF techniques, a larger diffusion coefficient D leads to a thicker equilibrium sample cloud that, on average, occupies a faster streamline of the parabolic flow profile and subsequently elutes at a shorter retention time t,. For well-retained samples analyzed by flow FFF, t, can be related to D and the hydrodynamic diameter d by... 

Looking at the schematic representation of the flow profile within the fiber shown in Fig. 3-39, it becomes apparent that a further sensitivity increase can be accomplished by changing the parabolic flow profile. When the fiber is packed with inert Nafion beads, the translational diffusion of ions is favored over longitudinal diffusion. This, in turn, improves the mass-transfer across the membrane which leads to a further increase in sensitivity, particularly pronounced in the case of orthophosphate as the salt of a weak acid. 

This is different from the distillation profiles generated earlier in this book, where the CMO assumption was employed those profiles had only x associated with every point. The reflux was held constant, and did not change down the length of the CS. In this scenario, however, both compositions and flows change down the length of the MGS. Hence, both x and change along a membrane column profile. 

To examine the transport-mode-transition region in more detail, simulations were run at different current densities [71]. The resultant membrane water profiles are shown in Figure 5.11 where a vapor-equilibrated membrane at unit activity has a water content of L = 8.8 as calculated by the modified chemical model (see Section 5.5.1). The profiles in the figure demonstrate that the higher the current density the sharper the transition from the liquid-equilibrated to the vapor-equilibrated mode as well as the lower the value of the water content at the anode GDL/membrane interface. The reason why the transition occurs at the same point in the membrane is that the electro-osmotic flow and the water-gradient flow are both proportional to the current... 

Phase changes occur at sohd (membrane)/hquid interfaces where precipitation occurs easily if there are nucleation sites. Precipitation occurs at a finite rate depending on the number of nucleation sites, the degree of saturation, temperature, pressure and time. Once the process of precipitation starts, the rate is controlled by the size of the sofid/Hq-uid interfacial area. If the particles are attached to a membrane surface, they grow in only one direction because the membrane hmits access to the adjacent surface. Thus particles grow as a sheet until they form a layer of precipitate on the membrane surface, and the membrane gets fouled. The solute concentration is maximum at the membrane surface and lower in the bulk hquid above the membrane surface the concentration profile is the reverse of a fluid flow profile in a channel or a tube, as shown in Figure 1.23. 

We now discuss that in the jetting regime radius selection is dominated by simple hydrodynamics [46]. We assume that the shear stress and the velocity difference across the membrane can be neglected. For the analysis of the hydrodynamic flow profile, we can hence focus on the fluid motion inside and outside of the tube. In the experiments shown in Figure 11.10, the outer silicate solution is confined to a glass cylinder of radius J cyi(l-1 cm)- Along its central axis, buoyant cupric sulfate solution ascends as a cyHndrical jet of radius R. The cylindrically symmetric velocity fields v(r), which solve the Navier-Stokes equations, are... 

The combined influence of elongation and shear rate induced by the geometry of spinnerets on membrane performance for gas separation has been studied as illustrated in Figure 31.6 (Cao et al., 2004). The flow profiles of dope solution and the elongation and... 

Temperature profiles, methane conversion, HRF and permeated flow are shown in Fig. 14.6. Figure 14.7 shows the membrane temperature profile and the permeation driving force along the reactor. Table 14.4 shows the permeation results and the product outcome, outlining that total hydrogen permeated is lower (26%) for the counter-current configuration. The calculated HRF and methane conversion ( CH4 ) in the counter-current flow... 

Fig. 4.6.2 ID velocity profiles of counterflow through the mini-hemodialyzer modules of type SMC (a) and SPAN (b), where velocity along z is plotted versus the x position axis (total width 9 mm), with z representing the flow direction. Positive velocities correspond to membrane-side flow (M) and negative velo-...

In practice, estimation of Laq requires information on the rate of solute removal at the membrane since aqueous resistance is calculated from experimental data defining the solute concentration profile across this barrier [7], Mean /.aq values calculated from the product of aqueous diffusivity (at body temperature) and aqueous resistance obtained from human and animal intestinal perfusion experiments in situ are in the range of 100-900 pm, compared to lumenal radii of 0.2 cm (rat) and 1 cm (human). These estimates will necessarily be a function of perfusion flow rate and choice of solute. The lower Laq estimated in vivo is rationalized by better mixing within the lumen in the vicinity of the mucosal membrane [6],... 

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