Channel height

Let H and L be two characteristic lengths associated with the channel height and the lateral dimensions of the flow domain, respectively. To obtain a uniformly valid approximation for the flow equations, in the limit of small channel thickness, the ratio of characteristic height to lateral dimensions is defined as e = (H/L) 0. Coordinate scale factors h, as well as dynamic variables are represented by a power series in e. It is expected that the scale factor h-, in the direction normal to the layer, is 0(e) while hi and /12, are 0(L). It is also anticipated that the leading terms in the expansion of h, are independent of the coordinate x. Similai ly, the physical velocity components, vi and V2, ai e 0(11), whei e U is a characteristic layer wise velocity, while V3, the component perpendicular to the layer, is 0(eU). Therefore we have... 

If the range of the channel height is limited to be above 10 pm, then the no-slip boundary condition can be adopted. Furthermore, with the assumptions of uniform inlet velocity, pressure, density, and specified pressure Pout at the outlet, the boundary conditions can be expressed as follows ... 

Figure 8.4 illustrates pressure-driven flow between flat plates. The downstream direction is The cross-flow direction is y, with y = 0 at the centerline and y = Y at the walls so that the channel height is 2Y. Suppose the slit width (x-direction) is very large so that sidewall effects are negligible. The velocity profile for a laminar, Newtonian fluid of constant viscosity is... 

Initial width of fluid channel Final width of fluid channel Height of fluid channel Metal plate thickness Metal thickness at the edges Channel length Volumetric polymer flowrate Polymer inlet temperature Temperature of Dowtherm Polymer density Polymer heat capacity Polymer thermal conductivity Metal thermal conductivity... 

The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

Figure 4.29 Voided channel height versus time for slug expulsion of Freon-113. (From Cronenberg et al., 1971. Copyright 1971 by Elsevier Science SA, Lausanne, Switzerland. Reprinted with permission.)...

Flow between parallel plates corresponds to the other limiting case of Eq. (27) for k -> 1 (r, = r0,de = 2h, where h = channel height) ... 

Probstein et al (10) Investigated the use of detached strip type turbulence promoters in the ultrafiltration of bovine serum albumin in laminar flow. His apparatus is shown in Figure 27 the detached strip type promoters tested were circular cylinders with a diameter (D) approximately one-half (0.46) of the channel height and were across the center of the channel cross-section, transverse to the flow. 

Albumin solutions (1, 2, and 5wt%) are continuously ultrafiltered through a flat plate filter with a channel height of 2 mm. Under cross-flow filtration with a transmembrane pressure of 0.5 MPa, steady-state filtrate fluxes (cm min ) are obtained as given in Table P9.2. 

Let s use the steady, two-dimensional flow in a thin channel or a narrow gap between solid objects as schematically represented in Fig. 5.11. The channel height or gap width... 

At the interface a small diffusive region exists where mixing takes place. This region is broader on the top and bottom of the channel, i.e. is smallest at half channel height. This is due to the higher velocity in the center of the channel, reducing the residence time. 

---