Flow channel thickness

A comprehensive difference model was developed by Madireddi et al. [71] to predict membrane fouling in commercial spiral-wound membranes with various spacers. This is a useful paper for experimental studies on the effect of flow channel thickness on flux and fouling. Avlonitis et al. [72] presented an analytical solution for the performance of spiral-wound modules with seawater as the feed. In a key finding they showed that it was necessary to incorporate the concentration and pressure of the feed into the correlation for the mass transfer coefficient. In a similar study, Boudinar et al. [73] developed the following relationship for calculating mass transfer coefficients in channels equipped with a spacer ... 

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... 

Cross-flow FFF or, as it was known in the past, FIFFF draws its name from the field type used to transport sample components across the channel thickness to the accumulation wall [3,8]. The... 

The ThFFF separation system is made up of a flat ribbon-like channel obtained by placing a trimming-spacer between two flat bars kept at different temperatures (at the upper wall) and (at the lower wall), with AT = Tg- The thickness of the spacer defines the channel thickness w. In the channel cross section, the thermal diffusion process pushes the analyte toward the so-called accumulation wall, usually the cold wall (thermophobic substances) the combination of the flow profile and the thermal diffusion produces the fractionation. 

The classical FEE retention equation (see Equation 12.11) does not apply to ThEEE since relevant physicochemical parameters—affecting both flow profile and analyte concentration distribution in the channel cross section—are temperature dependent and thus not constant in the channel cross-sectional area. Inside the channel, the flow of solvent carrier follows a distorted, parabolic flow profile because of the changing values of the carrier properties along the channel thickness (density, viscosity, and thermal conductivity). Under these conditions, the concentration profile differs from the exponential profile since the velocity profile is strongly distorted with respect to the parabolic profile. By taking into account these effects, the ThEEE retention equation (see Equation 12.11) becomes ... 

Drill core DDH-B shows the effect of a sill on a coal bed over a distance of 9 feet. McFarlane (12) noted that, except where the intrusion represents a flow channel or conduit, the coal is carbonized to anthracite only for distances ranging up to one-third the thickness of a sill, both above and below it. The sill in drill core DDH-B is just over 1 foot thick though the coal is coked for more than 4 feet from the contact, indicating that this sill must have been a flow channel or conduit. The greatest observable change in hydrogen, volatile matter, and reflectance occurs in samples 5.0 and 6.0 which are 4-5 feet from the sill, respectively. 

Figure 19.4. The spiral wound membrane module for reverse osmosis, (a) Cutaway view of a spiral wound membrane permeator, consisting of two membranes sealed at the edges and enclosing a porous structure that serves as a passage for the permeate flow, and with mesh spacers outside each membrane for passage of feed solution, then wound into a spiral. A spiral 4 in. dia by 3 ft long has about 60 sqft of membrane surface, (b) Detail, showing particularly the sealing of the permeate flow channel, (c) Thickness of membranes and depths of channels for flows of permeate and feed solutions.

In many situations, the system is primed with a buffer solution which is displaced by the protein solution of interest (Fig. 5 a). Assuming constant, laminar established flow, the velocity (V) in a rectangular flow channel of width (w), thickness (b), and length (1), where b 4 w has a characteristic parabolic profile, given by 36)... 

Figure 8. Comparison of the predicted residual carbon support in the cathode with the measured cathode thickness after 13 h run at 1.2 A/cm2 (70%H2/air counter-flow, 80 °C, 150 kPaabs, 40%/60% RH n) with partially filled flow channels as shown above the plot.

The boundary layer thickness 8 in Equation (4.11) is a function of the feed solution velocity u in the module feed flow channel thus, the term 8/D, can be expressed as... 

Cross-flow FFF (F1FFF) utilizes a second fluid flow to transport sample components across the channel thickness to the accumulation wall, and the position of individual species in the laminar carrier profile corresponds to their ordinary (Fick s) diffusion coefficient. As the particle size increases, the diffusion coefficient (decreases until it becomes a relatively insignificant transport process. For micron-size particles, the extent of protrusion into the channel becomes the decisive factor in determining the order of elution. 

This can be easily checked by assuming that the flow inside this section of the screw can be modeled using a simple shear flow, and that most of the conduction occurs through the channel thickness direction. For such a case, the energy equation in that direction, say the -direction, reduces to... 

The concentration polarization occurring in electrodialysis, that is, the concentration profiles at the membrane surface can be calculated by a mass balance taking into account all fluxes in the boundary layer and the hydrodynamic conditions in the flow channel between the membranes. To a first approximation the salt concentration at the membrane surface can be calculated and related to the current density by applying the so-called Nernst film model, which assumes that the bulk solution between the laminar boundary layers has a uniform concentration, whereas the concentration in the boundary layers changes over the thickness of the boundary layer. However, the concentration at the membrane surface and the boundary layer thickness are constant along the flow channel from the cell entrance to the exit. In a practical electrodialysis stack there will be entrance and exit effects and concentration... 

We note first that immediately following the injection of a sample at the head of the channel, the flow of carrier is stopped briefly to allow time for the sample particles to accumulate near the appropriate wall. As the particles concentrate near the wall, the growing concentration gradient leads to a diffusive flux which counteracts the influx of particles. Because channel thickness is small (approximately 0.25 mm), these two mass transport processes quickly balance one another, leading to an equilibrium distribution near the accumulation wall. This distribution assumes the exponential form... 

The reduction of the thickness of the flow channel, as discussed earlier, is equivalent to introducing more surface area per unit volume of medium. High surface areas inhibit all flow, including natural convective flow. One can increase relative surface areas by going to thinner tubes or channels, or by using a fine granular or porous support medium. Both approaches are used in electrophoresis as discussed in a subsequent chapter. 

The flow channel in FFF is typically 25-100 cm long and 1-3 cm in breadth. The thin dimension (thickness) of its ribbon-like structure is generally 50-500 /um, or 0.05-0.5 mm. The channel is open and unobstructed—it contains no packing material. There is no need for packing to support a stationary phase (as in chromatography) because there is no stationary phase retention is induced by the external field. 

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