Wall Concentration

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

A = 4.05 X lO " cm/(s-kPa)(4.1 X 10 cm/(s-atm)) and = 1.3 x 10 cm/s (4)//= 1 mPa-s(=cP), NaCl diffusivity in water = 1.6 x 10 cm /s, and solution density = 1 g/cm . Figure 4 shows typical results of this type of simulation of salt water permeation through an RO membrane. Increasing the Reynolds number in Figure 4a decreases the effect of concentration polarization. The effect of feed flow rate on NaCl rejection is shown in Figure 4b. Because the intrinsic rejection, R = 1 — Cp / defined in terms of the wall concentration, theoretically R should be independent of the Reynolds... 

Fig. 1. General dialysis is a process by which dissolved solutes move through a membrane in response to a difference in concentration and in the absence of differences in pressure, temperature, and electrical potential. The rate of mass transport or solute flux, ( ), is directly proportional to the difference in concentration at the membrane surfaces (eq. 1). Boundary layer effects, the difference between local and wall concentrations, are important in most...

A. Tubes, laminar, fuUy developed parabolic velocity profile, developing concentration profile, constant wall concentration... 

Since D/6 is the mass-transfer coefficient. For the portion of the operating cui ve in which flux is invariant, the wall concentration is apparently invariant. The mechanism governing why and how that occurs is the subject of a continuing debate in the hterature. 

Although allowing for axial variations in v with a constant wall concentration c yields Eq. (20-59), as a more rigorous expression applicable to higher concentrations [Trettin et al., Chem. Eng. Comm., 4, 507 (1980)] the form of Eq. (20-58) is convenient in analyzing a variety of complex behavior. 

In accordance with observed data, this model shows that water flux increases linearly with applied pressure AP, decreases with higher salt concentration through its impact on osmotic pressure Jt, increases with a smaller membrane thickness I, and increases with temperature through the temperature dependence of the water permeability P . The model also demonstrates that the solute or salt flux J, increases linearly with applied pressure AP, increases with higher salt concentration c , increases with a smaller membrane thickness I, and increases with temperature through the temperature dependence of the solute permeability Pj. Polarization, as described early in this section, causes the wall concentration c to exceed the bulk concentration ci,. 

At low flux these passages are equivalent. At high fluxes the wall concentration is high due to polarization and the observed passage increases, approaching 100 percent, regardless of the intrinsic passage. 

Figure 2. Foam drainage and water loss from the bubble wall concentrate is diluted into a premix, and the foam is generated from the premix. When foam is generated, hydrophobic R groups become part of the bubble wall (X = width of bubble wall). X decreases as water drains downward under the influence of gravity. The polymer strands prevent narrowing of bubble wall, decreasing drainage time and stabilizing the foam.

In the nltTa.filtTa.tion of a protein solution of concentration 0.01 kg/m3, analysis of data on gel growth rate and wall concentration Cw yields the second order relationship ... 

The gel growth rate as a function of the wall concentration, Cw, is given by ... 

Srinivasan and Tien (18) have made an analytical study on the mass-transfer characteristics of reverse osmosis in curved tubular membranes. The increase in mass-transfer due to secondary flow resulted in a substantial reduction in the wall concentration (the polarization modulus) for Np =100 and a/R=0.01 (see Figure 39). Further, the production capacity (permeation rate) was markedly increased (see Figure 40). 

Figure 39. Wall concentration (dimensionless) vs. axial coordinate for straight...

Stem Cell-Wall Concentration and Composition of Control and Transgenic Alfalfa Plants Averaged Across Two Harvests from 1999"... 

The observed rejection [Eq. (8a)j is clearly the important one for a practical separation operations, but it includes the confounding effects of concentration polarization. Since C k < Cj, the observed rejection is less than the intrinsic rejection and can be determined by estimating the solute wall concentration with Eq. (7). 

In the above equations, cjtS is the surface/wall concentration of species j, rw (intrinsic rate of surface reaction, Das is the reactor scale Damkohler number, which are given by... 

Occasionally, witnesses have mentioned that bodies were burned with liquid fuel in pits in Treb-linka II methyl alcohol and gasoline were allegedly used.107 In such a case, as in all open-air incinerations, only a small percentage of the energy released by the fuel in fact acts on the object to be burned, in contrast to suitable furnaces where insulated walls concentrate the heat in a small space. 

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