Concentration polarization Peclet number

Concentration Polarization, Peclet Number, and Enrichment Factor Show that Brian s equation (Equahon 8.8d) can be recast in the form... 

Concentration Polarization, Peclet Number, and Enrichment Factor... 

They recorded such a polarization curve for zinc, for copper in the presence of gelatin and for silver in nitrate solution. Under this mechanism, a negative fluctuation in concentration drives the current density up, resulting in further reduction in interfacial concentration. For this instability to be expressed, the surface concentration must be free to respond to variations in current. As a result, the instability is seen only far from the limiting current, where the interfacial concentration is pinned at zero. At high Peclet numbers, the concentration disturbance is propagated downstream by convection, and the striations follow the streamlines. 

Figure 5.15 shows streamlines and concentration contours calculated by Masliyah and Epstein (M6). Even in creeping flow, Fig. 5.15a, the concentration contours are not symmetrical. The concentration gradient at the surface, and thus Shjoc is largest at the front stagnation point and decreases with polar angle see also Fig. 3.11. The diffusing species is convected downstream forming a region of high concentration at the rear (often referred to as a concentration wake ) which becomes narrower at higher Peclet number.

In Equation (4.9) the balance between convective transport and diffusive transport in the membrane boundary layer is characterized by the term JvS/Di. This dimensionless number represents the ratio of the convective transport Jv and diffusive transport Dj/8 and is commonly called the Peclet number. When the Peclet number is large (./ 5>> D,/S), the convective flux through the membrane cannot easily be balanced by diffusion in the boundary layer, and the concentration polarization modulus is large. When the Peclet number is small (Jv <5C D,/8), convection is easily balanced by diffusion in the boundary layer, and the concentration polarization modulus is close to unity. 

Figure 4.7 Concentration polarization modulus ciolcih as a function of the Peclet number Jv8/Di for a range of values of the intrinsic enrichment factor E . Lines calculated through Equation (4.9). This figure shows that components that are enriched by the membrane (E0 > 1) are affected more by concentration polarization than components that are rejected by the membrane (E0 < 1) [13]...

Figure 4.8 Concentration gradients that form adjacent to the membrane surface for components (a) rejected or (b) enriched by the membrane. The Peclet number, characterizing the balance between convection and diffusion in the boundary layer, is the same JvS/Di = 1. When the component is rejected, the concentration at the membrane surface r, cannot be greater than 2.72 c,., irrespective of the membrane selectivity. When the minor component permeates the membrane, the concentration at the membrane surface can decrease to close to zero, so the concentration polarization modulus becomes very small...

Table 4.1 shows typical enrichments and calculated Peclet numbers for membrane processes with liquid feeds. In this table it is important to recognize the difference between enrichment and separation factor. The enrichments shown are calculated for the minor component. For example, in the dehydration of ethanol, a typical feed solution of 96 % ethanol and 4 % water yields a permeate containing about 80 % water the enrichment, that is, the ratio of the permeate to feed concentration, is about 20. In Figure 4.11, the calculated Peclet numbers and enrichments shown in Table 4.1 are plotted on the Wijmans graph to show the relative importance of concentration polarization for the processes listed. 

Process Typical enrichment, E Typical flux [in engineering units and as Jv (10-3 cm/s)] Diffusion coefficient (10 6 cm2/s) Peclet number, JvS/Di Concentration polarization modulus [Equation (4.9)]... 

Figure 4.11 Peclet numbers and intrinsic enrichments for the membrane separation processes shown in Table 4.1 superimposed on the concentration polarization plot of Wiimans et al. [13]...

In coupled transport and solvent dehydration by pervaporation, concentration polarization effects are generally modest and controllable, with a concentration polarization modulus of 1.5 or less. In reverse osmosis, the Peclet number of 0.3-0.5 was calculated on the basis of typical fluxes of current reverse osmosis membrane modules, which are 30- to 50-gal/ft2 day. Concentration polarization modulus values in this range are between 1.0 and 1.5. 

Gas separation process Pressure-normalized flux, P/i [10 6 cm3(STP)/ cm2 s cmHg] Pressure feed/permeate (atm/atm) Volume flux at feed pressure, JVf (10-3 cm3/cm2 s) Membrane selectivity, a Enrichment, E0 Feed gas diffusion coefficient at feed pressure (10-3 cm2/s) Peclet number, JVfS/Dj(yA04) Concentration polarization modulus [Equation (4.24)]... 

These relations can be used as rough estimates of steric rejection, if the solute and membrane pore dimensions are known. The derivation is based on a strictly model situation (see Figure 1) and a long list of necessary assumptions can be written. Apart from the simplified geometry (hard sphere in a cylindrical pore), it was also assumed that the solute travels at the same velocity as the surrounding liquid, that the solute concentration in the accessible parts of the pore is uniform and equal to the concentration in the feed, that the flow pattern is laminar, the liquid is Newtonian, diffusional contribution to solute transport is negligible (pore Peclet number is sufficiently high), concentration polarization and membrane-solute interactions are absent, etc. 

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