Membranes 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. 

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. 

Table 4.3 Calculated Peclet numbers for several important gas separations. The boundary layer thickness is assumed to be 2000 ixm. Permeant diffusion coefficients in the gas boundary layer are taken from tables for ambient pressure diffusion coefficients in Cussler [8] and corrected for pressure. Membrane enrichments E0 are calculated using Equation (4.21)... 

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)]... 

P P Po PCHE PCR PCS PDF PDMS Pe PEM PEMFC PET pH Power output Pressure Pressure drop of one SAR step Printed circuit heat-exchanger Printed circuit reactor Process control system Probability density function Poly-dimethylsiloxane Peclet-number Proton exchange membrane Proton exchange membrane fuel cell Poly-ethylene terephthalate Potentia Hydrogenii (measure for acid and base strength)... 

The Ha number (Ham) defined here for the mth sublayer of the membrane layer corresponds to the well-known Thiele modulus defined for catalyst particles, while the Peclet number corresponds to the often used Bodenstein number. 

The membrane reactor has an additional degree of freedom allowing to optimize the amount of the product removed. This can be conveniently described by a dimensionless Peclet number which relates convective flow through the reactor to transport through the membrane [50]. For each Da number there exists an optimal Pe number maximizing the conversion. For a given membrane material the Pe number is directly related to the membrane thickness. Curve (b) in Fig. 12.12 represents the theoretical behavior of a membrane reactor optimized with respect to membrane thickness. For low Da numbers the membrane should be very thick in order to keep the reactants in the reactor, and in this respect the membrane reactor is identical to the conventional fixed-bed reactor. In contrast, for high Da numbers the membrane reactor should possess a very thin wall for... 

Heat transfer through the membrane pores by convection can also be considered. It is the ratio of convective to conductive heat transfer rates within the membrane pores and is given by the Peclet number, Pe [39]... 

The magnitude of the Peclet number indicates the importance of the convective relative to the diffuse process for solute transport. The solute concentration profiles for representative values of Pe are illustrated in Fig. 12.2 according to Bungay [7]. When diffusion is dominant (Pe 0) the concentration varies nearly linearly in z. For large absolute values of the Peclet number, diffusion is significant only in a thin zone adjacent to the low pressure face of the membrane in which the concentration profile is very steep. For micro- and ultrafiltration membranes, the solute concentration varies little from the value at high pressure face. For nanofiltration the Peclet number can vary considerably depending on membrane characteristic almost dense or porous membranes. 

Fig. 12.2. Steady-state solute concentration profiles in simultaneous diffusion and convection across a membrane of uniform properties. Numbers adjacent to profiles indicate values of the Peclet number whose sign depends upon the direction of the volumetric flux relative to the extemed solution...

The influence of the Peclet number is shown in Fig. 14.14. Pe is reciprocally proportional to the membrane surface. Decreasing the Pe number increases hydrogen recovery and as a consequence the CO conversion. When more membrane surface is available, also more carbon dioxide and carbon monoxide permeates through the membrane and the carbon recovery decreases. 

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. 

Overall enzyme amount in the reactor, M Number of capillary membranes in a bundle Product concentration, M L 3 Peclet number... 

When the Peclet number, Pe, is small, Equation 5-15 reduces to the familiar expression for flux as a function of membrane permeability (recall Equation 5-2), where the membrane permeability is now P = HD /L. When Pe is large, Equation 5-15 reduces to the expression for ultrafiltration, where the commonly used reflection coefficient is a = I — W (see Section 5.5.1). 

Probability of an open site or bond in Chapter 4 Hydrostatic pressure (mmHg) in Chapters 5 and 6 Persistence length on polymer chain (cm) in Chapter 4 Membrane permeability (cm/s) in Chapter 5 Probability density function Critical probability Peclet number... 

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