Pore Peclet number

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. 

P clet number (3.1.143g), (7.3.34d) Pe number for dispersion of solute i (6.3.23a) pore Peclet number (6.3.145a) (z t/j/Dteffiz) (7.1.18h) 7phasePe2(8.1.92) vapor pressure of pure i and pure/, respectively, at system temperature value of Pf on a plane surface value of on a curved surface value of Pf for pure species i of molecular weight M amplitude of pressure wave (3.1.48) pressure of resin phase and external aqueous solution, respectively power number (6.4.976), power (3.1.47)... 

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

Using the methods of the theory of similarity, we studied the problem of flame propagation in narrow tubes, in pores and similar conditions. At the limit a specific value of the Peclet number occurs, constructed from the flame velocity, the characteristic tube dimension and the thermal properties of the combustion products. 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

The pore diffusivity used in this analysis was determined by the Renkin equation4, the axial dispersion coefficient calculated by assuming a constant Peclet number of 0.2, and the mass transfer coefficient from the bulk to the particle surface calculated by the correlation of Wakao and Kaguei. The product of the heat capacity and density of the solid phase was taken to be the same as that used by Raghavan and Ruthven17. The density of the fluid phase was assumed to be that of pure C02 and was calculated from data provided by the Dionix Corporation in their AI-450 SFC software. Constant pressure heat capacities for the mobile phase were also assumed to be that of pure C02 and were taken from Brunner3. 

Woodburn55 obtained gas-phase axial dispersion data at very high irrigation rales. He found that, over the range 15 < ReG < 500 and 126 < ReL < 1,321 (Reynolds numbers are defined in Sec. 8-3), the gas-phase Peclet number increased with the superficial gas rate. The data indicated that the gas-phase axial dispersion coefficient ZG was proportional to the gas pore velocity i.e., EZG cc C/G, where n > 1 for loading conditions and 0 < n 1 for subloading conditions. The data in the ranges 600 < ReG < 2,200 and 0 = ReL < 375 were well correlated by a correlation of Dunn ct al.,16 namely,... 

Dimensional analysis of the coupled kinetic-transport equations shows that a Thiele modulus (4> ) and a Peclet number (Peo) completely characterize diffusion and convection effects, respectively, on reactive processes of a-olefins [Eqs. (8)-(14)]. The Thiele modulus [Eq. (15)] contains a term ( // ) that depends only on the properties of the diffusing molecule and a term ( -) that includes all relevant structural catalyst parameters. The first term introduces carbon number effects on selectivity, whereas the second introduces the effects of pellet size and pore structure and of metal dispersion and site density. The Peclet number accounts for the effects of bed residence time effects on secondary reactions of a-olefins and relates it to the corresponding contribution of pore residence time. 

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 Peclet number means the ratio of the mass transfer rate by convection mechanism into the pore to the mass transport rate by a diffusion mechanism. At... 

Size can refer to volume, area, or length, and therefore pore-size distribution may be defined in terms of any one of these properties. In practice, the definition of size adopted is highly dependent upon the method of measurement. For example, the area size distribution of pores is often measured by image analysis of soil thin sections, while water retention data are usually interpreted in terms of the distribution of pore diameters (Bullock Thomasson, 1979). For consistency with the definition of the Peclet number, we have chosen to define size in terms of length, L. Dullien (1991) has proposed the following interrelationships between the different definitions of size L = VIS in three-dimensions or L=AJP in two-dimensions, where V is volume, S is surface area, A is cross-sectional area and P is perimeter. These relations can be used to compare pore-size distributions measured using different methods. 

Perea-Reeves and Stockman (1997) applied a lattice-gas cellular automaton model to study solute dispersion, including the effect of fluid buoyancy arising from solution density differences, in a pocketed channel. They found good agreement with the indented capillary model discussed in the Variable Shape, Discrete Pore Models section. For Peclet numbers smaller than 3 however, they found that K was actually smaller than the molecular diffusion coefficient. They attributed this to the restriction to diffusion in the direction of flow imposed by the pocket walls. They also observed that density differences between the existing and introduced fluids... 

Figure 7. Longitudinal dispersion (Dl) divided by the diffusion coefficient (Df) for tracers measured in column experiments as a function of the particle scale Peclet number (Npe). It is defined as the product of the average pore fluid velocity, u, and the grain diameter, d, divided by the free fluid diffusion coefficient, D/. The magnitude of the dispersion is independent of the pore fluid velocity (Vp) for very small Peclet numbers (or fluid velocities). Note that the effective diffusion coefficient in a porous media is smaller than the diffusion coefficient in a free fluid phase due to the tortuosity. The dispersion increases linearly with increasing flow velocity (increasing Peclet number). Modified from Appelo and Postma (1999).

The phenomena and processes described can be modeled by convective diffusion equations with chemical reactions. In the simplest model, we may apply these equations in a cylindrical capillary and by means of a capillary model to a porous medium. Assuming dilute solutions, rapid chemical reactions, the double-layer thickness to the soil pore radius and the Peclet number based on the pore radius both small, the overall transport rate for the ith species in a straight cylindrical capillary is... 

When the simplified mass transfer Peclet number is very small (i.e., <1), T 0.67 instead of unity because the numerator of T (i.e., lSA.eff. axial disp.) is based on unsteady-state pore diffusion without convection, whereas the denominator of T (i.e., a, ordinary) is measured in an unrestricted bulk fluid phase. In other words, the diffusivity in the numerator of T is reduced by porosity and tortuosity factors. 

Fourier s law and the interdiffusional fluxes are considered, but the diffusion-thermo (i.e., Dufour) effect is neglected in (30-17). Since contributions from convective transport are insignificant at extremely low Peclet numbers for heat and mass transfer within the catalytic pores, the previous balances reduce to... 

Moreover the Peclet number(Pe) characterizes radial and axial dispersions, which represent the convective and diffusive coefficients in the pores (see Chapter 24) ... 

Both diffusion coefficients and mass transfer are important, but they depend on the different solids, drainage (flow), and particles porosity. The effective diffusion involves Knudsen and convective diffusion, which depends on the phase of fluid (gas or liquid) and pore size (large or small). These coefficients are characterized by Peclet number (Pe), which depends on the axial or radial dispersion and diffusivity. Depending on the velocity profile, these coefficients can vary radially or axially. The diffusion and dispersion coefficients can also vary due to its dependence on the radial position. If the coefficients vary along the reactor, as in heterogeneous reactors, for example, the velocity is not constant. Thus, the axial dispersion occurs. 

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