Diffusion in capillaries

This value indicates that diffusion in capillary condensed liquid and the mass transfer through the gas-liquid interfacial film does not limit the overall rate of process under capillary condensation, unlike the situation in trickle-bed. Therefore, only diffusion in the gas phase (in large pores) or the catalytic reaction rate in the liquid (in small pores) is the rate limiting steps. 

Maxwell-Stefan formulation for bulk-Knudsen diffusion in capillary 470... 

Dietl, C. George, O. P Bansal, N. K. Modeling of Diffusion in Capillary Porous Materials During the Drying Process, Drying Technol. 1995,13(1 2), 267-293. 

The structure of the solid determines the mechanism for which internal liquid flow may occur. These mechanisms can include (1) diffusion in continuous, homogeneous solids, (2) capillary flow in granular and porous sohds, (3) flow caused by shrinkage and pressure gradients, (4) flow caused by gravity, and (5) flow caused by a vaporization-condensation sequence. 

Equation (11) accurately describes longitudinal diffusion in a capillary column where there is no impediment to the flow from particles of packing. In a packed column, however, the mobile phase swirls around the particles. This tends to increase the effective diffusivity of the solute. Van Deemter introduced a constant (y) to account... 

In summary, equation (13) accurately describes longitudinal dispersion in the stationary phase of capillary columns, but it will only be significant compared with other dispersion mechanisms in LC capillary columns, should they ever become generally practical and available. Dispersion due to longitudinal diffusion in the stationary phase in packed columns is not significant due to the discontinuous nature of the stationary phase and, compared to other dispersion processes, can be ignored in practice. 

The value of (q) takes into account the precise shape of the pool of stationary phase for a uniform liquid film as in a GC capillary column, q = 2/3. Diffusion in rod shaped and sphere shaped bodies (e.g., paper chromatography and LC) gives q=l/2 and 2/15, respectively [2]. 

Table 1. Dimensionless values of parameters in the Solutal Model for two cases studied here. The systems I and II are representative of the thermophysical properties of the succinonitrile-acetone systems with differing values of the dif-fusivity ratio Rm, temperature gradient G and capillary parameter F. System III corresponds to parameters for a Pb-Sb alloy with equal diffusivities in melt and crystal...

Natural convection depends strongly on cell geometry. No convection can arise in capillaries or in the thin liquid layers found in narrow gaps between electrodes. The rates of natural convective flows and the associated diffusion-layer thicknesses depend on numerous factors and cannot be calculated in a general form. Very rough estimates show that the diffusion-layer thickness under a variety of conditions may be between 100 and 500 pm. 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

The mobilities of alkylpyridines were modeled and predicted in capillary zone electrophoresis.35 The model predicted that compounds adopt a preferred orientation, and additionally predicted mobilities of structural isomers to within 4%, a higher degree of accuracy than can be obtained from simple considerations of van der Waal s radius. Quantitative prediction of the mobilities of some pyridines, such as alkenylpyridines, was not possible. Mobilities of small solutes in capillaries filled with oligomers of ethylene glycol were related to solution viscosity and the diffusion coefficient.36... 

It is essentially a 2-stage process, as shown in Figure 3-5. First, there is a dissociation (dissolution) of particles then the dissolved molecules or ions diffuse across capillary walls and are taken up by the blood. Immediately following dissolution, rapid absorption is observed. For some elements, rapid absorption does not occur because of binding to respiratory-tract components. In the absence of specific data for specific compounds, the model uses the following default absorption rate values for those specific compounds that are classified as Types F (fast), M (medium), S (slow), and V (instantaneous) ... 

Table 1 summarizes several of the experimental methods discussed in this chapter. A need exists for new or revised methods for transport experimentation, particularly for therapeutic proteins or peptides in polymeric systems. An important criterion for the new or revised methods includes in situ sampling using micro techniques which simultaneously sample, separate, and analyze the sample. For example, capillary zone electrophoresis provides a micro technique with high separation resolution and the potential to measure the mobilities and diffusion coefficients of the diffusant in the presence of a polymer. Combining the separation and analytical components adds considerable power and versatility to the method. In addition, up-to-date separation instrumentation is computer-driven, so that methods development is optimized, data are acquired according to a predetermined program, and data analysis is facilitated. 

It is probable that capillary flow of water contributes to transport in the soil. For example, a rate of 7 cm/year would yield an equivalent water velocity of 8 x 10-6 m/h, which exceeds the water diffusion rate by a factor of four. For illustrative purposes we thus select a water transport velocity or coefficient U6 in the soil of 10 x 10 6 m/h, recognizing that this will vary with rainfall characteristics and soil type. These soil processes are in parallel with boundary layer diffusion in series, so the final equations are... 

Eddy diffusion (the A term) is found in packed columns and derived from the different paths that analyte molecules must take through the column packing. It is minimized by using smaller particles and packing them efficiently. In capillary columns it is minimal enough to be neglected. 

FIGU RE 7.12 Representation of the diffuse double layer responsible for electroosmotic flow in capillary electrophoresis. 

Notwithstanding any particular structural model, water transport in PEMs, in general, should be considered a superposition of diffusion in gradients of activity or concentration and hydraulic permeation in gradients of liquid or capillary pressure. Hydraulic permeation is the predominant mechanism xmder conditions for which water uptake is controlled by capillary condensation, whereas diffusion contributes significantly if water strongly interacts with the polymeric host. The molar flux of liquid water in the membrane, N, is thus given by... 

First, porous membranes will be discussed. Gases can be separated due to differences in their molecular masses (Knudsen diffusion), due to interaction (surface diffusion, multilayer diffusion and capillary condensation) and due to their size (molecular sieving). All these mechanisms and their possibilities will be discussed. For the sake of simplicity, theoretical aspects are not covered in detail, but examples of separations in literature will be given. The next section deals with nonporous membranes. Here the separation mechanism is solution-diffusion, e.g. solution and diffusion of hydrogen through a platinum membrane. This section is followed by an outline of some new developments and conclusions. 

Finally it should be noted that inorganic membranes suitable for separation by multilayer diffusion and capillary condensation are also appropriate for performing pervaporation (a technique in which the feed is liquid and the permeate is gas) and distillation at reduced pressure (where gases with overlapping condensation regions are separated). 

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