A Capillary Model

Capillary flow systems of most practical interest are those that involve a solid, a liquid, and a second fluid phase. In the absence of other external forces, the net driving force for capillary flow in such a system will be controlled by 

---

Smith, T., A Capillary Model for Stress-corrosion Cracking of Metals in Fluid Media, Corros. Sci., 12, 45 (1972)... 

The pore geometry described in the above section plays a dominant role in the fluid transport through the media. For example, Katz and Thompson [64] reported a strong correlation between permeability and the size of the pore throat determined from Hg intrusion experiments. This is often understood in terms of a capillary model for porous media in which the main contribution to the single phase flow is the smallest restriction in the pore network, i.e., the pore throat. On the other hand, understanding multiphase flow in porous media requires a more complete picture of the pore network, including pore body and pore throat. For example, in a capillary model, complete displacement of both phases can be achieved. However, in real porous media, one finds that displacement of one or both phases can be hindered, giving rise to the concept of residue saturation. In the production of crude oil, this often dictates the fraction of oil that will not flow. 

The theoretical basis of the Hg-injection method is defined by Laplace law. By using a capillary model where the porous medium is assimilated to a bundle of cylindric capillary tubes the capillary pressure is Pc = y(l/Rci+l/Rc2) = 2y cos0 /Rc (3) where Pc is the capillary pressure Rd and Rc2 are mutually perpendiculcir radii of a surface segment R is the average pore-throat size (pm) 0is the angle between mercury menisc and pore wall (for mercury 0=140°) y is the interfacial tension (for mercury y = 0.480 N/m). 

Richards (R2, R3) and Klausner and Kraft (K6) developed mathematical relations for flow of liquids in a capillary model. Their equations have not been of much value in predicting internal resistance in drying. 

A capillary model presenting a system of channels with developed roughness ... 

Despite our use of a capillary model to characterize a porous medium, most porous beds employed for chromatographic purposes are random and generally the medium is isotropic. In such media, the effective solute dispersivity still arises from the nonuniform pore velocity coupled with molecular diffusion... 

Characterizing the porous bed by means of a capillary model of the interstitial space, the physical basis of the size separation procedure can be demonstrated through examination of the convection and Brownian diffusion of the colloidal particles in a liquid flow through a circular capillary. Figure 5.7.1 shows two freely-rotating spherical Brownian particles of different size sampling a nonuniform Poiseuille flow. The center of the larger particle in its travel... 

We observe here that in a capillary the volume flow rate due to a fixed pressure gradient is proportional to a Tra l8p. dpldx) for a circular capillary). The electroosmotic flow rate is proportional to U multiplied by the cross-sectional area TTa Therefore, the ratio of electroosmotic to hydraulic flow rate will be proportional to a. Thus, for example, if we employ a capillary model for a porous medium, it is evident that as the average pore size decreases electroosmosis will become increasingly effective in driving a flow through the medium, compared with pressure, provided... 

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... 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

To appreciate the questions raised by Knudsen s results, consider first the relation between molar flow and pressure gradient for a pure gas flowing through a porous plug, rather than a capillary. The form predicted by the dusty gas model can be obtained by setting = 1, grad = 0 in equation... 

The procedures described so far have all required a pore model to be assumed at the outset, usually the cylinder, adopted on the grounds of simplicity rather than correspondence with actuality. Brunauer, Mikhail and Bodor have attempted to eliminate the over-dejjendence on a model by basing their analysis on the hydraulic radius r rather than the Kelvin radius r . The hydraulic radius is defined as the ratio of the cross-sectional area of a tube to its perimeter, so that for a capillary of uniform cross-section r is equal to the ratio of the volume of an element of core to... 

Time, Cost, and Equipment Analysis time can vary from several minutes for samples containing only a few constituents to more than an hour for more complex samples. Preliminary sample preparation may substantially increase the analysis time. Instrumentation for gas chromatography ranges in price from inexpensive (a few thousand dollars) to expensive (more than 50,000). The more expensive models are equipped for capillary columns and include a variety of injection options and more sophisticated detectors, such as a mass spectrometer. Packed columns typically cost 50- 200, and the cost of a capillary column is typically 200- 1000. 

The density functional approach has also been used to study capillary condensation in slit-like pores [148,149]. As in the previous section, a simple model of the Lennard-Jones associating fluid with a single associative site is considered. All the parameters of the interparticle potentials are chosen the same as in the previous section. Our attention has been focused on the influence of association on capillary condensation and the evaluation of the phase diagram [42]. 

A theoretical model whereby maximum peak capacity could be achieved by the use of 3-D planar chromatographic separation was proposed by Guiochon and coworkers (23-27). Unfortunately, until now, because of technical problems, this idea could not be realized in practice. Very recently, however, a special stationary phase, namely Empore silica TLC sheets, has now become available for realization of 3-D PC. This stationary phase, developed as a new separation medium for planar chromatography, contains silica entrapped in an inert matrix of polytetrafluoroethy-lene (PTFE) microfibrils. It has been established that the separating power is only ca. 60% of that of conventional TLC (28) this has been attributed to the very slow solvent migration velocity resulting from capillary action. 

If it is known that a particular form of relation, such as the power-law model, is applicable, it is not necessary to maintain a constant shear rate. Thus, for instance, a capillary tube viscometer can be used for determination of the values of the two parameters in the model. In this case it is usually possible to allow for the effects of wall-slip by making measurements with tubes covering a range of bores and extrapolating the results to a tube of infinite diameter. Details of the method are given by Farooqi and Richardson. 21 ... 

The onset of flow instability in a heated capillary with vaporizing meniscus is considered in Chap 11. The behavior of a vapor/liquid system undergoing small perturbations is analyzed by linear approximation, in the frame work of a onedimensional model of capillary flow with a distinct interface. The effect of the physical properties of both phases, the wall heat flux and the capillary sizes on the flow stability is studied. A scenario of a possible process at small and moderate Peclet number is considered. The boundaries of stability separating the domains of stable and unstable flow are outlined and the values of the geometrical and operating parameters corresponding to the transition are estimated. 

The general features of two-dimensional flow with evaporating liquid-vapor meniscus in a capillary slot were studied by Khrustalev and Faghri (1996). Following this work we present the main results mentioned in their research. The model of flow in a narrow slot is presented in Fig. 10.16. Within a capillary slot two characteristic regions can be selected, where two-dimensional or quasi-one-dimensional flow occurs. Two-dimensional flow is realized in the major part of the liquid domain, whereas the quasi-one-dimensional flow is observed in the micro-film region, located near the wall. 

It has proven difficult to develop a theoretical model for multiple development under capillary flow controlled conditions and too little work has been done using forced-flow conditions for any concrete conclusions to be reached [135]. A model has been t... 

Fig. 2.6.11 Flow dispersion profiles obtained with (a) a capillary, (b) with a model microfluidic chip device containing a channel enlargement, directly connected to a capillary and (c) with the same microfluidic chip connected to a capillary via a small mixing volume. A sketch of the model microfluidic device is placed at the right side of each image, drawn to...

The concept of a T2 cut-off that partitions the relaxation time distribution between the pores which can be displaced and those that cannot does not always apply. An exception is when there is significant diffusional coupling between the micropores that retain water at a high capillary pressure and the macropores in close proximity to the microporous system [26, 27]. A spectral BVI model or a forward model has been suggested to interpret these systems [30, 31, 53]. 

Grossman, P. D., Colburn, J. C., and Lauer, H. H., A semiempirical model for the electrophoretic mobilities of peptides in tree-solution capillary electrophoresis, Anal. Biochem., 179, 28, 1989. 

---