Effective capillary cross sectional area

Fig. 7-21. Effective capillary cross-sectional area (ECCSA) of aspen wood as a function of pH (Stone, 1957). 1957. TAPPl. Reprinted from Tappi 40(7), p. 54, with permission.

If the rate of evaporative loss per unit gross area of surface— negative rainfall—is v (cm./day) and the effective total cross-sectional area of water in continuous soil capillaries is s, the mean linear rate of upflow in the soil water is v/s. If diffusion coefficient of the solute is D (sq. cm./sec.), it can be shown that the concentration at the steady state falls off with distance x (cm.) from the surface according to... 

Since capillary tubing is involved in osmotic experiments, there are several points pertaining to this feature that should be noted. First, tubes that are carefully matched in diameter should be used so that no correction for surface tension effects need be considered. Next it should be appreciated that an equilibrium osmotic pressure can develop in a capillary tube with a minimum flow of solvent, and therefore the measured value of II applies to the solution as prepared. The pressure, of course, is independent of the cross-sectional area of the liquid column, but if too much solvent transfer were involved, then the effects of dilution would also have to be considered. Now let us examine the practical units that are used to express the concentration of solutions in these experiments. 

Both injection techniques have been shown to suffer from the effects of analyte diffusion, specially when the clean capillary is initially introduced into the sample solution. Diffusion occurs across the boundary area between analyte and buffer, which is defined by the cross-sectional area of the capillary. Both techniques also suffer from the effects of inadvertent hydrodynamic flow that results from the reservoir liquid levels being at slightly different levels. While these effects are significant for the buffer-filled capillaries used in capillary zone electrophoresis, they are both much less important when the capillary is filled with a gel. 

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 void area fraction in (21-76) is based on the fractional area in a plane at constant x that is available for diffusion into catalysts with rectangular symmetry. A rather sophisticated treatment of the effect of g 6) on tortuosity is described by Dullien (1992, pp. 311-312). The tortuosity of a porous medium is a fundamental property of the streamlines or lines of flux within the individual capillaries. Tortuosity measures the deviation of the fluid from the macroscopic flow direction at every point in a porous medium. If all pores have the same constant cross-sectional area, then tortuosity is a symmetric second-rank tensor. For isotropic porous media, the trace of the tortuosity tensor is the important quantity that appears in the expression for the effective intrapellet diffusion coefficient. Consequently, Tor 3 represents this average value (i.e., trace of the tortuosity tensor) for isotropically oriented cylindrical pores with constant cross-sectional area. Hence,... 

In this case the porous media can be modeled by an array of N capillaries with the inner radius a and an effective length that differs from L, the actual length of the column. The effective length Le can be understood as the averaged path length of the fluid particles traveling from the entrance to the exit of the column as shown in Fig. 2. Then it is required that the volume AeLe should be the same as the total volume of the porous space, where = Nna is the total cross-sectional area of the collection of the... 

It only depends on the radii of the particles and the surface tension of the liquid. Neither does it depend on the actual radius of curvature of the meniscus nor does it depend on the vapor pressure. This at first sight surprising result is due to the fact that vith increasing vapor pressure the cross section of the meniscus I increases. At the same time, the capillary pressure decreases because r increases. The product of cross-sectional area and pressure difference, nfAP, remains constant and both effects compensate each other. 

The humidity dependence is also different from the humidity dependence between spheres. The capillary force at contact increases with humidity until it reaches a maximum at P/Po = 0.86. In this part ofthe curve, the meniscus only extends over the conical part. With increasing humidity, the meniscus increases in size and thus in circumference and cross-sectional area. This increase is more significant than the effect of the decrease in the Laplace pressure (in contrast to the sphere where both effects compensate each other). After the maximum at P/Po = 0.86, the capillary force steeply decreases. Here, the meniscus has reached the cylindrical part. Its circumference and the cross-sectional area remain constant but the curvature 1/r decreases. As a result, the force decreases steeply to a value of InyiTc = 0.09 pN for P/Po 1. 

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