Cylindrical capillary

In this case, the expression for the excess free energy is as follows  

Note that the disjoining pressure in this case is 

Let us assume that the transition zone profile does nof lend asymptotically to the equilibrium thickness but meets the film at the final point x = Xq. In this case, in the vicinity of this point, we approximate the disjoining pressure isotherm by a linear dependency H h) n(/tj - a h - hj, where a = - (/ ) is a positive value is a stable flat liquid film, and the derivative of the disjoining pressure should be negative and n(h ) = Pg. The liquid profile in this region has a low slope, which means Equation 2.23 can be rewritten as 

At jc = Xq, according to Equation 2.21, the following two boundary conditions should be satisfied  

Equation A1.1 and the two boundary conditions result in the following system of algebraic equations for the determination of the integration constants C, and C.  

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A slightly more general case is that in which the liquid meets the circularly cylindrical capillary wall at some angle 6, as illustrated in Fig. II-7. If the meniscus is still taken to be spherical in shape, it follows from simple geometric consideration that / 2 = r/cos 6 and, since R = / 2, Eq. II-9 then becomes... 

Geankoplis [54] fabricated a porous medium for which the values of K, and are known a priori. This was accomplished by sealing a bundle of identical parallel cylindrical capillary tubes between the two chambers of a Wlcke-Kallenbach apparatus. Then the relevant flux relations are those which apply to a single cylindrical capillary, rather than a porous medium, and these are obtained by setting... 

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

Use the model for the size exclusion of a spherical solute molecule in a cylindrical capillary to calculate for a selection of R/a values which... 

The correction factor for converting apparent shear rates at the wall of a circular cylindrical capillary to true shear rates is (3n + l)/4n, where n is the power law index of the polymer melt being extruded. 

L. L. Blyler and T. K. Kwei [39] proposed the direct opposite (to 4). In their reasoning, they proceeded from the known and generally acceptable Doolittle equation, which puts liquid viscosity in exponential dependence on the inverse value of the free volume of the latter. According to [39], gas has a volume of its own, the value of which it contributes to the free volume of the polymer when it dissolves therein as a result, viscosity falls. The theoretical formula obtained by the authors was experimentally confirmed in the same work. The authors measured pressure values at the entrance of cylindrical capillaries, through which melts of both pure polyethylene, and polyethylene with gas dissolved in it, extruded at a constant rate. 

The Alexander model allows a simple approach to this problem. Within this model, each tethered chain is, in effect, confined within a cylindrical capillary of diameter d. Combining Eq. 5 and 7, we can express the stretching energy as ... 

In the capillary method, the time required for a liquid to flow through a capillary tube is determined. The melt under investigation flows with a constant rate through a tube with a small, definite cross-sectional area, such as a cylindrical capillary. The viscosity can be measured in an absolute way from the pressure drop. This method can yield the most reliable absolute data, the viscosity being given by a modified Hagen-Poiseuille equation ... 

Rice, C., Whitehead, R., Electro kinetic flow in a narrow cylindrical capillary,... 

Influence of Soluble Surfactants on the Flow of Long Bubbles Through a Cylindrical Capillary... 

Figure 2. Flow of a single gas bubble through a liquid-filled cylindrical capillary. The liquid contains a soluble surfactant whose distribution along the bubble interface is sketched.

Realistically, the flow path through the seal volume will not consist of uniform cylindrical capillaries aligned normal to the seal. The actual leakage paths will be longer, less direct (convoluted), and of nonuniform cross-section. To account for these effects, the effective path length is increased by a tortuosity factor, t, typically having a value in the range of 2 to 3. 

The first model of porous space as a 2D lattice of interconnected pores with a variation of randomness and branchness was offered by Fatt [220], He used a network of resistors as an analog PS. Further, similar approaches were applied in a number of publications (see, e.g., Refs. [221-223]). Later Ksenjheck [224] used a 3D variant of such a model (simple cubic lattice with coordination number 6, formed from crossed cylindrical capillaries of different radii) for modeling MP with randomized psd. The plausible results were obtained in these works, but the quantitative consent with the experiment has not been achieved. 

From the physics point of view, cellulose fibers can indeed be considered as tiny roughly cylindrical capillary tubes of radius r and length h. Consequently, a wetting liquid placed into contact with this highly hydrophilic material penetrates it by capillary action. Actually, in capillaries with radii much smaller than the capillary length, gravity may be... 

Figure 4.5 shows a portion of a cylindrical capillary of radius Rc and length f. Our interest is in the flow of a viscous liquid through such a capillary. This arrangement has the same cylindrical symmetry as the viscometers we discussed in the last section and, once again, it is convenient to focus attention on a cylindrical shell of fluid of radius r and thickness dr, as shown in Figure 4.5. Because of the nonslip condition at the wall of the capillary, the liquid shell adjacent to that wall has a velocity equal to zero. The velocity increases for shells of...

FIG. 4.5 Flow in a cylindrical capillary (a) a volume element in the flowing liquid and (b) the parabolic flow profile. 

Adsorption hysteresis is often associated with porous solids, so we must examine porosity for an understanding of the origin of this effect. As a first approximation, we may imagine a pore to be a cylindrical capillary of radius r. As just noted, r will be very small. The surface of any liquid condensed in this capillary will be described by a radius of curvature related to r. According to the Laplace equation (Equation (6.29)), the pressure difference across a curved interface increases as the radius of curvature decreases. This means that vapor will condense... 

It will be noted that the importance of the correction for surface conductivity increases as Rc decreases and vanishes as Rc - oo. Equation (54) also suggests that the numerical evaluation of ks may be accomplished by studying electroosmosis in a set of capillaries identical in all respects except for variability in Rc. Finally, the expansion of Equation (50) to Equation (54) in correcting for surface conductivity explicitly assumes a cylindrical capillary. Experiments made with porous plugs cannot be corrected for surface conductivity by Equation (54), but the qualitative conclusion that the effect of surface conductivity increases as the pore radius decreases is valid in this case also. 

FIG. 12.10 Velocity profiles in electrophoresis cells (a) velocity (as time-1) of Klebsiella aerogenes particles as a function of their location in a rectangular electrophoresis cell (redrawn with permission from A. M. James, in Surface and Colloid Science, Vol. 11 (R. J. Good, and R. R. Stromberg, Eds.), Plenum, New York, 1979) (b) location of the surface of zero liquid velocity in a cylindrical capillary. 

For a Newtonian liquid D, is equal to the shear rate at the large side walls [the small side walls are disregarded in eqs. (1.10) and (1.11)]. For a non-Newtonian liquid an equation analogous to the well-known one derived by Rabinowitsch (24) for cylindrical capillaries is valid. It reads ... 

This equation (which corresponds with equation 17.53 in Volume 2, Chapter 17), however, cannot be directly applied to the majority of porous solids since they are not well represented by a collection of straight cylindrical capillaries. Everett(I0)... 

The cylindrical capillary model predicts that the size of the largest pore present in a membrane filter medium is inversely proportional to the pressure at which bulk flow of a test gas is not present. 

At 20°C the surface tension of benzene is 28.9 mN m 1 and its molar volume is 89.2 cm3 mol-1. Determine the relative pressures at which condensation of benzene vapour should begin in a cylindrical capillary of radius 10 nm if the capillary is (a) closed at one end, (b) open at both ends. Assume zero contact angle and neglect adsorption on the walls of the capillary. 

Although reverse osmosis, ultrafiltration and microfiltration are conceptually similar processes, the difference in pore diameter (or apparent pore diameter) produces dramatic differences in the way the membranes are used. A simple model of liquid flow through these membranes is to describe the membranes as a series of cylindrical capillary pores of diameter d. The liquid flow through a pore (q) is given by Poiseuille s law as ... 

Figure 2.36 shows the effect of the ratio r/k on the relative proportions of Knudsen to Poiseuille flow in a cylindrical capillary [57], When r/k is greater than one, Poiseuille flow predominates. Because the mean free path of gases at atmospheric pressure is in the range of 500-2000 A, for Knudsen flow to predominate and a separation to be obtained, the membrane pore radius must be less than 500 A. 

Flow in a thin rectangular channel (Figure 4.2), such as that used in field-flow fractionation, can be treated in a manner similar to that used for cylindrical capillary tubes. If the drag at the edges of the channel is neglected (infinite parallel plate model), then the force balance expression (corresponding to Eq. 4.5 for capillary tubes) becomes... 

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