Capillary equations

If the species is neutral, its chemical potential p% can be varied by changing its concentration and hence its activity ay. dpt — RT d nat. In this case the determination of the surface excesses offers no difficulty in principle. However, if a species is charged, its concentration cannot be varied independently from that of a counterion, since the solution must be electrically neutral. To be specific, we consider the case of a 1-1 electrolyte composed of monovalent ions A and D+. The electro capillary equation then takes the form ... 

The zeta potential and the thickness of the double layer (1/k) decrease rapidly with an increase in ionic strength or the valence of the electrolytes in the capillary (Equations (5) and (6)). Therefore, the ionic strength and the nature of the ions in the electrolyte solution are very important parameters determining the strength of the EOF. Careful control of the ionic... 

The fundamental units of dynamics are those of mass, length, and time. These units are denoted by the symbols [M], [L], and [T], respectively. The magnitude of these units may be fixed arbitrarily but all other units are derived from them, and depend upon them alone. The various derivative units are developed simply. For example, unit-density is unit-mass contained in a unit-volume. In terms of the fundamental units it is therefore [M/L8] or [ML-8]. Again, unit-velocity is unit-distance divided by unit-time, or [L/T]. By means of these units the terms in a given equation may be checked, inasmuch as all terms added to a given expression must be of the same kind and therefore of the same dimensions. If, for illustration, it is desired to determine the units of surface tension, a, from the capillary equation... 

These equations together with our capillary equation Eq (11-7) suffice to compute several important constants, as will now be shown. 

Work of Richards—L. A. Richards (1931) gave an excellent theoretical presentation of the factors involved in a study of capillary constants. He succeeded in establishing a general capillary equation and gave a method for determining conductivity. The conductivity was defined as the constant contained in Darcy s equation (Eq 13-2)... 

In the steady state, mass flux out of the capillary is equal to the rate of consumption in the tissue. The mass consumption rate in the tissue (mass per unit time) is equal to the volume times Mt, the mass consumed per unit volume per unit time. Similarly, if Mc is the rate of oxygen loss from the capillary expressed as mass per unit volume per unit time, then VCMC is the mass flux out of the capillary, where Vc is the volume of the capillary. Equating VtMt and VCMC, where V, is the volume of tissue, Mc is equal to VtMt/Vc. Thus in the steady state, Equation (8.15) becomes... 

These considerations bring us to the final statement of the electro capillary equation for our experimental system (1-4) ... 

For the Case 3 solution which included axial dispersion but neglected tissue effects, the mathematical equations reduced to the following Fetal capillary equation ... 

Equation (11) represents steady-state conditions within the maternal intervillous channel and is a nonlinear, partial difference equation with two independent variables. Since the dP/dr = 0 when r = R2> a special equation was also required at this position. The same techniques were used as in the fetal capillary equation. 

Similarly, the fetal capillary equation may be rearranged to give,... 

Assuming that the thickness of the electrical double layer is much smaller than the capillary radius (km 1), and, therefore, the streaming current takes place near the walls of the capillary. Equation 10.20 may be approximated taking x [Pg.164]

In the case of thick macroscopic capillaries. Equation 2.42 has three solutions, one of which corresponds to the stable equilibrium a-flhn with thickness h. The excess free energy of a-fllms is equal to zero, according to our choice in Equation 2.38. The second solution of Equation 2.42 in this case, /t , is unstable according to the stability condition (2.4, Section 2.1), and the third solution, h, is P-fllm, which is also stable according to the same stability condition (2.4, Section 2.1). It has been shown in Section 2.1 that P-films have higher excess free energy as compared with a-lihns, that is, P-lilms are less stable and eventually rupture to thinner and absolutely stable a-films. 

However, in thin capillaries. Equation 2.42 has only one solution (not shown in Figure 2.3), which is an absolutely stable a-film. 

This is the essential characteristic for every lubricant. The kinematic viscosity is most often measured by recording the time needed for the oil to flow down a calibrated capillary tube. The viscosity varies with the pressure but the influence of temperature is much greater it decreases rapidly with an increase in temperature and there is abundant literature concerning the equations and graphs relating these two parameters. One can cite in particular the ASTM D 341 standard. 

The capillary pressure can be related to the height of the interface above the level at which the capillary pressure is zero (called the free water level) by using the hydrostatic pressure equation. Assuming the pressure at the free water level is PI ... 

At first we tried to explain the phenomenon on the base of the existence of the difference between the saturated vapor pressures above two menisci in dead-end capillary [12]. It results in the evaporation of a liquid from the meniscus of smaller curvature ( classical capillary imbibition) and the condensation of its vapor upon the meniscus of larger curvature originally existed due to capillary condensation. We worked out the mathematical description of both gas-vapor diffusion and evaporation-condensation processes in cone s channel. Solving the system of differential equations for evaporation-condensation processes, we ve derived the formula for the dependence of top s (or inner) liquid column growth on time. But the calculated curves for the kinetics of inner column s length are 1-2 orders of magnitude smaller than the experimental ones [12]. 

Use now this equation to describe liquid film flow in conical capillary. Let us pass to spherical coordinate system with the origin coinciding with conical channel s top (fig. 3). It means that instead of longitudinal coordinate z we shall use radial one r. Using (6) we can derive the total flow rate Q, multiplying specific flow rate by the length of cross section ... 

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

As in the case of capillary rise, Sugden [27] has made use of Bashforth s and Adams tables to calculate correction factors for this method. Because the figure is again one of revolution, the equation h = a lb + z is exact, where b is the value of / i = R2 at the origin and z is the distance of OC. The equation simply states that AP, expressed as height of a column of liquid, equals the sum of the hydrostatic head and the pressure... 

Derive the equation for the capillary rise between parallel plates, including the correction term for meniscus weight. Assume zero contact angle, a cylindrical meniscus, and neglect end effects. 

While Eq. III-18 has been verified for small droplets, attempts to do so for liquids in capillaries (where Rm is negative and there should be a pressure reduction) have led to startling discrepancies. Potential problems include the presence of impurities leached from the capillary walls and allowance for the film of adsorbed vapor that should be present (see Chapter X). There is room for another real effect arising from structural peiturbations in the liquid induced by the vicinity of the solid capillary wall (see Chapter VI). Fisher and Israelachvili [19] review much of the literature on the verification of the Kelvin equation and report confirmatory measurements for liquid bridges between crossed mica cylinders. The situation is similar to that of the meniscus in a capillary since Rm is negative some of their results are shown in Fig. III-3. Studies in capillaries have been reviewed by Melrose [20] who concludes that the Kelvin equation is obeyed for radii at least down to 1 fim. 

Bianco and Marmur [143] have developed a means to measure the surface elasticity of soap bubbles. Their results are well modeled by the von Szyszkowski equation (Eq. III-57) and Eq. Ill-118. They find that the elasticity increases with the size of the bubble for small bubbles but that it may go through a maximum for larger bubbles. Li and Neumann [144] have shown the effects of surface elasticity on wetting and capillary rise phenomena, with important implications for measurement of surface tension. 

If the solid in question is available only as a finely divided powder, it may be compressed into a porous plug so that the capillary pressure required to pass a nonwetting liquid can be measured [117]. If the porous plug can be regarded as a bundle of capillaries of average radius r, then from the Laplace equation (II-7) it follows that... 

As an extension of Problem 11, integrate a second time to obtain the equation for the meniscus profile in the Neumann method. Plot this profile as y/a versus x/a, where y is the vertical elevation of a point on the meniscus (above the flat liquid surface), x is the distance of the point from the slide, and a is the capillary constant. (All meniscus profiles, regardless of contact angle, can be located on this plot.)... 

As illustrated in Fig. XU-13, a drop of water is placed between two large parallel plates it wets both surfaces. Both the capillary constant a and d in the figure are much greater than the plate separation x. Derive an equation for the force between the two plates and calculate the value for a 1-cm drop of water at 20°C, with x = 0.5, 1, and 2 mm. 

For some types of wetting more than just the contact angle is involved in the basic mechanism of the action. This is true in the laying of dust and the wetting of a fabric since in these situations the liquid is required to penetrate between dust particles or between the fibers of the fabric. TTie phenomenon is related to that of capillary rise, where the driving force is the pressure difference across the curved surface of the meniscus. The relevant equation is then Eq. X-36,... 

The Washburn equation has most recently been confirmed for water and cyclohexane in glass capillaries ranging from 0.3 to 400 fim in radii [46]. The contact angle formed by a moving meniscus may differ, however, from the static one [46, 47]. Good and Lin [48] found a difference in penetration rate between an outgassed capillary and one with a vapor adsorbed film, and they propose that the driving force be modified by a film pressure term. 

The Washburn model is consistent with recent studies by Rye and co-workers of liquid flow in V-shaped grooves [49] however, the experiments are unable to distinguish between this and more sophisticated models. Equation XIII-8 is also used in studies of wicking. Wicking is the measurement of the rate of capillary rise in a porous medium to determine the average pore radius [50], surface area [51] or contact angle [52]. 

The very considerable success of the BET equation stimulated various investigators to consider modifications of it that would correct certain approximations and give a better fit to type II isotherms. Thus if it is assumed that multilayer formation is limited to n layers, perhaps because of the opposing walls of a capillary being involved, one... 

The adsorption isotherms are often Langmuirian in type (under conditions such that multilayer formation is not likely), and in the case of zeolites, both n and b vary with the cation present. At higher pressures, capillary condensation typically occurs, as discussed in the next section. Some N2 isotherms for M41S materials are shown in Fig. XVII-27 they are Langmuirian below P/P of about 0.2. In the case of a microporous carbon (prepared by carbonizing olive pits), the isotherms for He at 4.2 K and for N2 at 77 K were similar and Langmuirlike up to P/P near unity, but were fit to a modified Dubninin-Radushkevich (DR) equation (see Eq. XVII-75) to estimate micropore sizes around 40 A [186]. 

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