Pressure Laplace capillary

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... 

Surfactants aid dewatering of filter cakes after the cakes have formed and have very Httle observed effect on the rate of cake formation. Equations describing the effect of a surfactant show that dewatering is enhanced by lowering the capillary pressure of water in the cake rather than by a kinetic effect. The amount of residual water in a filter cake is related to the capillary forces hoi ding the Hquids in the cake. Laplace s equation relates the capillary pressure (P ) to surface tension (cj), contact angle of air and Hquid on the soHd (9) which is a measure of wettabiHty, and capillary radius (r ), or a similar measure appHcable to filter cakes. 

In a small-diameter capillary tube, wetting forces produce a distortion of the free liquid surface, which takes a curvature. Across a curved liquid surface, a difference of pressure exists and the variation of pressure across the surface is the Laplace or capillary pressure, given by ... 

Liquid between the surface of two solid bodies gives rise to boundary forces. A pressure difference arises and is known as the capillary pressure (Pc). This can be calculated from Laplace s equation. 

If surface tension, analogous to that in liquids, really exists in solids, then also capillary pressure Pc must exist (Laplace 1805). The pressure at any point on the concave (convex) side of a curved interface would be by... 

Upon capillary condensation of water in PEMs, the relative humidity, f /P, determines capillary pressure, P , and capillary radius, via the Kelvin-Laplace equation ... 

The controlled drop tensiometer is a simple and very flexible method for measuring interfacial tension (IFI) in equilibrium as well as in various dynamic conditions. In this technique (Fig. 1), the capillary pressure, p of a drop, which is formed at the tip of a capillary and immersed into another immiscible phase (liquid or gas), is measured by a sensitive pressure transducer. The capillary pressure is related to the IFT and drop radius, R, through the Young-Laplace equation [2,3] ... 

For relatively thick films (higher than about 30 nm), the pressure drop at the film is the sum of the capillary pressures at the two film interfaces. In this case, the Young-Laplace equation for the film can be written as... 

The capillary pressure of interest in water-air-GDM systems is the difference between the pressures of the liquid and gas phases across static air-water interfaces within a GDM. This pressure difference is fundamentally related to the mean curvature H of the air-water interfaces through the well-known Young-Laplace equation 22... 

In prepared catalysts the pore sizes may be quite uniform. However, in most naturally occurring materials there is a wide range of pore sizes. The actual pore size distribution can be obtained from methods such as porosimetry, in which a nonwetting liquid (usually mercury) is pumped into a solid sample [12,13,15,26,30,55]. The solid is considered to be composed of a bundle of capillaries. For each capillary, the Laplace equation (see Section 3.2.2) gives the pressure drop across a curved liquid surface ... 

In primary oil recovery from underground reservoirs, the capillary forces described by the Young and Young-Laplace equations are responsible for retaining much of the oil (residual oil) in parts of the pore structure in the rock or sand. It is these same forces that any secondary or enhanced (tertiary) oil-recovery-process strategies are intended to overcome [2,133,421,690,691]. In an oil-bearing reservoir the relative oil and water saturations depend upon the distribution of pore sizes in the rock. The capillary pressure in a pore is... 

In spatially evolving multiphase media (e.g., during dissolution of a porous medium, or phase separation in a polymer blend), the mean curvature of the interface between two phases is of interest. Curvature is a sensitive indicator of morphological transitions such as the transition from spherical to rod-like micelles in an emulsion, or the degree of sintering in a porous ceramic material. Furthermore, important physicochemical parameters such as capillary pressure (from the Young-Laplace equation) are curvature-dependent. The local value of the mean curvature K — (1 /R + 1 /Ri) of an interface of phase i with principal radii of curvature Rx and R2 can be calculated as the divergence of the interface normal vector ,... 

Note that for 9 > 90°, ze is positive i.e., it corresponds to a depression for 9 < 90°, ze is negative and corresponds to a capillary rise. Equation (1.55) can also be derived by a mechanical approach, considering the hydrostatic pressure APh = pgz and the capillary pressure APC. Applying the Laplace equation (1.20) to the capillary configuration with R] = R2 = -r/cos0 (see Figure 1.37), APcis ... 

In the last decade, a variety of microporous and mesoporous materials have been developed for applications in catalysis, chromatography and adsorption. Great attention has been paid to the control of (i) pore surface chemistry and (ii) textural properties such as pore size distribution, pore size and shape. Recently, a new field of applications for these materials has been highlighted [1-3] by forcing a non-wetting liquid to invade a porous solid by means of an external pressure, mechanical energy can be converted to interfacial energy. The capillary pressure, Pc p, required for pore intrusion can be written from the Laplace-Washbum relation,... 

Young s law for capillary pressure see capillary pressure, Young and Laplace Z-average 1.7.63... 

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). 

From the liquid foam stability theory it follows that a liquid foam is most stable when the gas bubbles are strictly spherical in shape since, according to the Laplace and Plateau laws, the interface area and the capillary pressure have minimum values in this case. As it has been shown in Sect. 4.1 for the monodispersed... 

Oil Entrapment Mechanisms. Enhanced oil recovery processes depend in large part on the elimination or reduction of capillary forces. Capillary forces are the strongest that occur under typical reservoir conditions, and are most responsible for oil entrapment. Viscous forces, which act to displace oil, are composed of the applied pressure gradient, gravity, density differences between phases, and viscosity ratio. In a permeable medium, capillary forces result when the pores constrain the oil-water interface to a high degree of curvature. From the Laplace equation, the capillary pressure in a capillary tube can be derived ... 

The thermod niamical and mechanical interpretations of interfacial tensions point to two ways of measurement either the reversible work or the force required to extend cin interface should be determined. In practice it is often easier to balance the interfacial force against others, say against gravity. In general, the results can be interpreted in terms of the capillary pressure Ap created by the interfacial tension across curved interfaces. From sec. 1.2.23 we know that the required relation is given by the Young-Laplace equation (1.2.23.191... 

In the method of the Jailing meniscus a liquid-wetted tapering tube is placed vertically in a reservoir, as in fig. 1.26. Inside the tube liquid is held by the capillary pressure. The tube is now moved upwards - or the liquid in the vessel downwards - to increase the hydrostatic pressure head, and this is continued until the liquid in the capillary collapses. From the hydrostatic head the Laplace pressure is obtained and from that the surface tension. The method is very simple and may be considered as the counterpart of the maximum bubble pressure technique there are also similarities to the situation sketched in fig. 1.8a. The idea is rather old... 

This capillary pressure force can be approximated using the Laplace equation. 

In addition to the Laplace capillary pressure, three additional forces can operate at surfactant concentration below the cmc, namely electrostatic double layer repulsion Tj1, van der Waals attractions and steric (short-range) forces... 

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