Capillary radius

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. 

Capillary Suction Processes. The force needed to remove water from capillaries increases proportionately with a decrease in capillary radius, exceeding 1400 kPa (200 psi) in a 1-p.m-diameter capillary. Some attempts have been made to use this force as a way to dewater sludges and cakes by providing smaller dry capillaries to suck up the water (27). Sectors of a vacuum filter have been made of microporous ceramic, which conducts the moisture from the cake into the sector and removes the water on the inside by vacuum. Pore size is sufficiently small that the difference in pressure during vacuum is insufficient to displace water from the sector material, thus allowing a smaller vacuum pump to be effective (126). 

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

The simplest device for measuring ECC at mercury is Gouy s capillary electrometer (Eig. 10.5). Under the effect of a mercury column of height h, mercury is forced into the slightly conical capillary K. In the capillary, the mercury meniscus is in contact with electrolyte solution E. The radius of the mercury meniscus is practically equal to the capillary radius at that point. The meniscus exerts a capillary pressure Pk = directed upward which is balanced by the pressure = ftpegg of... 

The linear velocity of the liquid developing under the effect of this force is zero directly at the solid surface, and increases to some maximum value v at the distance X = 8o from the surface. Solution regions farther out lack the excess charges that could come under the effect of the external electric field hence, there is no further increase in liquid velocity (Fig. 31.4). When the layer (Sg) is much thinner than the capillary radius, 5g r, the assumption can be made that the bulk of the solution moves with a uniform velocity v. 

E is one of several elasticity numbers characterizing the stabilizing effect which adsorbed surfactant molecules have on an interface during mass-transfer processes (22). Note that E is inversely proportional to the capillary radius so that the effect of soluble surfactants on the bubble-flow resistance is larger for smaller capillary radii. 

Fig. 4.8 Schematic illustration of the working principle of the dynamic bubble pressure method. If the bubble radius equals the capillary radius, maximum pressure is detected. The pressure minimum occurs on bubble detachment.

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

Temperature and vapor pressure of the adjacent gas phase determine the capillary radius, r, up to which pores are swollen via Equation (6.13). 

What happens upon equilibration with liquid water instead of water vapor According to Equation (6.13), the capillary radius would go to infinity for PVP —> 1. Thus, in terms of external conditions, swelling would be thermodynamically unlimited, corresponding to the formation of an infinitely dilute aqueous solution of ionomer. However, the self-organized polymer is an effectively cross-linked elastic medium. Under liquid-equilibrated conditions, swelling is not controlled by external vapor... 

Rp M- membrane resistance (fl cm ) r equilibrium bond length (nm) r capillary radius (CCL modeling) (nm) f(, capillary radius (PEM modeling) (nm) r effective bead radius (nm)... 

I..J, and Rp are respectively, the separation factors for the porous matrix capillaries, and the Interstitial capillaries, given In each case by an expression In the form of Equations 1 to 3 (the upper limit radius R In this case refers to either the porous matrix capillary radius or the Interstitial capillary radius). 

Figure 3. Separation factor-particle diameter behavior computed from the pore-partitioning model showing the effect of the Hamaker constant at a low eluant ionic strength (O.OOl M). Other parameters are = 0.60, interstitial capillary radius = l6 fim, pore radius = fim,...

Figure 6. Separation factor-particle diameter tehavior as a function of packing diameter for the pore-partitioning model. Parameters are the same as in Figure 3 with the exception of the interstitial capillary radius which was computed from the hed hydraulic radius (Equation 11 (7.) with void fraction = 0.358).

With decreasing packing size in SEC columns, the probability of physical entrapment of macromolecules increases. To estimate the molecular weight limit above which ultrafiltration will occur, we must first calculate an average radius of the interstices formed in a packed bed. This is done by assuming that the packed column consists of a bundle of capillaries in which the capillary radius can be estimated from the bed hydraulic radius ... 

A simple —but incorrect — relationship between the height of capillary rise, capillary radius, contact angle, and surface tension is easily derived. At equilibrium the vertical component of the surface tension (2icRcy cos 0) equals the weight of the liquid column, approximated as the weight of a cylinder of height h and radius Rc. This leads to the approximation... 

How accurately should the values of liquid density and capillary radius be known if all of the figures in the measurements in Problem 20 are to be considered significant ... 

The vapor pressure of the liquid inside the pore decreases to P/f, with rc being the capillary radius at the point where the meniscus is in equilibrium. 

Many surfaces are not totally wetted, but they form a certain contact angle 0 with the liquid. In this case the radius of curvature increases. It is not longer equal to the capillary radius, but tor = rc/cos 0. 

Example 7.2. Water in trees rises in capillaries which are called xylem. They are typically 5-170 //,m in radius and are completely wetted (0 = 0). What is the maximum height water can rise in such capillaries To calculate an upper limit we use a capillary radius of 5 jim. Then... 

A liquid rises in a lyophilic (0 < 90°) capillary. The height increases with decreasing capillary radius. From lyophobic (0 > 90°) capillaries a liquid is expelled. 

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