Surface tension static force balance

Static holdup depends upon the balance between surface-tension forces tending to hold hquiciin the bed and gravity or other forces that tend to displace the liquid out of the bed. Estimates of static holdup (for gravity drainage) may be made from the following relationship of Shulman et al. [Am. Jn.st. Chem. Eng. J., 1, 259 (1955)] ... 

Bubble size at departure. At departure from a heated surface, the bubble size may theoretically be obtained from a dynamic force balance on the bubble. This should include allowance for surface forces, buoyancy, liquid inertia due to bubble growth, viscous forces, and forces due to the liquid convection around the bubble. For a horizontally heated surface, the maximum static bubble size can be determined analytically as a function of contact angle, surface tension, and... 

The static holdup can be correlated with the Eotvos number NBo as it results from a balance of surface tension and gravity forces on the liquid held up in the pores in absence of flow ... 

Surface tension. Assuming that a membrane stretches over each interface, the magnitudes of the interfacial tension between each pair of phases are the fluid-fluid interfacial tension Ogt, the wetting fluid-solid interfacial tension oft, and the nonwetting fluid-solid interfacial tension ogs. When in static equilibrium, the vectorial force balance at the line of contact (the law of Neumann triangle, Ref. 87) gives... 

Fig. 10.1.4. The liquid may spread freely over the surface, or it may remain as a drop with a specific angle of contact with the solid surface. Denote this static contact angle by 6. There must be a force component associated with the liquid-gas surface tension (t that acts parallel to the surface and whose magnitude is a cos 0. If the drop is to remain in static equilibrium without moving along the surface, it has to be balanced by other forces that act at the contact line, which is the line delimiting the portion of the surface wetted by the liquid, for example, a circle. It is assumed that the surface forces can be represented by surface tensions associated with the solid-gas and solid-liquid interfaces that act along the surface, and tr i, respectively. Setting the sum of the forces in the plane of the surface equal to zero, we have...

The viscosity force is important for the flow in the film region but negligible in the region of static liquid surface in the tank, where the shape of the meniscus is determined by the balance of surface tension and hydrostatic pressure. The transition from the static surface to the surface of the established film occurs through an intermediate surface curved in the shape of a meniscus. Since the established film has a small thickness, the film surface should be completely moistening and the contact angle should be equal to zero. If R is the curvature radius of the meniscus in the transition area, then the characteristic linear scale of the flow in this area is equal to R. Hence, the condition that the film thickness must be infinitesimal reduces to the inequality S R. The flow in the film can be regarded as almost parallel to the surface of the plate. 

Surface Tension, Capillarity, and Contact Angle, Fig. 3 A droplet on a hydrophobic surface. The static contact angle, 0o> is set by the balance of surface tension forces in the horizontal direction... 

In the absence of viscous forces, this equation provides a relationship for the change in pressure across a curved interface. It should be noted that pressure jumps across a fluid interface can only sustain normal stress jumps they do not result in a tangential stress jump. Therefore, tangential surface stresses can only be balanced by viscous stresses associated with fluid motion. Consequently, in the absence of other effects, it is not possible to have a static system in the presence of surface tension gradient. A flow induced by a gradient in the surface tension is called Marangoni flow. 

As the development of the glass drop is extremely slow, one can also consider the drop as quasi-static and perform a force balance on the constant property drop after Middleman . The volume of the drop having sufficient mass necessaiy to just balance the surface tension will have the maximum diameter. Consider the drop hanging from the inner diameter of a capillary tube as shown in figure 1. The angle 0, formed by the line tangent to the free surface at the capillary tube exit is used to determine the vertical component of the surface tension force. 

All of the methods described earlier can be classified as static, because all of the measurements are conducted under conditions of stable equilibrium. This condition corresponds to the minimum in potential energy, which in turn corresponds to the balance between the total surface energy and the energy in a field of gravitational or centrifugal force. There are also methods in which the measurements of surface tension are conducted under conditions of metastable equilibrium, corresponding to the maximum of the free energy of the system. Such methods can be classified as semistatic. 

A stand-alone static method is the popular Wilhelmy plate method (Figure 1.18). In this method, a completely wetted platinum plate is brought into contact with a liquid surface, and a pull force is applied to the plate. Equilibrium is achieved when that force, corrected by the buoyancy force acting on the immersed part of the plate, is balanced by the surface tension, that is, F + dbHpg = 2(d + b)a. The force F is measured with a sensitive dynamometer, which typically forms the core of modem surface tension meters. 

Residual gas-pressure drop hg This is believed to be largely the result of overcoming surface tension as the gas issues from a perforation. A balance of the internal force in a static bubble required to overcome surface tension is... 

The classical analysis of Young and Laplace of static wetting problems rests on the characterization of each interface by a macroscopic surface tension. At the intersection of three bulk phases, the three phase contact line is at rest only if the capillary forces represented by these surface tensions balance. When the three phases are a solid substrate S, a wetting liquid L and a vapor V, the mechanical equilibrium condition parallel to the solid gives the Young-Dupr6 equation for the contact angle Oq... 

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