Drops pressure difference across surface

A rotary drum filter is used to filter a slurry. The drum rotates at a rate of 3 min/cycle, and 40% of the drum surface is submerged in the slurry. A constant pressure drop at 3 psi is maintained across the filter. If the drum is 5 ft in diameter and 10 ft long, calculate the total net filtration rate in gpm that is possible for a slurry having properties as determined by the following lab test. A sample of the slurry was pumped at a constant flow rate of 1 gpm through 0.25 ft2 of the filter medium. After 10 min, the pressure difference across the filter had risen to 2.5 psi. The filter medium resistance may be neglected. 

The correction factor < > is required because on detachment (a) the drop does not completely leave the tip, (b) the surface tension forces are seldom exactly vertical and (c) there is a pressure difference across the curved liquid surface147. (f> depends on the ratio r/Vm. Values of have been determined empirically by Harkins and Brown148,149. It can be seen that values of r/Vm between about 0.6 and 1.2 are preferable (Figure 4.8). 

In order to describe a liquid meniscus and hence to obtain the interfacial tension from the shape of a drop or bubble the Laplace equation is used. This equation describes the mechanical equilibrium of two homogeneous fluids separated by an interface (Rusanov and Prokhorov 1996, Neumann and Spelt 1996) and relates the pressure difference across the interface to the surface tension and the curvature of the interface... 

Once the Weber number is increased from 1 to 10, the relative strength of the surface tension to the inertial forces decreases. Hence, less pressure difference across the interface can be tolerated (compared to the Wcg = 1 case) and the drop starts to deform, as shown in the left column (panel (a)) of Fig. 4.6. The deformed drop almost takes the shape of a flat ellipsoid and hence, the drop recirculating region in the downstream side of the drop grows larger. The enhanced vortex size... 

The equilibrium drop shape is related to the surface tension of the drop through its curvature. In other words, the tendency for the drop to assume a curved interfacial shape is due to its surface tension. A consequence of this is the existence of a pressure difference across the interface... 

It is found that there exists a pressure difference across the curved interfaces of liquids (such as drops or bubbles). For example, if one dips a tube into water (or any fluid) and applies a suitable pressure, then a bubble is formed (Figure 1.13). This means that the pressure inside the bubble is greater than the atmosphere pressure. It thus becomes apparent that curved liquid surfaces induce effects, which need special physicochemical analyses in comparison to flat liquid surfaces. It must be noticed that in this system a mechanical force has induced a change on the surface of a liquid. This phenomenon is also called capillary forces. Then one may ask, does this also require similar consideration in the case of solids The answer is yes, and will be discussed later in detail. For example, in order to remove liquid, which is inside a porous media such as a sponge, one would need force equivalent to these capillary forces. Man has been fascinated with bubbles for many centuries. As seen in Figure 1.13, the bubble is produced by applying a suitable pressure, AP, to obtain a bubble of radius R, where the surface tension of the liquid is y. 

Ap Capillary pressure jump across surface ij where i and j are different fluids Radius of antifoam entity (oil drop or oil/particle mixture)... 

Thus bubbles and drops with small radii have large excess pressures and vice versa. A plane surface of soap film or fluid can be considered as part of a spherical surface of large radius and so the excess pressure difference across the surface is zero. 

We will derive the Young-Laplace equation, which in general terms gives the pressure difference across a curved surface, for the specific case of a liquid spherical drop, having a radius R and a surface tension y. The pressure inside the droplet is designated as Pi and the pressure outside as P2. 

Ultimately, the surface energy is used to produce a cohesive body during sintering. As such, surface energy, which is also referred to as surface tension, y, is obviously very important in ceramic powder processing. Surface tension causes liquids to fonn spherical drops, and allows solids to preferentially adsorb atoms to lower tire free energy of tire system. Also, surface tension creates pressure differences and chemical potential differences across curved surfaces tlrat cause matter to move. 

The pressure drop through the filter is a function of two separate effects. The clean filter has some initial pressure drop. This is a function of filter material, depth of the filter, the superficial gas velocity, which is the gas velocity perpendicular to the filter face, and the viscosity of the gas. Added to the clean filter resistance is the resistance that occurs when the adhering particles form a cake on the filter surface. This cake increases in thickness as approximately a linear function of time, and the pressure difference necessary to cause the same gas flow also becomes a linear function with time. Usually, the pressure available at the filter is limited so that as the cake builds up the flow decreases. Filter cleaning can be based, therefore, on (1) increased pressure drop across the filter, (2) decreased volume of gas flow, or (3) time elapsed since the last cleaning. 

It will be shown here that, due to the presence of surface tension in liquids, a pressure difference exists across the curved interfaces of liquids (such as drops or bubbles). This capillary force will be analyzed later. 

Consider a homogeneous membrane with thickness Ax separating an outer solution 1 from an inner solution 2. The flow occurs along the x-coordinate perpendicular to the membrane surface. The zero point of x is on the surface in contact with solution 1. The electrolyte solutions are characterized by their electrochemical potentials. Within the membrane, the chemical potential is different from both p.n and p.i2. However, it is widely assumed that the potentials on the membrane surface are equal to those of the solutions they are in contact with. Across the membrane, we identify the mechanical pressure difference (AP Pt P2), the difference between concentration of species (Ac = cn — ci2), and the drop of electric potentials (A // i//, - ifi2). These differences are related to the electrochem-... 

To calibrate the SGR calculation, we need to compare the observed pressure drops at every part of the fault surface with the SGR at the same point. The pressure profiles shown in Fig. 11 have been input to the fault model and are stored at every grid-point for footwall and hangingwall. At each grid node, the difference between the footwall and hangingwall pressures is the in situ pressure drop at the fault. Fig. 12a shows a cross-plot of SGR against across-fault pressure difference for every fault grid node in the Brent-... 

It should be noted that the pressure is always greater on the concave side of the interface irrespective of whether or not this is a condensed phase.) The phenomena due to the presence of curved liquid surfaces are called capillary phenomena, even if no capillaries (tiny cylindrical tubes) are involved. The Young-Laplace equation is the expression that relates the pressure difference, AP, to the curvature of the surface and the surface tension of the liquid. It was derived independently by T. Young and P. S. Laplace around 1805 and relates the surface tension to the curvature of any shape in capillary phenomena. In practice, the pressure drop across curved liquid surfaces should be known from the experimental determination of the surface tension of liquids by the capillary rise method, detailed in Section 6.1. 

This equation is quite significant in explaining the properties of liquid surfaces and bubbles. First, Eq. 3.61 indicates that, in equilibrium, a pressure difference can be maintained across a curved surface. The pressure inside the liquid drop or gas bubble is higher than the external pressure because of the surface tension. The smaller the droplet or larger the surface tension, the larger the pressure difference that can be maintained. For a flat surface, r = 00, and the pressure difference normal to the interface vanishes. 

At the last step we have used also Eqs (35), (40), and (41). For two identical drops 7 y—> >, and then P reduces to the capillary pressure of the drops P = 2o/7 j = 2o/7 2 The condition II = P, see Eq. (42), means that at equilibrium the disjoining pressure H counterbalances the pressure difference applied across the film surface. In addition, the condition dlUdh < 0 guarantees that the equilibrium is stable (rather than unstable). 

For the stationary drop, there is a pressure difference between the liquid and gas phases across each interface. However, the liquid pressure between the two ends is balanced, that is, Pi,R PiA = 0- A. pressure difference is generated for liquid motion by manipulating the surface tension on one side of a liquid drop. The surface tension is modified by heating one drop interface. The decrease in surface tension because of increase in temperature can be expressed as... 

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