Spherical drops

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. 

Consider a spherical drop of radius r, and suppose this undergoes a virtual displacement such that r is changed to r + Sr. The work done in the displacement vanishes to the first order. 

In order to solve the complete problem of liquid flow around a single spherical drop or bubble in the presence of surface-active agents it is necessary to solve the equations for viscous liquid motion simultaneously with a conservation equation for dissolved surface-active agents ... 

Waslo and Gal-Or (Wl) recently generalized the Levich solution [Eq. (65)] by evaluating the effect of and y on the terminal velocity of an ensemble of spherical drops of bubbles. Their solution is... 

Mass transfer from a single spherical drop to still air is controlled by molecular diffusion and. at low concentrations when bulk flow is negligible, the problem is analogous to that of heat transfer by conduction from a sphere, which is considered in Chapter 9, Section 9.3.4. Thus, for steady-state radial diffusion into a large expanse of stationary fluid in which the partial pressure falls off to zero over an infinite distance, the equation for mass transfer will take the same form as that for heat transfer (equation 9.26) ... 

In a liquid-liquid extraction unit, spherical drops of solvent of uniform size are continuously fed to a continuous phase of lower density which is flowing vertically upwards, and hence countercurrently with respect to the droplets. The resistance to mass transfer may be regarded as lying wholly within the drops and the penetration theory may be applied. The upward velocity of the liquid, which may be taken as uniform over the cross-section of the vessel, is one-half of the terminal falling velocity of the droplets in the still liquid. 

Fig. 22. Spherical drop deformed into an ellipsoidal shape in a rotational shear field [76]...

The degree of deformation and whether or not a drop breaks is completely determined by Ca, p, the flow type, and the initial drop shape and orientation. If Ca is less than a critical value, Cacri the initially spherical drop is deformed into a stable ellipsoid. If Ca is greater than Cacrit, a stable drop shape does not exist, so the drop will be continually stretched until it breaks. For linear, steady flows, the critical capillary number, Cacrit, is a function of the flow type and p. Figure 14 shows the dependence of CaCTi, on p for flows between elongational flow and simple shear flow. Bentley and Leal (1986) have shown that for flows with vorticity between simple shear flow and planar elongational flow, Caen, lies between the two curves in Fig. 14. The important points to be noted from Fig. 14 are these ... 

For larger Reynolds numbers (1 < NRe < 500), Rivkind and Ryskind (see Grace, 1983) proposed the following equation for the drag coefficient for spherical drops and bubbles ... 

According to a newer theory83), see also84), the fracture energy 7 calculated from Eq. (43) or a similar relation has no connection with true surface energy. Energy is needed not to create a new surface but to deform the solid to its maximum strain. This rule is unquestionably correct for liquids. As was proved by Plateau (1869), a liquid cylinder can be extended by an external force until its length exceeds its circumference then it spontaneously splits into two spherical drops, and the combined area of the drops is smaller than that of the stretched cylinder. Thus, external work is required to stretch the cylinder, not to break it. 

COSILAB Combustion Simulation Software is a set of commercial software tools for simulating a variety of laminar flames including unstrained, premixed freely propagating flames, unstrained, premixed burner-stabilized flames, strained premixed flames, strained diffusion flames, strained partially premixed flames cylindrical and spherical symmetrical flames. The code can simulate transient spherically expanding and converging flames, droplets and streams of droplets in flames, sprays, tubular flames, combustion and/or evaporation of single spherical drops of liquid fuel, reactions in plug flow and perfectly stirred reactors, and problems of reactive boundary layers, such as open or enclosed jet flames, or flames in a wall boundary layer. The codes were developed from RUN-1DL, described below, and are now maintained and distributed by SoftPredict. Refer to the website http //www.softpredict.com/cms/ softpredict-home.html for more information. 

At the heart of the polarographic apparatus is a fine-bore capillary through which mercury flows at a constant rate. Mercury emerges from the end of the capillary as small droplets, which are formed at a constant, controlled rate of between 10-60 drops per minute. During each drop cycle , the spherical drop emerges, grows in diameter and then falls. ... 

Influenced by interfacial tension and centrifugal forces, spherical drops of various diameters originate at the holes. If we again assume the Sauter diameter, according to Eq. (9.1), as the mean diameter of the spectrum of particles, the following equation for heavy and light phases results from theoretical and experimental results [10] ... 

The surface molecules are under a different force field from the molecules in the bulk phase or the gas phase. These forces are called surface forces. A liquid surface behaves like a stretched elastic membrane in that it tends to contract. This action arises from the observation that, when one empties a beaker with a liquid, the liquid breaks up into spherical drops. This phenomenon indicates that drops are being created under some forces that must be present at the surface of the newly formed interface. These surface forces become even more important when a liquid is in contact with a solid (such as ground-water oil reservoir). The flow of liquid (e.g., water or oil) through small pores underground is mainly governed by capillary forces. Capillary forces are found to play a very dominant role in many systems, which will be described later. Thus, the interaction between liquid and any solid will form curved surface that, being different from a planar fluid surface, initiates the capillary forces. 

In order to allow for a variation of interfacial area from that of an equivalent sphere, the eccentricity of the ellipsoidal drop must be taken into account. The area ratio of Eq. (44) does not exceed unity by a serious amount until an eccentricity of 1.5 is attained. An experimental plot of eccentricity as ordinate vs. equivalent spherical drop diameter as abscissa may result in a straight line (G7, Kl, K3, S12). A parameter is yet to be developed by which the lines can be predicted without recourse to experiment. Eccentricity is not an accurate shape description of violently oscillating drops and should therefore be used only for drop size below the peak diameter (region B of Fig. 5). 

For formation or evaporation of a Hquid from vapor, the equilibrium shape of the Hquid is a spherical drop so the sphere approximation is more appropriate than for a solid particle, where we approximate it as a sphere to make the mathematics simple. 

The concentration profile for a reactant A which must migrate from a drop or bubble into the continuous phase to react might be as shown in Figure 12-10. There is a concentration drop around the spherical drop or bubble because it is migrating outward, but, as with a planar gas-liquid interface in the falling film reactor, there should be a discontinuity in at the interface due to the solubility of species A and a consequent equilibrium distribution between phases. 

When a fluid sphere exhibits little internal circulation, either because of high K = Pp/p or because of surface contaminants, the external flow is indistinguishable from that around a solid sphere at the same Re. For example, for water drops in air, a plot of versus Re follows closely the curve for rigid spheres up to a Reynolds number of 200, corresponding to a particle diameter of approximately 0.85 mm (B5). In fact, many of the experimental points used in Section II to determine the standard drag curve refer to spherical drops in gas streams, where high values of k ensure negligible internal circulation. 

J. Mass Transfer to a Continuous Phase from a Single Spherical Drop... 

The thickness of the diffusion layer, as first shown by Ilkovic, will be changed in that instead of Eq. (7.204), one has now for an expanding spherical drop... 

The influence of curvature on phase equilibria is most readily understood for liquids, for which the activity is measured by the vapor pressure of the liquid. Accordingly, suppose we consider the process of transferring molecules of a liquid from a bulk phase with a vast horizontal surface to a small spherical drop of radius Rs. 

Water is not strongly attracted to a wax surface, which is nonpolar. So as to minimize surface area, the water tends to form a sphere. Sitting on a solid surface, however, the spherical drop of water is squashed down into a bead by the force of gravity. 

Characteristic product shape thin slab extrusion granules spherical drops grains -in. particles... 

Stage II—Period of Liquid Combustion. The subject of combustion following ignition has been pursued rather extensively. The linear relationship between the square of the diameter and time, which is predicted theoretically to hold for the combustion of a spherical drop surrounded concentrically by a spherical flame, has been found to apply generally to the evaporation period in the combustion of pure compounds. 

The quasi-steady-state theory has been applied particularly where a condensed phase exists whose volume changes slowly with time. This is true, for example, in the sublimation of ice or the condensation of water vapor from air on liquid droplets (M3, M4). In the condensation of water vapor onto a spherical drop of radius R(t), the concentration of water vapor in the surrounding atmosphere may be approximated by the well-known spherically symmetric solution of the Laplace equation ... 

The diffusion rate into the drop of component B, therefore, can be calculated for the apparently stationary state with an over-the-whole-cross-section constant value of ha = k. Then for a spherical drop of radius R, the differential equation becomes... 

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