The spherical surface

The thermodynamics of curved surfaces is more subtle than that of planar, and we discuss only the most important case, the spherical surface, for which the two principal radii of curvature are equal. Hie extension of the argument to other curved surfaces is beset with difficulties into which we do not enter. The spherical surface, the bubble or the drop, is the only one that is stable in ffie absence of an external field. The original analysis of Gibbs was clarified and its consequences worked out by Tolman, whose work Koenig extended to multicomponent systems. Buff, Hill, and Kondo describe explicitly how the surface tension depends on the position of the dividing surface to which it is referred, or at which it is calculated. 

From a spherical drop of phase a surrounded by phase /3 we choose a system formed of a conical section of solid angle o (Fig. 2.2). The side 

The firat is a generalization of l.aplace s equation for a surface tension tr measured at, or referred to, an arbitrary dividing surface. Since Ap is invariant with respect to the choice of R, it follows that, in general, a must formally be a function of R, which we write a[li]. Moreover if we define the surface of tension as the radius at which the tension acts, in conformity with the model described in the opening paragraph of 2.2, then a comparison of (2.1) with (2.57) shows that the second term vanishes at this surface. That is, 

It follows that is an extremum, which, we shall see, is a minimum. The curvature term vanishes at this surface. 

The formal derivative [da/dK] can be expressed in terms of the adsorptions. Consider the change in F that follows from real isothermal changes in the variables of (2.52), 

---

The principle of the Brinell hardness test is that the spherical surface area of a recovered indentation made with a standard hardened steel ball under specific load is direcdy related to the property called hardness. In the following, HBN = Brinell hardness number, P = load in kgf,... 

Figure 5 shows conduction heat transfer as a function of the projected radius of a 6-mm diameter sphere. Assuming an accommodation coefficient of 0.8, h 0) = 3370 W/(m -K) the average coefficient for the entire sphere is 72 W/(m -K). This variation in heat transfer over the spherical surface causes extreme non-uniformities in local vaporization rates and if contact time is too long, wet spherical surface near the contact point dries. The temperature profile penetrates the sphere and it becomes a continuum to which Fourier s law of nonsteady-state conduction appfies. 

Our task is to derive an explicit expression for the potential U proceeding from this equation. This means that we have to take the function U out of this integral. With this purpose in mind consider the limiting value of the second integral, when the radius of the spherical surface r tends to zero. Since both the potential and its derivatives are continuous functions inside the volume, we have... 

Now we will establish a relationship between the potential U(p) at any point p of the volume V and its values on the spherical surface, surrounding all masses. Fig. 1.11. The reason why we consider this problem is very simple it plays the fundamental role in Stokes s theorem, which allows one to determine the elevation of the geoid with respect to the reference ellipsoid. 

Here the unit vector n and radius vector R have opposite directions. The volume V is surrounded by the surface S as well as a spherical surface with infinitely large radius. In deriving this equation we assume that the potential U p) is a harmonic function, and the Green s function is chosen in such a way that allows us to neglect the second integral over the surface when its radius tends to an infinity. The integrand in Equation (1.117) contains both the potential and its derivative on the spherical surface S. In order to carry out our task we have to find a Green s function in the volume V that is equal to zero at each point of the boundary surface ... 

By definition, any plane 0 — constant is a plane of symmetry. In other words, there are always two elementary masses, which are equal to each other, and located at opposite sides of this plane but at the same distance. As is seen from Fig. 1.5d, the sum of 0-components, caused by both masses is equal to zero. Representing the total mass as a sum of such pairs we conclude that the 0-component, gg, due to the spherical mass is absent at every point outside and inside the body. In the same manner we can prove that — 0. Of course, volume integration, Equation (1.6), can prove this fact, but this procedure is much more complicated. Thus, the attraction field has only a radial component, g, and the field is directed toward the origin, 0. In order to determine this component we will proceed from Equation (1.26) and consider a spherical surface with radius R, as is shown in Fig. 1.5c. Inasmuch as dS — dSiR and the scalar component g is constant at points of the spherical surface, we have for the flux ... 

Equations (2.24 and 2.27) look like as Clairaut s formulas which will be derived later. However, this similarity is superficial, since the former do not contain the flattening of the earth. A variation of the field magnitude, g with latitude. Equation (2.24), is caused by only a change of the centripetal acceleration on the spherical surface. 

As was shown in Chapter 1, these conditions uniquely define the function T. For determination of the disturbing potential we will make use of Poisson s integral, described in the Chapter 1, which allows one to find the harmonic function E outside the spherical surface of the radius R, Fig. 2.9b, if this function, E p), is known at points of this surface ... 

Here the point p belongs to the spherical surface A of radius R. In order to find the upper limit on the left hand side of this equality, let us recall that T is the disturbing potential. In other words, it is caused by the irregular distribution of masses whose sum is equal to zero. This means that its expansion in power series with Legendre s functions does not contain a zero term. The next term is also equal to zero, because the origin coincides with the center of mass. Therefore, the series describing the function T starts from the term, which decreases as r. This means that the product r T O if oo and... 

Making use of Equations (2.278 and 2.280) we can calculate the disturbing potential on the spherical surface and outside. In particular, at points of this surface we have... 

Thus, in general the pendulum moves along an ellipse on the spherical surface with radius equal to the string length, /. This motion has a periodic character and the period is that of the swinging, T — Injco). In particular, if the initial conditions are... 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

Convective diffusion to a growing sphere. In the polarographic method (see Section 5.5) a dropping mercury electrode is most often used. Transport to this electrode has the character of convective diffusion, which, however, does not proceed under steady-state conditions. Convection results from growth of the electrode, producing radial motion of the solution towards the electrode surface. It will be assumed that the thickness of the diffusion layer formed around the spherical surface is much smaller than the radius of the sphere (the drop is approximated as an ideal spherical surface). The spherical surface can then be replaced by a planar surface... 

The behavior of a rotating sphere or hemisphere in an otherwise undisturbed fluid is like a centrifugal fan. It causes an inflow of the fluid along the axis of rotation toward the spherical surface as shown in Fig. 1(a). Near the surface, the fluid flows in a spirallike motion towards the equator as shown in Fig. 1(b) and (c). On a rotating sphere, two identical flow streams develop on the opposite hemispheres. The two streams interact with each other at the equator, where they form a thin swirling jet toward the bulk fluid. The Reynolds number for the rotating sphere or hemisphere is defined as ... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

For large values of 6, Manohar [42], and Banks [4] solved the boundary layer Eqs. (2)-(5) numerically with a finite difference method. Manohar s results for the meridional and azimuthal velocity gradients on the spherical surface have been curve-fitted by Newman [45] and Chin [18] to follow the following equations in the regime of 0 < 9 < 7t/2 ... 

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

For the flow induced by the rotating hemisphere, the leading edge of the flow boundary layer occurs at the pole of rotation. The surface distance, x, from the leading edge is equal to ad. One may also take aQsinff as the characteristic velocity for every local point on the spherical surface. Thus, the quantities, Shx, Rex, and fx can be expressed as... 

Termolecular Reactions. If one attempts to extend the collision theory from the treatment of bimolecular gas phase reactions to termolecular processes, the problem of how to define a termolecular collision immediately arises. If such a collision is defined as the simultaneous contact of the spherical surfaces of all three molecules, one must recognize that two hard spheres will be in contact for only a very short time and that the probability that a third molecule would strike the other two during this period is vanishingly small. 

As the vapor flows in the direction along the spherical surface of the particle, a boundary layer coordinate ( , X, co) given in Fig. 21 is employed to describe the vapor-layer equation. In this coordinate, the continuity and momentum equations for incompressible vapor flows with gravitation terms neglected... 

Let us consider the uptake of a given species, either a nutrient or a pollutant heavy metal or an organic (macro)molecule, etc., which will be referred to as M. M is present in the bulk of the medium at a concentration, c, ar d we assume that the only relevant mode of transport from the medium to the organism s surface is diffusion. The internalisation sites are taken to be located on the spherical surface of the microorganism or in a semi-spherical surface of a specialised region of the organism with radius ro (see Figure 1). Thus, diffusion prescribes ... 

In a recent comprehensive study, Chhabra, Agarwal, and Sinha(27) have found that the most satisfactory characteristic linear dimension to use is the diameter of the sphere of equal volume and that the most relevant characteristic shape is the sphericity, (surface area of particle / surface area of sphere of equal volume). The limitation of this whole... 

The single-crystal structure of 31 clearly reflects the presence of an integral fulvene-type Jt-system on the spherical surface. The average bond length for the [5,6]-bond between C1-C2 and C3-C4 is 1.375 A, which is considerably shorter than a typical [5,6]-bond in CgQ. In contrast, the bond between C2 and C3 (1.488 A) is notably... 

A variant of the spherical surface-burning incendiary was the elongated projectile made by kneading a warm incendiary mix over a crossed iron frame which extended to approx twice the length of the desired diam. Typical incendiary components were green pitch, fine corned powd, oakum, tallow, and a small quantity of naphtha. A fuze of fine powd was inserted in the nose to ensure ignition... 

The rotational diffusion coefficient of the fuzzy cylinder can be formulated in a similar way. For the rotational diffusion process, it is convenient to imagine a hypothetical sphere which has the diameter equal to Lc, just encloses the test fuzzy cylinder, and moves with the translation of the fuzzy cylinder. If the test cylinder and the portions of surrounding fuzzy cylinders entering the sphere are projected onto the spherical surface as depicted in Fig. 15b (cf. [108]), the rotational diffusion process of the test cylinder can be treated as the translational diffusion process of a circle on the hypothetical spherical surface with ribbon-like obstacles. 

When the diffusion time is short enough, the translation on the spherical surface is approximately identical with that on the tangent plane to the spherical surface. The latter is the two-dimensional diffusion process treated by the Green function method in Appendix C, and we can use Eq. (C17) again. Since the rotational diffusion coefficient Dr is related to the translational diffusion coefficient D<2) in Eq. (Cl7) by Dr = D(2)/(Le/2)2, we have... 

---