The Laplace equation

If the solid in question is available only as a finely divided powder, it may be compressed into a porous plug so that the capillary pressure required to pass a nonwetting liquid can be measured [117]. If the porous plug can be regarded as a bundle of capillaries of average radius r, then from the Laplace equation (II-7) it follows that... 

For constant 6 and y (the contact angle was found not to be very dependent on pressure), one obtains from the Laplace equation. 

The Laplace equation, which defines tire pressure difference, AP, across a curved surface of radius, r. 

When the void space in an agglomerate is completely filled with a Hquid (Fig. Ic), the capillary state of wetting is reached, and the tensile strength of the wet particle matrix arises from the pressure deficiency in the Hquid network owing to the concave Hquid interfaces at the agglomerate surface. This pressure deficiency can be calculated from the Laplace equation for chcular capillaries to yield, for Hquids which completely wet the particles ... 

For isotropic homogeneous porous media (uniform permeability and porosity), the pressure for creeping incompressible single phase-flow may be shown to satisfy the LaPlace equation ... 

In the case of the free jet, the solution for the Aaberg exhaust system can be found by solving the Laplace equation by the method of separation of variables and assuming that there is no fluid flow through the surface of the workbench. At the edge of the jet, which is assumed to be at 0—0, the stream function is given by Eq. (10.113). This gives rise to... 

A similar mathematical model to that just described for bench slot exhausts can again be used, but in this case the Laplace equation should be employed in a cylindrical coordinate system (see Fig. 10.83), namely,... 

Thus the Laplace equation (Eq. (10.117)) has to be solved subiect to the following boundary conditions ... 

As the corrosion rate, inclusive of local-cell corrosion, of a metal is related to electrode potential, usually by means of the Tafel equation and, of course, Faraday s second law of electrolysis, a necessary precursor to corrosion rate calculation is the assessment of electrode potential distribution on each metal in a system. In the absence of significant concentration variations in the electrolyte, a condition certainly satisfied in most practical sea-water systems, the exact prediction of electrode potential distribution at a given time involves the solution of the Laplace equation for the electrostatic potential (P) in the electrolyte at the position given by the three spatial coordinates (x, y, z). 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

The numerical approaches to the solution of the Laplace equation usually demand access to minicomputers with fast processing capabilities. Numerical methods of this sort are essential when the electrolyte is unconfined, as for an off-shore rig or a submarine hull. However, where the electrolyte is confined, as within essentially cylindrical equipment such as pipework and heat-exchangers, or for restricted electrolyte depths, a simpler modelling procedure may be adopted in the case of electrolytes of good conductivity, such as sea-water . This simpler procedure enables computation to be carried out on small, desk-top microcomputers. 

The surface tension acting on the meniscus would pull the sphere toward the plane and give rise to an attractive pressure P over the contact region, which can be calculated in terms of the Laplace equation. 

G. Lippmann introduced the capillary electrometer to measure the surface tension of mercury (Fig. 4.10). A slightly conical capillary filled with mercury under pressure from a mercury column (or from a pressurized gas) is immersed in a vessel containing the test solution. The weight of the mercury column of height h is compensated by the surface tension according to the Laplace equation... 

The Laplace equation also applies to the distribution of electrical potential and current flow in an electrically conducting medium as well as the temperature distribution and heat flow in a thermally conducting medium. For example, if => E, V => i, and fi/K => re, where re is the electrical resistivity (re = RA/Ax), Eq. (13-22) becomes Ohm s law ... 

The purpose of this chapter is to introduce the effect of surfaces and interfaces on the thermodynamics of materials. While interface is a general term used for solid-solid, solid-liquid, liquid-liquid, solid-gas and liquid-gas boundaries, surface is the term normally used for the two latter types of phase boundary. The thermodynamic theory of interfaces between isotropic phases were first formulated by Gibbs [1], The treatment of such systems is based on the definition of an isotropic surface tension, cr, which is an excess surface stress per unit surface area. The Gibbs surface model for fluid surfaces is presented in Section 6.1 along with the derivation of the equilibrium conditions for curved interfaces, the Laplace equation. 

Equation (6.27) is the Laplace equation, or Young-Laplace equation, which defines the equilibrium condition for the pressure difference over a curved surface. In Section 6.2 we will examine the consequences of surface or interface curvature for some important heterogeneous phase equilibria. 

The Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal ... 

The fact that the curvature of the surface affects a heterogeneous phase equilibrium can be seen by analyzing the number of degrees of freedom of a system. If two phases a and are separated by a planar interface, the conditions for equilibrium do not involve the interface and the Gibbs phase rule as described in Chapter 4 applies. On the other hand, if the two coexisting phases a and / are separated by a curved interface, the pressures of the two phases are no longer equal and the Laplace equation (6.27) (eq. 6.35 for solids), expressed in terms of the two principal curvatures of the interface, defines the equilibrium conditions for pressure ... 

Here Vs and V1 are the molar volume of the two phases, but the subscript m is not used for simplicity. The pressure of the two phases is related by the Laplace equation (6.27), which for a spherical liquid droplet surrounded by its own vapour becomes, in differential form,... 

The pressure can be substituted by the mean curvature through the Laplace equation, for which... 

We will now consider a case where a spherical crystal with radius r of a single component solid phase is surrounded by a liquid with more than one component. The differential of the Laplace equation (6.27) is... 

Contents Introduction. - Basic Equations. -Diffusional Transport - Digitally. - Handling of Boundary Problems. - Implicit Techniques and Other Complications. - Accuracy and Choice. -Non-Diffusional Concentration Changes. - The Laplace Equation and Other Steady-State Systems. - Programming Examples. - Index. 

If we consider the pendant drop as shown in Figure 1, we can write from the Laplace equation (20) for the point P in terms of the pressure differences across the interface at a reference point B ... 

Consider Eq. (15.1) with no sources (Laplace) applied to a square plate with U defined everywhere on the boundary. If the problem specification is symmetric under interchange of the x and y directions, the Laplace equation may then be separable with solutions of the general form U(x, y) = X(x)L(y) namely,... 

Relaxation methods involve iteratively seeking a convergent solution to the Laplace equation. In the present case, for instance, if we rewrite the coefficient matrix A = I + E, where the latter matrix consists of elements that are all small compared to 1, the matrix Laplace equation takes the form = EU + b. One begins the calculation with values U = b [or, equivalently, U = 0] and iteratively computes successive values The calculation terminates when a specified limit of accuracy is achieved. One such measure involves calculating the proportional differences ... 

As is well known from elementary electrostatics, the equation governing the electrostatic potential, <1>, in the volume V is the Laplace equation ... 

The fundamental property of liquid surfaces is that they tend to contract to the smallest possible area. This property is observed in the spherical form of small drops of liquid, in the tension exerted by soap films as they tend to become less extended, and in many other properties of liquid surfaces. In the absence of gravity effects, these curved surfaces are described by the Laplace equation, which relates the mechanical forces as (Adamson and Gast, 1997 Chattoraj and Birdi, 1984 Birdi, 1997) ... 

The pressure applied produces work on the system, and the creation of the bubble leads to the creation of a surface area increase in the fluid. The Laplace equation relates the pressure difference across any curved fluid surface to the curvature, 1/radius and its surface tension y. In those cases where nonspherical curvatures are present, the more universal equation is obtained ... 

The Laplace equation is useful for analyses in a variety of systems ... 

Similar results can also be derived by using the Laplace equation (Equation 2.21) (1/radius = 1/R) ... 

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