Laplaces equation

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

Challenging. You will have to draw on your knowledge of all areas of ehemical engineering. You will use most of the mathematical tools available (differential equations, Laplace transforms, complex variables, numerical analysis, etc.) to solve real problems. 

For within-particle AT the simple analysis by Prater (1958) for any particle geometry and kinetics gives the desired expression. Since the temperature and concentration within the particle are represented by the same form of differential equation (Laplace equation) Prater showed that the T and distributions must have the same shape thus at any point in the pellet x... 

Scalar and vector fields that depend on more than one independent variable, which we write in the notation T fx, y), TCx, t), TCr), TCr, t), etc., are very often obtained as solutions to PDEs. Some classic equations of mathematical physics that we will consider are the wave equation, the heat equation, Laplace s equation, Poisson s equation, and the SchrOdinger equation for some exactly solvable quantum-mechanical problems. 

Analyses are based on three main equations Laplace s equation, solutions of which are used to predict the potential between the cathode and anode electrodes of the electroforming cell and particularly at the cathode where metal deposition takes place. Erom the potential solution so derived. Ohm s law can then be employed to find the current density at the cathode surface and consequently from Earaday s law the rate of electroforming is derived. Theoretical treatments need to take account of current efficiency effects to which are tied the electrolyte properties and periodic reversal of current (McGeough and Rasmussen 1981). 

The use of macroscopic equations, Laplace and Kelvin equations, appears quite convincing to quantify the variation of the adhesion force as the result of change of the geometrical parameters. However, several questions rise ... 

We have four partial differential equations and two equilibrium equations. Laplace transform is applied to solve four variables Ce(t), C e(t), Cm(r,t) and Ci(r,t). 

It is assumed from here on that the reader is familiar with differential equations, Laplace transforms, z-transforms, frequency respemse, and other dements SISO (single-input, singleoutput) control theory. 

Prescribed boundary condition on body-attached mesh is considered as a boundary condition on the mesh motion equation (Laplace equation). 

Equation [23] represents the GC solution for point ions. A key development in the theory of electrolytes was the introduction of a finite distance of closest approach of ions to a charged surface by Stern and further elaborated upon by Grahame. The layer of ions directly adsorbed onto the surface constitutes the inner Helmholtz layer those ions that make contact but do not adsorb define the abovementioned distance of closest approach and constitute the outer Helmholtz or Stern layer. These modifications still admit an analytical solution to the GC equation Laplace s equation is solved in the Stern layer with the (linear) potential and (constant) field matched at the polyelectrolyte surface and to the outer GC solution. The adsorbed ions serve to reduce the charge density of the surface. Identification of the inner and outer Helmholtz layers has been particularly helpful in improving agreement between GC theory and electrochemical data. If we assign a common radius a to all electrolyte ions, then the identification of the interface atx = a actually... 

Equation 11-3 is a special case of a more general relationship that is the basic equation of capillarity and was given in 1805 by Young [1] and by Laplace [2]. In general, it is necessary to invoke two radii of curvature to describe a curved surface these are equal for a sphere, but not necessarily otherwise. A small section of an arbitrarily curved surface is shown in Fig. II-3. The two radii of curvature, R and / 2[Pg.6]

There are a number of relatively simple experiments with soap films that illustrate beautifully some of the implications of the Young-Laplace equation. Two of these have already been mentioned. Neglecting gravitational effects, a film stretched across a frame as in Fig. II-1 will be planar because the pressure is the same as both sides of the film. The experiment depicted in Fig. II-2 illustrates the relation between the pressure inside a spherical soap bubble and its radius of curvature by attaching a manometer, AP could be measured directly. 

Returning to equilibrium shapes, these have been determined both experimentally and by solution of the Young-Laplace equation for a variety of situations. Examples... 

An approximate treatment of the phenomenon of capillary rise is easily made in terms of the Young-Laplace equation. If the liquid completely wets the wall of the capillary, the liquids surface is thereby constrained to lie parallel to the wall at the region of contact and the surface must be concave in shape. The... 

The exact treatment of capillary rise must take into account the deviation of the meniscus from sphericity, that is, the curvature must correspond to the AP = Ap gy at each point on the meniscus, where y is the elevation of that point above the flat liquid surface. The formal statement of the condition is obtained by writing the Young-Laplace equation for a general point (x, y) on the meniscus, with R and R2 replaced by the expressions from analytical geometry given in... 

Equations II-12 and 11-13 illustrate that the shape of a liquid surface obeying the Young-Laplace equation with a body force is governed by differential equations requiring boundary conditions. It is through these boundary conditions describing the interaction between the liquid and solid wall that the contact angle enters. 

This effect assumes importance only at very small radii, but it has some applications in the treatment of nucleation theory where the excess surface energy of small clusters is involved (see Section IX-2). An intrinsic difficulty with equations such as 111-20 is that the treatment, if not modelistic and hence partly empirical, assumes a continuous medium, yet the effect does not become important until curvature comparable to molecular dimensions is reached. Fisher and Israelachvili [24] measured the force due to the Laplace pressure for a pendular ring of liquid between crossed mica cylinders and concluded that for several organic liquids the effective surface tension remained unchanged... 

With the preceding introduction to the handling of surface excess quantities, we now proceed to the derivation of the third fundamental equation of surface chemistry (the Laplace and Kelvin equations, Eqs. II-7 and III-18, are the other two), known as the Gibbs equation. 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

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. 

Surface properties enter tlirough the Yoimg-Laplace equation of state for the surface pressure ... 

Due to the conservation law, the diffiision field 5 j/ relaxes in a time much shorter than tlie time taken by significant interface motion. If the domain size is R(x), the difhision field relaxes over a time scale R Flowever a typical interface velocity is shown below to be R. Thus in time Tq, interfaces move a distanc of about one, much smaller compared to R. This implies that the difhision field 6vj is essentially always in equilibrium with tlie interfaces and, thus, obeys Laplace s equation... 

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