Mathematical modeling of a field emitter with a hyperbolic shape

This article is devoted to modeling a field emission diode system. The emitter surface is a hyperboloid of rotation. The anode surface is a part of the hyperboloid of rotation, in a particular case, a circular diaphragm. A boundary value problem is formulated for the Laplace equation with non-axisymmetric boundary conditions of the first kind. A 3D solution was found by the variable separation method in the prolate spheroidal coordinates. The electrostatic potential distribution is presented in the form of the Legendre functions expansions. The calculation of the expansion coefficients is reduced to solving a system of linear equations with constant coefficients. All geometric dimensions of the system are the parameters of the problem.

Convective and Microwave Drying of Prolate Spheroidal Solids: Modeling and Simulation

This paper focuses some fundamental aspects of combined convective and microwave drying of prolate spheroidal solids. A transient mathematical modeling based on the diffusion theory (mass and heat balance equations) written in prolate spheroidal coordinates was derived and the importance of this procedure on the analysis of the drying process of wet porous solid, is also presented. Results pointed to the behavior of the moisture migration and heating of the solid with different aspect ratio. Solids with higher area/volume relationships dry and heat faster.

Green’s Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equation

A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.