Finite-differential time-domain method

A consequence of the complex interplay of the dielectric and thermal properties with the imposed microwave field is that both Maxwell s equations and the Fourier heat equation are mathematically nonlinear (i.e., they are in general nonlinear partial differential equations). Although analytical solutions have been proposed under particular assumptions, most often microwave heating is modeled numerically via methods such as finite difference time domain (FDTD) techniques. Both the analytical and the numerical solutions presume that the numerical values of the dielectric constants and the thermal conductivity are known over the temperature, microstructural, and chemical composition range of interest, but it is rare in practice to have such complete databases on the pertinent material properties. 

The finite-difference time-domain (FDTD) method [1] is one of the most frequently used techniques in electromagnetics. It involves the space and time discretization of the whole working space and the finite-difference approximation to Maxwell s differential equations. For the analysis of the... 

For brevity, further discussion is restricted to the spatial discretization used to obtain ordinary differential equations. Often the choice and parameters selection for this methods is left to the user of commercial process simulators, while the numerical (time) integrators for ODEs have default settings or sophisticated automatic parameter adjustment routines. For example, using finite difference methods for the time domain, an adaptive selection of the time step is performed that is coupled to the iteration needed to solve the resulting nonlinear algebraic equation system. For additional information concerning numerical procedures and algorithms the reader is referred to the literature. 

Linear first order parabolic partial differential equations in finite domains are solved using the Laplace transform technique in this section. Parabolic PDEs are first order in the time variable and second order in the spatial variable. The method involves applying the Laplace transform in the time variable to convert the partial differential equation to an ordinary differential equation in the Laplace domain. This becomes a boundary value problem (BVP) in the spatial direction with s, the Laplace variable as a parameter. The boundary conditions in x are converted to the Laplace domain and the differential equation in the Laplace domain is solved by using the techniques illustrated in chapter 3.1 for solving linear boundary value problems. Once an analytical solution is obtained in the Laplace domain, the solution is inverted to the time domain to obtain the final analytical solution (in time and spatial coordinates). Certain simple problems can be inverted to the time domain using Maple. This is best illustrated with the following examples. 

The independent variable in ordinary differential equations is time t. The partial differential equations includes the local coordinate z (height coordinate of fluidized bed) and the diameter dp of the particle population. An idea for the solution of partial differential equations is the discretization of the continuous domain. This means discretization of the height coordinate z and the diameter coordinate dp. In addition, the frequently used finite difference methods are applied, where the derivatives are replaced by central difference quotient based on the Taylor series. The idea of the Taylor series is the value of a function f(z) at z + Az can be expressed in terms of the value at z. 

Finite difference methods (FDM) are directly derived from the space time grid. Focusing on the space domain (horizontal lines in Fig. 6.6), the spatial differentials are replaced by discrete difference quotients based on interpolation polynomials. Using the dimensionless formulation of the balance equations (Eq. 6.107), the convection term at a grid point j (Fig. 6.6) can be approximated by assuming, for example, the linear polynomial. 

The differential equations presented in Section 5-2 describe the continuous movement of a fluid in space and time. To be able to solve those equations numerically, all aspects of the process need to be discretized, or changed from a continuous to a discontinuous formulation. For example, the region where the fluid flows needs to be described by a series of connected control volumes, or computational cells. The equations themselves need to be written in an algebraic form. Advancement in time and space needs to be described by small, finite steps rather than the infinitesimal steps that are so familiar to students of calculus. All of these processes are collectively referred to as discretization. In this section, disaetiza-tion of the domain, or grid generation, and discretization of the equations are described. A section on solution methods and one on parallel processing are also included. 

To analyze the airflow pattern, simulation of airflow was carried out using a fluid flow analysis package. Fluent 6.1 [1,6-12]. To solve the three-dimensional airflow field inside the nozzles, a CFD model was developed using the above software. Fluid flow and related phenomena can be described by partial differentiation equations, which caimot be solved analytically except in over-simplified cases. To obtain an approximate solution numerically, a discretization method to approximate the differential equations by a system of algebraic equations, which can be then numerically solved on a computer. The approximations were applied to small domains in space and/or time so the numerical solution provides results at discrete locations in space and time. Much of the accuracy depends on the quality of the methodology used, for which CFD is a powerful tool to predict the flow behavior of fluid inside any object. It provides various parameters such as air velocity profiles (axial, tangential, resultant etc.) and path lines trajectory, which are important for subsequent analysis. It was for those reasons that a CFD package. Fluent 6.1, which uses a Finite Volume (FV) method, was employed for airflow simulation. 

In order to find a numerical solution of the PBM the finite volume method has been applied. The investigated size domain has been discretized with equal-sized grid into 1000 intervals and partial differential equation has been represented as a system of ODE s. In Fig. 33, the time progressions of Sauter mean diameter of PSD for three different overspray rates are shown. 

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