Finite-differential time-domain

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. 

In this chapter, analytical solutions were obtained for linear hyperbolic and parabolic partial differential equations in finite domains using Laplace transform technique. In section 8.1.2, a linear hyperbolic partial differential equations was solved using the Laplace transform technique. First, the partial differential equation was converted to an ordinary differential equation by converting the PDF from the time domain to the Laplace domain. For hyperbolic partial differential equations this results in an initial value problem (IVP), which is solved analytically in the Laplace domain as illustrated in chapter 2.1. The analytical solution obtained in the Laplace domain was converted easily to the time domain using Maple s inbuilt Laplace transform package. For parabolic partial differential equations, the governing equation in the Laplace domain is a boundary value problem (BVP), which is solved analytically as in chapter 3.1. For certain simple parabolic partial differential equations, the Laplace domain solution can be inverted to time domain easily using Maple as illustrated in section 8.1.3. 

Thus, Eq. 12.131, Eq. 12.130 for i = 2,3,..., IV — 2, and Eq. 12.132 will form a set of N - 1 equations with AT - 1 unknowns (y, y2, , yv- )- This set of coupled ordinary differential equations can be solved by any of the integration solvers described in Chapter 7. Alternately, we can apply the same finite difference procedure to the time domain, for a completely iterative solution. This is essentially the Euler, backward Euler and the Trapezoidal rule discussed in Chapter 7, as we shall see. 

Since we introduced the Galerkin finite element approximations in space, the result (6.64) is an ordinary differential equation in the time-domain. 

The solution to the first problem is limited by the increase in time or the computer capacity available to solve more complete or more advanced equations. The second problem is even more difficult to acknowledge. It may be due to error accumulation through the nonlinear domain. The numerical solution of a differential equation is based on the approximation of time and, in the case of PDEs, space partial derivatives, by finite-difference equivalents. 

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. 

Diffusion layer — The diffusion layer is an imaginary layer of predominant occurrence of diffusion or a heterogeneous concentration by electrolysis, as shown in the Figure [i]. It is used conveniently when we want to conceptually separate a domain of charge transfer, a domain of diffusion, and a bulk. The diffusion layer thickness can be estimated (Dt) for the electrolysis time, t, and ranges from 0.01 mm to 0.1 mm. When current is controlled by diffusion, the thickness can also be estimated from the current density, j, through DcF/j, which is due to finite values of the differentiation in j = FD (r)c/r)x). The concept of the diffusion layer is im-... 

It is well known that, for the Navier-Stokes equations, the prescription of the velocity field or of the traction on the boundary leads to a well-posed problem. On the other hand, viscoeleistic fluids have memory the flow inside the domain depends on the deformations that the fluid has experienced before it entered the domain, and one needs to specify conditions at the inflow boundary. For integral models, an infinite number of such conditions are required. For differential models only a finite number of conditions are necessary (more and more as the number of relaxation times increases,. ..). The number... 

Here, Xjeg is the dimensionless concentration vector (normalized, e.g., by a feed concentration), R(X) is the corresponding rate vector, and is the dimensionless output concentration of the segregated flow system with a residence time 7. We allow the objective function, /, to be specified by the designer as any function of and t. One can see that the differential equation system can be uncoupled from the rest of the model and solved offline if the dimensionless feed concentration, Xq, is prespecified. Once the vectors Xseg are determined, we merely solve for/(r), which satisfies an additional set of linear constraints. If Gaussian quadrature on finite elements is applied to the above model over the domain [0, r axl we get... 

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. 

Our task is now to reduce this differential equation to mathematical stmctures that can be expressed finitely. Two steps are required for this (i) reducing the time dependence to the Laplace domain and (ii) integrating the relation expressed in x over the domain 0[Pg.116]

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. 

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