Generalized integral transform technique

The present section reviews the concepts behind the Generalized Integral Transform Technique (GITT) [35-40] as an example of a hybrid method in convective heat transfer applications. The GITT adds to tiie available simulation tools, either as a companion in co-validation tasks, or as an alternative approach for analytically oriented users. We first illustrate the application of this method in the full transformation of a typical convection-diffusion problem, until an ordinary differential system is obtained for the transformed potentials. Then, the more recently introduced strategy of... 

This work discusses hybrid numerical-analytical solutions and mixed symbolic-numerical algorithms for solving transient fully developed flow and transient forced convection in micro-channels, making use of the Generalized Integral Transform Technique (GITT) and the Mathematica system. 

All the theoretical work was performed by making use of mixed symbolic-numerical computation via the Mathematica 7.0 platform [22], and a hybrid numerical-analytical methodology with automatic error control, the Generalized Integral Transform Technique - GITT [23-26], in handling the governing partial differential equations. 

Equations (6) are now solved by the Generalized Integral Transform Technique, GITT, starting with the choice of an appropriate filtering solution that eliminates the non-homogeneous terms in the equation and boundary conditions ... 

Various formulations and methodologies have been suggested for describing combined heat and mass transfer problems, such as the integral transform technique, in the development of general solutions. In this chapter, cross phenomena or coupled heat and mass transfer are discussed using the linear nonequilibrium thermodynamics theory. 

This example summarizes all the necessary steps for the generalized integral transform. To recap the technique, we list below the specific steps to apply this technique. 

Extensive literature is available on general mathematical treatments of compartmental models [2], The compartmental system based on a set of differential equations may be solved by Laplace transform or integral calculus techniques. By far... 

The solution can be found conveniently by using Laplace transform technique with respect to time, but the solution takes somewhat complicated form, since the inversion integral includes the branch points corresponding to the critical frequency. The detail of the solution has been described by Toma Morioka [4]. Here we say only that the present result, generalized for arbitrary propagation direction, can be obtained by replacing M in [4] to m = ucos6/c. 

The analytical techniques proposed in the literature generally give reliable information on lipids present in the paint layer. However, the presence of lipid mixtures and of particular environmental conservation conditions may affect the lipid pattern to such an extent that their identification may be very difficult and sometimes erroneous. Thus, a multianalytical approach is recommended which integrates chromatographic data with techniques such as mapping based on Fourier transform infrared spectroscopy or SIM on cross-sections, in order to better understand the distribution of lipids in the various paint layers. 

Fick s second law (Eq. 18-14) is a second-order linear partial differential equation. Generally, its solutions are exponential functions or integrals of exponential functions such as the error function. They depend on the boundary conditions and on the initial conditions, that is, the concentration at a given time which is conveniently chosen as t = 0. The boundary conditions come in different forms. For instance, the concentration may be kept fixed at a wall located atx0. Alternatively, the wall may be impermeable for the substance, thus the flux at x0 is zero. According to Eq. 18-6, this is equivalent to keeping dC/dx = 0 at x0. Often it is assumed that the system is unbounded (i.e., that it extends from x = - °o to + °°). For this case we have to make sure that the solution C(x,t) remains finite when x -a °°. In many cases, solutions are found only by numerical approximations. For simple boundary conditions, the mathematical techniques for the solution of the diffusion equation (such as the Laplace transformation) are extensively discussed in Crank (1975) and Carslaw and Jaeger (1959). 

In order to obtain properly averaged results, either collision parameters for each trajectory must be selected by Monte Carlo methods or, when starting values are systematically chosen, the final results must be integrated over complete statistical distributions. The purpose of a Monte Carlo selection technique is to ensure that the distributions of each parameter within a sample of trajectories approach the true statistical distributions as the size of the sample grows. Some examples of how this can be done for different types of distribution function will be described below. Before starting the integration, it is generally necessary to transform the selected values of the collisional... 

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