Integral transform technique

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. 

In this section the result from the analyses of Bayazitoglu et al. [14-17] will be shown. They analytically solved the continuum version of the energy equation by the integral transform technique... 

The velocity profile was assumed to be fully-developed. The velocity distribution in a circular microchannel including the slip boundary condition was taken from the literature. However, for the other geometries, they derived the fully-developed velocity profiles from the momentum equation. It is straightforward for flow between parallel plates and flow in an annulus. They applied the integral transform technique to obtain the velocity in a rectangular channel. The problem was simplified by assuming the same amount of slip at all the boundaries. 

It is important to note that the last term in Eq. (5.1) which is the viscous generation term has been included in this analysis. At the microchannel level viscous generation is significantly more important. The integral transform technique is then applied. The appropriate integral transform pair is developed ... 

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 ... 

This linear partial differential equation was solved analytically in Chapters 10 and 11 using Laplace transform, separation of variables, and finite integral transform techniques, respectively. A solution was given in the form of infinite series... 

A direct analytical solution technique exists for certain cases, using the method of integral transforms. It provides a systematic approach to a very difficult problem and is beginning to appear more and more frequently in the technical literature, as can be seen in the review of the models in Section V. A full description of the integral transform technique for solution of differential equations is presented in the references cited, as well as in numerous mathematical texts, including the one by Snedden 87), Cleary and Adrian 14) present a very thorough version of the solution process, as does Yeh (765). 

Yeh and Tsai (104) present a model obtained by integral transform techniques. Their model includes a periodic velocity component as shown in Eq. (46). They assume that velocity is not a function of distance. Their solutions are too lengthy to present here, and numerical evaluation of the analytical expressions is required. A substantial computer effort is required for this, especially since the sine and cosine series are slow to converge. Benedict (5, 6) had noted the fact that coefficients determined by Yeh and Tsai (104) from fitting to field data are one to two orders of magnitude greater than typical values as presented in Section III. Care must therefore be exercised in model use. 

For the case of a linear sorption isotherm, an analytical solution to the above equations, (36) and (37) subject to conditions (28), (29), (38) has been obtained using the integral transform technique by Ramarao and Chatterjee [34], The solution is complex and will not be presented here. A numerical solution to the same equations when the isotherm is non-linear was also obtained by Bandyopadhyay et al. [40],... 

Inertial normal contact problems in three dimensions have been considered by Sabin (1975, 1987). By means of an integral transform technique, he reduces the problem to a set of dual Integral equations, which are in turn reduced to a single Volterra integral equation. This is solved numerically. He obtains the interesting result that the contact pressure is not significantly different from that in the non-inertial problem. A similar observation had been made earlier, in connection with the elastic problem, by Tsai (1971). 

If the kernel in (A4.1.1) has the form Kix-y), then it is possible to obtain a solution by means of integral transform techniques. In particular, by defining... 

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