Eigenfunction expansions for transient diffusion in heterogeneous media

Mass transfer simulation of biodiesel synthesis in microreactors

“…The Sturm-Liouville eigenvalue problems presented in Table 3 are solved by the GITT itself (Cotta, 1993;Naveira-Cotta, 2009). For this purpose, two auxiliary eigenvalue problems with known analytical solution are required and proposed according to Table 4:…”

“…The integral transformation for the main eigenvalue problems presented in Table 3 is accomplished by multiplying the respective normalized auxiliary eigenfunction given in Table 4 for each eigenvalue problem, followed by integration over the domain [0,1] in the Y direction, and then substituting the inversion formula for the original eigenfunctions (Cotta, 1993;Naveira-Cotta, 2009). After the described integral transformation, the following algebraic systems for calculating the eigenvalues and corresponding eigenvectors are obtained:…”

Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and Bayesian inference

“…The approach here employed in the direct problem solution for forced convection in micro-channels, is borrowed from a recent work on diffusion in heterogeneous media, with arbitrarily space variable thermophysical properties [13]. In this sense, the dimensionless velocity fields are mathematically equivalent to space variable thermal capacitances, and the formal solution procedure is here briefly described.…”

“…We consider a general formulation on steady or quasi-steady state hydrodynamically fully developed and thermally developing convection, that governs the temperature field T(x, z), dependent on the position x on the transversal plane and on the longitudinal coordinate z, defined in the transversal region V with boundary surface S. The formulation includes the convection term, the diffusion operator, a linear dissipation term, and an independent source term [13], as shown in problem (1) below. The coefficients w(x), k(x), and d(x), are responsible for the information related to the velocity field, geometry and eventually even heterogeneity of the medium.…”

“…For this purpose, a flexible approach was employed [13], based on formal solutions of the energy equation as obtained via the classical integral transform method [14], in association with the Generalized Integral Transform Technique, GITT [15e18], which was used for the solution of the required eigenvalue problem [19e21]. This method is here applied for illustration purposes in the integral transformation of the energy equation for thermally developing flow within parallel-plates micro-channels under the slip-flow regime.…”