Solução do problema de Graetz conjugado em um duto ortotrópico por transformada integral

Abstract

In the realm of simulation studies for heat transfer in ducts, this paper proposes a comparison between hybrid solution strategies for solving the conjugate heat transfer in a duct made of an orthotropic material with axisymmetric ow, which mechanical properties vary in the two main directions. The problem has a single domain. The permanent regime has been considered for this problem with no variaton of properties with the time. The work aims to analyze Nusselt number distribution for the simulated cases and analyze Nusselt convergence rate with Generalized Integral Transform Technique. The duct temperature profile is also determined. The axial diffusion effect was also studied and considered, varying Péclet number. Graphic results were created for better visualization of Nusselt behaviour profile. Several tables were created to indicate the modification of convergence rate according to variation of dimensionless parameters. Tables of comparison and validation purposes were created to compare with results published in literature. Through mathematical operations, a generalized formulation is defined and the results can be studied and analyzed. The formulation is simplified by hypothesis and dimensionless parameters. For the proposed work, some boundary conditions were adopted e and also replaced by dimensionless parameters, as the third kind condition, most known as Robin condition or convection boundary condition. There is also an initial temperature at the channel entrance, a finite temperature far from the entrance and no variation of temperature at the duct center due the axial symmetry of the problem. Using the Generalized Integral Transform Technique(GITT) in combination with an auxiliary problem, represented by Bessel eigenvalue problem, which the required solution is written in terms of orthogonal eigenfunctions of a Sturm-Liouville problem, the eigenvalues are calculated by a numerical method. Although the original problem transformation takes to a coupled ordinary differential equations system(ODE), in this work is used an analytical matrix method, which the solution is obtained in terms of eigenvalues and eigenvectors from a matrix involving the coupling coeficients of the ODEs system. Thus, only one numerical implementation is necessary, that is the calculation of eigenvalues and eigenvectors from the coeficient matrix mentioned. At the end, is possible to determine the results, to obtain the temperature profile and also the Nusselt number behaviour. By the results, GITT is an eficient technique according to the convergence analysis, worse rates were found in the channel entrance, Peclet value in uences the way as the ow develops thermally, as well as Biot and the duct aspect ratio, and as Peclet raises, Nusselt approaches the solution for the case with no axial diffusion, as expected.