EXTENDED GRAETZ SOLUTION FOR PREDICTING NONREACTING VAPOR SPECIES CONCENTRATION PROFILE AND WALL DEPOSITION RATE FOR NON-ISOTHERMAL LAMINAR FLOW THROUGH A PIPE

Abstract

Diffusive vapor deposition rates are predicted on the walls of a straight tube in which fully developed non-isothermal laminar flow exists. As a prerequisite to calculating the deposition rates at different axial positions on the wall, the species mass balance equation (convective diffusion) is solved, using the method of separation of variables. For the present calculations, fully developed Poiseuille flow is considered. The case of isothermal flow was solved by Graetz (1885). However, in most engineering systems, a highly non-isothermal environment exists where the center of the tube is at a temperature that is very different from the temperature at the wall. This results in Fickian diffusivity being a function of radial position, which can appreciably alter the species concentration profile in the tube and its deposition behavior. A method is presented here to predict the species concentration profile and its deposition rates on the tube wall, systematically, taking into account the variation in species Fickian diffusivity with radial position. Using the method of separation of variables, the problem is reduced to one of solving an ordinary differential equation, which belongs to the Sturm-Liouville class of equations. The first 10 eigenvalues, associated eigenfunctions, and relevant constants of the reduced equation have been computed and reported and are needed to estimate the species concentration profile and vapor deposition rates at the wall. Asymptotic eigenvalues are also reported and are seen to be in good agreement with the computed values.