Three-dimensional reaction-convection-diffusion analysis with temperature influence for biodiesel synthesis in micro-reactors

A three-dimensional nonlinear reaction-convection-diffusion model is proposed to analyze mass transfer with temperature effects, within micro-reactors of rectangular cross-section, towards the simulation of the transesterification process in biodiesel synthesis. The coupled nonlinear partial differential equations that govern the concentration of the species involved in the synthesis, is obtained from the general three-dimensional species balance equations for an isothermal and steady state system with constant physical properties. A kinetic model is considered to describe the variation of the species due to the transesterification process, assuming second order homogeneous and reversible chemical reactions. The velocity profile is analytically derived from the Navier-Stokes equations, considering fully developed stratified laminar flow of two immiscible Newtonian fluids in a rectangular cross section micro-reactor, with a plane interface between them and without field forces, and obtained by the Classical Integral Transform Technique (CITT). The nonlinear mass transport equations are then solved by the hybrid numerical-analytical approach known as the Generalized Integral Transform Technique (GITT). In the present GITT solution, two alternative eigenvalue problems are proposed, accounting or not accounting for the space variable velocity profile influence in the eigenfunction expansion basis, in order to investigate the differences in the convergence behavior of the solutions obtained through the two different auxiliary problems. The results for the concentration profiles of the biodiesel and triglyceride species are critically compared with literature data for the limiting case of parallel plates, showing a good agreement among them. The analysis also indicates that improved convergence rates are achievable through the eigenvalue problem with variable coefficients. The effects of governing parameters such as residence time, micro-reactor dimensions, and especially temperature, are investigated. It is then inspected the relative gains in triglycerides conversion rates at higher temperatures and residence times and lower micro-reactor hydraulic diameter.