Numerical investigation of generalized Graetz problem in circular tube with a mass transfer coupling between the solid and the liquid

Abstract

The unsteady diffusion equation in a tube and the advection/diffusion equation in laminar flow in a liquid of a solute are theoretically established. The most important parameters are the migration strength α and the so-called Graetz number Gr. The former dimensionless number is the product of the partition coefficient of the solute at the solid/liquid interface, the ratio of the tube thickness to the interior radius and the ratio of diffusion coefficients of the solute in the liquid and in the solid. The Graetz number Gr is the ratio of the diffusion time scale in the liquid based on the interior radius of the pipe to the advection time over the tube length. The problem is applied to the plasticizer migration from a polyvinyl chloride material to a liquid with the safety food applications in mind. The migration of the plasticizer is solved numerically for various tube sizes, flow conditions and partition coefficients of the plasticizer in the liquid. For comparison, the numerical results obtained in static condition are also provided.The average plasticizer concentration in the liquid behaves following two main regimes as a function of the migration strength. When α is much smaller than one, the average plasticizer concentration in the liquid is an algebraic function of the axial coordinate, z, at the power two third and increases linearly with time. Conversely, when α is much larger than one, the average plasticizer concentration is linear as a function of axial distance and increases as a function of the square root of time. Moreover, the concentration is much smaller of few orders of magnitude in dynamic condition than in static working showing that in the context of food contamination the dynamic approach is more appropriate and relevant for demonstration of compliance with the safety authorities requirements.