Mathematical modelling of mass transfer of paramagnetic ions through an inert membrane by the transient magnetic concentration gradient force. https://

Abstract

The objective of this work is to suggest a mathematical model for mass-transfer of a paramagnetic electrolyte, nickel(ii)chloride solution, through an inert, thin membrane from one chamber to another under the influence of magnetic fields which are applied perpendicular to the membrane. The model is based on the magnetic concentration gradient force, the Fick’s law of diffusion, and the Hagen-Poiseuille law for paramagnetic ion transport in the membrane. The magnetic concentration gradient force is found to be elusive and points in the direction of the magnetic field, in our case, the direction of the Fick diffusion flux. The reason is the gradient of the magnetic volume susceptibility for the electrolyte in the membrane, which decreases in the direction of the magnetic field. This is in accordance with the variable-reluctance principle. Mass balances for transport of Ni ions in distilled water through the membrane are derived and governed by a partial differential equation in one-dimensional space and time with specified initial and boundary conditions. The associated flux is superimposed on the pure Fick diffusion flux. The total flux is described by a nonlinear partial differential equation, which has not previously been used to describe transfer phenomena in paramagnetic solutions in magnetic fields. The simulated results were compared with experimental results and coincide approximately in all points for unstirred solutions. In stirred solutions, where the mass transfer coefficient at the membrane inlet approaches infinity if the mixing is ideal, no experimental or simulated effect was observed of the magnetic field.