1 Department of Applied Mathematics and Computer Science, Technical University of Denmark2 Mathematics, Department of Applied Mathematics and Computer Science, Technical University of Denmark3 Department of Chemical and Biochemical Engineering, Technical University of Denmark
This thesis concerns mathematical modelling of membrane separation. The thesis consists of introductory theory on membrane separation, equations of motion, and properties of dextran, which will be the solute species throughout the thesis. Furthermore, the thesis consist of three separate mathematical models, each with a different approach to membrane separation. The first model is a statistical model investigating the interplay between solute shape and the probability of entering the membrane. More specific the transition of solute particles from being spherical to becoming more elongated as prolate ellipsoids with the same volume. The porous membrane is assumed isotropic such that the model reduces to a two dimensional model. With this assumption ellipsoids with the same volume reduces to ellipses with the same area. The model finds the probability of entering the pore of the membrane. It is found that the probability of entering the pore is highest when the largest of the radii in the ellipse is equal to half the radius of the pore, in case of molecules with circular radius less than the pore radius. The results are directly related to the macroscopic distribution coefficient and the rejection coefficient. The second model is a stationary model for the flux of solvent and solute in a hollow fibre membrane. In the model we solve the time independent equations for transport of solvent and solute within the hollow fibre. Furthermore, the flux of solute and solvent through the membrane is coupled through the boundary conditions. The model investigates how the true and observed rejection coefficient depends on the transmembrane pressure, the average inlet velocity, and the molecular weight. Furthermore, the effect of concentration dependent viscosity on the rejection coefficients is investigated. The results show that the true rejection coefficient is increasing as a function of increasing transmembrane pressure, increasing inlet velocity, and decreasing molecular weight. Furthermore, it is found that a concentration dependent viscosity decreases the true rejection. The observed rejection is increasing for decreasing molecular weight and increasing inlet velocities. The observed rejection can be either increasing or decreasing as a function of increasing transmembrane pressure. Moreover, the observed rejection is reduced when the viscosity depends on the concentration. The study is a time dependent model of back-shocking. During back-shocking the pressure difference across the membrane is reversed for a given time. This implies that the concentration polarization at the membrane surface is flushed away. When the pressure is reversed back to normal the membrane performs better resulting in an increased average flux. Two models models of the problem was made. In a two dimensional model, limited to capture the dynamics close to the membrane, a positive effect was observed on both the observed rejection and the average solvent flux. Furthermore, an analytical upper estimate for the optimal back-shock time is given. In a three dimensional model, where the flow within the entire hollow fibre is modelled, the mentioned upper estimate is used to obtain a positive effect on both the observed rejection and the average solvent flux. Moreover, the effect of a concentration dependent viscosity was investigated. It was found that the average flux compared to the steady-state solution increased when the viscosity depends on the concentration.