Mathematical model of the process of ultrafiltration concentration of secondary milk raw materials in tubular membrane devices with filtering elements of BTU 05/2 type

For the qualitative application of ultrafiltration processes for the concentration and purification of food solutions, both experimental studies and a mathematical description of the processes of the membrane separation process of solutions from the standpoint of the development of computational mathematical models are required. In this work, by analytical solution of equations, that is, by the method of finite differences, mathematical equations are solved. To obtain the system, the flow continuity equations, convective diffusion equations, Navier-Stokes equations and flow equations with boundary conditions were solved in order to build a mathematical model of the process of ultrafiltration protein concentration in cheese whey in the production of rennet cheeses. As a result of the analytical solution of the equations, a system of mathematical equations was obtained that allows one to construct a profile of changes in the flow rates of the solution along the cross-section of the intermembrane channel and to determine the protein concentration in cheese whey along the length of the tubular ultrafiltration element BTU 05/2 of industrial type. The obtained mathematical model makes it possible to theoretically describe the process of ultrafiltration protein concentration in cheese whey along the entire length of the membrane channel of the tubular element under laminar and transient regimes of solution flow. The resulting system of mathematical equations makes it possible to find the numerical values of the mass flow rate of cheese whey, make it possible to calculate the specific output flow when the transmembrane pressure changes and to calculate the concentration of solutes in the secondary milk raw materials on the left and right ultrafiltration membrane of the intermembrane channel. The adequacy of the developed mathematical model was carried out by comparing the calculated and experimental data on the specific output flow when the transmembrane pressure in the intermembrane channel changes from 0.1 to 0.25 MPa with ultrafiltration concentration of cheese whey. The deviation of the calculated data found by the mathematical model from experimental studies obtained on a semi-industrial tubular ultrafiltration plant BTU 05/2 using semipermeable membranes, in which the active layer is made of fluoroplastic, hemisulphone and polyethersulfone, did not exceed 10%.