Mathematical model of the extrusion process of grain crops during non-isothermal flow of crop melting to the temperature at the beginning of the Maillard reaction

Abstract

To describe the non-isothermal flow of grain crops melt in the extruder, the equations of motion, the equation of continuity, the equation of energy (heat balance), and the rheological equation were chosen as initial equations. To solve the model, the following assumptions were made. The flow of a moving viscous medium is assumed to be laminar and steady. The forces of inertia and gravity are so small compared to the forces of friction and pressure that they can be neglected. A viscous medium (melt) is an incompressible liquid characterized by constant thermal conductivity and thermal diffusivity. The change in thermal conductivity in the longitudinal direction was neglected due to the fact that convective heat transfer in the direction of flow is higher than the heat transfer by thermal conductivity. Heat transfer in the direction perpendicular to flow of the melt occurs only due to thermal conductivity. To solve the system of equations taking into account convective heat transfer, a numerical finite difference method was used, according to which the considered area (extruder channel) is divided into computational cells using a grid. The grid consisted of rectangular cells with a constant step between nodes, which exactly lie on the boundaries of the integration region. In this case, the differential equations were transformed into difference equations by replacing the derivatives at a point with finite differences along the cell boundaries. As a result, a mathematical model of non-isothermal melt flow in the extruder channel was obtained. To solve a mathematical model of the grain crops extrusion process with a non-isothermal flow of their melts, a program in the programming language C++ was compiled. A non-isothermal mathematical model of the grain crops extrusion process at temperatures at the beginning of the Maillard reaction, i.e., up to 120-125 °C, was obtained, which makes it possible to reveal the nature of the temperature change along the length of the extruder. Comparative analysis of the results of the numerical solution and experimental data showed good convergence: the standard deviation did not exceed 12.7%.