Analytical Solution to One Problem of Cooling Fiber Mass

To analyze the cooling process of raw cotton, a mathematical model is proposed in the form of a boundary value problem for a system of thermal conductivity equations. The developed model makes it possible to describe the distribution and propagation of heat within a fibrous mass represented in the form of a ball. Such an approach provides an opportunity to investigate the thermal state of raw cotton during the cooling process and to determine the main regularities of heat transfer inside the fibrous material. The proposed mathematical formulation takes into account the thermal conductivity processes occurring within the fibrous mass and allows the temperature field to be determined as a function of spatial coordinates and time. The boundary and initial conditions are formulated in accordance with the physical characteristics of the cooling process. This makes it possible to obtain a mathematical description that adequately reflects the changes in temperature inside the considered fibrous medium during cooling. Using the method of separated variables, an analytical solution to the proposed initial-boundary value problem is obtained. The application of this method makes it possible to represent the solution in a convenient analytical form and to analyze the dependence of the temperature distribution on the main parameters of the process. The obtained analytical results provide a theoretical basis for studying the dynamics of heat propagation and evaluating the cooling efficiency of the fibrous mass. The results of the study can be used to improve and optimize technological processes associated with the cooling of raw cotton. In particular, the proposed mathematical model can contribute to the selection of appropriate cooling conditions and technological parameters aimed at reducing excessive temperature increases and minimizing possible losses in fiber quality. The obtained results may also be useful for developing more effective approaches to controlling thermal processes during the processing and storage of fibrous materials. The work is of interest to specialists in the fields of thermal physics, heat transfer, mathematical modeling, fibrous materials processing technology, cotton processing, and related scientific and engineering areas concerned with the analysis and control of thermal processes in fibrous materials.