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Methods for calculating the moistening of textile fibers in the process of yarn production when processing with conditioned air with certain technological parameters based on mathematical modeling and numerical methods are presented. There are appropriate mathematical models, adequately describing the process of distribution of the relative air humidity in the porous medium under consideration. A mathematical model of the process of moistening porous of a textile material in the form of the Cauchy problem for a second-order differential equation for the purpose of numerical modeling of the change in the relative humidity of conditioned air in the volume of a compactly formed semi-finished product of textile production. The classical instability of the problem is shown, methods of its solution are considered, the problem is numerically solved for various options of technological conditions for organizing the humidification process. Numerical calculations have shown that the relative humidity of the air flow decreases monotonically with distance from the surface to a certain limiting value. At the same time, an increase in the specific surface area of a capillary-porous medium leads to a more intense drop in the relative humidity of the conditioned air in a moving air stream. The results of numerical studies show that with an increase in the flow rate of conditioned air, the depth of penetration of moisture into the volume of a porous medium increases, i.e. more moisture settles on the last layers of the volume. Consequently, to optimize the process, one should choose an air flow rate that provides a sufficient intensity of material moistening with deep penetration of moisture into the volume of a porous medium. Mathematical models of changes in the air flow rate in the volume of a porous medium according to linear and exonential laws are presented. It is shown that taking into account the drop in velocity according to the exponential law allows one to obtain good agreement between the calculated and experimental data. Physical and mathematical modeling of the non-stationary process of moisture distribution in a textile material during its humidification with conditioned air for the case of diffusion kinetics of the humidification reaction in the form of a boundary value problem for the diffusion equation has been carried out. On the basis of physical concepts of the processes occurring at the boundary of a porous medium and in its volume, modeling equations and boundary conditions for the problem of calculating the unsteady distribution of moisture in a porous medium are formulated.

Regularities of distribution of the relative air humidity in the volume textile fiber material in the production of yarn