KINETICS OF NON-ISOTHERMAL ADSORPTION AT CONSTANT CONCENTRATION ON THE SURFACE OF THE THREAD

Abstract

To describe the process of dyeing textile materials, in the general case, it is not enough to consider the kinetics of dye adsorption by a single fiber. This assumption is valid in the case of low intensity low temperature processes, when the diffusion coefficient of the dye to the fiber is several orders of magnitude smaller than the diffusion coefficient in the space between the fibers. In this case, the mass transfer process is determined by the transfer in the fiber. As the temperature rises or with other methods of intensifying the process, there is a sharp increase in the diffusion coefficient in the fiber, while the diffusion coefficient in the liquid phase outside the fiber changes to an incomparably lesser extent, i.e. the difference in the values of the diffusion coefficients in the fiber and outside the fiber decreases sharply. If we take into account the significant difference in the linear dimensions of the fiber and thread, it becomes obvious that there comes a moment when the relaxation time of the diffusion in the fiber, defined as f f f r D 2 2 , becomes less than the relaxation time in the thread, defined as t t t r D 2 2 . Given these conditions for the comparability of the relaxation times of diffusion along the fiber and thread, it is completely incorrect to describe the process of dyeing a textile material by considering the kinetics of the process in a separate fiber. The following scheme of the process is more rigorous: transfer of the dye to the surface of the thread, diffusion in the space between the fibers, adsorption by the inner surface of the fiber, and chemical interaction, if any. Such a formulation of the problem has much in common with the problems of diffusion in granular porous media [1], which are of great importance for the quantitative description of the dynamics of sorption both in general theoretical terms and for describing various technological processes: filtration through a layer grains of various shapes, diffusion extraction of substances from porous media, etc. Existing equations that describe the processes of diffusion in porous systems containing microporous inclusions don’s take into account the restrictions on the shape of the microporous inclusion and, in general, on the geometry of the system. Naturally, the solution of these equations requires a certain specification both in relation to the shape of the microporous inclusion and in relation to the nature of diffusion. When considering such processes, it is necessary to proceed from the solution of the problem of diffusion into a microporous inclusion. The process of mass transfer in a fiber is considered as a diffusion process with an apparent diffusion coefficient . In the general case, this statement of the problem should be supplemented by taking into account the interaction of double electrical layers of the dye particle and fibers, which slows down (for the same surface charges) the process of dye transfer In the meantime, in order to simplify the problem, the following model of the process is proposed: we consider molecular diffusion into an infinitely long end-impermeable cylinder (thread) with many infinitely long microporous cylindrical inclusions (fibers) uniformly distributed over its cross section. In this case, a problem is considered that is symmetrical in the section. The kinetic problem is solved for four cases corresponding to the following process conditions: 1. Isothermal mode, constant concentration of dye on the surface of the thread. This task corresponds to dyeing from baths of constant concentration. 2. Isothermal mode, variable concentration on the surface. Corresponds to dyeing from baths of variable concentration, fixing from a film in an environment of saturated steam or in an environment of superheated steam, or heated air at relatively low temperatures (when the heating time of the material is much less than the fixation time). 3. Non-isothermal mode, constant concentration on the surface. Corresponds to dyeing from an aerosol medium. 4. Non-isothermal mode, variable surface concentration. This most complex case corresponds to high-intensity, high -temperature methods of fixing at temperatures of the material at the end of the process, close to the temperature of its softening or destruction. This article considers the third case, which corresponds to aerosol dyeing of fabrics.