In this work the modelling of a process of textile dyeing of a single cotton thread is presented. This thread moves at a constant velocity within a homogeneous dye solution under steady state conditions. The method of volume averaging is applied to obtain the mass transfer equations related to the diffusion and adsorption process inside the cotton thread on a small scale. The one-equation model is developed for the fiber and dye solution system, assuming the principle of local mass equilibrium to be valid. On a large scale, the governing equations for the cotton thread, including the expression for effective diffusivity tensor, are obtained. Solution of these equations permits the dye concentration profile for inside the cotton thread and in the dyeing batch to be obtained and the best conditions for the dyeing process to be chosen.

In this paper, the modeling of the mass transfer process in packed-bed reactors is presented and takes into account dispersion in the main fluid phase, internal diffusion of the reactant in the pores of the catalyst, and surface reaction inside the catalyst. The method of volume averaging is applied to obtain the governing equation for use on a small scale. The local mass equilibrium is assumed for obtaining the one-equation model for use on a large scale. The closure problems are developed subject to the length-scale constraints and the model of a spatially periodic porous medium. The expressions for effective diffusivity, hydrodynamic dispersion, total dispersion and the Darcy's law permeability tensors are presented. Solution of the set of final equations permits the variations of velocity and concentration of the chemical species along the packed-bed reactors to be obtained.