Dynamic Modelling of the Diffusion in a Dyeing Process

A common goal of all dyers is to achieve the correct shade as early as the first dyeing. But the disperse dyeing process is very complex, being characterized by the diffusion-controlled sorption of dyes that depends on several physicochemical parameters such as dye concentration and temperature of application. Wilson and Hill developed equations describing the uptake of disperse dyes by cylindrical fibres as a function of the diffusion coefficient and the nature of the dyebath. This coefficient is a physical term that permits comparison between different dyeing experiments. However, the literature reveals no equation that expresses the influence of experimental parameters on the diffusion coefficient value. Thus, this work proposes a mathematical model describing the evolution of this diffusion coefficient for disperses dyes as a function of time, initial dye concentration, and the nature of the dyebath (finite or infinite).