Mathematical Modeling of the Diffusion Coefficient for Disperse Dyes

Abstract

The kinetics of dyeing polyester fibers with disperse dyes is extremely complex. For a given dye/fiber system, the dye uptake rate is controlled by several experimental parameters such as temperature, initial dye concentration, and diffusion coefficients. Hill and Wilson established dyeing rate equations, but unlike them, we believe that the diffusion coefficient of a disperse dye into polyester fibers is not constant during the whole dyeing process. So using Wilson and Hill's equations and our own experimental results, we have computed the diffusion coefficient value at each sampling time throughout the dyeing process. In this paper, we discuss a mathematical model describing the evolution of the diffusion coefficient for disperse dyes as a function of time, initial dye concentration, and the nature of the dyebath (finite or infinite).