Mathematical modeling of the diffusion coefficient for disperse dyes

The kinetics of dyeing polyester fibers with disperse dyes is extremely com plex. For a given dye/fiber system, the dye uptake rate is controlled by se veral experimental parameters such as temperature, initial dye concentratio n, and diffusion coefficients. Hill and Wilson established dyeing rate equa tions, but unlike them, we believe that the diffusion coefficient of a disp erse dye into polyester fibers is not constant during the whole dyeing proc ess. So using Wilson and Hill's equations and our own experimental results, we have computed the diffusion coefficient value at each sampling time thr oughout 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 dyebat h (finite or infinite).