A semi-empirical solution of the diffusion equation is proposed for dyeing fibers with round cross sections from finite dye baths with soluble dyes whose adsorption isotherm is of the Langmuir type. This solution is valid during the early part of the dyeing cycle until about 45% of the dye has been exhausted. A method is described for measuring dyeing rates, which can be used to calculate diffusion coefficients. This method of determining diffusion coefficients is much faster than other methods described in the literature.
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