Polynomial Approximations Describing Rate of Dyeing from a Finite Bath

Three polynomial approximations to Wilson's equation were derived by means of Hastings's method. One of them is an approximation to Crank's equation with 3–figure accuracy and it approximates to Wilson's epuation very closely for high values of fractional equilibrium exhaustion (E), E > 0. 95. Two other polynomials approximate to Wilson's equation very well in the ranges 0. 95 > EX). 76 and 0. 76 > E > 0. 30 respectively. The polynomials derived here only include up to second order terms, so that by using the equations which are obtained by solving the polynomials for diffusion coefficient (D), D can easily be calculated from experimental dye uptake data. By using the polynomial approximation, the changes in rate–of–dyeing of a non–ionic dye on a nylon 6 fibre by addition of a dispersing agent were calculated. Results of the experiment agreed well with the calculated.