Washing of Fabrics

Part 2: Analysis of Material Transfer in Fiber Assembly Based on Diffusion Theory

Abstract

When a chemical is washed off from a fiber assembly such as cloth or a yarn, it migrates in the fiber and the water phase in a fiber assembly, and subsequently moves to water.
In such a process the diffusion coefficient of a chemical in a fiber differs largely from that in water. This was analyzed on the basis of a diffusion theory. Assuming that a single fiber is a column and a fiber assembly is a column or a plane board, the following theoretical relations, representing residual ratio of a chemical in a fiber assembly during washing, were obtained: y(t)=∑<∞><i=1>∑<∞><i=1>Aij exp(-Dfμij²/R²t) where t; washing time, sec y(t); residual ratio of a chemical in a fiber assembly D_f; ; diffusion coefficient of a chemical in a fiber, cm²/sec R; radius of a fiber, cm A_ij; ={2Qi/(1+α)β²μ_ij; ²}² 1/1+α/1+α J₁; ²(μ_ij; )/J₀; ²(μ_ij; ) …… (column) A_ij; =1/2{2Qi/(1+α)β²μ_ij; ²}² 1/1+α/1+α J₁; ²(μ_ij; )/J₀; ²(μ_ij; ) …… (plane board) Q_i; ; root of J₀; (Q_i; )=0 (column) root of cos(Q_i; )=0 (plane board) μ_ij; ; root of Q_i; =βμij√1+α 2J₁; (μij)/μijJ₀; (μij)/μijJ₀; (μij) J₀; J_i; ; Bessel functions of the 0th and the 1st order α, β: constants depending on the diffusion coefficient of a chemical in a fiber and in water, geometrical characteristics of a fiber assembly and so on.
The values calculated theoretically by using the above equations agree well with experimental values, and are very close to the results obtained by the simplified equation already reported.