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(-Dfμij
2/R
2t) where
t; washing time, sec
y(t); residual ratio of a chemical in a fiber assembly
Df; ; diffusion coefficient of a chemical in a fiber, cm
2/sec
R; radius of a fiber, cm A
ij; ={2Qi/(1+α)β
2μ
ij;
2}
2 1/1+α/1+α J
1;
2(μ
ij; )/J
0;
2(μ
ij; ) …… (column) A
ij; =1/2{2Qi/(1+α)β
2μ
ij;
2}
2 1/1+α/1+α J
1;
2(μ
ij; )/J
0;
2(μ
ij; ) …… (plane board)
Qi; ; root of
J0; (Qi; )=0 (column) root of cos(
Qi; )=0 (plane board) μ
ij; ; root of Q
i; =βμij√1+α 2J
1; (μij)/μijJ
0; (μij)/μijJ
0; (μij)
J0;
Ji; ; 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.
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