Page 58 - Advances in Textile Biotechnology

Developments in enzymatic textile treatments 37

Stagnant
core

Convective
shell

2.2 Yarn model with a stagnant core and a convective shell.

a still standing enzyme solution so all the liquid in the yarn is stagnant, then
the enzyme molecules will start to diffuse into the yarn. This process is
described by the diffusion equation of Fick (Bird et al., 1960):

∂C E,Y = D ∂ ⎛ ∂C E,Y ⎞
r
⎜ Y ⎟ [2.1]
∂t r Y ∂ ⎝ ∂r Y ⎠
r
−3
in which C E,Y is the enzyme concentration in the yarn (kg m ), r Y is the yarn
2 −1
radius (m), D is the diffusion coefficient of the enzymes (m s ) and t is the

time (s). This equation only describes the diffusion process in the radial
direction of the yarn. The solution of this equation is often graphically given
as in Fig. 2.3.
The variable E on the vertical axis is the dimensionless mean concentra-
tion in the yarn:
− C E,Y
C E,b
E = [2.2]
C E,b − C E,Y,0
−3
in which C E,b is the enzyme concentration in the enzyme solution (kg m ),
−3
C E,Y is the mean enzyme concentration in the yarn (kg m ) and C E,Y,0 is the
−3
mean initial enzyme concentration in the yarn (kg m ). The variable on the
horizontal axis is the Fourier number Fo and is a dimensionless expression
for the diffusion time:
Fo = Dt [2.3]
2
d Y
in which d Y is the yarn diameter (m).

© Woodhead Publishing Limited, 2010