Page 71 - Advances in Textile Biotechnology

50 Advances in textile biotechnology

capillary. C E is the enzyme concentration in this liquid. Γ S and Γ ES are the
surface concentrations of the available sites for enzyme adsorption and the
adsorbed enzymes, respectively.
The mass balance for the total amount of enzymes in the system is:

[2.38]
V C C E = V C C E,0 − A C Γ ES
where C E,0 is the initial concentration of enzymes in the solution.
Thus, the amount of enzymes in the solution equals the initial amount of
enzymes present in the solution minus the amount that has been adsorbed
at the capillary surface. The number of available sites for enzyme adsorption
can be related to the maximum amount of enzymes that can be adsorbed:

[2.39]
Γ S = Γ ES,max − Γ ES
The number of vacancies S, varies from A C Γ ES,max for Γ ES = 0 until 0 for
Γ ES = Γ ES,max . In the latter instance, all places are occupied by enzymes.
From equation [2.39], the equations [2.37] and [2.38] can be rewritten as:
d*
E *
ES = kV C C ( ES,max − * ES ) − k des * ES [2.40]
ads
dt
C E = C E,0 − A C * ES [2.41]
V C
Substitution of equation [2.41] into equation [2.40] and simplifying gives:
d* ES = kV C C E,0 * − k ( ads A C * ES,max + ads V C E,0 + k ) ES + kA * 2
des * k
dt ads ES,max C S ads C ES

[2.42]
This equation describes the dynamics of the adsorption–desorption
process of the enzymes in a capillary pore of fabrics. The equation takes
the form:
dΓ
γ
ES =+ β Γ ES +α Γ ES [2.43]
2
dt
where

ads
α = kA C
β =−(kA C Γ s,max + k V C E,0 + k des ) [2.44]
ads
ads
C
γ = kV C E, Γ ES,maax
C
ads
0
Equation [2.43] is the differential equation of Ricatti. Although the equa-
tion looks rather simple, solving it is quite complex. Here we will only give
the solution of the equation with the initial conditions t = 0, C E = C E,0 and
Γ ES = 0.
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