Biofilm kinetics
For an infinitesimal section of the biofilm, the following balance can be set up under
stationary conditions.
in
out + removed
aN
N = N + - · dx + rvf · dx
ax
where N
rvt
aN
-=-rvt ax
is the transport through the cross section
is the volumetric reaction rate of the biofilm.
(5.1)
As the transport through the cross section exclusively takes place by diffusion, we have:
N
asvf
- D · -
ax
aN
-D·
a2svf
ax
ax 2
a2Svt
rvf
ax 2
D
Quantitatively, this equation can be interpreted in such a way that the second
derivative of a concentration distribution expresses a curvature on the distribution.
If there is no reaction, the concentration distribution is rectilinear, for example by
diffusion in the water phase, see Section 5.3. If there is production, the distribution
curves upwards. If there is removal the distribution curves downwards as it appears
from the following pages.
This equation is made dimensionless by measuring relative to the characteristic parameters:
Svt
X
svf=s
s=I
82svf rvtL 2
(5.2)
DS2 = DS
As a solution to this second order differential equation, two cases are considered: A
zero and a first order reaction:
First order reaction:
rvf = klVf · Svf
where Svf
is the concentration in the biofilm
kiVf is a first order rate constant with the dimension d- 1
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