Biomedical Engineering Reference
In-Depth Information
S
KS
μ=⋅
max
+
S
(1)
Although there have been attempts to give a mechanistic explanation to the Monod
equation [2, 3] and although it has formal similarities to Michaelis-Menten kinetics [4], the
Monod equation remains essentially empirical, i.e. it is based on experimental observations
and curve fitting of experimental data.
In activated sludge models Monod type kinetics are used for mathematical convenience
rather than conformity to any fundamental rate law [5]. This has implications for the
interpretation of the kinetic parameters, in particular the Monod affinity constant, since
wastewater and activated sludge represents a multisubstrate/multispecies system. The value of
Monod's affinity constant in activated sludge systems has been discussed in the literature [6]
and some principle implications are given here. In single substrate single species systems the
affinity constant takes on values of a few mg/l [7]. In multisubstrate/multispecies systems the
value of K
S
is typically around 50 mg/l [8]. This difference can be attributed to diffusion
within and exterior to flocs [9]. It is thus important to note that diffusion plays an important
role in activated sludge processes, and that diffusional mass transport limitations affect the
value of Monod's affinity constant when modelling activated sludge systems. Since diffusion
is related to floc structure (see 1.2), it becomes apparent that floc structure plays an implicit
role when modelling activated sludge systems. The role of diffusion and floc structure in
activated sludge modelling needs further discussion (see chapter 2).
In biofilm models Monod kinetics are generally used to model the intrinsic kinetics of
biofilm systems, since diffusion is usually explicitly described using Fick's laws of diffusion.
Fick's Laws of Diffusion
Diffusion processes are governed by Fick's laws of diffusion. Fick's first law (Eq. 2) is
used in steady-state diffusion, i.e., when the concentration within the diffusion volume does
not change with respect to time. Fick's first law states, that the flux (J) of a given substance
(S) is proportional to the concentration gradient of the substance along the respective
dimension (x), the proportionality factor being the diffusion coefficient (D). Fick's second law
(Eq. 3) is used in non-steady or continually changing state diffusion, i.e., when the
concentration within the diffusion volume changes with respect to time (t).
S
JD
x
∂
=−
(2)
∂
2
∂
S
∂
S
=
D
(3)
2
∂
t
∂
x
In two or more dimensions Fick's second law can be formulated in generalized form
using the gradient operator ∇ (Eq. 4).
