Detailed model
hand, be described by detailed, deterministic modelling by which the door is
opened to a alienating world of complicated differential equations, complicated
functional expressions and solutions which are difficult to understand and which
can only be obtained by computer calculations. However, there is no doubt about
the need for the increased realism in the description and the need for such models
for scientific examinations and soon also for the practical design.
The following description mainly follows the description in /22/. Fig 5.24 shows an
element in the biofilm and the corresponding definitions.
The biofilm consists of several phases such as water, solids (and possibly air). The
concentration, Ckj, of a substance is stated as the amount of a given substance in a
unit volume of the phase concerned. The volume fraction, Ek, of a solid k is stated by
the volume of the solid in a unit volume of the biofilm containing all the phases k.
For the phases it applies:
(5.72)
For the concentration it correspondingly applies that the concentration distributed
on a total unit volume is Cfi = Ek · Ckj, where Cki is the definition of the model
concentration which has been used so far. It is necessary to make this distinction
when there are many phases; for example several phases of solids and different
types of bacteria.
The differential equation for an element in the biofilm can be written as:
(5.73)
where jki
is the flux of the substance i in phase k
rv,ki is the removal rate for the substance i in phase k per unit volume
Ek
is the volume fraction of phase k
cki
is the concentration of substance i in phase k.
The first term ahead of the equals sign is the local time derivative from the mean
concentration which, just after the first equals sign, is split into a time derivative for
the concentration and one for the volume fraction of a phase. The last term but one
is the local, geometrical derivative of the flux through a unit cross section. The last
term expresses the removal in the biofilm.
(5.74)
188
hand, be described by detailed, deterministic modelling by which the door is
opened to a alienating world of complicated differential equations, complicated
functional expressions and solutions which are difficult to understand and which
can only be obtained by computer calculations. However, there is no doubt about
the need for the increased realism in the description and the need for such models
for scientific examinations and soon also for the practical design.
The following description mainly follows the description in /22/. Fig 5.24 shows an
element in the biofilm and the corresponding definitions.
The biofilm consists of several phases such as water, solids (and possibly air). The
concentration, Ckj, of a substance is stated as the amount of a given substance in a
unit volume of the phase concerned. The volume fraction, Ek, of a solid k is stated by
the volume of the solid in a unit volume of the biofilm containing all the phases k.
For the phases it applies:
(5.72)
For the concentration it correspondingly applies that the concentration distributed
on a total unit volume is Cfi = Ek · Ckj, where Cki is the definition of the model
concentration which has been used so far. It is necessary to make this distinction
when there are many phases; for example several phases of solids and different
types of bacteria.
The differential equation for an element in the biofilm can be written as:
(5.73)
where jki
is the flux of the substance i in phase k
rv,ki is the removal rate for the substance i in phase k per unit volume
Ek
is the volume fraction of phase k
cki
is the concentration of substance i in phase k.
The first term ahead of the equals sign is the local time derivative from the mean
concentration which, just after the first equals sign, is split into a time derivative for
the concentration and one for the volume fraction of a phase. The last term but one
is the local, geometrical derivative of the flux through a unit cross section. The last
term expresses the removal in the biofilm.
(5.74)
188