Biomedical Engineering Reference
In-Depth Information
value,
C
bm
, and whenever the biomass density,
C
b
, grows larger than the
threshold value,
C
bm
, the extra biomass is redistributed giving rise to biofilm
volume expansion. Specific examples of growth algorithm of this type are
described in the following DPD simulation.
The original set of equations (7.1-7.5) can be rewritten in a dimensionless
form for the convenience of simulation and modeling by introducing the char-
acteristic length,
l
c
, and velocity,
v
c
, as the units of length and velocity and
substituting equation (7.2) into equation (7.5). The resulting dimensionless
equations are
2
c
s
P
e
−
∂c
s
/∂t
+
v
·∇
c
s
=
∇
k
1
k
4
c
b
c
s
/(
k
2
+
c
s
)
(7.6)
v
= 0
∇·
(7.7)
2
v
R
e
∂
v
/∂t
+
v
·∇
v
=
p
+
−∇
∇
(7.8)
dc
b
/
dt
=
k
1
c
b
{
c
s
/(
k
2
+
c
s
)
−
k
3
}
(7.9)
where the dimensionless numbers
k
1
−
k
4
are defined as,
k
1
=
µ
m
l
c
/
v
c
,
k
2
=
K
s
/
C
sm
,
k
3
=
Y
bs
m
s
/
µ
m
,
and
k
4
=
C
bm
/(
Y
bs
C
sm
).
C
sm
is the maximum substrate
concentration in the system, and it is used for normalization. The Reynolds
number is defined as,
R
e
=
v
c
l
c
/
ν
and the Peclet number is defined as
P
e
=
v
c
l
c
/
D
s
. The dimensionless velocity and pressure fields are defined as
v
=
V
/
v
c
and
p
=
P
ρv
c
. The dimensionless substrate concentration and
biofilm density are normalized by
c
s
=
C
s
/
C
sm
and
c
b
=
C
b
/
C
bm
, and they
both lie in the range 0-1. This set of dimensionless equations (7.6-7.9) can be
further reduced by introducing the new timescale
t
=
t
k
1
and velocity scale
v
=
v
/
k
1
. This gives
2
c
s
P
eb
−
∂c
s
/∂t
+
v
·∇
c
s
=
∇
k
4
c
b
c
s
/(
k
2
+
c
s
)
(7.10)
∇·
v
= 0
(7.11)
2
v
R
eb
∂
v
/∂t
+
v
·∇
v
=
−∇
p
+
∇
(7.12)
dc
b
/dt
=
c
b
{
c
s
/(
k
2
+
c
s
)
−
k
3
}
(7.13)
where the new set of equations contains only five dimensionless numbers
k
2
,
k
3
,
k
4
,
R
eb
=
µ
m
l
c
ν
, and
P
eb
=
µ
m
l
c
D
s
. The corresponding dimensionless
time, velocity, and pressure fields are defined as,
t
=
µ
m
t
,
v
=
V
/(
µ
m
l
c
) and
p
=
P/ρ
(
µ
m
l
c
)
2
.
Appropriate velocity, concentration, and pressure boundary conditions
associated with equations (7.10-7.13) must be provided to complete the defini-
tion of the model. The no-slip boundary conditions for the velocity are applied
on the interface between the liquid phase and the biofilm or solid substratum
phase. A variety of numerical approaches have been applied to solve the cou-
pled differential equations in biofilm modeling studies. A continuum model
similar to the phase-field approach has been developed by treating both the
liquid and the biofilm phase as continuous media, but with different diffusion
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