WOLFRAM|DEMONSTRATIONS PROJECT

Temperature Control of a Batch Fermentor

​
controller constant
5
time constant of controller
1
substrate and biomass
temperatures
cooling water flow rate
set-up figure
This Demonstration shows how one can control the temperature of a batch fermentor using a proportional-integral feedback controller. The governing equations are the following:
dX
dt
=μX
dS
dt
=
-μX
Y
μ=
μ
m
S
K
S
+S
d
T
R
dt
=
r
Q
ρ
C
p
-
UA
Vρ
C
p
(
T
R
-
T
C
)
d
T
C
dt
=
F
V
C
(
T
Cin
-
T
C
)+
UA
V
C
ρ
C
C
pc
(
T
R
-
T
C
)
F=max0,
F
0
+
K
p
ϵ+
1
τ
I
ϵdt
ϵ=
T
Rset
-
T
R
,
where
T
R
and
T
C
are the temperature of the fermentor and cooling water,
μ
and
μ
m
are the specific growth rate and its maximum value,
Y
and
Y
QS
are the yield coefficient and heat yield for substrate,
K
m
is the saturation coefficient,
K
p
and
τ
I
are the proportional gain and integral time constant of the controller,
U
and
A
are the heat transfer coefficient and area of the cooler,
T
Cin
is the inlet temperature of the cooling water,
F
is the flow rate of the cooling water,
V
and
V
C
are the volumes of the fermentor and the cooler,
ρ
and
ρ
C
are the densities of the fermentation media and cooling water,
T
Rset
is the fermentor's temperature set point (chosen equal to 25°C here),
X
and
S
are the biomass and substrate concentrations, and finally
C
p
and
C
pc
are the heat capacities of the fermentation media and cooling water.
For a batch reactor, the substrate will be depleted after a certain time and the biomass will reach a constant plateau. If
F
0
=0
and
K
p
=0
(i.e., no temperature control), the temperature of the fermentor will rise to a relatively large value (≈ 26.5°C) as can be seen in the snapshot. If control is applied, the fermentor's temperature will stay around 25°C for almost the entire growth period (up to
t=5hr
), then will decrease as the fermentor cools down and the substrate concentration reaches a value very close to zero.