Ghasem D. Najafpour. Biochemical Engineering and Biotechnology
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and diameter ratio (H/D) is in the range 2–6. The H:D ratio that is commonly used is 3:1; the
ratio especially used for baker’s yeast production is 6:1.
Hydrodynamics and mass transfer in bubble columns are dependent on the bubble size
and the bubble velocity. As the bubble is released from the sparger, it comes into contact
with media and microorganisms in the column. In sugar fermentation, glucose is converted
to ethanol and carbon dioxide:
(13.3.2.1)
The superficial gas velocity is the air flow rate per unit cross sectional area of the column.
The superficial gas velocity is less than 0.4 ms
1
and media flow rate is based on liquid
velocity. Then, the column may have a hydraulic retention time that is a function of fluid
velocity and column diameter. The liquid velocity is given by:
4
(13.3.2.2)
where, U
g
is the upward gas bubble velocity at the centre of column, the gas superficial
velocity is in the range of 0 U
g
0.4 ms
1
and the column diameter is in the range
0.1 D 7.5 m. The hydraulic retention time is obtained by division of column height and
the actual liquid velocity:
(13.3.2.3)
The mixing time, t
m
is given by:
2
(13.3.2.4)
where, H
L
is the height of bubble column.
Example 1
Let us have a bubble column with an H/D ratio of 3, diameter 0.5 m and a gas flow rate of
0.1 m
3
h
1
,
which gives a superficial gas velocity of 0.25 ms
1
. What would be the liquid
media flow rate?
Solution
Once superficial gas velocity is defined, then liquid velocity is given by (13.3.2.2).
t
H
D
gU
D
m
L
g
11
2
033
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
.
U
Q
A
g
g
Air flow rate
Cross-sectional area
UgDU
gL
09
033
.( )
.
CH O 2CHOH CO
26126 25
2
294 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY
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Equation (13.23) is used to calculate mixing time.
Based on mixing time, the liquid media flow rate is:
Example 2
Calculate the speed of an impeller and the power requirements of a production-scale biore-
actor with 60m
3
using two different methods. Also, match the volumetric mass transfer
coefficient. The following optimum conditions were obtained with a small-scale fermenter
of volume 0.3m
3
and 60% of the vessel working. The density of broth, r
broth
, 1200 kgm
3
,
working volume 0.18 m
3
, aeration rate of one volume of gas per volume of liquid (1 vvm),
oxygen transfer rate 0.25 kmolm
3
h
1
, liquid height inside the vessel, H
L
, 1.2D
t
. Two sets
of standard, flat-blade turbine impellers were installed.
Solution
Diameter of the vessel:
Diameter of the impeller:
DD
i,1 t
92 m
1
3
0 576
3
01
.
.
D
V
t
6 m
1
13
13
03
018
03
057
.
.
.
.
/
/
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
VDD D
1
23
4
12 0
p
p
tt t
3(. ) .
Q
V
L
3
mh
t
(.)(. )/
.
15 025 4
31
95 10
31
t
m
1 hours
11
15
05
9 81 0 255
05
3
2
033
.
.
(. )(. )
(.)
.
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
U
L
0.33 1
0.9 [(9.81)(0.5)(0.255)] 0.96 m s
U
Q
A
g
g
1
ms
401
05
0 255
(.)
.
.
BIOPROCESS SCALE-UP 295
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Height of liquid media was assumed to be 1.2 times the diameter of the fermentation
vessel.
Diameter of the larger vessel:
The impeller size for the larger vessel is:
and the liquid media height in the second fermenter is:
Assume the fermentation broth has the same viscosity as water:
Aeration rate for 1 vvm is:
F
1
1 0.18 0.018 m
3
.min
1
3 10
3
m
3
.s
1
Gas superficial velocity is:
Let us take average values for the partial pressure of oxygen
P
H
O
L
2
1 atm
m
10.3 m
atm
3 atm
1
2
021 021
Ê
Ë
Á
ˆ
¯
˜
..
U
S
018 60
4
0 576
41 4
2
1
.
.
.
()
5 m h
m
1
1 cp
H
L2
3 m 12 336 40.. .
D
i
2 m
2
336
3
11
.
.
D
V
t2
6 m
1
13 13
03
60
03
33
..
.
//
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
H
L1
m 1 2 0 576 0 691.. .
296 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY
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The oxygen transfer rate is
The mass transfer coefficient is
Use imperial correction based on the following equation for mass transfer in the bioreac-
tors.
1,2
The general equation for evaluation of K
L
a is
(E.2.1)
where x, y and z are empirical constants. For Newtonian fluids, non-coalescing broth and
gas bubbles, the following correlation is valid for a working volume of less than 4 m
3
and
a power per unit volume of 500–10,000 Wm
3
.
(E.2.2)
For the gassed power per unit volume is 630.7 W; the passed power, P
g
, was 0.15hp.
Since the flow regime is turbulent, the power number obtained from Figure 6.6, Power num-
ber versus Reynolds number, Re, reads P
no
6. For two sets of impellers, N
p
2(6) 12.
(E.2.3)
Using Michel and Miller’s correction factor for power calculation:
5,6
(E.2.4)
P
PD
g
i
1
1
2
11
3
1
056
045
05 .
.
.
h
m
Ï
Ì
Ô
Ó
Ô
¸
˝
Ô
˛
Ô
P
N
NN
1
1
3
5
1
3
4
1
12 1200 0 192
981
0 383 5 10=
()
.
.
.W hp
3
N
Pg
ND
P
C
i
1
1
35
r
P
V
g
Ê
Ë
Á
ˆ
¯
˜
1 174 0 002 41 45
07
05
.. .
.
.
P
V
g
Ê
Ë
Á
ˆ
¯
˜
()
Kv
P
V
Us 0 002
07
05
.
.
.
g
Ê
Ë
Á
ˆ
¯
˜
Ka x
P
V
U
L
Y
Z
g
s
Ê
Ë
Á
ˆ
¯
˜
()
Kv
1
31 1
025
0 213
117
.
.
. 4 kmol m h atm
().Kv P
O
2
kmol m h
025
3
1
BIOPROCESS SCALE-UP 297
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Knowing the power input, we can calculate the rotational speed:
The power input for ungassed system is
For constant power input based on geometric similarity of the vessels, agitation rate is
calculated.
(E.2.5)
(E.2.6)
For constant input velocity for a large system:
13.4 CSTR CHEMOSTAT VERSUS TUBULAR PLUG FLOW
The performance of a biochemical reactor is designed and evaluated based the reaction rate
equation. The rate of biomass generation is based on the Monod rate model:
(13.4.1)
qx
kC x
KC
kxS
aS
S
mS
NN
D
D
21
1
2
600
0 192
112
10
i
i
3 rpm
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
.
.
NN
V
V
D
D
21
2
1
13
1
2
53
13
600
60
03
0 192
1
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
//
/
.
.
.
i
i
112
1
53
Ê
Ë
Á
ˆ
¯
˜
/
85 rpm
N
N
V
V
D
D
2
1
3
2
1
5
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
i1
i2
rrND
V
ND
V
1
3
1
5
1
2
35
2
ii2
PN
P
P
1
4
1
3
43
5 10 5 10 10 0
015
05
03
() .
.
.
.
5 hp
g
015 05
5 10 0 192
310
42
1
3
23
3056
045
..
()()(.)
()
.
.
N
N
Ï
Ì
Ô
Ó
Ô
¸
˝
Ô
˛
Ô
11
1
10 rps
600 rpm
N
298 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY
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where are dimensionless, K
m
is the Monod rate constant, gl
1
, and
kx represents the maximum specific growth rate, gl
1
h
1
. The fresh media dilution rate
for cell production is D, h
1
(13.4.2)
The substrate concentration is defined as:
(13.4.3)
The yield is incorporated into (13.4.1).
(13.4.4)
Example 1
A special CSTR fermentation known as a chemostat bioreactor is used for microbial cell
growth. The rate of biomass generation is given by:
(E.1.1)
We wish to compare the performance of two reactor types: plug flow versus CSTR with a
substrate concentration of Cs
f
60 gm
3
and a biomass yield of Y
x/s
0.1. In a plug flow
bioreactor with volume of 1m
3
and volumetric flow rate of 2.5 m
3
h
1
, what would be the
recycle ratio for maximum q
x
compared with corresponding results and rate models pro-
posed for the chemostat?
Solution
To maximise q
x
, we need to take the derivative of the following function, if is
maximum, then the derivative must be set to zero
(E.1.2)
d
dS
SS
aS
Sa S S S
aS
()( )( )()
()
112 1
0
2
È
Î
Í
˘
˚
˙
SS
aS
()1
q
V
C
x
SX
S
4
34
g cells
mh
3
qx
kS
aS
SY C
XS Sf
()1
/
SSYC
XS Sf
()1
/
D
qx
x
q
CC
S
Sf S
S
C
C
a
K
C
s
sf
m
sf
and
BIOPROCESS SCALE-UP 299
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(E.1.3)
(E.1.4)
(E.1.5)
The quadratic equation is solved for a
Maximum cell production is obtained in a chemostat at:
(E.1.6)
Now, we can generate data by plug-in numbers in the above equation. We need to illustrate
inverse rate versus substrate concentration. After the data are plotted we can justify a suitable
type of bioreactor
The performance data for plug versus mix reactor were obtained. The data were collected as
the inverse of q
x
vs inverse of substrate concentration. Table E.1.1 shows the data based on
obtained kinetic data. From the data plotted in Figure E.1.1, we can minimise the volume of
the chemostat. A CSTR works better than a plug flow reactor for the production of biomass.
Maximum q
x
is obtained with a substrate concentration in the leaving stream of 12gm
3
.
Then, the substrate concentration is:
C
S
12 gm
3
S
C
S
1
560
xY C C
x
Xs f s
/
().().
.( ) .
0 1 60 3 5 7
01 55 55
1
4
3
4q
C
Cx
x
S
S
Saaa
2
2
1
15
1
15
1
15
1
5
Ê
Ë
Á
ˆ
¯
˜
a
K
C
Y
m
Sf
XS
1
15
4
60
01
/
.
Saaa
2
SaSa
2
20
()()()12 1 0Sa S S S
300 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY
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A tubular bioreactor design with operational may lead to a CSTR, having sufficient recycle
ratio for plug flow that behave like chemostat. The recirculation plug flow reactor is better
than a chemostat, with maximum productivity at C
s
3gm
3
. Combination of plug flow
with CSTR which behave like chemostat was obtained from the illustration minimised
curve with maximum rate at C
Sf
3gm
3
.
BIOPROCESS SCALE-UP 301
TABLE E.1.1. Substrate concentration versus inverse biomass
concentration
Substrate concentration Inverse of biomass rate 1/q
x
,
Cs, gm
3
m
3
g
1
h
1
3 0.310
5 0.250
10 0.210
15 0.210
20 0.225
25 0.250
30 0.280
35 0.330
40 0.410
45 0.540
50 0.810
55 1.610
FIG. E.1.1. Performance plot for plug versus mix reactor.
C
S
, g/m
3
0 102030405060
1/q
x
, m
3
/g.h
0.0
0.4
0.8
1.2
1.6
2.0
Ch013.qxd 11/21/2006 9:51 AM Page 301
In plug flow reactor the value for C
S
is reduced from 12 to 3 g·m
3
, then the retention
time and rate model with recycle ratio in a plug flow reactor can be written as:
(E.1.7)
(E.1.8)
where S
f
is the inlet to outlet substrate ratio, R is the recycle ratio, and f is the ratio of inlet
to outlet substrate concentration. Solve R by trial and error, with calculation for right- and
left-hand sides with the assumption of a value for R:
Let us plug in f 20,k
m
4/3 and R 2.5 into (E.1.9), then calculate kt
(E.1.9)
Example 2: Applied Calculation Method for Scale-up
The bioreactor will be scaled up by a factor of 125. It is necessary to discuss the effect of
operating variables resulting from constant power per unit volume, agitation rate, and speed
tip velocity, N
Re
. The data for a small-scale bioreactor are a 100 litres fermenter with 187.5
rpm and
(E.2.1)
P
V
constant
VV
12
3
125 12.5 m
k
k
t
t
(.)ln
.
.
/
ln
.()
.
25
35
25
43
20
60 2 5 3
25
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
˘
˚
˙
1
kR
R
R
K
f
CRC
R
m
Sf S
t
( ) ln ln
1
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
È
Î
Í
Í
˘
˚
˙
˙
43
60 3
60 2 5 3
253
25 1
25
25 1
25
4/
/
ln
.()
.()
ln
.
.
.
.
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
//
..
3
20 2 5
1
25
K
S
CRCs
RCs
R
R
R
R
K
fRR
m
f
Sf
m
ln ln
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
11 1
t
CV
R
C
C
C
Sf
O
S
SX S
S
CRC
R
C
Sf S
S
n
n
()
()
1
4
34
1
d
Ú
302 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY
Ch013.qxd 11/21/2006 9:51 AM Page 302
(E.2.2)
Factoring for scale-up, we make some assumptions and may use a few known correlations
to progress our calculations. Let us say PaV D
i
3
, H/D 1 and D/D
i
3. Also, use con-
stant power per unit volume for the small- and large-scale bioreactor.
(E.2.3)
Since we assumed power per unit volume is constant, it is possible (N
i
D
i
)
1
may not be equal
with (N
i
D
i
)
2
. Now calculate the agitation rate for a large vessel.
Now assume H⬇D, D 3D
i
and bioreactor volume is proportional to impeller diameter to
the power of V D
i
3
. The Reynolds number is written as:
(E.2.4)
The power for laminar flow is proportional to agitation rate, N
2
, and if the flow is turbulent
the power is proportional to ⬃ N
3
D
i
2
. Let us assume the mass transfer coefficients remain
constant (K
L
a unchanged):
Also assume the height of liquid media is 1.2 times the tank diameter.
H
D
VD
t
si
100 l
12
12
4
3
.
.
D
D
ND
i
t
i
0.5 m/s
033.
Re
ND
i
i
r
m
NN
D
D
H
V
ND
D
V
i
2
i
i
i
ii
i
4 rpm
1
1
2
23
23
3
3
600
1
5
20
Ê
Ë
Á
ˆ
¯
˜
Ê
Ë
Á
ˆ
¯
˜
/
/
a
HHD
D
D
2
3
3
44
9
4
i
P
V
ND ND⬇ () )
ii
2
Small i
3
i
2
Large
3
(
ND
ii
constant
BIOPROCESS SCALE-UP 303
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