Chapter 6 Mixing
Mixing, a physical process which aims at reducing non-uniformities
in fluids by eliminating gradients of concentration, temperature,
and other properties, is happening within every bioreactor. It is so
important that, in a very large extend, decides the performance of
a bioreactor. When mixing is beneficial to bioprocesses, for
example, the contact of substrate and other nutrients to cells
during cell culture, we should try to improve the mixing
performance of a bioreactor through all kinds of means. Otherwise,
we should avoid its negative effects.
1
6.1 Mixing Equipment
Fig. 6.1 A standard tank with a working volume of
100 M3 and used for penicillin production
2
Fig. 6.2 Typical configuration of a stirred tank
3
Fig. 6.3 Baffle arrangements
4
Fig. 6.4 Impeller designs
5
Viscosity (centipoise)
10
10
10
4
3
2
5
Helical ribbons
Helical screws
Gate anchors
10
paddles
6
Flat-blade turbines
10
propellers
7
Anchors
10
10
1
Impeller type
Fig. 6.5 Viscosity ranges for different impellers
6
6.2 Flow Patterns Developed in Agitated Tanks
Fig. 6.6 Circular flow in a unbaffled stirred tank
7
6.2.1 Radial-flow impeller
Fig. 6.7 Flow pattern produced by a radial-flow impeller in a baffled tank
8
6.2.2 Axial-flow impeller
Fig. 6.8 Pitched-blade turbine
9
Fig 6.9 Flow pattern produced by an axial-flow impeller in a baffled tank
10
6.3 Mechanism of Mixing
As illustrated before, large liquid-circulation loops developed in
stirred vessels make mixing performance poor. For mixing to be
effective, fluid circulated by the impeller must sweep the entire
vessel in a reasonable time. In addition, the velocity of fluid leaving
the impeller must be sufficient to carry material into the most
remote parts of the tank. Turbulence must also be developed in the
fluid; mixing is certain to be poor unless flow in the tank is turbulent.
All these factors are important in mixing, which can be described as
a combination of three physical processes: distribution, dispersion
and diffusion.
11
Fig 6.10 Flow pattern developed by a radial-flow impeller
12
The process whereby materials are transported to all regions of the
vessel by bulk circulation currents is called distribution. Distribution
is an important process in mixing, but can be relatively slow. In large
tank, the size of the circulation paths is also large and the time
taken to traverse them is long; this, together with the regularity of
liquid pumping at the impeller, inhibits rapid mixing. Accordingly,
distribution is often the slowest step in the mixing process.
However, if the rotational speed of the impeller is sufficiently high,
superimposed on the distribution process is turbulence. Turbulence
flow occurs when fluid no longer travels along streamlines but
moves erratically in the form of cross-currents.
13
The kinetic energy of turbulent fluid is directed into regions of
rotational flow called eddies; masses of eddies of various size
coexist during turbulent flow. Large eddies are continuously formed
by action of the stirrer; these break down into small eddies which
produce even smaller eddies. Eddies, like spinning tops, posses
kinetic energy. When the eddies become so small that they can no
longer sustain rotational motion, their kinetic energy is dissipated
as heat.
The process of breaking up bulk flow into smaller and smaller
eddies is called dispersion; dispersion facilitates rapid transfer of
material throughout the vessel. The degree of homogeneity as a
result of dispersion is limited by the size of the smallest eddies
which may be formed in a particular fluid.
14
This size is given approximately as the Kolmogorov scale of
mixing, or scale of turbulence, λ.
ν 3 1/4
λ=(
)
pm
(6.1)
Within eddies there is little mixing because rotating flow occurs in
streamlines. Therefore, to achieve mixing on a scale smaller than
the Kolmogorov scale, we must rely on diffusion. Molecular
diffusion is generally regarded as a slow process, however, over
small distances it can be accomplished quite rapidly. Within
eddies of 30~100 μm diameter, homogeneity is achieved in about
1 s for low-viscosity fluids. Consequently, if power input to a
stirred vessel produces eddies of this dimension, mixing on a
molecular scale is accomplished virtually simultaneously..
15
6.4 Assessing Mixing Effectiveness
Mixing time is a useful parameter for assessing mixing efficiency
and is applied to characterize bulk flow in fermenters. The mixing
time tm is the time required to achieve a given degree of
homogeneity starting from the completely segregated state. It can
be measured by injecting a tracer into the vessel and following its
concentration at a fixed point in the tank. Tracers in common use
include
acids,
bases
and
concentrated
salt
solutions;
corresponding detectors are pH probes and conductivity cells.
Mixing time can also be determined by measuring the temperature
response after addition of a small quantity of heated liquid.
16
Let us assume a small pulse of tracer is added to fluid in a stirred
tank already containing tracer material at concentration Ci. When
flow in the system is circulation, the tracer concentration
measured at some fixed point in the tank will follow a pattern
similar to that shown in Figure 6.10. Before mixing is complete, a
relatively high concentration will be detected every time the bulk
flow brings tracer to the measurement point. The peaks in
concentration will be separated by a period approximately equal
to the average time taken for fluid to traverse one bulk circulation
loop. In stirred tank this period is called the circulation time tc.
After several circulations the desired degree of homogeneity is
reached.
17
Tracer concentration
C
tc
0.1(C f - C i )
Cf
Ci
Time
tm t
Fig. 6.11 Concentration response after tracer is injected into a stirred tank
18
Definition of the mixing time tm depends on the degree of
homogeneity required. Usually, mixing time is defined as the time
after which the concentration of tracer differs from the finial
concentration Cf by less than 10% of the total concentration
difference (Cf − Ci). At tm the tracer concentration is relatively steady
and the fluid composition approaches uniformity. For a single-phase
liquid in a stirred tank with several baffles and small impeller, there
is an approximate relationship between mixing time and circulation
time
tm = 4tc
(6.2)
19
We can predict that mixing time in stirred tanks will depend on
variables such as the size of the tank and impeller, fluid properties
such as viscosity, and stirred speed. The relationship between
mixing time and several of these variables has been determined
experimentally for different impellers; results for a Rushton turbine
in a baffled tank are shown in Fig 6.11. The dimensionless number
Nitm is plotted as a function of the impeller Reynolds number (Re)i.
tm is the mixing time based on a 10% deviation from final
conditions, and Ni is rotational speed of the stirrer. Ni tm represents
the number of stirrer rotations required to homogenize the liquid.
20
Fig 6.12 Variation of mixing time with Reynolds number
for a six-blade Rushton turbine in a baffled tank
21
At low Reynolds number, Nitm increases significantly with decrease
of (Re)i. However, as Reynolds number is increased above about 5
× 103, Nitm approaches a constant value which persists at high
(Re)i. For Rushton turbines, this constant value can be estimated
using the following relationship.
Ni t m =
1.54V
Di
(6.3)
3
ND ρ
(Re)i = i i
μ
2
(6.4)
22
Example 6.1 Estimation of mixing time
A fermentation broth with viscosity 10−2Pa s and density 1000 kg
m−3 is agitated in a 2.7 m3 baffled tank using a Rushton turbine with
diameter 0.5 m and stirred speed 1 s−1. Estimate the mixing time.
Solution:
From Eq.(6.4):
(Re)i = 2.5 × 104
(Re)i > 5 × 103, therefore Nitm is constant and can be calculated
from Eq.(6.3):
Nitm = 33.3
Therefore:
tm = 33.3/1 = 33.3 s
23
6.5 Power Requirements for Mixing
Usually, electrical power is used to drive impellers in stirred tanks.
For a given stirred speed, the power required depends on the
resistance offered by the fluid to rotation of the impeller. Average
power consumption per unit volume for industrial bioreactors
ranges from 10 kW m−3 for small vessels to 1~2 kW m−3 for large
vessels. Friction in the stirrer motor gearbox and seals reduces the
energy transmitted to the fluid; therefore, the electrical power
consumed by stirrer motors is always greater than the mixing
power by an amount depending on the efficiency of the drive.
Energy costs for operation of stirrers in bioreactors are an
important consideration in process economics.
24
6.5.1 Ungassed Newtonian Fluids
Mixing power for non-aerated fluids depends on the stirrer speed,
the impeller diameter and geometry, and properties of the fluid
such as density and viscosity. The relationship between these
variables is usually expressed in terms of dimensionless numbers
such as the impeller Reynolds number (Re)i and the power number
Np. Np is defined as:
Np =
P
ρN i Di
3
5
(6.5)
and
P = NpρNi3Di5
(6.6)
25
Fig. 6.13 Correlation between power number and Reynolds number for
Rushton turbine, paddle and marine propeller without sparging
26
Fig 6.14 Correlation between power number and Reynolds number
for anchor and helical-ribbon impeller without sparging
27
™ Laminar region: The laminar regime corresponds to (Re)i < 10
for many impellers; for stirrers with very small wall-clearance
such as the anchor and helical-ribbon mixer, laminar flow
persists until (Re)i = 100 or greater. In the laminar regime:
Np ∝ 1/(Re)i
or
P = k1μNi2Di3
(6.7)
™ Turbulent regime: Power number is independent of Reynolds
number in turbulent flow. Therefore:
P = Np’ρNi3Di5
(6.8)
28
Table 6.1 Constants in Eq.(6.7) and (6.8)
Impeller type
k1 , (Re)i = 1
Np’, (Re)i = 105
Rushton turbine
70
5~6
Paddle
35
2
Marine propeller
40
0.35
Anchor
420
0.35
Helical ribbon
1000
0.35
29
Np’ for turbines is significantly higher than for most other impellers,
indicating that turbines transmit more power to the fluid than other
designs. Power required for turbulent flow is independent of the
viscosity of the fluid but proportional to fluid density. The turbulent
regime is fully developed at (Re)i > 103 or 104 for most small
impellers in baffled vessels.
™Transition regime: Between laminar and turbulent flow lies the
transition regime. Both density and viscosity affect power
requirements in this regime. There is usually a gradual transition
from laminar to fully-developed turbulent flow in stirred tanks; the
flow pattern and Reynolds-number range for transition depend on
system geometry.
30
Eqs.(6.7) and (6.8) express the strong dependence of power
consumption on stirrer diameter and, to a lesser extent, stirrer
speed. Small changes in impeller size have a large effect on power
requirements, as would be expected from dependency on impeller
diameter raised to the third or fifth power. In the turbulent regime, a
10% increase in impeller diameter increases the power required by
more than 60%; a 10% increase in stirrer speed rasies the power
required by over 30%.
31
Example 6.2 Calculation of power requirements
A fermentation broth with viscosity 10−2 Pa s and density 1000 kg
m−3 is agitated in a 50 m3 baffled tank using a marine propeller 1.3
m in diameter. The tank geometry is as specified in Figure 6.13.
Calculate the power required for a stirred speed of 4 s−1.
Solution:
From Eq.(6.4):
4 × 1.3 2 × 1000
5
(Re)i =
10
=
6.76
×
10 −2
From Figure 6.13, flow at this (Re)i is fully turbulent and
Np’ = 0.35
Therefore:
P = 0.35 × 1000 × 4 × 1.35 = 8.3 × 104 kgm2 s−3 = 83 kW
32
6.5.2 Ungassed Non-Newtonian Fluids
Impeller Reynolds number based on the apparent viscosity μa:
N i Di 2 ρ
(Re)i =
μa
(6.9)
For power-law fluids:
N i Di 2 ρ
(Re)i =
r −
K γn 1
(6.10)
For stirred tanks, an approximate relation for pseudoplastic fluids is
often used:
γ = kN i
（6.11)
Substituting Eq.(6.11) into (6.10) gives:
(Re)i =
Ni
2−n
Di ρ
Kk
2
n -1
(6.12)
33
NP =
50
P
ρ N i 3 Di 5
Newtonian
Non-Newtonian
10
1
1
10
100
1000
ρN i Di 2
ρN i Di 2
(Re)i =
or
μa
μ
Fig. 6.15 Correlation between power number and Renolds
number for a Rushton turbine in unaerated fluids
34
6.5.3 Gassed Fluids
All of the changes in hydrodynamic behavior duo to gassing are not
completely understood. Power consumption is strong controlled by
gas-cavities formation; because this process is discontinuous and
appears somewhat randomly, reduction in power consumption is
typically non-uniform. The random nature of gas dispersion in
agitated tanks means that it is difficult to obtain an accurate
prediction of power requirements. However, an expression for the
ratio of gassed to ungassed power as a function of operating
conditions has been obtained.
35
pg
P0
= 0.10(
Fg
N iV
) −0.25 (
2
N i Di
gWiV
4
−0.20
)
2/3
(6.13)
Fig. 6.16 Gas cavities formed behind the blades of a 7.6
cm nine-blade fat-disc turbine in water sparged with air
36
6.6 Scale-up of Mixing Systems
Design of industrial-scale bioprocess is usually based on the
performance of small-scale prototypes. Determining optimum
operating conditions at production scale is expensive and timeconsuming; accordingly, it is always better to know whether a
particular process will work properly before it is constructed in full
size. Ideally, scale-up should be carried out so that conditions in the
large vessels are as close as possible to those producing good
results in the small vessels. As mixing is an important function of
bioreactors, it would seem desirable to keep the mixing time
constant on scale-up. Unfortunately, as explained below, the
relationship between mixing time and power consumption makes
37
this rarely possible in practice. As the volume of mixing vessels is
increased, so too are the lengths of the flow paths for bulk
circulation. To keep the mixing time constant, the velocity of fluid in
the tank must be increased in proportion to the size. As a rough
guide, under turbulent conditions the power per unit volume is
proportional to the fluid velocity squared:
P/V ∝ v2
(6.14)
Suppose a cylindrical 1 m3 pilot-scale stirred tank is scaled up to
100 m3. If the tanks are geometrically similar, the length of the flow
path in the large tank is about 4.5 times that in the small one.
Therefore, to keep the same mixing time, fluid velocity in the large
tank must be approximately 4.5 times faster.
From Eq.(6.14) this would entail a 4.52 or 20-fold increase in power
38
per unit volume. So, if the power input to the 1 m3 pilot-scale vessel
is P, the power required for the same mixing time in the 100 m3
tank is about 2000P. This represents an extremely large increase in
power, much greater than is economically or technically feasible
with most equipment used for stirring.
Because the criterion of constant mixing time can hardly ever be
applied for scale-up, it is inevitable that mixing time increase with
scale. If instead of mixing time, P/V is kept constant during scaleup, mixing time can be expressed to increase in proportion to
vessel diameter raised to the power 0.67.
Reduced productivity and performance often accompany scale-up
of bioreactors as a result of lower mixing efficiency and subsequent
alteration of the physical environment. One way of improving the
design procedure is to use scale-down methods.
39
The
general
idea
behind
scale-down
is
that
small-scale
experiments to determine operating parameters are carried out
under conditions that can actually be realized, physical and
economically, at production scale. For example, if we decide that
power input to a large-scale vessel cannot exceed a certain limit,
we can calculate the corresponding mixing time and use an
appropriate power input to a small-scale bioreactor to stimulate
mixing conditions in the large-scale system. With this approach, as
long as the flow regime is the same in the small- and large-scale
fermenters, there is a better chance that results achieved in the
small-scale unit will be reproducible in the larger system.
40
6.7 Improving Mixing in Fermenters
Sometimes, it is impossible to reduce mixing times by simply
raising the power input into the stirrer. So, while increasing the
stirrer speed is an obvious way of improving fluid circulation, other
techniques may be required.
Mixing can be improved by changing the configuration of the
system. Baffles should be installed; this is routine for stirrer
fermenters and produces greater turbulence. For efficient mixing
the impeller should be mounted below the geometric center of the
vessel. In standard designs the impeller is located about one
impeller diameter, or one-third the tank diameter, above the bottom
of the tank.
41
Mixing is facilitated when circulation currents below the impeller are
smaller than those above; fluid particles leaving the impeller at the
same time instant then take different periods of time to return and
exchange material. Rate of distribution throughout the vessel is
increased
when
upper
and
lower
circulation
loops
are
asynchronous.
Another device for improving mixing is multiple impellers, although
this requires an increase in power input. Typical bioreactors used
for aerobic culture are tall cylindrical vessels with liquid depths
significantly greater than the tank diameter. This design produces a
higher hydrostatic pressure at the bottom of the vessel, and gives
rising air bubbles a longer contact time with liquid. Effective mixing
in tall fermenters requires more than one impeller.
42
Fig. 6.15 Multiple impellers in a tall fermenter
43
In ungassed systems with spacing between impellers of at least
one impeller diameter, the power dissipated by multiple impellers is
approximated by the following relationship:
Pn = nPs
(6.15)
Additional mixing problems can appear in fermenters when material
is fed into system during operation. If bulk distribution is slow,
fermenters operated continuously or in fed-batch mode may
develop highly localized concentrations of substrate or other added
material near the feed point.
44
This has been observed particularly in large-scale processes for
production of SCP (single-cell-protein) from methanol. Because high
levels of methanol are toxic to cell growth, biomass yields decrease
significantly when mixing of feed material into the broth is slow. Also
observed was that during animal cell culture when alkali such as
NaOH was used to control pH, high local pH value seriously affected
the growth of the cells, although experiment was carried out within a
small fermenter. Problems like this can be alleviated by using
multiple injection points to aid distribution of added material. It is
much less expensive to do this than to increase the fluid velocity and
power input.
45
6.8 Effect of Rheological Properties on Mixing
For effective mixing there must be turbulent conditions in the mixing
vessel. Intensity of turbulence is represented by the impeller’s
Reynolds number. As discussed before for a baffled tank with
turbine impeller, once (Re)i falls below criteria turbulence is
damped and mixing time increases significantly. (Re)i decreases in
direct proportion to increase in viscosity. Accordingly, non-turbulent
conditions and poor mixing are likely to occur during agitation of
highly viscous fluids. Increasing the impeller speed is an obvious
solution, but this requires considerable increase in power
consumption and therefore may not be feasible.
46
Most non-Newtonian fluids in bioprocessing are pseudoplastic.
Because the apparent viscosity of these fluids depends on the
shear rate, the rheological behavior of many culture broths
depends on shear conditions in the fermenters.
Pseoduplastic fluids are shear thinning, i.e. their apparent viscosity
decreases with increasing shear. Accordingly, in stirred vessels,
pseudoplastic fluids have relatively low apparent viscosity in the
high-shear zone near the impeller, and relatively high apparent
viscosity when the fluid is away from the impeller. As a result, flow
patterns similar to that illustrated below can develop.
47
Stagnant zones
Fig. 6.16 Mixing pattern for pseudoplastic in a stirred tank
48
The effects of local fluid thinning in pseudoplastic fluids can be
countered by modifying the geometry of the system or impeller
design. Stirrers of large diameter are recommended. For turbine
impellers, instead of the usual tank-to-impeller diameter ratio of 3:1
used with low viscosity fluids, this ratio is reduced to between 1.6 ~
2.0. Different impeller designs which sweep the entire volume of the
vessel are also recommended. The most common types used for
viscous mixing are helical impellers and gate- and paddle-anchors
mounted with small clearance between the impeller and tank wall.
Mixing with these stirrers is accomplished at low speed without highvelocity streams. Helical agitators have been successfully used to
reduce shear damage and improve mixing in viscous cell
suspensions.
49
Alternative impeller designs such as the helical ribbon and anchor
improve mixing in viscous fluids; however their application in
fermentaters is only possible when oxygen demand in culture is
relatively low. Although large-diameter impellers operating at
relatively slow speed give superior bulk mixing, high-shear systems
with small, high-speed impellers are preferable for breaking up gas
bubbles and promoting oxygen transfer to the liquid. in design of
fermenters for viscous cultures, a compromise is usually required
between mixing effectiveness and adequate mass transfer.
50
6.9 Role of Shear in Stirred Tank
Mixing in bioreactors must provide the shear conditions necessary
to disperse bubbles, droplets and cell flocs. Dispersion of gas
bubbles by agitation involves a balance between opposing forces.
Shear forces in turbulent eddies stretch and distort the bubbles and
break them into small sizes; at the same time, surface tension at
the gas-liquid interface tends to restore the bubbles to their
spherical shape. In the case of solid material such as cell flocs or
aggregates, shear forces in turbulent flow are resisted by the
mechanical strength of the particles.
While bubbles break-up is required in fermenters to facilitate
51
oxygen transfer, disruption of cell is undesirable. Different cell types
display different levels of shear sensitivity; insect, mammalian,
and plant cells are known to be particularly sensitive to mechanical
forces. Bioreacters used for culture for those cells must limit the
intensity of shear while still providing adequate mixing and mass
transfer. At the present time, the effects of shear on cells are not
well understood. Cell disruption is an obvious outcome of high
shear forces; however more subtle changes such as retardation of
growth and product synthesis, denaturation of extracellular
proteins, change in morphology, and thinning of the cell wall, may
also occur.
B ecause there is significant spatial variation in shear
52
intensity
in
stirred
vessels,
the
precise
shear
conditions
experienced by cells are poorly defined. There have been many
publications in recent years addressing the problem of shear
damage, especially in inset- and mammalian-cell cultures. Several
mechanisms have been considered in terms of their contribution to
cell damage:
™ Interaction between cells and turbulent eddies;
™ Collision between cells, collision of cells with the impeller, and
collision of cells with stationary surfaces in the vessel;
™ Generation of shear forces in the boundary layers and wakes
near solid objects in the reactor, especially the impeller;
53
™ Generation of shear forces as bubbles rise through liquid; and
™ Bursting of bubbles at the liquid surface.
In general, when gas bubbles are not present in the liquid,
interactions between cells and turbulent eddies are considered
most likely to damage cells. However, if the vessel is sparged with
air, shear damage can occur at much lower impeller speeds due to
shear effects associated with bubbles.
54
6.9.1 Interaction Between Cells and Eddies
Hydrodynamic effects have been studied mainly with animal cells
because shear damage is a significant problem in large-scale
culture. Many animal cells used in bioprocessing are anchoragedependent; this means that the cells must be attached to a solid
surface for survial.
In bioreactors, the surface area required for cell attachment is
provided very effectively by microcarrier beads, which range in
diameter from 80 ~ 200 μm. Cells cover the surface of the beads
which are then suspended in nutrient medium.
55
Fig. 6.17 Chinese hamster ovary (CHO) cells
attached to microcarriers
56
There are many benefits associated with the use of microcarriers;
however, a disadvantage is that cells attached to microcarriers
cannot easily change position or rotate in response to shear forces
in the fluid. This, coulped with the lack of a protective cell wall,
make animal cells on microcarriers especially susceptible to shear
damage.
Interactions between microcarriers and eddies in turbulent flow
have the potential to cause mechanical damage to cells. The
intensity of shear associated with these interactions is dependent
on the relative sizes of the eddies and microcarriers. If the particles
are small relative to the eddies, they tend to be captured or
entrained in the eddies as shown below.
57
(b)
(a)
Eddy
streamlines
Microcarrier
Microcarrier
paths
Microcarrier
High shear
zone
Eddy
streamlines
Fig. 6.18 Eddy-microcarrier interactions
58
As fluid motion within eddies is laminar, if the density of the
microcarriers is about the same as the suspending fluid, there is little
relative motion of the particles. Accordingly, the velocity difference
between the fluid streamlines and the microcarriers is small, except
for brief periods of acceleration when the bead enters a new eddy.
On average, therefore, if the particles are smaller than the eddies, the
shear effects of eddy-cell interactions are minimal.
If the stirred speed is increased and the average eddy size reduced,
interactions between eddies and microcarriers can occur in two
possible ways. A single eddy that cannot fully engulf the particle will
act on part of its surface and cause the particle to rotate in the fluid;
59
this will result in a relatively small level of shear at the surface of
the bead.
However, much higher shear stress result when several eddies
with opposing rotation interact with the particle simultaneously. It
has been found experimentally that detrimental effects start to
occur when the Kolmogorov scale for eddy size drops below 2/3 ~
1/2 the diameter of the microcarrier beads. Excessive agitation
leads to formation of eddies with size small enough and of sufficient
energy to cause damage to the cells. These findings for cell on
microcarroers apply also to freely suspended cells; however,
because cells are smaller than microcarriers, eddy size causing
shear damage are also small.
60
Example 6.3 Operating conditions for turbulent shear damage
Microcarrier beads 120 μm in diameter are used to culture
recombinant CHO cells for growth hormone. It is proposed to use a
6-cm turbine impeller to mix the culture in a 3.5 liter stirred tank. Air
and carbon dioxide are supplied by flow through the bioreactor
headspace. The microcarrier suspension has a density of
approximately 1010 kg m−3 and a viscosity of 1.3 × 10−3 Pa s.
Estimate the maximum allowable stirrer speed which avoids
turbulent shear damage of the cells.
61
Solution:
Damage due to eddies is avoided if the Kolmogorov scale remains
greater than 2/3 ~ 1/2 the diameter of the beads. Let us determine
the stirrer speed required to create eddies with size λ = 2/3 × 120 =
80 μm = 8 × 10−5 m. The stirrer power producing eddies of this
dimension can be calculated:
ν 3 (1.29 × 10 −6 ) 3
2 -3
pm = 4 =
=
0
.
052
m
s
−5 4
λ
(8 × 10 )
where
μ 1.3 ×10−3
= 1.29 ×10−6 m2 s −1
ν= =
ρ
1010
62
Fluid mass in the impeller zone is roughly equal to ρDi3 where ρ is
fluid density and Di is impeller diameter. Therefore, the stirrer
power P is equal to pm multiplied by ρDi3:
P = 0.052 × 1010 × (6 × 10−2)3 = 1.13 × 10−2 kg m2 s−3 = 1.13 × 10−2 W
Np’ is about 5 for a turbine impeller operating in the turbulent
regime, depending on the geometry of the tank. The stirrer speed
corresponding to these conditions can be calculated:
−2
1
.
13
×
10
-3
Ni 3 =
=
=
2
.
89
s
'
5
5 × 1010 × (6 × 10 −2 ) 5
N p ρDi
P
63
Ni = 1.42 s−1 = 85.5 rpm
Flow is just turbulent with (Re)i = 4 × 103. This analysis indicates
that shear damage from turbulent eddies is not expected until the
stirrer speed exceeds about 85 rpm. If the culture were sparged
with gas, it is possible that shear damage would happen due to
other mechanisms, e.g. bursting bubbles.
If the viscosity of the liquid is increased, the size of the smallest
eddies also increases. Increasing the fluid viscosity should,
therefore, reduce shear damage in bioreactors. This effect has
been demonstrated by addition of thickening agents to animal-cell
growth medium; moderate increase in viscosity have been shown
to significantly reduce turbulent cell death.
64
6.9.2 Bubble Shear
When liquid containing shear-sensitive cells is sparged with air,
other damaging mechanisms come into play. From experiments
conducted so far, these appear to be associated primarily with
bubbles bursting at the surface of the liquid. breakage of the thin
bubble film and rapid flow from the bubble rim back into the liquid
generate high shear forces capable of damaging certain types of
cell.
65
Summary
After the study of this chapter, you should be:
™ familiar with equipment used for mixing in stirred tanks;
™ able to describe the mechanisms of mixing and their effect on
mixing time;
™ able to understand the effects of scale-up on mixing;
™ able to know how liquid properties, gas sparging, impeller size
and stirrer speed affect power consumption in stirred vessels; and
™ able to understand how cells can be damaged by shear in stirred
fermenters.
66