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Chapter 7 Mass TransferMass transfer occurs in mixtures containing local concentration
variation. For example, when dye is dropped into a cup of water,
mass-transfer processes are responsible for the movement of dyemolecules through the water until equilibrium is established and the
concentration is uniform. Mass is transferred from one place to
another under the influence of a concentration difference or
concentration gradient in the system.
Gas-liquid mass transfer is extremely important in bioprocessing
because many processes are aerobic, oxygen must first be
transferred from gas bulk through a series of steps onto the
surfaces of cells before it can be utilized.
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The solubility of oxygen within broth is very poor. Therefore, the
enhancement of gas-liquid mass transfer during aerobic cultures
and fermentations is always put into priority.
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7.1 Basic Knowledge of Mass Transfer
7.1.1 Molecular Diffusion
Molecular diffusion is the movement of component molecules in a
mixture under the influence of a concentration difference in the
system. Diffusion of molecules occurs in the direction required to
destroy the concentration gradient. If the gradient is maintained byconstantly supplying material to the region of high concentration
and removing it from the region of low concentration, diffusion will
be continuous. This situation is often exploited in mass-transfer
operations and bioreaction system.
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Fig. 7.1 Concentration gradient of component A inducing mass transfer
C
Direction of mass transfer
Distance, y
Concen
trationofA, a
A
CA1
CA2
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Ficks law of diffusion:
7.1.2 Role of Diffusion in Bioprocessing
Mixing As discussed before, turbulence in fluids produces bulk
mixing on a scale equal to the smallest eddy size. Within the
smallest eddies, flow is largely streamline so that further mixing
must occur by diffusion of fluid components. Mixing on a
molecular scale therefore completely relies on diffusion as thefinal step in the mixing process.
J A = =a
NA
dy
dC
D
A
AB (7.1)
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Solid-phase reaction In biological systems, reactions are
sometimes mediated by catalysts in solid form, e.g. clumps,
flocs and films of cells and immobilized-enzyme and -cell
particles. When cells or enzyme molecules are clumped
together into a solid particle, substrates must be transported into
the solid before reaction can take place. Mass transfer within
solid particles is usually unassisted by bulk fluid convection; theonly mechanism for intraparticle mass transfer is molecular
diffusion. As the reaction proceeds, diffusion is also responsible
for removing of product molecules away from the site of
reaction, this will be discussed more fully in heterogeneous
bioreaction kinetics. When reaction is coupled with diffusion,
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the overall reaction rate can be significantly reduced if diffusion
is low.
Mass transfer across a phase boundary Mass transfer
between phases occurs often in bioprocesses. Oxygen transfer
from gas bubbles to fermentation broth, penicillin recovery from
aqueous to organic liquid, and glucose transfer from liquid
medium into mould pellets are typical examples. When differentphases come into contact, fluid velocity near the phase interface
is significantly decreased and diffusion becomes crucial for
mass transfer across the phase interface.
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7.1.3 Film Theory
Fig. 7.2 Two mass-transfer films formed within two phases
Phase boundary
Phase 2
Phase 1
Film 2 Film 1
CA1
CA1i
CA2i
A2C 12
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7.1.4 Mass Transfer Equation
Rate of mass transfer is directly proportional to the driving force for
transfer, and the area available for the transfer process to take
place, that is:
Transfer rate transfer area driving force
The proportional coefficient in this equation is called the mass-
transfer coefficient, so that:
Transfer rate = mass-transfer coefficient
transfer area driving force
NA =kaCA =ka(CAoCAi) (7.2)
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Mass transfer coupled with fluid flow is a more complicated processthan diffusive mass transfer. The value of the mass-transfer
coefficient reflects the contribution to mass transfer from all the
processes in the system that affect the boundary layer. k depends
on the combined effects of flow velocity, geometry of equipment,
and fluid properties such as viscosity and diffusivity. Because the
hydrodynamics of most practical systems are not easily
characterized, k cannot be calculated reliably from theoretical
equations. Instead, it is measured experimentally or estimated
using correlations available from the literatures. In general,
reducing the thickness of the boundary layer or improving the
diffusion coefficient in the film will result in enhancement ofk and
improvement in the rate of mass transfer.
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7.1.5 Liquid-Solid Mass Transfer
Fig. 7.3 Concentration gradient for liquid-solid mass transfer
Solid-liquid
CAo
CAi
interface
Solid
liquid film
NA =kLaCA =kLa(CAoCAi) (7.3)
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7.1.6 Liquid-Liquid Mass Transfer
Liquid-liquid mass transfer between immisible solvents is most
often encountered in the product-recovery stages of bioprocessing.
Organic solvents are used to isolate antibodies, steroids andalkaloids from fermentation broths; two-phase aqueous systems
are used in protein purification.
The rate of mass transfer NA in each liquid phase can be obtained:
NA1 =kL1a(CA1 CA1i) (7.4)
and
NA2 =kL2a(CA2i CA2) (7.5)
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At steady state, there is no accumulation of component A at the
interface or anywhere else in the system, and component A
transported through liquid 1 must be transported through phase 2,
that is NA1 =NA2 =NA.
IfCA1i and CA2i are equilibrium concentrations, they can be related
using the distribution coefficient m.
Therefore:
m=iA
iA
C
C
2
1 or CA1i =mCA2i (7.6)
2121
)1
( AALL
A CCak
m
akN =+ (7.7)
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and
Here we define two overall mass-transfer coefficients:
and
Therefore:
21
21
)11
( AA
LLA C
m
C
akamkN =+ (7.8)
ak
m
akaK LLL 211
11+= (7.9)
akamkaKL LL 212
111+= (7.10)
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and
These two Eqs indicate that the rate of mass transfer between two
phases is not dependent simply on the concentration difference;the equilibrium relationship is also an important factor. The driving
force for transfer component A out of liquid 1 is the difference
between the bulk concentration CA1 and the concentration of
component A in liquid 1 which would be in equilibrium with
concentration CA2 in liquid 2.
NA =KL1a(CA1mCA2) (7.11)
NA =KL2a(mCA1 CA2) (7.12)
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7.1.7 Gas-Liquid Mass Transfer
Fig 7.4 Concentration gradient for gas-liquid mass transfer
Phase boundary
Liquid phase
Gas phase
Liquid film Gas film
CAG
CAGiCALi
ALC 12
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The rate of mass transfer of component A through the gasboundary layer is:
and the rate of mass transfer of component A through the liquid
boundary layer is:
If we assume that equilibrium exists at the interface, CAGi and CALi
can be related. For dilute concentration of most gases and for a
wide range of concentration for some gases, equilibriumconcentration in the gas phase is a linear function of liquid
concentration. Therefore:
NAG =kGa(CAGCAG i) (7.13)
NAL =kLa(CALiCAL) (7.14)
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Therefore,
and
The overall gas-phase mass-transfer coefficient KG is defined by:
CAGi =mCALi (7.15)
ALAGLG
A mCCak
m
akN =+ )
1( (7.16)
ALAG
LGA C
m
C
akamkN =+ )
11( (7.17)
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and the overall liquid-phase mass-transfer coefficient KL is
defined as:
Thus:
ak
m
akaK LGG+=
11(7.18)
akamkaK LGL111 += (7.19)
NA =KGa(CAGmCAL) (7.20)
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and
Usually
and
NA =KLa(m
CAG CAL) (7.21)
NA =KGa(CAGCAG*) (7.22)
NA =KLa(CAL*CAL) (7.23)
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When solute A is very soluble in the liquid, for example, ammonia,
the liquid-phase resistance is small compared with that posed by
the gas interfacial film, therefore,
Conversely, if component A is poorly soluble in the liquid, e.g.
oxygen, the liquid-phase mass-transfer resistance dominates andkGa is much larger than kLa, thus:
NA =kGa(CAGCAG*) (7.24)
NA =kLa(CAL*CAL) (7.25)
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7.2 Oxygen Uptake in Cell Culture (contd)Cells in aerobic culture take up oxygen from broth. The rate of
oxygen transfer from gas to liquid is therefore of prime important,
especially at high cell densities when cell growth is likely to be
limited by availability of oxygen.
The solubility of oxygen in aqueous solutions at ambient temperature
and pressure is only about 10 ppm. This amount of oxygen is quickly
consumed in aerobic cultures and must be constantly replaced by
sparging. This is not an easy task because the low solubility of
oxygen guarantees that the concentration difference (CAL* CAL) is
always very small. Design of fermenters for aerobic operation musttake these factors into account and provide optimum mass-transfer
conditions.
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7.2.1 Factors Affecting Cellular Oxygen Demand
The rate at which oxygen is consumed by cells in fermenters
determines the rate at which it must be transferred from gas to
broth. Many factors influence oxygen demand; the most important
of these factors are cell species, culture growth phase, and nature
of the carbon source in the medium. In batch culture, rate of
oxygen uptake varies with time. The reasons for this are twofolds.
First, the concentration of cells increases during the course ofbatch culture and the total rate of oxygen uptake is proportional to
the number of cell present. In addition, the rate of oxygen
consumption per cell, known as the specific oxygen uptake rate,
also varies.
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Typically, specific oxygen demand passes through a maximum in
early exponential phase as illustrated below, even though the cell
concentration is relatively low at that time
200
150
50
100
0
100
80
40
60
0
20
0 20 40 60 80 100
Time, h
q
,g
h
g
(ce
lldryw
t)
o
-1
-1
Dry
we
ightx,
gl-1
x
qo
Fig 7.5 Variation in specific rate of oxygen consumption
and biomass concentration during batch culture
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IfQO is the oxygen uptake rate per volume of broth and qO is the
specific oxygen uptake rate:
The inherent demand of an organism for oxygen depends primarily
on the biochemical nature of the cell and its nutritional environment.
However, when the level of dissolved oxygen in the broth falls
below a certain point, the specific rate of oxygen uptake is also
dependent on the oxygen concentration in the broth.
QO =qOx (7.26)
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Fig 7.6 Relationship between specific oxygen uptake
rate and dissolved-oxygen concentration
Dissolved-oxygen concentration, CAL
CcritSpecif
icoxygen-u
pta
kera
te,
qO
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To eliminate dissolved oxygen limitations and allow cell metabolism
to function at its optimum, the dissolved oxygen concentration at
every point in the fermenter must be above Ccrit. The exact value of
Ccrit depends on the organism, but under average operationconditions usually falls between 5~10% of air saturation. For cells
with relatively high Ccrit level, the task of transferring sufficient
oxygen to maintain CLA >Ccrit is always more challenging than for
cultures with low Ccrit.
Choice of substrate for the fermentation can also significantly affect
oxygen demand. Because glucose is generally consumed more
rapidly than other sugars or carbon-containing substrates, rates of
oxygen demand are higher when glucose is used.
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For example, maximum oxygen-consumption rates of 5.5, 6.1 and
12.0 mmol l1 h1 have been observed for Penicillium mould
growing on lactose, sucrose and glucose, respectively.
7.2.2 Oxygen Transfer from Gas Bubble to Cell
In aerobic fermentation, oxygen molecules must overcome a series
of transport resistances before being utilized by the cells. Eight
mass-transport steps involved in transport of oxygen from the
interior of gas bubbles to the site of intracellular reaction are
represented diagrammatically
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Fig 7.7 Steps for oxygen transport from gas bubble to cell
Gas bubble
1 5 6
7
2 3 4
Stagnant region
Gas-liquid interface
Immobilized or aggregate cells
Solid-liquid interface
Cells
8
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Transfer through the bulk gas phase in the bubble is relatively fast.
The gas-liquid interface itself contributes negligible resistance.
The liquid f ilm around is a major resistance to oxygen transfer.
In a well mixed fermenter, concentration gradients in the bulk liquid
are minimized and mass-transfer resistance in this region are small.
Because single cells are much smaller than gas bubbles, the liquidfilm surrounding each cell is much thinner than that around the
bubbles and its effect on mass transfer can generally be neglected.
On the other hand, if the cells form large clumps, liquid-film
resistance can be significant.
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Resistance at the cell-liquid interface is generally neglected.
When the cells are in clumps, intraparticle resistance is likely to
be significant as oxygen has to diffuse through the solid pelletsto reach the interior cells. The magnitude of this resistance
depends on the size of the clumps.
Intracellular oxygen-transfer resistance is negligibile because ofthe small distances involved.
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Mass balance for oxygen at steady-state:
We can use this Eq. to predict the response of the fermenter to
changes in mass-transfer operating conditions. For example, if the
rate of cell metabolism remains unchanged but kLa is increased by
raising the stirrer speed to reduce the thickness of the boundarylayer around the bubbles, the dissolved-oxygen concentration CAL
must rise in order for the left-hand side to remain equal to the right-
hand side. Similarly, if the rate of oxygen consumption by the cells
accelerates while kLa is unaffected, CAL must decrease.
kLa(CAL*CAL) =qOx (7.27)
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Further, we can deduce some important relationship for fermenteroperations. First, let us estimate the maximum cell concentration
that can be supported by the fermenters oxygen-transfer system.
For a given set of operating conditions, the maximum rate of
oxygen transfer occurs when the concentration-difference driving
force (CAL* CAL) is highest, i.e. when the concentration of
dissolved oxygen CAL is zero. Therefore, the maximum cell
concentration that can be supported by the mass-transfer function
of the reactor is:
O
ALL
q
aCk
x
*
max = (7.28)
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Another important parameter is the minimum kLa required tomaintain CAL >Ccrit in the fermenter. This can also be determined
as:
Example 7.1 Cell concentration in aerobic culture
A strain ofAzotobacter vinelandii is cultured in a 15 m3
stirredfermenter for alginate production. Under current operating
conditions kLa is 0.17 s1. Oxygen solubility in the broth is
approximately 8 103 kg m3.
(a) The specific rate of oxygen uptake is 12.5 mmol g1 h1. What is
the maximum possible cell concentration?
critAL
O
critL CC
xq
ak = *)( (7.29)
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(b) The bacteria suffer growth inhibition after copper sulphate is
accidentally added to the fermentation broth. This causes areduction in oxygen uptake rate to 3 mmol g1 h1. What maximum
cell concentration can now be supported by the fermenter?
Solution:
(a) From Eq.(7.28):
(b) Assume that addition of copper sulphate does not affect CAL*
and kLa. If qO is reduced by a factor of 12.5/3 = 4.167, xmax is
increased to:
xmax' = 4.167 12 = 50 g l1
To achieve the calculated cell concentrations all of other conditions
must be favorable, e.g. sufficient substrate and time.
1-3-43
max lg12mg102.1325.12
10001000360010817.0==
=
x
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7.3 Measuring Dissolved-Oxygen Concentration
Liquid film
Membrane
CathodeA
node
Electrolyte
solution
Bulk fluid
Fig 7.8 Polarographic electrodes
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The electrode response time can be determined by quickly
transferring the probe from a beaker containing medium saturated
with nitrogen to one saturated with air. The response time is defined
as the time taken for the probe to indicate 63% of the total change indissolved-oxygen level. For commercially-available steam-
sterilisable electrodes, response times are usually 10 ~ 100 s.
Polarographic electrodes measure the partial pressure of dissolved
oxygen or oxygen tension in the fermentation broth, not the true
dissolved-oxygen concentration, it is necessary to know the solubility
of oxygen in the broth at the temperature and pressure of
measurement.
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7.4 Estimating Oxygen Solubility
Table 7.1 the oxygen solubility of pure oxygen and air in water (1atm)
Temperature
C
Pure oxygen solubility
kg m3Henrys constant
atm m3 kg1Air oxygen solubility
kg m3
0 7.03 102
14.2 1.48 102
10 5.49 102 18.2 1.15 102
15 4.95 102 20.2 1.04 102
20 4.50 102 22.2 9.45 103
25 4.14 102 24.2 8.69 103
26 4.07 102 24.6 8.55 103
27 4.01 102 24.9 8.42 103
28 3.95 102 25.3 8.29 103
29 3.89 102 25.7 8.17 103
30 3.84 102 26.1 8.05 103
35 3.58 102 27.9 7.52 103
40 3.37 102 29.7 7.07 103
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7.4.1 Effect of Temperature
7.4.2 Effect of Solutes
CAL* = 14.161 0.3943T+ 7.71 103T2 6.46 105T3
Table 7.2 Solubility of oxygen in NaCl solution under 1 atmoxygen pressure
Concentration
M
Oxygen solubility
kg m3
00.51.02.0
4.14 102
3.43 102
2.91 102
2.07 10
2
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Table 7.3 Solubility of oxygen in sugar solutions under 1 atmoxygen pressure
Sugar Concentrationgmol per kg H2O
TemperatureC
Oxygen solubility
kg m3
Glucose0
0.71.53.0
20202020
4.50 102
3.81 102
3.18 102
2.54
10
2
Sucrose0
0.40.9
1.2
151515
15
4.95 102
4.25 102
3.47 102
3.08 102
Table 7.3 Solubility of oxygen in sugar solutions under 1 atmoxygen pressure
Sugar Concentrationgmol per kg H2O
TemperatureC
Oxygen solubility
kg m3
Glucose0
0.71.53.0
20202020
4.50 102
3.81 102
3.18 102
2.54
10
2
Sucrose0
0.40.9
1.2
151515
15
4.95 102
4.25 102
3.47 102
3.08 102
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Quicker et al have developed an empirical correlation to correct
values of oxygen solubility in water for the effects of cations, anions
and sugars:
lo +=j
jLji
iLii
AL
AL CKCzHC
C 2*
*0 5.0 (7.31)
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7.5 Mass-Transfer Correlations
In general, there are two approaches to evaluating kL and a:
calculation using empirical correlations, and experimental
measurement. In both cases, separate determination ofkL and a is
laborious and sometimes impossible. It is convenient therefore to
directly evaluate the product kLa; the combined term kLa is often
referred to as the mass-transfer coefficient rather than just kL and a.
kLa= Gu
V
P)( (7.32)
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7.6 Measurement ofkLa
7.6.1 Dynamic Method
CAL1
AL2C
critC
Air off
Air on
t0 t1 t2
Time, t
ALC
CAL
Fig. 7.9 Variation of oxygen tension for dynamic measurement ofkLa
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During the re-oxygenation, the system is at an unsteady state. Therate of change in dissolved-oxygen concentration is equal to the
rate of oxygen transfer from gas to broth, minus the rate of oxygen
uptake by the cells:
where qOx is the rate of oxygen consumption. We can determine an
expression for qOx by considering the final steady dissolved-oxygen
concentration. When dCAL/dt = 0, therefore:
xqCCakdt
dCOALALL
AL = )(* (7.33)
qOx=kLa(CAL* ALC ) (7.34)
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thus,
Integrating:
)( ALALLAL CCakdt
dC= (7.35)
kLa=12
2
1 )ln(
tt
CCCCALAL
ALAL
(7.36)
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7.6.2 Oxygen-Balance Method
Mass balance at steady-state:
or
NA = ])()[(1
oAGgiAGgL
CFCFV
(7.37)
NA = ])()[(1
oAGg
iAGg
L T
pF
T
pF
RV (7.38)
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Summary
At the end of this chapter, you should:
know the two-film theory of mass transfer between phases and
the Ficks law;
be able to identity which steps are most likely to be majorresistances to oxygen mass transfer from bubbles to cells;
know the importance of the critical oxygen concentration;
understand how oxygen mass-transfer kLa can limit the biomass
density in fermenters;
know how temperature, total pressure, oxygen partial pressure
and presence of dissolved material in the broth affect oxygen
solubility and rates of oxygen mass transfer in fermenters; and know the techniques of dynamic method for experimental
determination ofkLa for oxygen transfer.