1

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 dye molecules 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 bioprocessingbecause 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.

2

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.

3

7.1 Basic Knowledge of Mass Transfer

7.1.1 Molecular DiffusionMolecular diffusion is the movement of component molecules in a mixture under the influence of a concentration difference in thesystem. Diffusion of molecules occurs in the direction required to destroy the concentration gradient. If the gradient is maintained by constantly 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.

4

Fig. 7.1 Concentration gradient of component A inducing mass transfer

C

Direction of mass transfer

Distance, y

Con

cent

ratio

n of

A,

a

A

CA1

CA2

5

Fick’s law of diffusion:

7.1.2 Role of Diffusion in BioprocessingMixing 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 the final step in the mixing process.

JA = =a

N A −dy

dCD AAB (7.1)

6

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; the only 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,

7

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 fromaqueous to organic liquid, and glucose transfer from liquid medium into mould pellets are typical examples. When different phases come into contact, fluid velocity near the phase interface is significantly decreased and diffusion becomes crucial for mass transfer across the phase interface.

8

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

CA1iCA2i

A2Cδδ 12

9

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 = kaΔCA = ka(CAo − CAi) (7.2)

10

Mass transfer coupled with fluid flow is a more complicated process than 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, which 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 orestimated 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 of kand improvement in the rate of mass transfer.

11

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 = kLaΔCA = kLa(CAo − CAi) (7.3)

12

7.1.6 Liquid-Liquid Mass TransferLiquid-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 and alkaloids 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)

13

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.

If CA1i and CA2i are equilibrium concentrations, they can be related using the distribution coefficient m.

Therefore:

m = iA

iA

CC

2

1 or CA1i = mCA2i (7.6)

2121

)1( AALL

A CCak

mak

N −=+ (7.7)

14

and

Here we define two overall mass-transfer coefficients:

and

Therefore:

21

21)11( A

A

LLA C

mC

akamkN −=+ (7.8)

ak

makaK LLL 211

11+= (7.9)

akamkaKL LL 212

111 += (7.10)

15

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 transferring 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(CA1 − mCA2) (7.11)

NA = KL2a(m

CA1 − CA2) (7.12)

16

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

ALCδδ 12

17

The rate of mass transfer of component A through the gas boundary 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 awide range of concentration for some gases, equilibrium concentration in the gas phase is a linear function of liquid concentration. Therefore:

NAG = kGa(CAG − CAG i) (7.13)

NAL = kLa(CALi − CAL) (7.14)

18

Therefore,

and

The overall gas-phase mass-transfer coefficient KG is defined by:

CAGi = mCALi (7.15)

ALAGLG

A mCCak

mak

N −=+ )1( (7.16)

ALAG

LGA C

mC

akamkN −=+ )11( (7.17)

19

and the overall liquid-phase mass-transfer coefficient KL is defined as:

Thus:

akm

akaK LGG+=

11 (7.18)

akamkaK LGL

111+= (7.19)

NA = KGa(CAG − mCAL) (7.20)

20

and

Usually

and

NA = KLa(m

CA G − CAL) (7.21)

NA = KGa(CAG − CAG*) (7.22)

NA = KLa(CAL* − CAL) (7.23)

21

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 and kGa is much larger than kLa, thus:

NA = kGa(CAG − CAG*) (7.24)

NA = kLa(CAL* − CAL) (7.25)

22

7.2 Oxygen Uptake in Cell Culture

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 must take these factors into account and provide optimum mass-transfer conditions.

23

7.2.1 Factors Affecting Cellular Oxygen DemandThe rate at which oxygen is consumed by cells in fermentersdetermines 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.

24

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 100Time, h

q ,

g h

g

(cel

l dry

wt)

o-1

-1

Dry

wei

ght x

, g l-1

x

qo

Fig 7.5 Variation in specific rate of oxygen consumption

and biomass concentration during batch culture

25

If QO 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)

26

Fig 7.6 Relationship between specific oxygen uptake

rate and dissolved-oxygen concentration

Dissolved-oxygen concentration, CAL

Ccrit

Spec

ific

oxyg

en-u

ptak

e ra

te, q

O

27

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 operation conditions 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.

28

For example, maximum oxygen-consumption rates of 5.5, 6.1 and 12.0 mmol l−1 h−1 have been observed for Penicillium mould growing on lactose, sucrose and glucose, respectively.

7.2.2 Oxygen Transfer from Gas Bubble to CellIn aerobic fermentation, oxygen molecules must overcome a seriesof 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

29

Fig 7.7 Steps for oxygen transport from gas bubble to cell

Gas bubble1 5 6

7

2 3 4

Stagnant region

Gas-liquid interface

Immobilized or aggregate cells

Solid-liquid interface

Cells

8

30

Transfer through the bulk gas phase in the bubble is relatively fast.

The gas-liquid interface itself contributes negligible resistance.

The liquid film 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 liquid film 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.

31

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 pellets to reach the interior cells. The magnitude of this resistance depends on the size of the clumps.

Intracellular oxygen-transfer resistance is negligibile because of the small distances involved.

32

Mass balance for oxygen at steady-state:

We can use Eq. (7.27) 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 boundary layer 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)

33

Further, we can deduce some important relationship for fermenteroperations. First, let us estimate the maximum cell concentration that can be supported by the fermenter’s 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

qaCk

x*

max = (7.28)

34

Another important parameter is the minimum kLa required to maintain CAL > Ccrit in the fermenter. This can also be determined as:

Example 7.1 Cell concentration in aerobic culture

A strain of Azotobacter vinelandii is cultured in a 15 m3 stirred fermenter for alginate production. Under current operating conditions kLa is 0.17 s−1. Oxygen solubility in the broth is approximately 8 × 10−3 kg m−3.

(a) The specific rate of oxygen uptake is 12.5 mmol g−1 h−1. What is the maximum possible cell concentration?

critAL

OcritL

CCxq

ak−

= *)( (7.29)

35

(b) The bacteria suffer growth inhibition after copper sulphate is accidentally added to the fermentation broth. This causes a reduction in oxygen uptake rate to 3 mmol g−1 h−1. 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 l−1

To achieve the calculated cell concentrations all of other conditions must be favorable, e.g. sufficient substrate and time.

1-3-43

max l g 12m g 102.1325.12

10001000360010817.0=×=

××××××

=−

x

36

7.3 Measuring Dissolved-Oxygen Concentration

Liquid film

MembraneC

atho

deAno

deElectrolytesolution

Bulk fluid

Fig 7.8 Polarographic electrodes

O2 + 4 e− + 2 H2O 4 OH−

37

The electrode response time can be determined by quickly transferring the probe from a beaker containing medium saturatedwith 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 in dissolved-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.

38

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 m−3

Henry’s constantatm m3 kg−1

Air oxygen solubility kg m−3

0 7.03 × 10−2 14.2 1.48 × 10−2

10 5.49 × 10−2 18.2 1.15 × 10−2

15 4.95 × 10−2 20.2 1.04 × 10−2

20 4.50 × 10−2 22.2 9.45 × 10−3

25 4.14 × 10−2 24.2 8.69 × 10−3

26 4.07 × 10−2 24.6 8.55 × 10−3

27 4.01 × 10−2 24.9 8.42 × 10−3

28 3.95 × 10−2 25.3 8.29 × 10−3

29 3.89 × 10−2 25.7 8.17 × 10−3

30 3.84 × 10−2 26.1 8.05 × 10−3

35 3.58 × 10−2 27.9 7.52 × 10−3

40 3.37 × 10−2 29.7 7.07 × 10−3

39

7.4.1 Effect of Temperature

7.4.2 Effect of Solutes

CAL* = 14.161 − 0.3943T + 7.71 × 10−3T 2 − 6.46 × 10−5T 3

Table 7.2 Solubility of oxygen in NaCl solution under 1 atm oxygen pressure

ConcentrationM

Oxygen solubilitykg m−3

00.51.02.0

4.14 10−2

3.43 10−2

2.91 × 10−2

2.07 × 10−2

××

40

Table 7.3 Solubility of oxygen in sugar solutions under 1 atm oxygen pressure

Sugar Concentrationgmol per kg H2O

Temperature°C

Oxygen solubilitykg m−3

Glucose0

0.71.53.0

20202020

4.50 × 10−2

3.81 × 10−2

3.18 × 10−2

2.54 × 10−2

Sucrose0

0.40.91.2

15151515

4.95 × 10−2

4.25 × 10−2

3.47 × 10−2

3.08 × 10−2

Table 7.3 Solubility of oxygen in sugar solutions under 1 atm oxygen pressure

Sugar Concentrationgmol per kg H2O

Temperature°C

Oxygen solubilitykg m−3

Glucose0

0.71.53.0

20202020

4.50 × 10−2

3.81 × 10−2

3.18 × 10−2

2.54 × 10−2

Sucrose0

0.40.91.2

15151515

4.95 × 10−2

4.25 × 10−2

3.47 × 10−2

3.08 × 10−2

41

Quicker et al have developed an empirical correlation to correct values of oxygen solubility in water for the effects of cations, anions and sugars:

log ∑+∑=j

jLji

iLiiAL

AL CKCzHCC 2

*

*0 5.0 (7.31)

42

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 of kL 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 = γβα GuVP )( (7.32)

43

7.6 Measurement of kLa

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 of kLa

44

During the re-oxygenation, the system is at an unsteady state. The rate 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)

45

thus,

Integrating:

)( ALALLAL CCak

dtdC

−= (7.35)

kLa =12

2

1 )ln(

ttCCCC

ALAL

ALAL

−−−

(7.36)

46

7.6.2 Oxygen-Balance Method

Mass balance at steady-state:

or

NA = ])()[(1oAGgiAGg

LCFCF

V− (7.37)

NA = ])()[(1o

AGgi

AGg

L TpF

TpF

RV− (7.38)

47

Summary

At the end of this chapter, you should:know the two-film theory of mass transfer between phases and the Fick’s law;be able to identity which steps are most likely to be major resistances 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; andknow the techniques of dynamic method for experimental determination of kLa for oxygen transfer.