mass transfer ch 7

7/30/2019 mass transfer ch 7

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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.

mass transfer ch 7

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