Chapter 6
Transport Phenomena In
Biochemical Reactors
Mass Transfer Criteria for Simple
and Complex Systems
(1)
参考阅读：
,生化反应动力学与反应器,
戚以政 汪淑雄 编著
第 6章 生化反应器的传递过程
第 5章 生化反应器的设计与分析
第 7章 生化反应器的流动模型与放大
dy
dv x
yx
Chapter 5 Basic Concepts of Transport
Phenomena In Bio-engineering and Chemical
Engineering
5.1 Transport phenomena and reaction are the basic
phenomena in the nature,Chemical Engin,and Bio-
engin.
If there is velocity gradient,temperature gradient,or
concentration gradient,there must be momentum
transfer,heat transfer,and mass transfer,
For momentum transfer,there is Newton’s Law of
Viscosity as follows:
-----shear stress,-------viscosity of the fluid,
Momentum flux = - viscosity velocity gradient
Fluids that behave in this fashion are termed Newtonian
fluids,Fluids that do not obey this law are referred to as non-
Newtonian fluids,The subject of non-Newtonian flow is a
subdivision of Rheology.
For the heat transfer,there is Fourier’s Law of Heat
Conduction as follows:
q/A -----The local heat flow per unit area; heat flux
k ----- thermal conductivity.
Heat flux = thermal conductivity temperature gradient
dy
dTk
A
q
jA------the molar diffusion flux of A in a binary system;
DAB ----- mass diffusivity; ------- mass density;
Molar diffusion flux = mass diffusivity density gradient
Assumption of Constant C and DAB in a binary system with
chemical reaction,the molar diffusion differential equation:
RA ------ the molar rate of production A
dy
dDj A
ABA
For the mass transfer,there is Fick’s First Law:
A
AAA
AB
A R
z
c
y
c
x
cD
Dt
Dc?
)(
2
2
2
2
2
2
If the above situation without reaction in a stable system:
This is Fick’s second Law.
5.2 Importance of Study on Transport phenomena
There are microbes,substrate and metabolism products
effect on the viscosity of the bio-reaction systems,and then
effect on the momentum transfer,heat transfer,and mass
transfer,Further the transport factors effect on the bio-
reactions,design of reactors and general output of the
whole bio-process,
)( 2
2
2
2
2
2
z
c
y
c
x
cD
Dt
Dc AAA
AB
A
Viscosity
Fluid kinetics/power required
Heat transfer Mss transfer
Raw material feed
(Pumping,
heating,cooling
and mixing)
Fermentation
(cells dispersion,
oxygen dissolved,
temperature )
Purification and
recovery of
products
(Pumping,
heating,cooling
and separation)
Cells growth products formed morphology
Viscosity
Design and economics
5.3 Rheological Propertics of The Process Materials
(1) Pure viscous fluids,
a)Newtonian fluids
= (=constant),shear stress,N/m2
b)non-Newtonian fluids,shear rate,s-1
= ( constant),viscosity,kg/ms
(2) viscoelastic fluids
= f (,extent of deformation)
Most non-Newtonian fluids follows the power-law model
= K( )n
dilatant
Newtonian
pseudoplastic
plastics
Bingham
Fig.5.1 General shear behavior of
rheologically time-independent fluid
classes
5.4 Basic Dispersion Concepts
The oxygen transfer rate from the gas bubble to the medium is
largely determined by kL and the interfacial area a,
Main variables which influence a,
bubble size dB,the terminal velocity of the bubble UB
and the gas hold-up?
Basic correlation,
For small,rigid interface bubbles follow Stokes equation:
UB= dB2 (5-1)
which is valid for Re 1.
For mobile interface bubbles,
UB= dB2 (5-2)
At higher bubble Reynolds numbers:
UB= (5-3A)
When the gravity stresses are higher than the surface
tension stresses,UB= (5-3B)
18
g
16
g
2
2 B
B
gd
d
2
Bgd
Chapter 6 Gas-Liquid Mass Transfer
6.1 Oxygen Transfer in Metabolism Process of
Microorganism
Oxygen solubility in a fermentation medium is very
low,while its demand for the growth of aerobic
microorganisms is high,
For example,when the oxygen is provided from air,the
typical maximum concentration of oxygen in aqueous
solution is on the order of 6 to 8 mg /L.Oxygen
requirement of cells is,although it can vary widely
depending on microorganisms,on the order of 1 g /L h,
Even though a fermentation medium is fully
saturated with oxygen,the dissolved oxygen will be
consumed in less than one minute by organisms if
not provided continuously,Adequate oxygen supply
to cells is often critical in aerobic fermentation,
Even temporary depletion of oxygen can damage
cells irreversibly.Therefore,gaseous oxygen must be
supplied continuously to meet the requirements for
high oxygen needs of microorganisms,and the
oxygen transfer can be a major limiting step for cell
growth and metabolism,
Mass-Transfer Path:The path of gaseous substrate
from a gas bubble to an organelle in a
microorganism can be divided into several steps
(Figure 6.1),as follows:
(1)Transfer from bulk gas in a bubble to a
relatively unmixed gas layer
(2)Diffusion through the relatively unmixed gas
layer
(3) Diffusion through the relatively unmixed liquid
layer surrounding the bubble
(4)Transfer from the relatively unmixed liquid
layer to the bulk liquid
(5)Transfer from the bulk liquid to the relatively
unmixed liquid layer surrounding a microorganism
(6)Diffusion through the relatively unmixed liquid
layer
(7)Diffusion from the surface of a microorganism to
an organelle in which oxygen is consumed
4
1
Gas bubble Microorganism
Fig.6.1 Schematic diagram of the path of a gaseous
substrate to an organenelle in a cell
2
3 5
6
6.2 Basic Mass-Transfer Concepts
6.2.1 Molecular Diffusion in Liquids
When the concentration of a component varies from one
point to another,the component has a tendency to flow in
the direction that will reduce the local differences in
concentration,Molar flux of a component A relative to the
average molal velocity of all constituent JA is proportional
to the concentration gradient dCA/dz as
Which is Fick’s first law written for the z-direction,The
DAB in Eq,(9.1) is the diffusivity of
Z
A
ABA d
dCDJ
(6.1)
component A through B,which is a measure of its
diffusive mobility,Molar flux relative to stationary
coordinate NA is equal to
Z
A
ABBA
A
A D
dCDNN
C
CN
(6.2)
Where C is total concentration of components A and B
and NB is the molar flux of B relative to stationary
coordinate,The first term of the right hand side of
Eq.(9.2) is the flux due to bulk flow,and the second
term is due to the diffusion,For dilute solution of A,
AA JN?
(6.3)
Diffusivity,The kinetic theory of liquids is much less
advanced than that of gases,Therefore,the correlation for
diffusivities in liquids is not as reliable as that for gases.
Among several correlations reported,the Wilke-Chang
correlation (Wilke and Chang,1955) is the most widely
used for dilute solutions of nonelectrolytes,( or see
戚以政 ‘ book 6-47)
6.01.1
5.016101 7 3.1
bA
B
AB V
TMD
ξ?
(6.4)
When the solvent is water,and the solute is small
molecular substance,DO2 is about 0.5～ 2.0× 10-5cm2/s,
Skelland(1974) recommends the use of the correlation
developed by Othmer and Thakar(1953),
6.01.1
13101 1 2.1
bA
AB VD?
(6.5)
The preceding two correlations are not dimensionally
consistent; therefore,the equations are for use with the
units of each term as SI unit as follows:
ABD
=diffusivity of A in B,in a very dilute solution,m2/s
MB=molecular weight of component B,kg/kmol
T=temperature,K
μ=solution viscosity,kg/m s
VbA=solute molecular volume at normal boiling point,
m3/kmol:0.0256m3/kmol for oxygen [See Perry and Chil-
ton (p.3-233,1973)for extensive table]
ξ=association factor for the solvent,2.26 for water,1.9 for
methanol,1.5 for ethanol,1.0 for unassociated solvents,
such as benzene and ethyl ether
Example 6.1
Estimate the diffusivity for oxygen in water at
25℃,Compare the predictions from the Wilke-Chang and
Othmer-Thakar correlations with the experimental value
of 2.5*10-9 m2/s (Perry and Chilton,p,3-225,1973).
Convert the experimental value to that corresponding to a
temperature of 40℃,
Solution:
Oxygen is designated as component A,and water,
component B,The molecular volume of oxygen VbA is
0.0256 m3/kmol,The association factor for water ξ is 2.26,
The viscosity of water at 25℃ is 8.904*10-4kg/ms (CSC
Handbook of chemistry and Physics,p,F-38,1983),In
Eq.(6.4)
smD AB /1025.20 2 5 6.0109 0 4.8
2 9 81826.2101 7 3.1 29
6.04
5.016
In Eq.(6.5)
smD AB /1027.20 2 5 6.010904.8
10112.1 29
6.01.14
13
If we define the error between these predictions and the
experimental value as
1 0 0% ale x p e r i m e n t
ale x p e r i m e n tp r e d i c t e d
AB
ABAB
D
DD
e r r o r
The resulting errors are –9.6 percent and –9.2 percent for
Eqs,(6.4) and (6.5),respectively,Since the estimated
possible error for the experimental values from both
equations are satisfactory.
Eq,(6,4) suggests that the quantity DoABμ/T is constant
for a given liquid system,Though this is an
approximation,we may use it here to estimate the
diffusivity at 40℃,
C40a t /10529.6 4?mskg
smD AB /1058.3
2 9 8
3 1 3
105 2 9.6
109 0 4.8102,5C40a t 29
4
4
9-?
If we Eq,(6.5),DABoμ1.1 is constant,
smD AB /1052.3
10529.6
10904.8102,5C40a t 29
1.1
4
4
9-?
CGi
CLi
CL
CG
LiquidGas
CL CLi
CG
CGi
G
Lkk?
6.2.2 Mass-Transfer Cofficient
The mass flux,the rate of mass transfer qG per unit area,
is proportional to a concentration difference,If a solute
transfers from the gas to the liquid phase,its mass flux
from the gas phase to the interface NG is
GiGGG CCk
A
qGN (6.6)
Where CG and CGi is the gas-side concentration at the
bulk and the interface,respectively,as shown in Figure
6.3,kG is the individual mass-transfer coefficient for the
gas phase and A is the interfacial area.
Similarly,the liquid-side phase mass flux NL is
LLiLL CCk
A
qLN (6.7)
Where kL is the amount of solute transferred from the gas
phase to the interface must equal that from the gas phase
to the interface must equal that from the interface to the
liquid phase,
LG NN?
(6.8)
Substitution of Eqs.(9.6) and (9.7) into Eq,(9.8) gives
G
L
LiL
GiG
k
k
CC
CC
(6.9)
Which is equal to the slope of the curve connecting the
(CL,CG) and (CLi,CGi),as shown in Figure 9.3.
It is hard to determine the mass-transfer coefficient
according to Eq.(6.6) or Eq.(6.7) because we cannot
measure the interfacial concentrations,CLi or CGi.
Therefore,it is convenient to define the overall mass-
transfer coefficient as follows:
LLLGGGLG CCKCCKNN(6.10)
Where CG* is the gas-side concentration which would be
in equilibrium with the existing liquid phase
concentration,Similarly,CL* is the liquid-phase
concentration,These can be easily read from the
equilibrium curve as shown in Figure 9.4,The newly
defined KG and KL are overall mass-transfer coefficients
for the gas and liquid sides,respectively.
Example 6.2
Derive the relationship between the overall mass-transfer
coefficient for liquid phase KL and the individual mass-
transfer coefficients,kL and kG,How can this relationship
be simplified for sparingly soluble gases?
Solution:
According to Eqs.(6.7) and (6.10),
LLLLLiL CCKCCk
(6.11)
Therefore,by rearranging Eq,(6.11)
LLi
LiL
LL
LLi
LiLLLi
L
LLi
LL
LL
CC
CC
kk
CC
CCCC
k
CC
CC
kK
11
1
11
(6.12)
Since
GiGGLLiL CCkCCk (6.13)
By substituting Eq.(6.13) to Eq.(6.12),we obtain
and
Further the correlation as follows can be obtained:
(6.14)
MkKCC
CC
kkK GLGiG
LL
GLG
11111 *
LGG HkkK
111
LGL kk
H
K
11
Which is the relationship between KL,kL,and kG,M is
the slope of the line connection (CLi,CGi) and (CL*,CG) as
shown in Figure 6.4.
For sparingly soluble gases,the slope of the equilibrium
curve is very steep; therefore,M is much greater than 1
and from Eq.(6.14)
LL kK?
(6.15)
Similarly,for the soluble gases,gas-phase mass-
transfer coefficient,
GG kK?
(6.16)
Which H is the Henry
coefficient.LG HKK?
6.2.3 Mechanism of Mass Transfer
Several different mechanisms have been proposed to
provide a basis for a theory of interphase mass transfer.
The three best known are the two-film theory,the
penetration theory,and the surface renewal theory.
The two-film theory supposes that the entire resistance
to transfer is contained in two fictitious films on either
side of the interface,in which transfer occurs by
molecular diffusion,This model leads to the conclusion
that the mass-transfer coefficient kL is proportional to the
film thickness zf as
zf
Dk AB
l?
(6.17)
Penetration theory (Higbie,1935) assumes that
turbulent eddies travel from the bulk of the phase to the
interface where they remain for a constant exposure
time te,The solute is assumed to penetrate into a given
eddy during its stay at the interface by a process of
unsteady-state molecular diffusion,This model predicts
that the mass-transfer coefficient is directly
proportional to the square root of molecular diffusivity
21
2
e
AB
L t
D
k
(6.18)
Surface renewal theory (Danckwerts,1951) proposes
that there is an infinite range of ages for elements of the
surface and the surface age distribution function Φ (t)
can be expressed as
ststt
(6.19)
Where s is the fractional rate of surface renewal,This
theory predicts that again the mass-transfer coefficient
is proportional to the square root of the molecular
diffusivity
21ABL sDk?
(6.20)
All these theories require knowledge of one unknown
parameter,the effective film thickness zf,the exposure
time te,or the fractional rate of surface renewal s,Little
is known about these properties,so as theories help us to
visualize the mechanism of mass transfer at the interface
and also to know the exponential dependency of
molecular diffusivity on the mass-transfer coefficient.
6.3 Correlation for Mass-Transfer Coefficient
Mass-transfer coefficient is a function of physical
properties and vessel geometry,Because of the
complexity of hydrodynamics in multiphase mixing,it
is difficult,if not impossible,to derive a useful
correlation based on a purely theoretical basis,It is.
common to obtain an empirical correlation for the mass-
transfer coefficient by fitting experimental data,The
correlations are usually expressed by dimensionless
groups since they are dimensionally consistent and also
useful for scale-up processes.
In mass transfer coefficient correlations of gas-liquid
dispersion,the following dimensionless groups are
often employed.
AB
32
Sh
t r a n s f e rm a s s d i f f u s i v e
t r a n s f e rm a s s t o t a l
Nn u m b e r,S h e r w o o d
D
Dk L
(6.21)
2
c
c
3
32
Gr
c
c
SC
f o r c e s v i s c ou s
f o r c e sn g r a v i t a t i o
Nn u m b e r,G r a s h o f
yd i f f u s i v i t m a s s
yd i f f u s i v i t m o m e n t u m
Nn u m b e r,Sc h m i d t
gD
D
AB
(6.22)
(6.23)
Where μ c and ρ c are the viscosity and density of the
continuous phase (liquid phase),respectively,Sauter-
mean diameter D32 can be calculated from measured
drop-size distribution from the following relationship,
2n
1i
3n
1i
32
ii
ii
Dn
Dn
D
(6.24)
Earlier studies in mass transfer between the gas-liquid
phase reported the volumetric mass-transfer
coefficient kLa,Since kLa is the combination of two
experimental parameters,mass-transfer coefficient and
interfacial area,it is difficult to identify which
parameter is responsible for the change of kLa when
we change the operating condition of a fermenter.
Calderbank and Moo-Young (1961) separated kLa by
measuring interfacial area and correlated mass-transfer
coefficients in gas-liquid dispersions in mixing vessels,
and sieve and sintered plate column,as follows:
1.For small bubbles less than 2.5mm in diameter,
3
1
2
C323.0
c
SCL
g
Nk
(6.25)
Or
313131.0
GrScSh NNN?
(6.26)
The more general forms which can be applied for both
small rigid sphere bubble and suspended solid particle are
3
1
2
C32
32
31.0
2
c
SC
AB
L
g
N
D
D
k
(6.27)
or
313131.00.2
GrScSh NNN
(6.28)
Eqs,(9.27) and (9.28) were confirmed by Calderbank
and Jones (1961),for mass transfer to and from
dispersions of low-density solid particles in agitated
liquids which were designed to simulate mass transfer
to microorganisms in fermenters.
2.For bubbles larger than 2.5 mm in diameter,
3
1
2
C2142.0
c
SCL
g
Nk
(6.29)
or
312142.0
GrScSh NNN?
(6.30)
Based on the three theories reviewed in the previous
section,the dependency of molecular diffusivity on the
mass-transfer coefficient is expected to be some value
between 1/2 and 1,It is interesting to note that the
exponential dependency of molecular diffusivity in the
preceding correlations is 2/3 or 1/2,which is within the
range predicted by the theories.
Example 6.3
Estimate the mass-transfer coefficient for the oxygen
dissolution in water 25℃ in a mixing vessel equipped
with flat-blade disk turbine and sparger by using
Calderbank and Moo-young’s correlations.
Solution:
The diffusivity of oxygen in water 25℃ is 2.5*10-9 m2/s
(Example 9.1),The viscosity and density of water at
25℃ is 8.904*10-4 kg/ms (CRC Handbook of Chemistry
and Physics,p,F-38,1983) and 997.08 kg/m3 (Perry and
Chilton,p,3-71,1973),respectively,The density of air
can be calculated from the ideal gas law,
3
3
5
k g / m 186.1
29810314.8
291001325.1?
RT
PM
a i r
Therefore the Schmidt number,
2.3 5 7105.208.9 9 7
109 0 4.8
9
4
AB
Sc DN
Substituting in Eq,(6.29) for small bubbles,
m / s1027.1
08.9 9 7
81.9109 0 4.81 8 6.108.9 9 7
2.3 5 731.0
4
31
2
4
32
Lk
m / s1058.4
08.9 9 7
81.9109 0 4.81 8 6.108.9 9 7
2.3 5 742.0
4
31
2
4
21
Lk
Therefore,for the air-water system,Eqs,(6.29) predict that
the mass-transfer coefficients for small and large bubbles
are 1.27*10-4 and 4.58*10-4 m/s,respectively,which are
independent of power consumption and gas-flow rate.
Eq,(6.29) predicts that the mass-transfer coefficient for
the oxygen dissolution in water 25℃ in a mixing vessel is
4.58*10-4m/s,regardless of the power consumption and
gas-flow rate as illustrated in the previous example
problem,Lopes De Figueiredo and Calderbank (1978)
reported later that the value of kL varies from 7.3*10-4 to
3.4*10-3 m/s,depending on the power dissipation by
impeller per unit volume (Pm/τ ) as
33.0
mL
P
k
(6.31)
This dependence of kL on Pm/v was also reported by
Prasher and Wills (1973) base on the absorption of
CO2 into water in an agitated vessel,as follows:
25.0
c
21592.0
m
ABL
P
Dk (6.32)
Which is dimensionally consistent,However,this
equation is limited in its use because the correlation is
based on only one gas-liquid system.
Akita and Yoshida (1974) evaluted the liquid-phase
mass-phase mass-transfer coefficient based on the
oxygen absorption into several liquids of different
physical properties using bubble columns without
mechanical agitation,Their correlation for kL is
83
C
3
32
41
2
C
3
32
21
AB
c
AB
32L 5.0
gDgD
DD
Dk (6.33)
Where vc and σ are the kinematic viscosity of the
continuous phase and interfacial tension,respectively,
Eq,(9.33) is applicable for column diameters of up to
0.6 m,superficial gas velocities up to 25 m/s,and gas
hold-up to 30 percent,
6.4 Measurement of Interfacial Area
To calculate the gas absorption rate qL for Eq.(6.7),
we need to know the gas-liquid interfacial area,which
can be measured employing several techniques such
as photography,light transmission,and later optics,
The interfacial area per unit volume can be calculated
from the Sauter-mean diameter D32 and the volume
fraction of gas-phase H,as follows:
32
6
D
a
(6.34)
参见戚以政书 (6-49)
6.5 Correlations for α and D32
6.5.1 Gas Sparging with No Mechanical Agitation
Leaving the vicinity of a sparger,the bubbles may break
up or coalesce with others until an equilibrium size
distribution is reached,A stable size is achieved when
turbulent fluctuations and surface tension forces are in
balance (Calderbank,1959),Akita and Yoshida (1974)
determined the bubble-size distribution in bubble columns
using a photographic technique,The gas was sparged
through perforated plates and single orifices,while various
liquids were used,The following correlation was proposed
12.012.0
2
250.02
32 26
C
S
L
CCC
C gD
VgDgD
D
D (6.36)
and for the interfacial area
13.1
1.0
2
350.02
3
1
H
gDgD
aD
L
CLC
C
(6.37)
Where DC is bubble column diameter and Vs is
superfical gas velocity,which is gas flow rate Q divided
by the tank cross-sectional area,Eqs.(6.36) and (6.37)
are based on data in columns of up to 0.3m in diameter
for the Sauter-mean diameter:
and up to superficeal gas velocities of about 0.07 m/s.
6.5.2 Gas Sparging with Mechanical Agitation
Calderbank (1958) correlated the interfacial areas for
the gas-liquid dispersion agitated by a flat-blade disk
turbine as follows:
1.For Vs <0.02 m/s,
21
6.0
2.04.0
44.1
t
scm
V
VP
a
(6.38)
for
2 0,0 0 0
3.0
7.0
Re
s
I
i V
ND
N
Where NRei is the impeller Reynolds number defined as
c
cI NDN
i?
2
Re?
(6.39)
The interfacial area for NRei0.7(NDI/Vs)0.3>20,000 can
be calculated from the interfacial area α 0 obtained
from Eq,(6.38) by using the following relationship.
3.0
7.0
Re
5
0
10 1095.1
3.2
l o g
S
I
V
ND
N
a
a
i
(6.40)
2.For Vs>0.02 m/s,Miller (1974) modified Eq.(6.38)
by replacing the aerated power input by mechanical
agitation Pm with the effective power input Pe,and the
terminal velocity Vt with the sum of the superficial gas
velocity and the terminal velocity Vs+Vt,The effective
power input Pe combines both gas sparging and
mechanical agitation energy contributions,The modified
equation is
21
6.0
2.04.0
0 44.1
St
Sce
VV
VP
a
(6.41)
Calderbank (1958) also correlated the Sauter-mean
diameter for the gas liquid dispersion agitated by a flat-
blade disk turbine impeller as follows:
1.For dispersion of air in pure water,
45.0
2.04.0
6.0
32 100.915.4
cvmP
D
(6.42)
2.In electrolyte solutions (NaCl,Na2SO4,and Na3PO4),
25.0
4.0
2.04.0
6.0
32 25.2
c
d
cvmP
D
(6.43)
3.In alcoholic solution (aliphatic alcohols),
25.0
65.0
2.04.0
6.0
32 90.1
c
d
cvmP
D
(6.44)
where all constants are dimensionless except 9.0*10-4m
in Eq,(6.42),Again the preceding three equations can
be modified for high gas flow rate (Vs>0.02m/s) by
replacing Pm by Pe,as suggested by Miller (1974).
6.6 Gas Hold-Up εor (H)
gas hold-up is one of the most important parameters
characterizing the hydrodynamics in a fermenter,Gas
hold-up depends mainly on the superficial gas velocity
and the power consumption,and often is very sensitive to
the physical properties of the liquid,Gas hold-up can be
determined easily by measuring level of the aerated liquid
during operation ZF and that of clear liquid ZL,Thus,the
average fractional gas hold-up H( ε) is give as
tS
S
VV
V
(6.45)
Gas Sparging with No Mechanical Agitation,In a two-
phase system where the continuous phase remains in
place,the hold-up is related to superficial gas velocity Vs
and bubble rise velocity Vt (Sridhar and Potter,1980):
F
LF
Z
ZZ?
(6.46)
Akita and Yoshida (1973) correlated the gas hold-up for
the absorption of oxygen in various aqueous solutions in
bubble columns,as follows:
C
SCCC
gD
VgDgD
1212812
4 2.01
(6.47)
Gas Sparging with Mechanical Agitation,Calderbank
(1958) correlated gas hold-up for the gas-liquid
dispersion agitated by a flat-blade disk turbine impeller as
follows:
21
6.0
2.04.0
4
21
1016.2
t
Scvm
t
S
V
VP
V
V
(6.48)
(87)
Where 2.16*10-4 has a unit (m) and Vt=0.265 m/s when
the bubble size is in the range of 2-5 mm diameter,The
preceding equation can be obtained by combining Eqs.
(6.38) and (6.42) by means of Eq,(6.34).
For high superficial gas velocities (Vs >0.02m/s),replace
Pm and Vt of Eq,(9.48) with effective power input Pe and
Vt+Vs,respectively (Miller,1974).
5.3 Gas Flow Effects on Bubble Swarms
Two regions in any gas sparged tank need to be
considered:
5.3.1 Bubble Sizes Generated at Orifice
when Re0 < 2000,0.1< dB < 1.0cm
dB =0.18d01/2Re01/3 (6.49)
The following equation is recommended for design
purposes:
=3.23Re0L-0.1Fr00.21 (6.50)
where Re0L=,Fr0= (6.51)0
d
dB
0
4
d
Q
L
L
gd
Q
5
0
2
When Re0 > 10000,bubble size in equilibrium only
relate to Re0:
dBe = 0.71Re0-0.05 (dBe in cm ) (6.52)
5.3.2 Bubble Size Far from the Orifice
The bubble size may vary,depending on the liquid
properties and the liquid motions generated by the rising
gas stream,It is very complicated,An energy balance on
the gas and liquid phases can be used to determine the
liquid velocity.
5.4 Bubble Coalescence and Break-up
Bubble Coalescence rising in a line takes place in several
stages:
A.the approach of the
following bubbles to the
leading bubble,
B,the trailing bubble moves
in the vortex of the leading
bubble until the bubbles are
separated only by a thin
interface.
C,final thinning and rupture
of the film between bubbles.
In gas-sparged vessels the power per unit volume can be
found form an energy balance on the gas phase
Fig.6.1 Bubble Coalescence
The kinetic energy of the gas leaving the vessel is generally
negligible,and the kinetic energy term is generally small for
most values of Q,Hence,the equation reduces to,
= ln (6.53)
The resulting mean bubble size can be found by inserting this
value of P/V into
dBe = 0.7
(dBe in m) (6.54)
The equilibrium size can predicted by
dBe = 0.71Re0-0.05 (dBe in cm) (6.55)
1.0
2.04.0
6.0
)(
)( G
a
LV
P?
)(
21
2
PP
PgH
V
Q
2
1
P
P
V
P
)(
21
2
PP
P
V
Q
V
P
ln
V
VgH
P
P 1
2
2
0
2
1
6.7 Power Consumption
The power consumption for mechanical agitation can be
measured using a torque table as shown in Figure 6.7,The
torque table is constructed by placing a thrust bearing
between two circular plates,and the force required to
prevent rotation of the turntable during agitation F is
measured,The power consumption P can be calculated
by the following formula
NFP 2
Where N is the agitation speed,and r is the distance
from the axis to the point of the force measurement.
Power consumption by agitation is a function of
physical properties,operating condition,and vessel and
impeller geometry,Dimensional analysis provides the
following relationship:
(6.56)
r
bearing
gauge
,,,,,
22
53
I
W
II
II
I D
D
D
H
D
Dr
g
DNNDf
DN
P (6.57)
The dimensionless group in the left-hand side of Eq,(6.57)
is known as power number NP,which is the ratio of drag
force on impeller to inertial force,The first term of the
right-hand side of Eq.(6.57) is the impeller Reynolds
number NRei,which is the ratio of inertial force to viscous
force,and the second term is the Froude number NFr,
which takes into account gravity forces,The gravity force
affects the power consumption due to the formation of the
vortex in an agitating vessel,The vortex formation can be
prevented by installing baffles.
PN
For fully baffled geometrically similar systems,the
effect of the Froude number on the power consumption
is negligible and all the length ratios in Eq.(6.57) are
constant,Therefore,Eq,(6.57) is simplified to
i
NN P Re
For the normal operating condition of gas-liquid contact,
the Reynolds number is usually larger than 10000,For
example,for a 3-inch impeller with an agitation speed of
150 rpm,the impeller Reynolds number is 16225 when
the liquid is water,Therefore,Eq,(6.58) is simplified to
10 00 0f or c o n s t a n t Re iNN P
(6.58)
(6.59)
参见戚以政书 P312-313,(6-61) –(6-66)
The power required by an impeller in a gas sparged
system Pm is usually less than the power required by the
impeller operating at the same speed in a gas-free liquids
Pmo,The Pm for the flat-blade disk turbine can be
calculated from Pmo,The Pm for the flat-blade disk turbine
can be calculated from Pmo (Nagata,1975),as follows:
3
96.121 1 5.0238.4
10 1 9 2l o g
I
D
D
II
T
I
mo
m
ND
Q
g
NDND
D
D
p
P T
I
Example 6.4
A cylindrical tank (1.22m diameter) is filled with water
to an operating level equal to the tank diameter,The tank
is equipped with four equally spaced baffles whose width
is 1/10 of the tank diameter,The tank is agitated
参见 P314-315,(6-71) –(6-76) (6.60)
with a 0.36 m diameter flat six-blade disk turbine,The
impeller rotational speed is 2.8 rps,The air enters through
an open-ended tube situated below the impeller and its
volumetric flow rate is 0.00416 m3/s at 1.08 atm and 25℃,
Calculate the following properties and compare the
calculated values with those experimental data reported
by Chandrasekharan and Calderbank (1981),Pm=697W;
=0.02; kLa=0.0217s-1.
a.Power requirement Pm
b.Gas hold-up ε
c.Sauter-mean diameter D32
d.Interfacial area a
e.Volumetric mass-transfer coefficient kLa
Solution,
Power requirement,The viscosity and density of water
at 25 ℃ is 8.904*10-4kg/ms (CRC Handbook of
Chemistry and Physics,p.F-38,1983) and 997.08 kg/m3
(Perry and Chilton,p.3-71,1973),respectively,Therefore,
the Reynolds number is
4 0 6 3 5 7
109 0 4.8
36.08.208.9 9 7
4
22
Re
INDN
Which is much larger than 10000,above which the
power number is constant at 6,Thus,
W7 9 436.08.208.9 9 766 5353 Imo DNP
I
mo
P DN
PN
53 1
10
102
1 10 102 103 104 105 106 INDN
2
Re?
Fig.6.8
(参见图 6-19)
3
22.1
36.096.1
21 1 5.0
7
238.4
10 36.08.2
0 0 4 1 6.0
81.9
8.236.0
1093.8
8.236.0
22.1
36.01 9 2
7 9 4
l o g Pm
Therefore,
W6 8 7?mP
b.Gas hold-up,The interfacial tension for the air-water
interface is 0.07197kg/s2 (CRC Handbook of Chemistry
and Physics,p,F-33,1983),The volume of the dispersion is
32 m43.11,2 222.14
The superficial gas velocity is
m / s0 0 3 5 6.0
22.1
0 0 4 1 6.044
22
T
s D
QV
The power required in the gas-sparged system is from Eq,
(6.60)
Substituting these values into Eq.(6.48) gives
21
6.0
2.04.0
4
21
2 6 5.0
0 0 3 5 6.0
0 7 1 9 7.0
08.9 9 743.16 8 71016.2
2 6 5.0
0 0 3 5 6.0?
The solution of the preceding equation for ε gives
023.0
c,Sauter-mean dimeter,In Eq.(6.42)
mm9.300 36 6.0
100.902 3.0
08.99 743.168 7
07 19 7.0
15.4 45.0
2.04.0
6.0
32
D
d,Interfacial area α,In Eq.(6.34)
1
32
m7.37
0,0 0 3 6 6
0,0 2 36
D
6a
e.Volumetric mass-transfer coefficient,Since the average
size of bubbles is 4mm(＞ 2.5),we should use Eq.(6.29),
then,from Example 6.3
m / s1058.4 4Lk
Therefore,
-14 0.0 17 s37,71058.4Lak
The preceding estimated values compare well with
those experimental values,The percent errors as defined
in Example 9.1 are –1.4 percent for the power
consumption,15 % for the gas hold-up,and –21.7 % for
the volumetric mass-transfer coefficient.
6.8 Determination of Oxygen-Absorption Rate
To estimate the design parameters for oxygen uptake in a
fermenter,you can use the correlations presented in the
previous sections,which can be applicable to a wide
range of gas-liquid systems in addition to the air-water
system,However,the calculation procedure is lengthy
and the predicted value from those correlations can vary
widely,Sometimes,you may be unable to fimd suitable
correlations which will be applicable to your type and
size of fermenters,In such cases,you can measure the
oxygen-transfer rate yourself of use correlations base on
those experiments.
The oxygen absorption rate per unit volume qα/v can be
estimated by
LLLLLLa CCakCCaKq
(6.61)
Since the oxygen is sparingly soluble gas,the overall
mass-transfer coefficient KL is equal to the individual
mass-transfer coefficient kL,Our objective in fermenter
design is to maximaize the oxygen transfer rate with the
minimum power consumption necessary to agitate the
fluid,and also minimum air flow rate,To maximize the
oxygen absorption rate,we have to maximize kL,α,CL*-
CL,However,the concentration difference is quite
limited for us to control because the value of CL* is
limited by its very low maximum solubility,Therefore,
the main parameters of interest in design are the mass-
transfer coefficient and the interfacial area.
T e m p e r a t u r e
℃
S o l u b i l i t y
m m o l O
2
/l
Mg O
2
/l
0 2,1 8 6 9,8
10 1,7 0 5 4,5
15 1,5 4 4 9,3
20 1,3 8 4 4,2
25 1,2 6 4 0,3
30 1,1 6 3 7,1
35 1,0 9 3 4,9
40 1,0 3 3 3,0
Table 6.1 Solubility of Oxygen
in water at 1 atm.a
Table6.1 lists the solubility of oxygen at 1 atm in water at
various temperatures,The value is the maximum
concentration of oxygen in water when it is in
equilibrium with pure oxygen,This solubility decreases
with the addition of acid or salt as shown in Table 6.2.
C on c.
m ol / l
S ol ub il ity,m m ol O
2
/ l
H Cl H
2
SO
4
N aC l
0,0 1,2 6 1,2 6 1,2 6
0,5 1,2 1 1,2 1 1,0 7
1,0 1,1 6 1,1 2 0,8 9
2,0 1,1 2 1,0 2 0,7 1
Table 6.2,Solubility of Oxygen in Solution of
Salt or Acid at 25℃,a
Normally,we use air to supply the oxygen demand of
fermenters,The maximum concentration of oxygen in
water which is in equilibrium with air CL* at atmospheric
pressure is about one fifth of the solubility listed,
according to the Henry’s law,
THo
po
C L
2
2
(6.62)
Where pO2 is the partial pressure of oxygen and HO2 (T)
is Henry’s law constant of oxygen at a temperature,T,
The value of Henry’s law constant can be obtained
from the solubilities listed in Table6.1,For example,
at 25℃,CL* is 1.26mmol/l and pO2 is 1 atm because it
is pure oxygen,By substituting these values into
Eq,(6.55),we obtain HO2 (25℃ ) is 0.793 atm
l/mmol,Therefore,the equilibrium concentration
of oxygen for the air-water contact at 25℃ will
be
lmglom m o
m m oa t m l
a t m
C L
/43.8/l264.0
l/793.0
209.0
2
Ideally,oxygen-transfer rates should be measured in a
fermenter which contains the nutrient broth and
microorganisms during the actual fermentation process,
However,it is difficult to carry out such a task due to the
complicated nature of the medium and the ever changing
rheology during cell growth,A common strategy is to use
a synthetic system which approximates fermentaiton
conditions.
6.8.1 Sodium Sulfite Oxidation Method
The sodium sulfite oxidation method (Cooper et al.,
1944) is based on the oxidation of sodium sulfite in the
presence of catalyst (Cu++ or Co++) as
42
Coo r Cu
22132 SONaOSONa
(6.63)
This reaction has following characteristics to be qualified for
the measurement of the oxygen-transfer rate:
1.The rate of this reaction is independent of the
concentration of sodium sulfite within the range of 0.04 to 1
N.
2.The rate of reaction is much faster than the oxygen
transfer rate; therefore,the rate of oxidation is controlled
by the rate of mass transfer alone.
Example 6.5
To measure kLa,a fermenter was filled with 10l of 0.5M
sodium sulfite solution containing 0.003M Cu++ ion and
the air sparger was turned on,After exactly 10 minutes,
the air flow was stopped and a 10 ml sample was taken
and titrated,The concentration of the sodium sulfite in
the sample was found to be 0.21 mol/l,The experiment
was carried out at 25℃ and 1 atm,Calculate the oxygen
uptake and kLa.
Solution:
The amount of sodium sulfite reacted for 10 minutes is
lmo /l29.021.05.0
According to the stoichiometric relation,Eq,(6.63),the
amount of oxygen required to react 0.29mol/l is
lmo /l1 4 5.0
2
129.0
Therefore,the oxygen uptake is
lg
s
mogOlOm o l e /1073.7
6 0 0
l/32/ 1 4 5.0 32
2
The solubility of oxygen in equilibrium with air can be
estimated by Eq.(6.62) as
lg
g/mol
Omoa t m
TH
p
C L
/1043.8
32m o l 1l/ 7 9 3 a t m
a i r l209.0 1
3
2
0
O
2
2
Therefore,the value of kLa is,according to the Eq,(6.61),
1
3
3
9 1 7.0
0/1043.8
/1073.7?
s
lg
lsg
CC
q
k
LL
a
La
6.8.2 Dynamic Gassing-out Technique
This technique (Van’t Riet,1979) monitors the change of
the oxygen concentration while an oxygen-rich liquid is
deoxygenated by passing nitrogen through it.
Polarographic electrode is usually used to measure the
concentration,The mass balance in a vessel gives
tCCk
dt
tdC
LLLa
L
(6.64)
Intergration of the preceding equation between t1 and
t2 results in
12
2
1In
tt
tCC
tCC
k
LL
LL
La
(6.65)
From which kLa can be calculated based on the measured
values of CL(t1) and CL(t2).
6.8.3 Direct measurement
In this technique,we directly measure the oxygen content
of the gas stream entering and leaving the fermenter by
using gaseous oxygen analyzer,The oxygen uptake can
be calculated as
o u t,in,22 Oo u tOina CQCQq (6.66)
Where Q is the gas flow rate.
Once the oxygen uptake is measured,the kLa can be
calculated by using Eq.(9.54),where CL is the oxygen
concentration of the liquid in a fermenter and CL* is the
concentration of the oxygen which would be in
equilibrium with the gas stream,The oxygen concentration
of the liquid in a fermenter can be measured by an on-line
oxygen sensor,If the size of the fermenter is rather small
(less than 50l),the variation of the CL*-CL in the fermenter
is fairly small,However,if the size of a fermenter is very
large,the variation can be significant,In this case,the log-
mean value of CL*-CL of the inlet and outlet of the gas
stream can be used as
o u tLLinLL
o u tLLinLL
LMLL CCCCIn
CCCC
CC
/
(6.67)
6.8.4 Dynamic Technique
By using the dynamic technique (Taguchi and
Humphrey,1966),we can estimate the kLa value for the
oxygen transfer during an actual fermentaton process with
real culture medium and microorganisms,This technique
is based on the oxygen material balance in an aerated batch
fermenter while microorganisms are actively growing as
摄氧率,可在稳态下,由进气和排气中氧的分压求出 。
XOLLL CrCCk
dt
dC
2La
(6.68)
Xo Cr 2 0?
dt
dC L
Where rO2 is cell respiration rate [g O2/g cell h].
While the dissolved oxygen level of the fermenter is
steady,if you suddenly turn off the air supply,the
oxygen concentration will be decreased (Figure 6.9)
with the following rate
XO
L Cr
dt
dC
2?
(6.69)
Since kLa in Eq,(6.68) is equal to zero,Therefore,by
measuring the slope of the CL vs,T curve,we can
estimate rO2CX,If you turn on the air flow again,the
dissolved oxygen concentration will be increased
according to Eq,(6.68),which can be rearranged to result
in a linear relationship as
XO
L
LL Crdt
dC
k
CC 2
La
1
(6.70)
参见戚以政书 (6-58 )
air off
air on
dt
dCt
CL
t
Fig,6.9 Dynamic technique for the determination of kLa
( 参见戚以政书 P307 )
CL Lak1?
Xo
L Cr
dt
dC
2?
*LC
所谓动态法是根据停止向培养溶液中通气时，
培养液中溶氧浓度的变化率求出摄氧率，当溶氧浓度下降到一定程度时（不低于临界溶氧浓度），再恢复通气，则培养液中氧浓度又逐渐升高，直至恢复到原先的水平，见 Fig,
6.9 左图。右图中，截距为饱和溶氧浓度，斜率为 -
1/kLa.
溶氧法的优点是只需测定溶氧浓度 CL 随时间变化曲线，非常方便地求出 kLa.
当稳态时，
)( *2 LLLXo CCakCr
0dtdCOL
OLOL
o
La CC
r
k
* 2
The plot of CL versus (dCt/dt +ro2CX) will in a straight line
which has the slope of -1/(kLa) and the y-intercept of C*L,
6.9 Correlation for kLa
6.9.1 Bubble Column
Akita and Yoshida (1973) correlated the volumetric mass-
transfer coefficient kLa for the absorption of oxygen in
various aqueous solutions in bubble columns,as follows:
(参见戚书 P331)
1.193.017.0
62.0
12.05.0
6.0?
gD
c
Dk TcABLa
(6.71)
Which applies to columns with less effective
sparkers.
In bubble columns,for 0＜ Vs＜ 0.15m/s and100
＜ Pg/v＜ 1100W/m3,Botton et al,(1980) correlated
the kLa as
75.0
8 0 0
/
08.0?
vPk g
La
(6.72)
Where Pg is the gas power input,which can be
calculated from
75.0
1.0
800?
sg V
v
P (6.73)
6.9.2 Mechanically Agitated Vessel
For aerated mixing vessels in an aqueous solution,
the mass-transfer coefficient is proportional to the
power consumption (Lopes De Figueiredo and
Calderbank,1978)as
33.0/ vPk mL?
(6.74)
The interfacial area for the aerated mixing vessel
is a function of agitation conditions,Therefore,
according to Eq.(6.38),
5.0
4.0
s
m V
v
P
a?
Therefore,by combining Eqs,(6.74)and (6.75),kLa
will be
5.0
77.0
s
m
La V
v
P
k?
(6.75)
(6.76)
Numerous studies for the correlations of kLa have
been reported and their results have the general
form as
3
2
1
b
s
b
m
La V
v
P
bk?
(6.77)
Where b1,b2,and b3 vary considerably depending
on the geometry of the system,the range of
variables covered,and the experimental method
used,The values of b2and b3 are generally
between 0 to 1 and 0.43 to 0.95,respectively,as
tabulated by Sideman et al.(1966).
Van,t Riet (1979) reviewed the data obtained by
various investigators and correlated them as follows:
1,For,coalescing” air-water dispersion,
5.0
4.0
0 2 6.0 smLa V
v
P
k?
(6.78)
参见书 (6-85)
2,For,noncoalescing” air-electrolyte solution
dispersions,
2.0
7.0
0 0 2.0 smLa V
v
P
k?
(6.79)
Both of which are applicable for the volume up to
2.6m3;for a wide variety of agitator types,sizes,and
DI/DT ratios; and 500＜ P m/v＜ 10,000W/m3.These
correlations are accurate within approximately 20
percent to 40 percent.
Example 6.6
Estimate the volumetric mass-transfer coefficient kLa
for the gas-liquid contactor described in Example 6.4
by using the correlation for kLa in this section.
Solution:
For Example 6.4,the reactor volume v is 1.43m3,
the superficial gas velocity Vs is 0.00356 m/s,and
power consumption Pm is 687 W.By substituting
these values into Eq.(6.71),
15.0
4.0
0 1 8.00 0 3 5 6.0
43.1
6 8 70 2 6.0
sk
La
Problems
6.1 Derive the relationship between the overall mass
transfer coefficient for gas phase KG and the
individual mass-transfer coefficients,kL and kG.How
can this relationship be simplified for sparingly and
soluble gases
6.2 Prove that Eq.(6.25)is the same with Eq,(6.26),
and Eq.(6.27) is the same with Eq.(6.28).
6.3 The power consumption by impeller P in
geometrically similar fermenters is a function of the
diameter DI and speed N of impeller,density ρand
Viscosity μ of liquid,and acceleration due to gravity
g,Determine the functional relationship between
appropriate dimensionless parameters,which can
relate the power consumption,by applying
dimensional analysis using the Buckingham-Pi
theorem.
6.4 A cylindrical tank(1.22m diameter) is filled with
water to an operating level equal to the tank diameter,
The tank is equipped with four equally spaced
baffles,the width of which is one tenth of the tank
diameter,The tank is agitated with a 0.36m
diameter,flat-blade disk turbine,The impeller
Rotational speed is 4.43 rps,The air enters through
an open ended tube situated below the impeller and
its volumetric flow rate is 0.0217m3/s at 1.08 atm
and 25℃,Calculate:
a,Power requirement
b,Gas hold-up
c,Sauter-mean diameter
d,Interfacial area
e,Volumetric mass-transfer coefficient
Compare the preceding calculated results with those
Experimental values reported by
Chandrasekharan and Calderbank (1981),Pm =
2282W; ε = 0.086; kLa = 0.0823 s-1
6.5 Estimate the volumetric mass-transfer
coefficient kLa for the gas-liquid contactor
described in Problem 9.4 by using a correlation
for kLa and comp9are the result with the
experimental value.
6.6 The power consumption by an agitator in an
unbaffled vessel can be expressed as
2
53
I
I
mo NDf
DN
P
Can you determine the power consumption and
impeller speed of a 1,000gallon fermenter based on
findings of the optimum condition from a one-gallon
vessel by using the same fluid system? Is your
conclusion reasonable? Why or why not?
6.7 The optimum agitation speed for the cultivation
of plant cells in a 3-l fermenter equipped with four
baffles was found to be 150 rpm.
a,What should be the impeller speed of a
geometrically similar 1,000 -l fermenter if you
scale up based on the same power consumption per
unit volume.
b,When the impeller speed by part (a) was
employed for the cultivation of a 1,000 -l
fermenter,the cells do not seem to grow well due
to the high shear generated by the impeller even
though the impeller speed is lower than 150 rpm of
the model system,This may be due to the higher
impeller tip speed which is proportional to NDI,Is
this true? Justify your answer with the ratio of the
Impeller tip speed of the prototype to the model
fermenter.
c,If you use the impeller tip speed as the
criteria for the scale-up,what will be the impeller
speed of the prototype fermenter?
Transport Phenomena In
Biochemical Reactors
Mass Transfer Criteria for Simple
and Complex Systems
(1)
参考阅读：
,生化反应动力学与反应器,
戚以政 汪淑雄 编著
第 6章 生化反应器的传递过程
第 5章 生化反应器的设计与分析
第 7章 生化反应器的流动模型与放大
dy
dv x
yx
Chapter 5 Basic Concepts of Transport
Phenomena In Bio-engineering and Chemical
Engineering
5.1 Transport phenomena and reaction are the basic
phenomena in the nature,Chemical Engin,and Bio-
engin.
If there is velocity gradient,temperature gradient,or
concentration gradient,there must be momentum
transfer,heat transfer,and mass transfer,
For momentum transfer,there is Newton’s Law of
Viscosity as follows:
-----shear stress,-------viscosity of the fluid,
Momentum flux = - viscosity velocity gradient
Fluids that behave in this fashion are termed Newtonian
fluids,Fluids that do not obey this law are referred to as non-
Newtonian fluids,The subject of non-Newtonian flow is a
subdivision of Rheology.
For the heat transfer,there is Fourier’s Law of Heat
Conduction as follows:
q/A -----The local heat flow per unit area; heat flux
k ----- thermal conductivity.
Heat flux = thermal conductivity temperature gradient
dy
dTk
A
q
jA------the molar diffusion flux of A in a binary system;
DAB ----- mass diffusivity; ------- mass density;
Molar diffusion flux = mass diffusivity density gradient
Assumption of Constant C and DAB in a binary system with
chemical reaction,the molar diffusion differential equation:
RA ------ the molar rate of production A
dy
dDj A
ABA
For the mass transfer,there is Fick’s First Law:
A
AAA
AB
A R
z
c
y
c
x
cD
Dt
Dc?
)(
2
2
2
2
2
2
If the above situation without reaction in a stable system:
This is Fick’s second Law.
5.2 Importance of Study on Transport phenomena
There are microbes,substrate and metabolism products
effect on the viscosity of the bio-reaction systems,and then
effect on the momentum transfer,heat transfer,and mass
transfer,Further the transport factors effect on the bio-
reactions,design of reactors and general output of the
whole bio-process,
)( 2
2
2
2
2
2
z
c
y
c
x
cD
Dt
Dc AAA
AB
A
Viscosity
Fluid kinetics/power required
Heat transfer Mss transfer
Raw material feed
(Pumping,
heating,cooling
and mixing)
Fermentation
(cells dispersion,
oxygen dissolved,
temperature )
Purification and
recovery of
products
(Pumping,
heating,cooling
and separation)
Cells growth products formed morphology
Viscosity
Design and economics
5.3 Rheological Propertics of The Process Materials
(1) Pure viscous fluids,
a)Newtonian fluids
= (=constant),shear stress,N/m2
b)non-Newtonian fluids,shear rate,s-1
= ( constant),viscosity,kg/ms
(2) viscoelastic fluids
= f (,extent of deformation)
Most non-Newtonian fluids follows the power-law model
= K( )n
dilatant
Newtonian
pseudoplastic
plastics
Bingham
Fig.5.1 General shear behavior of
rheologically time-independent fluid
classes
5.4 Basic Dispersion Concepts
The oxygen transfer rate from the gas bubble to the medium is
largely determined by kL and the interfacial area a,
Main variables which influence a,
bubble size dB,the terminal velocity of the bubble UB
and the gas hold-up?
Basic correlation,
For small,rigid interface bubbles follow Stokes equation:
UB= dB2 (5-1)
which is valid for Re 1.
For mobile interface bubbles,
UB= dB2 (5-2)
At higher bubble Reynolds numbers:
UB= (5-3A)
When the gravity stresses are higher than the surface
tension stresses,UB= (5-3B)
18
g
16
g
2
2 B
B
gd
d
2
Bgd
Chapter 6 Gas-Liquid Mass Transfer
6.1 Oxygen Transfer in Metabolism Process of
Microorganism
Oxygen solubility in a fermentation medium is very
low,while its demand for the growth of aerobic
microorganisms is high,
For example,when the oxygen is provided from air,the
typical maximum concentration of oxygen in aqueous
solution is on the order of 6 to 8 mg /L.Oxygen
requirement of cells is,although it can vary widely
depending on microorganisms,on the order of 1 g /L h,
Even though a fermentation medium is fully
saturated with oxygen,the dissolved oxygen will be
consumed in less than one minute by organisms if
not provided continuously,Adequate oxygen supply
to cells is often critical in aerobic fermentation,
Even temporary depletion of oxygen can damage
cells irreversibly.Therefore,gaseous oxygen must be
supplied continuously to meet the requirements for
high oxygen needs of microorganisms,and the
oxygen transfer can be a major limiting step for cell
growth and metabolism,
Mass-Transfer Path:The path of gaseous substrate
from a gas bubble to an organelle in a
microorganism can be divided into several steps
(Figure 6.1),as follows:
(1)Transfer from bulk gas in a bubble to a
relatively unmixed gas layer
(2)Diffusion through the relatively unmixed gas
layer
(3) Diffusion through the relatively unmixed liquid
layer surrounding the bubble
(4)Transfer from the relatively unmixed liquid
layer to the bulk liquid
(5)Transfer from the bulk liquid to the relatively
unmixed liquid layer surrounding a microorganism
(6)Diffusion through the relatively unmixed liquid
layer
(7)Diffusion from the surface of a microorganism to
an organelle in which oxygen is consumed
4
1
Gas bubble Microorganism
Fig.6.1 Schematic diagram of the path of a gaseous
substrate to an organenelle in a cell
2
3 5
6
6.2 Basic Mass-Transfer Concepts
6.2.1 Molecular Diffusion in Liquids
When the concentration of a component varies from one
point to another,the component has a tendency to flow in
the direction that will reduce the local differences in
concentration,Molar flux of a component A relative to the
average molal velocity of all constituent JA is proportional
to the concentration gradient dCA/dz as
Which is Fick’s first law written for the z-direction,The
DAB in Eq,(9.1) is the diffusivity of
Z
A
ABA d
dCDJ
(6.1)
component A through B,which is a measure of its
diffusive mobility,Molar flux relative to stationary
coordinate NA is equal to
Z
A
ABBA
A
A D
dCDNN
C
CN
(6.2)
Where C is total concentration of components A and B
and NB is the molar flux of B relative to stationary
coordinate,The first term of the right hand side of
Eq.(9.2) is the flux due to bulk flow,and the second
term is due to the diffusion,For dilute solution of A,
AA JN?
(6.3)
Diffusivity,The kinetic theory of liquids is much less
advanced than that of gases,Therefore,the correlation for
diffusivities in liquids is not as reliable as that for gases.
Among several correlations reported,the Wilke-Chang
correlation (Wilke and Chang,1955) is the most widely
used for dilute solutions of nonelectrolytes,( or see
戚以政 ‘ book 6-47)
6.01.1
5.016101 7 3.1
bA
B
AB V
TMD
ξ?
(6.4)
When the solvent is water,and the solute is small
molecular substance,DO2 is about 0.5～ 2.0× 10-5cm2/s,
Skelland(1974) recommends the use of the correlation
developed by Othmer and Thakar(1953),
6.01.1
13101 1 2.1
bA
AB VD?
(6.5)
The preceding two correlations are not dimensionally
consistent; therefore,the equations are for use with the
units of each term as SI unit as follows:
ABD
=diffusivity of A in B,in a very dilute solution,m2/s
MB=molecular weight of component B,kg/kmol
T=temperature,K
μ=solution viscosity,kg/m s
VbA=solute molecular volume at normal boiling point,
m3/kmol:0.0256m3/kmol for oxygen [See Perry and Chil-
ton (p.3-233,1973)for extensive table]
ξ=association factor for the solvent,2.26 for water,1.9 for
methanol,1.5 for ethanol,1.0 for unassociated solvents,
such as benzene and ethyl ether
Example 6.1
Estimate the diffusivity for oxygen in water at
25℃,Compare the predictions from the Wilke-Chang and
Othmer-Thakar correlations with the experimental value
of 2.5*10-9 m2/s (Perry and Chilton,p,3-225,1973).
Convert the experimental value to that corresponding to a
temperature of 40℃,
Solution:
Oxygen is designated as component A,and water,
component B,The molecular volume of oxygen VbA is
0.0256 m3/kmol,The association factor for water ξ is 2.26,
The viscosity of water at 25℃ is 8.904*10-4kg/ms (CSC
Handbook of chemistry and Physics,p,F-38,1983),In
Eq.(6.4)
smD AB /1025.20 2 5 6.0109 0 4.8
2 9 81826.2101 7 3.1 29
6.04
5.016
In Eq.(6.5)
smD AB /1027.20 2 5 6.010904.8
10112.1 29
6.01.14
13
If we define the error between these predictions and the
experimental value as
1 0 0% ale x p e r i m e n t
ale x p e r i m e n tp r e d i c t e d
AB
ABAB
D
DD
e r r o r
The resulting errors are –9.6 percent and –9.2 percent for
Eqs,(6.4) and (6.5),respectively,Since the estimated
possible error for the experimental values from both
equations are satisfactory.
Eq,(6,4) suggests that the quantity DoABμ/T is constant
for a given liquid system,Though this is an
approximation,we may use it here to estimate the
diffusivity at 40℃,
C40a t /10529.6 4?mskg
smD AB /1058.3
2 9 8
3 1 3
105 2 9.6
109 0 4.8102,5C40a t 29
4
4
9-?
If we Eq,(6.5),DABoμ1.1 is constant,
smD AB /1052.3
10529.6
10904.8102,5C40a t 29
1.1
4
4
9-?
CGi
CLi
CL
CG
LiquidGas
CL CLi
CG
CGi
G
Lkk?
6.2.2 Mass-Transfer Cofficient
The mass flux,the rate of mass transfer qG per unit area,
is proportional to a concentration difference,If a solute
transfers from the gas to the liquid phase,its mass flux
from the gas phase to the interface NG is
GiGGG CCk
A
qGN (6.6)
Where CG and CGi is the gas-side concentration at the
bulk and the interface,respectively,as shown in Figure
6.3,kG is the individual mass-transfer coefficient for the
gas phase and A is the interfacial area.
Similarly,the liquid-side phase mass flux NL is
LLiLL CCk
A
qLN (6.7)
Where kL is the amount of solute transferred from the gas
phase to the interface must equal that from the gas phase
to the interface must equal that from the interface to the
liquid phase,
LG NN?
(6.8)
Substitution of Eqs.(9.6) and (9.7) into Eq,(9.8) gives
G
L
LiL
GiG
k
k
CC
CC
(6.9)
Which is equal to the slope of the curve connecting the
(CL,CG) and (CLi,CGi),as shown in Figure 9.3.
It is hard to determine the mass-transfer coefficient
according to Eq.(6.6) or Eq.(6.7) because we cannot
measure the interfacial concentrations,CLi or CGi.
Therefore,it is convenient to define the overall mass-
transfer coefficient as follows:
LLLGGGLG CCKCCKNN(6.10)
Where CG* is the gas-side concentration which would be
in equilibrium with the existing liquid phase
concentration,Similarly,CL* is the liquid-phase
concentration,These can be easily read from the
equilibrium curve as shown in Figure 9.4,The newly
defined KG and KL are overall mass-transfer coefficients
for the gas and liquid sides,respectively.
Example 6.2
Derive the relationship between the overall mass-transfer
coefficient for liquid phase KL and the individual mass-
transfer coefficients,kL and kG,How can this relationship
be simplified for sparingly soluble gases?
Solution:
According to Eqs.(6.7) and (6.10),
LLLLLiL CCKCCk
(6.11)
Therefore,by rearranging Eq,(6.11)
LLi
LiL
LL
LLi
LiLLLi
L
LLi
LL
LL
CC
CC
kk
CC
CCCC
k
CC
CC
kK
11
1
11
(6.12)
Since
GiGGLLiL CCkCCk (6.13)
By substituting Eq.(6.13) to Eq.(6.12),we obtain
and
Further the correlation as follows can be obtained:
(6.14)
MkKCC
CC
kkK GLGiG
LL
GLG
11111 *
LGG HkkK
111
LGL kk
H
K
11
Which is the relationship between KL,kL,and kG,M is
the slope of the line connection (CLi,CGi) and (CL*,CG) as
shown in Figure 6.4.
For sparingly soluble gases,the slope of the equilibrium
curve is very steep; therefore,M is much greater than 1
and from Eq.(6.14)
LL kK?
(6.15)
Similarly,for the soluble gases,gas-phase mass-
transfer coefficient,
GG kK?
(6.16)
Which H is the Henry
coefficient.LG HKK?
6.2.3 Mechanism of Mass Transfer
Several different mechanisms have been proposed to
provide a basis for a theory of interphase mass transfer.
The three best known are the two-film theory,the
penetration theory,and the surface renewal theory.
The two-film theory supposes that the entire resistance
to transfer is contained in two fictitious films on either
side of the interface,in which transfer occurs by
molecular diffusion,This model leads to the conclusion
that the mass-transfer coefficient kL is proportional to the
film thickness zf as
zf
Dk AB
l?
(6.17)
Penetration theory (Higbie,1935) assumes that
turbulent eddies travel from the bulk of the phase to the
interface where they remain for a constant exposure
time te,The solute is assumed to penetrate into a given
eddy during its stay at the interface by a process of
unsteady-state molecular diffusion,This model predicts
that the mass-transfer coefficient is directly
proportional to the square root of molecular diffusivity
21
2
e
AB
L t
D
k
(6.18)
Surface renewal theory (Danckwerts,1951) proposes
that there is an infinite range of ages for elements of the
surface and the surface age distribution function Φ (t)
can be expressed as
ststt
(6.19)
Where s is the fractional rate of surface renewal,This
theory predicts that again the mass-transfer coefficient
is proportional to the square root of the molecular
diffusivity
21ABL sDk?
(6.20)
All these theories require knowledge of one unknown
parameter,the effective film thickness zf,the exposure
time te,or the fractional rate of surface renewal s,Little
is known about these properties,so as theories help us to
visualize the mechanism of mass transfer at the interface
and also to know the exponential dependency of
molecular diffusivity on the mass-transfer coefficient.
6.3 Correlation for Mass-Transfer Coefficient
Mass-transfer coefficient is a function of physical
properties and vessel geometry,Because of the
complexity of hydrodynamics in multiphase mixing,it
is difficult,if not impossible,to derive a useful
correlation based on a purely theoretical basis,It is.
common to obtain an empirical correlation for the mass-
transfer coefficient by fitting experimental data,The
correlations are usually expressed by dimensionless
groups since they are dimensionally consistent and also
useful for scale-up processes.
In mass transfer coefficient correlations of gas-liquid
dispersion,the following dimensionless groups are
often employed.
AB
32
Sh
t r a n s f e rm a s s d i f f u s i v e
t r a n s f e rm a s s t o t a l
Nn u m b e r,S h e r w o o d
D
Dk L
(6.21)
2
c
c
3
32
Gr
c
c
SC
f o r c e s v i s c ou s
f o r c e sn g r a v i t a t i o
Nn u m b e r,G r a s h o f
yd i f f u s i v i t m a s s
yd i f f u s i v i t m o m e n t u m
Nn u m b e r,Sc h m i d t
gD
D
AB
(6.22)
(6.23)
Where μ c and ρ c are the viscosity and density of the
continuous phase (liquid phase),respectively,Sauter-
mean diameter D32 can be calculated from measured
drop-size distribution from the following relationship,
2n
1i
3n
1i
32
ii
ii
Dn
Dn
D
(6.24)
Earlier studies in mass transfer between the gas-liquid
phase reported the volumetric mass-transfer
coefficient kLa,Since kLa is the combination of two
experimental parameters,mass-transfer coefficient and
interfacial area,it is difficult to identify which
parameter is responsible for the change of kLa when
we change the operating condition of a fermenter.
Calderbank and Moo-Young (1961) separated kLa by
measuring interfacial area and correlated mass-transfer
coefficients in gas-liquid dispersions in mixing vessels,
and sieve and sintered plate column,as follows:
1.For small bubbles less than 2.5mm in diameter,
3
1
2
C323.0
c
SCL
g
Nk
(6.25)
Or
313131.0
GrScSh NNN?
(6.26)
The more general forms which can be applied for both
small rigid sphere bubble and suspended solid particle are
3
1
2
C32
32
31.0
2
c
SC
AB
L
g
N
D
D
k
(6.27)
or
313131.00.2
GrScSh NNN
(6.28)
Eqs,(9.27) and (9.28) were confirmed by Calderbank
and Jones (1961),for mass transfer to and from
dispersions of low-density solid particles in agitated
liquids which were designed to simulate mass transfer
to microorganisms in fermenters.
2.For bubbles larger than 2.5 mm in diameter,
3
1
2
C2142.0
c
SCL
g
Nk
(6.29)
or
312142.0
GrScSh NNN?
(6.30)
Based on the three theories reviewed in the previous
section,the dependency of molecular diffusivity on the
mass-transfer coefficient is expected to be some value
between 1/2 and 1,It is interesting to note that the
exponential dependency of molecular diffusivity in the
preceding correlations is 2/3 or 1/2,which is within the
range predicted by the theories.
Example 6.3
Estimate the mass-transfer coefficient for the oxygen
dissolution in water 25℃ in a mixing vessel equipped
with flat-blade disk turbine and sparger by using
Calderbank and Moo-young’s correlations.
Solution:
The diffusivity of oxygen in water 25℃ is 2.5*10-9 m2/s
(Example 9.1),The viscosity and density of water at
25℃ is 8.904*10-4 kg/ms (CRC Handbook of Chemistry
and Physics,p,F-38,1983) and 997.08 kg/m3 (Perry and
Chilton,p,3-71,1973),respectively,The density of air
can be calculated from the ideal gas law,
3
3
5
k g / m 186.1
29810314.8
291001325.1?
RT
PM
a i r
Therefore the Schmidt number,
2.3 5 7105.208.9 9 7
109 0 4.8
9
4
AB
Sc DN
Substituting in Eq,(6.29) for small bubbles,
m / s1027.1
08.9 9 7
81.9109 0 4.81 8 6.108.9 9 7
2.3 5 731.0
4
31
2
4
32
Lk
m / s1058.4
08.9 9 7
81.9109 0 4.81 8 6.108.9 9 7
2.3 5 742.0
4
31
2
4
21
Lk
Therefore,for the air-water system,Eqs,(6.29) predict that
the mass-transfer coefficients for small and large bubbles
are 1.27*10-4 and 4.58*10-4 m/s,respectively,which are
independent of power consumption and gas-flow rate.
Eq,(6.29) predicts that the mass-transfer coefficient for
the oxygen dissolution in water 25℃ in a mixing vessel is
4.58*10-4m/s,regardless of the power consumption and
gas-flow rate as illustrated in the previous example
problem,Lopes De Figueiredo and Calderbank (1978)
reported later that the value of kL varies from 7.3*10-4 to
3.4*10-3 m/s,depending on the power dissipation by
impeller per unit volume (Pm/τ ) as
33.0
mL
P
k
(6.31)
This dependence of kL on Pm/v was also reported by
Prasher and Wills (1973) base on the absorption of
CO2 into water in an agitated vessel,as follows:
25.0
c
21592.0
m
ABL
P
Dk (6.32)
Which is dimensionally consistent,However,this
equation is limited in its use because the correlation is
based on only one gas-liquid system.
Akita and Yoshida (1974) evaluted the liquid-phase
mass-phase mass-transfer coefficient based on the
oxygen absorption into several liquids of different
physical properties using bubble columns without
mechanical agitation,Their correlation for kL is
83
C
3
32
41
2
C
3
32
21
AB
c
AB
32L 5.0
gDgD
DD
Dk (6.33)
Where vc and σ are the kinematic viscosity of the
continuous phase and interfacial tension,respectively,
Eq,(9.33) is applicable for column diameters of up to
0.6 m,superficial gas velocities up to 25 m/s,and gas
hold-up to 30 percent,
6.4 Measurement of Interfacial Area
To calculate the gas absorption rate qL for Eq.(6.7),
we need to know the gas-liquid interfacial area,which
can be measured employing several techniques such
as photography,light transmission,and later optics,
The interfacial area per unit volume can be calculated
from the Sauter-mean diameter D32 and the volume
fraction of gas-phase H,as follows:
32
6
D
a
(6.34)
参见戚以政书 (6-49)
6.5 Correlations for α and D32
6.5.1 Gas Sparging with No Mechanical Agitation
Leaving the vicinity of a sparger,the bubbles may break
up or coalesce with others until an equilibrium size
distribution is reached,A stable size is achieved when
turbulent fluctuations and surface tension forces are in
balance (Calderbank,1959),Akita and Yoshida (1974)
determined the bubble-size distribution in bubble columns
using a photographic technique,The gas was sparged
through perforated plates and single orifices,while various
liquids were used,The following correlation was proposed
12.012.0
2
250.02
32 26
C
S
L
CCC
C gD
VgDgD
D
D (6.36)
and for the interfacial area
13.1
1.0
2
350.02
3
1
H
gDgD
aD
L
CLC
C
(6.37)
Where DC is bubble column diameter and Vs is
superfical gas velocity,which is gas flow rate Q divided
by the tank cross-sectional area,Eqs.(6.36) and (6.37)
are based on data in columns of up to 0.3m in diameter
for the Sauter-mean diameter:
and up to superficeal gas velocities of about 0.07 m/s.
6.5.2 Gas Sparging with Mechanical Agitation
Calderbank (1958) correlated the interfacial areas for
the gas-liquid dispersion agitated by a flat-blade disk
turbine as follows:
1.For Vs <0.02 m/s,
21
6.0
2.04.0
44.1
t
scm
V
VP
a
(6.38)
for
2 0,0 0 0
3.0
7.0
Re
s
I
i V
ND
N
Where NRei is the impeller Reynolds number defined as
c
cI NDN
i?
2
Re?
(6.39)
The interfacial area for NRei0.7(NDI/Vs)0.3>20,000 can
be calculated from the interfacial area α 0 obtained
from Eq,(6.38) by using the following relationship.
3.0
7.0
Re
5
0
10 1095.1
3.2
l o g
S
I
V
ND
N
a
a
i
(6.40)
2.For Vs>0.02 m/s,Miller (1974) modified Eq.(6.38)
by replacing the aerated power input by mechanical
agitation Pm with the effective power input Pe,and the
terminal velocity Vt with the sum of the superficial gas
velocity and the terminal velocity Vs+Vt,The effective
power input Pe combines both gas sparging and
mechanical agitation energy contributions,The modified
equation is
21
6.0
2.04.0
0 44.1
St
Sce
VV
VP
a
(6.41)
Calderbank (1958) also correlated the Sauter-mean
diameter for the gas liquid dispersion agitated by a flat-
blade disk turbine impeller as follows:
1.For dispersion of air in pure water,
45.0
2.04.0
6.0
32 100.915.4
cvmP
D
(6.42)
2.In electrolyte solutions (NaCl,Na2SO4,and Na3PO4),
25.0
4.0
2.04.0
6.0
32 25.2
c
d
cvmP
D
(6.43)
3.In alcoholic solution (aliphatic alcohols),
25.0
65.0
2.04.0
6.0
32 90.1
c
d
cvmP
D
(6.44)
where all constants are dimensionless except 9.0*10-4m
in Eq,(6.42),Again the preceding three equations can
be modified for high gas flow rate (Vs>0.02m/s) by
replacing Pm by Pe,as suggested by Miller (1974).
6.6 Gas Hold-Up εor (H)
gas hold-up is one of the most important parameters
characterizing the hydrodynamics in a fermenter,Gas
hold-up depends mainly on the superficial gas velocity
and the power consumption,and often is very sensitive to
the physical properties of the liquid,Gas hold-up can be
determined easily by measuring level of the aerated liquid
during operation ZF and that of clear liquid ZL,Thus,the
average fractional gas hold-up H( ε) is give as
tS
S
VV
V
(6.45)
Gas Sparging with No Mechanical Agitation,In a two-
phase system where the continuous phase remains in
place,the hold-up is related to superficial gas velocity Vs
and bubble rise velocity Vt (Sridhar and Potter,1980):
F
LF
Z
ZZ?
(6.46)
Akita and Yoshida (1973) correlated the gas hold-up for
the absorption of oxygen in various aqueous solutions in
bubble columns,as follows:
C
SCCC
gD
VgDgD
1212812
4 2.01
(6.47)
Gas Sparging with Mechanical Agitation,Calderbank
(1958) correlated gas hold-up for the gas-liquid
dispersion agitated by a flat-blade disk turbine impeller as
follows:
21
6.0
2.04.0
4
21
1016.2
t
Scvm
t
S
V
VP
V
V
(6.48)
(87)
Where 2.16*10-4 has a unit (m) and Vt=0.265 m/s when
the bubble size is in the range of 2-5 mm diameter,The
preceding equation can be obtained by combining Eqs.
(6.38) and (6.42) by means of Eq,(6.34).
For high superficial gas velocities (Vs >0.02m/s),replace
Pm and Vt of Eq,(9.48) with effective power input Pe and
Vt+Vs,respectively (Miller,1974).
5.3 Gas Flow Effects on Bubble Swarms
Two regions in any gas sparged tank need to be
considered:
5.3.1 Bubble Sizes Generated at Orifice
when Re0 < 2000,0.1< dB < 1.0cm
dB =0.18d01/2Re01/3 (6.49)
The following equation is recommended for design
purposes:
=3.23Re0L-0.1Fr00.21 (6.50)
where Re0L=,Fr0= (6.51)0
d
dB
0
4
d
Q
L
L
gd
Q
5
0
2
When Re0 > 10000,bubble size in equilibrium only
relate to Re0:
dBe = 0.71Re0-0.05 (dBe in cm ) (6.52)
5.3.2 Bubble Size Far from the Orifice
The bubble size may vary,depending on the liquid
properties and the liquid motions generated by the rising
gas stream,It is very complicated,An energy balance on
the gas and liquid phases can be used to determine the
liquid velocity.
5.4 Bubble Coalescence and Break-up
Bubble Coalescence rising in a line takes place in several
stages:
A.the approach of the
following bubbles to the
leading bubble,
B,the trailing bubble moves
in the vortex of the leading
bubble until the bubbles are
separated only by a thin
interface.
C,final thinning and rupture
of the film between bubbles.
In gas-sparged vessels the power per unit volume can be
found form an energy balance on the gas phase
Fig.6.1 Bubble Coalescence
The kinetic energy of the gas leaving the vessel is generally
negligible,and the kinetic energy term is generally small for
most values of Q,Hence,the equation reduces to,
= ln (6.53)
The resulting mean bubble size can be found by inserting this
value of P/V into
dBe = 0.7
(dBe in m) (6.54)
The equilibrium size can predicted by
dBe = 0.71Re0-0.05 (dBe in cm) (6.55)
1.0
2.04.0
6.0
)(
)( G
a
LV
P?
)(
21
2
PP
PgH
V
Q
2
1
P
P
V
P
)(
21
2
PP
P
V
Q
V
P
ln
V
VgH
P
P 1
2
2
0
2
1
6.7 Power Consumption
The power consumption for mechanical agitation can be
measured using a torque table as shown in Figure 6.7,The
torque table is constructed by placing a thrust bearing
between two circular plates,and the force required to
prevent rotation of the turntable during agitation F is
measured,The power consumption P can be calculated
by the following formula
NFP 2
Where N is the agitation speed,and r is the distance
from the axis to the point of the force measurement.
Power consumption by agitation is a function of
physical properties,operating condition,and vessel and
impeller geometry,Dimensional analysis provides the
following relationship:
(6.56)
r
bearing
gauge
,,,,,
22
53
I
W
II
II
I D
D
D
H
D
Dr
g
DNNDf
DN
P (6.57)
The dimensionless group in the left-hand side of Eq,(6.57)
is known as power number NP,which is the ratio of drag
force on impeller to inertial force,The first term of the
right-hand side of Eq.(6.57) is the impeller Reynolds
number NRei,which is the ratio of inertial force to viscous
force,and the second term is the Froude number NFr,
which takes into account gravity forces,The gravity force
affects the power consumption due to the formation of the
vortex in an agitating vessel,The vortex formation can be
prevented by installing baffles.
PN
For fully baffled geometrically similar systems,the
effect of the Froude number on the power consumption
is negligible and all the length ratios in Eq.(6.57) are
constant,Therefore,Eq,(6.57) is simplified to
i
NN P Re
For the normal operating condition of gas-liquid contact,
the Reynolds number is usually larger than 10000,For
example,for a 3-inch impeller with an agitation speed of
150 rpm,the impeller Reynolds number is 16225 when
the liquid is water,Therefore,Eq,(6.58) is simplified to
10 00 0f or c o n s t a n t Re iNN P
(6.58)
(6.59)
参见戚以政书 P312-313,(6-61) –(6-66)
The power required by an impeller in a gas sparged
system Pm is usually less than the power required by the
impeller operating at the same speed in a gas-free liquids
Pmo,The Pm for the flat-blade disk turbine can be
calculated from Pmo,The Pm for the flat-blade disk turbine
can be calculated from Pmo (Nagata,1975),as follows:
3
96.121 1 5.0238.4
10 1 9 2l o g
I
D
D
II
T
I
mo
m
ND
Q
g
NDND
D
D
p
P T
I
Example 6.4
A cylindrical tank (1.22m diameter) is filled with water
to an operating level equal to the tank diameter,The tank
is equipped with four equally spaced baffles whose width
is 1/10 of the tank diameter,The tank is agitated
参见 P314-315,(6-71) –(6-76) (6.60)
with a 0.36 m diameter flat six-blade disk turbine,The
impeller rotational speed is 2.8 rps,The air enters through
an open-ended tube situated below the impeller and its
volumetric flow rate is 0.00416 m3/s at 1.08 atm and 25℃,
Calculate the following properties and compare the
calculated values with those experimental data reported
by Chandrasekharan and Calderbank (1981),Pm=697W;
=0.02; kLa=0.0217s-1.
a.Power requirement Pm
b.Gas hold-up ε
c.Sauter-mean diameter D32
d.Interfacial area a
e.Volumetric mass-transfer coefficient kLa
Solution,
Power requirement,The viscosity and density of water
at 25 ℃ is 8.904*10-4kg/ms (CRC Handbook of
Chemistry and Physics,p.F-38,1983) and 997.08 kg/m3
(Perry and Chilton,p.3-71,1973),respectively,Therefore,
the Reynolds number is
4 0 6 3 5 7
109 0 4.8
36.08.208.9 9 7
4
22
Re
INDN
Which is much larger than 10000,above which the
power number is constant at 6,Thus,
W7 9 436.08.208.9 9 766 5353 Imo DNP
I
mo
P DN
PN
53 1
10
102
1 10 102 103 104 105 106 INDN
2
Re?
Fig.6.8
(参见图 6-19)
3
22.1
36.096.1
21 1 5.0
7
238.4
10 36.08.2
0 0 4 1 6.0
81.9
8.236.0
1093.8
8.236.0
22.1
36.01 9 2
7 9 4
l o g Pm
Therefore,
W6 8 7?mP
b.Gas hold-up,The interfacial tension for the air-water
interface is 0.07197kg/s2 (CRC Handbook of Chemistry
and Physics,p,F-33,1983),The volume of the dispersion is
32 m43.11,2 222.14
The superficial gas velocity is
m / s0 0 3 5 6.0
22.1
0 0 4 1 6.044
22
T
s D
QV
The power required in the gas-sparged system is from Eq,
(6.60)
Substituting these values into Eq.(6.48) gives
21
6.0
2.04.0
4
21
2 6 5.0
0 0 3 5 6.0
0 7 1 9 7.0
08.9 9 743.16 8 71016.2
2 6 5.0
0 0 3 5 6.0?
The solution of the preceding equation for ε gives
023.0
c,Sauter-mean dimeter,In Eq.(6.42)
mm9.300 36 6.0
100.902 3.0
08.99 743.168 7
07 19 7.0
15.4 45.0
2.04.0
6.0
32
D
d,Interfacial area α,In Eq.(6.34)
1
32
m7.37
0,0 0 3 6 6
0,0 2 36
D
6a
e.Volumetric mass-transfer coefficient,Since the average
size of bubbles is 4mm(＞ 2.5),we should use Eq.(6.29),
then,from Example 6.3
m / s1058.4 4Lk
Therefore,
-14 0.0 17 s37,71058.4Lak
The preceding estimated values compare well with
those experimental values,The percent errors as defined
in Example 9.1 are –1.4 percent for the power
consumption,15 % for the gas hold-up,and –21.7 % for
the volumetric mass-transfer coefficient.
6.8 Determination of Oxygen-Absorption Rate
To estimate the design parameters for oxygen uptake in a
fermenter,you can use the correlations presented in the
previous sections,which can be applicable to a wide
range of gas-liquid systems in addition to the air-water
system,However,the calculation procedure is lengthy
and the predicted value from those correlations can vary
widely,Sometimes,you may be unable to fimd suitable
correlations which will be applicable to your type and
size of fermenters,In such cases,you can measure the
oxygen-transfer rate yourself of use correlations base on
those experiments.
The oxygen absorption rate per unit volume qα/v can be
estimated by
LLLLLLa CCakCCaKq
(6.61)
Since the oxygen is sparingly soluble gas,the overall
mass-transfer coefficient KL is equal to the individual
mass-transfer coefficient kL,Our objective in fermenter
design is to maximaize the oxygen transfer rate with the
minimum power consumption necessary to agitate the
fluid,and also minimum air flow rate,To maximize the
oxygen absorption rate,we have to maximize kL,α,CL*-
CL,However,the concentration difference is quite
limited for us to control because the value of CL* is
limited by its very low maximum solubility,Therefore,
the main parameters of interest in design are the mass-
transfer coefficient and the interfacial area.
T e m p e r a t u r e
℃
S o l u b i l i t y
m m o l O
2
/l
Mg O
2
/l
0 2,1 8 6 9,8
10 1,7 0 5 4,5
15 1,5 4 4 9,3
20 1,3 8 4 4,2
25 1,2 6 4 0,3
30 1,1 6 3 7,1
35 1,0 9 3 4,9
40 1,0 3 3 3,0
Table 6.1 Solubility of Oxygen
in water at 1 atm.a
Table6.1 lists the solubility of oxygen at 1 atm in water at
various temperatures,The value is the maximum
concentration of oxygen in water when it is in
equilibrium with pure oxygen,This solubility decreases
with the addition of acid or salt as shown in Table 6.2.
C on c.
m ol / l
S ol ub il ity,m m ol O
2
/ l
H Cl H
2
SO
4
N aC l
0,0 1,2 6 1,2 6 1,2 6
0,5 1,2 1 1,2 1 1,0 7
1,0 1,1 6 1,1 2 0,8 9
2,0 1,1 2 1,0 2 0,7 1
Table 6.2,Solubility of Oxygen in Solution of
Salt or Acid at 25℃,a
Normally,we use air to supply the oxygen demand of
fermenters,The maximum concentration of oxygen in
water which is in equilibrium with air CL* at atmospheric
pressure is about one fifth of the solubility listed,
according to the Henry’s law,
THo
po
C L
2
2
(6.62)
Where pO2 is the partial pressure of oxygen and HO2 (T)
is Henry’s law constant of oxygen at a temperature,T,
The value of Henry’s law constant can be obtained
from the solubilities listed in Table6.1,For example,
at 25℃,CL* is 1.26mmol/l and pO2 is 1 atm because it
is pure oxygen,By substituting these values into
Eq,(6.55),we obtain HO2 (25℃ ) is 0.793 atm
l/mmol,Therefore,the equilibrium concentration
of oxygen for the air-water contact at 25℃ will
be
lmglom m o
m m oa t m l
a t m
C L
/43.8/l264.0
l/793.0
209.0
2
Ideally,oxygen-transfer rates should be measured in a
fermenter which contains the nutrient broth and
microorganisms during the actual fermentation process,
However,it is difficult to carry out such a task due to the
complicated nature of the medium and the ever changing
rheology during cell growth,A common strategy is to use
a synthetic system which approximates fermentaiton
conditions.
6.8.1 Sodium Sulfite Oxidation Method
The sodium sulfite oxidation method (Cooper et al.,
1944) is based on the oxidation of sodium sulfite in the
presence of catalyst (Cu++ or Co++) as
42
Coo r Cu
22132 SONaOSONa
(6.63)
This reaction has following characteristics to be qualified for
the measurement of the oxygen-transfer rate:
1.The rate of this reaction is independent of the
concentration of sodium sulfite within the range of 0.04 to 1
N.
2.The rate of reaction is much faster than the oxygen
transfer rate; therefore,the rate of oxidation is controlled
by the rate of mass transfer alone.
Example 6.5
To measure kLa,a fermenter was filled with 10l of 0.5M
sodium sulfite solution containing 0.003M Cu++ ion and
the air sparger was turned on,After exactly 10 minutes,
the air flow was stopped and a 10 ml sample was taken
and titrated,The concentration of the sodium sulfite in
the sample was found to be 0.21 mol/l,The experiment
was carried out at 25℃ and 1 atm,Calculate the oxygen
uptake and kLa.
Solution:
The amount of sodium sulfite reacted for 10 minutes is
lmo /l29.021.05.0
According to the stoichiometric relation,Eq,(6.63),the
amount of oxygen required to react 0.29mol/l is
lmo /l1 4 5.0
2
129.0
Therefore,the oxygen uptake is
lg
s
mogOlOm o l e /1073.7
6 0 0
l/32/ 1 4 5.0 32
2
The solubility of oxygen in equilibrium with air can be
estimated by Eq.(6.62) as
lg
g/mol
Omoa t m
TH
p
C L
/1043.8
32m o l 1l/ 7 9 3 a t m
a i r l209.0 1
3
2
0
O
2
2
Therefore,the value of kLa is,according to the Eq,(6.61),
1
3
3
9 1 7.0
0/1043.8
/1073.7?
s
lg
lsg
CC
q
k
LL
a
La
6.8.2 Dynamic Gassing-out Technique
This technique (Van’t Riet,1979) monitors the change of
the oxygen concentration while an oxygen-rich liquid is
deoxygenated by passing nitrogen through it.
Polarographic electrode is usually used to measure the
concentration,The mass balance in a vessel gives
tCCk
dt
tdC
LLLa
L
(6.64)
Intergration of the preceding equation between t1 and
t2 results in
12
2
1In
tt
tCC
tCC
k
LL
LL
La
(6.65)
From which kLa can be calculated based on the measured
values of CL(t1) and CL(t2).
6.8.3 Direct measurement
In this technique,we directly measure the oxygen content
of the gas stream entering and leaving the fermenter by
using gaseous oxygen analyzer,The oxygen uptake can
be calculated as
o u t,in,22 Oo u tOina CQCQq (6.66)
Where Q is the gas flow rate.
Once the oxygen uptake is measured,the kLa can be
calculated by using Eq.(9.54),where CL is the oxygen
concentration of the liquid in a fermenter and CL* is the
concentration of the oxygen which would be in
equilibrium with the gas stream,The oxygen concentration
of the liquid in a fermenter can be measured by an on-line
oxygen sensor,If the size of the fermenter is rather small
(less than 50l),the variation of the CL*-CL in the fermenter
is fairly small,However,if the size of a fermenter is very
large,the variation can be significant,In this case,the log-
mean value of CL*-CL of the inlet and outlet of the gas
stream can be used as
o u tLLinLL
o u tLLinLL
LMLL CCCCIn
CCCC
CC
/
(6.67)
6.8.4 Dynamic Technique
By using the dynamic technique (Taguchi and
Humphrey,1966),we can estimate the kLa value for the
oxygen transfer during an actual fermentaton process with
real culture medium and microorganisms,This technique
is based on the oxygen material balance in an aerated batch
fermenter while microorganisms are actively growing as
摄氧率,可在稳态下,由进气和排气中氧的分压求出 。
XOLLL CrCCk
dt
dC
2La
(6.68)
Xo Cr 2 0?
dt
dC L
Where rO2 is cell respiration rate [g O2/g cell h].
While the dissolved oxygen level of the fermenter is
steady,if you suddenly turn off the air supply,the
oxygen concentration will be decreased (Figure 6.9)
with the following rate
XO
L Cr
dt
dC
2?
(6.69)
Since kLa in Eq,(6.68) is equal to zero,Therefore,by
measuring the slope of the CL vs,T curve,we can
estimate rO2CX,If you turn on the air flow again,the
dissolved oxygen concentration will be increased
according to Eq,(6.68),which can be rearranged to result
in a linear relationship as
XO
L
LL Crdt
dC
k
CC 2
La
1
(6.70)
参见戚以政书 (6-58 )
air off
air on
dt
dCt
CL
t
Fig,6.9 Dynamic technique for the determination of kLa
( 参见戚以政书 P307 )
CL Lak1?
Xo
L Cr
dt
dC
2?
*LC
所谓动态法是根据停止向培养溶液中通气时，
培养液中溶氧浓度的变化率求出摄氧率，当溶氧浓度下降到一定程度时（不低于临界溶氧浓度），再恢复通气，则培养液中氧浓度又逐渐升高，直至恢复到原先的水平，见 Fig,
6.9 左图。右图中，截距为饱和溶氧浓度，斜率为 -
1/kLa.
溶氧法的优点是只需测定溶氧浓度 CL 随时间变化曲线，非常方便地求出 kLa.
当稳态时，
)( *2 LLLXo CCakCr
0dtdCOL
OLOL
o
La CC
r
k
* 2
The plot of CL versus (dCt/dt +ro2CX) will in a straight line
which has the slope of -1/(kLa) and the y-intercept of C*L,
6.9 Correlation for kLa
6.9.1 Bubble Column
Akita and Yoshida (1973) correlated the volumetric mass-
transfer coefficient kLa for the absorption of oxygen in
various aqueous solutions in bubble columns,as follows:
(参见戚书 P331)
1.193.017.0
62.0
12.05.0
6.0?
gD
c
Dk TcABLa
(6.71)
Which applies to columns with less effective
sparkers.
In bubble columns,for 0＜ Vs＜ 0.15m/s and100
＜ Pg/v＜ 1100W/m3,Botton et al,(1980) correlated
the kLa as
75.0
8 0 0
/
08.0?
vPk g
La
(6.72)
Where Pg is the gas power input,which can be
calculated from
75.0
1.0
800?
sg V
v
P (6.73)
6.9.2 Mechanically Agitated Vessel
For aerated mixing vessels in an aqueous solution,
the mass-transfer coefficient is proportional to the
power consumption (Lopes De Figueiredo and
Calderbank,1978)as
33.0/ vPk mL?
(6.74)
The interfacial area for the aerated mixing vessel
is a function of agitation conditions,Therefore,
according to Eq.(6.38),
5.0
4.0
s
m V
v
P
a?
Therefore,by combining Eqs,(6.74)and (6.75),kLa
will be
5.0
77.0
s
m
La V
v
P
k?
(6.75)
(6.76)
Numerous studies for the correlations of kLa have
been reported and their results have the general
form as
3
2
1
b
s
b
m
La V
v
P
bk?
(6.77)
Where b1,b2,and b3 vary considerably depending
on the geometry of the system,the range of
variables covered,and the experimental method
used,The values of b2and b3 are generally
between 0 to 1 and 0.43 to 0.95,respectively,as
tabulated by Sideman et al.(1966).
Van,t Riet (1979) reviewed the data obtained by
various investigators and correlated them as follows:
1,For,coalescing” air-water dispersion,
5.0
4.0
0 2 6.0 smLa V
v
P
k?
(6.78)
参见书 (6-85)
2,For,noncoalescing” air-electrolyte solution
dispersions,
2.0
7.0
0 0 2.0 smLa V
v
P
k?
(6.79)
Both of which are applicable for the volume up to
2.6m3;for a wide variety of agitator types,sizes,and
DI/DT ratios; and 500＜ P m/v＜ 10,000W/m3.These
correlations are accurate within approximately 20
percent to 40 percent.
Example 6.6
Estimate the volumetric mass-transfer coefficient kLa
for the gas-liquid contactor described in Example 6.4
by using the correlation for kLa in this section.
Solution:
For Example 6.4,the reactor volume v is 1.43m3,
the superficial gas velocity Vs is 0.00356 m/s,and
power consumption Pm is 687 W.By substituting
these values into Eq.(6.71),
15.0
4.0
0 1 8.00 0 3 5 6.0
43.1
6 8 70 2 6.0
sk
La
Problems
6.1 Derive the relationship between the overall mass
transfer coefficient for gas phase KG and the
individual mass-transfer coefficients,kL and kG.How
can this relationship be simplified for sparingly and
soluble gases
6.2 Prove that Eq.(6.25)is the same with Eq,(6.26),
and Eq.(6.27) is the same with Eq.(6.28).
6.3 The power consumption by impeller P in
geometrically similar fermenters is a function of the
diameter DI and speed N of impeller,density ρand
Viscosity μ of liquid,and acceleration due to gravity
g,Determine the functional relationship between
appropriate dimensionless parameters,which can
relate the power consumption,by applying
dimensional analysis using the Buckingham-Pi
theorem.
6.4 A cylindrical tank(1.22m diameter) is filled with
water to an operating level equal to the tank diameter,
The tank is equipped with four equally spaced
baffles,the width of which is one tenth of the tank
diameter,The tank is agitated with a 0.36m
diameter,flat-blade disk turbine,The impeller
Rotational speed is 4.43 rps,The air enters through
an open ended tube situated below the impeller and
its volumetric flow rate is 0.0217m3/s at 1.08 atm
and 25℃,Calculate:
a,Power requirement
b,Gas hold-up
c,Sauter-mean diameter
d,Interfacial area
e,Volumetric mass-transfer coefficient
Compare the preceding calculated results with those
Experimental values reported by
Chandrasekharan and Calderbank (1981),Pm =
2282W; ε = 0.086; kLa = 0.0823 s-1
6.5 Estimate the volumetric mass-transfer
coefficient kLa for the gas-liquid contactor
described in Problem 9.4 by using a correlation
for kLa and comp9are the result with the
experimental value.
6.6 The power consumption by an agitator in an
unbaffled vessel can be expressed as
2
53
I
I
mo NDf
DN
P
Can you determine the power consumption and
impeller speed of a 1,000gallon fermenter based on
findings of the optimum condition from a one-gallon
vessel by using the same fluid system? Is your
conclusion reasonable? Why or why not?
6.7 The optimum agitation speed for the cultivation
of plant cells in a 3-l fermenter equipped with four
baffles was found to be 150 rpm.
a,What should be the impeller speed of a
geometrically similar 1,000 -l fermenter if you
scale up based on the same power consumption per
unit volume.
b,When the impeller speed by part (a) was
employed for the cultivation of a 1,000 -l
fermenter,the cells do not seem to grow well due
to the high shear generated by the impeller even
though the impeller speed is lower than 150 rpm of
the model system,This may be due to the higher
impeller tip speed which is proportional to NDI,Is
this true? Justify your answer with the ratio of the
Impeller tip speed of the prototype to the model
fermenter.
c,If you use the impeller tip speed as the
criteria for the scale-up,what will be the impeller
speed of the prototype fermenter?
