8 Aeration and mixing

241

8

AERATION AND MIXING

Aeration and mixing of nutrient media and microorganisms which grow in these media should ensure: a) a sufficient amount of oxygen for aerobic microorganisms, b) concentration and temperature homogeneity of the culture. Oxygen is the final acceptor of electrons and hydrogen in microorganisms which obtain their energy via aerobic respiration; at the same time it serves as a substrate because, through the action of oxygenases, it is incorporated into carbon substrates in the course of their dissimilation. Oxygen acts also as a regulatory factor which either induces aerobic enzymes or inhibits the activity of these enzymes during glycolysis (Pasteur effect) and the synthesis of enzymes during dissimilation of energy sources under anaerobic conditions (oxygen effect). The rate of oxygen consumption in aerobic microorganisms is so high that the oxygen supply after maximum air saturation of the culture can support cell metabolism for a mere 15 s. Consequently, interruption of oxygen supply for an interval even as short as 15 s can seriously impair the respiration of cells of highly aerobic microorganisms such as Acetobaeter. Although such a dramatic effect of a short suspension of oxygen supply is not frequent, an insufficient oxygen level in the culture leads in any case to lowered yields of cells or metabolites. Considerations of oxygen supply in microbial processes include simultaneously aeration and mixing because these two processes exert a common effect on the oxygen level. In some processes using unicellular microorganisms, culture mixing by impeller systems is obviated provided the air dispersers (spargers) are correctly placed in the bioreactor. On the other hand, in cultivations of filamentous microorganisms such as actinomycetes and fungi, mixing is indispensable for a successful process. The main problems in aeration and mixing of submerged microbial cultures are: a) achieving the correct mixing intensity, b) finding reasonable ways of evaluating the effect of aeration and mixing.

242

8.1

THEORY OF OXYGEN TRANSFER

Transfer of oxygen from the gas phase proceeds by molecular diffusion. The double-film theory assumes the formation of a gas-and-liquid film at the gas-liquid interface. All resistance to the transport of oxygen from the gaseous into the liquid phase is assumed to be concentrated in the liquid part, i.e. in the liquid film (Fig. 8.1). gas P

tiquid

I I

I I

I I I I I I

~--

dl

C

d2

Fig. 8.1 Scheme of oxygen transfer from gas to liquid d~ -- film thickness, d2 -- thickness of liquid film, C § -- equilibrium oxygen concentration (a function of T, Po2 and physical properties of the liquid), C - momentary concentration of dissolved oxygen, A C - driving force

When the oxygen concentration profile in the film is linear, no reaction takes place in the film and the overall process is a physical oxygen absorption. In the case of chemical absorption the concentration profile is no longer linear. The magnitude of resistance of the liquid film depends on the physicochemical properties of the liquid and its thickness is affected by mixing intensity. The double-film theory presumes an instantaneous equilibration of oxygen concentrations in the gas and the liquid following the contact of the two phases P He

where

-

c

[8.~1

is Henry's constant, C --momentary concentration of dissolved oxygen. The solution of a diffusion process according to the double-film theory is based on the 1st Fick's law He

where N D d C/dx

x d

is diffusion flow, -- diffusivity of oxygen, - - concentration gradient, -- distance, --thickness of the diffusion layer.

243

Solution of the equation has the form D

---d ( C + - C)

[8.3]

N = kL (C + - C)

[8.4]

N= or

where kL is an oxygen transfer coefficient. When the rate of oxygen absorption is referred to the total area of dispersion, A, in a volume VL it can be seen that dC

v , --d7 =

(C + -

dC - kLa(C +dt

C)

[8.5]

[8.6]

C)

where kLa is the volume coefficient of oxygen transfer characterizing the aeration efficiency of the device, a -- specific interface area for mass transfer, C + --equilibrium oxygen concentration. In a system with microorganisms, equation [8.6] has the form dC dt

- kLa(C +-

C)-

rX

[8.7]

where r is the respiration rate of the microorganism, X - - cell concentration. When the rate of oxygen consumption exceeds the rate of absorption the whole system enters, after a certain period, a region of oxygen limitation. Equations [8.6] and [8.7] hold for an ideally mixed system, i.e. for ideally mixed gaseous and liquid phases. In practice this condition is rarely encountered since different oxygen concentrations can be determined in different locations in the liquid phase inside bioreactors; the condition of an ideally mixed system is then not fulfilled. If the oxygen concentration in the liquid is zero equation [8.6] is reduced to the form dC = kLa C + dt

[8.8]

which defines the maximum rate of oxygen transfer under given conditions.

244

8.2

METHODS OF AERATION CAPACITY (kLa) DETERMINATION

The methods for determination of kLa can in practice be divided depending on what system they can be used in, i.e. methods applicable in the system gas-liquid (model systems) and those that can be used in the system gas-liquid-microorganism (culture systems). 8.2.1

Methods applicable to model systems

The sulphite method The oldest method for measuring kLa, elaborated by Cooper, Fernstrom and Miller (1944), is based on chemical monitoring of the decrease in oxygen concentration via oxidation of sulphite to sulphate in the absence or presence of a catalyst (Cu 2+, Co 3+ in a concentration of 10-3-10 -5 kmol m -3) according to the equation 3+ Co

SO~- + 1 / 2 0 2

2-?" S02Cu

Since in this system the concentration of oxygen in the liquid, C-- O, the aeration capacity can be described by the equation

kea =

(dfNa2s~

[8.9]

2C +

The rate of absorption is increased by the chemical reaction and it is therefore necessary to know the acceleration factor of the method. This can be determined as the ratio of chemical and physical absorption values according to the formula

kea f = kOa

[8.101

Since usually f > 1 the measured kLa > ~a and the calculated values are higher than the actual ones. The method is suitable only for comparing the efficiency of individual culture devices. Because of the consumption of the relatively expensive sulphite it cannot be used for testing large-volume bioreactors. It is simple but time taking; one determination may take up to 3 h depending on aeration and mixing intensity. The accuracy of the method drops substantially in the presence of even very low concentrations of surface active compounds such as proteins, amino acids, fatty acids and lipids (Bell and Gallo, 1971).

245

The static and volumetric methods Integration of equation [8.6] gives the formula In

C

+

C+-C

= kLa t

[8.11]

and kLa is evaluated from an aeration experiment by determining suitably the oxygen concentration C during the aeration interval t. Various methods differ in the technique of determining the concentration of dissolved oxygen. In the static method the oxygen concentration is measured by the oxygen electrode; air is brought into the system for an interval t~, the aeration is stopped and steady state oxygen concentration is then measured. The whole system is then freed of oxygen, and air is again brought in for an interval t2(t2 > t~). This technique is used to monitor the whole range of bioreactor liquid saturation throughout the whole time interval t. In the volumetric method the amount of dissolved oxygen is determined via desorption by a carbon dioxide stream and absorption of excess carbon dioxide into a solution of potassium hydroxide in a gas burette.

Gassing-out method The method is based on monitoring, by an oxygen electrode, the exponential change in oxygen concentration in a liquid following a transient displacement of the oxygen by an inert gas, usually nitrogen. The gassing-out phase is followed by removal of the nitrogen gas from the space above the liquid using an air stream. This step is necessary to avoid inadvertent nitrogen intake during the exponential change (Fig. 8.2).

C "~

100 .-.

8O 60

E

40 2O 0 0

I

I

I

I

I

20

40

60

80

100

O, scd.,u.rcttion (%)

Fig. 8.2 Time course of dissolved oxygen concentration in the gassing-out method

246

Integration of equation [8.6] gives the formula 1

kLa -

t - to

In

C + - Co

C+- C

[8.12]

When the change in oxygen concentration is plotted against time the aeration capacity value is obtained as the slope of the line (Fig. 8.3). This method is often used to test bioreactors. It is limited, however, by nitrogen consumption and is not applicable in large volumes because the measurement requires

200

100

b b

~

7o

~

50

~3 9

30 20

10

I 2

I 6

I 10 ~--

I 14

I 18

I 22

I 26

30

time (s)

Fig. 8.3 Determination of kLa by the gassing-out method

a sudden change in both aeration and the impeller power input, which is not feasible in large-volume devices. When both the measurement and calculation are carried out in this way the resulting value of kLa has a tentative character and does not provide information about the absolute aeration capacity value. 8.2.2

M e t h o d s applicable to cultivation systems

The balance method

This method is based on monitoring of changes in oxygen concentration in the gas phase. Measurement of changes in the partial oxygen pressure in the liquid makes it possible to calculate also the value of kLa. Taking into account the respiration of carbon dioxide, and referring the air flow to an inert gas such as nitrogen, the rate of absorption Na can be defined by the equation

247

Na -

v[

R VL

where V is the volume of VL -- volume of the R -- gas constant, Y~ molar fraction Y~c -- molar fraction Y2 -- molar fraction Y2c -- molar fraction The rate of absorption ters of oxygen transfer

T~(1- Y l - Y~c) - T2( 1 -

,22

Y 2 - Y2c )

]

[8.131

the inert gas, liquid, of oxygen at the inlet, of carbon dioxide at the inlet, of oxygen at the outlet, of carbon dioxide at the outlet. is a basic parameter for calculating other parameNa kLa = C+ - C

[8.14]

The value of C + has to be corrected for oxygen concentration at the outlet and pressure change at the outlet; another quantity to be known is the change in oxygen solubility in the liquid caused by the disappearance of substrate and the formation of products. The specific rate of oxygen consumption is given by the equation QQ-

Na

X

[8.151

where X is biomass concentration. The requirement for oxygen in the linear growth phase can be defined by the equation Yo/,-

Na Pr

[8.16]

where Pr is productivity. The following equation holds for the exponential growth phase Y0/, -

Qo2 ~t

[8.17]

where ~t is the specific growth rate. The respiration coefficient is given as follows R~=

COz y , _ y:

[8.18]

This method can conveniently be used for scale-up from laboratory to pi-

248

lot plant scale, although it is limited to closed bioreactors. It makes it possible to measure oxygen transfer in the course of a cultivation without interfering with culture aeration and mixing.

Integral balance m e t h o d

The method provides an assessment of utilization of dissolved oxygen by microorganisms and its impact on both the gas and the liquid phase. The oxygen balance is given by the formula dt

-

--H-~eY2 - C

-

Q(C, t)

[8.191

The term Q(C, t) has to be determined from growth experiments. The first term on the right-hand side of the equation represents a change in oxygen concentration during growth, the second term corresponds to oxygen level change during formation of intermediates. Aeration capacity can be determined by computer evaluation of the parameters of a model which embodies data from an experimentally established course of biomass production, concentration of dissolved oxygen, substrate consumption, time course of oxygen concentration at the outlet or other quantities characterizing cumulative oxygen consumption. The advantage of the integral balance method is that it allows one to obtain not only parameters of oxygen transfer but, in addition, other kinetic parameters of growth and it can be used also in the range of oxygen limitation. Another advantage is that it eliminates all effects accompanying the dynamic measurement by oxygen electrode, so that only the quasi-concentration, i.e. steady-state concentration of oxygen has to be measured during the cultivation.

The d y n a m i c m e t h o d

The method is based on monitoring exponential changes in oxygen concentration during cultivation, using an oxygen electrode. The changes are brought about by suspending and restoring aeration and mixing in the bioreactor (Fig. 8.4). Changes in oxygen concentration in the culture are described by equation [8.17] which is reduced in region II, following suspension of aeration and mixing, to the form dC = -rX dt

[8.20]

249

Equations [8.7] and [8.20] hold as long as the oxygen concentration does not drop below a concentration critical for given microorganism, i.e. below the concentration at which the respiration rate begins to depend on the dissolved oxygen concentration. The respiration rate can be determined from the slope of the electrode trace in region II, aeration capacity from the slope of the plot of oxygen concentration against - d C / d t + rX. The limitation of this

L C§ . . . . t.-,+_ "~

r =d

r =x

b_ ............

II

y I

r }K L(:1

C1

0

time (s) Fig. 8.4 Time course of oxygen concentration in a culture after interruption (1) and renewal (2) of mixing and aeration

method is the same as in the gassing-out method" it cannot be used in the region of oxygen limitation and for testing large-volume bioreactors because of the sudden energy load on the impeller during restart.

8.3

FACTORS AFFECTING AERATION

CAPACITY(kLa)

The value of k L a in the culture device is affected by a number of factors. Among them are air flow rates, mixing intensity, rheological properties of the culture, presence of defoamers and some organic substances, ionic strength, concentration of carbon dioxide, temperature and pressure. A feature which is important for process scale-up is that the optimum value of kLa determined on the small scale should be applicable also on the large scale. This value can be achieved, although the equipment of culture devices varies widely. Quantitation of relationships between operational variables and kLa should enable a forecast of process conditions that would ensure the attainment of an appropriate k L a value (Stanbury and-Whitaker, 1986).

250

8.3.1

Critical oxygen concentration and oxygen consumption

As seen from equation [8.6] the rate of oxygen transfer is maximal at a zero concentration of dissolved oxygen. However, a decrease of oxygen concentration below a certain threshold, the so-called critical concentration Cc, brings about a decrease in the respiration rate which, under these conditions, becomes a function of oxygen concentration (Fig. 8.5). The critical oxygen 0.04

& "i-" I/1 t::: o

0.03 ri col concentration

0.02 (1; 1=: r o

,.., 0.01 0

t",4

Io

I

0

1 ---.-~. C L

I

i

2 (gO2

3 [-3)

Fig. 8.5 Dependence of specific rate of oxygen consumption (Qo2) on dissolved oxygen concentration (C) in a yeast culture (Finn, 1967)

concentrations for some selected microorganisms are given in Table 8.1. Like other concentration limits, oxygen limitation is manifested by a lowering of the growth rate. In addition, a prolonged residence of microorganisms in a medium with zero oxygen concentration can cause irreversible changes in the respiratory system. Oxygen limitation is therefore undesirable in practice

Table 8.1 Critical oxygen concentrations in some microorganisms (Finn, 1967) Microorganism

Azotobacter vinelandii Escherichia coli Serratia marcescens Pseudomonas denitrificans Yeasts

Penicillium chrysogenum Aspergillus oryzae

Temperature (~ 30.0 37.8 15.0 31.0 30.0 34.8 20.0 24.0 30.0 30.0

Cr (mM l -I) 0.018 - - 0.049 0.0082 0.0031 -- 0.015 -- 0.009 0.0046 0.0037 -- 0.022 --- 0.009 -- 0.020

251

and the concentration of dissolved oxygen has to be maintained at a level higher than the critical concentration. Critical concentrations are relatively low (of the order of units of per cent saturation) but in non-homogeneous systems such as culture broths of filamentous microorganisms can display local oxygen shortages which are reflected in increased values of critical concentrations (the so-called apparent critical concentration Cc). The values of apparent critical concentration for some microorganisms can be relatively high. The following values were given by Phillips and Johnson (196 l) for Aspergillus niger growing in a synthetic medium (compared with analogous values for Escherichia colt):

Microorganism

c

Aspergillus niger Escherichia coli

65

(gmol 1-l) 81

455 65

The apparent critical concentration C c corresponding to 455 t,tmol 1-~ represents, at an oxygen solubility in the medium C + = 5.8 mg 1-~, a value of C = 1.93 mg 1-~, in other words 33.3 % saturation. Commonly encountered critical oxygen concentrations C~ correspond to 5-10 % saturation. Under steady state conditions the rate of oxygen transfer into the culture equals the rate of oxygen consumption by the microorganisms kLa (C § - C) = Qo2x

[8.21]

or

c = c +-

Qo2X kLa

[8.21a]

At a constant temperature, culture composition and a constant partial pressure of oxygen in the gaseous phase, the steady state concentration of dissolved oxygen does not change and it is affected by Qo2X and by kLa.

8.3.2 Aeration and mixing intensity The rate of air flow has a relatively small effect on kLa values in usual mixed systems (Fig. 8.6). The commonly used range of air flow rates is 0.5-1.5 air volume per culture volume per 1 min (volume air flow rate) irrespective of the bioreactor size. The air flow can give rise to foam formation because, while the volume air flow rate increases with the third power, the outflow of inlet air is related to the culture surface area in the culture vessel and rises

252

therefore with the second power. The surface rate of oxygen transfer, i.e. the volume air flow rate referred to the bioreactor cross-section, can sometimes be kept constant during scale-up but this may bring about oxygen limitation. The choice of an optimum volume air flow rate therefore represents a compromise. Mixing has a positive effect on the rate of oxygen transfer because the kLa value is increased for the following reasons: a) finer air dispersion leads to increased interface area a; b) increased turbulence brings about a reduction of the thickness of the liquid film at the interface and an increase in the value of kL; C) the bubble holdup time in the bioreactor increases; d) mixing hampers air bubble coalescence.

0

__1

t

S I

0

0.5 ----air flow votume rate

1.0

Fig. 8.6 Effect of air flow rate on in a stirred and aerated vessel

kLa

The kLa value in stirred vessels increases with rising air surface velocity (i.e. with the amount of air supplied per bioreactor cross-section) to a certain critical value. This value corresponds to the point of impeller choking, i.e. a situation in which the impeller revolves essentially only on an air cushion. In a certain range the kua value is relatively independent of the bubble size because large bubbles, which have a relatively small surface area, create a larger turbulence in their vicinity, decreasing thereby the interface liquid film thickness and increasing kL. On the other hand, small bubbles have a larger surface area and bring about a lowering of turbulence (Finn, 1964). Only in very small bubbles (diameter below 1 mm) an increase in kLa can be observed. Because the bubble diameter is about 10-fold greater than the diameter of the opening through which air is fed into the bioreactor, these small bubbles can be produced only by using sintered glass spargers or ceramic candles.

253

Cooper, Fernstrom and Miller (1944) measured kca in aerated and stirred bioreactors with a single impeller and derived the following formula kca = k

[8.221

V~

where Pg is power input, V -- liquid volume in the vessel, V~- surface velocity of oxygen, k -- constant. Equation [8.221 implies that kca is nearly directly proportional to the power input per unit volume. However, Bartholomew (1960) showed that this proportionality depends on the vessel size and the exponent in the P g / V t e r m varies with the process scale: Process scale

Value of the

Laboratory Pilot plant Industrial

Pg/Vexponent

0.95 0.67 0.50

It should be noted, however, that the bioreactors used by Bartholomew for his measurements contained multiple impellers whereas those used by Cooper, Fernstrom and Miller were fitted with a single impeller. Top impellers are likely to have consumed more power to achieve the same values of oxygen transfer than bottom impellers, and this may have affected the value of the exponent. Richards (1961) pointed out that any relationship between kLa and the power input per unit volume is affected by a large number of variables which are not included in the above equations (impeller speed, size and shape, surface velocity of air and culture rheology). He proposed a formula which includes one of these variables kca = a

( )04

N

[8.23]

where Nis the impeller speed. Other relationships between kca and power input were verified during microbial cultivations. Taguchi et al. (1968) derived the formula kca = k

( )033

V~

'8 4,

254

while the formula derived by Steel and Maxon (1962) has the form kLa = k p0.46 g

[8.25]

Wang et al. (1979) proposed that in addition to a constant value of kLa which should be maintained during scale-up by changing the impeller diameter relative to bioreactor diameter, the circumferential speed of the impeller should also be kept constant during the scale-up.

8.3.3

Physical factors

Temperature Culture temperature affects oxygen solubility, i.e. steady state concentration of dissolved oxygen, as well as its diffusivity (transfer coefficient) in the culture fluid. Solubility of oxygen in water decreases with temperature while its diffusivity in the liquid phase rises nearly linearly with absolute temperature. The effect of temperature on the rate of oxygen transfer depends on the temperature range. At lower temperatures (10 ~ < T < 40 ~ the rate of oxygen transfer increases with increasing temperature owing to increasing oxygen diffusivity. At higher temperatures (40 ~ < T < 90 ~ the oxygen solubility decreases significantly, which negatively affects kLa.

Pressure Partial tension of oxygen in the culture affects mainly oxygen solubility and thereby also the driving force for oxygen transfer. The dependence of oxygen solubility on oxygen tension is given by the Henry law (Kargi and Moo-Young, 1985) Po2 =PcYo2 = He C~:

[8.26]

where He is the Henry constant, Pc -- total air pressure, Y o 2 - molar fraction of oxygen in the gas phase. Increase in Po2 (at a constant temperature) brings about an increase in + C 02 and a consequent rise in (C § - C) and in kLa.

Medium and culture rheology The rheology of nutrient media and cultures significantly affects the kLa value. A nutrient medium containing polysaccharides, whether as substrates or products, and mycelium exhibits a non-Newtonian behaviour. The rheolog-

255

ical properties of such media change during the cultivation (Tuffile and Pinho, 1970; Le Duy, Marsan and Coupal, 1974; Roels, van den Berg and Voncken, 1974) with changing concentrations of individual components of the culture broth (substrates, products, cell matter). Most bacterial and yeast cultivations take place in relatively low-viscosity Newtonian fluids in which turbulent flow can readily be achieved. The non-Newtonian behaviour of fungal and actinomycetal cultures poses problems with frequent limitation of growth and product formation by oxygen. The difference between oxygen consumption profile in unicellular and mycelial cultures is illustrated in Fig. 8.7. The

o

lb

I----oxygen I I

'

[imitation

I t---- oxygen II limitation

1

I

\\

\X\,/ i /

/

'I

", 'l

f

time

~

t... time

_

Fig. 8.7 Effect of oxygen limitation on the rate of oxygen consumption by a bacterial (a) and fungal (b) culture (Banks, 1977) 1 -- concentration of dissolved oxygen, 2 -- rate of oxygen consumption

100

QJ

.~_

.~'50 0

t I

1

I

0.4 0.8 1.2 ---..-concentration of mycetium (%w/v)

1.6

Fig. 8.8 Effect of concentration of Penicillium chrysogenum mycelium on kLa in a stirred bioreactor (Deindoerfer and Gaden, 1955)

256

overall oxygen consumption profile in the two cultures is the same in the exponential growth phase until the point of oxygen limitation is achieved. In the region of oxygen limitation the oxygen consumption in unicellular cultures is constant while in mycelial cultures it declines. The effect of mycelium concentration o n kLa in a Penicillium chrysogenum culture is shown in Fig. 8.8. 8.3.4

Chemical factors

Surface active agents Surface active compounds can both increase and decrease the kLa. Increase in kLa can be caused by a relative increase in bubble surface area, lowering of kLa c a n reflect accumulation of surface active substances at the gasliquid interface which enhances the resistance to oxygen diffusion. It has been experimentally established that the addition of 0.02 % (volume) sodium lauryl sulphate enhanced the rate of oxygen transfer by some 200 % whereas the addition of 0.01% silicon oil or 0.02 % Tween reduced the rate by about 50 %. Table 8.2 gives the effect of different surfactants on the kLa value.

Ionic strength The value of kLa depends considerably on the ionic strength of the culture broth. Robinson and Wilke (1973) derived a formula for the relationship between kLa and the ionic strength in Newtonian liquids

kLa

Pg V m --- " ~ T -s

p~533 /-)2/3 "-" 02 0.0.6 ,/.21/3

[8.27]

where A, is a function of ionic strength, t7 -- surface tension, Do2-- oxygen diffusivity, PL - - density of the fluid, ~uL - - v i s c o s i t y of the fluid. Parameters )~ and m are in an empirical relationship with the ionic strength I of the solution 1

2

I = ~- E Z~ C~ where Z~ is the charge on particle i, C; -- concentration of particle i.

[8.28]

Table 8.2 Effect of surfactants on the value of kL (Aiba, Humphrey and Millis, 1973)

Experimental conditions Surfactant d, (cm)

Q (cm 3 min-l)

kc (cm s-')

HL (cm)

C, (ppm)

kLat C ~ = 0

kLmin

Dioctyl sulphosuccinate

0.028

0.012

Alkylbenzenesulphonate

0.017

0.007

Pentapropylbenzenesulphonate Laurylsulphate

0.55 0.12

Synthetic detergent Laurylsulphate

daQ HE-Q -o- --

0.017 0.14

40

0.16 0.70

bubble diameter air flow volume rate height of liquid concentration of surface-active agent surface tension (l.tN cm -1)

ll0

128 0.85

40

70

0.027

0.014

0.017

0.056

0.013

0.042

0.005

300-

600

0.016

C~ at kLmin

10

20 20

0.016

0.039

0.028

Cs at O'rnin

15 1000

l0 50

100

25

258

8.4.

RHEOLOGY OF FLUIDS

Scale-up of microbial processes and the study of processes of motion, heat and mass transfers in industrial bioreactors requires information about the flow properties of nutrient media and cultures. These properties have a crucial influence on the overall effectivity of the process (Charles, 1978). The flow of fluids is the subject of rheology, which explores the relationship between different deformations of matter (elastic, plastic, flov~ deformation) and their causes. Rheology also explores the macroscopic properties of fluids as related to their microstructure. These relationships then serve for predicting the type and magnitude of deviations from Newtonian behaviour. The current state of rheology of non-Newtonian fluids has so far not permitted the description of their behaviour by a single flow equation. The method of choice is therefore the use of empirical relationships which best describe the experimental data. However, these purely empirical formulas bear no relationship to the actual mechanisms underlying the deviations from Newtonian behaviour. A number of fundamental definitions and terms will be presented, for example the definition of a Newtonian fluid, and the flow behaviour of various types of Newtonian fluids and the possibilities of a mathematical description of the flow of these fluids will be discussed. A fluid is a physical body capable of considerable resistance against external compression forces, and providing a much smaller resistance to small shape changes. Its particles are highly mobile. A fluid is envisaged as a continuum, i.e. a system with continuous distribution of matter and physical quantities characterizing its state. This concept is warranted in real systems whose geometrical dimensions are much greater than the dimensions of individual molecules; the fluid then behaves macroscopically as a continuous body. A fluid particle (element) is defined as a fluid volume which is sufficiently large relative to molecular dimensions and can thus behave externally as a continuum, but is relatively small in regard of the dimensions of the whole system. Fluid flow is the result of forces acting on the fluid. These forces can be divided into two groups: forces whose magnitude is proportional to the mass of a fluid element, such as weight, exert the same effect from the outside on a fluid element and on the whole fluid volume, and they are called volume forces. Forces of the other group act within the body of the fluid. They represent external forces with regard to individual fluid elements but internal forces with regard to the whole fluid volume. They are assumed to act only at a distance corresponding to the distance between neighbouring atoms (molecules) at the two boundaries of an ideal plane delineating a given volume element.

259 They are therefore termed plane forces. They can be divided into forces acting at right angles to the surface (normal forces such as pressure) and forces acting tangentially to the surface (tangential forces).

8.4.1

Viscosity and Newtonian fluids

An important basic concept indispensable for the description of fluid flow is the shear deformation. Let us assume, in agreement with Fig. 8.9, that individual layers of the material studied slide on each other without a volume change. At the same time the body is subject to the sole force P which acts tangentially to the contact area A of the two layers.

A

_~p

__.t dx I,__

T

Fig. 8.9 Model of shear deformation

The tension component acting in the direction of the relative motion of the two layers is called tangential tension, r. Let us introduce the term "relative deformation" y as the mutual displacement of the two material layers referred to their thickness. Hook's law P dx - r = G?' = G ~ A dy

[8.29]

holds for elastic deformation and can be expressed in words as follows" the deformation of a body is proportional to the force acting thereon. G is the socalled modulus of elasticity in shear. Hook's Law does not hold for plastic deformation and the deformation ?' increases as long as the tangential tension r acts. If the force stops acting the rise in deformation ceases and the body keeps its deformed shape. Viscous deformation (or flow) is characterized by increasing deformation under the action of the tension, the rate of deformation being proportional to the acting force. For most fluids the shear rate or

260

the rate of shear deformation D is a linear function of the tangential tension and obeys Newton's law d~'_ r = ~t - - ~ - #D

r

[8.301

where the proportionality coefficient ~t is the dynamic viscosity (viscosity in the following text) of the fluid. Fluids whose flow obeys Newton's law are called Newtonian fluids and their viscosity depends only on their molecular structure and on quantities of state (temperature and pressure). Viscosity is a measure of consistency of the fluid, i.e. its resistance against flow, or a measure of intrinsic friction of molecules. Along with density it represents a fundamental hydrodynamic parameter. Let us envisage a situation as shown in Fig. 8.10. The space between the immobile plane 1 and the mobile plane 2 is filled with a fluid with viscosity #. 2 IIIIIIIII/111~1l

[11111 II IIIII

I III

/1111111111/

IIII/11/I/111111111 1

V

Fig. 8.10 Scheme depicting the shear rate I -- immobile plate, 2 - mobile plate

The fluid adhering to plane 1 is immobile whereas the fluid adhering to plane 2 is carried along at a velocity v. If the width of the gap between the two planes is dy, equations [4.29] and [4.30] can be used for expressing the shear velocity as a function of the translational velocity v (under the provision of independence of derivatives) D = --d-t

=--~y -d--t = dy

[8.31]

The shear velocity D is thus equal to the velocity increment or gradient. In a Newtonian fluid the velocity gradient of individual fluid layers in the direction perpendicular (normal) to the acting tension is directly proportional to this tension while both the tangential tension and the shear velocity are at the same time causes and results of mutual displacement of fluid layers. On constructing flow curves (rheograms) for two Newtonian fluids whose

261

flows are described by equations [8.30] we obtain curves as shown in Fig. 8.11. As implied by the definition equation, viscosity # is the proportionality constant in the linear dependence r =/.tD and in the plot it is determined by the line slope, for example the tangent of the angle a. The viscosity of fluid 1 apparently exceeds that of fluid 2. If the two fluids are flowing, e.g., through a tubing under the action of the same tangential tension r~ = r2, proportional to the pressure gradient in the tubing and thus to the energy driving the flow, the fluid 2 with a lower viscosity will acquire a higher shear velocity 1)2 > D~, in other words it will flow faster and more easily than fluid 1. Rheograms of Newtonian fluids are rectilinear; the viscosity # of these fluids is independent Newfonian fluid

'E

G# I_ ..1-II1

4i---(lJ

m~ - ~2_ -

-

. . . . . . . . . I I

~'2.

2 I I

r.4--

7"..v"" ', \ !n~

//'~

----> shear velocity Is q ) r-

(,4

i!D2 Fig. 8.11 Rheogram of a non-Newtonian fluid

of the value of the shear velocity D and is constant for given physical conditions (temperature and pressure). 8.4.2

Non-Newtonian fluids

The flow of non-Newtonian fluids does not obey Newton's law (equation [8.30]), although the law can be formally written also for these fluids in the form r = #~D, where ~t~ is the so-called apparent viscosity. Index D denotes the dependence of apparent viscosity on the shear velocity D. In contrast to the Newtonian viscosity #, which is solely a function of quantities of state and represents a physical constant describing the properties of given material, the apparent viscosity #~ is a complex structural function which depends under given conditions on actual flow relations and has no significance as a physical constant.

262

The large majority of non-Newtonian fluids exhibit rheograms such as the curve I in Fig. 8.12. Such fluids are called pseudoplastic and their apparent viscosity is the lower, the higher is the tangential tension acting on them. Their consistence in the resting state is several orders of magnitude more viscous than under the action of high tangential tensions. These fluids include solutions and melts of high molecular weight substances (plastics, rubbers), solutions of soaps, detergents, lubricants and diluted suspensions. pseudoptostic fluid Ii/I

lf/i

t.-,

."

I

Ill I,...

A_Z___A_V

',

.,.,,,, ,.I,-. eQ,I eI:1 .4-.

/,I !X.-q .f

ii

!

I

t

I ~-

I ,

~ ~shear

I , velocity

DI D

(S-1)

Fig. 8.12 Rheogram of a pseudoplastic fluid

Curve 1 in Fig. 8.12 illustrates a pseudoplastic fluid, curve 2 denotes a Newtonian one. Let us follow the flow behaviour of the two fluids. When they are subject to a low tangential tension q the behaviour of the pseudoplastic fluid .1 will be defined by point A and the corresponding shear velocity Dl whereas the behaviour of the Newtonian fluid 2 is defined by point A' and the corresponding shear velocity D~. The following equation is seen to hold /~ ~ tg a < #o~ ~ tg a~

[8.32]

Under the action of tangential tension r~, the apparent viscosity of the pseudoplastic fluid #o~ is higher than the viscosity of the Newtonian fluid #, i.e. for r~ the Newtonian fluid flows better than the pseudoplastic one. On further increasing the tangential tension an opposite situation may obtain" at a sufficiently high tangential tension r2 the pseudoplastic fluid flows better than the Newtonian one. The apparent viscosity of pseudoplastic fluids decreases with increasing r (tg a2 < tg a~). Another major group of non-Newtonian fluids is represented by plastic or Bighamian fluids in which the shear velocity increases when the initial ten-

263

ptastic fluid ,.;., (,/I

u o'I

I/I

(:I,I r .4..._, .aG: G: I:I .,i-.

-

!i~gC~ltu de

Fig. 8.13 Rheogram of a plastic fluid

> sheor vetocify D(s -I)

sion attains the value r0 > 0 (Fig. 8.13). When the fluid in the vicinity of an immobile wall is subjected to a tangential tension r < r0 it retains its resting condition or flows with a flat velocity profile, i.e. at a zero shear velocity. Among these fluids are concentrated industrial and waste sludges, and paste-like suspensions of chalk and lime. Less common are dilatant fluids which exhibit a behaviour opposite to that of pseudoplastic ones. Their apparent viscosity rises with increasing tangential tension. Figure 8.14 compares the rheograms of a dilatant fluid 1 and a Newtonian fluid 2. The same consideration as with pseudoplastic fluids leads to the conclusion that at lower values of tangential tension r~ the dilatant fluid 1 can flow better than the Newtonian fluid 2 whereas at high values of the tangential tension r2 the opposite may hold true. di[afant fluid

1

..-... 'o'1

I

..._..

gl Or) OJ t.. ,4I/1 ...., ,4-r-QJ

r tr

..I--

--,-- shear vetocity D (s-1)

Fig. 8.14 Rheogram of a dilatant fluid

264

Rheograms of pseudoplastic and dilatant fluids sometimes exhibit a nonzero value of initial tension r0 and can therefore be viewed as plastic fluids with a nonlinear dependence of r on D. The above types of fluids (Newtonian, pseudoplastic, plastic and dilatant) have one property in common, namely the time independence of the tangential tension on the shear velocity D. However, this independence is a limit case because a time dependence of r / D is a frequently encountered phenomenon.

8.5

MIXING

Mixing is the blending of two or more different substances which aims at obtaining a homogeneous mixture. The term also denotes the maintenance of a given medium in an intensive motion which aids in mass or heat transfer or in the formation of a suspension. The method of choice involves achieving a high level of turbulence which ensures efficient transfer of heat and mass in a mixing device. A measure of mixing efficiency is the degree of homogeneity I defined by the formula I = 1 ---1 ~ n

i=1

C~--Ck

[8.33]

CO--Ok

For a given time the degree of homogeneity is determined by simultaneous sampling at n representative points in the mixing vessel, determining the concentrations of the component q under study and inserting the values into equation [8.33], where Cois the initial and Ckthe final concentration of the component. Mixing intensity is a frequently used, but hitherto imprecisely defined, term because different definitions of mixing intensities usually refer to a particular mixing system and a particular process. Three types of mixing are basically used in microbial processes: mechanical, pneumatic and hydraulic mixing. Mechanical mixing is carried out by mechanical mixers or impellers. Impellers impart energy to the fluid either by the action of blades on the fluid or due to tangential friction arising when the fluid comes into contact with the impeller surface. Mechanical impellers can be classified by a number of criteria such as the revolution velocity, direction of flow, and others. In devices using pneumatic mixing air is brought into the vessel from the bottom and it is distributed in the fluid in the form of bubbles. The bubbles rise and therefore bring along parts of the fluid. The contents of the vessel are

265

thus mixed and blended. Depending on the type of gas distribution the fluid flow in the vessel can be either unorganized or directed. Hydraulic mixing is based on a circulation of the fluid by a pump. The inlet and outlet tubings should be as far apart as possible. The outlet tubing ends in a nozzle immersed in the fluid. The conically shaped stream coming out of the nozzle displaces and carries along the ambient fluid (Fig. 8.15a).

0J

E

|

\

O .m

_~ ) .

.

.

.

,1 '

N N 0 r

I

r

Fig. 8.15 O u t f l o w from an i m m e r s e d nozzle with a circular cross-section (a) flow regions: 1 -- flow core, 2 - - mixing zone, 3 -- zone of rest; (b) streamline and longitudinal velocity rate profiles

Vortices are formed at the rim of the stream; the fluid in the vessel is gradually brought into accelerated motion and is thereby mixed (Fig. 8.15b).

8.5.1

M e c h a n i c a l mixing - - impellers

Mechanical impellers can in principle be divided to high-speed and lowspeed ones. The circumferential speed of low-speed impellers varies in the range vc c (0.5 ; 1.5) m-~ s-~, revolution frequency n c (20; 60) min -~. The impeller diameter is near the diameter of the stirred vessel D; this prevents both the formation of incrustations on the walls and local overheating. Low-speed impellers are used for blending highly viscous fluids. Figure 8.16 illustrates the basic types of low-speed impellers. Their dispersion efficiency is very low and their use in the microbial industry is therefore limited, although they have been tested for culturing filamentous microorganisms (Steel and Maxon, 1966). Among low-speed impellers belong also spiral impellers which operate at higher revolution frequencies (n c (30; 250) min-~). They are suitable for

266

blending of highly viscous fluids; fluid circulation is achieved at very low power inputs because these impellers generate a low level of turbulence. Lowspeed impellers include also blade impellers which usually consist of two right-angle or skewed blades with a diameter d ~ (0.5; 0.8) D. Their shortcoming is a low mixing intensity and a large proportion of the tangential component of fluid circulation.

i.

i!, l ii '.-lJ a

b

c

: d

Fig. 8.16 Low-speed impellers (a) horseshoe, (b) comb-shaped, (c) flame impellers, (d) flame impellers with blades

Circulation of the fluid during mixing by high-speed impellers depends on the type of the impeller used and on the presence or absence of baffles. High-speed impellers can be divided into radial, axial and disc types. A typical representative of radial impellers is the turbine impeller or Rushton turbine impeller fitted with a separating disc with six right-angle blades. Its diameter is usually d ~ (0.25; 0.33) D and revolution frequency n is chosen so as to ensure a circumferential speed Vc in the interval vc (3; 9) m s -~. Turbine impellers can be used in a broad range of viscosities of culture fluids (# ~ ( 10-3, 20) kg m-~s-t). In large-volume bioreactors the highest applicable viscosity value is somewhat lower. The turbine impeller has an excellent dispersion efficiency and has found a wide application in microbial industry. The dispersion efficiency and the level of turbulence generated by the impeller increases with decreasing impeller diameter while the circulation efficiency decreases. Turbine impellers with a separating disc can have straight, slanting or curved blades (Fig. 8.17). A representative of axial impellers is the propeller impeller which has the shape of a ship's propeller. Its dispersion efficiency is low but its circulation efficiency is high. For this reason it is sometimes used together with a turbine impeller which serves as disperser. Propeller impellers can be used in a broad range of viscosities (# ~ (10-3; 10) kg m-is -l) and their revolution frequency is chosen so as to ensure a circumferential speed in the interval Vc

267

(6; 15) m -~ s -~. The same function as the propeller impeller is fulfilled by a blade impeller with slanted blades. It is cheaper but has a lower pumping efficiency. The diameter of a blade impeller with slanted blades corresponds to that of a propeller impeller. Disc impellers are shaped like rotating discs and they impart energy to the fluid via viscous friction. They generate considerable shear tensions (high shear velocity gradients). They are therefore sometimes used in microbial industry as dispersers and foam breakers. They operate at high circumferential speeds (Vc c (5; 35) m -~ s -~) and cause only slight circulation of the fluid. Among these impellers are" a) bare disc, b) modified disc in which the disc rim is fabricated as in a circular saw, e) modified blade impeller with blade heights tapering off towards the impeller circumference, d) modified turbine impeller with a toothed ring at the rim. The self-suction impeller is firmly attached to a suction cylinder. Rotation of the impeller, which can be either a closed turbine element or a mere curved tube, creates a negative pressure which drives air, foam and fluid from the space above the fluid surface. The advantage of self-suction impellers is in

a

b

c

Fig. 8.17 Turbine impellers (a) straight blades, (b) slanting blades, (c) curved blades

that they obviate air supply otherwise necessary for air distribution, and foam is removed by back suction into the fluid. However, the rate of air suction is not very high and the fluid level in the vessel has to be kept at a constant height. Schematics of a self-suction impeller consisting of a rotating cylinder fitted in its upper part with openings are shown in Fig. 8.18 (K~ov/tk, Salvet and Sikyta, 1984). The fluid passes gradually through three working spaces that fill up the whole volume of the mixed vessel.This ensures a large contact area

268

and avoids the generation of large shear velocities. The impeller is suitable also for blending highly viscous fluids. In contrast to rotating impellers, vibrating impellers execute an axial motion. The impeller has the shape of discs fitted with openings all over their area. These are positioned on an oscillating shaft. The oscillations are produced by an electromagnet. In the upper part of the disc the openings form cones tapering upwards which give rise to directed fluid flow in the vessel. The oscillation amplitude is 0.5 to 3 mm, the frequency being about 50 Hz.

7

i i

2

6

Fig. 8.18 S c h e m e of a self-suction i m p e l l e r with a r o t a t i n g c y l i n d e r 1 - - rotor, 2 - - r o t o r b o t t o m , 3 - - o p e n i n g in r o t o r b o t t o m , 4 - - r o t o r shaft, 5 - - p a r a b o l o i d o f a rising fluid, 6 - - vessel, 7 drive, 8 r o t o r extension, 9 - o p e n i n g s , 1 0 - s u p p o r t s

The impeller is easy to mount in the vessel because it has a simple seal. It has a low power input and its operation does not produce too much foam. On the other hand, its use is limited to bioreactors with relatively small volumes.

8.5.2

Oxygen supply underneath the impeller

In small mixed vessels, gas is usually brought in via an axially positioned tube underneath the impeller. Impellers of smaller diameter have a high revolution frequency and a very high dispersion efficiency. This efficiency decreases with increasing impeller diameter, i.e. with decreasing revolution frequency. For this reason, gas supply in large volume devices is therefore ensured by a distribution ring with a diameter equal to, or slightly larger than, the impeller diameter d. The number and diameter of distributor openings can be determined from the equation (Leibson et al., 1956)

269

~Ps Ret = N~dt/,z~'

Ret c (500; 2500)

[8.341

where ~ is the gas volume flow, N~ -- number of nozzles, a~ -- nozzle diameter, s - - i n d e x denoting the liquid. The diameter of bubbles in the close vicinity of nozzles is given by the equation (Lehrer, 1971) a~ = 0.8205

(Vg)04 ~

[8.35]

At the same time the condition a~ =< 0.75 X~ has to hold true; Xt is the distance between two neighbouring openings on the distribution ring. When this condition is not fulfilled it is necessary to insert into equation [8.35] an effective number of openings N~,ef 0.75 ( U t - 1) Xt Nt,e f --

8.5.3

dt

+

1

[8.36]

Dimensionless criteria and geometrical similarity

Application of the theory of similarity to the dimensionless forms of differential equations describing fluid flow and subsequent modification of these equations has yielded the following dimensionless criteria characterizing the relations in the mixing vessel"

L Reynolds criterion for mixing ReM ReM- ndZp It

[8.37]

where n is the impeller revolution frequency, d - impeller diameter, p - specific weight of the fluid batch, ~ t - dynamic viscosity. Calculation of the Reynolds criterion for mixing of non-Newtonian fluids involves the computation of the mean shear velocity (DM) value for the given impeller revolution frequency (n) in the stirred vessel according to the equation (Metzner and Taylor, 1960) DM = 11.5 n

[8.38]

270

This correlation holds for d/D < 0.6 and m > 1. The apparent viscosity po corresponding to the calculated value DM is obtained from the rheogram of the stirred fluid and is inserted into equation [8.37].

II. Euler criterion for mixing EUM P

EuM =

n3dSp

[8.39]

where P is impeller power input. The Euler criterion describes the dimensionless impeller power input.

IlL Fround criterion for mixing FrM FrM =

nZd g

[8.40]

represents the ratio of inertial forces to gravitational forces. It is used for describing processes in which gravitational forces play an important role.

I

t

[3' r

u -c L I'

J

Fig. 8.19 Geometrical similarity of turbine impellers (a) and stirred vessels (b)

271

IV. Flow number Kp

V nd 3

Kp-

[8.41]

where V is the volume flow of the stirred fluid. A number of values of Kp have been published for turbine impellers, most of them in the interval Kp ~ (0.5; 0.8). V. Flow number for aeration Kpg

This parameter is defined analogously to Kp"

Vg Kpg-

[8.421

nd 3

where I28 is the gas flow supplied underneath the impeller. It is advantageous to introduce the term geometric similarity of two systems (impellers or stirred vessels) for cases when one of the systems can be transformed into the other by a simple multiplication of all dimensions by a proportionality constant. Figure 8.19a illustrates geometrically similar tur-

I

curve I. .

.

I I

1 oo0

.curve.

.

.

curve 3 .

.

curve

I

I

[

2

-t

t

e4

I~

C

~

l

z%

100

___B:O__

50

N t=EUM - ,

" 9 #z LLI

~'

1>,-!

l

\

I

.

•

.

-

o~

1

d

5

1 1 d 4 np=4 -

b= 0

''

~--I

'I

.

10

h d b d

. .

~,

1 5 1 10

-

''

.

1

h _ 1 d 8 b I d 12

rip= 4

--;4"-'::::3

~

~-~"

--t~

--~

I-- ~ ~.,~_.---- + - - - ' ~ " T - -

I

-'~I

'

h

' 1

1

I

I

h 1 d 5 b = 1 D 10

. . . . . . . . . . . . .

~.

I

h

D

\

10

F

i

I

1 _ 1 d 4 np= 0

X\.~

5

I

r-

+

"

I 10 2 ~--,,-

--

F

._..1

I 10 3

.....

np=4

B:Oj,:Eu..~ .....2'4_...

e=

L

-=

1Lto(:JR~

40

1 10 4

10 5

Re M = nd 2Q

Fig. 8.20 Power input characteristics of turbine impellers placed in a vessel with dimensions D = 3d, H - D and H 2 = d (Rushton, Costich and Everett, 1950)

272

bine impellers and Fig. 8.19b geometrically similar stirred vessels, including commonly used symbols for individual dimensions.

Power input characteristics Application of dimensionless criteria makes it possible to describe some phenomena occurring during mixing, independent of the size of the stirrer or impeller; the only necessary prerequisite is geometric similarity. Power input characteristics for turbine impellers with and without baffles are given in Fig. 8.20. For ReM < 10 the mixing is in the laminar range whereas for ReM e <10; 5000) it is in the transient region. For ReM > 5000 the mixing is in the turbulent region with a constant value of EuM. This region is also called the automodulation range. When ReM is expressed as shown above, the power input

I

N

I

\

I

I I

o-

z ~% 10 I

I

I

I I 1

1

II

1

E Ill

" ~'1, ' t ' -

~---

"1~

2

o o o o A^

A~

j

_1

;,,..,. I

'11 ""J

o

illl

10

lOO

1000

'1

ndZq Re M =

Fig. 8.21 Power input characteristics of a turbine impeller during mixing of non-Newtonian fluids for d~ 6 (0.2" 1.5> and/1o 6 (0.1" 18> kg m-~s -~ (Metzner et aL, 1961). Different symbols denote different experiental runs

characteristic under these conditions holds also for the mixing of non-Newtonian fluids. Figure 8.21 shows that for ReM ~ ( 10; 300) the power input is lowered as compared with Newtonian fluids due to the extension of the laminar region up to ReM = 30. The impeller power input in an aerated system is determined

273

from appropriate graphs or correlations. Figure 8.22 shows a typical decrease in the power input of a turbine impeller during aeration of Newtonian fluidgas systems. These results can also be used for aerated non-Newtonian systems. A frequently recommended correlation has the form

PZnd3 )0.45 ~56

Pg : 0.706

[8.43]

and holds for p c (800; 1650) kg m -3 and # ~ (0.0009; 0.1) kg m-' s -~.

1.0 0.9 0.8 0.7 o Z ........

c~

Z

o'~ o 0.6 0.5

E1E FT'~~+ ] N ' S~T ~ _

t b

+II [3 ~ 9

0.4

0

[] a

oA

_

0.3

o

+ Z~

oy

I 0.01

I 0.03

1 0.05

I 0.07

0 A

A o

1 0.09

1 0.11

Vg / nd 3

Fig. 8.22 Decrease in turbine impeller power input during aeration of Newtonian fluids (Calderbank and Moo-Young, 1961). Different symbols denote different experimental runs

8.5.4

Bubble holdup in an aerated vessel

Numerous data and correlations have been published on the problem of determining the mean bubble diameter dBM, gas holdup q)and interface area in a stirred vessel (Tsao and Lee, 1977; Blanch, 1979; Lee and Luk, 1983). However, they have for the most part been limited to the given geometry of a small-volume vessel and to physical properties of pure Newtonian liquids such as water or salt solutions. The absolute purity of the fluids is crucial because even a small addition of a surface active agent substantially affects the surface tension and thereby also the bubble size. Utmost care should be exercised when these correlations are extrapolated to actual devices because they

274

were derived from measurements in pure fluid-gas systems; even in these systems deviations of the order of tens of per cent are often found. The gas holdup in a stirred vessel 9 is defined as

(i):Vg V- v

[8.44]

where Vg is culture volume during aeration, V -- liquid phase volume. These values can be estimated from various complex correlations (Calderbank, 1958) which hold for pure and low-viscosity fluids. Determination of a mean bubble diameter dBM is subject to analogous limitations. The correlation given by Calderbank (1958) can be used for fluids approximating water -0.4

dBM = 4.15

ps0.2 o-O.6q~0.5 + 0.0009

[8.45]

where Ps is the density of the stirred fluid. However, the value dBM calculated in this way cannot be used in systems with different physical properties because of the different bubble coalescence in the culture. The range of bubble sizes is broader in aerated non-Newtonian fluids, especially in cultures of filamentous microorganisms, because very small bubbles are trapped among the filaments and larger bubbles are simultaneously formed by increased coalescence (Yoshida, 1982). The following correlation (Calderbank, 1958) is often recommended for calculation of interface area and total bubble surface area a in a stirred vessel referred to the volume of the aerated fluid 0"4.-.0 2

a = 1.44 ( Pg / V) p~ 0-0.6

@~

[8.46]

which is subject to the above limitations.

8.5.5

Flow pattern in a stirred vessel during aeration

Let us consider flow patterns in a stirred vessel with baffles (Fig. 8.23A). Unless air is brought underneath the impeller c via tubing e, two circulation loops d are formed in the vessel. The stream of fluid directed from the impeller towards the vessel wall carries along large vortices generated by the impeller. In this region the turbulence attains the highest value and the conditions for mass transfer are the best. The impact of the fluid against the wall gives

275 rise to pseudoturbulences and the fluid circulates in the vessel with a substantially lower level of turbulence. For this reason a stirred vessel with a turbine impeller is sometimes modelled in a simplified m a n n e r as an ideal mixer in which perfect blending is achieved immediately, with a circulation zone in which there is almost no heat and mass transfer. The highest velocities of the fluid are attained in the stream from the rotating impeller and along the walls.

Fig. 8.23 Flow patterns in stirred vessels during aeration a -- impeller axis. b -- surface of fluid, c - impeller, d - circulation loop, e - gas inlet, f - - baffle, g - gas bubbles; for A to E see the text

At other points in the vessel the flow velocities are lower and attain m i n i m u m values in the centres of circulation loops. It can be readily seen from Fig. 8.23A why the use of n turbine impellers on a c o m m o n shaft does not bring a power input equal to an n-multiple of a single-impeller input. The fluid flowing in a single-impeller vessel towards the centre of the vessel is retarded at the bottom by friction, whereas at the surface, mechanical energy is consumed and e x p e n d e d for n o n u n i f o r m ebbing of the fluid. No such losses occur in the space between two impellers; in fact, the streams due to circulation loops from the two impellers mutually accelerate each other. The energy re-

276

quired for return of the fluid from the wall towards the centre is also lower at the point of contact of the two streams than at any other point in the vessel. The total power input P2 in the case of two impellers is therefore equal to only = 1.414 times the power input of a single impeller Pl. In general we may write P,, = P, (0.586 + 0.414 n)

[8.471

When a small quantity of gas is fed underneath the impeller (Fig. 8.23B) it becomes perfectly dispersed, small bubbles follow fluid particles continuously and are even carried along in the upper circulation loop back towards the impeller. Small bubbles are essentially unaffected by the centrifugal force resulting from fluid rotation in the vessel; at higher gas flow rates this force (Fig. 8.23 C) causes a displacement of bubbles towards the centre of the vessel owing to the pressure of fluid of a higher density. Another circulation loop d with an opposite flow direction is consequently formed at the liquid surface. With increasing gas flow the impeller forms larger bubbles which are rapidly decelerated by the fluid and are then sucked back into the impeller (Fig. 8.23D). The transfer of other bubbles to the centre of the vessel is then more conspicuous and the circulation loop with the opposite flow direction grows. The fluid flow velocity is so low that practically no gas is carried along into the lower circulation loop. Further increase in gas flow causes choking of the impeller (Fig. 8.23 E) and the gas rises to the surface. The only motion imparted to the fluid is by rising bubbles which carry the fluid along. A single circulation loop exists in the vessel; the impeller has no effect on mass transfer and the rate of mass transfer in a vessel with a choked impeller is therefore very low. Efficient aeration aims at attaining the largest possible homogeneity of bubbles in the stirred fluid; the most suitable circumstances complying with this requirement are given in Fig. 8.23B and 8.23 C. For the majority of situations the gas flow (Fig. 8.23 B) will not be sufficient and the flow pattern in an appropriately aerated vessel will approach that shown in Fig. 8.23 C. The situation illustrated in Fig. 8.23 D is no longer appropriate because of the low dispersion efficiency of the impeller and a shortage of dissolved gas in the lower circulation loop.

8.5.6

Homogenization time

The time of homogenization T is defined as the time necessary for a perfect blending of the culture. The simplest method for its determination consists in adding a tracer compound into the mixing device and monitoring the time course of concentration changes. Depending on the type of the tracer

277

compound used, its momentary concentration at a given representative point is determined conductometrically, thermically or optically. The time of homogenization is given by the equation T=

CV V 1)" = C V"n'~"'S-~,bu,

= CTp

[8.48]

where V is the volume of the stirred vessel, I? volume flow of the stirred fluid through the impeller, Tp-- mean time of primary circulation, i.e. mean interval between two passages of a fluid particle through the impeller region. A value of C = 1 is expected for an ideal mixer (a single passage through the impeller region suffices for satisfactory.homogenization) whereas C = 5 is recommended for practical situations. Mixing with turbine impellers in a highly turbulent region (ReM = 50 000) can be defined by the correlation (Norwood and Metzner, 1960) T = 4.1

HD 3

(n4dli

),/,

[8.491

where His the height of fluid in the bioreactor.

8.5.7

Scale-up

The calculation of some parameters of the mixing process (e.g. impeller power input in an aerated and nonaerated system and in part also the time of homogenization) poses no serious problems during scale-up because these parameters can be described (for geometrically similar systems) by numerical values of dimensionless criteria. However, the hydrodynamic similarity achieved during the scale-up does not measure up to the geometric similarity because of the impossibility of maintaining the same values of all dimensionless criteria. The problems of scale-up are the most marked in aerobic microbial processes where the impeller fulfills two functions -- a pump for culture circulation and gas disperser. The working conditions of the impeller are therefore subject to numerous constraints (Einsele, 1978). Certain correlations exist for the immediate vicinity of the impeller (e.g. for power input in an aerated system or for impeller choking) but not for the whole bioreactor volume. These problems have to be addressed with regard to the local values of parameters in the whole volume, obtained by studying suitable mathematical models (Miura, 1976).

278

A number of criteria are recommended for microbial process scale-up" a constant value of the specific power input P/V, constant value of Reynolds criterion ReM, constant value of the circumferential speed of the impeller ltdn, a constant value of the term ndD -~ and others. However, none of the criteria are universal and the use of any of them depends on the particular scaled-up process (Oldshue, 1983). Scale-up often involves a change in the geometry of the system, which further complicates matters. Moreover, most of available data have been acquired by measurements on laboratory devices. Data from production-scale devices are rare and so are studies addressing their relation to data obtained on laboratory units for the same process. The scale-up of a microbial process is therefore still more of an art than an exact procedure.

10

o~

o

P / V " V "~

E >

0

1.0

0

_

0

0,1 0.1

1, 1 -----,--

I 10

~

1 100

"1000

V(m 3)

Fig. 8.24 Relationship between specific power input and bioreactor volume

Einsele (1976) made an attempt to compare data from small-scale and large-scale bioreactors for processes with both unicellular and filamentous microorganisms. His work implies that the specific power input decreases with increasing bioreactor size irrespective of the type of the process (Fig. 8.24). The values of specific power input for a nonaerated system are typically in the interval P / V e (1000; 3000) W m -3 and the following relationship holds P - - ~ V-~ V

[8.50]

Most production units work with relative impeller sizes d/D e (0.3; 0.45) and the height of the fluid in bioreactors is equal to twice the impeller diame-

279 1

1

I

I 10

I 100

10 Re = qnd2.... V 0.35

"2 "T

E "-"

1.0

oi

0,1 0,1

I 1 .,.,,,.

V

1000

[m3~

Fig. 8.25 Relationship between Reynolds criterion and bioreactor volume

ter H = 2D. Two turbine impellers are placed, as a rule, on a single shaft. The value of the Reynolds criterion ReM increases with increasing size of the device according to the formula

ReM ~ 1/0.35

[8.51]

(see Fig. 8.25). The times of homogenization increase approximately with the same exponent T-~ 1/0.3 I

I

[8.52]

I

1 000 T '-'V ~

.--... th I--

100

--

10 0.1

o

I

1

!

1

10

100

--4-

V (m 3)

ooo

Fig. 8.26 Relationship between the period of homogenization and bioreactor volume

280

and in large bioreactors they reach values of up to 100 s -~ (Fig. 8.26). On the other hand, the circumferential speed of the impeller remains constant irrespective of the size of the device and the type of microorganism (i.e. the rheological properties of the fluid; Fig. 8.27) ~tnd ~ (5; 6) m-' s-'

[8.531

The working conditions during scale-up should therefore be chosen according to this equation. Einsele (1976) found no production process with an oxygen consumption higher than No: = 8.3 mol-lm -3.

I

I

I

100

~T

E "13 e-

10 o o _.__.____.--~-

ll >

o

I

0.1

1

o

I

10 V (m 3 )

!

100

1 000

Fig. 8.27 Relationship between the circumferential speed of impeller and bioreactor volume

REFERENCES Aiba, S., Humphrey, A. E., Millis, N. F. (1973) Biochemical Engineering, Tokyo. Banks, G. T. (1977) Aeration of molds and streptomycete culture fluids. In: Topics in Enzyme and Fermentation Biotechnology 1, Chichester. Bartholomew, W. H. (1960) Adv. Appl. Microbiol. 2, 289. Bell, G. H., Gallo, M. (1971) Process Biochem. 6(4), 443. Blanch, H. W. (1979) Ann. Rep. Ferm. Proc. 3, 47. Calderbank, P. H. (1958) Trans. Inst. Chem. Engrs. 37, 443. Calderbank, P. H., Moo-Young, M. B. (1961) Trans. Inst. Chem. Engrs. 39, 337. Charles, M. (1978) Adv. Biochem. Eng. 8, 1. Cooper, C. M., Fernstrom, G. A., Miller, S. A. (1944) Ind. Eng. Chem. 36, 504. Deindoerfer, F. H., Gaden, E. L. (1955) Appl. Microbiol. 3, 253. Einsele, A. (1976) Scaling-up of Bioreactors. 5th Int. Ferm. Symp., Berlin.

281 Einsele, A. (1978) Process Biochem. 13(7), 13. Finn, R. K. (1964) Bact. Rev. 18, 254. Finn, R. K. (1967) Biochem. Biol. Eng. Sci. 1, 69. Kargi, F., Moo-Young, M. (1985) Transport phenomena in bioprocesses. In: Comprehensive Biotechnology 2, New York. K~ov~.k, P., Salvet, M., Sikyta, B. (1984) Biotechnol. Lett. 6, 307. Le Duy, A., Marsan, A., Coupal, B. (1974) Biotechnol. Bioeng. 16, 61. Lee, Y. H., Luk, S. (1983) Ann. Rep. Ferm. Proc. 6, 101. Lehrer, I. H. (1971) I.E.C. Proc. Des. Develop. 10, 37. Leibson, I., Holcomb, E. G., Cacoso, A. G., Jacmic, J. J. (1956) AIChE Journal 2, 296. Metzner, A. B., Taylor, J. S. (1960) AIChE Journal 6, 109. Metzner, A. B., Feehs, R. H., Rmos, H. L., Otto, R. E., Tuthill, J. D. (1961) AIChE Journal7, 3. Miura, Y. (1976) Adv. Biochem. Eng. 4, 3. Norwood, K. W., Metzner, A. B. (1960) AIChE Journal 6, 432. Oldshue, J. Y. (1983) Ann. Rep. Ferm. Proc. 6, 75. Phillips, H. D., Johnson, M. J. (1961) J. Biochem. Microbiol. Technol. Eng. 3, 277. Richards, J. W. (1961) Progr. Ind. Microbiol. 3, 143. Robinson, C. W., Wilke, C. R. (1973) Biotechnol. Bioeng. 15, 755. Roels, J. A., Van Den Berg, J., Voncken, R. (1974) Biotechnol. Bioeng. 16, 181. Rushton, J. H., Costich, E. W., Everett, H. J. (1950) Chem. Eng. Progr. 49, 467. Stanbury, P. F., Whitaker, A. (1986) Principles of Fermentation Technology, Oxford. Steel, R., Maxon, W. (1962) Biotechnol. Bioeng. 4, 231. Steel, R., Maxon, W. (1966) Biotechnol. Bioeng. 8, 109. Taguchi, H., Imanaka, T., Teramoto, S., Takatsu, M., Sato, M. (1968) J. Ferm. Technol. 44, 823. Tsao, G. T., Lee, Y. H. (1977) Ann. Rep. Ferm. Proc. 1, 115. Tuffile, C. M., Pinho, F. (1970) Biotechnol. Bioeng. 12, 849. Wang, D. I. C., Cooney, C. L., Demain, A. L., Dunnill, P., Humphrey, A. E., Lilly, M. D. (1979) Fermentation and Enzyme Technology, New York. Yoshida, F. (1982) Ann. Rep. Ferm. Proc. 5, 1.