Hydrodynamics of bubble columns in the liquid circulation regime

Hydrodynamics of bubble columns in the liquid circulation regime

che,,icaI E,@,,eeri,qg Science. Printed in Great Britain. Vol. 41. No. 10, pp. HYDRODYNAMICS 268-2692. 0009-2509/86 Pergamon 1986. OF BUBBLE C...

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che,,icaI E,@,,eeri,qg Science. Printed in Great Britain.

Vol.

41. No.

10, pp.

HYDRODYNAMICS

268-2692.

0009-2509/86 Pergamon

1986.

OF BUBBLE CIRCULATION

COLUMNS REGIME

IN THE

E3.00 ~0.00 Journals Ltd.

LIQUID

0. MOLERUS and M. KURTIN Lehrstuhl fiir Mechanische Verfahrenstechnik der Universitlt Erlangen:Ntirnberg, Erlangen, F.R.G. (Received

11 Mnrrh 1985)

Abstract-A physical model based on integral flow parameters is used to describe the hydrodynamics of bubble columns. Comparison with experiments from several different systems shows applicability of the model over the Froude number range investigated. Furthermore. the model allows more detailed information about the bubble dynamics to be predicted.

INTRODUCTION

In

a previous

paper

[l],

we

derived

relations

and

prediction

of the hydrodynamics

the liquid

circulation

diagrams which describe the hydrodynamics of bubble columns in the uniform bubbling regime at low superficial gas velocities. In particular, theoretical

ASSUMPTIONS

In

results predict maximum gas throughput which cannot be exceeded in the uniform bubbling regime. Experimental results normally reveal a transition to a liquid circulation regime at higher superficial gas velocities. The key role of bubble coalescence in this phenomenon is exhibited by the fact that with completely inhibited bubble coalescence, froth flow instead of liquid circulation is observed [ 11. Bubble coalescence can be manipulated to a large extent by adding small portions of surfactants (for example, 0.5 o/obutanol to water). From this fact it follows that the behaviour of an actual system (= particular mixture of gas-liquid and column geometry) is not predictible on the basis of

order

considerations, are inevitable:

to

is applied

in the present

paper

AND

reduce

the

in

DEFINITIONS

complexity

the following

of

simplifying

theoretical

assumptions

compressibility effects with respect to gas flow are neglected; ascend in the core region of the (ii) large bubbles column; (iii) monosized spherical small bubbles ascend homogeneously distributed relative to the circulating liquid; defining the resistance against liquid (iv) a relation circulation will be derived on the basis of reasonably defined mean values; is assumed. (v) no net liquid throughput In addition,

a Newtonian

liquid

is assumed

through-

out. The The

definitions and notation are given in Fig. overall void Fraction is defined by V,

The void fraction of the small reasonably defined as

&’=v,

1.

(1)

&= vs + v;

4).

bubble columns. Such a scheme

columns

(i)

hydrodynamic considerations alone. The aim of the present paper is to predict the system behaviour for industrially relevant processes realized in bubble columns, using data obtained from simple expansion measurements, i.e. the measurement of void Fraction E of the gas phase as a function of superficial gas velocity The state-of-the-art is characterized by empirical correlations for the gas content as a function of the material properties and the superficial velocities of the gas and liquid phases, respectively. A review has been given by Linneweber [Z]. Several authors [3-73 derived models for the prediction ofthe liquid circulation pattern. The model of Riquarts and Pilhofer [8], which takes bubble motion into account, results in an empirical correlation of- restricted accuracy. Owing to their empirical nature, such correlations ignore variations in system behaviour due to different physicochemical properties. In modelling the gas phase in the liquid circulation regime, two bubble class models were used by Beinhauer [9] and Joseph and Shah ClO]. Joseph et al. [ 1 l] applied a two bubble class model to reactions in

of bubble

regime.

bubbles

(subscript

1) is

VI

because the drag of the small bubbles depends on their concentration in the liquid phase. On the other hand, the void fraction of the large bubbles (subscript 2) is defined as

V2

-~

E2-

V*+Q

since the large bubbles ascend liquid and small bubbles. From

relations

(l)-(3) E2

to the 2685

relative

it follows =

E--E, I_El

to the mixture that

of

0. MOLERUSand M. KURTIN

2686

Taking into account relations (2), (3), (4) and (lo), it follows from eq. (11) that

EI(~ -E)

(12)

hence a nondimensional gas throughput due to the rise of small bubbles can be defined as

El(l --4

y’ =3@5i

1 -El

The nondimensional relative velocity of the small bubbles is defined by Fig. 1. Bubblecolumn, definitionsand notation in the liquid circulationregime.

The ratio of the core diameter to the column diameter is a==%.

E, = E,

(6)

and the void fraction of the gas phase in the core region is E--E1 (7) E E =gf+slFrom eqs (6) and (7) it folIows in particular for the difference E, - E, that =

1

(14)

O1 =3J-zJThus it follows from eq. (13) that VI =

l--E I-_EICl

(15)

Cl.

(5)

From restriction (iii) it follows that the void fraction of the gas phase in the outer annular region is

&,-&,

%I

E-E, p.

(8)

a:

The total gas throughput u,, consists of three parts: u0 = 111+u*+u,.

The superficial velocity, attributed to the rise of large bubbles, is thus p2 U2 = xD2/4

(10) In eq. (lo), H denotes the expanded height of the gas-liquid mixture. The corresponding superficial velocity is thus defined by

= m

V2

VR12’ (17)

Taking into account relations (3) and (4), it follows from eq. (17) that E-&1

(9)

The parts ul, u2 originate in the rise of small and large bubbles, respectively. u, results from the transport of gas bubbles by the circulating liquid. With known velocity u,,, I of the small bubbles relative to the surrounding liquid it follows that the volumetric flow rate of the gas due to the rise of the small bubbles is

fi, ui=Ao2/4.

Correspondingly, for a given relative velocity u,,2 of the large bubbles, the volumetric flow rate due to the rise of large bubbles relative to the surrounding gas-liquid mixture becomes

u2 = I-_El

%I 2’

For the liquid circulation, the superficial velocity w, is defined by

i;

= ;D2 w,.

With the liquid volume in the annulus v,,=

(l-s&

n(D2 - D,3 H 4

and taking into account relations (5), (6) and (19) it follows that the mean downward liquid velocity in the annulus is

j;H

(11)

W,=---=

VI,

(1 -E*;;ll-6,).

Hydrodynamics Correspondingly, the core region

of bubble columns

the mean (upward) liquid velocity in is found from eqs (7) and (19) to be

ti’H

w =l=VI, c

(l-el)~;~(E-E1)-

(21)

2687

GAS THROUGHPUT

DUE

The superficial gas velocity due to liquid circulation -sgwa.

(22)

Taking into account relations (6), (7), (8) and (21), it follows the superficial gas velocity due to liquid circulation is E--E1

%=

[(l-&i)6:-(E-.SE1)](1-E,)

WI_

(23)

With eqs (15), (18) and (23), the three different contributions u1 (i.e. rl), u2 and U, to the total gas throughput are related to parameters of physical significance, namely: nondimensional relative velocity ~~ of the small bubbles, relative velocity v,,,~ of the large bubbles and (superficial) velocity w, of the liquid circulation. Thus the physics of the flow provide relations which govern these three variables. In the following paragraphs, these relations will be derived.

Ub

;JsR,

=

-

In eq. (26), R designates the radius of curvature of the spherical cup bubble. Introduction of the diameter de of the sphere with equivalent volume gives vb

=

0.71&d,.

DUE

TO THE

(27)

Equation (27) holds for bubbles with diameters less than 0.3 D, if D designates the diameter of a column containing stagnant liquid. For large bubbles with d > 0.3 D, the rise velocity remains constant with increasing bubbIe volume, since the regime of slug flow is attained. This type of flow has been investigated by Dumitrescu [13]. In analogy to the Davies-Taylor equation, the rise velocity with slug flow is v, = 0.495

J

i.e.

g; (28)

V, = 0.35 GAS THROUGHPUT

RISE OF LARGE

The velocity of large bubbles rising relative to a liquid is, according to Davies and Taylor [12],

is u, = &,6,2W,-&,(l

TO THE

BUBBLES

&Z

.

RISE OF SMALL

BUBBLES

From restrictions (iii) and (v), i.e. homogeneous distribution of small bubbles and no net liquid throughput, it follows immediately that independent of any particular liquid circulation pattern there exists no net transport of small bubbles due to liquid circulation. Thus, for the contribution of the small bubbles to the total gas throughput, the results derived in [l] are applicable, if the relationships listed in Table 1 are considered. In particular, a nondimensional diameter of the small bubbles is introduced via B1 = 3J(I-_)$d,. In the homogeneous bubbling regime, the nondimensional superficial velocity is defined by y = EU, thus the corresponding variable in the liquid circulation regime is si ol. Thus for predictions, Fig. 4 of [ 1] can be used if the variables E, /I, y are replaced by E1, B 1, s101. Taking into account eq. (15), a relation

According to eq. (28), the bubble size corresponds to the size of the Davies-Taylor model with d, x 0.3 D, i.e. continuous transition from bubble flow without wall effects [eq. (27)] to slug flow [eq. (ZS)] is satisfied. Hills and Darton [ 143 investigated the influence of the concentration of small bubbles on the rise velocity of large bubbles. In contradiction to simple considerations (reduced buoyancy), increasing content of small bubbles increases the rise velocity of large bubbles of given size. This increase in rise velocity is attributed to the influence of the turbulent moving liquid-gas mixture on the surface geometry of the large bubbles. As a rough estimate, the measurements reported in [ 147 indicate a 60 % increase in rise velocity with small bubble void fractions s1 > 0.08. Thus, with large bubbles ascending in the core region and recirculation of a gas-liquid mixture in the outer annulus, the following holds for large bubbles with d, 2 0.3 6,D, taking into account eq. (28): Urel 2 =

1.6 x

0.35 ,/g

6, D

i.e.

(29) u,CI2 = 0.56 m.

is defined.

Introduction of eq. (29) into eq. (18) gives Table 1. Corresponding variables, uniform bubbling regime [I]

and rise of small bubbles (present paper)

Uniform bubbling regime [l] Rise of small bubbles (present paper)

GAS THROUGHPUT

DUE

TO LIQUID

CIRCULATION

Equation (30) reflects only that part of the gas throughput carried by the large bubbles which is attributed to the motion of the bubbles relative to the

2688

0. MOLERUS and M. KURTIN

surrounding gas-liquid mixture. Due to the liquid circulation, however, the absolute velocity of the large bubbles is higher than u,,~. Since the large bubbles ascend only in the core region, there exists a distinct contribution of the liquid circulation to the gas transport by large bubbles. Thus, for a treatment which is based on the physics of flow, the liquid circulation must first be modelled. With liquids of not too high a viscosity and for column diameters typically used in practice, measured liquid circulation patterns give Reynolds numbers WI D Re = ~ ”

%- 1.

The balance

APnCD2 - Df) 4 -

core region and of downward velocities in the outer annulus (Fig. 1). The boundary conditions for the timeaveraged flow field are thus w = 0 at r = DC/2 and r = D/Z. Owing to the velocity gradients, shear stresses are transmitted at the solid-liquid boundary r = D/2 as well as at the liquid-liquid interface r = DJ2. For the sake of simplicity, the shear stresses at both positions are assumed to be equal. For an element of length AL, a balance of forces for the core (= pressure drop AP, weight of the gas-liquid mixture and shear stress r,+,) reads (compare Fig. 2): A+

=

s&+(1

--=)P,

AP

1

AL pig

I-&=

4%

E, - E, =

6,)

Equation (34) reflects the fact that the liquid circulation is driven by the density differences of the gas-liquid mixtures in the core and in the annulus, respectively, and is retarded by the action of the wall shear stresses. With turbulent flow, the mean hydraulic diameter concept can be applied to the time-averaged field of flow. Thus, for the shear stress acting on the annulus one can assume that 5,

A(Re

)PI

(35)

=Yyw”z

with a friction factor I., which in the turbulent flow regime shows a weak dependence on the Reynolds based on the mean hydraulic it

diameter,

= (1 -~,)Dw, V

i.e. (36)

Insertion of eq. (35) into eq. (34), and using eqs (8) and (20), yields an expression for w,: (a-&,)(1

-&,)‘(l

-a;)‘(1

-6,)

~(Re,)6,

Fig. 2. Balance of forces for the core region.

(33)

(34)

Da,(l -

Re

(32)

form

Elimination of the pressure gradient, i.e. equating of eqs (32) and (33) gives

(31) in the nondimen-

+ D,)AL

4%

$AL+~,D,~AL.

Rearrangement of eq. (31) results sional representation

4

P,gD(l-6,)’

number

1

r,x(D

reads (Fig. 3):

9

%Psf(l-%)P,

and in the nondimensional

From this fact it is reasonable to assume that the liquid circulation is turbulent, i.e. with dominating inertial effects. If one ignores time-dependent fluctuations of velocities as well as of liquid-liquid interfaces, the time-averaged profile of the liquid circulation shows a continuous distribution of upward velocities in the

14D2 -0:) AL

of forces for the annulus

Fig. 3. Balance of forces for the annulus.

Hydrodynamicsof bubble columns

2689

Insertion of eq. (37) into eq. (23) yields

(38)

NONDIMENSIONAL AMOUNT

REPRESENTATION

OF GAS

CARRIED

BY LARGE

OF THE BUBBLES

Measurements recently published by Nottenkamper et al. [15] and experimental data compiled by Zehner [7] indicate that in the liquid circulation regime for different column diameters and with variation of gas throughput, 6, = 0.75 is observed. This result is at least plausible since it implies nearly equal areas for upward and downward flow of the liquid phase. With this result, eq. (30) becomes u2 = 0.485Z

(39)

,&.

As described, one can expect only a weak dependence R = I(Re,). Considering the other simplifying assumptions, it is reasonable to assume that dm x constant. With S, = 0.75 and all constant parameters summarized into one constant C, it follows from eq. (38), with ps e p, that u, =

c

1 -si

(&-

El,‘ I’_

- 1.78

JDS

(E-Ed)



(40)

Addition of eqs (39) and (40) yields the nondimensional representation of the amount of gas which is carried by large bubbles in terms of a Froude number:

% + UC

Fr,.,, = ____ fi +c

DETERMINATION

= 0.485 6l-E, * (E -El

1 -or

)3’2

(41)

- 1.78 (s--&r)

OF THE

CONSTANT

C FROM

EXPERIMENTS

For a given gas throughput, the overall void fraction E of the gas phase is easily determined from expansion measurements. Since the rise velocities of the large bubbles are high in comparison to those of the small bubbles, sudden shutdown of the gas supply provides an accurate measurement of the void fraction sr of the small bubbles. In Fig. 4, the pressure difference AP* between the static head of pure liquid and the actual pressure drop over a distance of AL = 1 m is depicted vs. time t elapsed after shutdown. The pressure fluctuations observed for short times correspond to the passage of the last large bubbles. Thereafter a nearly constant value of AP* is recorded which corresponds to the void fraction er of the small bubbles. After the passage of the last small bubbles at the lower probe position, the pressure difference falls

Fig. 4. Pressuredrop vs. elapsed time after shutdownof the gas supply. off linearly to AP* = 0, which corresponds to the liquid phase at rest and free from gas bubbles. Thus simple expansion measurements after passage of the last large bubble give E, [eq. (2)]. Measurement procedures as described in [ 11, for example, two point tip probe measurements, yield the size of the small bubbles. Furthermore, one can assume that the bubble sizes observed in the homogeneous bubbling regime also provide a good estimate for the size of small bubbles in the liquid circulation regime. Thus, by simple expansion measurements at low superficial gas velocities and mapping of the data into Fig. 4 of [l], the nondimensional bubble size j3i is obtained. Taking into account the relationships given in Table 1, and using the actual value er in the liquid circulation regime, one obtains from Fig. 4 of [l] the value &ror and from eq. (15) the nondimensional gas throughput y1 attributed to small bubbles. The absolute value ur then follows from eq. (13). Subtraction of u, from u,, gives the sum u2 + u, according to eq. (9). By evaluation of simple measurements, all the variables in eq. (41) are known and, hence, the constant C can be determined. Using data taken from our own experiments in three different bubble columns (D = 0.1,0.19,0.5m), the constant C was determined to be C = 1.5 by a least-squares fitting procedure. With 0.1 m < D < 0.5 m, the geometrical factor in eq. (41)

was varied as

fi,

i.e. by

a

factor

of

Js x 2.24. The accuracy of the determination of C was checked by substitution of the void fraction s1 of the small bubbles using eq. (41) and comparison with values measured after a sudden shutdown of the gas supply (Fig. 5). Less than 10 o/0deviation was observed in nearly all cases. STATE

DIAGRAM

REGIME:

FOR THE

COMPARISON

LIQUiD WITH

CIRCULATION EXPERIMENTS

By evaluation of eq. (41), i.e. plotting of E over Fr,,G, with .sl as parameter, a state diagram for the liquid circulation regime is established (Fig. 6). The accuracy of the prediction can be seen from Fig. 7, in which the measurements from the three investigated columns are depicted.

2690

MOLERUS and M. KIJRTIN

0.

Fig. 5. Void fraction eI of small bubbles: comparison of prediction and measurement for the air-water system.

In order to show the physical significance of the state diagram given in Fig. 6, experimental results obtained with different liquids are presented in Fig. 8. Vermeer and Krishna [ 161 investigated a column of D = 0.19 m using as the liquid turpentine 5 (pI = 761 kgm-3; p = 0.91 x 10m3 Nsm-‘). Godbole et al. [17] published measurements of the gas content using an organic liquid (pl = 751 kgme3; p = I.3 x 10e3 Nsm-‘)inacolumnofD = 0.305m.As Fig. 8 reveals, again a significant deviation is observed only for one measurement point at Fr,,,, 5 0.06. The measurement data indicate applicability of the underlying model for all Reynolds numbers [eq. (36)] investigated, i.e. for Re, z lo*.

USE OF THE STATE 0.5

I

E 0.4

0.3 0.2 0.1

0.0 0.0

0.1

0.2

0.4

0.3 Frtc

OS

-

Fig. 6. State diagram for the Iiquid circulation regime.

0.5

I

E 0.1.

DIAGRAM

FOR PREDICTIONS

With these results, obtained from simple expansion measurements, i.e. measurements of E = E(z(~) for a given system (gas, liquid, column diameter), relevant process data can be determined as follows. At low superficial gas velocities uO, the uniform bubbling regime prevails. Mapping of the measurement data (6,~) in the state diagram for uniform bubbling (Fig. 4 of [l]) yields a nondimensional bubble diameter 8, which can also be regarded as a good estimate of the nondimensional bubble diameter /?I of the small bubbles in the liquid circulation regime. For higher gas throughputs, the liquid circulation regime prevails. As will be shown, with higher gas throughputs the contributions u2 and uCrwhich originate in the rise of the large bubbles and the liquid circulation respectively, exceed the contribution u1 of the small bubbles. Thus, as a first approximation, one can calculate the Froude number from measured superficial gas velocities, uO:

0.3 0.2 0.1 0.0 0.0

0.1

0.2

0.3

0.4

0s

Fh< -

With this value one can determine a point in the state diagram (Fig. 6) using measured data E and Fr,, . From Fig. 6 one can read off a first approximation E~ for the void fraction of the small bubbles. With known data &I and /.?I, Table 1 can be used to read off from Fig. 4 in

Fig. 7. State diagram: comparison with measurements for the air-water system.

The filled symbols in Fig. 7 indicate the positions according to measured values of E; the thin bars (where necessary) indicate the positions according to measured values Ed. Significant deviations between these two results are observed only for Frl,,. 5 0.07. These deviations have to be attributed to the fact that with low Froude numbers the liquid circulation pattern is not yet fully developed. This statement is supported by the observation that measured values of E~ are higher than those predicted for Fr,.,. 5 0.07.

0.0

0.r

0.2

0.3

0.4

Fr,,

-

Fig. 8. State diagram (measurementstaken from [16, 171). Representation of data as in Fig. 7.

Hydrodynamics

of bubble columns

2691

1

H

t* =

%eu+Wc=0485+15

X

04

t

0

0.3

0-L yImw---

Fig. 9. Relative contributions t(,/uO. u~/u,,, uC/uo vs. superficial gas velocity uO. air-water system, D = 0.19 m.

[l]avaluecI~l. Withknowndatas,sl andaI,onecan then calculate from eq. (15) the nondimensional gas throughput yl, hence from eq. (13) the absolute value u1 due to the rise of the small bubbles. Subtraction of u1 from u0 gives an improved value Frl.c.. With this value an improved point (Fr,,=,, E) in the state diagram (Fig. 6) is defined. Thus, from Fig. 6 an improved value Ed can be read off. A converging iteration is established which finally gives from a measured set of data uO, E more detailed information, namely ul, u2, uc, E, cl and

Bl .

As an example, from the measured relation E = .s(uc) in Fig. 9 the relative contributions uI /u,, u2 /u,, u,/uO are depicted vs. the superficial gas velocity uO_ With known data cI and E,, the gas-liquid interface of the small bubbles can be calculated, which defines the overwhelming part of the total gas-liquid interface. From eq. (13) it follows that the relative velocity of the small bubbles becomes

&-~)gv&+.

v,,,~ =

H tmin

(44)

From eqs (21), (23) and (40), the mean liquid velocity in the core region with C = 1.5 can then be expressed as

(1 -s1)JG 1 -sr - 1.78(c-cc,)

H

C

4

D 9

H AL

AP AP*

U vb

ureI OS

V P

constant equivalent spherical diameter, m diameter, m gravity, m SK’ height, m length, m pressure difference, N rnpz pressure difference, N m- 2 radius, m radius, m hold-up, s shortest gas residence time, s superficial gas velocity, m s- ’ rising velocity of large bubbles, ms-’ liquid circulation velocity, m s- ’ relative velocity, m s- r rising velocity of slugs, m s - r volume, m3 volumetric flow rate, m3 s- 1 liquid velocity, m s- 1

(45)

Since the large bubbles ascend only in the core region, with u,rz and w, known from eqs (44) and (45), the hold-up tz of the large bubbles can be calculated as

(47)

(i) contributions of small bubbles, large bubbles and liquid circulation to the total gas throughput; (ii) gas-liquid interface due to the presence of small bubbles; (iii) hold-ups for small bubbles, large bubbles and shortest residence time.

W

-1.78@--1)

0.485 + 3

CONCLtJSIONS

UC

l-s1

%,Iz+2w, =

With the aid of a phenomenological model calibrated from simple expansion measurements in several test cases, detailed information about the flow pattern in bubble columns can be predicted:

tmin

From eq. (29) the relative velocity of the large bubbles (6, = 0.75) follows:

w, = 1.5

1

A%

t

0.485 fi.

=

x-.

&

=

(46)

JG’

NOTATlON

(42)

With no net liquid throughput and homogeneous distribution of the small bubbles, the liquid circulation does not affect the hold-up of the small bubbles. For an expanded height H of the gas-liquid mixture, it follows from eq. (42) that the hold-up of the small bubbles can be expressed as

Vrel2

H

According to measurements published in the literature [ 13, 181, the liquid circulation profile in the core region is parabolic. Thus, the maximum liquid velocity is w, Inax= 2 wc and hence the shortest gas residence time reduces to

J 0.2

0.1

(l-Q),/= . 1 -E, - 1.78(s--E,)

Greek

6, “n

letters

diameter ratio void fraction friction factor

2692

cc V

P 7,

O.MOLERUS

dynamic viscosity, kg m- ’ skinematic viscosity, mz s- ’ density, kg m- 3 shear stress,

and



N m - *

Nondimensional groups Fr Froude number Reynolds number Rt?

B Y 0

nondimensional bubble diameter nondimensional gas throughput nondimensional relative velocity

Subscripts 0 total gas flow

1 2 a exp.

small bubbles large bubbles annulus core experimental value

g 1 l.c. theor.

gas liquid liquid circulation regime theoretical prediction

C

REFERENCES

Cl1 Molerus 0. and Kurtin M., Hydrodynamics of bubble

columns in the uniform bubbling regime. Chem. Engng Sci. (in press). c21 Linneweber K.-W., tirtliche Gehalte an Gas sowie an Gas und Feststoff in Blasenslulen. Dissertation, TU Munchen 1981. c31 Rietema K. and Ottengraph lr., Laminar liquid circulation and bubble street formation in a gas-liquid system. Trans. Inst. Chem. Engrs 1970 48 T54T62. c41 Freedman W. and Davidson J. F., Hold-up and liquid circulation in bubble columns. Trans. Inst. Chem. Engrs. 1969 47 T251-T262.

M. KURTIN c51 Ueyama K. and Miyauchi T., Properties of recirculating

turbulent two phase flow in gas bubble columns. A.I.Ch.E. J. 1979 25 258. C61Joshi J. B. and Sharma M. M.. A circulation cell model for bubble columns. Trans. Inst. Chem. Engrs. 1979 57 244. E71 Zehner P., lmpuls-, Stoff- und Wlrmetransport in Blasens%ulen. Teil 1: Stromungsmodell der Blasentiule und Fliissigkeitsgeschwindigkeiten. Yer-ahrenstechnik 1982 16 347-351. PI Riquarts H.-P. and Pilhofer Th., Model1 des heteroin genen Strdmungszustandes Blasensaulen. Verfahrenstechnik 1978 12 77-80. PI Beinhauer R., Dynamische Messungen des relativen Gasgehaltes in Blasensgulen mittels Absorption von Rontgenstrahlen. Dissertation, TU Berlin 1971. Cl01 Joseph S. and Shah Y. T., A two bubble class model for churn turbulent bubble column slurry reactor. ACS Symp. Ser. 1984 237 149167. Cl11 Joseph S. Shah Y. T. and Carr N. L., Two bubble class model for mass transfer in a bubble column with internals. Int. Symp. them. React. Engng 1984 8 223230. Cl21 Davies R. M. and Taylor G., The mechanics of large bubbles rising through extended liquids and through liquids in tubes. Proc. R. Sot. 1950 375390. t131 Dumitrescu D. T., Stromung an einer Luftblase im senkrechten Rohr. 2. angew. Math. Mech. 1943 23 139. Cl41 Hills J. H. and Darton R. C., The rising velocity of a large bubble in a bubble swarm. Trans. Inst. Chem. Engrs 1976 54 2588264. Cl51 NottenkBmper R., Steiff A. and Weinspach P.-M., Experimental investigation of hydrodynamics of bubble columns. Ger. them. Engng 1983 6 147-155. L161 Vermeer D. J. and Krishna R., Hydrodynamics and mass transfer in bubble columns operating in the churnturbulent regime. Ind. Engng Chem. Proc. Des. Dev. 1981 20 475. Cl71 Godbole S. P.. Joseph S., Shah Y. T. and Carr N. L.. Hydrodynamics and mass Transfer in a bubble column with an organic liquid. Can. J. them. Engng 1984 62 440. Cl81 Hills J. H., Radial non-uniformity of velocity and voidage in a bubble column. Trans. Inst. Chem. Engrs 1974 52 1-9.