Mixing and mass transfer in tall bubble columns

Chemical

Engineering

Science, 1974,Vol. 29, pp. 2177-2188. Pergamon Press.

MIXING

AND MASS TRANSFER BUBBLE COLUMN?

W.-D. DECKWER, Institut

Printed in Great Britain

R. BURCKHART

IN TALL

and G. ZOLL

fiir Technische Chemie, Technische Universitst Berlin, 1 Berlin 12, Germany (Received 5 April 1974; accepted 2

k,a were obtained adjusting the experimental

predictions the axial dispersed both columns differ by a factor

flow model.

various liquid phases which only depend

the different gas

two. Correlations

the

INTRODUCTION

Bubble columns (BC) are frequently applied as gas liquid reactors for the manufacture of important products in the process industries [l-6]. In petrochemical applications bubble columns showed certain distinct advantages over fixed bed reactors, namely, high space time yields and improved selectivity resulting from the excellent temperature control which was possible in this type of reactor[l,4,6]. Mixing in the reactor is obtained from the sparging action of the gas alone, and thus, a major advantage of this form of reactor is its simplicity of design which provides ease of operation and maintenance owing to the absence of any mechanical parts of operation. Quite recently bubble columns gain increasing importance in the area of biotechnology particularly in fermentations and waste water treatment because bubble columns provide favourable mixing and mass transfer properties combined with low shear stressing of the biological material. In these processes the mass transfer rates in bubble columns from the gaseous to the liquid phase are sufficiently high. The present paper reports the results of experimental studies on tall bubble columns with cocurrent flow of the phases. The objective of these investigations had been to measure gas holdup, mixing of the liquid phase and volumetric mass transfer coefficients for oxygen. Furthermore, it was intended to determine from the measured concentration profiles along the column whether the mixing and mass transfer coefficients were axially dependent or constant. Water, aqueous solutions of

are

profiles with the values

data

for

the

velocity. electrolytes, and molasses liquid phases.

solutions

were used as

APPARATUS

Measurements were performed with two bubble columns of different dimensions. Bubble column I (BC I) has a diameter of 20 cm and a total height of 723 cm. A cross with 56 nozzles of 1 mm dia was used as gas sparger. Bubble column II (BC II) has a diameter of 15 cm and its length is 440 cm. In BC II the gas was sparged by a glass sintered porous plate with a mean pore diameter of 150 pm. The porous plate reduces the cross sectional area of the tube to 14 per cent, while the cross of nozzles diminishes the cross sectional area of BC II to 72 per cent only. Both columns were made from glass and provided with numerous side connections (Schott & Gen., Mainz). Although bubble columns in industrial applications have larger diameters they are usually equipped with baffles and heat installments. Therefore the columns used in the present study are.well suited to simulate conditions which may prevail approximately in industry. All measurements were carried out with cocurrent flow of the phases. The mean temperature was 16°C. The linear gas velocities applied were smaller than 5 cm/set for the main part of the measurements, because at higher gas through puts the formation of large coalesced bubbles and slugging was observed, which decreases the efficiency of bubble columns considerably [7,8]. The liquid phases used were: tap water, aqueous solutions of sodium sulfate (0.225 N and 0.7 N) and sodium chloride (0.17 N) as well as aqueous solutions of

2177

2178

W.-D.

DECKWER et al.

r

molasses with various concentrations. The sugar content of the molasses is about 50 per cent. The pH-value of the molasses solutions was adjusted to 3.5 with sulfuric acid in order to prevent infections. Finally hexanol was added (0.1 per cent) to depress foaming.

0 0

0

x 0 0

molasses 0,50 ‘I. l,oo %

1.93 v. 131 % 5.00 ‘I. IO.0 %

BC I

SalIs

GAS HOLDUP

The fractional gas holdup is highly sensitive to the gas distribution system[9-111 and to trace impurities, as was shown by Anderson and Quinn[l2]. After bubbling for about 1 hr a good reproducibility of holdup was obtained for tap water and solutions of electrolytes. As transient effects and day-to-day fluctuations were not observed the concentration of contaminants is obviously at a high enough level where the flow characteristics of the air-water dispersions are influenced no more. Contrary to this a strong scattering of the holdup data was found for the solutions of molasses (Figs. 2 and 3). Holdup data are shown in Fig. 1 for tap water. The maximum of the holdup in BC II is typical for bubble columns with porous spargers[7-91. It corresponds with. the transition from the pseudohomogeneous flow regime with almost ordered bubbling to the heterogeneous flow regime with large spherical cap bubbles [7]. Additionally in Fig. 1 values of l are cylindrical bubble column with a diameter of 20 cm and a length of 250 cm and a cross of nozzles (1 mm) as gas sparger too. As these data agree well with those of BC I, it can be concluded that the liquid height does not effect the holdup in bubble columns with nozzles as gas spargers. This is in Contras t to columns with porous plates as gas Ec 0.2:

;r

0

l

Na+O,,0,70N

v

Na,SO, ,422SN

o

q

’

50

Fig. 2. Gas holdup

U&

LcmM

in BC I.

distributors [8,13]. The holdup data presented in Fig. 1 concur satisfactorily with those given in [9,14] which were obtained in similar columns. Figures 2 and 3 show the holdups of the solutions of molasses and electrolytes. In comparison to water the gas holdup is increased,. particularly in BC I for the solutions of molasses. In BC II the holdup falls below that of water at a linear gas velocity of about 3 cm/set. At this velocity the sudden formation of spherical cap bubbles could be observed visually which happens at gas velocities of about 5 cm/set at first when using tap water as liquid phase. This fact indicates that the transition from the homogeneous to the heterogeneous flow regime, above which bubble columns are less efficient, occurs at lower gas velocities in electrolyte solutions. A dependence of the holdup on the concentration of electrolyte cannot be ascertained from the measured data. This conclusion coincides with the observations of Lessard and Zieminski [15] since the concentrations applied in the present study are always higher than the critical concentration above which coalescence is prevented almost completely at low bubble densities. MIXING

0

3

6

9

Fig. 1. Gas holdup

12

15

uGO [cm/s1

for tap water.

The mixing in the liquid phase was determined for tap water only. The mixing properties of bubble columns are usually described by the axial dispersed plug flow model. In order to determine the dispersion coefficients of BC I and BC II the

Mixing and mass transfer in tall bubble columns EG

0,ZE

;r

0 x II v

molasses 0,50 7. I,00 % 1.93 ‘I. 3.37 v. 5.w % IO,0 ‘I.

l

salts : Na2SOL

, 0.70

.

N+SO,

, 0,225

+

NaCl

, 0.17 N

0 A

0

2179

to0

2P

N N

3,O

Fig. 3. Gas holdup in BC II.

stationary methods was used. Thereby a tracer is introduced and its profile is measured upwards the bulk tlow of liquid. The dispersion coefficients are calculated from this profile. Electrolyte, dye and heat were applied as tracer, each of them yielded identical dispersion coefficient [13]. As the stationary method is rather time consuming it is more convenient to use an instationary technique. Such a technique represents the determination of the mixing time which is frequently used to characterize the mixing properties of stirred vessels and fermenters [16-191. The mixing time is defined as that time which is needed to distribute an amount of tracer (introduced as Dirac pulse) uniformly or to a distinct degree of homogenity The mixing time and the C/CO, respectively. dispersion coefficient are related by the solution of Fick’s second law[20,9]. The curve in Fig. 4 is computed from this solution and shows the concentration as function of the dimensionless group Dt /L *. The points in Fig. 4 are obtained from mixing time measurements for a distance between the tracer source and the measuring point of 700 cm and a dispersion coefficient known from measurements with the stationary technique. They agree well with the curve if one neglects the small deviations at low degrees of homogenity. Curves of the type shown in Fig. 4 are useful for a fast determination of the dispersion coefficient with the transient method. Dispersion coefficients measured with both methods in BC I are plotted vs the linear gas velocity in Fig. 5. The scattering at gas velocities

-7 Fig. 4. Concentration

D.1

as function of the group Dt/L *.

of about 5 cmlsec can be attributed again to the transition from the homogeneous to the heterogeneous regime of flow. It appears remarkable that no constant dispersion coefficient is achieved at higher gas velocities. Owing to the liquid flow through the annulus between the column tube and the gas distribution plate the dispersion in BC II is not uniform at low gas velocities u. < 5 cm/sec[l3,21,22]. A region of lower dispersion can be observed which extends to about 1 m above the gas distributor. At gas velocities of above 10 cm/set the dispersion coefficients of BC II are almost constant (D = 300 cm/set). The splitting into zones with different mixing properties is restricted to bubble columns

W.-D. DECKWER et al. D

c

1000

I

I

I

I

I

Icm2/sl

I

800

LOO-

600 t

.

BC I

(IhIs work1

_

300 / 200 L 0

A instationary

3

6

9

-

12

15 0

uGJcmlsl

1

2

3

L

5

6

Fig. 5. Dispersion coefficient in BC I.

1O-2dlL G Fig. 6. Correlation for dispersion coefficients in bubble

with porous gas spargers and should be regarded as end effect. Plenty of measurements were carried out in BC II to determine the influence of gas and liquid flow rates and of the column height. A more detailed analysis is given elsewhere [ 131. On the whole the measured dispersion coefficients of the liquid phase in BC I and BC II can be well correlated with the data of other authors [9, 13,22-291 by an expression similar to that proposed by Towell and Ackermann[28]. This correlation is

columns with different diameters.

D = 2.7 d’.4UG0’3.

(1)

The mean error is 30 per cent at a significance level of 9.5 per cent (Fig. 6). DETERMINATION OF LIQUn, PHASE MASSTRANSFER COEFFICIENTS When measuring the oxygen transfer the bubble columns were connected and the liquid was recycled. The main part of the measurement were carried out as shown in Fig. 7. In BC I oxygen was absorbed by the liquid phase from filtered air, in BC II the oxygen dissolved in the liquid phase was desorbed with pure nitrogen. By cooling the buffer tanks (Bl, B2) the temperature was maintained at approximateIy 16°C in both bubble columns. At different positions of the columns (M) a liquid flow was removed and its oxygen concentration was measured with a polarographic electrode. Oxygen saturated tap water was always used to check the electrodes. Therefore the oxygen concentrations measured in the solutions of molasses and electrolytes are relative values only which differ from the true concentrations by a factor which considers the different solubilities of oxygen. This factor needs

not to be known

when evaluating below.

because it can be canceled out the mass transfer data as outlined

EVALUATION

OF DATA

Steady state measurements of mass transfer rates in bubble columns were usually analysed by applying the NTU method[l4,30-371, which regards the bubble column as tube with plug flow of the phases. On the other hand the evaluation of transient measurements is carried out with the assumption the bubble column being a well stirred tank[ 10, 11,3tiO]. The flow pattern in bubble columns lies actually between these extreme cases. When using the NTU concept the mass transfer data are to small, while comprehending the bubble column as stirred tank the evaluated data are too high. Mass transfer data determined by one of these ways are less reliable for scaling up purposes if the dispersion changes. But it will be shown yet, that the effect of changes in dispersion is comparatively small. In order to avoid this inaccuracy the volumetric liquid phase transfer coefficients kLa were calculated from the measured concentration profiles in the liquid phase. As to Fig. 8 the mass balance equations of each phase have to consider convection, dispersion and mass transfer between the phases. In addition the linear pressure decrease in the bubble column must be taken into account[41-43]. Since the solubility of oxygen in the liquid phases used is low the change of the oxygen concentration in the gas phase is small. Therefore as a first approximation the mole fraction

Mixing and mass transfer

in tall bubble

2181

columns

M

-1, 62 Ii

P2

filtered

Fig. 7. Experimental

set-up for mass transfer

measurements.

Liquid

-D (%I x.dx

t

t

t

4

"G,x.dx

"L,x.dx

4

dx

-

t

1

t -D(f$i)

UL,X

x

Dispersion e

Q(l-Et)-*QCG

x=0 t

Fig. 8. Schematic

representation

of bubble

columns.

air

W.-D.

2182

DECKWER

of oxygen in the gas phase was assumed to be constant. Then the oxygen balance equation for the liquid becomes

(2) which yields in dimensionless

+&g

form:

+sqc*-c)=O,

(3)

with Pe =

ULOL

(4)

DC1 - EG)

and St = kra-&.

(5)

Owing to the linear pressure gradient the equilibrium concentration is axially dependent: c*=a+bx,

(6)

where (Y and p are obtained from the holdup, the pressure at the top of the column, the solubility and the column height. The solution of Eqs. (3) and (6)

et

al.

with consideration of the appropriate boundary conditions is given in [41,42] and in the appendix. The determination of the liquid phase mass transfer coefficient kLa resulted from adjusting profiles predicted from the analytical solution with experimentally obtained profiles using an optimization technique[42]. It was possible to determine simultaneously the optimum Peclet and Stanton number as well as the Stanton number alone since the Peclet number could be calculated from the independently measured dispersion coefficients. A comparison of the optimized and measured dispersion coefficients reveals that they may differ up to 100 per cent, while the Stanton numbers obtained from both methods do not differ more than 3 per cent. This result permits the conclusion that the dispersion does not influence the concentration profiles remarkably if the dispersion is relative high and exceeds a distinct value (D > lOOcm’/sec). Owing to the weak dependence on the dispersion there was no cause to improve the model for BC II, in which zones of different dispersion were found experimentally. Computed profiles and measured concentrations are presented in Fig. 9. It can be concluded that the experimentally obtained profiles are described well by the dispersion model with consideration of the

1 bW1

1 12,0

W

BC

95 v.

desorption

BC II

Fig. 9. Oxygen concentration in liquid phase.

Mixing and mass

transfer in tall bubble columns

linear pressure gradient. However, a systematic deviation between predicted measured profiles could be observed for the top of the column when the liquid flow rate is high and the gas flow rate is low. This can be recognized from Fig. 10. The deviation indicates that the neglect of the variation of the gas composition is not justified. However, when the model is improved by considering the mass balance of the gas phase, it is necessary to introduce a variable gas velocity too. Thus the problem becomes rather complicated and involves an iterative scheme to solve the system of equations which are nonlinear now [43]. Therefore a simpler approach was applied in the present study to obtain the mass transfer data from the measured profiles. The overall oxygen balance reveals that the variation of the mole fraction in the gas phase may be regarded as small as it does not exceed a value of 0.04. Now, if one assumes approximately that the mole fraction varies linearly with the axial coordinate one obtains y = yo + AY,x,

(7)

where y0 is the oxygen mole fraction at the column entry and Ay, is the total change of the gas composition which is determined by an overall oxygen balance. With the approximation (7) a

0

Fig. 10. Oxygen

CES Vol. 29 No. II-D

concentration

2183

closed solution of Eq. (3) can be achieved yet, as is performed in the appendix. The optimum profiles predicted from this solution are shown as dotted lines in Fig. 10. The measured profiles are now described with sufficient accuracy over the whole column length. For extreme cases (low gas and high liquid flow rate) the kLa -data are increased up to 20 per cent compared with the values determined without considering a linear variation in gas composition. Several investigators [ 14,33,37,45] found that kLa varies with column height. These studies were dispersion carried out usually at various heights 137,451, concentration profiles were not determined. When fitting the experimental profiles of this study on the model predictions, there was no necessity to consider a variable kLa. Therefore the higher kLa values at short columns can be explained with the increased mass transfer during the phase of bubble formation. However, the formation of bubbles should be regarded as end effect, which will be constant for a given gas distributor. The axial dependence of kLa obtained by Shulman and Molstad[14] yields from the application of the NTU method whereby the concentration jump at reactor entry, which is due to the mixing in the column, is interpreted with increased mass transfer.

92

in liquid phase with consideration

W

of a linear variation

in gas phase, Eq. (7).

W.-D.

2184

DECKWER

RESULTS

Values of kLa for tap water are shown in Fig. 11. They do not differ for desorption and absorption as was realized from several test measurements in both columns. Experiments with tap water were carried out at various liquid flow rates (04-6 m’/hr), an influence on kLa could not be noticed. There is a remarkable difference between the mass transfer coefficients of both columns. The values in BC II amount to double of those in BC I. This effect is attributed to the gas spargers used which provide for different fractional holdups with different bubble size distributions. Hence different specific interfacial areas are obtained. Akita and Yoshida[39] obtained kLa data for oxygen transfer from a column with a single orifice as gas distributor. Of course, their values are somewhat lower, because the gas distributor is less efficient. Chang[42] investigated the oxygen transfer in a bubble column with 10.2 cm dia. and a height of 256 cm using a porous plate (150-20 pm) as gas sparger. His data agree well with those of BC II. Some of the desorption experiments in BC II were evaluated using the NTU method[l4]. The obtained data for the mass transfer coefficients are lower, however the error does not exceed more than 20 per cent as can be seen from Fig. 11. About the same error was predicted by Watson and Cochran from theoretical computations [44]. Obtained values of kLa for the salt solutions are plotted vs the gas velocity in Fig. 12. The data of the 0.7 N Na2S04 solution are about the same as kLa

0,20

0,002

Fig.

1

I 92

I

et al.

those for water, whereas the data of the less concentrated salt solutions are increased. In Fig. 13 results are presented for molasses. In BC I kLa is decreased distinctly compared with water. No dependence on the concentration of molasses seems to exist in BC I, while in BC II such a dependence is found. But this influence of the concentration on kLa is not clearly perceptible. The data presented in Fig. 13 should be regarded as preliminary results, further experiments are needed to clarify this point. Using log-log plots the dependence of the liquid kL=

b'l 0,lO -

I

qos

402

0,Ol

0,005 _

+ NaCl 0,17N /*

/$O

Na2SOL 0,225 N Na2S0,

0.70 N

o,oo?l~ l

43

q5

l,o

2,0 -

Fig. 12. Liquid

I

I

phase mass electrolyte

I

I

I

I

I

I

0,5

1,O

2p

5x0

10,o

11.Liquid phase mass transfer

.

coefficients

5.0

UGo [cmlsl

transfer coefficients solutions.

I

1 20,o

“Go [cmlsl k,a for tap water.

IO,0

k,_a for

Mixing and mass transfer in tall bubble columns

2185

the 0.17 NaCl 0.225 N Na2S04 solutions are about the same, a single correlation is also proposed for both. DISCUSSION From

a 3,31% 0 5.00%

0)

q2

q5

l#o

2,o -

5P

140

++,, [cmisl

Fig. 13. Liquid phase mass transfer coefficient kra for aqueous molasses.

phase oxygen transfer coefficient kLa on the gas velocity can be described fairly well by straight lines for all liquids studied in this work. Hence the data can be presented by the relation kLa = bous.

(8)

The coefficients b. and b, for the various liquids were determined from a regression analysis. They are compiled in Table 1, additionally the range of UC, the relative mean deviation and the standard deviation is given. The correlation for tap water in BC II includes the data of Chang. As the values of

the relation

where d, is the mean Sauter diameter the specific interfacial area can be estimated. As d, is decreased to about 1 mm or even below for the solution of electrolytes[46] and molasses, the interfacial area is increased by a factor equal or larger than 5. Since kLa is increased at most by a factor of 1.5 for the aqueous solutions of NaCl (0.17 N) and Na2S04 (0.225 N) the mass transfer coefficient kL appears to be decreased strongly. For the solutions of molasses this can be explained with the addition of hexanol and surface active compounds which are inherent in molasses. These surface active compounds yield an additional resistance to the interfacial mass transfer. Calderbank[47] found that for small bubbles which behave as rigid spheres the coefficient kL is smaller as for larger bubbles. The results of this study indicate that an additional influence of the electrolyte concentration exists, as the kLa values of the O-7 N solution of Na2S04 are lower than that of the O-225 N solution. This observation denotes a marked contradiction to the results of Zieminski and coworkers [ 15,481, which obtained an increase of the interfacial area with increasing salt concent-

Table 1. Constants of relation (8) for the various liquids

Liquid BCI

BCII

ho

b,

Rel. mean deviation (o/o)

Standard deviation (k)

Range of UC (cmlsec)

Water NaCl 0.17 N NaZS04 0.225 N NaZSOI + NaCl Molasses (05-10 % wt.)

00X6 0.0125 0.0115 0.0123

0.785 0.825 0.785

14.9 4.2 7.3 6.1

0+)077 0.0021 0.0038 0.0029

04-10 04-8 04-8 0+8

04044

0.723

17.0

0.0027

0.25-5

Water NaCl 0.17 N NaSO, 0.225 N Na,SO, + NaCl Molasses (% wt) 0.5 1.0 1.93 3.37 5.00 10.0

0.0274 0.0370 0.0370 0.0374

0.8 0.8 1.0 0.88

14.6 7.2 16.1 12.2

0.0105 0.0045 0.0135 0.0098

0.25-8 0.4-4 04-l 04-4

0.025 0.015 0.012 0.019 0.026 0.043

1.0 0.93 0.78 1.21 1.28 1.12

7.7 7.6 17.0 8.4 15.7 10.7

0.0019 0.0008 0.0016 0.0017 0.0052 0.0057

0.25-1.5 0.25-1.5 0.25-1.5 0.25-1.5 0.25-1.5 0.25-1.5

om4

W.-D.

2186

DECKWER et al.

ration only, but the decrease of kL was below 10 per cent for the range of concentrations they used (up to 0.24 g-mol/l). The effects observed by Zieminski were obtained at very low bubble densities, In the present study of tall bubble columns the bubble density was high and it appears questionable to take over the results of Zieminski to gas liquid dispersion with fractional gas holdups prevailing in bubble columns. From this work it can be suspected that the mass transfer coefficient is apparently decreased with increasing concentration of electrolyte. It is assumed that this effect can be attributed to the occurence of an electric double layer at the gas-liquid interface. This double layer provides an ordered structure near the‘interface which makes the interfacial mass transfer more difficult. The degree of order at the interface will increase with the concentration of electrolyte. Hence the transfer coefficient kL decreases.

d D i kL L P Pt Pe St t ; X *; z

Greek symbols E holdup p

SUMMARY

With regard to the measurements of the oxygen transfer rates it can be concluded: (1) The measured concentration profiles in the liquid phase can be described well using the dispersion model with consideration of the linear pressure decrease and a linear approximation of the gas profile. An axial dependence of kLa found by some authors was not observed. The sensitivity of the profiles on changes in dispersion is rather small. (2) The kLa data obtained for tap water are not effected by the liquid velocity. The mass transfer rates in both columns differ remarkably which is attributed to the different gas distributors. (3) The proposed correlations for the kLa data of the various liquids indicate that kLa increases approximately linearly with the gas velocity for the range studied (b, = 1). (4) The transfer rates for the solutions of molasses are lower than for tap water. Since the transfer rates for the electrolyte solutions do not increase with the same degree as the interfacial areas increase, a decrease of the mass transfer coefficients kL is assumed. Acknowledgement-This grant

of the Deutsche

study was supported in part by a Forschungsgemeinschaft. NOTATION

A a .z

interfacial area, cm* specific interfacial area, cm-’ concentration of oxygen, mg/l equilibrium concentration of oxygen, py/H, mdl

column diameter, cmdispersion coefficient, cm’/sec gravitational acceleration Henry’s constant, atm l/mg liquid phase mass transfer coefficient, cmlsec length of column, cm pressure, atm pressure at top of column, atm Peclet number, defined by Eq. (4) Stanton number, defined by Eq. (5) time, set linear flow velocity, cmlsec volume throughput, m3/hr dimensionless axial coordinate, z/L mole fraction in gas phase total change in gas phase composition axial coordinate, cm

density, glcm’

Indices 0 refers to column

entry G refers to gas phase L refers to liquid phase REFERENCES

[I]

KBlbel H. and Ackermann P., Chem. Ing. Techn. 1956 28 381. [2] Harders M., Heller G. and Laurer P. R., Chem. Ing.

Techn. 1963 35 405. [3] Smidt I., Hafner W., Jira R., Sedlmeier J., Sieber R., Rtlttinger R. and Kojer H., Angew. Chem. 1959 71 176. [4] Kijlbel H., Hammer H. and Meisl U., Proc. 3rd Eur. Sympos. Chem. React. Engng., Suppl. Chem. Engng. Sci. 1965 20 115. [5] Barona N. and Prengle H. W., Hydrocarbon Processing 1973 52 63. [6] Dubil H. and Gaube J., Chem. Ing. Techn. 1973 45 529. [7] Beinhauer R., doctoral thesis 1971, Technische Universitlt Berlin. [8] Kolbel H., Beinhauer R. and Langemann H., Chem. Ing. Techn. 1972 44 697. [9] Ohki Y. and Inoue H., Chem. Engng Sci. 1970 25 1. [lo] Sideman S., Hortscu 0. and Fulton J. W., Ind. Engng Chem. 1966 58 32. [ 1 I] Yoshida F. and Akita K., A. I. Ch. E. JI 1965 119. [12] Anderson J. L. and Quinn J. A., Chem. Engng Sci. 1970 25 373. 1131 Badura R., Deckwer W. D., Warnecke H. J. and Langemann H., Chem. Ing. Techn. 1974 46 399. [14] Shulman H. L. and Molstad M. C., Ind. Engng Chem. 1950 42 1058. [15] Lessard R. R. and Zieminski S. A., Ind. Engng Chem. Fund. 1971 10 260. 1161 Hansford G. S. and Humphrey A. E., Biotechn. Bioengng 1966 8 65.

Mixing and mass transfer in tall bubble columns [17] Hoogendorn

C. J. and Den Hartog A. P., Chem.

2187

be constant (= 0.21) combining of Eqs. (3) and (6) yields

Engng Sci. 1967 22 1689.

[18] Novak V. and Rieger F., Trans Inst. Chem. Engr. 1969 41 335. [19] Goldstein A. M., Chem. Engng Sci. 1973 28 1021. [20] Siemes W. and Weiss W., Chem. kg. Techn. 1957 29 727. [21] De Nevers N., A.LCh.E.Jl 1968 14 222. 1221 Deckwer W. D., Graeser U., Serpemen Y. and Langemann H., Chem. Engng Sci. 1973 28 1223. [23] Argo, W. B. and Cova, D. R., Ind. Engng Chem. Proc. Des. Dev. 1965 4 352. [241 Reith T., Renken S. and Israel B. A., Chem. Engng Sci. 1968 23 619. [25] Aoyama Y., Ogushi K., Koide K. and Kubota H., J. Chem. Engng Japan 1968 1 158. [26] Kunugita E., Ikura M. and Otake T., J. Chem. Engng Japan 1970 3 24. thesis 1972. Technische [271 Gahler P., doctoral Universitgt Berlin. [281 Towell. G. D. and Ackermann G. H.. Proc. 5th Eur. 2nd In;. Symp. on Chemical React& Engineering, Amsterdam 1972. [291 Tadaki T. and Maeda S., Kagaku Kogaku 1964 28 270. [301 Guyer A., Richarz W. and Guyer A. Jr., Helv. Chimica Acta 1955 38 1193. t311 Metzner A. B. and Brown L. F., lnd. Engng Chem. 1956 48 2041. [321 Houghton G., McLean A. M. and Ritchie P. D., Chem. Engng Sci. 1957 7 26. 1965 7 [331 Kling G., Fortschritte der Verfahrenstechnik 268. [341 Voyer R. D. and Miller A. I., Can. J. Chem. Engng 1968 46 335. 1351 Mashelkar R. A. and Sharma M. M., Trans. Inst. Chem. Engr. 1970 48 162. WI Mashelkar R. A., Brit. Chem. Engng 1970 15 1297. [371 Coulon G., Chem. Ing. Techn. 1971 43 280. t381 Eckenfelder W. W. and Barnhart E. L., A. I. Ch. E. J1 1961 7 631. [391 Akita K. and Yoshida F., Ind. Engng Chem. Proc. Des. DeveIoD. 1973 12 76: 1974 13 84. 1401 Towel1 G. c., Strand C. P. and Ackermann G. H., Mixing-Theory Related to Practice, Rottenburg P. A. (Ed.) p. 97 1965. App. 1968 92 [411 Langemann H., Chem. Ztg.-Chem. 845. ~421 Chang C. C., doctoral thesis 1970, Technische Universitat Berlin. M., doctoral thesis 1970, Technische [431 Reuss Universitst Berlin. [441 Watson J. S. and Cochran H. D., Ind. Engng Chem. Proc. Des. Develop. 1971 10 83. [45] Oestergaard K., doctoral thesis 1969, Technical University of Denmark, Lyngby. [46] Marrucci G. and Nicodemo L., Chem. Engng Sci.

(Al)

Q =j$(pc

b=-

642)

& pge,L.

With the boundary conditions c(o)=c

’

_ldco

(A4)

Pe dx

and

de(l) _ o

(A5)

dx

the solution of Eq. (Al) is[41]: c=A,e’l‘+A,e’z”+a+bx--

b

St

(A6)

where (A7) A, = (BrZe5 - br,)/N

(‘48)

A, = (- Br,e’l + brJ/N

(A9)

B=

(AlO)

and N = r 2e” - r 22e1.

(All)

If the change in the gas phase composition is considered approximately by a linear relation as indicated in Fig. 14 Eq. (7) is introduced in Eq. (5) which yields explicitely c*=a’y,+(a’hy,+b’yo)x+b’hy,x2

(-412)

a’ = (pt + pgeLL)/H

(Al3)

with

and b’

= pgeLL /If.

1967 22 12.57. [47] Calderbank P. H. and Moo-Young

M. B., Chem. Engng Sci. 1961 16 39. [48] Zieminski S. A. and Whittemore R. C., Chem. Engng Sci. 1971 26 509. APPENDIX

Solution of y in

(A3)

Fig. 14.

W.-D.

2188 Writing

DECKWER et al. with

Eq. (A12) in the form: c*=ff+px+yxz

the differential 1 d*c Fedx-z-

equation dc

(A14)

becames:

B,=

Stc=-.S~t(o!+~x+~x~).

Considering the boundary condition following solution is obtained:

B, = - [Per,e’fc,-

[

Per,ey

c = B,e’l” + B2e)ix + B, + B4x + Bsx2

+ r,(B,+2B5)]/N

1

co-B,+$)+ (&++I

(A% the

Bn=p-5 L41f3

B,= y.

(A17)

r2(B4+2B5) IN

(A13 B,=$

(A4) and

(

B,+$

-2 2Y

(Al@ (At9)

(A201