Ind. Eng. Chem. Res. 1987,26,933-937
The apparatus has been used for measuring adsorption data on a newly available attrition-resistant activated carbon, Kureha beads (GP Grade).
Acknowledgment
I am grateful to Exxon Research and Engineering Company for the permission to present and publish this work. Registry No. CH,, 74-82-8; CH3CH3, 74-84-0; H,C=CH,, 74-85-1; H&CHzCH,, 74-98-6; C, 7440-44-0.
Literature Cited
933
Danner, R. P.; Choi, E. C. F. Ind. Eng. Chem. Fundam. 1978,17,248. Danner, R. P.; Wenzel, L. A. AIChE J. 1969,15, 515. Hyan, S. H.; Danner, R. P. AIChE National Meeting, Orlando, FL, February 28-March 3, 1982. Kaul, B. K. Ind. Eng. Chem. Process Des. Deu. 1984, 23, 711. Kaul, B. K.; Sweed, N. H. Paper Presented at an International Conference on Fundamentals of Adsorption, Schloss Elmau, West Germany, May 1983. Ray, G. C.; Box, E. 0. Ind. Eng. Chem. 1950,42, 1315. Reich, R.; Ziegler, W. T.; Rogers, K. A. Ind. Eng. Chem. Process Des. Deu. 1980, 19 (3), 336. Suwanayuen, S. Ph.D. Dissertation, The Pennsylvania State University, University Park, 1979. Suwanayuen, S.; Danner, R. P. AIChE J. 1980,26, 68, 76.
Al-Ameeri, R. S. Ph.D. Dissertation, The Pennsylvania State University, University Park, 1979. Cochran, T. W. M.S. Dissertation, The Pennsylvania State University, University Park, 1982.
Received for review January 14, 1985 Revised manuscript received April 28, 1986 Accepted December 22, 1986
Theoretical Prediction of Gas Hold-Up in Bubble Columns with Newtonian and Non-Newtonian Fluids Yoshinori Kawase and Murray Moo-Young* Department of Chemical Engineering, University o f Waterloo, Waterloo, Ontario, Canada N2L 3G1
A theoretical model for gas hold-up in bubble columns with Newtonian and non-Newtonian fluids has been developed on the basis of the concept of a characteristic turbulent kinematic viscosity in bubble columns. Gas hold-ups in a 40-L bubble column and a 1000-L pilot plant fermenter with Newtonian fluids (water, glycerine, dextrose, and fermentation media) and non-Newtonian fluids ((carboxymethyl)cellulose,carboxypolymethylene, and polyacrylamide) were measured. Predictions were compared with the present data and other experimental data and correlations available in the literature, over a wide range of conditions. A satisfactory agreement was found. Gas hold-up is one of the most important parameters for the design and scaleup of bubble columns. Therefore, fairly extensive literature on gas hold-up in bubble columns is available. It has been known that gas hold-up depends mainly on the superficial gas velocity and often is very sensitive to the physical properties of the liquid, and a number of correlations have been proposed (Shah et al., 1982). However, no single unified correlation is available. While experimental study of gas hold-up has been extensively carried out, there are few theoretical studies. Most of the early work made use of empirical curve-fitting analyses. Azbel and Zeldin (1971) derived a correlation for the gas hold-up using the principle of minimum total energy. Despite the complicated analysis, the resulting correlation is simple. In their correlation, the gas hold-up is given as a function of superficial gas velocity and liquid height. It should be noted, however, that most of the correlations in the literature do not include functions of the liquid height. The equation of motion was solved to obtain a theoretical correlation for gas hold-up by Ueyama and Miyauchi (1979). This correlation is rather complicated, and the experimental results for the slip velocity of a bubble relative to the liquid and a turbulent kinematic viscosity are required to calculate the gas hold-up. Gas hold-up by their correlation is approximately proportional to the square root of the superficial gas velocity and slightly decreases with an increase of column diameter. Walter and Blanch (1983) analyzed the effect of the liquid velocity on gas hold-up. In their analysis, the liquid above the gas sparger is divided into two regions: the bottom region and the upper region. Their resulting *To whom all correspondence should be directed. 0888-5885/87/2626-0933$01.50/0
complex equation includes the liquid velocity and the bubble rising velocity which must be estimated from prior experimental data. Recently, Viswanathan and Rao (1984) derived an expression for gas hold-up based on the cell model proposed by Whalley and Davidson (1974). In their simple correlation, the rising velocity of a single isolated bubble is included, and it is assumed to be 0.12 and 0.235 m/s at low and high superficial gas velocities, respectively. Their correlation predicts that the gas hold-up decreases with increasing column diameter. The theoretical correlations mentioned above are complex or include adjustable parameters (except for the correlation of Azbel and Zeldin (1971)). It is obvious that there are no simple theoretical correlations which can provide a reasonable prediction of gas hold-up in bubble columns. In addition, despite the wide occurrence of non-Newtonian fluids in the chemical and biochemical industries, the attention of research on gas hold-up in bubble columns with non-Newtonian fluids has been narrow and has occurred only in recent years, and very little is known about the effect of non-Newtonian flow behavior on the gas hold-up. Furthermore, the reports are restricted to experimental studies, and no theoretical work has been reported yet in the literature. In this paper, a theoretical model for gas hold-up in bubble columns with Newtonian and non-Newtonian fluids is developed. The gas hold-ups in a 40-L bubble column and a 1OOO-Lfermenter pilot plant are also measured. The capacility of the proposed model is examined using the present data and the available experimental data and correlations in the literature.
Theoretical Analysis The equation of motion for the liquid phase in a bubble column may be written as (Ueyama and Miyauchi, 1979) 0 1987 American Chemical Society
934 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987
I d -- -(r7) = -(z r dr
- t)pg
where 7
= -(u
du + ut)p -&
and t
= z( %N)(
1-
($)
We assume that the non-Newtonian flow behavior of the liquid can be represented by a power-law model 7
= -KT"
(4)
When the liquid is turbulent, the molecular kinematic viscosity, u, is negligible compared with the turbulent kinematic viscosity, ut. Many reports on the gas hold-up profile in the literature indicate a value for N of 2 (Ueyama and Miyauchi, 1979). Therefore, a general solution of eq 1 is written as
u=
-(
z&! 1 r4 - - - :rz) ut 8 R 2
+ A , In r + A2
(5)
The constants of integration AI and Az are determined using appropriate boundary conditions. Despite the fluctuations in the local liquid velocity, relatively stable velocity profiles are found in bubble columns. The liquid rises with the bubbles in the center of the column and flows downward in the outer annular region. The liquid velocity is maximum a t the center of the column and decreases in a radial direction. At high Reynolds numbers, a transition point a t which the timeaverage velocity is zero occurs a t around r/R = 0.7, and large velocity, the magnitude of which is 50-100% of the velocity a t the column axis, is found very close to the column wall. On the basis of these observations, we have the following boundary conditions u=uo
u =0
at r = O
(64
a t r = 0.7R
(6b)
u = (-o.5--1.0)u0 at r = R (64 I t may be reasonable to assume that the liquid velocity profile in non-Newtonian fluids at high Reynolds numbers is similar to that in Newtonian fluids described above (Walter and Blanch, 1983). More measurements of liquid velocity profile in non-Newtonian fluids may be necessary to discuss the validity of this assumption. The constants of eq 5 are determined by using only boundary condition 6a, and we have (7) Kawase and Moo-Young (1986b) suggested that condition 6b is better than condition 6c in order to represent a liquid velocity profile in the core region of the column which may characterize the mixing in a whole bubble column. Substitution of eq 6b into eq 7 yields an expression for the average gas hold-up
In order to calculate gas hold-up by using the above equation, expressions for the turbulent kinematic viscosity, v,, and the liquid velocity a t the column axis, uo,must be known.
Table I. Details of the Columns 40-L bubble column height, m 1.22 diameter, m 0.23 gas sparger orifice plate" type no. of holes 20 diameter of holes, m 0.001 draft tube no height, m diameter, m
1000-L pilot plant fermenter 3.21 0.76 ring sparger* 100 0.003 Yes 2.1 0.406
An inverted conical bottom. The orifices are located on a circle of 0.215-m diameter centered on the fermenter axis, and a 90° cone aligned on the fermenter center line is secured through its base to the orifice plate. bDiameter of ring, 0.7 m; diameter of tube, 0.0127 m; bottom clearance, 0.2 m.
The turbulent kinematic viscosity may vary with the distance from the wall. Since no reliable data for ut in a gas-liquid two-phase flow is available, we use a characteristic or average turbulent kinematic viscosity, 3t, instead of ut to estimate the average gas hold-up. Kawase and Moo-Young (1986b,c) defined a characteristic kinematic viscosity in a bubble column on the basis of the Prandtl mixing length theory zit
x2
=p ) R
where
x
= 0.4n
(10)
The validity of this definition was discussed for Newtonian fluids (Kawase and Moo-Young, 1986b). If this equation is combined with the energy balance in a bubble column, the following expression for uois obtained (Kawase and Moo-Young, 1986~): uo = 0.787n-2/3g'13D,1/3u~~1~3
(11)
Substituting eq 9 and 11 into eq 8, we obtain z = 1.07n2/3Fr1/3 (12) The present theory predicts that the gas hold-up is proportional to two-thirds power of superficial gas velocity and an inverse of one-third power of column diameter. In addition, it suggests a decrease of the gas hold-up caused by the increase of pseudoplasticity of the liquid (or the decrease of the flow index, n).
Experimental Section Hold-up measurements were carried out in a 40-L bubble column and a 10o0-L pilot plant fermenter. Details of the two devices are listed in Table I. The location of the orifices causes liquid circulation up in the outer annular region and down in the core region which is opposite to that in a conventional bubble column described previously. Since bubbles have a tendency to migrate toward the center of the column, the liquid flow is not well-defined as compared with that in a conventional column. However, Kawase and Moo-Young (1986b,c) showed that the characteristics of overall liquid-phase mixing in these columns are almost equivalent to each other. The gas hold-ups were determined by the difference between the clear liquid and dispersion heights, and the gas flow rates were metered by means of a calibrated rotameter. Column operation was batch with respect to the liquid phase. The Newtonian liquids studied in this work are water, glycerine, dextrose aqueous solution, and three kinds of fermentation media. Aqueous solutions of the (carboxy-
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 935 Table 11. Liquid-Phase Physical Properties at Test Conditions consistflow ency index, density, Pa s" kg m-3 index water 1.0 0.00090 991 1.0 0.00135 1009 glycerine 1.0 0.00095 1004 dextrose solution fermentation medium (glucose + 1.00 0.0010 995 mineral salt) fermentation medium (molasses + 992 mineral salt)" fermentation medium (Alpha-floc 999 + mineral salt)* CMC-1 0.770 0.034 991 CMC-2 0.677 0.160 991 CMC-3 0.504 1.65 993 0.816 0.0282 992 Carbopol-1 Carbopol-2 0.656 0.320 992 0.889 0.0094 991 Separan 10 Total solids (upon evaporation) = 26.4 g/L. = 9.05 g/L.
I I
I nGodbde et a) (19821 Dc-aMsm mMiy4iChl RM 5hy11(1970) Dc-Olm
I
IO'
Oat0 ty Kalooko el Ul 11979) Oc=a5m,aIr-water
"I
10.'
5
1
10"
5
[-I Figure 2. Comparison of the present model with the available experimental data for Newtonian fluid systems in the literature.
* Filterable solids
1 10'3
1
I
loz
I
Fr
I
/
Ur,
I
Wwk
I
I 5
I
I -Present
[mal
Figure 3. Gas hold-up in the 5.5-m-diameter column (data by Kataoka et al. (1979)).
102
[-I Figure 1. Gas hold-up in Newtonian fluid systems. Fr
methy1)cellulose (CMC7H4, Hercules Inc.), carboxypolymethylene (Carbopol 941, Goodrich Chemical Co.), and polyacrylamide (Separan NP10, Dow Chemical Co.) were used to study the effect of non-Newtonian flow behavior on the gas hold-up. Their pseudoplasticities were represented by the power-law model, eq 4. The rheological properties of the non-Newtonian liquids were measured with a concentric cylinder viscometer (Fann, Model 35A) a t shear rates of 1.1 1021 s-l. Those of the Newtonian fluids were measured with a CannonFenske viscometer. The results are summarized in Table 11.
-
Discussion In order to assess the predictive capability of the proposed model, we carried out a number of tests by comparing eq 12 with the experimental data in this work and the available data and correlations in the literature. Figure 1compares the proposed model with the experimental data in this work for Newtonian fluid systems. The proposed model is seen to be in good agreement with the data in the 40-L bubble column with water and aqueous solutions of glycerine and dextrose. Although liquid-phase mixing in a bubble column with a draft tube is sometimes different from that in a bubble column without a draft tube, the average gas hold-ups in these columns are usually very similar (e.g.: Koide et al., 1983; Weiland, 1984). Therefore, the data in the 1000-L pilot plant fermenter with water and fermentation media are compared with the proposed model in Figure 1. For comparison, the data for water and an electrolyte solution in the same fermenter obtained by Andre (1982) are also plotted. I t is seen that the data in the 1OOO-L pilot plant
0.35
I I I I I Data by Kolde etaL(19791 Dc.55m
I
1
,
0.05
't O l d
0;
'/R
0;
[-I
ola
L1
Figure 4. Gas hold-up distribution profile in the 5.5-m-diameter column (data by Koide et al. (1979)).
fermenter are in reasonable agreement with the proposed model. It can also be seen in Figure 2 that the present theory, eq 12, is in reasonable agreement with the available data for Newtonian fluids over a wide range of column diameter (D,= 0.1-1.07 m). As shown in Figure 3, the experimental data for a large bubble column (D, = 5.5 m) obtained by Kataoka et al. (1979) lie significantly above eq 12. In solving the equation of motion, we assumed that N in eq 3 is 2. However, the value of N in this large bubble column is 15 rather than 2 as illustrated in Figure 4. Using N = 15, we obtain the following equation instead of eq 8
Substitution of eq 9 and 11 into eq 13 yields z = 3.38Fr1I3
(14)
936 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 I
1
I
I
36 3
1c'
5
5
102
i IO'
5
.
[vel
llsg
Figure 5. Comparison of the present model with the available correlations for Newtonian fluid systems in the literature.
This equation successfully correlates the data in the 5.5-m bubble column (Figure 3). For more precise discussion on the effect of column diameter, reliable data for gas hold-up distribution profiles in large columns are required. The present theory is compared with the empirical correlations proposed by Hughmark (1967), Shyu and Miyauchi (1971),Akita and Yoshida (1973),and Hikita and Kikukawa (1974) in Figure 5. In these correlations, the gas hold-up is given as independent of the column diameter. The present result for Dc = 0.5 m agrees reasonably well with the correlations of Hughmark (1967) and Akita and Yoshida (1973). A correlation of Riquarts and Pilhofer (1978) for an air-water system may be written as (15) This is also in good agreement with the prediction of the present model for Dc = 0.1 m. The correlation obtained on the basis of the cell model by Viswanathan and Rao (1984) may be written as
This equation predicts a dependency of the gas hold-up on the column diameter. This dependency is similar to that in the proposed model but somewhat smaller. Godbole et al. (1982) proposed the following empirical correlation for highly viscous (carboxymethy1)cellulose(CMC) solutions: = 0 23gU 0.6340 -0.5 C (17)
.
si?
The dependency of 5: on D, in this correlation is larger than that in the proposed model. However, they expected the diameter effect to level off in columns larger than 0.3 m in diameter. Equation 16 predicts lower values of the gas hold-up as compared with eq 1 2 for uBg< 0.05 m/s. A comparison of the proposed model with our data for non-Newtonian fluids obtained in the 40-L bubble column is given in Figure 6. The model correlates the given data reasonably well. For the highly viscous non-Newtonian fluid systems, we observed the formation of large spherical cap bubbles. They were not formed in the Newtonian fluid systems, where a fairly uniform bubble size was observed. The large spherical cap bubbles may have only a small influence on 5: when the time-averaged values are taken into account.
z = O.24n4.(
%)
kDc)0.5
(5)
0.84-0.14n
0.07
(18)
where n-1
Pa
v, = P
This correlation suggests that 5: = K4.14. In the proposed theory, on the other hand, the gas hold-up in non-Newtonian fluid systems is independent of the consistency index, K. The decrease in the gas hold-up with respect to the consistency index indicated by this empirical correlation, which was obtained using a wide range of experimental data, may include many factors, e.g., experimental error, sparger design, other physical properties. It should be pointed out that the concept of an average shear rate in a bubble column, which has been widely used to correlate gas hold-up data, may possibly hide the actual influence of the consistency index. Following Nishikawa et al. (1978), the average shear rate is related to superficial gas velocity by = 5000u,, (20)
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 937 The apparent viscosities defined by eq 19 and 20 are functions of superficial gas velocity and their dependency increases with decreasing the flow index, n. It is probable that the combined effects of the flow index and the superficial gas velocity due to the uncertainty of definitions of the apparent viscosity in a bubble column overshadow the actual influence of the consistency index on the gas hold-up. In the present model, the surface tension is not considered. It may have effects on bubble size and gas hold-up. In fact, the correlations proposed by Hughmark (1967) and Hikita and Kikukawa (1974) suggest that the gas hold-up decreases with increasing surface tension. In order to develop a more precise correlation, the effect of surface tension should be considered.
Conclusions We have developed a theoretical model for gas hold-up in Newtonian and non-Newtonian fluids. It is relatively simple, compared with the prior art, and has no adjustable parameters. The predictions of the proposed model were compared with experimental data over a wide range of conditions. The comparison was found to be satisfactory. It may be concluded that the theoretical model developed in this work provides a simple but reasonably accurate means of estimating the gas hold-up in bubble columns with Newtonian and non-Newtonian fluids.
Nomenclature Al, A2 = constants in eq 5 D, = column diameter, m Fr = Froude number, usg2/D,g,dimensionless g = gravitational acceleration, m HL = clear liquid height, m K = consistency index in a power-law model, Pa sn N = exponent in eq 3, dimensionless n = flow index in a power-law model, dimensionless R = column radius, m r = radial coordinate, m u = velocity, m s-l Ubr = bubble rising velocity, m s-l uSg= superficial gas velocity, m uo = liquid velocity at column axis, m s-l Greek Symbols y = shear rate, s-l yav= average shear rate, s-l e
= gas hold-up, dimensionless
7=
average gas hold-up, dimensionless
= apparent viscosity, Pa s v = molecular kinematic viscosity, m2 s-l v, = apparent kinematic viscosity, m2 s-l vt = turbulent kinematic viscosity, m2 s-l ij = characteristic kinematic viscosity, m2 s-] p = liquid density, kg m-3 T = shear stress, Pa x = constant in eq 10, dimensionless M~
Literature Cited Akita, K.; Yoshida, F. Ind. Eng. Chem. Process Des. Deu. 1973,12, 76. Andre, G. Ph.D. Thesis, University of Waterloo, Ontario, Canada, 1982. Azbel, D. S.; Zeldin, A. N. Teor. Osn. Khim. Tekhnol. 1971,5,863. Bello, R. A. Ph.D. Thesis, University of Waterloo, Ontario, Canada, 1981. Buchholz, H.; Buchholz, R.; Lucke, J.; Schugerl, K. Chem. Eng. Sci. 1978,33, 1061. Capuder, E.; Koloini, T. Chem. Eng. Res. Des. 1984,62,255. Deckwer, W.-D.; Nguyen-Tien, K.; Schumpe, A.; Serpemen, Y. Biotechnol. Bioeng. 1982,24,461. Franz, K.; Buchholz, R.; Schugerl, K. Chem. Eng. Commun. 1980, 5, 187. Godbole, S . P.; Honath, M. F.; Shah, Y. T. Chem. Eng. Commun. 1982,16,119. Godbole, S. P.; Schumpe, A.; Shah, Y. T.; Carr, N. L.AIChE J. 1984, 30,213. Hikita, H.; Asai, S.; Tanigawa, K.; Segawa, K.; Kitaeo, M. Chem. Eng. J . 1980,20,59. Hikita, H.; Kikukawa, H. Chem. Eng. J . 1974,8,191. Hughmark, G. A. Ind. Eng. Chem. Process Des. Deu. 1967,6, 218. Kataoka, H.; Takeuchi, H.; Nakao, K.; Yagi, H.; Tadaki, T.; Otake, T.; Miyauchi, T.; Washimi, K.; Watanabe, K.; Yoshida, F. J. Chem. Eng. Jpn. 1979,12,105. Kawase, Y.; Moo-Young, M. Chem. Eng. Commun. 1986a,40,67. Kawase, Y.; Moo-Young, M. Chem. Eng. J. 1986b,in press. Kawase, Y.; Moo-Young, M. Chem. Eng. Sci. 1986c,41, 1969. Koide, K.; Morooka, S.; Ueyama, K.; Matsuura, A,; Yamashita, F.; Iwamoto, S.; Kato, Y.; Inoue, H.; Shigeta, M.; Suzuki, S.; Akehata, T. J. Chem. Eng. Jpn. 1979,12,98. Koide, K.;Krematsu, K.; Iwamoto, S.; Iwata, Y.; Horibe, K. J. Chem. Eng. Jpn. 1983,16,413. Miyauchi, T.; Shyu, C. N. Kagaku Kogaku 1970,34,957. Nishikawa, M.; Kato, H.; Hashimoto, K. Ind. Eng. Chem. Process Des. Deu. 1978,16,13. Nottenktimper, R.; Steiff, A.; Weinspach, P.-M. Ger. Chem. Eng. 1983,6,147. Riquarta, H.-P.; Pilhofer, T. Verfahrenstechnik 1978,12(2),77. Shah, Y. T.; Kelkar, B. G.; Godbole, S. P.; Deckwer, W.-D. AIChE J . 1982,28, 353. Shyu, C. N.; Miyauchi, T. Kagaku Kogaku 1971,35,663. Towell, G. D.; Ackerman, G. H. h o c . 5th Eur. 2nd Int. Symp. Chem. Reaction Eng. Amsterdam 1972,B3-1. Ueyama, K.; Miyauchi, T. Kagaku Kogaku Ronbunshu 1977,3,19. Ueyama, K.; Miyauchi, T. AIChE J. 1979,25,258. Ulbrecht, J. J.; Kawase, Y.; Auyang, K.-F. Chem. Eng. Commun. 1985,35, 175. Viswanathan, K.; Rao, D. S. Chem. Eng. Commun. 1984,25,133. Walter, J. F.;Blanch, H. W. Chem. Eng. Commun. 1983,19,243. Weiland, P. Ger. Chem. Eng. 1984,7,373. Wendt, R.;Steiff, A.; Weinspach, P.-M. Ger. Chem. Eng. 1984,7, 267. Whalley, P. B.; Davidson, J. F. Proc. Symp. Two-Phase Flow System, Symp. Ser. 1974,36,55.
Received for review June 26, 1985 Accepted December 15, 1986
