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A simple method of predicting bubble size in bubble columns
Chemical Engineering and Processing, 33 (1994)
l-5
A simple method of predicting bubble size in bubble columns R. H. S. Winterton School of Manufacturing and Mechanical Engineering, University of Birmingham, PO Box 363, Birmingham B15 2TT (UK) (Received May 28, 1993; in final form September 6, 1993)
Abstract A basic parameter in two-phase gas-liquid flow is the size of the bubbles. From the point of view of calculating the interfacial area for mass transfer, the size of the bubbles is as important as the holdup. Many empirical expressions for bubble diameter have been proposed. The purpose of this paper is to see whether recent work on bubble diameter in developed flow in pipes can also be used for bubble columns. The new expression for bubble diameter, which is based on an analogy with the structure of single-phase turbulent flows, is simpler to use and comparable in accuracy with the previous empirical expressions, in spite of not having any adjustable constants in it.
Introduction The gas-liquid interfacial area A available for mass transfer is related to the number of bubbles n per unit volume and their diameters d, by: A = f
ndi2
i=l
while the holdup
E is given by:
E = (x/6) i di3 i=l With a mean diameter d = !,
di3/$,
(2) d defined by:
di2
the relationship between surface area and holdup becomes simply E = Ad/6
then 10 -
(4)
and this is the reason for choosing the definition of d in eqn. (3). The quantity d is the Sauter mean diameter. Other terms used are d,, (volume/surface) or dx2 (3 dimensions/2 dimensions). A number of empirical correlations exist in the literature for predicting the value of d. Recently, a new approach has been suggested for calculating bubble diameter in two-phase flow in pipes [ 11. It was considered that the theory only applied to developed flow, i.e. to points a long way downstream from the point of introduction of the bubbles, where an equilibrium had been reached between coalescence and breakup processes. The flow in a bubble column reactor cannot be considered fully developed in this sense; the height of
0255-2701/94/$7.00 SSDI 0255-2701(93)00481-I
the column is often not very much greater than the diameter. Also the flow is more complex. Unlike the essentially one-directional flow in a pipe, there is recirculation; the liquid in the centre is carried up by the bubbles and returns downwards close to the walls. The detailed flow pattern can be quite complex (see, for example, the figures in ref. 2). There did not seem to be any simple way of resolving these problems, so it was decided to compare the original unmodified expression for bubble diameter given in ref. 1 directly with bubble column data, even though this expression had been derived for pipe flow. The fact that in many cases in bubble column experiments the flow cannot be considered to be in equilibrium can be seen in Fig. 1, taken from ref. 3. The results from just one of the experiments, with a given
q
.
6E
E
6-
& 5 E
4-
2 n
;___‘-__:_--‘_-__*_
2-
q
o! 0
.
1
1
.
,
q .
2
,
.
3 u(gas)
,
.
4
,
5
.
6
cm/s
Fig. 1. Data from ref. 3 for water with 1% n-butanol for three different bubble distributors. Also shown is the prediction of the new theory, eqn. (7).
0 1994 -
Elsevier
Sequoia.
All rights
reserved
2
method of introducing the bubbles, might be considered in rather poor agreement with the theory (details of the new theory are given later). However, when the results of all three experiments are considered, the theory is seen to be a reasonable match. None of the various correlations mentioned later allow for a variation with the method of introducing the bubbles. A search of the literature revealed a number of expressions for calculating d. These expressions themselves incorporate to some extent the correlation of the earlier experimental measurements. It was felt that the most useful comparisons would be between the various equations and the more recent experimental data. In fact, in the limited literature survey, we did not encotmter any other comparisons of this type, so a subsidiary purpose of the paper became to compare the existing correlations with one another and the recent data, regardless of the new equation.
d = 0.040
where D is the pipe diameter. Although derived for pipe flow, this expression is also assumed to apply to bubble columns with D being the bubble column diameter.
Other correlations The following correlations were found in a search of the literature. In many cases the equations found were already in basic SI units, or even in dimensionless form. In a few cases, minor changes have been made so that all quantities that appear are in basic SI units; the only exception is the pressure p which (when needed) is in MPa. Akita and Yoshida [ 51 suggested, based on a correlation of a large number of experimental results, that
d/D = 26(gD2p,/a) The new theory Inside a turbulent single-phase flow in a pipe, there exist turbulent eddies. The structure of the flow is known to be stable with these eddies present. We postulate that a two-phase gas-liquid flow is also stable with gas bubbles present in some of the eddies, or even replacing an eddy completely. The diameter of the bubble is therefore considered to be similar to that of the eddies. To obtain a quantitative measure of the average bubble diameter, we simply assume that it is equal to the average value of the mixing length. This is reasonable since the mixing length is the distance over which a lump of single-phase flow retains its identity and velocity. Nikuradse (as reported in ref. 4) showed that the mixing length 1 in a circular pipe varies with distance y from the wall, being zero at the wall and 0.14R in the centre of the pipe (radius R). The variation can be represented by the following (strictly this expression is for Reynolds numbers greater than 100,000; since we are only interested in the average value of 1, we assume the result can be used in any turbulent flow).
(5) Since at the moment we are only interested in average bubble size, we require the average value of I weighted according to the flow area. This gives the bubble diameter as: 0 0 d =
s tR
i.e.
127c(R - JJ) d(R - y)
i’s
R
27@
- Y) d(R -
Y)
(6)
(7)
-“.5(gD3/v,2) -“~lz(up’/{gD}) -‘.06 (8)
There is a dependence on physical properties, p, the liquid density, D the surface tension and vi the kinematic viscosity of the liquid. In addition, the gas superficial velocity ug appears. Idogawa et al. [6] suggested
d = 0.00391p,-0~07(a/0.072)0~22ex~(-p)
(9)
A similar equation, apparently due to the same authors, is quoted in ref. 7, i.e. d = 0.0031p,-0.029( 1000#‘.0@
=P(-6’)
(10)
The next equation has some theoretical justification, in that bubble breakup is seen as a balance of buoyancy and surface tension forces [8, 91. This leads to:
d = 0.5(90/{Apg})“.5
(11)
In fact, the constant is specified as ranging from 0.4 to 0.6, but a single value is needed for the comparisons later.
Comparisons with recent data A survey of the recent literature revealed four papers by different research groups, covering a range of fluids and conditions. Representative data were taken from all the papers. The only data to be excluded were those in ref. 3 relating to non-Newtonian liquids. To make the graphs easier to understand, the predictions of eqn. (10) are not shown; in all cases they are similar to the predictions of eqn. (9). Physical property data are taken from the experimental papers, except that pp is calculated from the perfect gas law. A problem was encountered with the value of the surface tension in one case (see later).
J
5-
P8:‘Lm 6
z
Z
E .E
a
4-
q8
0
8
3-
4-
? 0 _----_---a_--
0
_'-n--"
0-n
9-
2-
0
2_
8
11
PI
-___-___-_-_________,
l-
O_
0
12
u(gas)
I
1
0
14
Pressure
cm/s
Fig. 2. Data from ref. 3 for n-butanol. The theoretical predictions in this and subsequent figures are shown by lines; that for the new theory, eqn. (7), by a dashed line. Bukur and Pate1 [3] report bubble diameters in a 0.05-m diameter column for butanol, water with 0.5% butanol and water with 1% butanol. The bubble diameters were deduced indirectly from the decay in the holdup value once the gas supply was cut off (dynamic gas disengagement method). The results for water with 1% butanol cover a limited range (they have already been shown in Fig. 1). The other results are directly comparable in that the same 1.85mm orifice was used and the same range of ug. Figure 2 shows the experimental data for butanol and the predictions of eqns. (7), (8), (9) and ( 11). Only eqn. (8) gives a good representation of the data. Figure 3 shows the results for water with 0.5% butanol. Equation (7) is the closest to the average of the results but does not reproduce the trend with ug. Oyevaar et al. [lo] used water with 2.0-2.2 mol kg-’ of diethanolamine in a 0.0855-m diameter column. Interfacial area was measured by a chemical method and the Sauter mean diameter deduced from eqn. (4). Results are reported for a range of Us values as a function
Fig. 4. Data diethanolamine
2
MPa
from ref. 10 for water with 2.0-2.2mol kg-’ and a gas superficial velocity of 1 cm SC’.
of
of the pressure p. The physical property values needed for the predictions are given directly in the paper apart from the surface tension. Working backwards from a prediction on one of the graphs, we deduced a 0 value of 0.020 N m-‘. In view of the slightly approximate nature of this calculation and the fact that the resulting value of o seemed rather low, we used g = 0.025 N m-i for the predictions; this improves the prediction of eqns. (9), (10) and (11) slightly relative to eqn. (7) (the effect on the overall performance of the different equations is negligible). Figure 4 shows the results for ug = 0.01 m SK’. Equations (7), (9) and (10) all give good results. Figure 5 shows the results for ug = 0.03 m SK’. None of the equations is particularly close but eqn. (8) is the best. The equations that include a variation with pressure fail to predict the trend with pressure shown in Fig. 5. The authors also report data for ug = 0.02 m s-i, which is intermediate in character, and limited results for higher superficial velocities which continues the trend seen in Figs. 4 and 5. Idogawa et al. [6] report a number of values of the bubble diameter. Of interest for the present comparison 10 ,
10 0
8-
8 : $8
ti 5
; z
E
6-
4-
.!? 0
:" P
0
m
w
EI
q
_------
_-_---_
2-
2 0
8
i 0
11
0. I
2
.
I
4
*
I
6
u(gas) Fig. 3. Data
7
.
I
.
8
,
10
.
,
12
0.0
.
14
cm/s
from ref. 3 for water
with 0.5% n-butanol.
0.2
0.4
0.6 Pressure
0.8
1.0
1.2
1.4
MPa
Fig. 5. Data from ref. 10 for water with 2.0-2.2 mol kg -’ of diethanolamine and a gas superficial velocity of 3 em s-l.
4
is that they used a number of liquids with a range of physical properties. In particular, the surface tension range was large and well covered. A 0.05-m diameter bubble column was used and the bubble diameter was measured by an electrical resistivity probe. Our comparison is based on their results for d versus surface tension at the highest and lowest pressures used. Since some of the equations require other properties in addition to surface tension, it becomes difficult to show all the predictions on the graphs. So the experimental points, and the predictions, relate, in order of increasing surface tension, to methanol, water with 30% ethanol, water with 1% isoamyl alcohol, water with 8% ethanol and water itself. The experimental points and predictions not shown relate to acetone and ethanol. These have nearly the same surface tension as methanol (and showed nearly the same d values) but viscosities that are lower and higher respectively. Figure 6 shows the results for p = 0.1 MPa. Equation (9) gives a particularly close prediction; eqn. (11) is good as well. Figure 7 shows the results for p = 5 MPa. All of the predictions are now quite good apart from that of eqn. (8).
10 7 - ____
6-
01 0
2
----____-____
4
6 u(gas)
6
10
cm/s
Fig. 8. Data from ref. 2 for water at atmospheric pressure.
The results of Wolff et al. [2] are for tap water and air in a 0.2-m diameter column. Bubble diameters were measured using an optical fibre probe. The comparison is shown in Fig. 8. Equation (8) gives the closest fit, despite displaying the wrong trend as the superficial velocity increases. Discussion
z
E Z? P
4-
9 11
---_-----__-
2-
01 0.01
.
, 0.02
.
--_--
,
,
0.03
0.04
Surface
.
I
0.05
.
7
I.
I.
0.06
tenslon
0.07
I 0.06
NI m
Fig. 6. Data from ref. 6 for various liquids with us = 3 cm s-l and p = 0.1 MPa.
I
0.01
0.02
I
I
0.03
0.04
Surface
,
0.05
tension
,
.
0.06
0.07
,
0.06
N/m
Fig. 7. Data from ref. 6 for various liquids with us = 3 cm s-l and p = 5 MPa.
A disappointing feature of the results is that there seem to be inconsistencies between one data set and another or between data and the correlating equations. Since the correlating equations themselves are based on data, the problem is entirely of apparent inconsistency between different data sets. Sometimes the bubble diameter increases with the superficial gas velocity, sometimes it decreases. The same applies to the variation with pressure. It may well be that the ideal situation mentioned in the Introduction, where the processes of bubble coalescence and breakup have had a chance to come to equilibrium, does not apply. If the data are being affected by, for example, the method of introducing the bubbles or the height of the column, then none of the equations can be expected to give particularly good results. Even the best of the equations, in this comparison, disagree with individual data sets to the extent of 50 or 60%. A small part of the error may arise from the fact that we had to read values off figures in the original references; this is not thought to be significant. Little information on experimental errors in determining the bubble diameter is given in the original references, but values as high as 50 or 60”/ do not seem plausible. To try to make the comparison more objective, all the data used so far, i.e. in Figs. 2-8, were used to calculate mean errors in the various equations, viz. mean values of ldexP- d,,,J/d,,, . Since this choice of data is slightly arbitrary, a word or two of explanation is needed. All of the data of ref. 2 are included. All of
5
the data of ref. 3 are used apart from Fig. 1 which was included in the Introduction to make a particular point; as is clear from Fig. 1, they support eqn. (7). In the other references, a small amount of extra data is available. To include these as well would unbalance the comparison. As things stand, a similar number of data points are being used from each source. The result of this quantitative comparison is that four of the equations for predicting the bubble diameter [eqns. (7), (9), (10) and (ll)] give similar errors in the range 31-37% with eqn. (9) being the best; eqn. (8) has a mean error of 74%. However, it could reasonably be argued that this comparison is unfair since the data of ref. 6 were themselves used in producing the correlation, eqn. (9). If the data of ref. (6) are excluded from the comparison, the data base is of course reduced but the performance of the equations becomes more uniform. The same four equations now give errors in the range 37-42% with eqns. (7) and (10) being the best; eqn. (8) has a mean error of 59%. However, it is hard to argue that eqn. (8) should be rejected. Its poor performance now is entirely due to its failure to predict the water/OS% butanol data of ref. 3. It is interesting to speculate on the limits of applicability of the new equation [eqn. (7)]. It does not seem reasonable that it would apply in indefinitely large columns, but we are not aware of any data for very large diameter columns. A number of the authors of the experimental studies make the point that the size of the bubbles produced by the sparger is not the same as those in the bubble column. It seems that, in practice, equilibrium is approached in bubble columns, making possible reasonable, but not very accurate, predictions that ignore the method of introducing the bubbles.
Conclusions
None of the equations under all conditions.
gives an accurate
prediction
Any of the equations can be used to give reasonable estimates of bubble diameter, but errors as high as 50-60% or even higher are possible. The equation based on the new theory, eqn. (7), d = 0.040, has the advantage of being mathematically simpler and not requiring any physical property values. Equation (7) is the only one not to have been fitted to experimental data. This suggests that this equation, and the ideas that lie behind it, are a promising starting point for any future attempt at correlation.
References Prediction of bubble size in 1 P. Obry and R. H. S. Winterton, two phase bubble flow in ducts - a new approach, Int. J. Multiphase Flow, submitted for publication. 2 C. Wolff, F. U. Briegleb, J. Bader, K. Hektor and H. Hammer, Measurements with multipoint microprobes: effect of suspended solids on the hydrodynamics of bubble columns, Chem. Eng. Technol., 13 (1990) 172-184. studies with 3 D. B. Bukur and S. A. Pate], Hydrodynamic foaming and non-Newtonian solutions in bubble columns, Can. J. Chem. Eng., 67 (1989) 741-751. 4 H. Schlichting, Boundury Layer Theory, 7th edn., McGrawHill, New York, 1987. Bubble size, interfacial area and 5 K. Akita and F. Yoshida, liquid-phase mass-transfer coefficient in bubble columns, Ind. Eng. Chem., Prncess Des. Dev., 13 (1974) X4-91. 6 K. Idogawa, K. Ikeda, T. Fukuda and S. Morooka, Effect of gas and liquid properties on the behaviour of bubbles in a column under high pressure, Int. Chem. fig., 27( 1987) 93-99. Mass transfer phenom7 M. H. Oyevaar and K. R. Westerterp, ena and hydrodynamics in agitated gas-liquid reactors and bubble columns at elevated pressures: state of the art, Chem. Eng. Process., 25 (1989) 85-98. and coalescence of bubbles and droplets, 8 E. Blass, Formation Int. Chem. Eng., 30 (1990) 206-221. Auslegung und Massstabsvergrosserung von 9 A. Mersmann, Blasen- und Tropfensiulen, Chem.-Ing.-Tech., 49 (1977) 679691. T. de la Rie, C. L. van der Sluijs and K. R. 10 M. H. Oyevaar, Westerterp, Interfacial areas and gas holdups in bubble columns and packed beds at elevated pressures, Chem. Eng. Process., 26(1989) l-14.
l-5
A simple method of predicting bubble size in bubble columns R. H. S. Winterton School of Manufacturing and Mechanical Engineering, University of Birmingham, PO Box 363, Birmingham B15 2TT (UK) (Received May 28, 1993; in final form September 6, 1993)
Abstract A basic parameter in two-phase gas-liquid flow is the size of the bubbles. From the point of view of calculating the interfacial area for mass transfer, the size of the bubbles is as important as the holdup. Many empirical expressions for bubble diameter have been proposed. The purpose of this paper is to see whether recent work on bubble diameter in developed flow in pipes can also be used for bubble columns. The new expression for bubble diameter, which is based on an analogy with the structure of single-phase turbulent flows, is simpler to use and comparable in accuracy with the previous empirical expressions, in spite of not having any adjustable constants in it.
Introduction The gas-liquid interfacial area A available for mass transfer is related to the number of bubbles n per unit volume and their diameters d, by: A = f
ndi2
i=l
while the holdup
E is given by:
E = (x/6) i di3 i=l With a mean diameter d = !,
di3/$,
(2) d defined by:
di2
the relationship between surface area and holdup becomes simply E = Ad/6
then 10 -
(4)
and this is the reason for choosing the definition of d in eqn. (3). The quantity d is the Sauter mean diameter. Other terms used are d,, (volume/surface) or dx2 (3 dimensions/2 dimensions). A number of empirical correlations exist in the literature for predicting the value of d. Recently, a new approach has been suggested for calculating bubble diameter in two-phase flow in pipes [ 11. It was considered that the theory only applied to developed flow, i.e. to points a long way downstream from the point of introduction of the bubbles, where an equilibrium had been reached between coalescence and breakup processes. The flow in a bubble column reactor cannot be considered fully developed in this sense; the height of
0255-2701/94/$7.00 SSDI 0255-2701(93)00481-I
the column is often not very much greater than the diameter. Also the flow is more complex. Unlike the essentially one-directional flow in a pipe, there is recirculation; the liquid in the centre is carried up by the bubbles and returns downwards close to the walls. The detailed flow pattern can be quite complex (see, for example, the figures in ref. 2). There did not seem to be any simple way of resolving these problems, so it was decided to compare the original unmodified expression for bubble diameter given in ref. 1 directly with bubble column data, even though this expression had been derived for pipe flow. The fact that in many cases in bubble column experiments the flow cannot be considered to be in equilibrium can be seen in Fig. 1, taken from ref. 3. The results from just one of the experiments, with a given
q
.
6E
E
6-
& 5 E
4-
2 n
;___‘-__:_--‘_-__*_
2-
q
o! 0
.
1
1
.
,
q .
2
,
.
3 u(gas)
,
.
4
,
5
.
6
cm/s
Fig. 1. Data from ref. 3 for water with 1% n-butanol for three different bubble distributors. Also shown is the prediction of the new theory, eqn. (7).
0 1994 -
Elsevier
Sequoia.
All rights
reserved
2
method of introducing the bubbles, might be considered in rather poor agreement with the theory (details of the new theory are given later). However, when the results of all three experiments are considered, the theory is seen to be a reasonable match. None of the various correlations mentioned later allow for a variation with the method of introducing the bubbles. A search of the literature revealed a number of expressions for calculating d. These expressions themselves incorporate to some extent the correlation of the earlier experimental measurements. It was felt that the most useful comparisons would be between the various equations and the more recent experimental data. In fact, in the limited literature survey, we did not encotmter any other comparisons of this type, so a subsidiary purpose of the paper became to compare the existing correlations with one another and the recent data, regardless of the new equation.
d = 0.040
where D is the pipe diameter. Although derived for pipe flow, this expression is also assumed to apply to bubble columns with D being the bubble column diameter.
Other correlations The following correlations were found in a search of the literature. In many cases the equations found were already in basic SI units, or even in dimensionless form. In a few cases, minor changes have been made so that all quantities that appear are in basic SI units; the only exception is the pressure p which (when needed) is in MPa. Akita and Yoshida [ 51 suggested, based on a correlation of a large number of experimental results, that
d/D = 26(gD2p,/a) The new theory Inside a turbulent single-phase flow in a pipe, there exist turbulent eddies. The structure of the flow is known to be stable with these eddies present. We postulate that a two-phase gas-liquid flow is also stable with gas bubbles present in some of the eddies, or even replacing an eddy completely. The diameter of the bubble is therefore considered to be similar to that of the eddies. To obtain a quantitative measure of the average bubble diameter, we simply assume that it is equal to the average value of the mixing length. This is reasonable since the mixing length is the distance over which a lump of single-phase flow retains its identity and velocity. Nikuradse (as reported in ref. 4) showed that the mixing length 1 in a circular pipe varies with distance y from the wall, being zero at the wall and 0.14R in the centre of the pipe (radius R). The variation can be represented by the following (strictly this expression is for Reynolds numbers greater than 100,000; since we are only interested in the average value of 1, we assume the result can be used in any turbulent flow).
(5) Since at the moment we are only interested in average bubble size, we require the average value of I weighted according to the flow area. This gives the bubble diameter as: 0 0 d =
s tR
i.e.
127c(R - JJ) d(R - y)
i’s
R
27@
- Y) d(R -
Y)
(6)
(7)
-“.5(gD3/v,2) -“~lz(up’/{gD}) -‘.06 (8)
There is a dependence on physical properties, p, the liquid density, D the surface tension and vi the kinematic viscosity of the liquid. In addition, the gas superficial velocity ug appears. Idogawa et al. [6] suggested
d = 0.00391p,-0~07(a/0.072)0~22ex~(-p)
(9)
A similar equation, apparently due to the same authors, is quoted in ref. 7, i.e. d = 0.0031p,-0.029( 1000#‘.0@
=P(-6’)
(10)
The next equation has some theoretical justification, in that bubble breakup is seen as a balance of buoyancy and surface tension forces [8, 91. This leads to:
d = 0.5(90/{Apg})“.5
(11)
In fact, the constant is specified as ranging from 0.4 to 0.6, but a single value is needed for the comparisons later.
Comparisons with recent data A survey of the recent literature revealed four papers by different research groups, covering a range of fluids and conditions. Representative data were taken from all the papers. The only data to be excluded were those in ref. 3 relating to non-Newtonian liquids. To make the graphs easier to understand, the predictions of eqn. (10) are not shown; in all cases they are similar to the predictions of eqn. (9). Physical property data are taken from the experimental papers, except that pp is calculated from the perfect gas law. A problem was encountered with the value of the surface tension in one case (see later).
J
5-
P8:‘Lm 6
z
Z
E .E
a
4-
q8
0
8
3-
4-
? 0 _----_---a_--
0
_'-n--"
0-n
9-
2-
0
2_
8
11
PI
-___-___-_-_________,
l-
O_
0
12
u(gas)
I
1
0
14
Pressure
cm/s
Fig. 2. Data from ref. 3 for n-butanol. The theoretical predictions in this and subsequent figures are shown by lines; that for the new theory, eqn. (7), by a dashed line. Bukur and Pate1 [3] report bubble diameters in a 0.05-m diameter column for butanol, water with 0.5% butanol and water with 1% butanol. The bubble diameters were deduced indirectly from the decay in the holdup value once the gas supply was cut off (dynamic gas disengagement method). The results for water with 1% butanol cover a limited range (they have already been shown in Fig. 1). The other results are directly comparable in that the same 1.85mm orifice was used and the same range of ug. Figure 2 shows the experimental data for butanol and the predictions of eqns. (7), (8), (9) and ( 11). Only eqn. (8) gives a good representation of the data. Figure 3 shows the results for water with 0.5% butanol. Equation (7) is the closest to the average of the results but does not reproduce the trend with ug. Oyevaar et al. [lo] used water with 2.0-2.2 mol kg-’ of diethanolamine in a 0.0855-m diameter column. Interfacial area was measured by a chemical method and the Sauter mean diameter deduced from eqn. (4). Results are reported for a range of Us values as a function
Fig. 4. Data diethanolamine
2
MPa
from ref. 10 for water with 2.0-2.2mol kg-’ and a gas superficial velocity of 1 cm SC’.
of
of the pressure p. The physical property values needed for the predictions are given directly in the paper apart from the surface tension. Working backwards from a prediction on one of the graphs, we deduced a 0 value of 0.020 N m-‘. In view of the slightly approximate nature of this calculation and the fact that the resulting value of o seemed rather low, we used g = 0.025 N m-i for the predictions; this improves the prediction of eqns. (9), (10) and (11) slightly relative to eqn. (7) (the effect on the overall performance of the different equations is negligible). Figure 4 shows the results for ug = 0.01 m SK’. Equations (7), (9) and (10) all give good results. Figure 5 shows the results for ug = 0.03 m SK’. None of the equations is particularly close but eqn. (8) is the best. The equations that include a variation with pressure fail to predict the trend with pressure shown in Fig. 5. The authors also report data for ug = 0.02 m s-i, which is intermediate in character, and limited results for higher superficial velocities which continues the trend seen in Figs. 4 and 5. Idogawa et al. [6] report a number of values of the bubble diameter. Of interest for the present comparison 10 ,
10 0
8-
8 : $8
ti 5
; z
E
6-
4-
.!? 0
:" P
0
m
w
EI
q
_------
_-_---_
2-
2 0
8
i 0
11
0. I
2
.
I
4
*
I
6
u(gas) Fig. 3. Data
7
.
I
.
8
,
10
.
,
12
0.0
.
14
cm/s
from ref. 3 for water
with 0.5% n-butanol.
0.2
0.4
0.6 Pressure
0.8
1.0
1.2
1.4
MPa
Fig. 5. Data from ref. 10 for water with 2.0-2.2 mol kg -’ of diethanolamine and a gas superficial velocity of 3 em s-l.
4
is that they used a number of liquids with a range of physical properties. In particular, the surface tension range was large and well covered. A 0.05-m diameter bubble column was used and the bubble diameter was measured by an electrical resistivity probe. Our comparison is based on their results for d versus surface tension at the highest and lowest pressures used. Since some of the equations require other properties in addition to surface tension, it becomes difficult to show all the predictions on the graphs. So the experimental points, and the predictions, relate, in order of increasing surface tension, to methanol, water with 30% ethanol, water with 1% isoamyl alcohol, water with 8% ethanol and water itself. The experimental points and predictions not shown relate to acetone and ethanol. These have nearly the same surface tension as methanol (and showed nearly the same d values) but viscosities that are lower and higher respectively. Figure 6 shows the results for p = 0.1 MPa. Equation (9) gives a particularly close prediction; eqn. (11) is good as well. Figure 7 shows the results for p = 5 MPa. All of the predictions are now quite good apart from that of eqn. (8).
10 7 - ____
6-
01 0
2
----____-____
4
6 u(gas)
6
10
cm/s
Fig. 8. Data from ref. 2 for water at atmospheric pressure.
The results of Wolff et al. [2] are for tap water and air in a 0.2-m diameter column. Bubble diameters were measured using an optical fibre probe. The comparison is shown in Fig. 8. Equation (8) gives the closest fit, despite displaying the wrong trend as the superficial velocity increases. Discussion
z
E Z? P
4-
9 11
---_-----__-
2-
01 0.01
.
, 0.02
.
--_--
,
,
0.03
0.04
Surface
.
I
0.05
.
7
I.
I.
0.06
tenslon
0.07
I 0.06
NI m
Fig. 6. Data from ref. 6 for various liquids with us = 3 cm s-l and p = 0.1 MPa.
I
0.01
0.02
I
I
0.03
0.04
Surface
,
0.05
tension
,
.
0.06
0.07
,
0.06
N/m
Fig. 7. Data from ref. 6 for various liquids with us = 3 cm s-l and p = 5 MPa.
A disappointing feature of the results is that there seem to be inconsistencies between one data set and another or between data and the correlating equations. Since the correlating equations themselves are based on data, the problem is entirely of apparent inconsistency between different data sets. Sometimes the bubble diameter increases with the superficial gas velocity, sometimes it decreases. The same applies to the variation with pressure. It may well be that the ideal situation mentioned in the Introduction, where the processes of bubble coalescence and breakup have had a chance to come to equilibrium, does not apply. If the data are being affected by, for example, the method of introducing the bubbles or the height of the column, then none of the equations can be expected to give particularly good results. Even the best of the equations, in this comparison, disagree with individual data sets to the extent of 50 or 60%. A small part of the error may arise from the fact that we had to read values off figures in the original references; this is not thought to be significant. Little information on experimental errors in determining the bubble diameter is given in the original references, but values as high as 50 or 60”/ do not seem plausible. To try to make the comparison more objective, all the data used so far, i.e. in Figs. 2-8, were used to calculate mean errors in the various equations, viz. mean values of ldexP- d,,,J/d,,, . Since this choice of data is slightly arbitrary, a word or two of explanation is needed. All of the data of ref. 2 are included. All of
5
the data of ref. 3 are used apart from Fig. 1 which was included in the Introduction to make a particular point; as is clear from Fig. 1, they support eqn. (7). In the other references, a small amount of extra data is available. To include these as well would unbalance the comparison. As things stand, a similar number of data points are being used from each source. The result of this quantitative comparison is that four of the equations for predicting the bubble diameter [eqns. (7), (9), (10) and (ll)] give similar errors in the range 31-37% with eqn. (9) being the best; eqn. (8) has a mean error of 74%. However, it could reasonably be argued that this comparison is unfair since the data of ref. 6 were themselves used in producing the correlation, eqn. (9). If the data of ref. (6) are excluded from the comparison, the data base is of course reduced but the performance of the equations becomes more uniform. The same four equations now give errors in the range 37-42% with eqns. (7) and (10) being the best; eqn. (8) has a mean error of 59%. However, it is hard to argue that eqn. (8) should be rejected. Its poor performance now is entirely due to its failure to predict the water/OS% butanol data of ref. 3. It is interesting to speculate on the limits of applicability of the new equation [eqn. (7)]. It does not seem reasonable that it would apply in indefinitely large columns, but we are not aware of any data for very large diameter columns. A number of the authors of the experimental studies make the point that the size of the bubbles produced by the sparger is not the same as those in the bubble column. It seems that, in practice, equilibrium is approached in bubble columns, making possible reasonable, but not very accurate, predictions that ignore the method of introducing the bubbles.
Conclusions
None of the equations under all conditions.
gives an accurate
prediction
Any of the equations can be used to give reasonable estimates of bubble diameter, but errors as high as 50-60% or even higher are possible. The equation based on the new theory, eqn. (7), d = 0.040, has the advantage of being mathematically simpler and not requiring any physical property values. Equation (7) is the only one not to have been fitted to experimental data. This suggests that this equation, and the ideas that lie behind it, are a promising starting point for any future attempt at correlation.
References Prediction of bubble size in 1 P. Obry and R. H. S. Winterton, two phase bubble flow in ducts - a new approach, Int. J. Multiphase Flow, submitted for publication. 2 C. Wolff, F. U. Briegleb, J. Bader, K. Hektor and H. Hammer, Measurements with multipoint microprobes: effect of suspended solids on the hydrodynamics of bubble columns, Chem. Eng. Technol., 13 (1990) 172-184. studies with 3 D. B. Bukur and S. A. Pate], Hydrodynamic foaming and non-Newtonian solutions in bubble columns, Can. J. Chem. Eng., 67 (1989) 741-751. 4 H. Schlichting, Boundury Layer Theory, 7th edn., McGrawHill, New York, 1987. Bubble size, interfacial area and 5 K. Akita and F. Yoshida, liquid-phase mass-transfer coefficient in bubble columns, Ind. Eng. Chem., Prncess Des. Dev., 13 (1974) X4-91. 6 K. Idogawa, K. Ikeda, T. Fukuda and S. Morooka, Effect of gas and liquid properties on the behaviour of bubbles in a column under high pressure, Int. Chem. fig., 27( 1987) 93-99. Mass transfer phenom7 M. H. Oyevaar and K. R. Westerterp, ena and hydrodynamics in agitated gas-liquid reactors and bubble columns at elevated pressures: state of the art, Chem. Eng. Process., 25 (1989) 85-98. and coalescence of bubbles and droplets, 8 E. Blass, Formation Int. Chem. Eng., 30 (1990) 206-221. Auslegung und Massstabsvergrosserung von 9 A. Mersmann, Blasen- und Tropfensiulen, Chem.-Ing.-Tech., 49 (1977) 679691. T. de la Rie, C. L. van der Sluijs and K. R. 10 M. H. Oyevaar, Westerterp, Interfacial areas and gas holdups in bubble columns and packed beds at elevated pressures, Chem. Eng. Process., 26(1989) l-14.