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Flow in bubble columns: some numerical experiments
Chemical Printed
Er~,~ineeriw in Great
Science,
Vol.
47, No.
8, pp.
1857F1869,
1992.
ooo!-250!?/92
Britain
0
FLOW
IN BUBBLE
COLUMNS: SOME EXPERIMENTS+ V.
5 September
1991;
+ 0.00
Reap
Ltd
NUMERICAL
V. RANADE
Chemical Engineering Division, National (Received
35.00
1992 Pergamon
Chemical Laboratory,
acceptedfor
publication
Pune 411008,
India
29 Nooember 199 1)
Abstract-Bubble columns are being widely used in the chemical and petrochemical industry. It has become increasingly important to develop theoretically sound models for scaling up and designing of these reactors for more complicated reaction systems. Such reactor models would require detailed knowledge of the flow characte’ristics of the reactor. This paper presents a numerical simulation model for predicting the details of turbulent gas-liquid flow encountered in bubble columns. The time-averaged Navier-Stokes equations for the liquid phase are solved, using a finite-volume solution algorithm. The turbulent stresses are obtained from the k-e model. The bubble slip velocities were specified externally to the solution, avoiding the need for the solutions of the gas-phase momentum equations. The results of various numerical experiments are described to highlight the influences of key parameters on the characteristics of flow in bubble columns.
1. INTRODUCTION
in bubble columns has been the topic of a large number of publications in the last two decades. However, no general flow model has been developed until now. This is because of the lack of understanding about complicated interactions between numerous hydrodynamic parameters. Internal liquid circulation due to the non-uniform radial hold-up profile is a commonly encountered flow structure in bubble columns. Though the importance of internal liquid circulation is recognised, no theory based on first principles is available. Most of the published flow models are restricted to one-dimensional approximations. These models require the radial gas hold-up profile and eddy viscosity as model inputs. Different constitutive equations have been proposed for the eddy viscosity, ranging from purely empirical correlations (Ueyema and Miyayuchi, 1979) to turbulence models based on single-phase mixing length (Clark et al., 1987; Anderson and Rice, 1989; Gasche et al., 1991). These one-dimensional models have not addressed the problem of axial development (variation) of the radial hold-up profile with specific sparging conditions. This problem of development of the radial hold-up profile is still the least understood aspect of the gas-liquid flow in bubble columns. Recently, detailed experimental data about flow in bubble columns have been published {Devanathan et al., 1990; Yao et al., 1990; Menzel et ai., 1990; Yu and Kim, 1991), which clearly indicate the complex interactions of bubble slip velocity, radial hold-up profile and internal circulation in bubble columns. For the interpretation and understanding of these data, more advanced models are necessary. In the present paper+ we have developed a numerical model for simulating flows in bubble columns. This model requires onljr Flow
sparging pattern as an external input and is capable of simulating the development 6f the hold-up profile and the corresponding internal circulation in bubble columns. However, before embarking on a detailed simulation exercise, it is essential to understand the key features of the flow problem. This will enable one to pose/reframe the problem, keeping only the essentials. Therefore, in this paper, we have restricted the scope of the simulations to two dimensions and concentrated on highlighting the influences of various key parameters on flow structure. 2 MATHEMATICAL
l
‘NCL Communication
MODEL
2.1. Model equations and boundary conditions The sparging of gas in a stagnant pool of liquid generates complex, three-dimensional turbulent flow. Therefore, it is essential to use an appropriate turbulence model to simulate this flow. Previously developed one-dimensional flow models use various turbulent viscosity profiles (recently reviewed by Joshi et al., 1991). Recently, Torvik and Svendsen (1990) have used the k-e model of turbulence to simulate flow in a bubble column slurry reactor. Schwarz and Turner (1988) have also applied the standard k-s model to simulate flow in gas-stirred baths. Therefore, in the present work, we have used the k-e model to estimate turbulent viscosity. The case of gas-liquid flow in a cylindrical bubble column is considered. The cylindrical coordinate system has been used with the origin located at the bottom centre. A schematic diagram of the column with the coordinate system is shown in Fig. 1. In order to highlight the key features of internal circulation in bubble columns, we have eliminated non-essential aspects from the problem using following assumptions:
No. 5266. 1857
Gas and liquid are considered as incompressible fluids.
V. V. RANADE
1858
forces, the calculated radial slip velocities (using the drag law suggested by Schwarz and Turner, 1988) are less than 0.05 m/s for an air-water system. Therefore, the assumption of zero radial bubble slip can be considered as a reasonable assumption for the qualitative investigation like the present one. Inertial and viscous terms in the gas-phase momentum equations are negligible. Interphase drag terms in the liquid-phase momentum equations can be obtained from the simplified gasphase momentum equations, since the slip velocity is known. This arrangement conveniently eliminates the need for the correct formulation of the interphase forces (drag, lift etc.) No mass transfer occurs between gas and liquid phases.
m The flow has a symmetry in the tangential direction and the tangential component of the liquidphase velocity is zero everywhere. l The slip velocity of gas bubbles with respect to the liquid phase will be specified, without consideringthe axial variation. The slip velocity in the radial direction is assumed to be zero. This assumption eliminates the need for solving gasphase momentum equations. Recently, Wachi and Yates (1991) have discussed the role of Magnus forces and centrifugal forces in the development of the radial hold-up profile. Both of these forces are relevant to the estimation of the radial bubble slip velocity. Preliminary estimates of these forces from the available data suggest that both would be of the same order of magnitude, but with opposite signs. In the absence of both of
w=o
FREE
SLIP
BC
-
-
these
All the transport equations (momentum and mass) for a bubble column, simplified using the above assumptions, can be written as a general transport equation for phase j : a(Ej+J)lar
+o
Yo SLIP
u=o
aeG
BC
i
_
ar-O
IJ=O
2
I
I
r _-
+
+
ak,
wj4jm
+
wjr,,wjmm
Wjr~jf$j)lrar =
f%+-,jafb,,/w/rar +
BC,W=O
(1)
r&j denotes an effective turbulent exchange coefficient (including the molecular contribution) for variable Q of phase j, and S, denotes the volumetric growth or decay in C#I because of internal or external sources. The corresponding I’,, and Sei terms for different 4 of the liquid phase are listed in Table 1. All the terms appearing in these equations are dimensionless. The pressure, P, is the reduced pressure. The X appearing in the radial momentum equation is the hydrostatic head defined as
H
NO SUP
s,.
’
x=
R
sI
E& dz.
(2)
Interphase transport terms can be obtained from the Fig. 1. Solution
domain.
Table 1. The details of eq. (1)
1 u W k E
0 P’m
aCr(~.,r/oo)aE,/arl/rar + w,,,i~c)waw~ - Q~P/& - e,aX/& + F’,, - +3Pia2 + EL+4 + FL, G--s .z(C,G - C,E)/k
@Cff
where
flefrl~c &fff me G = r,{2~(au/ar)*
+
(aw/ag
+ (LJ/~)~I+ (au/a2 + aw/a+j
Perr= P + Pt jit = C,k2/s Model parameters c, = Qh =
0.09 1.0
c, = 1.43 0, = 1.3
C, = 1.92 Do = 1.0
1859
Flow in bubble columns gas-phase momentum F&z F;,
=
balance:
E,axiar
-F&
=
-
~~a~idt
-
=
-
EGaP/aZ + E~E=~.
F&
-
(3) (4)
The boundary conditions used in the present simulation are shown in Fig. 1. Wall functions have been used to impose a no-slip boundary condition at an impermeable wall. The sparging of gas at the bottom of the column was accounted via source terms in the gas-phase continuity equations for those computational cells which contain sparger holes at their bottom face. With this arrangement, various sparging patterns can be accommodated conveniently into the model. It should be noted, however, that the model assumes that the gas enters in the form of bubbles, and therefore cannot account for the gas jetting etc. The model parameters used in the present work are also listed in Table 1.
0.
2.2. Solution of model equations The set of conservation equations described by eq. (1) are highly non-linear and elliptic. In the present work, they were solved using a finite-volume method. Recently, Ranade et al. (1991) have described a code, FIAT, for single-phase turbulent flows using nonuniform grids. This work uses a two-phase flow version of that code, called TPFLOW [details of the numerical method and the computer code, TPFLOW, are given by Ranade (1991)-j. All the flow computations were carried out for a bubble column with a height-to-diameter ratio of one unless otherwise mentioned. The mean velocity, fractional hold-up and pressure need to underrelax to achieve convergence. The iterations-for Aow computations were continued till the mass residue becomes less than lo-“. Figure 2 presents the computed results for 5 x 10 and 10 x 20 grids (all the radial profiles shown in the figures correspond to approximately half of the column
8GRIDS
CURVE
_--
0. 6-
10x20
0. 5-
0. 4-
0. 3-
0 .2-
0 .l-
o-
-0
.l
-
-0
-2 -
-0
3-
0.1
0.3
DIMENSIONLESS
0.5
RADIAL
0.7
COORDINATE,
0.9
r/R
Fig. 2. Influenceof grid size on predicted flow characteristics.Column diameter = 0.3 m, column height = 0.3 m. sparging area = 64%, superticial gas velociiy = O.O64m/s. axial bubble slip velocity = 0.3075 m/s.
1860
V. V.
RANADE
height unless otherwise mentioned). Since the difference is not very significant for the purpose of developing a qualitative understanding of the flow system, all the other computations were carried out using the 5 x 10 grids unless otherwise mentioned. 3.
RESULTS
AND
DISCUSSION
As mentioned in Section 2, in this work, we have specified the bubble slip velocity externally (without solving the gas-phase momentum equations). Therefore, the specification of the uniform slip velocity across the column with uniform sparging will not lead to internal circulation in the column. Since the objective is to understand the internal circulation in bubble columns, we have restricted sparging of gas only to the central portion of the column (circulation will be more or less similar to those columns operated with draft tubes). Figure 3 shows the predicted results for the cases of sparging in 64% and 34% column area. The profiles look qualitatively similar. Therefore, all the results described in Section 3.1 are for gas sparging restricted to the central 64% of the area.
\ 0.9-
‘\
3.1. With radially uniform slip velocity 3.1 .l. Influence of gas velocity and column diameter. It is necessary to verify that the numerical simulation model correctly predicts the experimentally observed trends in circulation velocity with respect to column diameter and superficial gas velocity. Figure 4 shows the predicted variation of the maximum axial velocity with these parameters. It can be seen that the column diameter does not affect the average gas hold-up, whereas it is almost linearly proportional to the superficial gas velocity. The maximum axial velocity varies roughly as the cube root of the diameter and gas velocity. This proves the applicability of the model to a broad range of operation modes, as the trends are represented correctly. Results of some numerical experiments are described below to enable us to find out the sensitive parameters which control and dominate the fluid-dynamic behaviour. 3.1.2. Influence of bubble slip velocity. The imposed bubble slip velocity is one of the key parameters, which determines the average gas hold-up and inter-
CURVE
----
-0.2
SPARGING
AREA
64
%
36
%
-
DIMENSIONLESS
RADIAL
COORDINATE,
r/R
Fig. 3. Influence of sparging pattern on predicted flow characteristics (other details as in Fig. 2).
1861
Flow in bubble columns COLUMN
DIAMETER,
0.5
o-4
0.3
2
D m 0.6
O-6-
E-l
3
2: k v s 2
o-5-
o-4-
2 ij
o-3-
!s H s
0.2-
a I
o-1 -
I
OO
I
0.01
I
I
SUPERFICIAL
I
0 -07
O-05
O-03 GAS
VELOCITY,
V,
I
0.09
<
m/s
Fig. 4. Trends of variation of the predicted maximum axial velocity and average gas hold-up with respect to column diameter and gas superficial velocity (other details as in Fig. 2).
in the column. The predicted influence of the bubble slip velocity on the maximum axial velocity and the gas hold-up is shown in Fig. 5. The decrease in the bubble slip velocity will result in less drag on the liquid phase, and therefore lower the liquid circulation velocities. Gasche et al. (1991) also report similar trends in the predictions of their onedimensional model of bubble columns. nal circulation
3.1.3. InJuence of the k--E model parameters. Schwarz and Turner (1988) have examined the applicability of the standard k--E model for the simulation of flow in gas-stirred baths. They have concluded that with the proper boundary conditions, the standard k--E model can adequately simulate key fluid-dynamic characteristics. However, Abujelala and Lilley (1984) have recommended values different from the standard model parameters for recirculating flows. Therefor-e, to test the parametric sensitivity of the predicted results, we have used parameter values recommended by Abujelala and Lilley (1984) in our numerical simulation model. A decrease in the value of C2 from 1.92 to 1.6 causes an increase in the predicted circulation in the bubble column (Fig. 6). The average gas hold-up and the hold-up profile are not affected significantly by this change in the value of CZ (average gas hold-up decreases from 0.18 to 0.175). The lower value of C2
results in a decrease in the predicted values of turbulent kinetic energy (Fig. 6) and dissipation rates. Thus, it can be concluded that C2 is one of the most important model parameters which strongly influences the predictions. We have also tested the influence of the value of C,, keeping CZ = 1.6. A change in the value of C, from 0.09 to 0.125 obviously resulted in larger turbulent viscosities, and therefore in a decrease in the circulation. However, the influence of the value of C, is not as strong as that of C7. also
3.1.4. InJluence of dispersion of gas bubbles. In all the results discussed till now, the turbulent Schmidt number for gas bubbles was assumed to be equal to one. However, the dispersion of gas bubbles in turbulent flow is a complicated phenomenon dependent on the bubble diameter and the energy spectrum/length scales of turbulence. A general model for the estimation of the dispersion coefficient for gas bubbles is not yet available. Therefore, it would be worthwhile to examine the sensitivity of the simulated results to the value of the turbulent Schmidt number for gas bubbles. Figure 7 shows the comparison of the predicted axial velocity profiles for different Schmidt numbers. It can be seen that a very high Schmidt number (that is, almost no dispersion of gas bubbles) results in steeper gas hold-up profiles. However, the
1862
V. V.
RANADE
- 0.6
0.7
04
O.!
BUBBLE
SLIP
VELOCITY,
V,
m/s
Fig. 5. Influenceof bubble slip velocity on predictedaxial velocity and averagegas hold-up (other details as in Fig. 2).
liquid circulation in the column is almost the same as that for a Schmidt number of one_ This might be because of the higher turbulence in the absence of dispersion of gas bubbles. An increase in the turbulent viscosity compensates the increased density-driving forces in the absence of dispersion, resulting in almost the same circulation velocities. On the other hand, a lower Schmidt number results in flatter hold-up profiles. A ten-fold decrease in the Schmidt number decreases the maximum liquid circulation velocity only by 17%.
3.2. With radial variation of slip velocity All the results discussed till now use a radially uniform bubble slip velocity. However, it would be very interesting to investigate the influences of the radial profile of the bubble slip velocity on the internal circulation in bubble columns. Measurements of the bubble diameter (Yao et al., 1990; Yu and Kim, 1991) indicate that larger bubbles exist in the central region of the column as compared to those of the near-wall region. This would suggest a similar profile, that is a higher slip velocity at the column centre and a lower one near the wall for the bubble slip velocity. However, the subtraction of the measured local liquid velocity from the local bubble velocity (Yao er al+ 1990) show the opposite trend. Yao et al. (1990) have reported increasing bubble slip velocity towards the wall of the column. Thus, this issue of the radial
profile of the slip velocity certainly needs further investigation. We have used various radial profiles of the bubble slip velocity, keeping the area-averaged bubble slip velocity the same. Figure 8 shows the predicted profiles of the axial velocity with various slip velocity profiles. In all these simulations, gas was sparged only in the central 44% area. It can be seen that an increase in the bubble slip velocity towards the column wall leads to higher circulation in the column. It is very interesting to note that inverse profiles of the bubble slip velocity, i.e. decreasing slip velocity towards the wall, result in the inverse circulation in the bubble column. Such profiles lead to a downward flow in the central region and an upward flow near the wall. Such inverse circulation in the presence of central sparging of gas has not been reported in the literature. Curve 6 in Fig. 8 shows a normal circulation pattern despite the inverse profile of the bubble slip velocity. However, this behaviour is due to the non-uniform sparging of gas at the bottom of the column. If the gas is sparged uniformly at the bottom of the column, any inverse bubble slip velocity profile will lead to inverse circulation in the column. These results, showing the influence of the bubble slip velocity profiles, are for the non-uniform sparging of gas. We had considered the case of non-uniform sparging mainly to understand the circulation in the column with a radially uniform slip velocity. However, if there is a radial variation in the bubble slip
Flow
in bubble columns
1863
CURVE
---_ -.-
=2
CD
1.92
0.09
1.60 1.60
0.09 0.125
O-6
-
o-5
-
0.4
-
0.3
-
50
0.2
-
50
0.1
-
o-
-0.1
-0.2
-
-
DIMENSIONLESS Fig. 6. Effect
of
RADIAL
model parameters on
flow
velocity, internaI circulation will develop even with the uniform gas sparging. Figure 9 shows a comparison of the predicted flow characteristics with and without uniform gas sparging (without and with uniform bubble slip velocity, respectively). It can be seen that the flow characteristics are qualitatively similar. Numerical simulation models can generate much more detailed information about the system than the results described thus far. These results can help to understand the intricacies and the complicated interactions of the various hydrodynamic parameters. The axial variation and the development of the radial profiles of various key parameters is of considerable interest. Figures 10 and 11 show the axial variation of the radial profiles of gas hold-up and turbulent viscosity respectively. Figure 11 also shows the experimentally obtained values of turbulent viscosity by Menzel et al. (1990) for two gas velocities. The simulation model fails to reproduce the double-peaked CES
47:8-E
COORDINATE, characteristics
(other
r/R details
as in Fig. 2).
profile of turbulent viscosity. However, agreement can be said to be reasonable.
the overall
3.2.1. Infhence of height-to-diameter ratio. Experimental results show that the column height does not affect the gross hydrodynamic characteristics (average hold-up, centre-line velocity of liquid etc.) of the gas-liquid flow in the column. Joshi and Sharma (1979) and Zehner (1986) have proposed the existence of multiple circulation cells (each of height equal to the diameter of the column) along the height of the column. With such an assumption, the predicted centre-line velocities, using the mode1 of Joshi and Sharma (1979), agree with the experimental data. However, recent visual observations in two-dimensional bubble columns by Chen et al. (1989) or the tracking techniques used by Devanathan et al. (1990) do not indicate the existence of the multiple cells. Visual observations of Chen et al. (1989) suggest the
1864
V. V.
-0-3
I o-1
I
I 0.3
DIMENSIONLESS
Fig. 7.
Influence
I
RANADE
I 0.5
I
RADIAL
I 0.7
I
I 0.9
COORDINATE
of turbulent Schmidt number of gas bubbles on predicted flow (other details as in Fig. 2).
existence of two rows of staggered circulation cells similar to the vortices shed behind circular cylinders, whereas Devanathan et al. (1990) have observed only a single circulation cell in the column with a heightto-diameter ratio of 2.5. The observed independence of the liquid centre-line velocity with respect to height-to-diameter ratio is rather difficult to explain (using existing models) if only single circulation cells form in the bubble columns. Detailed numerical simulations are expected to throw some light on this controversy. In the present paper, our aim is limited to showing the influence of the height-to-diameter ratio on flow using crude grids. It should be noted however that, for tall bubble columns f > 20 m) operated under non-elevated pressures, the assumption of an incompressible gas phase, as used in the present simulations, would not be realistic. Figure 12 shows the axial profiles of the mean axial velocity for three different height-to-diameter ratios. It is interesting to note that the H/D ratio does not affect the magnitude of the maximum axial velocity. Experimental data also do not show any effect of the H/D ratio on the axial velocity. The location of the
axial velocity in terms of the normalised height is however substantially different for the three H/D ratios. Even for higher H/D ratios, the absolute location of the maximum axial velocity is the same as that for H/D equal to one. The examination of the simulated results show that the hold-up profile in the upper region of the higher H/D ratio column is almost flat. Therefore, there is only a weak circulation above a height more than one column diameter. Flatter hold-up profiles may be the result of an incorrect slip velocity profile or of an incorrect turbulent Schmidt number for the gas bubbles. The results shown in Fig. 12 are obtained for the uniform radial profile of turbulent Schmidt number for gas bubbles. However, significant radial variation of the turbulent length scales exists in the bubble columns_ Especially, the estimation of dispersion of gas bubbles in the near-wall region, where turbulent length scales are smaller than the bubble size, is not clear. More detailed models are necessary to account for such finer points in the simulations. To get an idea of the influence of bubble dispersion for taller columns, Fig. 12 also shows the predicted axial profile of the
1865
FIow in bubble columns
CURVE Flo. 1 2 3 4 5- 6 -
6LtP VELOCITY PROFtLE 0.205 [l+(r/Rf] 0.1645 [lHr/R)] O-3076 0.9225 0.61 5 0.46125
[I-(r/R) [l-(r/R) [l-b/R
I]
p]
3
DIMENSIONLESS
RADIAL
COORDINATE
Fig. 8. Influence of radial profiles of bubble slip velocity (other
axial velocity using very high turbulent Schmidt numbers for gas bubbles (equivalent to no dispersion of bubbles)_ It can be seen that, with no dispersion of gas bubbles, the radial hold-up profiles do not become flat, and therefore considerable circulation exists in the upper portion of the column with an H/D ratio of four. However, even in the case of no dispersion, no multiple cells were observed in the column with an H/D ratio of four (with 5 x 40 grids).
Discussion In order to understand the details of the flow in bubble columns and to enhance the effectiveness of future simulations, the results described in Sections 3.1 and 3.2 deserve some comments. It can be seen that these simulated results indicate flatter gas hold-up profiles than those observed experimentally 3.3.
,r/R
details as in Fig. 2).
[e.g. Yao et al. (1990) J. This disagreement needs to be rectified for better simulations of bubble columns. The previously discussed numerical experiments can provide some clues to achieve this. Section 3.1.4 shows that the decrease in the dispersion coefficient for gas bubbles does not affect the steepness of the hold-up profiles significantly. The use of a radially varying dispersion coefficient of bubbles may improve the agreement to some extent. However, the real mechanism of the maintenance of the hold-up profile in the bubble column needs to be further investigated. It should be noted that, in spite of the flatter hold-up profiles predicted, the predicted maximum axial velocity is overestimated as compared to the predictions of the available correlations (Joshi and Sharma, 1979). The incorporation of the radial slip velocity into the simulation model may improve the predictions of the hold-up profiles. However, a solu-
V.
1866
V.
RANADE
0.7.
CURVE
__--_
3
SPARGING AREA
SLIP VELOCITY PROFILE
100 5%
0205
64
l/.
[l + ( r/R ,*]
0.3075
0-2-
s w > ;r’
x a
O-l-
0
I
1 0.2
I
I o-4
DIMENSIONLESS
1
RADIAL
I 0.6
I
COORDINATE,
I O-6
t
r/R
Fig. 9. Comparison of predicted flow I’m uniform sparging (with non-uniform slip velocity) and for nonuniform sparging (with uniform slip velocity) (other details as in Fig. 2).
tion of the gas-phase momentum equations to estimate the slip velocities would require knowledge of the bubble-size distribution across the column. The results described in Section 3.2 clearly show that slip velocity profiles which increase with radius give the observed circulatory pattern with uniform gas sparging. If the observed bubble-size distribution (which shows larger bubbles in the central region) is used with the drag law (Schwarz and Turner’s (1988) drag law is independent of the bubble diameter, which is valid for the air-water system in the bubble diameter range 2-8 mm), the predicted axial slip velocity will decrease with radius. Such a slip velocity profile will lead to inverse circulation in the column. Yao et al. (1990) have reported the usual circulation pattern for the bubble size profile, with a central peak and an axial bubble slip velocity profile with a peak near the wall. This points to the need for a further investigation to reconcile the apparently inconsistent bubble size and slip velocity profiles. Specific experiments to understand the influences of the variation in the sur-
rounding liquid flow field on bubble rise velocity need to be undertaken. Another very important aspect, which needs further investigation, is the verification of the existence of multiple circulation cells in bubble columns. This is crucial for the performance modelling of the bubble column reactor, since mixing in gas and liquid phase and residence time distributions will IX determined by the internal, coherent flow structures. In bubble columns, circulation is developed due to the gas holdup profile. Somewhat analogously, internal circulation develops in the vertical slot (of which two vertical sides are maintained at two different temperatures) due to the temperature profile. Lee and Korpela (1983) have demonstrated the existence of multiple circulation cells in such tall cavities using their numerical simulations. Whether such cells also exist in bubble columns needs to be found out. Present simulations did not show such cells for the columns with an H/D ratio of four. However, these results were obtained using crude grids (5 x 40). Further simu-
Flow
CWRVE
in bubble
No.
Z/H
0.025
1 2
0.375
3
O-675
4
o-975
I o-4
I o-2
DIMENSIONLESS
Fig. 10. Axial
variation
1867
cohunns
1 O-6
I
RADIAL
I
I 0.6
COORDINATE,
I
0
r/R
of radial profile of gas hold-up. Sparging area = 100%. = 0.205*[1. + (r/R)**23 (other details as in Fig. 2).
axial bubble
slip velocity
0
m
l
0
l
4’
,
o-2
I
I
o-4
DIMENSIONLESS
0.6
RAOIAL
0-e
I
COORDINATE,r/R
Fig. 11. Axial variation of radial profile of turbulent kinematic viscosity (other details as in Fig. 10). (0) Superficial gas velocity = 0.072 (In/s) Menzel et al. (1990) (0) superficial gas velocity = 0.096 (m/s) Menzel et al. (1990).
V. V.
1868
RANADE
6-
,5-
,4-
.3-
-2-
o-1
o-2
0.3
o-4
DIMENSIONLESS
Fig. 12. Influence of H/D
AXIAL
0.6
0.6
0-S
COORDINATE,
0.9
t /t-l
ratio on predicted centre-line axial velocity (other details as in Fig. 10).
lations using finer grids and better differencing schemes (we have used the power-law differencing scheme of Patankar, 1980) to reduce the numerical diffusion need to be developed to clarify the issue of multiple cells. More experimental data on identifying flow structures in tall bubble columns (similar to Devanathan et al., 1990) need to be generated. Before closing, it would be worthwhile to discuss the use of detailed flow simulation models for design and scale-up of bubble columns. The most commonly used models for designing bubble column reactors are the dispersion models. These models use dispersion coefficients (axial and/or radial) to account for the mixing in the reactor. However, because of the complex flow structure in the bubble columns, no general relationship for the dispersion coefficients has been found. Dispersion models give no deeper understanding of what is going on inside the reactor, and therefore design and scale-up based on these models will be rather uncertain. In contrast to this, a detailed flow simulation model, which gives a better understanding of the flow structure inside the reactor, can be used to develop better mixing models. Ranade et al. (1991) and Ranade and Bourne (1991) have developed such mixing and reaction models based on detailed flow simulations for stirred-vessel reactors. Torvik and Svendsen (1990) have used flow simulations for developing a reaction model of a three-phase slurry reactor. Such models allow a spatially distributed description of kinetics and imply that fundamental kinetic data
and mass transfer data together with a flow simulation model can be used for more accurate design and scale-up. However, much better flow simulation models (and systematic model validation using experimental data) are needed before one starts developing reactor models. 4.
CONCLUSIONS
The present work demonstrates the numerical simulation approach for investigating flow in bubble columns. A mathematical model for the two-phase flow in bubble columns has been developed. Numerical experiments were carried out to highlight the key variables in determining the flow characteristics of bubble columns. These numerical experiments can provide much more detailed understanding about the relative importance of each variable for flow than that obtained experimentally. The results described in this work can be used as a starting point for the more detailed simulations to investigate the fluid mechanics of bubble columns. Acknowledgenzents-This research is supported by the Department of Science and Technology, Government of India (Sanction No. III 4(31)/9O/ET). NOTATION CD
Cl, D
c2
constant linking the eddy viscosity to k constants in the dissipation equations column diameter
Flow in bubble columns F&L
interphase in radial
FI,L
G H k P r R % u u.w W X Z
drag
force acting
on liquid
phase
direction
drag force acting on liquid phase in axial direction gravitational constant gravitational constant dimensionless (= sRlU2,) turbulence energy generation rate per unit volume height of the bubble column turbulent kinetic energy per unit mass pressure radial coordinate column radius source term in the &-equation mean radial velocity reference velocity mean axial velocity hydrostatic head defined by eq. (2) axial coordinate interphase
Greek letters effective transport coefficient for 4 l-4 phase hold-up E & turbulent energy dissipation rate per unit mass molecular viscosity of liquid P effective viscosity kff turbulent viscosity Pl turbulent Schmidt number for Q Q# general transport variable d Subscripts G of gas phase of phase j j L of liquid phase REFERENCES
Abujelala, M. T. and Lilley, D. G., 1984, Limitations and empirical extensionsof the k-~ model as applied to turbulent confined swirling flows. Gem. Engng Commun. 31, 22%236. Andersqn, K. G. and Rice, R. G., 1989, Local turbulence model for predicting circulation rates in bubble columns. A.Z.Ch.E. J. 35, 514-518.
1869
Chen, J. J. J., Jamialahmadi, M. and Li, S. M., 1989, Effect of liquid depth on circulation in bubble columns. Chem. Engng Res. Des. 67, 203-207. Clark, N. N., Atkinson, C. M. and Flemmer, R. L. C., 1987, Turbulent circulation in bubble columns. A.I.Ch.E. J. 33, 575-578. Devanathan. N., Moslemian, D. and Dudukovik, M. P., 1990, Flow mapping in bubble columns using CARPT. Chem. Engng Sci. 45, 2285-2291. Gasche, H. E., Edinger, C., Kompel, H. and Hofmann, H., 1990, A fluiddynamically based model of bubble column reactor. Chem. Engng Technol. 13, 341-349. Joshi, J. B., 1991, Chem. Engng Res. Des. 69. 253. Joshi, J. B. and Sharma, M. M., 1979, A circulation cell model for bubble columns. Trans. Znstn them. Engrs 57, 244-251. Joshi, J. B., Ranade, V. V., Gharat, S. D. and Lele, S. S., 1990, Sparged Ioop reactors. Can. J. them. Engng 68, 705-741. Lee, Y. and Korpela, S. A., 1983, Multicellular natural convection in a vertical slot. J. Fluid Mech. 126, 91-121. Men& T., Weide, T., Staudacher, 0.. Wein, 0. and Onken, U., 1990, Reynolds shear stress for modelling of bubble column reactors. Znd. Engng them. Res. 29, 988-994. Patankar, S. V., 1980, Numerical Heat Transfer and FIuid Flow. Hemisphere, Washington, DC. Ranade, V. V., 1991, TPFLOW: a code for simulating three dimensional two phase flows. NCL Internal Report. Ranade, V. V., Bourne, J. R. and Joshi, J. B., 1991, Fluid mechanics and blending in agitated tanks. Chem. Engng Sci. 46, 1883-1893. Ranade, V. V. and Bourne, J. R., 1991, Reactive mixing in agitated tanks. Chem. Engng Commun. 99, 33. Schwarz, M. P. and Turner, W. J., 1988, Applicability of the standard k--E turbulence model to gas stirred baths. Appl. Math. Modell. 12, 273-279. Torvik, R. and Svendsen, H. F., 1990, Modelling of slurry reactors. A fundamental approach. Chem. Engng Sci. 45, 23252332. Ueyama, K. and Miyauchi, T., 1979, Properties of recirculatina turbulent two phase flow in aas bubble columns. A.I.&E. J. 25, 258-2-65. Wachi, S. and Yates, J. G.. 1991. Comments on modellinr! of slurry reactors. A fundamental approach. Chem. En&zg Sci. 46, 1528-l 529. Yao, B. P., Zheng, C., Gasche, H. E. and Hofmann, H., 1990, Bubble bchaviour and flow structure of bubble columns (preprint). Yu, Y. H. and Kim, S. D., 1991, Bubble properties and local liquid velocity in the radial direction of cocurrent gas-liquid flow. Chem. Engng Sci. 46, 313-320. Zehner, P., 1986, Momentum, mass and heat transfer in bubble columns. Znt. them. Engng 26, 22-35.
Er~,~ineeriw in Great
Science,
Vol.
47, No.
8, pp.
1857F1869,
1992.
ooo!-250!?/92
Britain
0
FLOW
IN BUBBLE
COLUMNS: SOME EXPERIMENTS+ V.
5 September
1991;
+ 0.00
Reap
Ltd
NUMERICAL
V. RANADE
Chemical Engineering Division, National (Received
35.00
1992 Pergamon
Chemical Laboratory,
acceptedfor
publication
Pune 411008,
India
29 Nooember 199 1)
Abstract-Bubble columns are being widely used in the chemical and petrochemical industry. It has become increasingly important to develop theoretically sound models for scaling up and designing of these reactors for more complicated reaction systems. Such reactor models would require detailed knowledge of the flow characte’ristics of the reactor. This paper presents a numerical simulation model for predicting the details of turbulent gas-liquid flow encountered in bubble columns. The time-averaged Navier-Stokes equations for the liquid phase are solved, using a finite-volume solution algorithm. The turbulent stresses are obtained from the k-e model. The bubble slip velocities were specified externally to the solution, avoiding the need for the solutions of the gas-phase momentum equations. The results of various numerical experiments are described to highlight the influences of key parameters on the characteristics of flow in bubble columns.
1. INTRODUCTION
in bubble columns has been the topic of a large number of publications in the last two decades. However, no general flow model has been developed until now. This is because of the lack of understanding about complicated interactions between numerous hydrodynamic parameters. Internal liquid circulation due to the non-uniform radial hold-up profile is a commonly encountered flow structure in bubble columns. Though the importance of internal liquid circulation is recognised, no theory based on first principles is available. Most of the published flow models are restricted to one-dimensional approximations. These models require the radial gas hold-up profile and eddy viscosity as model inputs. Different constitutive equations have been proposed for the eddy viscosity, ranging from purely empirical correlations (Ueyema and Miyayuchi, 1979) to turbulence models based on single-phase mixing length (Clark et al., 1987; Anderson and Rice, 1989; Gasche et al., 1991). These one-dimensional models have not addressed the problem of axial development (variation) of the radial hold-up profile with specific sparging conditions. This problem of development of the radial hold-up profile is still the least understood aspect of the gas-liquid flow in bubble columns. Recently, detailed experimental data about flow in bubble columns have been published {Devanathan et al., 1990; Yao et al., 1990; Menzel et ai., 1990; Yu and Kim, 1991), which clearly indicate the complex interactions of bubble slip velocity, radial hold-up profile and internal circulation in bubble columns. For the interpretation and understanding of these data, more advanced models are necessary. In the present paper+ we have developed a numerical model for simulating flows in bubble columns. This model requires onljr Flow
sparging pattern as an external input and is capable of simulating the development 6f the hold-up profile and the corresponding internal circulation in bubble columns. However, before embarking on a detailed simulation exercise, it is essential to understand the key features of the flow problem. This will enable one to pose/reframe the problem, keeping only the essentials. Therefore, in this paper, we have restricted the scope of the simulations to two dimensions and concentrated on highlighting the influences of various key parameters on flow structure. 2 MATHEMATICAL
l
‘NCL Communication
MODEL
2.1. Model equations and boundary conditions The sparging of gas in a stagnant pool of liquid generates complex, three-dimensional turbulent flow. Therefore, it is essential to use an appropriate turbulence model to simulate this flow. Previously developed one-dimensional flow models use various turbulent viscosity profiles (recently reviewed by Joshi et al., 1991). Recently, Torvik and Svendsen (1990) have used the k-e model of turbulence to simulate flow in a bubble column slurry reactor. Schwarz and Turner (1988) have also applied the standard k-s model to simulate flow in gas-stirred baths. Therefore, in the present work, we have used the k-e model to estimate turbulent viscosity. The case of gas-liquid flow in a cylindrical bubble column is considered. The cylindrical coordinate system has been used with the origin located at the bottom centre. A schematic diagram of the column with the coordinate system is shown in Fig. 1. In order to highlight the key features of internal circulation in bubble columns, we have eliminated non-essential aspects from the problem using following assumptions:
No. 5266. 1857
Gas and liquid are considered as incompressible fluids.
V. V. RANADE
1858
forces, the calculated radial slip velocities (using the drag law suggested by Schwarz and Turner, 1988) are less than 0.05 m/s for an air-water system. Therefore, the assumption of zero radial bubble slip can be considered as a reasonable assumption for the qualitative investigation like the present one. Inertial and viscous terms in the gas-phase momentum equations are negligible. Interphase drag terms in the liquid-phase momentum equations can be obtained from the simplified gasphase momentum equations, since the slip velocity is known. This arrangement conveniently eliminates the need for the correct formulation of the interphase forces (drag, lift etc.) No mass transfer occurs between gas and liquid phases.
m The flow has a symmetry in the tangential direction and the tangential component of the liquidphase velocity is zero everywhere. l The slip velocity of gas bubbles with respect to the liquid phase will be specified, without consideringthe axial variation. The slip velocity in the radial direction is assumed to be zero. This assumption eliminates the need for solving gasphase momentum equations. Recently, Wachi and Yates (1991) have discussed the role of Magnus forces and centrifugal forces in the development of the radial hold-up profile. Both of these forces are relevant to the estimation of the radial bubble slip velocity. Preliminary estimates of these forces from the available data suggest that both would be of the same order of magnitude, but with opposite signs. In the absence of both of
w=o
FREE
SLIP
BC
-
-
these
All the transport equations (momentum and mass) for a bubble column, simplified using the above assumptions, can be written as a general transport equation for phase j : a(Ej+J)lar
+o
Yo SLIP
u=o
aeG
BC
i
_
ar-O
IJ=O
2
I
I
r _-
+
+
ak,
wj4jm
+
wjr,,wjmm
Wjr~jf$j)lrar =
f%+-,jafb,,/w/rar +
BC,W=O
(1)
r&j denotes an effective turbulent exchange coefficient (including the molecular contribution) for variable Q of phase j, and S, denotes the volumetric growth or decay in C#I because of internal or external sources. The corresponding I’,, and Sei terms for different 4 of the liquid phase are listed in Table 1. All the terms appearing in these equations are dimensionless. The pressure, P, is the reduced pressure. The X appearing in the radial momentum equation is the hydrostatic head defined as
H
NO SUP
s,.
’
x=
R
sI
E& dz.
(2)
Interphase transport terms can be obtained from the Fig. 1. Solution
domain.
Table 1. The details of eq. (1)
1 u W k E
0 P’m
aCr(~.,r/oo)aE,/arl/rar + w,,,i~c)waw~ - Q~P/& - e,aX/& + F’,, - +3Pia2 + EL+4 + FL, G--s .z(C,G - C,E)/k
@Cff
where
flefrl~c &fff me G = r,{2~(au/ar)*
+
(aw/ag
+ (LJ/~)~I+ (au/a2 + aw/a+j
Perr= P + Pt jit = C,k2/s Model parameters c, = Qh =
0.09 1.0
c, = 1.43 0, = 1.3
C, = 1.92 Do = 1.0
1859
Flow in bubble columns gas-phase momentum F&z F;,
=
balance:
E,axiar
-F&
=
-
~~a~idt
-
=
-
EGaP/aZ + E~E=~.
F&
-
(3) (4)
The boundary conditions used in the present simulation are shown in Fig. 1. Wall functions have been used to impose a no-slip boundary condition at an impermeable wall. The sparging of gas at the bottom of the column was accounted via source terms in the gas-phase continuity equations for those computational cells which contain sparger holes at their bottom face. With this arrangement, various sparging patterns can be accommodated conveniently into the model. It should be noted, however, that the model assumes that the gas enters in the form of bubbles, and therefore cannot account for the gas jetting etc. The model parameters used in the present work are also listed in Table 1.
0.
2.2. Solution of model equations The set of conservation equations described by eq. (1) are highly non-linear and elliptic. In the present work, they were solved using a finite-volume method. Recently, Ranade et al. (1991) have described a code, FIAT, for single-phase turbulent flows using nonuniform grids. This work uses a two-phase flow version of that code, called TPFLOW [details of the numerical method and the computer code, TPFLOW, are given by Ranade (1991)-j. All the flow computations were carried out for a bubble column with a height-to-diameter ratio of one unless otherwise mentioned. The mean velocity, fractional hold-up and pressure need to underrelax to achieve convergence. The iterations-for Aow computations were continued till the mass residue becomes less than lo-“. Figure 2 presents the computed results for 5 x 10 and 10 x 20 grids (all the radial profiles shown in the figures correspond to approximately half of the column
8GRIDS
CURVE
_--
0. 6-
10x20
0. 5-
0. 4-
0. 3-
0 .2-
0 .l-
o-
-0
.l
-
-0
-2 -
-0
3-
0.1
0.3
DIMENSIONLESS
0.5
RADIAL
0.7
COORDINATE,
0.9
r/R
Fig. 2. Influenceof grid size on predicted flow characteristics.Column diameter = 0.3 m, column height = 0.3 m. sparging area = 64%, superticial gas velociiy = O.O64m/s. axial bubble slip velocity = 0.3075 m/s.
1860
V. V.
RANADE
height unless otherwise mentioned). Since the difference is not very significant for the purpose of developing a qualitative understanding of the flow system, all the other computations were carried out using the 5 x 10 grids unless otherwise mentioned. 3.
RESULTS
AND
DISCUSSION
As mentioned in Section 2, in this work, we have specified the bubble slip velocity externally (without solving the gas-phase momentum equations). Therefore, the specification of the uniform slip velocity across the column with uniform sparging will not lead to internal circulation in the column. Since the objective is to understand the internal circulation in bubble columns, we have restricted sparging of gas only to the central portion of the column (circulation will be more or less similar to those columns operated with draft tubes). Figure 3 shows the predicted results for the cases of sparging in 64% and 34% column area. The profiles look qualitatively similar. Therefore, all the results described in Section 3.1 are for gas sparging restricted to the central 64% of the area.
\ 0.9-
‘\
3.1. With radially uniform slip velocity 3.1 .l. Influence of gas velocity and column diameter. It is necessary to verify that the numerical simulation model correctly predicts the experimentally observed trends in circulation velocity with respect to column diameter and superficial gas velocity. Figure 4 shows the predicted variation of the maximum axial velocity with these parameters. It can be seen that the column diameter does not affect the average gas hold-up, whereas it is almost linearly proportional to the superficial gas velocity. The maximum axial velocity varies roughly as the cube root of the diameter and gas velocity. This proves the applicability of the model to a broad range of operation modes, as the trends are represented correctly. Results of some numerical experiments are described below to enable us to find out the sensitive parameters which control and dominate the fluid-dynamic behaviour. 3.1.2. Influence of bubble slip velocity. The imposed bubble slip velocity is one of the key parameters, which determines the average gas hold-up and inter-
CURVE
----
-0.2
SPARGING
AREA
64
%
36
%
-
DIMENSIONLESS
RADIAL
COORDINATE,
r/R
Fig. 3. Influence of sparging pattern on predicted flow characteristics (other details as in Fig. 2).
1861
Flow in bubble columns COLUMN
DIAMETER,
0.5
o-4
0.3
2
D m 0.6
O-6-
E-l
3
2: k v s 2
o-5-
o-4-
2 ij
o-3-
!s H s
0.2-
a I
o-1 -
I
OO
I
0.01
I
I
SUPERFICIAL
I
0 -07
O-05
O-03 GAS
VELOCITY,
V,
I
0.09
<
m/s
Fig. 4. Trends of variation of the predicted maximum axial velocity and average gas hold-up with respect to column diameter and gas superficial velocity (other details as in Fig. 2).
in the column. The predicted influence of the bubble slip velocity on the maximum axial velocity and the gas hold-up is shown in Fig. 5. The decrease in the bubble slip velocity will result in less drag on the liquid phase, and therefore lower the liquid circulation velocities. Gasche et al. (1991) also report similar trends in the predictions of their onedimensional model of bubble columns. nal circulation
3.1.3. InJuence of the k--E model parameters. Schwarz and Turner (1988) have examined the applicability of the standard k--E model for the simulation of flow in gas-stirred baths. They have concluded that with the proper boundary conditions, the standard k--E model can adequately simulate key fluid-dynamic characteristics. However, Abujelala and Lilley (1984) have recommended values different from the standard model parameters for recirculating flows. Therefor-e, to test the parametric sensitivity of the predicted results, we have used parameter values recommended by Abujelala and Lilley (1984) in our numerical simulation model. A decrease in the value of C2 from 1.92 to 1.6 causes an increase in the predicted circulation in the bubble column (Fig. 6). The average gas hold-up and the hold-up profile are not affected significantly by this change in the value of CZ (average gas hold-up decreases from 0.18 to 0.175). The lower value of C2
results in a decrease in the predicted values of turbulent kinetic energy (Fig. 6) and dissipation rates. Thus, it can be concluded that C2 is one of the most important model parameters which strongly influences the predictions. We have also tested the influence of the value of C,, keeping CZ = 1.6. A change in the value of C, from 0.09 to 0.125 obviously resulted in larger turbulent viscosities, and therefore in a decrease in the circulation. However, the influence of the value of C, is not as strong as that of C7. also
3.1.4. InJluence of dispersion of gas bubbles. In all the results discussed till now, the turbulent Schmidt number for gas bubbles was assumed to be equal to one. However, the dispersion of gas bubbles in turbulent flow is a complicated phenomenon dependent on the bubble diameter and the energy spectrum/length scales of turbulence. A general model for the estimation of the dispersion coefficient for gas bubbles is not yet available. Therefore, it would be worthwhile to examine the sensitivity of the simulated results to the value of the turbulent Schmidt number for gas bubbles. Figure 7 shows the comparison of the predicted axial velocity profiles for different Schmidt numbers. It can be seen that a very high Schmidt number (that is, almost no dispersion of gas bubbles) results in steeper gas hold-up profiles. However, the
1862
V. V.
RANADE
- 0.6
0.7
04
O.!
BUBBLE
SLIP
VELOCITY,
V,
m/s
Fig. 5. Influenceof bubble slip velocity on predictedaxial velocity and averagegas hold-up (other details as in Fig. 2).
liquid circulation in the column is almost the same as that for a Schmidt number of one_ This might be because of the higher turbulence in the absence of dispersion of gas bubbles. An increase in the turbulent viscosity compensates the increased density-driving forces in the absence of dispersion, resulting in almost the same circulation velocities. On the other hand, a lower Schmidt number results in flatter hold-up profiles. A ten-fold decrease in the Schmidt number decreases the maximum liquid circulation velocity only by 17%.
3.2. With radial variation of slip velocity All the results discussed till now use a radially uniform bubble slip velocity. However, it would be very interesting to investigate the influences of the radial profile of the bubble slip velocity on the internal circulation in bubble columns. Measurements of the bubble diameter (Yao et al., 1990; Yu and Kim, 1991) indicate that larger bubbles exist in the central region of the column as compared to those of the near-wall region. This would suggest a similar profile, that is a higher slip velocity at the column centre and a lower one near the wall for the bubble slip velocity. However, the subtraction of the measured local liquid velocity from the local bubble velocity (Yao er al+ 1990) show the opposite trend. Yao et al. (1990) have reported increasing bubble slip velocity towards the wall of the column. Thus, this issue of the radial
profile of the slip velocity certainly needs further investigation. We have used various radial profiles of the bubble slip velocity, keeping the area-averaged bubble slip velocity the same. Figure 8 shows the predicted profiles of the axial velocity with various slip velocity profiles. In all these simulations, gas was sparged only in the central 44% area. It can be seen that an increase in the bubble slip velocity towards the column wall leads to higher circulation in the column. It is very interesting to note that inverse profiles of the bubble slip velocity, i.e. decreasing slip velocity towards the wall, result in the inverse circulation in the bubble column. Such profiles lead to a downward flow in the central region and an upward flow near the wall. Such inverse circulation in the presence of central sparging of gas has not been reported in the literature. Curve 6 in Fig. 8 shows a normal circulation pattern despite the inverse profile of the bubble slip velocity. However, this behaviour is due to the non-uniform sparging of gas at the bottom of the column. If the gas is sparged uniformly at the bottom of the column, any inverse bubble slip velocity profile will lead to inverse circulation in the column. These results, showing the influence of the bubble slip velocity profiles, are for the non-uniform sparging of gas. We had considered the case of non-uniform sparging mainly to understand the circulation in the column with a radially uniform slip velocity. However, if there is a radial variation in the bubble slip
Flow
in bubble columns
1863
CURVE
---_ -.-
=2
CD
1.92
0.09
1.60 1.60
0.09 0.125
O-6
-
o-5
-
0.4
-
0.3
-
50
0.2
-
50
0.1
-
o-
-0.1
-0.2
-
-
DIMENSIONLESS Fig. 6. Effect
of
RADIAL
model parameters on
flow
velocity, internaI circulation will develop even with the uniform gas sparging. Figure 9 shows a comparison of the predicted flow characteristics with and without uniform gas sparging (without and with uniform bubble slip velocity, respectively). It can be seen that the flow characteristics are qualitatively similar. Numerical simulation models can generate much more detailed information about the system than the results described thus far. These results can help to understand the intricacies and the complicated interactions of the various hydrodynamic parameters. The axial variation and the development of the radial profiles of various key parameters is of considerable interest. Figures 10 and 11 show the axial variation of the radial profiles of gas hold-up and turbulent viscosity respectively. Figure 11 also shows the experimentally obtained values of turbulent viscosity by Menzel et al. (1990) for two gas velocities. The simulation model fails to reproduce the double-peaked CES
47:8-E
COORDINATE, characteristics
(other
r/R details
as in Fig. 2).
profile of turbulent viscosity. However, agreement can be said to be reasonable.
the overall
3.2.1. Infhence of height-to-diameter ratio. Experimental results show that the column height does not affect the gross hydrodynamic characteristics (average hold-up, centre-line velocity of liquid etc.) of the gas-liquid flow in the column. Joshi and Sharma (1979) and Zehner (1986) have proposed the existence of multiple circulation cells (each of height equal to the diameter of the column) along the height of the column. With such an assumption, the predicted centre-line velocities, using the mode1 of Joshi and Sharma (1979), agree with the experimental data. However, recent visual observations in two-dimensional bubble columns by Chen et al. (1989) or the tracking techniques used by Devanathan et al. (1990) do not indicate the existence of the multiple cells. Visual observations of Chen et al. (1989) suggest the
1864
V. V.
-0-3
I o-1
I
I 0.3
DIMENSIONLESS
Fig. 7.
Influence
I
RANADE
I 0.5
I
RADIAL
I 0.7
I
I 0.9
COORDINATE
of turbulent Schmidt number of gas bubbles on predicted flow (other details as in Fig. 2).
existence of two rows of staggered circulation cells similar to the vortices shed behind circular cylinders, whereas Devanathan et al. (1990) have observed only a single circulation cell in the column with a heightto-diameter ratio of 2.5. The observed independence of the liquid centre-line velocity with respect to height-to-diameter ratio is rather difficult to explain (using existing models) if only single circulation cells form in the bubble columns. Detailed numerical simulations are expected to throw some light on this controversy. In the present paper, our aim is limited to showing the influence of the height-to-diameter ratio on flow using crude grids. It should be noted however that, for tall bubble columns f > 20 m) operated under non-elevated pressures, the assumption of an incompressible gas phase, as used in the present simulations, would not be realistic. Figure 12 shows the axial profiles of the mean axial velocity for three different height-to-diameter ratios. It is interesting to note that the H/D ratio does not affect the magnitude of the maximum axial velocity. Experimental data also do not show any effect of the H/D ratio on the axial velocity. The location of the
axial velocity in terms of the normalised height is however substantially different for the three H/D ratios. Even for higher H/D ratios, the absolute location of the maximum axial velocity is the same as that for H/D equal to one. The examination of the simulated results show that the hold-up profile in the upper region of the higher H/D ratio column is almost flat. Therefore, there is only a weak circulation above a height more than one column diameter. Flatter hold-up profiles may be the result of an incorrect slip velocity profile or of an incorrect turbulent Schmidt number for the gas bubbles. The results shown in Fig. 12 are obtained for the uniform radial profile of turbulent Schmidt number for gas bubbles. However, significant radial variation of the turbulent length scales exists in the bubble columns_ Especially, the estimation of dispersion of gas bubbles in the near-wall region, where turbulent length scales are smaller than the bubble size, is not clear. More detailed models are necessary to account for such finer points in the simulations. To get an idea of the influence of bubble dispersion for taller columns, Fig. 12 also shows the predicted axial profile of the
1865
FIow in bubble columns
CURVE Flo. 1 2 3 4 5- 6 -
6LtP VELOCITY PROFtLE 0.205 [l+(r/Rf] 0.1645 [lHr/R)] O-3076 0.9225 0.61 5 0.46125
[I-(r/R) [l-(r/R) [l-b/R
I]
p]
3
DIMENSIONLESS
RADIAL
COORDINATE
Fig. 8. Influence of radial profiles of bubble slip velocity (other
axial velocity using very high turbulent Schmidt numbers for gas bubbles (equivalent to no dispersion of bubbles)_ It can be seen that, with no dispersion of gas bubbles, the radial hold-up profiles do not become flat, and therefore considerable circulation exists in the upper portion of the column with an H/D ratio of four. However, even in the case of no dispersion, no multiple cells were observed in the column with an H/D ratio of four (with 5 x 40 grids).
Discussion In order to understand the details of the flow in bubble columns and to enhance the effectiveness of future simulations, the results described in Sections 3.1 and 3.2 deserve some comments. It can be seen that these simulated results indicate flatter gas hold-up profiles than those observed experimentally 3.3.
,r/R
details as in Fig. 2).
[e.g. Yao et al. (1990) J. This disagreement needs to be rectified for better simulations of bubble columns. The previously discussed numerical experiments can provide some clues to achieve this. Section 3.1.4 shows that the decrease in the dispersion coefficient for gas bubbles does not affect the steepness of the hold-up profiles significantly. The use of a radially varying dispersion coefficient of bubbles may improve the agreement to some extent. However, the real mechanism of the maintenance of the hold-up profile in the bubble column needs to be further investigated. It should be noted that, in spite of the flatter hold-up profiles predicted, the predicted maximum axial velocity is overestimated as compared to the predictions of the available correlations (Joshi and Sharma, 1979). The incorporation of the radial slip velocity into the simulation model may improve the predictions of the hold-up profiles. However, a solu-
V.
1866
V.
RANADE
0.7.
CURVE
__--_
3
SPARGING AREA
SLIP VELOCITY PROFILE
100 5%
0205
64
l/.
[l + ( r/R ,*]
0.3075
0-2-
s w > ;r’
x a
O-l-
0
I
1 0.2
I
I o-4
DIMENSIONLESS
1
RADIAL
I 0.6
I
COORDINATE,
I O-6
t
r/R
Fig. 9. Comparison of predicted flow I’m uniform sparging (with non-uniform slip velocity) and for nonuniform sparging (with uniform slip velocity) (other details as in Fig. 2).
tion of the gas-phase momentum equations to estimate the slip velocities would require knowledge of the bubble-size distribution across the column. The results described in Section 3.2 clearly show that slip velocity profiles which increase with radius give the observed circulatory pattern with uniform gas sparging. If the observed bubble-size distribution (which shows larger bubbles in the central region) is used with the drag law (Schwarz and Turner’s (1988) drag law is independent of the bubble diameter, which is valid for the air-water system in the bubble diameter range 2-8 mm), the predicted axial slip velocity will decrease with radius. Such a slip velocity profile will lead to inverse circulation in the column. Yao et al. (1990) have reported the usual circulation pattern for the bubble size profile, with a central peak and an axial bubble slip velocity profile with a peak near the wall. This points to the need for a further investigation to reconcile the apparently inconsistent bubble size and slip velocity profiles. Specific experiments to understand the influences of the variation in the sur-
rounding liquid flow field on bubble rise velocity need to be undertaken. Another very important aspect, which needs further investigation, is the verification of the existence of multiple circulation cells in bubble columns. This is crucial for the performance modelling of the bubble column reactor, since mixing in gas and liquid phase and residence time distributions will IX determined by the internal, coherent flow structures. In bubble columns, circulation is developed due to the gas holdup profile. Somewhat analogously, internal circulation develops in the vertical slot (of which two vertical sides are maintained at two different temperatures) due to the temperature profile. Lee and Korpela (1983) have demonstrated the existence of multiple circulation cells in such tall cavities using their numerical simulations. Whether such cells also exist in bubble columns needs to be found out. Present simulations did not show such cells for the columns with an H/D ratio of four. However, these results were obtained using crude grids (5 x 40). Further simu-
Flow
CWRVE
in bubble
No.
Z/H
0.025
1 2
0.375
3
O-675
4
o-975
I o-4
I o-2
DIMENSIONLESS
Fig. 10. Axial
variation
1867
cohunns
1 O-6
I
RADIAL
I
I 0.6
COORDINATE,
I
0
r/R
of radial profile of gas hold-up. Sparging area = 100%. = 0.205*[1. + (r/R)**23 (other details as in Fig. 2).
axial bubble
slip velocity
0
m
l
0
l
4’
,
o-2
I
I
o-4
DIMENSIONLESS
0.6
RAOIAL
0-e
I
COORDINATE,r/R
Fig. 11. Axial variation of radial profile of turbulent kinematic viscosity (other details as in Fig. 10). (0) Superficial gas velocity = 0.072 (In/s) Menzel et al. (1990) (0) superficial gas velocity = 0.096 (m/s) Menzel et al. (1990).
V. V.
1868
RANADE
6-
,5-
,4-
.3-
-2-
o-1
o-2
0.3
o-4
DIMENSIONLESS
Fig. 12. Influence of H/D
AXIAL
0.6
0.6
0-S
COORDINATE,
0.9
t /t-l
ratio on predicted centre-line axial velocity (other details as in Fig. 10).
lations using finer grids and better differencing schemes (we have used the power-law differencing scheme of Patankar, 1980) to reduce the numerical diffusion need to be developed to clarify the issue of multiple cells. More experimental data on identifying flow structures in tall bubble columns (similar to Devanathan et al., 1990) need to be generated. Before closing, it would be worthwhile to discuss the use of detailed flow simulation models for design and scale-up of bubble columns. The most commonly used models for designing bubble column reactors are the dispersion models. These models use dispersion coefficients (axial and/or radial) to account for the mixing in the reactor. However, because of the complex flow structure in the bubble columns, no general relationship for the dispersion coefficients has been found. Dispersion models give no deeper understanding of what is going on inside the reactor, and therefore design and scale-up based on these models will be rather uncertain. In contrast to this, a detailed flow simulation model, which gives a better understanding of the flow structure inside the reactor, can be used to develop better mixing models. Ranade et al. (1991) and Ranade and Bourne (1991) have developed such mixing and reaction models based on detailed flow simulations for stirred-vessel reactors. Torvik and Svendsen (1990) have used flow simulations for developing a reaction model of a three-phase slurry reactor. Such models allow a spatially distributed description of kinetics and imply that fundamental kinetic data
and mass transfer data together with a flow simulation model can be used for more accurate design and scale-up. However, much better flow simulation models (and systematic model validation using experimental data) are needed before one starts developing reactor models. 4.
CONCLUSIONS
The present work demonstrates the numerical simulation approach for investigating flow in bubble columns. A mathematical model for the two-phase flow in bubble columns has been developed. Numerical experiments were carried out to highlight the key variables in determining the flow characteristics of bubble columns. These numerical experiments can provide much more detailed understanding about the relative importance of each variable for flow than that obtained experimentally. The results described in this work can be used as a starting point for the more detailed simulations to investigate the fluid mechanics of bubble columns. Acknowledgenzents-This research is supported by the Department of Science and Technology, Government of India (Sanction No. III 4(31)/9O/ET). NOTATION CD
Cl, D
c2
constant linking the eddy viscosity to k constants in the dissipation equations column diameter
Flow in bubble columns F&L
interphase in radial
FI,L
G H k P r R % u u.w W X Z
drag
force acting
on liquid
phase
direction
drag force acting on liquid phase in axial direction gravitational constant gravitational constant dimensionless (= sRlU2,) turbulence energy generation rate per unit volume height of the bubble column turbulent kinetic energy per unit mass pressure radial coordinate column radius source term in the &-equation mean radial velocity reference velocity mean axial velocity hydrostatic head defined by eq. (2) axial coordinate interphase
Greek letters effective transport coefficient for 4 l-4 phase hold-up E & turbulent energy dissipation rate per unit mass molecular viscosity of liquid P effective viscosity kff turbulent viscosity Pl turbulent Schmidt number for Q Q# general transport variable d Subscripts G of gas phase of phase j j L of liquid phase REFERENCES
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