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Hydrodynamic simulations of laboratory scale bubble columns fundamental studies of the Eulerian–Eulerian modelling approach
Chemical Engineering Science 54 (1999) 5091}5099
Hydrodynamic simulations of laboratory scale bubble columns fundamental studies of the Eulerian}Eulerian modelling approach D. P#eger*, S. Gomes, N. Gilbert, H.-G. Wagner Process Engineering Department ZAT/EA, BASF AG, D-67056 Ludwigshafen, Germany
Abstract The Eulerian}Eulerian model is used for the hydrodynamic simulation of a two-phase gas}liquid #ow in a laboratory scale bubble column. The behaviour of the air}water system characterises a test case for bubbly #ow with low gas void fractions. A dynamic test case with a centred gas sparger is chosen for validation of the simulation models. Long-time-averaged liquid velocity pro"les and time series at speci"c points are compared with experimental data. The main focus lays on the in#uence of turbulence modelling. Laminar and turbulent simulations are carried out. A standard k}e model is used to describe turbulence occurring in the continuous #uid. Additionally turbulent dispersion of the gas bubbles can be taken into consideration. The results show that a turbulent model has to be considered to gain correct results. The laminar model shows a chaotic behaviour and not the harmonic oscillations observed in experiments. In contrast good agreement of the results can be obtained for three-dimensional calculations including turbulence. Distinct modelling of turbulent dispersion seems not to be necessary for the chosen test case. Furthermore, it can be concluded that a three-dimensional simulation with a su$cient "ne resolution is necessary for accurate results. The test column depth, which is the determining length scale, must be resolved meticulously to receive accurate turbulence intensity in the bubble column. The conclusion concerning grid re"nement and independence still needs further evaluations but fails up to now due to the limited computational power. Finally, two-phase bubbly #ow calculations are carried out successfully with commercial CFD software in a transient way with the two-#uid model. The computational calculations still need very high resources. Further developments and model discussions are strongly recommended. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Bubble columns; CFD; Euler}Euler model; Hydrodynamics; Multiphase #ow; Turbulence
1. Introduction Two-phase #ows are present in numerous processes in the chemical process industry. The design tools for the transport of such #ows in pipes are developed quite well in opposite to the situation in process and reactor design. Detailed knowledge of the hydrodynamics in chemical reactors is often not available and not accessible but important for the e$ciency of processes. A method to gain more knowledge and detailed physical understanding is computational #uid dynamics (CFD). In the last years more and more applications are taken into consideration (Casey, Lang, Mack, Schlegel & Wehrli, 1998).
No basic problems occur as long as the calculation of the complex hydrodynamics stays or can be regarded as single phase. The applications cover #ows like in extruders, static mixers, zyklons or stirred vessels. The recent situation for two-phase #ows is fundamentally di!erent because the #uid dynamics are in#uenced by the complex interactions between the various phases concerning mass, momentum, and energy transfer. Some of the basic physics are not well understood up to now. In this publication we will present fundamental work which focuses on the advanced development of such CFD models for bubble columns.
2. The hydrodynamic model * Corresponding author. Tel.: 00-49-621-60-52457; fax: 00-49-621-6052426. E-mail address: dirk.p#[email protected] (D. P#eger)
The commercial Computational Fluid Dynamics software package CFX 4.2 from AEA Technology plc is used
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 2 6 1 - 4
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for modelling the hydrodynamics of bubble columns (Aea, 1998). The code bases on the "nite volume technique. The so-called two-#uid model for Eulerian} Eulerian description can be applied for two-phase #ows. Within this approach both phases are described as continuous #uids which interpenetrate each other. Basics of the two-#uid model can be found in Ishii (1975). Several authors use the Eulerian}Eulerian approach with di!erent simpli"cations and assumptions (Boisson & Malin, 1996; Delnoij, Kuipers & van Swaaij, 1997; Simonin, Deutsch & Minier, 1993; Sokolichin & Eigenberger, 1994,1999). The state of the art including the various approaches of the several research groups has been summarised by Jakobsen, Sannaes, Grevskott and Svendsen (1997). 2.1. Mass conservation equation The continuity equation describes the mass #ux into and out of a control volume and its internal change of mass. The fundamental form of the equation for multiphase #ows with N phases reads for phase a: N * * ,M\ (o r )# (o r u )" (m !m )#r S . ?@ @? ? ? *t ? ? *x ? ? ? G G @
(1)
The left-hand side describes the internal change of mass over time and the convective #ux crossing the boundaries of the volume. On the right-hand side the "rst term describes mass transfer from phase a to b and vice versa while the second term includes additional source terms. In case of neglected mass transfer and source terms Eq. (1) simpli"es to * * (o r )# (o r u )"0. ? ? *t *x ? ? ? G G
(2)
This is the standard continuity equation used if not noted otherwise, r is the volume fraction of phase a, which ? needs to satisfy the relation ,M r "1. G G
(3)
In the case of a two-phase #ow with one liquid and one gas phase the phase volume fraction requirement reduces to r #r "1. J E Mass transfer is not taken into consideration and such terms of Eq. (1) are therefore neglected. The source term of the continuity equation can be used for several models, which are explained in detail further below. 2.2. Momentum transfer equations In analogy to the mass conservation the momentum conservation for multiphase #ows is described by the
Navier}Stokes equation expanded by the phase volume fraction * * (o r u )# (o r u u ) *t ? ? ? G *x ? ? ? G ? H H *u *p * *u ? G# ? H #o r g #M . "!r # rk (4) ?*x *x ? ? *x ?? G ? G *x H G H G The terms on the right-hand side describe all forces acting on the phase a #uid element in the control volume. These are the overall pressure gradient, the viscous stresses, the gravitational force and interphase momentum forces combined in M . The pressure is de"ned equal in ? G both phases. The e!ective viscosity k of the viscous ? stress term consists of the laminar viscosity and an additional turbulent part in case of turbulence. Only the drag force is used in the hydrodynamic model while other forces are neglected so far. Lot of di!erent approaches can be found in literature for drag correlations of gas bubbles in liquid #ow. Our model bases on Clift, Grace and Weber (1978):
3 1 M " C r o "u !u " (u !u ). (5) ? G 4 B @ ?d @ ? @ G ? G @ The drag coe$cient is set constant to C "0.66 while B a constant bubble diameter of 2 mm is used in the simulations. The experimental observations show that bubbles in the original con"guration are between 1 and 5 mm large. The slip velocity for these bubble diameters lies around 20 cm/s which is reproduced in the simulations by the given settings. 2.3. Turbulence equations Turbulence can be taken into consideration for both phases. While the continuous phase is modelled laminar or turbulent in the following investigations the dispersed gas phase is always modelled laminar. The well-known single-phase turbulence models are usually used to model turbulence of the liquid phase in Eulerian}Eulerian multiphase simulations. In the present case the standard k}e model published by Launder and Spalding (1972) is used. Its conservation equations for the turbulent kinetic energy k and turbulent dissipation e are:
* * * k *k ? (r o k )# (r o u k )! r k # ? *t ? ? ? *x ? ? ? G ? *x ? ? p *x G G I G "r (G !o e )#S , (6) ? ? ?? ? I * * k *e * (r o e )# (r o u e )! r k # ? ? *t ? ? ? *x ? ? ? G ? *x ? ? p *x G G C G e "r ? (C G !C o e )#S . (7) ?k CJ ? C ? ? ? C ? This standard model is taken without any further modi"cations. The source terms S and S on the right-hand ? I ? C side of the equations are not considered yet. A possible
D. Pyeger et al. / Chemical Engineering Science 54 (1999) 5091}5099
and by physics explainable modi"cation concerns the production term of the turbulent kinetic energy. This term models in single-phase #ows the production of turbulence by the local shear. In two-phase #ows it is imaginable that energy caused by bubble wakes is transferred into turbulent kinetic energy. This can be taken into account by additional turbulence production, which is de"ned as bubble-induced turbulence (BIT) (Kataoka & Serizawa, 1989; Sato, Sadatomi & Sekoguchi, 1981). The e!ective viscosity of phase a in Eq. (4) is combined by k k "k # ? . (8) ? ? p I Using the standard k}e model the turbulent viscosity of the continuous phase is calculated by k k "C o A . (9) ? I Ae A All constants of Eqs. (8) and (9) are summarised in Table 1. The main objectives of our investigations are the in#uences of the turbulence model approaches on the hydrodynamic results. Beside the simple use of the k}e model further thoughts have to be made concerning additional turbulent e!ects. Under these is the mentioned bubble-induced turbulence caused by the wakes of the dispersed second phase. Bubble-induced turbulence seems to have an important role for correct simulation results (Sato et al., 1981). This objective is still under investigations and yet not included in this publication. Turbulent dispersion as another turbulent e!ect is taken into consideration that seems to be necessary for accurate results as shown by Sokolichin, Eigenberger, Lapin and LuK bbert (1997).
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gradient-di!usion model is used for the turbulent dispersion term in the continuity equation: S " (C ) r )" (D ) (o r )). (10) ? ? ? ? ?? The dispersion coe$cient of a two-phase #ow C is ana? logous to the di!usion coe$cient D of a binary mixture. It is de"ned using the turbulent viscosity and a turbulent Prandtl number p : R ? k C " ? . (11) ? p R ? Due to the fact that only the continuous phase is modelled turbulent, a de"nition has to be made for the turbulent viscosity of the dispersed phase. Grienberger (1992) suggests a coupling by the reciprocal density ratio: o B. k "k (12) B A o A This results in a full set of continuity equations for mass, momentum, and turbulence energy plus closure terms, which can be used for numerical solution. Our aim is to discuss especially the turbulent models and their e!ects on the simulation results for bubbly gas}liquid #ow.
3. The test case: laboratory-scale bubble column
Looking at the continuity Eq. (1) the right-hand side term S can be used to model additional sources such as ? turbulent dispersion. This e!ect describes the di!usion of gas bubbles caused by turbulent #uctuations. Applying the Reynolds' averaging procedure to the multiphase systems (Ishii, 1975) leads to additional terms caused by the appearance of volume fraction #uctuations. Such terms have to be modelled by a useful approach. The
A rectangular bubble column with the #uid system air}water is chosen as a simple test case. Its dimensions are summarised in Table 2. A rectangular cross-section is set up to simplify the non-intrusive experimental evaluation with laser techniques (LDA) and image processing methods (PTV, PIV). The experimental evaluations are done parallel in the project to gain a detailed database for the validation of the CFD models. Detailed descriptions of the investigations and results with the experimental set-up can be found in di!erent papers of Becker, De Brie & Sweeney, 1998, Becker, Sokolichin and Eigenberger (1998), and Gomes, Gilbert, P#eger & Wagner, 1998. The bottom plates of the column are exchangeable to set up various spatial positions of the gas distributor. All gas spargers have a set of 8 holes in a rectangular con"guration for the gas deliverance. The holes are located in the centre position of the bottom plate in test case A, while they are moved to the left side in test case B. Test case C consists of a sparger plate with three gas inlet
Table 1 Used constants in the k}e model
Table 2 Test case dimensions and operating conditions
2.4. Turbulent dispersion model
C I
C CJ
C C
p I
0.09
1.44
1.92
1.0
p C
i i
(C !C )(C C C I
0.4187
Height 45 cm (Fluid level) Width 20 cm Depth 5 cm
Gas volume #ux
20}90 l/h
Continuous phase Dispersed phase
Tap water at 253C Air at 253C
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Fig. 1. Laboratory bubble column with sketch of di!erent sparger positions.
sections. Fig. 1 shows the bubble column con"guration with a sketch of the bubble swarm for all three test cases and a photography of the experimental set-up. The operating conditions are summarised in Table 2. The air volume #ux is varied between 20 and 90 l/h. If not otherwise mentioned the standard set up is 48 l/h for the comparisons of the simulation and experimental results. The #uid data are taken at room temperature (253C) and handled isothermal and saturated. Therefore, heat and mass transfer is neglected in the simulations. The commercial software package CFX 4.2 is used for modelling the test bubble column. The simulations are set up in two and three dimensions. The geometry is divided into volumes of roughly 0.5;0.5;0.5 cm for a standard grid. Thereof it consists of +40.000 cells. A higher-order TVD scheme (total variation diminishing) is used for the discretisation of the convective #uxes (Sokolichin et al., 1997). Because of the transient #uid #ow in the bubble column all simulations are made in a transient mode for several hundred real-time seconds. The time step is chosen to 0.1 s. This setting re#ects a useful time scale of the investigated hydrodynamics. Smaller time steps result in similar results with the drawback of additional computation time while greater time steps do not show all essential time behaviour. A simulation over 400 real-time seconds needs for the standard grid roughly 94 h. The simulations start with gas injection through the distributor. The continuous phase is at rest at that moment. Fig. 2 shows the transient evolution of test case A with a gas volume #ux of 48 l/h. It can be seen how the gas bubbles enter the water at the sparger position. The gas bubbles start rising to the surface as a reaction to an upward directed force. Buoyancy (gravity) and drag forces are the two main components of the vertical force
balance. An essential parameter to the resulting force per balance element (control volume) is the local gas void. The local quantity of gas in#uences the `meana density and furthermore buoyancy e!ects. The movement of the continuous phase is a result of the acting drag forces to the #uid elements. Two vortices develop behind the gas front on the left and right side of the bubble hose. After the surface break-through the hose stays quite in the centre of the column for a while. Temporal instabilities in the side vortices cause a horizontal non-homogeneity in the force balance and let the hose move sidewards after a few seconds. That movement is intensi"ed by vortex development at the free surface. The vortices move downward with time alternately on the left and right side of the gas hose. These phenomena are well known and have been described by various authors Becker et al. (1999). Lin, Reese, Hong and Fan (1996), and Mudde, Lee, Reese and Fan (1997). Due to the lack of transient experimental data at numerous spatial locations the simulations are time averaged over a range of several hundred seconds which correspond to numerous hose periods. These time-averaged #uid velocity pro"les are compared at various heights beside the discussion of time-dependent data at a "xed point for di!erent model approaches.
4. Results 4.1. Two-dimensional versus three-dimensional simulations Fig. 2 shows the result of a three-dimensional simulation on the standard grid. The dynamic movement of the bubble hose is similar to the experimental observations.
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Fig. 2. Transient evolution of test case A with standard operating conditons. Velocity vectors of the water phase are shown underlying the volume fraction of the gas phase at speci"c time steps of 1, 5, 9, 60, 70 and 80 s. The both most right pictures show the experimental LDA data and the long-time averaged result over a time range of 220 s.
The periodic behaviour is also expressed by Fig. 3 looking at the time development of the horizontal #uid velocity at the centre point of the test column. The bubble swarm begins its swinging after a certain start period. This speci"c movement of test case A is not covered by a two-dimensional simulation. Herein the bubble movement is only upwards directed and does not show any horizontal swing. A linear course is shown in Fig. 3 by the two-dimensional simulation opposite to that of the 3D case. The steady 2D result can be explained by a higher turbulent viscosity in the continuous phase, which dampens totally the bubble movement. The 2D viscosity values can be 5}10 times higher than the values of the 3D cases. The viscosity rise corresponds with a higher turbulent kinetic energy. A proof of that fact is given by Fig. 4.
A high constant k value is given over the depth (zdirection) of the bubble column due to the symmetrical boundary conditions applied to the front walls for 2D simulations. All 3D pro"les reach their maximum k value in the middle plane at z"2.5 cm. The turbulent intensity is decreasing towards both walls. Furthermore the in#uence of the grid resolution can be taken out of the "gure. The coarsest grid (45;22;5) shows a clear di!erence to the "ner grids. Especially, the z-direction is not resolved su$ciently by "ve grid points. An increased number of seven or more points in z-direction results in higher pro"les of the turbulent kinetic energy. Further grid re"nement in x- and y-direction does not change the result additionally. A repeated re"nement in the depth does not change the maximum value at z"2.5 cm but increases slightly k in the near wall areas. The
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Fig. 3. Time development of #uid velocity at centre point (h/H", w/=", d/D") of the bubble column for the 2D and 3D case of a standard grid (90;44;1 or 10).
experimental data point lies between the simulation results. After all it can be said that the turbulence is much stronger in 2D than in 3D simulations due to the abovementioned e!ects. The results are agree with the conclusions of Sokolichin and Eigenberger (1999). A comparison of the Reynolds numbers supports the thesis of stronger turbulence because Re is 2}4 times " higher than Re (estimated with a reference gas velocity " at the centre point of bubble column): o ) u ) w 1000 kg/m ) 0.3 m/s ) 0.2 m Re " J E " +67.000, " k 0.9 ) 10\ Pas J (13) o ) u ) d 1000 kg/m ) 0.4 m/s ) 0.05 m Re " J E " " 0.9 ) 10\ Pas k J +22.000.
Fig. 4. Turbulent kinetic energy pro"le over column depth for two and three-dimensional simulations on various grid sizes at the centre line (h/H", w/=").
The discussion can be summarised with two major conclusions: E The kind of the simulation dimension (2D versus 3D) with their speci"c boundary conditions, length scales, and resulting Reynolds numbers have an essential impact on the turbulent intensity. The hydrodynamics in the test case di!er signi"cantly as a consequence of the turbulence present in the #uid. E The grid resolution in z-direction * the determining length scale * plays an important role for the turbulent kinetic energy level in 3D simulations of the laboratory scale bubble column. 4.2. Laminar versus turbulent approaches
(14)
The conclusion is that a 2D simulation with a reduced gas volume #ux, a smaller gas inlet velocity (+factor 1/5), and therefore a smaller Reynolds number (Re +Re ) should also show a dynamic bubble hose " " movement like the ones observed in 3D simulations caused by a smaller turbulence level. The latter fact can be seen in Fig. 4. Opposite to the mentioned thesis the dynamic behaviour did not occur in 2D tests with smaller gas #uxes because of the much smaller energy input compared to the #uxes in 3D. Other 2D tests with a gas volume #ux of the 3D cases and an e!ective viscosity of the liquid phase scaled by the factor 1/10 show again the dynamic behaviour but on a weaker level. The tests proof the thesis that the turbulence intensity is the determining factor for the dynamic behaviour. To simulate correctly the real physical behaviour of the bubble column it is recommended to use the three-dimensional model.
Fig. 5 shows a comparison of long-time-averaged simulation results with experimental data. On the right side the #ow structure of the liquid phase is drawn as a vector plot for the centre plane of the rectangular bubble column. Contours of gas void fraction are added. We recognise the centre up#ow of the bubble hose and the circulation of the #uid on both sides of the hose. Vertical #uid velocity pro"les versus the column width are drawn in detail on the left side for the three heights of 0.13/0.25/0.37 m which represent the ratios h/H", , . Experimental data points of the LDA measurements are drawn beside the simulation results. The di!erent curves represent the three model approaches for the description of turbulence in the liquid phase. The laminar model is one possible approach (Sokolichin & Eigenberger, 1994) and marks the starting point of the evaluation. The turbulent approach with a basic k}e model follows and results "nally in a turbulent model with additional turbulent dispersion.
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Fig. 5. Comparison of long time averaged results with experimental data at various heights for a centre sparger (gas volume #ux 48 l/h).
All three simulations show a qualitatively correct picture of the overall #uid circulation. They describe the strong up#ow in the centre range above the gas sparger. The down#ow at the column walls is represented as well. The laminar result shows exaggerated values of the #uid velocity at nearly all points. A good agreement can only be found close to the corners of the bottom area. The velocity at the centre line of h/H" is fairly as double as high as in the experiments. Furthermore, the velocity gradient shows a steep increase to the velocity maximums, which is unusual and not observed in the experiments. A correctly formed velocity pro"le can be found with the turbulent description of the liquid phase. The turbulent result "ts very well with the experiment in the upper and medium height. Some di!erences can be found in the close neighbourhood of the gas sparger. Not only the qualitative form but also the quantitative values "t reasonably. Cause for the better agreement is a more realistic value of the e!ective viscosity of the liquid phase. Due to turbulence the value increases by two orders. The higher viscosity dampens all movement of the #uids in the bubble column and reduces so also the liquid velocity. The consideration of turbulent dispersion in the simulation results in a further decrease of liquid velocity and #at pro"les at all heights. Reason for this more and more homogenous up#ow is a greater and more equal distribution of bubbles over a cross section due to the dispersion. The simulation result and the discrepancies to
the measured data show that the dispersion is too big in these cases. Already a simulation without any additional turbulent dispersion term gives good results with a su$cient bubble distribution that is probable a result of numerical di!usion due to the geometric grid. A "rst test with a re"ned grid supports this thesis. Further detailed investigations on "ner grids are a focus of actual research and were up to now not possible due to the great computational e!ort and long calculation times of days to weeks. The in#uence of turbulence modelling cannot only be seen in long-time-averaged velocity and gas void pro"les but also in the discussion of the time behaviour at "xed positions (here centre point of bubble column at h/H"w/="d/D"). The time series of the horizontal liquid velocity is very well suitable for this purpose because this component shows clearly the oscillation of the bubble hose and of the joining liquid phase. Results of experiments and various simulations are summarised in Fig. 6 for a time range of 100 s. The periodic movement of the bubble hose can be studied very well in the experimental diagram. The overlapping high frequent oscillations of the velocity express the turbulent #uctuations. Such observations are very well known and described for this test case with more experimental details in Becker, De Brie and Sweeney (1999). The mean period of the carrier wave takes roughly 15 s. The velocity amplitudes are in a range of $0.20 m/s. The laminar simulation
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Fig. 6. Time series of the horizontal #uid velocity at the centre point of bubble column (h/H"w/="d/D").
result shows a chaotic, high frequent oscillation with velocity peaks smaller than 0.1 m/s. The dynamic behaviour is not comparable with the experimental observation. In contrast, the result of the turbulent simulation shows a periodic movement. The period lies between 15 and 20 s. The cycle of the time series reproduces the reality quite well. Additionally, the velocity range between $0.20 m/s agrees very well. The turbulent #uctuations of the experiment are not reproducible in the simulation caused by the use of Reynolds' averaged conservation equations (see Section 2). The fourth diagram in Fig. 6 shows the result of a turbulent simulation with the additional dispersion term. A periodic behaviour can be seen here, too. Further high frequent oscillations join the mean carrier wave. The mean period is greater than 40 s and much longer than the experimental value. The overall time behaviour of the curve shows clear di!erences. With both comparisons and discussions * the timeaveraged velocity pro"les as well as the time series at the centre point * it can be concluded that the results of the simulations with k}e model without additional dispersion term reproduce most exactly the measured data.
5. Conclusions The numerical modelling of simple bubble columns with transient two-phase hydrodynamics can success-
fully be carried out by a commercial CFD code. The choice of the appropriate models has a signi"cant in#uence on the simulation results of bubbly two-phase #ows. Simulations with three dimensions on a su$cient "ne grid are necessary even the #ow structure looks two dimensional. Two-dimensional simulations are not able to show the periodic bubble hose movement observed in reality due to an over-prediction of the turbulent energy in the #uid. Especially, the resolution of the determining length scale has to be checked very meticulously because boundary e!ects on the front walls could not be neglected. Laminar simulations also do not reproduce the behaviour of the test case. A turbulence model has to be considered. Using the standard single-phase k}e model for the continuous phase is still a good choice. The turbulence dampens the dynamic of the bubble hose and changes signi"cantly the velocity pro"les. Implementation of a turbulent dispersion model for the gas phase seems to be not necessary for this test case but needs further detailed investigations. As well the complex correlations of additional momentum exchange terms or bubble-induced turbulence are not solved up to now. Further research in the area of CFD modelling of two-phase gas}liquid #ows is strongly necessary to understand in detail the physical background of the complex hydrodynamic behaviour.
D. Pyeger et al. / Chemical Engineering Science 54 (1999) 5091}5099
BE95 } 2039). We thank all partners for the scienti"c discussion and the productive co-operation.
Notation C B C ,C ,C ,C I C C C d @ d, D D g G h, H k M
N . p r ? Re S t u u, v, w w, = x, y, z
drag coe$cient, dimensionless constants in k}e model, dimensionless bubble diameter, m column depth, m di!usion coe$cient, m/s gravity acceleration, m/s turbulence production term, J/(ms) column height, m turbulent kinetic energy, m/s momentum interphase transfer term, N/m mass #ux per control volume, kg/(m s) number of phases, dimensionless pressure, N/m gas volume fraction, dimensionless Reynolds number, dimensionless source term, various time, s velocity vector, m/s spatial velocity components, m/s column width, m spatial coordinates, m
Greek letters e C i k o p,p I C p R
turbulent energy dissipation, m/s dispersion coe$cient, kg/(m s) constant in k}e model, dimensionless dynamic viscosity, kg/(m s) density, kg/m constant in k}e model, dimensionless turbulent Prandtl number, dimensionless
Subscripts a, b b c, d i, j g, l lam, tur e!
phases bubble continuous, disperse spatial directions gas, liquid laminar, turbulent e!ective
Abbreviations LTA
long time averaged
m
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Acknowledgements The authors acknowledge "nancial support provided by the Commission of the European Union under the BriteEuram programme (ADMIRE project } Contract
References Aea (1998). CFX 4 } Flow Solver User Manual, Harwell, GB. Becker, S., Sokolichin, A., & Eigenberger, G. (1994). Gas}liquid #ow in bubble columns and loop reactors: Part II. Comparison of detailed experiments and #ow simulations. Chemical Engineering Science, 49(24B), 5747}5762. Becker, S., De Brie, H., & Sweeney, J. (1999). Dynamic #ow behaviour in bubble columns. Submitted to the Fourth conference on gas}liquid and gas}liquid}solid reactor engineering, Delft, August 23}25, 1999. Boisson, N., & Malin, M. R. (1996). Numerical prediction of two-phase #ow in bubble columns. International Journal for Numerical Methods in Fluids, 23, 1289}1310. Casey, M., Lang, E., Mack, R., Schlegel, R., & Wehrli, M. (1998). Applications of computational #uid dynamics for process engineering at Sulzer Innotec. Speedup, 12 (l) p. 12}21. Clift, R., Grace, J. R. Weber, M. E. (1978). Bubbles, drops and particles. New York, USA: Academic Press. Delnoij, E., Kuipers, J. A. M., & van Swaaij, W. P. M. (1997). Computational #uid dynamics applied to gas}liquid contactors. Chemical Engineering Science, 52(21/22), 3623}3638. Gomes, S., Gilbert, N., P#eger, D., & Wagner, H.-G., 1998. Experimental investigations of two-phase gas}liquid #ows in bubble column reactors. Proceedings of Me}technik fu( r stationa( re und transiente Mehrphasenstro( mungen, FZR-241, 75-82, Forschungszentrum Rossendorf, D. Grienberger, J. (1992). Untersuchung und Modellierung von Blasensa( ulen. Dissertation, UniversitaK t Erlangen-NuK rnberg. Ishii, M. (1975). Thermo-#uid dynamic theory of two-phase #ow. Paris, F: Eyrolles. Jakobsen, H. A., Sannaes, B. H., Grevskott, S., & Svendsen, H. F. (1997). Modelling of vertical bubble-driven #ows. Industrial Engineering Chemical Research, 36, 4052}4074. Kataoka, I., & Serizawa, A. (1989). Basic equations of turbulence in gas}liquid two-phase #ow. International Journal of Multiphase Flow, 15(5), 843}855. Launder, B. E., & Spalding, D. B. (1972). Mathematical models of turbulence. London, GB: Academic Press. Lin, T. -J., Reese, J., Hong, T., & Fan, L. -S. (1996). Quantitative analysis and computation of two-dimensional bubble columns. A.I.Ch.E. Journal, 42(2), 301}318. Mudde, R. F., Lee, D. J., Reese, J., & Fan, L. S. (1997). Role of coherent structures on Reynolds stresses in a 2-D bubble column. A.I.Ch.E. Journal, 43(4), 913}926. Sato, Y., Sadatomi, M., & Sekoguchi, K. (1981). Momentum and heat transfer in two-phase bubble #ow, Part I theory. International Journal of Multiphase Flow, 7, 167}177. Simonin, O., Deutsch, E., & Minier, J. P. (1993). Eulerian prediction of the #uid particle correlated motion in turbulent two-phase #ows. Applied Science Research, 51(1-2), 275}283. Sokolichin, A., & Eigenberger, G. (1994). Gas}liquid #ow in bubble columns and loop reactors: Part I. Detailed modelling and numerical simulation. Chemical Engineering Science, 49(24B), 5735}5746. Sokolichin, A., Eigenberger, G., Lapin, A., & LuK bbert, A. (1997). Dynamic numerical simulation of gas}liquid two-phase #ows. Chemical Engineering Science, 52(4), 611}626. Sokolichin, A., Eigenberger, G. (1999). Applicability of the standard k}e turbulence model to the dynamic simulation of bubble columns * Part I: Detailed numerical simulations. Chemical Engineering Science, 54, 2273}2284.
Hydrodynamic simulations of laboratory scale bubble columns fundamental studies of the Eulerian}Eulerian modelling approach D. P#eger*, S. Gomes, N. Gilbert, H.-G. Wagner Process Engineering Department ZAT/EA, BASF AG, D-67056 Ludwigshafen, Germany
Abstract The Eulerian}Eulerian model is used for the hydrodynamic simulation of a two-phase gas}liquid #ow in a laboratory scale bubble column. The behaviour of the air}water system characterises a test case for bubbly #ow with low gas void fractions. A dynamic test case with a centred gas sparger is chosen for validation of the simulation models. Long-time-averaged liquid velocity pro"les and time series at speci"c points are compared with experimental data. The main focus lays on the in#uence of turbulence modelling. Laminar and turbulent simulations are carried out. A standard k}e model is used to describe turbulence occurring in the continuous #uid. Additionally turbulent dispersion of the gas bubbles can be taken into consideration. The results show that a turbulent model has to be considered to gain correct results. The laminar model shows a chaotic behaviour and not the harmonic oscillations observed in experiments. In contrast good agreement of the results can be obtained for three-dimensional calculations including turbulence. Distinct modelling of turbulent dispersion seems not to be necessary for the chosen test case. Furthermore, it can be concluded that a three-dimensional simulation with a su$cient "ne resolution is necessary for accurate results. The test column depth, which is the determining length scale, must be resolved meticulously to receive accurate turbulence intensity in the bubble column. The conclusion concerning grid re"nement and independence still needs further evaluations but fails up to now due to the limited computational power. Finally, two-phase bubbly #ow calculations are carried out successfully with commercial CFD software in a transient way with the two-#uid model. The computational calculations still need very high resources. Further developments and model discussions are strongly recommended. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Bubble columns; CFD; Euler}Euler model; Hydrodynamics; Multiphase #ow; Turbulence
1. Introduction Two-phase #ows are present in numerous processes in the chemical process industry. The design tools for the transport of such #ows in pipes are developed quite well in opposite to the situation in process and reactor design. Detailed knowledge of the hydrodynamics in chemical reactors is often not available and not accessible but important for the e$ciency of processes. A method to gain more knowledge and detailed physical understanding is computational #uid dynamics (CFD). In the last years more and more applications are taken into consideration (Casey, Lang, Mack, Schlegel & Wehrli, 1998).
No basic problems occur as long as the calculation of the complex hydrodynamics stays or can be regarded as single phase. The applications cover #ows like in extruders, static mixers, zyklons or stirred vessels. The recent situation for two-phase #ows is fundamentally di!erent because the #uid dynamics are in#uenced by the complex interactions between the various phases concerning mass, momentum, and energy transfer. Some of the basic physics are not well understood up to now. In this publication we will present fundamental work which focuses on the advanced development of such CFD models for bubble columns.
2. The hydrodynamic model * Corresponding author. Tel.: 00-49-621-60-52457; fax: 00-49-621-6052426. E-mail address: dirk.p#[email protected] (D. P#eger)
The commercial Computational Fluid Dynamics software package CFX 4.2 from AEA Technology plc is used
0009-2509/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 2 6 1 - 4
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for modelling the hydrodynamics of bubble columns (Aea, 1998). The code bases on the "nite volume technique. The so-called two-#uid model for Eulerian} Eulerian description can be applied for two-phase #ows. Within this approach both phases are described as continuous #uids which interpenetrate each other. Basics of the two-#uid model can be found in Ishii (1975). Several authors use the Eulerian}Eulerian approach with di!erent simpli"cations and assumptions (Boisson & Malin, 1996; Delnoij, Kuipers & van Swaaij, 1997; Simonin, Deutsch & Minier, 1993; Sokolichin & Eigenberger, 1994,1999). The state of the art including the various approaches of the several research groups has been summarised by Jakobsen, Sannaes, Grevskott and Svendsen (1997). 2.1. Mass conservation equation The continuity equation describes the mass #ux into and out of a control volume and its internal change of mass. The fundamental form of the equation for multiphase #ows with N phases reads for phase a: N * * ,M\ (o r )# (o r u )" (m !m )#r S . ?@ @? ? ? *t ? ? *x ? ? ? G G @
(1)
The left-hand side describes the internal change of mass over time and the convective #ux crossing the boundaries of the volume. On the right-hand side the "rst term describes mass transfer from phase a to b and vice versa while the second term includes additional source terms. In case of neglected mass transfer and source terms Eq. (1) simpli"es to * * (o r )# (o r u )"0. ? ? *t *x ? ? ? G G
(2)
This is the standard continuity equation used if not noted otherwise, r is the volume fraction of phase a, which ? needs to satisfy the relation ,M r "1. G G
(3)
In the case of a two-phase #ow with one liquid and one gas phase the phase volume fraction requirement reduces to r #r "1. J E Mass transfer is not taken into consideration and such terms of Eq. (1) are therefore neglected. The source term of the continuity equation can be used for several models, which are explained in detail further below. 2.2. Momentum transfer equations In analogy to the mass conservation the momentum conservation for multiphase #ows is described by the
Navier}Stokes equation expanded by the phase volume fraction * * (o r u )# (o r u u ) *t ? ? ? G *x ? ? ? G ? H H *u *p * *u ? G# ? H #o r g #M . "!r # rk (4) ?*x *x ? ? *x ?? G ? G *x H G H G The terms on the right-hand side describe all forces acting on the phase a #uid element in the control volume. These are the overall pressure gradient, the viscous stresses, the gravitational force and interphase momentum forces combined in M . The pressure is de"ned equal in ? G both phases. The e!ective viscosity k of the viscous ? stress term consists of the laminar viscosity and an additional turbulent part in case of turbulence. Only the drag force is used in the hydrodynamic model while other forces are neglected so far. Lot of di!erent approaches can be found in literature for drag correlations of gas bubbles in liquid #ow. Our model bases on Clift, Grace and Weber (1978):
3 1 M " C r o "u !u " (u !u ). (5) ? G 4 B @ ?d @ ? @ G ? G @ The drag coe$cient is set constant to C "0.66 while B a constant bubble diameter of 2 mm is used in the simulations. The experimental observations show that bubbles in the original con"guration are between 1 and 5 mm large. The slip velocity for these bubble diameters lies around 20 cm/s which is reproduced in the simulations by the given settings. 2.3. Turbulence equations Turbulence can be taken into consideration for both phases. While the continuous phase is modelled laminar or turbulent in the following investigations the dispersed gas phase is always modelled laminar. The well-known single-phase turbulence models are usually used to model turbulence of the liquid phase in Eulerian}Eulerian multiphase simulations. In the present case the standard k}e model published by Launder and Spalding (1972) is used. Its conservation equations for the turbulent kinetic energy k and turbulent dissipation e are:
* * * k *k ? (r o k )# (r o u k )! r k # ? *t ? ? ? *x ? ? ? G ? *x ? ? p *x G G I G "r (G !o e )#S , (6) ? ? ?? ? I * * k *e * (r o e )# (r o u e )! r k # ? ? *t ? ? ? *x ? ? ? G ? *x ? ? p *x G G C G e "r ? (C G !C o e )#S . (7) ?k CJ ? C ? ? ? C ? This standard model is taken without any further modi"cations. The source terms S and S on the right-hand ? I ? C side of the equations are not considered yet. A possible
D. Pyeger et al. / Chemical Engineering Science 54 (1999) 5091}5099
and by physics explainable modi"cation concerns the production term of the turbulent kinetic energy. This term models in single-phase #ows the production of turbulence by the local shear. In two-phase #ows it is imaginable that energy caused by bubble wakes is transferred into turbulent kinetic energy. This can be taken into account by additional turbulence production, which is de"ned as bubble-induced turbulence (BIT) (Kataoka & Serizawa, 1989; Sato, Sadatomi & Sekoguchi, 1981). The e!ective viscosity of phase a in Eq. (4) is combined by k k "k # ? . (8) ? ? p I Using the standard k}e model the turbulent viscosity of the continuous phase is calculated by k k "C o A . (9) ? I Ae A All constants of Eqs. (8) and (9) are summarised in Table 1. The main objectives of our investigations are the in#uences of the turbulence model approaches on the hydrodynamic results. Beside the simple use of the k}e model further thoughts have to be made concerning additional turbulent e!ects. Under these is the mentioned bubble-induced turbulence caused by the wakes of the dispersed second phase. Bubble-induced turbulence seems to have an important role for correct simulation results (Sato et al., 1981). This objective is still under investigations and yet not included in this publication. Turbulent dispersion as another turbulent e!ect is taken into consideration that seems to be necessary for accurate results as shown by Sokolichin, Eigenberger, Lapin and LuK bbert (1997).
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gradient-di!usion model is used for the turbulent dispersion term in the continuity equation: S " (C ) r )" (D ) (o r )). (10) ? ? ? ? ?? The dispersion coe$cient of a two-phase #ow C is ana? logous to the di!usion coe$cient D of a binary mixture. It is de"ned using the turbulent viscosity and a turbulent Prandtl number p : R ? k C " ? . (11) ? p R ? Due to the fact that only the continuous phase is modelled turbulent, a de"nition has to be made for the turbulent viscosity of the dispersed phase. Grienberger (1992) suggests a coupling by the reciprocal density ratio: o B. k "k (12) B A o A This results in a full set of continuity equations for mass, momentum, and turbulence energy plus closure terms, which can be used for numerical solution. Our aim is to discuss especially the turbulent models and their e!ects on the simulation results for bubbly gas}liquid #ow.
3. The test case: laboratory-scale bubble column
Looking at the continuity Eq. (1) the right-hand side term S can be used to model additional sources such as ? turbulent dispersion. This e!ect describes the di!usion of gas bubbles caused by turbulent #uctuations. Applying the Reynolds' averaging procedure to the multiphase systems (Ishii, 1975) leads to additional terms caused by the appearance of volume fraction #uctuations. Such terms have to be modelled by a useful approach. The
A rectangular bubble column with the #uid system air}water is chosen as a simple test case. Its dimensions are summarised in Table 2. A rectangular cross-section is set up to simplify the non-intrusive experimental evaluation with laser techniques (LDA) and image processing methods (PTV, PIV). The experimental evaluations are done parallel in the project to gain a detailed database for the validation of the CFD models. Detailed descriptions of the investigations and results with the experimental set-up can be found in di!erent papers of Becker, De Brie & Sweeney, 1998, Becker, Sokolichin and Eigenberger (1998), and Gomes, Gilbert, P#eger & Wagner, 1998. The bottom plates of the column are exchangeable to set up various spatial positions of the gas distributor. All gas spargers have a set of 8 holes in a rectangular con"guration for the gas deliverance. The holes are located in the centre position of the bottom plate in test case A, while they are moved to the left side in test case B. Test case C consists of a sparger plate with three gas inlet
Table 1 Used constants in the k}e model
Table 2 Test case dimensions and operating conditions
2.4. Turbulent dispersion model
C I
C CJ
C C
p I
0.09
1.44
1.92
1.0
p C
i i
(C !C )(C C C I
0.4187
Height 45 cm (Fluid level) Width 20 cm Depth 5 cm
Gas volume #ux
20}90 l/h
Continuous phase Dispersed phase
Tap water at 253C Air at 253C
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Fig. 1. Laboratory bubble column with sketch of di!erent sparger positions.
sections. Fig. 1 shows the bubble column con"guration with a sketch of the bubble swarm for all three test cases and a photography of the experimental set-up. The operating conditions are summarised in Table 2. The air volume #ux is varied between 20 and 90 l/h. If not otherwise mentioned the standard set up is 48 l/h for the comparisons of the simulation and experimental results. The #uid data are taken at room temperature (253C) and handled isothermal and saturated. Therefore, heat and mass transfer is neglected in the simulations. The commercial software package CFX 4.2 is used for modelling the test bubble column. The simulations are set up in two and three dimensions. The geometry is divided into volumes of roughly 0.5;0.5;0.5 cm for a standard grid. Thereof it consists of +40.000 cells. A higher-order TVD scheme (total variation diminishing) is used for the discretisation of the convective #uxes (Sokolichin et al., 1997). Because of the transient #uid #ow in the bubble column all simulations are made in a transient mode for several hundred real-time seconds. The time step is chosen to 0.1 s. This setting re#ects a useful time scale of the investigated hydrodynamics. Smaller time steps result in similar results with the drawback of additional computation time while greater time steps do not show all essential time behaviour. A simulation over 400 real-time seconds needs for the standard grid roughly 94 h. The simulations start with gas injection through the distributor. The continuous phase is at rest at that moment. Fig. 2 shows the transient evolution of test case A with a gas volume #ux of 48 l/h. It can be seen how the gas bubbles enter the water at the sparger position. The gas bubbles start rising to the surface as a reaction to an upward directed force. Buoyancy (gravity) and drag forces are the two main components of the vertical force
balance. An essential parameter to the resulting force per balance element (control volume) is the local gas void. The local quantity of gas in#uences the `meana density and furthermore buoyancy e!ects. The movement of the continuous phase is a result of the acting drag forces to the #uid elements. Two vortices develop behind the gas front on the left and right side of the bubble hose. After the surface break-through the hose stays quite in the centre of the column for a while. Temporal instabilities in the side vortices cause a horizontal non-homogeneity in the force balance and let the hose move sidewards after a few seconds. That movement is intensi"ed by vortex development at the free surface. The vortices move downward with time alternately on the left and right side of the gas hose. These phenomena are well known and have been described by various authors Becker et al. (1999). Lin, Reese, Hong and Fan (1996), and Mudde, Lee, Reese and Fan (1997). Due to the lack of transient experimental data at numerous spatial locations the simulations are time averaged over a range of several hundred seconds which correspond to numerous hose periods. These time-averaged #uid velocity pro"les are compared at various heights beside the discussion of time-dependent data at a "xed point for di!erent model approaches.
4. Results 4.1. Two-dimensional versus three-dimensional simulations Fig. 2 shows the result of a three-dimensional simulation on the standard grid. The dynamic movement of the bubble hose is similar to the experimental observations.
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Fig. 2. Transient evolution of test case A with standard operating conditons. Velocity vectors of the water phase are shown underlying the volume fraction of the gas phase at speci"c time steps of 1, 5, 9, 60, 70 and 80 s. The both most right pictures show the experimental LDA data and the long-time averaged result over a time range of 220 s.
The periodic behaviour is also expressed by Fig. 3 looking at the time development of the horizontal #uid velocity at the centre point of the test column. The bubble swarm begins its swinging after a certain start period. This speci"c movement of test case A is not covered by a two-dimensional simulation. Herein the bubble movement is only upwards directed and does not show any horizontal swing. A linear course is shown in Fig. 3 by the two-dimensional simulation opposite to that of the 3D case. The steady 2D result can be explained by a higher turbulent viscosity in the continuous phase, which dampens totally the bubble movement. The 2D viscosity values can be 5}10 times higher than the values of the 3D cases. The viscosity rise corresponds with a higher turbulent kinetic energy. A proof of that fact is given by Fig. 4.
A high constant k value is given over the depth (zdirection) of the bubble column due to the symmetrical boundary conditions applied to the front walls for 2D simulations. All 3D pro"les reach their maximum k value in the middle plane at z"2.5 cm. The turbulent intensity is decreasing towards both walls. Furthermore the in#uence of the grid resolution can be taken out of the "gure. The coarsest grid (45;22;5) shows a clear di!erence to the "ner grids. Especially, the z-direction is not resolved su$ciently by "ve grid points. An increased number of seven or more points in z-direction results in higher pro"les of the turbulent kinetic energy. Further grid re"nement in x- and y-direction does not change the result additionally. A repeated re"nement in the depth does not change the maximum value at z"2.5 cm but increases slightly k in the near wall areas. The
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Fig. 3. Time development of #uid velocity at centre point (h/H", w/=", d/D") of the bubble column for the 2D and 3D case of a standard grid (90;44;1 or 10).
experimental data point lies between the simulation results. After all it can be said that the turbulence is much stronger in 2D than in 3D simulations due to the abovementioned e!ects. The results are agree with the conclusions of Sokolichin and Eigenberger (1999). A comparison of the Reynolds numbers supports the thesis of stronger turbulence because Re is 2}4 times " higher than Re (estimated with a reference gas velocity " at the centre point of bubble column): o ) u ) w 1000 kg/m ) 0.3 m/s ) 0.2 m Re " J E " +67.000, " k 0.9 ) 10\ Pas J (13) o ) u ) d 1000 kg/m ) 0.4 m/s ) 0.05 m Re " J E " " 0.9 ) 10\ Pas k J +22.000.
Fig. 4. Turbulent kinetic energy pro"le over column depth for two and three-dimensional simulations on various grid sizes at the centre line (h/H", w/=").
The discussion can be summarised with two major conclusions: E The kind of the simulation dimension (2D versus 3D) with their speci"c boundary conditions, length scales, and resulting Reynolds numbers have an essential impact on the turbulent intensity. The hydrodynamics in the test case di!er signi"cantly as a consequence of the turbulence present in the #uid. E The grid resolution in z-direction * the determining length scale * plays an important role for the turbulent kinetic energy level in 3D simulations of the laboratory scale bubble column. 4.2. Laminar versus turbulent approaches
(14)
The conclusion is that a 2D simulation with a reduced gas volume #ux, a smaller gas inlet velocity (+factor 1/5), and therefore a smaller Reynolds number (Re +Re ) should also show a dynamic bubble hose " " movement like the ones observed in 3D simulations caused by a smaller turbulence level. The latter fact can be seen in Fig. 4. Opposite to the mentioned thesis the dynamic behaviour did not occur in 2D tests with smaller gas #uxes because of the much smaller energy input compared to the #uxes in 3D. Other 2D tests with a gas volume #ux of the 3D cases and an e!ective viscosity of the liquid phase scaled by the factor 1/10 show again the dynamic behaviour but on a weaker level. The tests proof the thesis that the turbulence intensity is the determining factor for the dynamic behaviour. To simulate correctly the real physical behaviour of the bubble column it is recommended to use the three-dimensional model.
Fig. 5 shows a comparison of long-time-averaged simulation results with experimental data. On the right side the #ow structure of the liquid phase is drawn as a vector plot for the centre plane of the rectangular bubble column. Contours of gas void fraction are added. We recognise the centre up#ow of the bubble hose and the circulation of the #uid on both sides of the hose. Vertical #uid velocity pro"les versus the column width are drawn in detail on the left side for the three heights of 0.13/0.25/0.37 m which represent the ratios h/H", , . Experimental data points of the LDA measurements are drawn beside the simulation results. The di!erent curves represent the three model approaches for the description of turbulence in the liquid phase. The laminar model is one possible approach (Sokolichin & Eigenberger, 1994) and marks the starting point of the evaluation. The turbulent approach with a basic k}e model follows and results "nally in a turbulent model with additional turbulent dispersion.
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Fig. 5. Comparison of long time averaged results with experimental data at various heights for a centre sparger (gas volume #ux 48 l/h).
All three simulations show a qualitatively correct picture of the overall #uid circulation. They describe the strong up#ow in the centre range above the gas sparger. The down#ow at the column walls is represented as well. The laminar result shows exaggerated values of the #uid velocity at nearly all points. A good agreement can only be found close to the corners of the bottom area. The velocity at the centre line of h/H" is fairly as double as high as in the experiments. Furthermore, the velocity gradient shows a steep increase to the velocity maximums, which is unusual and not observed in the experiments. A correctly formed velocity pro"le can be found with the turbulent description of the liquid phase. The turbulent result "ts very well with the experiment in the upper and medium height. Some di!erences can be found in the close neighbourhood of the gas sparger. Not only the qualitative form but also the quantitative values "t reasonably. Cause for the better agreement is a more realistic value of the e!ective viscosity of the liquid phase. Due to turbulence the value increases by two orders. The higher viscosity dampens all movement of the #uids in the bubble column and reduces so also the liquid velocity. The consideration of turbulent dispersion in the simulation results in a further decrease of liquid velocity and #at pro"les at all heights. Reason for this more and more homogenous up#ow is a greater and more equal distribution of bubbles over a cross section due to the dispersion. The simulation result and the discrepancies to
the measured data show that the dispersion is too big in these cases. Already a simulation without any additional turbulent dispersion term gives good results with a su$cient bubble distribution that is probable a result of numerical di!usion due to the geometric grid. A "rst test with a re"ned grid supports this thesis. Further detailed investigations on "ner grids are a focus of actual research and were up to now not possible due to the great computational e!ort and long calculation times of days to weeks. The in#uence of turbulence modelling cannot only be seen in long-time-averaged velocity and gas void pro"les but also in the discussion of the time behaviour at "xed positions (here centre point of bubble column at h/H"w/="d/D"). The time series of the horizontal liquid velocity is very well suitable for this purpose because this component shows clearly the oscillation of the bubble hose and of the joining liquid phase. Results of experiments and various simulations are summarised in Fig. 6 for a time range of 100 s. The periodic movement of the bubble hose can be studied very well in the experimental diagram. The overlapping high frequent oscillations of the velocity express the turbulent #uctuations. Such observations are very well known and described for this test case with more experimental details in Becker, De Brie and Sweeney (1999). The mean period of the carrier wave takes roughly 15 s. The velocity amplitudes are in a range of $0.20 m/s. The laminar simulation
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Fig. 6. Time series of the horizontal #uid velocity at the centre point of bubble column (h/H"w/="d/D").
result shows a chaotic, high frequent oscillation with velocity peaks smaller than 0.1 m/s. The dynamic behaviour is not comparable with the experimental observation. In contrast, the result of the turbulent simulation shows a periodic movement. The period lies between 15 and 20 s. The cycle of the time series reproduces the reality quite well. Additionally, the velocity range between $0.20 m/s agrees very well. The turbulent #uctuations of the experiment are not reproducible in the simulation caused by the use of Reynolds' averaged conservation equations (see Section 2). The fourth diagram in Fig. 6 shows the result of a turbulent simulation with the additional dispersion term. A periodic behaviour can be seen here, too. Further high frequent oscillations join the mean carrier wave. The mean period is greater than 40 s and much longer than the experimental value. The overall time behaviour of the curve shows clear di!erences. With both comparisons and discussions * the timeaveraged velocity pro"les as well as the time series at the centre point * it can be concluded that the results of the simulations with k}e model without additional dispersion term reproduce most exactly the measured data.
5. Conclusions The numerical modelling of simple bubble columns with transient two-phase hydrodynamics can success-
fully be carried out by a commercial CFD code. The choice of the appropriate models has a signi"cant in#uence on the simulation results of bubbly two-phase #ows. Simulations with three dimensions on a su$cient "ne grid are necessary even the #ow structure looks two dimensional. Two-dimensional simulations are not able to show the periodic bubble hose movement observed in reality due to an over-prediction of the turbulent energy in the #uid. Especially, the resolution of the determining length scale has to be checked very meticulously because boundary e!ects on the front walls could not be neglected. Laminar simulations also do not reproduce the behaviour of the test case. A turbulence model has to be considered. Using the standard single-phase k}e model for the continuous phase is still a good choice. The turbulence dampens the dynamic of the bubble hose and changes signi"cantly the velocity pro"les. Implementation of a turbulent dispersion model for the gas phase seems to be not necessary for this test case but needs further detailed investigations. As well the complex correlations of additional momentum exchange terms or bubble-induced turbulence are not solved up to now. Further research in the area of CFD modelling of two-phase gas}liquid #ows is strongly necessary to understand in detail the physical background of the complex hydrodynamic behaviour.
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BE95 } 2039). We thank all partners for the scienti"c discussion and the productive co-operation.
Notation C B C ,C ,C ,C I C C C d @ d, D D g G h, H k M
N . p r ? Re S t u u, v, w w, = x, y, z
drag coe$cient, dimensionless constants in k}e model, dimensionless bubble diameter, m column depth, m di!usion coe$cient, m/s gravity acceleration, m/s turbulence production term, J/(ms) column height, m turbulent kinetic energy, m/s momentum interphase transfer term, N/m mass #ux per control volume, kg/(m s) number of phases, dimensionless pressure, N/m gas volume fraction, dimensionless Reynolds number, dimensionless source term, various time, s velocity vector, m/s spatial velocity components, m/s column width, m spatial coordinates, m
Greek letters e C i k o p,p I C p R
turbulent energy dissipation, m/s dispersion coe$cient, kg/(m s) constant in k}e model, dimensionless dynamic viscosity, kg/(m s) density, kg/m constant in k}e model, dimensionless turbulent Prandtl number, dimensionless
Subscripts a, b b c, d i, j g, l lam, tur e!
phases bubble continuous, disperse spatial directions gas, liquid laminar, turbulent e!ective
Abbreviations LTA
long time averaged
m
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Acknowledgements The authors acknowledge "nancial support provided by the Commission of the European Union under the BriteEuram programme (ADMIRE project } Contract
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