Large eddy simulation of the Gas Liquid ow in a square cross-sectioned bubble column

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Chemica Engineering Science 56 (2001) 6341 6349 www.esevier.com/ocate/ces Large eddy simuation of the Gas Liquid ow in a square cross-sectioned bubbe coumn N. G. Deen, T. Soberg, B. H.

1 Chemica Engineering Science 56 (2001) Large eddy simuation of the Gas Liquid ow in a square cross-sectioned bubbe coumn N. G. Deen, T. Soberg, B. H. Hjertager Chemica Engineering Laboratory, Aaborg University Esbjerg, Nies Bohrs Vej 8, DK-6700 Esbjerg, Denmar Abstract In this wor the use of arge eddy simuations (LES) in numerica simuations of the gas iquid ow in bubbe coumns is studied. The Euer Euer approach is used to describe the equations of motion of the two-phase ow. It is found that, when the drag, ift and virtua mass forces are used, the transient behaviour that was observed in experiments can be captured. Good quantitative agreement with experimenta data is obtained both for the mean veocities and the uctuating veocities. The LES shows better agreement with the experimenta data than simuations using the mode.? 2001 Esevier Science Ltd. A rights reserved. Keywords: Large eddy simuation; Euer Euer approach; Bubbe coumn; Gas iquid ow: Bubby ow; Computationa uid dynamics 1. Introduction Bubbe coumns are widey used in the chemica and biochemica process industry. In order to deveop design toos for engineering purposes, a arge amount of research has been carried out in the area of computationa uid dynamics (CFD) modeing of gas iquid ows. Two approaches are mosty used to simuate the ow in bubbe coumns: the Euer Lagrange and the Euer Euer approach. In the Euer Lagrange approach, the continuous iquid phase is described as a continuum in an Euerian framewor. The dispersed gas phase on the other hand is treated in a Lagrangian way; that is, the individua bubbes in the system are traced by soving Newton s second aw for the forces acting on the bubbes. This approach was amongst others used by Denoij, Lammers, Kuipers, and van Swaaij (1997) and Sooichin, Eigenberger, Lapin, and Lubbert (1997). An advantage of this approach is that bubbe bubbe interactions are easiy accounted for. A disadvantage is that the arger the system gets, the more equations need to be soved, i.e. one for every bubbe. Another disadvantage is that the size of the numerica grid-ces is imited by the size of the bubbes, in order to determine the interface forces acting on the bubbe correcty. This means that, uness the bubbes are Corresponding author. Fax: E-mai address: nd@aue.auc.d (N. G. Deen). very sma, one cannot use arge eddy simuations (LES) to mode the turbuence of the iquid phase. In the Euer Euer approach bubbes are not traced individuay, but the dynamics of the dispersed phase are ensembe averaged, to obtain a set of Euerian equations, simiar to the equations for the continuous phase. Sooichin et a. (1997) showed that provided that the equations for the gas phase are discretized in an adequate way, the Euer Euer approach gives simiar resuts as the Euer Lagrange approach. The advantage of this approach is that the computationa demands are much ower compared to the Euer Lagrange approach, for systems with higher dispersed void fractions. For further information of both approaches, the reader is referred to the detaied reviews of Kuipers and van Swaaij (1997), Jaobsen, Sannaes, Grevsott, and Svendsen (1997) and Hjertager (1998). In this wor, the Euerian approach wi be used for the description of both phases. The turbuence of the continuous phase can be incorporated through dierent modes, ie the mode or LES. The atter mode has been identied as a good way to mode the turbuence by, amongst others, Jaobsen et a. (1997). This mode has however so far hardy been used for the modeing of gas iquid ows. To our nowedge ony Smith and Miei (1998) have used a combination of the Euer Euer approach and a LES mode for the turbuence of the iquid phase. In this wor, we wi tae a simiar approach. Euer Euer simuations of the gas iquid ow /01/$ - see front matter? 2001 Esevier Science Ltd. A rights reserved. PII: S (01)

2 6342 N. G. Deen et a. / Chemica Engineering Science 56 (2001) in a bubbe coumn using a LES turbuence mode wi be compared with simuation incuding a mode by Deen, Soberg, and Hjertager (2000b) and experimenta data from Deen, Hjertager, and Soberg (2000a). 2. Theory The equations of motion for phase in an Euerian Euerian simuation are generay given as ) + ( u u ) + ( u u ( ) P + g + M I; : (2) The terms on the right-hand side of Eq. (2) are, respectivey, representing the stress, the pressure gradient, gravity and the ensembe averaged momentum exchange between the phases, due to interface forces. The stress term and the interface term wi be discussed ater on. The veocities in Eqs. (1) and (2) are dened as foows: u = ũ u : (3) Here, u is the part of the veocity for phase that wi be resoved in the numerica simuations, ũ is the instantaneous veocity and u is the unresoved part of the numerica simuations. The interpretation of the terms u and u depends on the method of derivation. Most authors derive Eqs. (1) and (2) through ensembe averaging. Then u and u represent the mean veocity and the uctuating veocity. When Eqs. (1) and (2) are obtained through a tering operation, these terms are, respectivey, the grid scae (GS) and the sub-grid scae (SGS) veocities. From now on, when speaing about veocities, we refer to u, uness mentioned otherwise. When either ensembe averaging or tering is used, uncosed parts occur in the stress term and the interface forces. In this wor, the uncosed part of the interface forces wi be negected. The stress term of phase is described as foows: ( = e; u +( u ) T 2 3 I( u ) ) ; (4) where e; is the eective viscosity. The eective viscosity of the iquid phase is composed of three contributions; the moecuar viscosity, the turbuent viscosity and an extra term due to bubbe induced turbuence e; = L; + T; + BI; : (5) The cacuation of the eective gas viscosity is based on the eective iquid viscosity as foows: e;g = g e; (6) as was proposed by Jaobsen et a. (1997). There are severa modes avaiabe to tae account of the turbuence induced by the movement of the bubbes. In this study the mode proposed by Sato & Seoguchi (1975) was used: BI; = C ;BI g d B u g u (7) with a mode constant C ;BI which is equa to 0.6. In this wor, we use two dierent modes to cacuate the turbuent viscosity: LES and the mode. When appying LES, the mode proposed by Smagorinsy (1963) is used to cacuate the turbuent viscosity T;. The turbuent viscosity, i.e. the SGS viscosity, is formuated as foows: T; = (C S ) 2 S ; (8) where C S is a mode constant with a vaue of 0.1 and S is the characteristic tered rate of strain and =( i j ) 1=3 is the ter width. When the mode is used, the turbuent viscosity, is formuated as foows: T; = C 2 : (9) The turbuent inetic energy and its energy dissipation rate are cacuated from their conservation ) + ( u ( ) T; + (G ); = + ( u ) ( ) ( T; + C 1 G C 2 2 ) (11) with the standard mode constants C =0:09; =1:00; =1:30; C 1 =1:44 and C 2 =1:92. The term G in Eqs. (10) and (11) is the production of turbuent inetic energy and described by G = : u : (12) The term M I; in Eq. (2) describing the interface forces, is given as foows: M I; = M I;g = M D; + M L; + M VM; ; (13) where the terms on the right-hand side of Eq. (13) are respectivey forces due to drag, ift and virtua mass. For an extensive discussion of these forces the reader is referred to the wors of Jaobsen et a. (1997) and Denoij et a. (1997). The drag force is given as foows: M D; = 3 4 C D g u g u (u g u ): (14) d b Ishii and Zuber (1979) give the foowing expression for the drag coecient C D in the case of distorted bubbes: C D = 2 3 Eo1=2 ; (15)

3 N. G. Deen et a. / Chemica Engineering Science 56 (2001) where Eo is the dimensioness Eotvos number (Eo = gd 2 B=). In this wor, a bubbe size of 4:0 mm is used, giving Eo =2:2 and C D =1:0. The bubbe size is in correspondence with the observations of Deen et a. (2000a). The eect of shearing motion in the iquid phase on the movement of the gas phase is modeed through the ift force M L; = g C L (u g u ) u ; (16) where C L is a mode constant, which is set to 0.5. The acceeration of the iquid in the wae of the bubbe is taen into account through the virtua mass force, which is given by M VM; = g C VM ( Dug Dt Du ) Dt (17) with C VM a mode constant with a vaue of 0.5. The D=Dt operators denote the substantia derivatives in the two phases. 3. Case studies The ow in a square bubbe coumn was simuated, using seven dierent settings, as summarized in Tabe 1. The eects of both the dierent interface forces and the dierent turbuence modes are investigated. The simuations are compared with the predictions performed by Deen et a. (2000b) and with experiments performed by Deen et a. (2000a). A simuations were carried out with the commercia CFD code CFX 4.3. The Smagorinsy mode and the averaging routines were impemented in user Fortran foowing the wor of Jacobsen (1997). In this section a short description of the experimenta set-up and the numerica impementation wi be given Experiments Deen et a. (2000a) performed measurements in a 3-D bubbe coumn ed with distied water. In order to obtain a non-coaescing system, 4 g of itchen sat per itre water was used. The coumn has a square cross-section (W D) of0:15 0:15 m 2 and a height (L) of1m.the coumn was initiay ed with water up to a height of 0:45 m. Air was introduced into the system through a perforated pate. The distributor pate contained 49 hoes, with a diameter of 1 mm, which were positioned in the midde of the coumn at a square pitch of 6:25 mm. A two-camera partice image veocimetry (PIV) system was used to simutaneousy measure the gas and iquid veocities Numerica detais The boundary conditions were dened as foows. At the inet the gas veocity was given by v g;in = v g;swd ; (18) g A in where v g;s the supercia gas veocity and W D denotes the cross sectiona area of the coumn. In the case of the coarse grid, the gas inet area A in was impemented in a centra area of 3 3 gridces (i.e. A in =0:03 0:03 m 2 ). A supercia gas veocity of 4: ms 1 eads to a gas veocity at the inet of 0:12 m s 1. In the case of the ne grid 8 8 gridces (i.e. A in =0:038 0:038 m 2 ) were used for the gas inet, eading to v g;in =0:078 m s 1. Aong the was, wa functions were appied for the mode, whie no-sip boundary conditions were adopted for LES. At the outet of the coumn, a pressure boundary was used. In the case of the mode, the free surface was captured in the simuations. The free surface was omitted for the LES, because of convergence probems for the cases were the ift and=or the virtua mass force were incuded. In the simuations, a bounded third order accurate QUICK scheme was used for the discretization of the convection terms. The ow was simuated for a period of 150 s. The data was time averaged over the ast 140 s. For the LES on the ne grid, the simuations too about 90 h rea time on a SGI Origin 2000 machine, using 4 processors. 4. Resuts The gas iquid ow in a bubbe coumn has been modeed as described in the previous section with the use of Tabe 1 Overview of modes and gridspacings Case i, (mm) j (mm) t (s) e M I , BIT M D LES, BIT M D LES, BIT M D M L LES, BIT M D M VM LES, BIT M D M L M VM LES M D M L M VM LES, BIT M D M L M VM

6344 N. G. Deen et a. / Chemica Engineering Science 56 (2001) 6341 6349 Fig. 1.

4 6344 N. G. Deen et a. / Chemica Engineering Science 56 (2001) Fig. 1. Comparison of simuated and experimenta proes of the axia iquid veocity for dierent interface forces, at a height of 0:25 m. Fig. 2. Time history of the axia iquid veocity at the centerine of the coumn, at a height of 0:25 m. CFX 4.3. In this section we wi discuss the resuts of the simuations. First we wi focus on the inuence of the interface forces. In that case the LES turbuence mode was used. Then we wi discuss the eect of turbuence modes. In that evauation a interface forces were incorporated in the simuations Interface forces The eect of the dierent interface forces for the LES can be inferred from Fig. 1. When ony the drag force was incorporated the bubbe pume woud rise right up to the top of the coumn, without transversa spreading. Fig. 3. Snapshot of the instantaneous isosurface of g =0:04 and iquid veocity ed after 100 s, for the mode. This is in contrast to the experiment, where the bubbe pume moves around in the bubbe coumn in a random fashion. Accordingy, the predicted iquid veocity proe shows a strong pea in the centre, where the gas is rising and moderate downow aongside the pume. When the virtua mass force is incuded, the behaviour is basicay the same. The itte eect of the virtua mass force can be expained by the fact that the simuations yied a quasi-stationary state. In a quasi-stationary state there is ony itte acceeration, so the inuence of the virtua

N. G. Deen et a. / Chemica Engineering Science 56 (2001) 6341 6349 6345 Fig. 4.

5 N. G. Deen et a. / Chemica Engineering Science 56 (2001) Fig. 4. Snapshots of the instantaneous isosurfaces of g =0:04 and iquid veocity eds after 30 and 35 s, for the LES mode at the coarse grid (case 5). mass force is sma. When both the drag force and the ift force are incorporated, the pume is spread across the coumn, due to the nature of the ift force. When aso the virtua mass force is added, the dierences are sma. The veocity proe for case 5 is ony sighty ower than for case Turbuence modes A time history pot of the axia iquid veocity at one point in the coumn is shown in Fig. 2. From this gure it becomes cear that, provided that drag, ift and virtua mass forces are taen into account, the LES mimics the transient behaviour observed in the experiments of Deen et a. (2000a). Both the time and veocity scaes are in agreement with the experimenta data. In the mode a the sma-scae veocity uctuations are contained in the turbuent inetic energy. In this case ony the arge-scae uctuations are resoved. These uctuations appear to be very sma in magnitude. In Figs. 3 5 snapshots of gas fraction isosurfaces and iquid veocity eds are dispayed for cases 1, 5 and 7,

6346 N. G. Deen et a. / Chemica Engineering Science 56 (2001) 6341 6349 Fig. 5. Snapshots of the instantaneous isosurfaces of g =0:04 and iquid veocity eds after 30 and 35 s, for the LES mode at the ne grid (case 7).

6 6346 N. G. Deen et a. / Chemica Engineering Science 56 (2001) Fig. 5. Snapshots of the instantaneous isosurfaces of g =0:04 and iquid veocity eds after 30 and 35 s, for the LES mode at the ne grid (case 7). respectivey. We can see a cear dierence between the resuts for the two turbuence modes. In accordance with the resuts shown in Fig. 2, the LES mode resoves much more of the detais of the ow. Large vortices are observed aong side the bubbe pume. Due to the high turbuent viscosity, a quasi-stationary state is obtained for the mode. That is, the transient detais are not resoved but impicity contained in the turbuent inetic energy. It is seen that a arge stationary vortex is obtained next to the bubbe pume. The presence of this vortex produces an asymmetric soution. A more quantitative comparison between the two turbuence modes can be obtained from Figs. 6 and 7. These gures show the average veocity proes for, respectivey, the iquid and the gas phase. The average veocities are cacuated on the y according to u ;n = n n 0 1 u ;n u ;n ; (19) n n 0 n n 0 where the averaging is started at timestep n 0. The predictions with the mode produce asymmetric resuts, though in the right order of magnitude. For both phases the maximum average veocities are predicted somewhat too high. The LES mode yieds more symmetric veocity proes. The simuations on both grids predict the same trends. In Figs. 6 and 7 the eect of the mode for the

7 N. G. Deen et a. / Chemica Engineering Science 56 (2001) Fig. 6. Comparison of simuated and experimenta proes of the axia iquid veocity for dierent grid sizes and turbuence modes, at a height of 0:25 m. Fig. 7. Comparison of simuated and experimenta proes of the axia gas veocity for dierent grid sizes and turbuence modes, at a height of 0:25 m. turbuence induced by the bubbes can aso be seen. The dierences between the cases with and without bubbe induced turbuence are margina. In the mode the veocity uctuations are not resoved but contained in the turbuent inetic energy,. Assuming oca isotropy of the turbuence, the veocity uctuations in each direction can be derived as foows: u 2 = v 2 = 2 3: (20) In the LES the veocity uctuations are resoved and are cacuated during the cacuation, using Eq. (21). u 2 ;n = n n 0 1 u 2 n n ;n (u ;n u ;n ) 2 : (21) 0 n n 0 Proes of the radia and axia uctuations of the iquid veocity are shown in Figs. 8 and 9. It is apparent from these gures that the assumption of isotropy is not vaid. The radia and axia uctuations predicted by the mode are respectivey too high and too ow. The LES performs much better in that respect. Both proes show good agreement with the experimenta data. A na comparison can be made by means of the turbuent inetic energy,. In the LES is cacuated through = (u + v 2 + w 2 ): (22) Deen et a. (2000a) ony measured the veocity uctuations in the radia and the axia direction. The turbuent inetic energy was therefore estimated as = (2u + v 2 ): (23) Proes of the turbuent inetic energy are shown in Fig. 10. It is seen that both the simuations with the mode and the coarse LES mode capture the trend reasonaby. The simuation with the ne LES mode however shows overa the best agreement with the PIV data. Fig. 8. Comparison of simuated and experimenta proes of the axia iquid veocity uctuations for dierent grid sizes, at a height of 0:25 m. The dierences between the simuations with and without the bubbe induced turbuence (cases 5 and 6) are sma. 5. Concusions The gas iquid ow in a square cross-sectioned bubbe coumn has been successfuy simuated. The ow was described with the Euer Euer approach, where the drag, ift and virtua mass forces were taen into account. The turbuence viscosity in the iquid phase was modeed with

8 6348 N. G. Deen et a. / Chemica Engineering Science 56 (2001) Notation Fig. 9. Comparison of simuated and experimenta proes of the radia iquid veocity uctuations for dierent grid sizes, at a height of 0:25 m. A area, m 2 C mode constant mode, dimensioness C ;BI mode constant bubbe induced turbuence, dimensioness C 1 mode constant mode, dimensioness C 2 mode constant mode, dimensioness C D drag coecient, dimensioness C L ift force mode constant, dimensioness C S sub-grid scae mode constant, dimensioness C VM virtua mass force mode constant, dimensioness D coumn depth, m d B bubbe diameter, m Eo Eotvos number, dimensioness g gravitationa constant, m 2 s 1 G production of turbuent inetic energy, g m 1 s 3 I unity tensor, dimensioness turbuent inetic energy, m 2 s 2 L coumn height, m M D drag force, N m 3 M L ift force, N m 3 M VM virtua mass force, N m 3 M I tota interfacia force, N m 3 n time step, dimensioness n 0 time step at which time averaging starts, dimensioness P pressure, N m 2 S characteristic tered rate of strain, s 1 u radia veocity, m s 1 u mean or grid scae veocity vector, m s 1 u uctuating or sub-grid scae (SGS) veocity, ũ v W w ms 1 instantaneous veocity, m s 1 axia veocity, m s 1 coumn width, m atera veocity, m s 1 Gree etters Fig. 10. Comparison of simuated and experimenta proes of the turbuent inetic energy for dierent grid sizes and turbuence modes, at a height of 0:25 m. the Smagorinsy LES mode. An extra contribution in the eective viscosity for the turbuence induced by bubbes was taen into account. The resuts of the LES simuations were compared with simuations of Deen et a. (2000b) and with the experiments of Deen et a. (2000a). Contrary to the simuations of Deen et a. (2000b) the LES simuations captured the strong transient movement of the bubbe pume, which were observed in the experiment. Furthermore, both the veocity and the veocity uctuations are in quantitative agreement with the measurement data. i j L T e voume fraction, dimensioness tering width, m grid spacing in radia direction, m grid spacing in axia direction, m grid spacing in atera direction, m energy dissipation, m 2 s 3 density, g m 3 mode constant mode, dimensioness mode constant mode, dimensioness surface tension, N m 1 stress tensor, g m 1 s 2 inematic viscosity, m 2 s 1 turbuent viscosity, m 2 s 1 eective viscosity, m 2 s 1

9 N. G. Deen et a. / Chemica Engineering Science 56 (2001) Subscripts g in n s gas phase inet either phase iquid phase time step supercia References Deen, N. G., Hjertager, B. H., & Soberg, T. (2000a). Comparison of PIV and LDA measurement methods appied to the gas iquid ow in a bubbe coumn. Presented at the 10th internationa symposium on appications of aser techniques to uid mechanics, Lisbon, Portuga. Deen, N. G., Soberg, T., & Hjertager, B. H. (2000b). Numerica simuation of the gas iquid ow in a square cross-sectioned bubbe coumn. CHISA 14th internationa congress of chemica and process engineering, Praha, Czech Repubic. Denoij, E., Lammers, F. A., Kuipers, J. A. M., & van Swaaij, W. P. M. (1997). Dynamic simuation of dispersed gas iquid two-phase ow using a discrete bubbe mode. Chemica Engineering Science, 52, Hjertager, B. H. (1998). Computationa uid dynamics (CFD) anaysis of mutiphase chemica reactors. Trends in Chemica Engineering, 4, Ishii, M., & Zuber, N. (1979). Drag coecient and reative veocity in bubby, dropet or particuate ows. AIChE Journa, 25(5), Jacobsen, C. B. (1997). Large eddy simuation of conned swiring ow. Ph.D. Thesis, Aaborg University. Jaobsen, H. A., Sannaes, B. H., Grevsott, S., & Svendsen, H. F. (1997). Modeing of vertica bubbe-driven ows. Industria and Engineering Chemistry Research, 36, Kuipers, J. A. M., & van Swaaij, W. P. M. (1997). Computationa uid dynamics appied to chemica reaction engineering. Reviews in Chemica Engineering, 13, 3. Sato, Y., & Seoguchi, K. (1975). Liquid veocity distribution in two-phase bubbe ow. Internationa Journa of Mutiphase Fow, 2, 79. Smagorinsy, J. (1963). Genera circuation experiments with the primitive equations. Monthy Weather Review, 91, Smith, B. L., & Miei, M. (1998). An investigation of conned bubbe pumes. Presented at the third internationa conference on mutiphase ow, ICMF 98, Lyon, France. Sooichin, A., Eigenberger, G., Lapin, A., & Lubbert, A. (1997). Dynamic numerica simuations of gas iquid two-phase ows, Euer=Euer versus Euer=Lagrange. Chemica Engineering Science, 52,

Large Eddy Simulation of the Gas-Liquid Flow in a Square Cross-sectioned Bubble

Large Eddy Simulation of the Gas-Liquid Flow in a Square Cross-sectioned Bubble Large Eddy Simuation of the Gas-Liquid Fow in a Square Cross-sectioned Bubbe Coumn 1 N.G. Deen 2, T. Soberg and B.H. Hjertager Chem. Eng. Lab., Aaborg Univ. Esbjerg; Nies Bohrs Vej 8; DK-6700 Esbjerg;