14 th European Conference on Mixing Warszawa, 10-13 Septemer 2012 APPLICATION OF THE CONDITIONAL QUADRATURE METHOD OF MOMENTS FOR THE SIMULATION OF COALESCENCE, BREAKUP AND MASS TRANSFER IN GAS-LIQUID STIRRED TANKS M. Petitti a, M. Vanni a, D.L. Marchisio a, A. Buffo a, F. Podenzani, a Politecnico di Torino, Dip. Scienza Applicata e Tecnologia, corso Duca degli Aruzzi 24, 10129 Torino, Italy ENI S.p.a., Divisione Refining and Marketing, via Felice Maritano 26, 20097 San Donato Milanese, Italy miriam.petitti@polito.it Astract. Mass transfer has een investigated in a gas-liquid stirred reactor y considering the asorption of oxygen in the liquid phase (water. A multivariate population alance model (PBM, coupled with an Eulerian multi-fluid approach, has een employed to descrie the spatial and temporal evolution of the ules size and composition distriutions. The PBM has een solved y adopting the Conditional Quadrature Method of Moments (CQMOM, that has een implemented through userdefined functions (UDF and scalars (UDS in the commercial Computational Fluid Dynamics code Ansys Fluent 12. Model predictions were compared with experimental data of ule size distriutions and then with oxygen accumulation in the liquid phase. The simulation results show that, although coalescence and reakup tend to homogenize ule composition, a multivariate PBM is necessary to properly descrie mass transfer. In fact, due to the fact that smaller ules, generally located in regions characterized y high turulent dissipation rates, exchange mass with a much faster rate than igger ules, oth ule size and ule composition should e included as internal coordinates. Keywords: Gas-liquid stirred tank, Population alance, Multiphase flow, Mass transfer, Bule size distriution, Conditional Quadrature Method of Moments (CQMOM. 1. INTRODUCTION An accurate description of fluid-dynamics and mass transfer in gas-liquid systems requires the evaluation of ule size and composition distriutions, oth varying in time and space, due to gas ule reakage, coalescence and mass exchange with the liquid phase. To this purpose a multivariate population alance model (PBM, coupled with an Eulerian multi-fluid approach, has een here employed to descrie the spatial and temporal evolution of ules size and composition distriutions in a gas-liquid system. The system examined is a 194 liter stirred-tank, previously studied y Laakkonen et al. [1] and then y Petitti et al. [2], agitated via a Rushton turine and operating with air and tap water. Gas-liquid mass transfer is investigated y considering the asorption of oxygen in the liquid phase, for which many experimental data are availale, in many different operating conditions (stirring speeds and gassing rates. 371
For the numerical solution of the multivariate PBM the Conditional Quadrature Method of Moments (CQMOM, the multivariate extension of the Quadrature Method of Moments (QMOM to two internal coordinates (ule size and composition, was implemented through user-defined functions (UDF in the commercial computational fluid dynamics (CFD code Ansys Fluent 12. The implemented algorithm was proven to e numerically stale in all the operating conditions examined. Validation was carried out y comparing first the model predictions with experimentally measured ule size distriutions and then with oxygen accumulation in the liquid phase, otaining satisfactory agreements. 2. REACTOR INVESTIGATED The stirred reactor investigated is a 194 liter four-affle reactor, agitated y a six-lade Rushton turine, previously investigated y Laakkonen et al. [1] and y Petitti et al. [2]. The gas is introduced through a metal porous sparger with pores of 15 µm in diameter. The asorption of oxygen in the liquid phase is examined and compared with experimental data in different operating conditions: the stirring speeds are in the range 155-250 rpm while the gassing rates in the range 0.018 0.093 vvm (vessel volumes per minute. 3. MODEL DESCRIPTION 3.1 Multiphase modeling The Eulerian multi-fluid approach (in this case two-fluid was adopted and oth phases (continuous and dispersed were considered incompressile. The drag force and the uoyancy force were the only forces taken into account for the dispersed phase; additional forces like lift and virtual mass were neglected ecause in stirred reactors the velocity field is dominated y the turine rotation (on the contrary of ule columns where these forces are usually accounted for. The drag coefficient for the gas-liquid interaction was evaluated from ule terminal velocity, y assuming a unique value for the terminal velocity of all the ules, equal to 12 cm/s, taking thus into account the damping effect of turulence on ule slip velocity (for details see [2]. Turulence was modeled with the k-ε standard mixture model since it represents a good trade-off etween accuracy and computational costs. 3.2 Population alance modeling The population of ules dispersed in a turulent liquid and characterized y different properties such as size (L and chemical composition (φ, representing the internal coordinates of the population, can e modeled through a smooth and differentiale Numer Density Function (NDF, as explained y Fox [3]. The NDF is defined in such a way that the following quantity: n( L, φ ;x, t dl dφ dx (1 represents the expected numer of ules in the physical volume dx=dx 1 dx 2 dx 3 around the physical point x = (x 1, x 2, x 3 and with ule size in the infinitesimal range etween L and L + dl and composition etween φ and φ + dφ. From the theoretical point of view, there is no limitation on the numer of internal coordinates (other possile coordinates might e temperature, velocity, etc. ut here only size and composition were examined. In fact, the system investigated is isothermal and, thanks to the low hold-up values, the effects of ules collision, reakage and coalescence on momentum exchange can e neglected. 372
The evolution of the NDF follows the Generalized Population Balance Equation (GPBE (for details see [4], [5]: ( n u + ( n G + ( n φ = H( L, φ ;x, t n + t x L φ (2 where u is the ule velocity; G represents the rate of continuous change of ule size generally, descriing ule continuous growth or shrinkage due to molecular phenomena, like the addition or the depletion of single molecules (evaporation, condensation, mass transfer processes; φ represents the rate of continuous change of ule composition (with respect to the different chemical components caused y chemical reactions and/or mass transfer processes etween the phases. The term appearing on the right hand side of the previous equation instead takes into account discontinuous events, accounting for instantaneous change of size and composition due to ule collisions, coalescence and reakage. The φ term is generally a vector ecause many chemical components may e considered, ut in the system here investigated the gas phase is made up of only two chemical components (nitrogen and oxygen, with only one active component (oxygen transferring etween the phases and thus φ is a scalar, representing the numer of moles of oxygen within one single ule; consequently in the following expressions it will e expressed as φ. In the present work all the ules of the distriution are assumed to have the same velocity. The local value of ules velocity (u is calculated from the momentum alance equation, using the local ule Sauter diameter (d 32 calculated from the multivariate PBM for the computation of drag and uoyancy forces. As regards the rates of continuous change of size and composition, the expressions for G and φ are calculated y considering only the effect of oxygen on mass transfer, through a simple mass alance on a single (spherical ule: 2kLM O φ 2 G = ψc HO 2 3 ρ kvl (3 2 φ φ = kl π L ψc H O (4 2 3 kvl where M O2 is the molecular weight of oxygen, ψ c is the concentration of oxygen in the continuous phase, H O2 is Henry constant and ρ is the density of the gas in the ule. The mass transfer coefficient k L has een evaluated with the expression used y Lamont and Scott [6], ased on the local value of the turulent dissipation rate: k L = c D 1 / 2 ερc μc 1 / 4 (5 where D is oxygen molecular diffusivity, ε the turulent dissipation rate, ρ c the density of the continuous phase and µ c the viscosity of the continuous phase. Different values were tested for the coefficient c appearing in Eq. (5 and the theoretical value of 1.13 proposed y Kawase et al. [7] gave promising results, as shown in section 4. 373
By assuming that reakage and coalescence kernels depend only on ule size and y considering that in the present prolem φ is a scalar representing the numer of moles of oxygen within one single ule, the discontinuous event term can e expressed as follows: H( L, φ ; x,t = n( L, φ ; x,t L φ a ( L n( L, φ ; x,t 00 1 2 0 0 a 3 3 ( 1 L λ, λ n( λ, φ ; x,t n ( L λ 1 3 3 3 3,( φ φ ( L, λ n( λ, φ ; x,t dλ dφ + ( λ,λ,λ,λ L φ P( L, φ λ, φ,λ,λ, x,t dλ dφ n( λ, φ ; x,t +,λ,λ + (6 where the first two terms represent respectively the irth and death due to ules coalescence, whereas the last two terms the irth and death caused y ules reakage. The coalescence kernel (a, the reakage kernel ( and the daughter distriution function (P derive from turulence theory and were descried in detail in our previous work [2]. The prolem is here solved in terms of the moments of the NDF, expressed as: k l M k, l = n( L,φ L φdl dφ (7 ΩΩ L φ where Ω L and Ω φ are the phase spaces generated y the internal coordinates. By applying the moment transform to Eq. (2 the transport equation for a generic moment of the NDF can e otained. The population alance is solved y resorting to the Conditional Quadrature Method of Moments (CQMOM, where the closure is achieved y means of a quadrature approximation through which the NDF is represented as a weighted summation of Dirac delta functions centered on proper nodes: N1 φ = wiδ, i= 1 j= 1 N2 ( L Li wi jδ ( φ φ;i,j n( L, (8 The numer densities w i,j and the compositions φ ;i,j are conditioned on the value of the first internal coordinate L i with numer density w i.the total numer of nodes is given y N 1 N 2, where N 1 is the numer of nodes used for ule size and N 2 the numer of nodes used for ule composition. In other words each one of the N 1 groups of ules with size L i is sudivided into N 2 groups of ules characterized y different compositions φ,i,j.. In order to limit the computational costs in the present work the values adopted are N 1 =2; N 2 =1. This requires tracking the following six moments: M 0,0, M 1,0, M 2,0, M 3,0, M 0,1, M 1,1. The mean ule size (d 32 is then availale via the ratio of M 3,0 and M 2,0, whereas the mean oxygen concentration of the ules is calculated as the ratio etween M 0,1 and k v M 3,0, where k v is the volume shape factor of the ules (for details see [8]. 4. SIMULATION RESULTS The Multi Reference Frame method was adopted and the simulations were performed in non-stationary conditions on only one half of the reactor (485,974 cells including a region aove the liquid level occupied y the gas phase only. In fact, the use of a computational domain that extends eyond the initial liquid level turned out to e essential in simulating mass transfer. This is done y including a region aove the liquid in which there is only the gas phase; this is necessary in order to keep within the domain oundaries the whole liquid present in the system, whose level increases a little as a 374
consequence of the gas fed into the tank. In this way any loss of liquid, where the oxygen accumulates, is avoided and any liquid ackflow with unknown concentration is prevented. The ules entering the reactor were modeled with fixed oxygen concentration (saturation value and with a lognormal size distriution, with a mean ule size (d 32 calculated with Kazakis s correlation for porous spargers [9]. The PBM was solved only where the local gas volume fraction was in the range 0.001 0.99, elsewhere the local d 32 was set equal to the inlet value. Oxygen adsorption in the liquid was monitored and its temporal evolution was compared with the experiments y Laakkonen conducted under different operating conditions. Different values were tested for the coefficient c appearing in Lamont and Scott s correlation. The more promising results were otained with the values reported in Fig. 1. Figure 1. Comparison of the predicted temporal evolution of the dimensionless oxygen concentration in the liquid with the experimental data, in the monitor point, for the operating conditions investigated. The theoretical value of 1.13, proposed y Kawase et al. [7] gives the est agreement with the experimental data, especially at the lowest stirring speed and gassing rate. The stirring speeds investigated guarantee a good mixing in the reactor and a rather uniform concentration of oxygen in the liquid is oserved at the end of the simulations. The predictions for k L a reported in Fig. 2 highlight that the highest values are reached near the impeller where the ules are smaller due to significant reakage. A satisfactory agreement is then otained for the comparison of the predicted d 32 with the experimental data, in selected points on the planes at 45 etween the affles, as shown in Fig. 3. 375
Figure 2. Contours of the local kla predicted for the case at 250 rpm and 0.052 vvm. Figure 3. Comparison of the predicted d32 (mm with the experimental data (in green, 250 rpm and 0.093 vvm. The value zero is plotted for the d32 where the PBM is not solved. 5. REFERENCES [1] Laakkonen M., Alopaeus V., Aittamaa J., 2006. Validation of ule reakage, coalescence and mass transfer models for gas-liquid dispersion in agitated vessel, Chem. Eng. Sci., 61, 218-228. [2] Petitti M., Nasuti A., Marchisio D.L., Vanni M., Baldi G., Podenzani F., Mancini N., 2010. Bule size distriution modelling in stirred gas-liquid reactors with QMOM augmented y a new correction, AIChE J., 56, 36-53. [3] Fox R.O., 2007. Introduction and fundamentals of modeling approaches for polydisperse multiphase flows, in: Multiphase Reacting Flows, (D.L. Marchisio, R.O. Fox, eds, vol. 492 of CISM International Centre for Mechanical Sciences, Springer, Wien, pp. 1-40. [4] Ramkrishna D., 2000. Population Balances, Academic Press, New York. [5] Marchisio D.L., 2007. Quadrature method of moments for poly-disperse flows, in: Multiphase Reacting Flows, (D.L. Marchisio, R.O. Fox, eds, vol. 492 of CISM International Centre for Mechanical Sciences, Springer, Wien, pp. 41-77. [6] Lamont J.C., Scott D.S., 1970. An eddy cell model of mass transfer into the surface of a turulent liquid, AIChE J., 16, 513-519. [7] Kawase Y., Halard B., Moo-Young M., 1987. Theoretical prediction of volumetric mass transfer coefficients in ule columns for Newtonian and non-newtonian fluids, Chem. Eng. Sci., 42, 1609-1617. [8] Buffo A., Vanni M., Marchisio D.L., Fox R.O., 2011. Comparison etween different methods for turulent gas-liquid systems y using multivariate population alances, Proc. 8th Int. Conf. on CFD in Oil and Gas, Metallurgical Process Industries (Trondheim, 21-23 June, pp. 10. [9] Kazakis N.A., Mouza A.A., Paras S.V., 2008. Experimental study of ule formation at metal porous spargers: Effect of liquid properties and sparger characteristics on the initial ule size distriution, Chem. Eng. J., 137, 265-281. 376