14th European Conference on Mixing Warszawa, 10-13 September 2012 APPLICATION OF THE CONDITIONAL QUADRATURE METHOD OF MOMENTS FOR THE SIMULATION OF COALESCENCE, BREAKUP

AND MASS TRANSFER IN GAS-LIQUID STIRRED TANKS

M. Petittia, M. Vannia, D.L. Marchisioa, A. Buffoa, F. Podenzanib,

a Politecnico di Torino, Dip. Scienza Applicata e Tecnologia, corso Duca degli Abruzzi 24, 10129 Torino, Italy

b ENI S.p.a., Divisione Refining and Marketing, via Felice Maritano 26, 20097 San Donato Milanese, Italy

miriam.petitti@polito.it

Abstract. Mass transfer has been investigated in a gas-liquid stirred reactor by considering the absorption of oxygen in the liquid phase (water). A multivariate population balance model (PBM), coupled with an Eulerian multi-fluid approach, has been employed to describe the spatial and temporal evolution of the bubbles size and composition distributions. The PBM has been solved by adopting the Conditional Quadrature Method of Moments (CQMOM), that has been implemented through user-defined functions (UDF) and scalars (UDS) in the commercial Computational Fluid Dynamics code Ansys Fluent 12. Model predictions were compared with experimental data of bubble size distributions and then with oxygen accumulation in the liquid phase. The simulation results show that, although coalescence and breakup tend to homogenize bubble composition, a multivariate PBM is necessary to properly describe mass transfer. In fact, due to the fact that smaller bubbles, generally located in regions characterized by high turbulent dissipation rates, exchange mass with a much faster rate than bigger bubbles, both bubble size and bubble composition should be included as internal coordinates. Keywords: Gas-liquid stirred tank, Population balance, Multiphase flow, Mass transfer, Bubble size distribution, 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 bubble size and composition distributions, both varying in time and space, due to gas bubble breakage, coalescence and mass exchange with the liquid phase.

To this purpose a multivariate population balance model (PBM), coupled with an Eulerian multi-fluid approach, has been here employed to describe the spatial and temporal evolution of bubbles size and composition distributions in a gas-liquid system.

The system examined is a 194 liter stirred-tank, previously studied by Laakkonen et al. [1] and then by Petitti et al. [2], agitated via a Rushton turbine and operating with air and tap water.

Gas-liquid mass transfer is investigated by considering the absorption of oxygen in the liquid phase, for which many experimental data are available, in many different operating conditions (stirring speeds and gassing rates).

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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 (bubble 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 be numerically stable in all the operating conditions examined.

Validation was carried out by comparing first the model predictions with experimentally measured bubble size distributions and then with oxygen accumulation in the liquid phase, obtaining satisfactory agreements.

2. REACTOR INVESTIGATED The stirred reactor investigated is a 194 liter four-baffle reactor, agitated by a six-blade

Rushton turbine, previously investigated by Laakkonen et al. [1] and by Petitti et al. [2]. The gas is introduced through a metal porous sparger with pores of 15 µm in diameter.

The absorption 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 both phases

(continuous and dispersed) were considered incompressible. The drag force and the buoyancy force were the only forces taken into account for the

dispersed phase; additional forces like lift and virtual mass were neglected because in stirred reactors the velocity field is dominated by the turbine rotation (on the contrary of bubble columns where these forces are usually accounted for).

The drag coefficient for the gas-liquid interaction was evaluated from bubble terminal velocity, by assuming a unique value for the terminal velocity of all the bubbles, equal to 12 cm/s, taking thus into account the damping effect of turbulence on bubble slip velocity (for details see [2]).

Turbulence was modeled with the k-ε standard mixture model since it represents a good trade-off between accuracy and computational costs.

3.2 Population balance modeling

The population of bubbles dispersed in a turbulent liquid and characterized by different properties such as size (L) and chemical composition (φb), representing the internal coordinates of the population, can be modeled through a smooth and differentiable Number Density Function (NDF), as explained by Fox [3]. The NDF is defined in such a way that the following quantity: x),x;,( dddLtLn bb φφ (1) represents the expected number of bubbles in the physical volume dx=dx1dx2dx3 around the physical point x = (x1, x2, x3) and with bubble size in the infinitesimal range between L and L + dL and composition between φb and φb + dφb.

From the theoretical point of view, there is no limitation on the number of internal coordinates (other possible coordinates might be temperature, velocity, etc.) but 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 bubbles collision, breakage and coalescence on momentum exchange can be neglected.

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The evolution of the NDF follows the Generalized Population Balance Equation (GPBE) (for details see [4], [5]):

( ) ( ) ( ) ),x;,(ux b tLHnGn

Ln

tn

bbb

φφφ

=∂∂

+∂∂

+∂∂

+∂∂ (2)

where ub is the bubble velocity; G represents the rate of continuous change of bubble size generally, describing bubble continuous growth or shrinkage due to molecular phenomena, like the addition or the depletion of single molecules (evaporation, condensation, mass transfer processes); bφ represents the rate of continuous change of bubble composition (with respect to the different chemical components) caused by chemical reactions and/or mass transfer processes between 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 bubble collisions, coalescence and breakage.

The φb term is generally a vector because many chemical components may be considered, but 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 between the phases and thus φb is a scalar, representing the number of moles of oxygen within one single bubble; consequently in the following expressions it will be expressed as φb.

In the present work all the bubbles of the distribution are assumed to have the same velocity. The local value of bubbles velocity (ub) is calculated from the momentum balance equation, using the local bubble Sauter diameter (d32) calculated from the multivariate PBM for the computation of drag and buoyancy forces.

As regards the rates of continuous change of size and composition, the expressions for G and bφ are calculated by considering only the effect of oxygen on mass transfer, through a simple mass balance on a single (spherical) bubble:

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 32

22

LkHψ

ρMk

Gv

bOc

b

OL φ (3)

⎟⎟⎠

⎞⎜⎜⎝

⎛−= 3

22 Lk

HψLπkv

bOcLb

φφ (4)

where MO2 is the molecular weight of oxygen, ψc is the concentration of oxygen in the continuous phase, HO2 is Henry constant and ρb is the density of the gas in the bubble.

The mass transfer coefficient kL has been evaluated with the expression used by Lamont and Scott [6], based on the local value of the turbulent dissipation rate:

41

21/

c

c/L Dck ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

μερ (5)

where D is oxygen molecular diffusivity, ε the turbulent dissipation rate, ρc the density of the continuous phase and µc 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 by Kawase et al. [7] gave promising results, as shown in section 4.

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By assuming that breakage and coalescence kernels depend only on bubble size and by considering that in the present problem φb is a scalar representing the number of moles of oxygen within one single bubble, the discontinuous event term can be expressed as follows:

( ) ( ) ( )

( ) ( )

( ) )t,;,L(nb

)t,;,(n),|,L(Pbd)t,;,(n,La)t,;,L(n

dt,,,Ln)t,;,(n,La)t,;,L(H

L

L

x

xxx

xxx

b

λb,λb,bλb,λb,b

λb,λb,bλb,b

L

λλλdλλλ

dλλλλλ

b

b

φ

φφφφφφ

φφφφφ

φ

φ

−

++−

+⎟⎠

⎞⎜⎝

⎛−−⎟

⎠

⎞⎜⎝

⎛−=

∫ ∫∫ ∫

∫ ∫∞ ∞∞ ∞

0 0

31

33

0 0

31

33

21

(6)

where the first two terms represent respectively the birth and death due to bubbles coalescence, whereas the last two terms the birth and death caused by bubbles breakage. The coalescence kernel (a), the breakage kernel (b) and the daughter distribution function (P) derive from turbulence theory and were described in detail in our previous work [2]. The problem is here solved in terms of the moments of the NDF, expressed as:

∫ ∫Ω Ω

=L b

blb

kblk ddLLLnM

φ

φφφ ),(, (7)

where ΩL and Ωφb are the phase spaces generated by the internal coordinates. By applying the moment transform to Eq. (2) the transport equation for a generic moment of the NDF can be obtained. The population balance is solved by resorting to the Conditional Quadrature Method of Moments (CQMOM), where the closure is achieved by means of a quadrature approximation through which the NDF is represented as a weighted summation of Dirac delta functions centered on proper nodes:

( ) ( )∑∑==

−−=21

11

N

jji,b;bji

N

iiib wLLwLn φφδδφ ,),( (8)

The number densities wi,j and the compositions φb;i,j are “conditioned” on the value of the first internal coordinate Li with number density wi .The total number of nodes is given by N1⋅ N2, where N1 is the number of nodes used for bubble size and N2 the number of nodes used for bubble composition. In other words each one of the N1 groups of bubbles with size Li is subdivided into N2 groups of bubbles characterized by different compositions φb,i,j.. In order to limit the computational costs in the present work the values adopted are N1=2; N2=1. This requires tracking the following six moments: M0,0, M1,0, M2,0, M3,0, M0,1, M1,1. The mean bubble size (d32) is then available via the ratio of M3,0 and M2,0, whereas the mean oxygen concentration of the bubbles is calculated as the ratio between M0,1 and kvM3,0, where kv is the volume shape factor of the bubbles (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 above the liquid level occupied by the gas phase only. In fact, the use of a computational domain that extends beyond the initial liquid level turned out to be essential in simulating mass transfer. This is done by including a region above the liquid in which there is only the gas phase; this is necessary in order to keep within the domain boundaries the whole liquid present in the system, whose level increases a little as a

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consequence of the gas fed into the tank. In this way any loss of liquid, where the oxygen accumulates, is avoided and any liquid backflow with unknown concentration is prevented. The bubbles entering the reactor were modeled with fixed oxygen concentration (saturation value) and with a lognormal size distribution, with a mean bubble size (d32) 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 d32 was set equal to the inlet value. Oxygen adsorption in the liquid was monitored and its temporal evolution was compared with the experiments by 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 obtained 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 by Kawase et al. [7] gives the best 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 observed at the end of the simulations. The predictions for kLa reported in Fig. 2 highlight that the highest values are reached near the impeller where the bubbles are smaller due to significant breakage. A satisfactory agreement is then obtained for the comparison of the predicted d32 with the experimental data, in selected points on the planes at 45° between the baffles, as shown in Fig. 3.

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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 bubble breakage, 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. “Bubble size distribution modelling in stirred gas-liquid reactors with QMOM augmented by 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 turbulent liquid”, AIChE J., 16, 513-519. [7] Kawase Y., Halard B., Moo-Young M., 1987. “Theoretical prediction of volumetric mass transfer coefficients in bubble 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 between different methods for turbulent gas-liquid systems by using multivariate population balances”, 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 bubble formation at metal porous spargers: Effect of liquid properties and sparger characteristics on the initial bubble size distribution ”, Chem. Eng. J., 137, 265-281.

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