Modelling local bubble size distributions in agitated vessels

Chemical Engineering Science 62 (2007) 721 – 740 www.elsevier.com/locate/ces

Modelling local bubble size distributions in agitated vessels Marko Laakkonen ∗ , Pasi Moilanen, Ville Alopaeus, Juhani Aittamaa Laboratory of Chemical Engineering, Helsinki University of Technology, P.O. Box 6100, FIN-02015 HUT, Finland Received 16 January 2006; received in revised form 6 October 2006; accepted 6 October 2006 Available online 18 October 2006

Abstract Photography and capillary suction probe were used to measure local bubble size distributions (BSDs) from Rushton turbine agitated (14/200 L) air–tap water and CO2 –n-butanol dispersions. A multiblock stirred tank model with population balances (PBs) for bubbles was created to describe local BSDs in agitated vessels. Unknown parameters in breakage and coalescence models were adjusted by comparing the predicted and measured local BSDs. The BSDs from both investigated systems and varying vessel-operating conditions were included simultaneously to the fitting. The adjusted models were incorporated to MUSIG PB model in CFX-5.7 and tested for the laboratory stirred tanks. The multiblock model showed to be an optimal trade-off between the accuracy and CPU time for the investigation of gas–liquid hydrodynamics and validation of closure models. As a result of fitting, the adjusted model seems to describe local BSDs more accurately in agitated vessels than the model of Lehr et al. [2002. Bubble-size distributions and flow fields in bubble columns. A.I.Ch.E. Journal 48, 2426–2443], which has been successful in bubble column studies. This shows that phenomenological breakage and coalescence closures need experimental validation for various flow environments. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Bubble size distribution; Mixing; Multiphase flow; Parameter identification; Population balance

1. Introduction Agitated gas–liquid reactors are used in many fields of industry. Hydrogenations, oxidations and aerobic fermentations are typical examples of stirred tank processes. In large gas–liquid reactors dispersion is often inhomogeneous due to non-ideal mixing, heat and mass transfer limitations. Detailed reactor design and scale-up need information about local hydrodynamics, mass transfer and reaction conditions. Local gas–liquid mass transfer areas can be estimated most accurately from local bubble size distributions (BSDs) based on population balance (PB) modelling. PBs are also a natural choice for computational fluid dynamic (CFD) calculations. Bubble breakage and coalescence are important phenomena in agitated gas–liquid reactors and need to be modelled to close PBs (Prince and Blanch, 1990; Luo and Svendsen, 1996; Lehr et al., 2002; Hagesaether et al., 2002; Wang et al., 2003, 2005). ∗ Corresponding author. Tel.: +358 9 4512642; fax: +358 9 4512694.

E-mail addresses: marko.laakkonen@tkk.fi, marko.laakkonen@hut.fi (M. Laakkonen). 0009-2509/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2006.10.006

PBs have been applied to the investigation of bubble columns (Jakobsen et al., 2005), but there are few studies available for the agitated vessels (Venneker et al., 2002). Recently Kostoglou and Karabelas (2005) reviewed available bubble breakage models. In their opinion, the current state of art in the development of breakage functions is unsatisfactory. They concluded that similar physical arguments used by various authors lead to quite different forms of breakage model thus resulting in great uncertainties. There is clear need for experimental validation studies. The measurement of bubble breakage and coalescence separately is difficult, but their combined effects can be investigated by measuring local BSDs. The measurement of local BSDs from the agitated tanks is a demanding task (Machon et al., 1997; Alves et al., 2002; Laakkonen et al., 2005a). The inaccuracy of measurement techniques, the sensitivity of bubble size to small amounts of contaminants and varying vessel geometries are reasons to the differences between available bubble size measurements. Optical techniques like phase Doppler anemometry (PDA) (Schäfer et al., 2000) and photography (Machon et al., 1997) incorporated to multiphase particle image velocimetry (Deen et al.,

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2002) are versatile and can produce information about flow fields, but are suitable only for transparent and lean dispersions. Capillary probe techniques (Barigou and Greaves, 1992; Alves et al., 2002; Laakkonen et al., 2005b) are straightforward to use, but cannot detect bubbles smaller than the capillary diameter. This is why Pacek and Nienow (1995) and Machon et al. (1997) recommended photography for the measurement of BSDs from agitated vessels. A benefit of photography compared to the capillary and PDA is that both small and large bubbles can be detected from high-resolution photographs. A viewing chamber technique combines the benefits of photography and capillary technique (Grau and Heiskanen, 2002). In this technique, bubbles are sucked through a sampling tube into a chamber where they are photographed. Hernandez-Aquilar et al. (2004) compared viewing chamber and capillary techniques and concluded that overall they are in good agreement. Possible bias errors in these techniques are the nonisokinetic sampling of bubbles, the coalescence of bubbles before the analysis and the disturbances of sampling tube to the flow field. In most empirical studies bubble sizes have been measured from lean dispersions. There is need for accurate experimental information from dense dispersions, which are common in industrial gas–liquid reactors. The algorithms for bubble identification from the photographs typically fail with dense dispersions, although there are studies where this problem has been addressed (Honkanen and Saarenrinne, 2003). The reasons for poor performance are that bubbles overlap in the images and the separation of bubbles from uneven image background is difficult. The difficulties to arrange backlighting and the bubbles out of focus cause blurred image background. A narrow depth of field (DOF) can minimize these problems to some extent but causes another bias, namely, that large bubbles probably are more in focus than small bubbles (Chigier, 1991). The background lighting seems to be the best method for producing sharp bubble contours in the photographs but is difficult to use with dense dispersions. Identification algorithms are improving, but manual analysis still seems more reliable for dense dispersions. In the present work, bubble breakage and coalescence closures were adjusted against local BSD measurements from two laboratory stirred tanks. The aim was to investigate bubble breakage and coalescence in dense, turbulent dispersions and to validate a set of closure models for agitated gas–liquid reactors. A multiblock stirred tank model was used to consider the inhomogeneity of dispersion in the fitting. The adjusted models were then incorporated to MUSIG PB model (Lo, 2000) in CFX-5.7 and tested for laboratory stirred tanks. The present study extends the authors’ earlier experimental and modelling work with lean air–water dispersions (Laakkonen et al., 2006). The adjusted model is also compared to earlier experimental studies and the model of Lehr et al. (2002). 2. Experimental 2.1. Experimental setup Local BSDs were measured from two geometrically similar, transparent 14 and 200 L laboratory vessels (T = 0.26, 0.63 m)

(X,Y)=(0,0)

D(0,58) E(72,74)

C(0,116)

X T/10

T

Y

B(0,174) T/3 A(0,217)

T/3

T = 260 mm or 630 mm Fig. 1. The dimensions of agitated vessel and the locations of bubble size experiments (A–E) in the 200 L tank in millimetres.

agitated by Rushton turbine (DI = T /3). The gas was injected through a ring sparger below the impeller. The sparger includes 12 holes of diameter 1 and 2 mm in the small and large vessels, respectively. The vessels are fully baffled including also surface baffles to prevent the surface aeration. The vessel dimensions are presented in Fig. 1. The experiments were carried out at laboratory temperature 22 ◦ C and ambient pressure. The gas feed varied from 0.1 to 0.9 vvm and was measured with a set of calibrated rotameters. The agitation speed varied in the range 340–700 rpm in the small and 155–500 rpm in the large vessel and was measured with an optical tachometer. Two chemical systems, air–tap water in both vessels and CO2 –n-butanol in the small vessel were investigated. 2.2. Photographing experiments Local BSDs were measured with digital photography through transparent vessel wall. The photographs were taken with a 6.5 Mpix Canon 300D digital camera, Canon 550EX flash, and Sigma 28–135 mm/f3.8–5.6 macro-objective. The 14 L stirred tank was placed into a rectangular container filled with water to avoid optical reflections from the rounded wall. The rectangular container was not used with 200 L vessel as the curvature of vessel wall was considered to be small enough. Due to dense dispersion, the photographing was possible only near the wall. Four measurement positions A–D were located between the bottom and surface in the mid-plane between baffles (Fig. 1). At the gas inlet, BSDs had to be investigated without mixing to get a disturbance-free optical access to the sparger.

M. Laakkonen et al. / Chemical Engineering Science 62 (2007) 721 – 740

All measurement positions (A–D, Fig. 1) were investigated only at few agitation conditions. In most operating conditions, local BSDs were analysed from one or two positions. The surface aeration was minimal due to surface baffling and gassing. The optics was adjusted to the smallest distance of focus, which produced a viewing area of 30 × 50 mm. The smallest detectable bubble size was 0.08 mm (∼ 5 pixels) with this viewing area. The exposure time of 1 ms was adequate to freeze the bubble motion. The aperture values varied from 10 to 13. According to the calibration experiments the DOF was 8 mm with the aperture of 9 and 9.5 mm with the aperture of 11. The biasing of BSDs due to the fact that large bubbles touch the DOF more probably than small bubbles (Chigier, 1991) was corrected by introducing scaling factors that consider the dependence of effective DOF on the measured bubble size (s + ai ) w(ai ) = NB , j =1 (s + aj )

(1)

where s is the DOF obtained from the calibration experiments. Bubbles were identified manually from the photographs. At first, a small region with 100–200 bubbles was selected from the centre of image. The bubbles in focus were marked with ellipsoids. After subtracting the background, image was thresholded. The particle analysis freeware ImageJ-1.32 was used to identify the minor and major axes of ellipsoids. Bubbles were assumed to be oblate ellipsoids. The diameter of a spherical bubble, which has a volume equal to the identified oblate ellipsoid, was selected to be the characteristic bubble size. Overall, 600–1000 bubbles were identified for each BSD. The identification was made at least from five randomly taken photographs to minimize transient effects in the measured BSD. The results were calibrated by placing a ruler to the measurement position. 2.3. Capillary experiments Capillary suction probe was used to investigate BSDs in location E (Fig. 1) from air–tap water dispersion in the 200 L tank. The experiments were carried out with a 1.2 mm capillary, which allowed the detection of 1.2–6 mm bubbles. The procedure of capillary experiments was presented in detail in our earlier study (Laakkonen et al., 2005b). 2.4. Gassed power consumption and gas holdup Gassed power consumption was needed as an input data for the modelling. The resulting model should also predict the overall gas holdup accurately. Gassed power consumption of mixing and overall gas holdup were measured from air–water dispersion in the 200 L tank. The power consumption of mixing was measured from torque on impeller. Overall gas holdup was investigated by measuring the difference between the gassed and ungassed height of dispersion. The liquid level was read from a submerged burette, which was

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connected to the vessel through a narrow pipe to smooth out the fluctuation of liquid surface inside the burette. 3. Modelling 3.1. The multiblock stirred tank model Multiblock (‘network of mixed zones’, ‘compartmental’) models have been used to investigate gas–liquid hydrodynamics in numerous studies including the authors’ earlier papers (Alopaeus et al., 1999; Laakkonen et al., 2006). In the multiblock modelling, agitated vessel is divided into a limited number of ideally mixed subregions, which are connected to each other. The main benefit of this approach is that local gas–liquid hydrodynamics can be modelled with a small computational effort. In some recent studies multiblock models have been incorporated to CFD simulators (Bezzo et al., 2003; Wells and Ray, 2005). Multiblock models were created for the 14 and 200 L laboratory stirred tanks. The vessel was divided into 21 subregions based on the analysis of CFD simulation results (Fig. 2). The model assumes rotational symmetry elsewhere but in the impeller region, where volume is divided into regions behind and in front of impeller blades. The division into subregions is based on the following criteria: • The subregions are arranged so that their number is minimal while still large enough to allow accurate description of gas and liquid flow patterns. • The inhomogeneity of dispersion should be minimal inside a subregion. • The variable gradients should be nearly constant along the subregion interfaces. • The model should be applicable for varying gas feeds. Liquid flow rates and local turbulent dissipation rates are needed as input data for the multiblock model. Due to lack of experimental data they were obtained from the CFD simulations. The subregions were defined as volumes and the interfaces were defined as planes in the CFX-5.7 post-processor for this purpose. A 180◦ segment of the vessel was simulated with CFX-5.7 by using a coarse and fine grid. The coarse mesh was equal-sized with 25 × 52 × 24 = 31 200 (radial × axial × rotational) volume elements. The fine grid included 39 × 79 × 36 = 110 916 volume elements. The impeller was modelled as a frozen assuming negligible interaction with baffles (multiple reference of frames method). All surfaces were modelled without thickness. The liquid surface was defined as frictionless and flat. The standard k-epsilon turbulence model was applied. Single-phase CFD simulations indicated that even the coarse grid can predict liquid flow fields fairly well, although turbulent energy dissipations and pumping capacity of impeller are smaller compared to the simulations with finer grid. CFD simulations showed that ungassed flow numbers (F ∗ = F /N/DI3 ) are independent on the stirring speed. This is expected as the vessels operate in the turbulent flow regime. The flow number of impeller was determined from the CFD results at distance

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Fig. 2. Subregions of multiblock model and the effect of gassing on liquid flow field in the 200 L tank. Dimensionless velocities UL /Utip and local flow numbers (Fij∗ ). Without gassing (left), Q = 0.3 vvm (middle), Q = 0.7 vvm (right).

0.05 · DI from the blade tip  +B/2   1.1 · DI −B/2 0 L UL,r d dz ∗ FI,CFD = , ND3

(2)

I

where B is the blade width. The predicted flow number was 0.53 with the coarse and 0.58 with the fine grid for water in the 200 L tank. These are notably smaller than 0.75 that has been found in experimental studies for Rushton turbines (Revill, 1982; Costes and Couderc, 1988; Ranade and Joshi, 1990). Ungassed flow numbers obtained from the CFD were hence scaled before passing them to the multiblock model. The scaling factor is Cflow =

0.75 , ∗ FI,CFD

(3)

where 0.75 is an experimental flow number for Rushton turbines (Revill, 1982). The change of flow fields due to gassing was considered in the multiblock model. The effects of gassing on the flow pattern were investigated by making gas–liquid CFD simulations. Details are given in Section 3.5. Liquid flow rates were determined similarly as from the single-phase simulations. After this, the change of flow pattern was considered by relating liquid flow numbers to the gas feed by Fij∗ ,g = Fij∗ ,u (1 + cij Q),

(4)

where Fij∗ ,u is local flow number without gassing. The parameters cij were adjusted by comparing to the CFD predictions at varying gas feeds. Flow numbers from Eq. (4) are constrained so that the sum of inflows is equal to the sum of outflows for each subregion (Fij∗ ,g = Fj∗i,g ). It is also required that the

decrease of impeller pumping capacity due to gassing is proportional to the power consumption mixing (Bakker and Van den Akker, 1994) Fij∗ ,g = Fij∗ ,u

Np,g . Np,u

(5)

Fig. 2 shows the predicted flow field of multiblock model at three gas feeds. The change due to gassing is most significant near the wall in the vicinity of liquid surface, where upward liquid flow turns earlier towards the centre of vessel at high gas feed. This agrees with experimental observations (Nienow et al., 1977; Khopkar et al., 2005). High flow numbers in the impeller region result mainly from the rotational flow between the subregions behind and in front of impeller blades. The transportation of bubbles between subregions consists of the convective and slip velocity Uk =

Fij ,g + Uslip,k , Aij

(6)

where Fij ,g is the volumetric liquid flow from subregion i to j and Aij is the flow area. Uslip,k is the bubble slip velocity, which is solved from the force balance on bubble motion as will be presented below. The rise of bubbles is assumed to cause a backward liquid flow of equivalent volume, which is summed to the mixing-induced liquid flow. Care is needed in the implementation of Eq. (6), because different sized bubbles can be transported to different directions. Local turbulence energy dissipations are obtained from the CFD by averaging over the volumes of subregions as  V L C  dV CFD,i =  i . (7)   dV V

L C

M. Laakkonen et al. / Chemical Engineering Science 62 (2007) 721 – 740

It has been observed that CFD and turbulence models underpredict turbulent energy dissipation (Gentric et al., 2005). Local dissipation rates from Eq. (7) are hence scaled with the measured power consumption of mixing based on the assumption that mixing energy converts to turbulent energy and dissipates to heat in the liquid phase. From the CFD results relative dissipation rates i for each subregion are obtained as CFD,i , (8) i = NB i=1 L,i CFD,i where L,i is a fraction of the total liquid mass in a subregion. Local dissipation rates due to mixing are related to the integral mixing energy mix,g , which is obtained from the gassed power consumption measurements mix,g,i = i mix,g .

(9)

Local dissipation rates are calculated by summing energy inputs due to mixing, rising bubbles and kinetic energy of gas injection as NC Yk |Fdrag,k ||Uslip,k | i,g = mix,g,i + k=1 C,i (1 − G,i ) +

2 Qi D,i Usparger,i

2Vi C,i (1 − G,i )

.

(10)

In an agitated vessel impeller motion is a major source of turbulent energy. The turbulence due to rising bubbles becomes significant at high gas feeds. The gas injection creates turbulence in a limited area near the sparger at high gas feeds. Local BSDs are modelled based on discretized PBs for bubbles (Ramkrishna, 2000). Bubble breakage, coalescence and slip are assumed to be the most important phenomena and are described by phenomenological models. 3.2. Bubble breakage Bubble breakage depends on the balance between external stresses that disrupt the bubble and surface/viscous stresses that resist the bubble deformation (Jakobsen et al., 2005). The breakage can be related to the arrival of turbulent eddies on the surface of bubble (Narsimhan et al., 1979; Lee et al., 1987a,b; Luo and Svendsen, 1996; Lehr et al., 2002; Hagesaether et al., 2002; Wang et al., 2003). The model of Luo and Svendsen (1996) has been the basis of many later works. Some of its shortcomings have been avoided by adding either capillary (Lehr et al., 2002) or kinetic energy density constraints (Hagesaether et al., 2002) or both of them (Wang et al., 2003). Especially kinetic energy density constraint increases the complexity of model and, hence, the computational cost. Computation times become a critical issue in the CFD calculations, where high number of closure model calls is needed to obtain a converged solution. It has been noted in many studies that high CPU demand limits the combination of CFD and PBs at the moment. There is need for simple while still accurate breakage/coalescence closures. Narsimhan et al. (1979) developed a model based on the assumption that arrival of eddies on the surface of bubble is a Poisson process. This means that breakage events are independent

725

on the history, i.e., they occur instantly. Risso and Fabre (1998) confirmed experimentally that instant bubble breakup occurs most frequently. Lee et al. (1987a) developed a model for the bubble breakage based on the ideas of Narsimhan et al. (1979), Alopaeus et al. (2002) extended the model of Narsimhan et al. (1979) by including viscous stresses as a resisting force for the breakage of droplet ⎛

 g(aj ) = C1 1/3 erfc ⎝ C2

C 2/3 aj5/3

+ C3 √

C

C D 1/3 aj4/3

⎞ ⎠.

(11) The assumptions of Eq. (11) should be valid for gas–liquid systems, because the mechanisms of bubble and droplet breakage are similar (Hesketh et al., 1991). The dispersed phase viscosity is replaced with the viscosity of continuous phase in the viscous term of Eq. (11) by assuming that viscous stresses that resist the breakage are proportional to the viscosity of liquid surrounding the bubble rather than viscosity of gas, which is small. The observations of Walter and Blanch (1986) support this assumption. Besides Eq. (11) daughter size distribution (DSD) is needed to calculate breakage rates from a category to another. Both phenomenological and purely mathematical relations have been used. Many physical models have been suggested. Due to lack of reliable experimental information, there is no general agreement which one is the most accurate one. Some physical models predict preferably equal-sized breakage (Martínez-Bazán et al., 1999) while some others predict non-equal sized breakage (Wang et al., 2003). The DSD may also depend on the size of mother bubble, physical properties and turbulence energy dissipation (Lehr et al., 2002; Wang et al., 2003), but these dependences need further experimental validation. Binary breakage is a common assumption in the breakage models (Wang et al., 2005; Kostoglou and Karabelas, 2005). The experiments of Risso and Fabre (1998) showed that binary breakage occurs most frequently (48% of the events), but 3–7 fragments are evolved in 37% and even more than 10 fragments in 15% of the events. These observations can be interpreted so that multibreakage events (more than 3 fragments) are a major source of bubbles born by breaking. Lee et al. (1987b) investigated BSDs from air–tap and air–salt water systems in airlift column and found best agreement to the experiments by using a multibreakage assumption. They suggested that turbulent eddies seldom provide the exact minimum work against the surface tension required by breakage. It was found that DSD influences strongly on the steady-state BSD. The DSD is calculated from the following mathematical relation: 1 (ai , aj ) = (1 + C4 )(2 + C4 )(3 + C4 )(4 + C4 ) 2 2

C4 ai2 ai3 ai3 × 1− 3 , aj3 aj3 aj

(12)

where C4 is an adjustable parameter. The integration of Eq. (12) from zero to the size of mother bubble results into 43 + C4 /3

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daughter bubbles. The breakage is binary when C4 is 2.0. The C4 –dependent scaling factor of Eq. (12) may seem peculiar but results analytically from the requirement that gas volume must conserve in breakage, i.e.,  aj (a, aj )a 3 da = aj3 . (13) 0

DSDs similar to Eq. (12) have been applied to droplets (Hsia and Tavlarides, 1983; Alopaeus et al., 2002) and bubbles (Lee et al., 1987b). Depending on the adjustable parameter C4 unequal or equal-sized breakage can be described. The fitting can then give information about the DSD. Eq. (12) predicts zero probability as the daughter bubble size approaches zero. In contrast to some more elaborated models (Hagesaether et al., 2002; Wang et al., 2003) it needs no numerical evaluation of double or triple integrals. Eq. (12) is also independent on the hydrodynamics and physical properties. In consequence, it can be calculated beforehand and needs not to be updated during the simulation. This can produce substantial savings in computation times. Lehr et al. (2002) developed a breakage model which has no adjustable parameters. It is tested as an alternative to the above-presented model. According to Wang et al. (2005) both the model of Lehr et al. (2002) and a more complicated model of Wang et al. (2003) predict BSDs successfully in bubble columns. The breakage frequencies are calculated in Lehr’s model from

5/3 7/5 √ ai3 6/5 9/5 ai C 19/15 C g(ai ) = 0.5 · . (14) exp − 2

7/5

9/5 The DSD is obtained from ⎧ 3/5 exp(− 49 [ln(22/5 (ai C 2/5 / 3/5 ))]2 ) 6 ⎪ ⎪ ⎪ , ⎪ 3/2 3 ⎪ 2/5 / 3/5 ))]  ai 1 + erf[ 3 ln(21/15 (aj 3/5 ⎪  ⎪ C 2 ⎨ a3 v (ai , aj ) = 3 j , 0 a ⎪ i ⎪ ⎪ 2 ⎪ ⎪ 3 ⎪ ⎪ ⎩ (a 3 − a 3 , a 3 ), aj a 3 a 3 , v j i j i j 2 (15) which assumes binary breakage. The DSD depends on rate of turbulent energy dissipation, physical properties and size of mother bubble. This also means that it must be calculated for each location of vessel and needs to be updated during the simulation. 3.3. Bubble coalescence The coalescence is modelled based on the bubble collision frequency and the efficiency of coalescence based on a macroscopic approach (Jakobsen et al., 2005). The collision frequency is a product of collision cross-sectional area and relative velocity between bubbles. At extremely low gas fractions, the length scale of bubble fluctuations due to turbulence may be smaller than is the free distance between bubbles. Based on this Wang et al. (2005) introduced a correction to the coalescence frequency. The need for this correction is not, however, evident

as the collision frequencies are related to local gas fractions through bubble number density. Numerical tests showed that turbulence is the dominant driving force for collisions in an agitated vessel. The incorporation of buoyant collisions (Prince and Blanch, 1990) increased local Sauter mean bubble diameter approximately 10%. This should be the maximum effect, because the efficiency of buoyant collisions was set to unity in these simulations. Actual efficiencies are apparently lower, but their evaluation would require a new closure model. Buoyant collisions are hence neglected. This assumption has been made also in most bubble column studies (Jakobsen et al., 2005) even though turbulent collisions are less dominating in bubble columns than in agitated vessels. Turbulent collision frequencies are calculated based on the kinetic gas theory from the model of Coulaloglou and Tavlarides (1977) with a small algebraic correction (Prince and Blanch, 1990; Alopaeus et al., 1999) 2/3

h(ai , aj ) = C5 · 1/3 (ai + aj )2 (ai

2/3

+ aj )1/2 (ai , aj ), (16)

where theoretical values of C5 vary from 0.28 to 1.11 depending on the effective collision cross-sectional area and the expression for turbulent fluctuation velocities (Jakobsen et al., 2005). Coalescence efficiency (ai , aj ) is described based on film drainage. The coalescence occurs if collided bubbles remain in contact for sufficient time so that the liquid film between them drains out until a critical film thickness for rupture is reached. The film drainage depends on the mobility of bubble surface and the approach velocity. Prince and Blanch (1990) developed the following model assuming that inertial and surface tension forces control the drainage of a fully mobile bubble surface:

1/2 C 1/3

(ai , aj ) = exp −C6 , (17) (1/ai + 1/aj )5/6 1/2 where initial and critical film thicknesses are lumped to C6 , which is of magnitude 2.3 (Prince and Blanch, 1990). The model should be applicable for low viscosity liquids such as tap water and n-butanol used in the present study. Many coalescence efficiency models predict a similar decreasing trend of coalescence efficiency with increasing bubble size. This trend can be modified altering the value of C6 . Lehr et al. (2002) investigated bubble collisions with a highspeed camera and obtained the following model for coalescence rates:   m h(ai , aj ) = (ai + aj )2 min u , 0.08 4 s ⎛

2 ⎞  1/3  G,max × exp ⎝− (18) − 1 ⎠, G where G,max = 0.6. The model assumes that only gentle collisions lead to coalescence. For the air bubbles in distilled water a critical approach velocity of 0.08 m/s was obtained from the measurements. The critical velocity was suggested to be constant and independent of the physical properties for pure liquids. In liquids with additives or in liquid mixtures critical velocity was expected to be lower than in distilled water. The velocity

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difference between colliding bubbles in Eq. (18) is calculated from  √ 2/3 2/3 u = max( 21/3 ai + aj , |U slip,i − U slip,j |), (19) which includes the effect of both turbulent and buoyant collisions. 3.4. Bubble slip velocity The relative velocities (slip) between bubbles and liquid are solved from the force balance on bubble motion, where hydrostatic and dynamic pressure gradients are the driving forces and drag is the resisting force (Ranade, 2002):   ∇p 1 − g zˆ = ah CD C U slip,k |U slip,k |. (20) Vb (D − C ) C 2 Dynamic pressure gradients at subregion interfaces are obtained from the CFD. In the multiblock model they are scaled with the second power of stirring speed according to mechanical energy balance ∇p∗ = ∇p/N 2 .

(21)

CFD simulations at varying stirring speeds showed that these scaled pressure gradients are relatively independent on the agitation speed. As a result of dynamic pressure variation gas accumulates to the low-pressure regions such as behind the impeller blades and in the middle of upper circulation loop. According to the analysis of Khopkar et al. (2005) other forces such as lift force and virtual mass force can be neglected in the case of agitated vessel. An empirical correlation is needed for bubble drag coefficients. Correlations have been developed mostly for individual bubbles in stagnant liquids. Drag coefficients depend also on swarm (Ishii and Zuber, 1979) and turbulence (Poorte and Biesheuvel, 2002) effects. The measurement of these effects is difficult due to bubble breakage and coalescence. Ishii and Zuber (1979) developed a correction, which predicts the increase of drag for small bubbles and the reduction of drag for large bubbles with increasing gas fraction. Test simulations using this correlation, however, showed no significant difference compared to the correlations for isolated bubbles. A reason is that swarm effects become significant only at relatively high gas fractions (> 0.2). The correlation of Tomiyama (1998) for isolated bubbles in slightly contaminated liquids is used     24 72 8 Eo CD = max min (1 + 0.15Re0.687 ), , . Re Re 3 Eo + 4 (22) Many experimental (Brucato et al., 1998; Poorte and Biesheuvel, 2002) and modelling studies (Bakker and Van den Akker, 1994; Spelt and Biesheuvel, 1997; Alves et al., 2002; Lane et al., 2005; Khopkar et al., 2005) have shown the need of turbulent drag correction. For solid particles settling velocities have been observed to reduce as low as 15% of the settling velocity in stagnant liquids

727

(Brucato et al., 1998). In bubbly flows, rise velocities may be 35% of those in stagnant liquids (Poorte and Biesheuvel, 2002). It therefore seems that turbulent drag correction is needed for the modelling of agitated gas–liquid reactors. Brucato et al. (1998) developed an empirical correction for turbulent drag based on particle settling measurements. Khopkar et al. (2005) modified it for the gas–liquid stirred tank simulations. Lane et al. (2005) proposed that turbulent drag could be related to the ratio of bubble/particle relaxation and the integral turbulent time scale, but calculation procedure for these time scales was not presented. The correction of Bakker and Van den Akker (1994) based on effective viscosity concept is adopted with a slight modification similarly as in the authors’ earlier study (Laakkonen et al., 2006). From the definition of kinetic energy concentration k = 0.5 · u 2 , and the relation of turbulent fluctuation velocity u = 1.4 · 1/3 · a 1/3 the following expression is obtained for the effective viscosity:  = C + C7 C 1/3 a 4/3 ,

(23)

which combines the molecular viscosity and a term proportional to turbulent viscosity. Bubble drag coefficients are calculated from Eq. (22) by using the effective viscosity. Adjustable parameter C7 is of magnitude 0.02 (Bakker and Van den Akker, 1994). The predicted turbulence dampening of slip is significant especially for bubbles smaller than 5 mm. This seems reasonable, because large bubbles fluctuate even in stagnant liquids so that turbulent fluctuations should have small additional effect on their drag. It is noted that rather than comparing simulated and measured gas volume fractions from agitated vessels, turbulent drag closures should be developed based on slip velocity measurements from turbulent flows (Poorte and Biesheuvel, 2002). 3.5. Gas–liquid CFD simulations Bubble breakage and coalescence models were incorporated to MUSIG PB model in CFX-5.7 through user routines. The MUSIG model (Lo, 2000) is a framework in which PBs are solved simultaneously with Navier–Stokes equations for Eulerian gas and liquid phases. The major assumption of MUSIG model is that all bubble size groups share a common velocity field. The drag on bubbles is calculated based on local Sauter mean bubble diameter. The CFD model was used to predict local BSDs in the laboratory stirred tanks. The investigated operating conditions were 0.7 vvm/300 rpm, 0.7 vvm/390 rpm and 0.3 vvm/390 rpm for the 200 L tank. In the case of 14 L tank, air–water and CO2 –n-butanol dispersions were investigated at operating condition 0.7 vvm/700 rpm. The use of coarse grid (∼ 31 000 elements) was accepted due to high computational cost and small difference between the simulations with coarse and fine grid (∼ 111 000). Local dissipation rates of turbulent energy were scaled with the ratio of measured and predicted gassed power consumption before passing them to the breakage and coalescence models. This avoids the underestimation of turbulent dissipation and should minimize the dependence of results on the

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simulation grid. Bubble drag coefficients were calculated from Eq. (22) with the turbulence correction of drag (Eq. (23)). Liquid surface was defined as a degassing boundary. The maximum number of bubble size discretization categories was limited to 20 in the MUSIG model. A geometric, mass based discretization of bubble size was used according to 3 3 − amin 3 amax . (24) 4 2NC−i Numerical tests showed that with 20 size classes and the discretization from Eq. (24) arithmetic mean diameter (a10 ) is overpredicted by ∼ 1.6% and Sauter mean diameter (a32 ) by ∼ 4% compared to the accurate solution with 400 classes. With 85 size classes, which was used in all multiblock simulations, a10 and a32 were overpredicted approximately 0.5% compared to the accurate solution. The largest bubble size was set to 10 and 15 mm in the CFD simulation of 14 and 200 L vessels, respectively, to minimize the discretization error, although up to 25 mm bubbles were observed at gas inlet. In the multiblock model simulations, denser discretization with more size categories for small bubbles was used according to the formulas presented in the authors’ previous work (Laakkonen et al., 2006).

3 + ai3 = amin

The fitting was made at two stages. Initially all parameters C1 –C7 were included in the fitting. Based on the preliminary analysis the correlating parameters were fixed to reasonable values to decrease the parameter space. The remaining parameters were fitted using the Levenberg–Marquardt (LM) algorithm. The simulated and measured local mean diameters, number and volume BSDs were compared by means of least squares. All measured BSDs from different positions of vessel in the air–water and CO2 –n-butanol systems at all investigated agitation conditions were included simultaneously to the fitting. The multiblock stirred tank model was solved numerically for each experiment within each LM step. The dynamic balance equations were solved to steady state with a fourth order Runge–Kutta ODE solver. The integration was continued for each operating condition, until relative residuals of population and mass balances were smaller than 0.01%. The overall number of vessel operating conditions in the fitting was 19 from which 11 were air–water and 8 CO2 –n-butanol experiments. The overall number of local BSDs was 35 from which 20 were from the air–water and 15 from the CO2 –n-butanol dispersion.

5. Results and discussion

4. Parameter fitting

5.1. Bubble size experiments

The unknown parameters in the bubble breakage and coalescence models were adjusted by comparing the measured and predicted local BSDs. The influence of vessel geometry on the adjusted parameter values should be avoided, because multiblock model considers the inhomogeneity of dispersion. Bubble breakage, coalescence and turbulent drag are related to local gas–liquid hydrodynamics and physical properties and there are no vessel geometry dependent parameters in them. Therefore these parameters should be global and applicable to an arbitrary vessel size or geometry. The multiblock model was used in the parameter fitting instead of CFD, because it is computationally less expensive. The flow rates in the multiblock model were obtained initially from the single-phase phase CFD simulations but they were updated according to the procedure in Section 3.1 based on gas–liquid CFD simulations with MUSIG. The final multiblock simulation results were calculated with the updated flow fields (Fig. 2). Bubble breakage rates were calculated from Eqs. (11) and (12), coalescence rates from Eqs. (16) and (17) and drag from Eq. (22) including the turbulence correction (Eq. (23)). Gas was injected below the impeller (Fig. 2). The measured surface tension against air was 0.069 N/m for water and 0.023 N/m for n-butanol. Available experimental facilities allowed the measurement of surface tension for air–n-butanol interface only. This may deviate from the surface tension for CO2 –n-butanol interface but was accepted, because the solubility of CO2 to nbutanol is fairly low and the surface tension depends mostly on liquid properties at low pressures (Poling et al., 2000). The densities and viscosities were obtained from the physical property databases of Flowbat flowsheet simulator program (Aittamaa and Keskinen, 2005).

The resolution of camera limited the minimum detectable bubble size to 0.08 mm. Smaller than 0.08 bubbles probably existed (Machon et al., 1997; Laakkonen et al., 2005b), but they are expected to include a minor fraction of the gas–liquid interfacial area and gas volume. Only bubbles with sharp edges were identified from the photographs, because the blurred ‘halo’ effect causes the biasing of bubble size (Chigier, 1991). The DOF correction based on Eq. (1) shifted Sauter mean bubble diameters by few percent towards small bubble size. The correction is small due to relatively large DOF (∼ 9 mm). Bubbles were identified from the photographs as oblate ellipsoids. For bubbles smaller than 4 mm this is a good assumption. Larger bubbles were more distorted, but due to manual analysis it was possible to identify them as equivalent ellipsoidal bubbles. Bubbles larger than 7 mm were observed rarely in the samples of 600–1000 bubbles, but in a large set of photographs even 10 mm bubbles were observed in the air–water system at high gas feeds. Due to dense dispersion camera lens had to be focused to the vessel wall to get sharp images. This may have caused a slight biasing of BSDs, because the largest bubbles may not touch the DOF due to wall effects. The measured number BSDs were strongly skewed towards small bubble size in both investigated systems in all measurement positions. Arithmetic mean bubble diameters (a10 ) varied from 0.2 to 0.9 mm being on average smaller in the CO2 –nbutanol than in the air–water dispersion. The majority of gas volume was in 1–5 mm sized bubbles. Sauter mean bubble size a32 varied between 1.2 and 4.1 mm in the air–water and between 0.8 and 2.6 mm in the CO2 –n-butanol system. Lower surface tension and higher liquid viscosity apparently explain the smaller bubble diameters in the CO2 –n-butanol dispersion.

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Local volumetric BSDs showed systematic trends with the measurement position, stirring speed and gas feed. In the air–water system, bubbles were generally smallest in the impeller discharge flow (location B, Fig. 1), where high turbulent energy dissipation favours the breakage. Bubbles were largest in the positions A and E, where they were trapped by the downward liquid flow. In the air–water dispersion bubble size increased from the impeller plane towards the surface (from location C to D) in both 14 and 200 L vessels while in the CO2 –n-butanol dispersion bubble size decreases slightly from positions C to D. A similar observation was made earlier in the capillary probe experiments from the CO2 –n-butanol dispersion at low gas feeds (Laakkonen et al., 2005b). Machon et al. (1997) and Schäfer et al. (2000) have made similar observations. It is possible that large bubbles in the impeller discharge flow originate from the gas vortices behind impeller blades or from the gas sparger. Near the wall in the vicinity of liquid surface gas fractions are low, because large bubbles move towards impeller shaft (Khopkar et al., 2005). This apparently increased the proportion of small bubbles in secondary liquid circulation loops near the wall (Nienow et al., 1977).

5.2. BSDs at gas inlet The BSDs at gas inlet were needed for the stirred tank simulations. The photographing technique was used, because the use of capillary suction probe was not possible due to high gas fraction. The experiments had to be carried out without agitation to get a disturbance-free optical access to the gas sparger. The agitation may have influence on the feed size distribution, because transverse/co-current flow of liquid eases the detachment of bubbles from the sparger (Kulkarni and Joshi, 2005). It is noted that high gas velocities, which varied from few metres per second up to 77 m/s, generated liquid flow around the sparger even without agitation. The measured mean bubble diameters are compared to the correlation of Geary and Rice (1991) in Fig. 3. The agreement is good, although gas jetting made the accurate identification of bubbles from the photographs difficult at high gas feeds. Small bubbles were observed at high gas feeds, but they most probably resulted from the breakage after the injection. This motivated to incorporate energy dissipation term due to gas injection in Eq. (10). The measured BSDs were close to a normal distribution. The standard deviation followed approximately the dependence 0.16 · a32 . The measurement of BSDs at gas inlet would have been laborious at all gassing rates. The BSDs at gas inlet were approximated from the normal distribution with a mean bubble size from the correlation of Geary and Rice (1991) and the standard deviation of 0.16 · a32 .

5.3. Gassed power consumption of mixing Gassed power consumption of mixing was needed to update liquid flow rates (Eq. (5)) and turbulent dissipations (Eq. (10)). The power consumption measurements from the 200 L vessel

Fig. 3. Measured (markers) and predicted (Geary and Rice, 1991) mean bubble diameters at gas inlet at varying gas injection speeds.

were correlated by (Hassan and Robinson, 1977) Pg = Cp W e−0.25 Fl−0.38 , Pu

(25)

where the best value for parameter Cp was found to be 1.03 as a result of fitting to the experiments. From the 14 L vessel, gassed power consumption was not measured, but the predicted power numbers from Eq. (25) seemed to be too high at low gassing rates. Therefore, another literature correlation was preferred. Midoux and Charpentier (1984) made a literature survey of over 40 studies and obtained

0.45  (Pu /W)2 (N/s−1 )(DI /m)3 Pg /W = 0.34 Np,u . (26) (Q/m3 /s)0.56 The measured power number of 5.8 for the ungassed liquid in the 200 L vessel was used in Eq. (26). 5.4. The adjusted model parameters The measured BSDs were time-averaged so that the fitting of parameters in the breakage and coalescence models relies on the spatial inhomogeneity of BSDs, varying vessel operating conditions and physical properties of dispersion. The fitting of seven unknown parameters C1 –C7 simultaneously was not feasible. The parameters C2 and C3 were included in the fitting initially, but due to correlation with C1 it was reasonable to fix them. At the final stage of fitting it was checked that modifying C2 or C3 did not improve predictions. The parameter C2 that considers the surface tension as a resisting force for breakage was set to 0.04. The effect of viscous forces on bubble breakage was considered qualitatively

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Table 1 The model parameters Breakage, Eqs. (11) and (12)

C1 = 2.52 C2 = 0.04a C3 = 0.01a C4 = 18.25

Coalescence, Eqs. (16) and (17)

C5 = 2.65 C6 = 5.17

Turbulence dampening of slip, Eq. (23)

C7 = 0.06a

a Not

fitted at the final stage.

setting C2 to 0.01. The validation of this dependence was not possible due to low liquid viscosities (water 1 mPa s, n-butanol 2.7 mPa s). It is noted that C2 and C3 must also have such values that surface tension and viscosity terms have notable effects on the complementary error function (Eq. (11)). The fitting was further simplified by adjusting the turbulence dampening of slip parameter C7 (Eq. (23)) independently. At the final stage, it was checked that C7 had such value that the simulated and measured gas volume fractions matched. The resulting parameter values are presented in Table 1. The parity plots of local arithmetic and Sauter mean bubble diameters are shown in Figs. 4a and b. The adjusted model describes the magnitude of arithmetic mean diameters successfully. The simulated and measured local Sauter mean diameters are in better agreement with mostly smaller than ±20% deviation (Fig. 4b). The fitting shows similarly as the measurements that bubbles are smaller in the CO2 –n-butanol than in the air–water dispersion. The adjusted parameters deviate from those in the authors’ earlier study (Laakkonen et al., 2006). This shows that available models are not fully predictive at the moment. It is noted that the set of models is different than in our earlier study. The parameters may be correlated, because it is difficult to make experiments, which allow their identification accurately. The breakage and coalescence rates can be separated only if transient or local BSDs from varying vessel-operating conditions are included in the fitting. The present fitting based on multiblock modelling is an improvement in that direction. It is noted that experimental data are included from two chemical systems and a wide range of hydrodynamic conditions allowing more accurate parameter identification. The fitting also covers a range of hydrodynamic conditions in which many industrial gas–liquid reactors operate. In addition, the model is adjusted to describe both number and volume BSDs. This issue has been neglected in most studies by comparing number BSDs or Sauter mean diameters only. The adjusted collision rate constant C5 = 2.65 is larger compared to the theoretical values 0.28–1.11 (Jakobsen et al., 2005). This may indicate that assumptions of kinetic gas theory or turbulent theory are uncertain for bubbles. The actual diameter of collision tube may be larger than ai + aj , because large bubbles are non-spherical and fluctuate even under quiescent conditions. Another reason could be that adjusted parameters C5 and C6 are correlated. Larger coalescence rates are balanced by rather low coalescence efficiencies with C6 = 5.17 in

Eq. (16). Laari and Turunen (2005) adjusted coalescence efficiency parameters based on bubble persistence time and bubble column experiments. Their results showed similarly as the present fitting that C6 should be much larger than 2.3, which was suggested by Prince and Blanch (1990). It is noted that bubble collision times are approximated in the model of Prince and Blanch (1990) based on purely dimensional considerations. The model of Alopaeus et al. (2002) (Eq. (11)) predicts that breakage rates approach asymptotically a constant value while the models (Hagesaether et al., 2002; Lehr et al., 2002; Wang et al., 2003) originating from the work of Luo and Svendsen (1996) predict more closely exponential increase of breakage rates with increasing bubble size. The model of Alopaeus et al. (2002) showed to be more suitable for predicting a wide range of bubble sizes, which was observed in the experiments. It is noted that the comparison of breakage models is not straightforward, because breakage rates and hence the BSDs are related strongly to DSD (Lee et al., 1987b). The fitting of Eq. (12) indicates that bubbles break into 7.4 unequal-sized fragments. This shows some agreement to the measurements of Risso and Fabre (1998). Multiblock and CFD predictions by using the adjusted model are compared next to the measurements. Due to high computational cost CFD simulations were made at few operating conditions. The multiblock model was solved within few minutes while one CFD simulation took approximately 48 h CPU time. This highlights the potential of multiblock model as an efficient tool for the testing and validation of PB closures. 5.5. Gas volume fractions in the 200 L tank The adjusted turbulence dampening parameter of slip C7 is larger (Table 1) than 0.02 used by Bakker and Van den Akker (1994) but still smaller than the proposed maximum value C =0.07 in the k-epsilon turbulence model. Apparently, the adjusted C7 considers not only the turbulence dampening of slip, but also compensates the uncertainties of flow field modelling. Turbulence dampening correction was necessary, because neglecting it would have roughly halved the overall gas fraction compared to the measurements. The overall gas fraction must be predicted accurately, because gas fractions and BSDs are related complicatedly to each other through bubble slip, coalescence rates and turbulence. Figs. 5a and b present the comparison of fitted model to the measurements from the 200 L vessel under varying agitation conditions. For reference, the correlation of Calderbank (1958) is included to the comparison  G =

U s G Ut

0.5 + 0.000216 · 

0.4

  0.6  U 0.5 C

s



Ut

,

(27)

where all values are in SI units and Ut is 0.265 m/s. The comparison shows that measurements, multiblock and CFD simulations are in the same range being also in agreement to Calderbank’s correlation except at high mixing intensities. Even the ideal mixing assumption with the adjusted model works well producing slightly higher values compared to the multi-

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Fig. 4. The predicted (multiblock model) vs. measured local arithmetic (left) and Sauter mean bubble diameters (right). Measurement locations • A,  B,  C,  D,  E, air–water dispersion with closed markers and CO2 –n-butanol with open markers.

Fig. 5. Overall gas holdup vs. (a) mixing intensity, (b) gas feed in the 200 L tank.

block predictions. A reason is that overall gas faction is related strongly to the bubble rise velocities. In ideal mixing case, turbulence dampening of slip is calculated by using the vesselaveraged dissipation rate while in the multiblock model rise velocities are calculated based on dissipation rates in surface blocks. Turbulent energy dissipations are smaller near the liquid surface than vessel-averaged dissipations, which results in larger bubble rise velocities and lower gas volume fractions. An example of local gas volume fractions in the 200 L tank based on multiblock and CFD model is presented in Fig. 6. The averaged CFD results for each subregion are compared to the multiblock simulations. Both modelling approaches predict largest gas fraction near the sparger. Gas accumulates to the region near the wall in the downward liquid flow and to the centre of upper liquid circulation loop. The smallest gas fractions are

found below the impeller in the middle and close to the bottom. In the impeller discharge flow, gas holdup decreases towards the wall. The simulations also showed that gas becomes distributed more uniformly at high stirring speed and at low gas feed, as expected. The simulated trends of gas distribution are very similar to the observations of Barigou and Greaves (1996) with a conductivity probe technique. It is interesting to note that the multiblock model, which is computationally less intensive, predicts very similar distribution of gas as CFD. There are some differences as well. The multiblock simulations predict larger gas fraction near the wall in the vicinity of surface and near the gas sparger than CFD. CFD predictions agree better with visual observations, namely, that rising bubbles move towards the centre of vessel. Above the impeller and below the gas sparger multiblock simulations

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Fig. 6. Predicted local gas fractions (vol%) in the air–water dispersion, N = 300 rpm, Q = 0.7 vvm, 200 L tank (multiblock simulation left, CFD right). Numerical values: multiblock (parentheses) and CFD (brackets), overall gas fraction: experimental 4.5 vol%, multiblock 4.1 vol%, CFD 3.8 vol%. *Behind; # in front of impeller blade.

predict lower gas fraction than CFD. In the CFD model the gas sparger is located closer to the bottom of tank. Due to low number of subregions and the fact that most of the gas rises directly to the impeller it was more realistic to feed the gas in the upper subregion in the multiblock model. 5.6. Local BSDs in the 200 L tank The multiblock and CFD simulations are compared to the measured local Sauter mean bubble diameters from air–tap water system in the 200 L tank in Fig. 7. The block averaged CFD results are compared to the multiblock simulations. The local trends of multiblock and CFD model are similar. The largest bubbles are found above the gas sparger. Bubbles rise from the sparger to the impeller. In the impeller region gas fractions are relatively high. As a consequence, bubble size decreases due to breakage not until in the impeller discharge flow. The smallest bubbles are found below the impeller near the bottom of tank. Above the impeller bubble size increases from the wall towards impeller shaft. This agrees with the capillary measurement. Near the impeller shaft and near the wall below the impeller bubbles are trapped in the downward liquid flow generating high local gas fractions that favour coalescence. The simulated trends of local bubble size variation in Fig. 7 agree with the observations of Barigou and Greaves (1992), who measured bubble sizes from a Rushton turbine agitated vessel (T = 1 m) with a capillary suction probe. Gas feed

and mixing intensity varied in the same range as in the present work. The Sauter mean bubble diameters varied generally in the range 3–4.5 mm above the impeller, but the actual variation may have been larger due to limited detectable size range of capillary. In the impeller discharge flow Barigou and Greaves (1992) observed smaller bubbles compared to the predictions of adjusted model. It is noted that the sampling of large bubbles into capillary is more difficult in that region compared to more quiescent regions of vessel, because breakage occurs more easily at the nose of capillary in intense liquid flow. There is some difference between multiblock and CFD simulations in Fig. 7. The CFD results indicate that bubble size first increases and then decreases near the wall from the impeller plane towards the surface. This disagrees with the multiblock simulations and experiments. The difference apparently results from the underestimation of liquid flow rates and the assumption of common velocity field for different sized bubbles in the CFD model. Local volume BSDs simulated by using the adjusted model and the closure models of Lehr et al. (2002) (breakage: Eqs. (14), (15); coalescence: Eqs. (18), (19)) are compared to measurements from air–water system in Fig. 8. Turbulence correction of slip (Eq. (23)) was included in the simulations with Lehr’s model. Due to smaller bubble sizes turbulent drag parameter C7 had to be set to 0.038 to these simulations to predict reasonable gas holdups compared to the measurements. Lehr’s model is included in the comparison because it has been

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Fig. 7. Local Sauter mean bubble diameters a32 in the air–water dispersion, N = 390 rpm, Q = 0.7 vvm, 200 L tank. (multiblock simulation left, CFD right), Numerical values in millimetres: measured (bold), multiblock (parentheses), CFD (brackets), *Behind, # in front of impeller blade.

successful in bubble column studies (Lehr et al., 2002; Wang et al., 2005). This allows the evaluation of how phenomenological breakage and coalescence closures can be transferred from a flow environment to another, which should be possible if the models are truly fundamental. The comparison in Fig. 8 shows that Lehr’s model predicts small bubble sizes compared to the present measurements elsewhere but in location E. As a result fitting, the present model shows better agreement. This indicates that care is needed, when breakage and coalescence models are applied to a new flow environment. In location E, the present fitting predicts larger size range of bubbles compared to the capillary measurement. It is noted that only bubbles from approximately 1.2–6 mm can be detected with a capillary. This means that actual BSD may be wider than the measurement indicates. The adjusted model predicts larger tails in the BSDs near the wall compared to the photographing measurement. The reason for these tails is that some bubbles pass the impeller region without breaking. Actually, the photographing measurement from location B (Fig. 8) shows some bimodality as well. Up to 10 mm bubbles were found, when large number of photographs was checked. As the proportion of tiny small bubbles was extremely high in the investigated systems, the analysis of 600–1000 bubbles for one BSD is not statistically significant for large bubbles. Even one 10 mm bubble influences significantly the volume BSD. It is also noted that due to dense dispersion the camera lens had to be focused on the wall of the vessel. The amount of large bubbles may be smaller near the

wall due to wall effects. The CFD simulations show larger tails in the volume BSDs than multiblock model. The use of only 20 size classes and the uncertainties of flow field prediction are apparent reasons for this. Testing showed that with 20 classes bubble size is overpredicted by few percent compared to the accurate solution with 400 classes. 5.7. Local BSDs in the 14 L tank The adjusted and measured local BSDs and mean bubble diameters from air–water and CO2 –n-butanol dispersions in 14 L tank are compared in Figs. 9 and 10. The BSDs and arithmetic mean bubble diameters from the CFD are from exact experimental locations (Fig. 1) while Sauter mean bubble diameters are averaged for subregion volumes to make a comparison to the multiblock simulations. The volume BSD at gas inlet is presented to show that breakage and coalescence control local BSDs in the bulk region of vessel. The adjusted model describes strong skewing of number BSDs towards small bubble size successfully. A majority of gas volume in larger bubbles produce more symmetrical volume BSDs. Takahashi and Nienow (1992) investigated local BSDs from lean air–water dispersion with photography and observed similarly that number BSDs are skewed heavily towards small bubble size. In their experiments arithmetic mean bubble diameter varied from 0.5 to 1.0 mm while Sauter mean diameters varied from 0.5 mm in the impeller up to 4.7 mm in the bulk region above the impeller. Also Machon et al. (1997) observed

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experiments due to reasons discussed in the previous section. The deficiencies of closure models, flow field or primary bubble size estimation are other possible causes, but specific reason cannot be found. Lehr’s model predicts again narrow size distributions with too few small and too few large bubbles compared to the measurements especially in the case of air–water dispersion. The comparison of Figs. 9 and 10 shows that bubbles are smaller in the CO2 –n-butanol than air–water dispersion at corresponding vessel-operating conditions. This agrees with the measurements. As a result of smaller bubble size and higher liquid viscosity (2.7 mPa s) slip velocities are smaller, dispersion is more homogeneous and gas fractions are larger in the CO2 –n-butanol than air–water system. The predicted inhomogeneities of bubble size in the 14 L tank are similar as in the 200 L tank (Fig. 7). There is some difference between multiblock and CFD simulations. The multiblock simulations show that bubble size increases from the impeller plane (location b) towards the surface in both investigated systems while CFD predicts a maximum in location c. The multiblock simulations are in better agreement with the measurements from air–water dispersion than CFD. In the case of CO2 –n-butanol system, CFD shows better agreement with the measurements. 5.8. Comparison to earlier experimental studies An independent comparison to earlier experimental bubble size studies from agitated tanks is made to evaluate the predictive capabilities of the adjusted model. The comparison is extended outside the range of hydrodynamic conditions investigated in the present experiments. The model of Lehr et al. (2002) is included in the comparison. Fig. 8. Local volumetric bubble size distributions. Photographing (a–d) and capillary (e) measurement (markers), multiblock (solid line) and CFD (dashed line) predictions using the present model, multiblock prediction using the model of Lehr et al. (2002) (dotted line), Sauter mean diameters a32 (mm): measured (bold), multiblock (parentheses), CFD (brackets), Lehr et al. (underlined).

that number probability density has a pronounced peak of small bubbles with a long tail of large bubbles in dense air–water dispersion (Q= ∼ 1.1 vvm). Bubble sizes varied in their experiments in the range 0.04–5 mm. Even 50% of the bubbles were smaller than 0.4 mm. The arithmetic mean bubble diameters varied from 0.6 to 0.75 mm and Sauter mean diameters from 2.2 to 3.2 mm. The present model predicts similar BSDs. The tails in the volume BSDs predicted by the adjusted model are larger compared to the present measurements especially in the case of air–water system (Fig. 9). Similarly as in the case of 200 L tank (Fig. 8) the measured volume BSD includes a secondary peak of 5 mm bubbles in location B. This indicates that all bubbles do not break as they pass the impeller region. The simulations show a similar peaking of 5 mm sized bubbles, although there is also a fraction of 8–10 mm primary bubbles. It is possible that these large bubbles were not observed in the

5.8.1. The measurements of Laakkonen et al. (2005a,b) Fig. 11 presents a comparison of predicted local BSDs to the measurements from lean (G = ∼ 0.01) air–water dispersions at low stirring speeds (Laakkonen et al., 2005a,b). The multiblock simulations in Fig. 11a show that the model of Lehr et al. (2002) predicts narrow BSDs and smaller bubble sizes compared to the capillary measurements from 200 L vessel ( ≈ 0.8 W/kg). In the case of 14 L vessel at low mixing intensity ( ≈ 0.35 W/kg) (Fig. 11b) it succeeds better. This is expected as the model was validated for bubble columns where turbulent energy dissipations are generally smaller (< 1 W/kg). This raises a question, whether the breakage and coalescence models, in which turbulence is assumed to be the driving force, can be validated against experiments under such mild turbulence conditions. Turbulence energy dissipation is an important parameter in the breakage and coalescence closures. In agitated vessels, turbulence originates mainly from the impeller motion and can be related to power consumption. In bubble columns, rising bubbles are a major source of turbulence. A part of the hydrostatic and kinetic energy of gas may be converted directly to heat in the viscous sublayer at bubble surface, which complicates the estimation of turbulence energy dissipation rate. In addition,

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735

Fig. 9. Local bubble size distributions in the air–water dispersion, 14 L tank, N = 700 rpm, Q = 0.7 vvm. Measured (markers), the multiblock (solid line) and CFD (dashed line) simulation using the present model, multiblock simulation using Lehr’s model (dotted line), feed size distribution (dashed dot line), mean diameters (mm): measured (bold), multiblock (parentheses), CFD (brackets), Lehr et al. (underlined). *Behind and # in front of impeller blade.

low levels of turbulence other than turbulent bubble–bubble interactions may be significant in bubble columns. It is noted that many validation studies are based on the CFD simulations, which often underestimate turbulent dissipation rate by 20–50% unless extremely fine grids are used. The use of fine grids with the solution of PBs is computationally intensive at the moment. These issues could partly explain the results of Chen et al. (2005), namely, that breakage rates had to be multiplied by a factor of 10 to get reasonable agreement between bubble column simulations and experiments. The current fitting predicts the existence of large bubbles in both simulation cases (Fig. 11), although the proportion of small bubbles is overestimated compared to the measurements. It is noted again that the diameter of capillary limits the detection of small bubbles. Many experimental studies including the present work (Takahashi and Nienow, 1992; Machon et al., 1997; Laakkonen et al., 2005a) have shown that the majority of bubbles are smaller than 0.5 mm in agitated vessels in both lean and dense dispersions. The simulations indicate that primary bubbles do not break immediately in the vicinity of impeller but more gradually in the impeller discharge flow, when they escape as trails from the gas cavities behind the im-

peller blades. The measured BSDs near the wall of 14 L tank (Fig. 11b) are bimodal as well thus giving some support to the model predictions. 5.8.2. The measurements of Hu et al. (2005) Hu et al. (2005) measured bubble sizes under extremely intense agitation ( = 25–40 W/kg) and at constant gas volume fraction of 1 vol% in a closed tank. The measurements were carried out with a photographing microscope from several chemical systems. Sauter mean bubble diameters were correlated by a 32 /DI = e · W ef .

(28)

The adjusted and Lehr’s model are compared to the results of Hu et al. (2005) for pure systems in Table 2. Eq. (28) was adjusted to the PB simulation results from the dissipation range ( = 25–40 W/kg). The steady-state BSDs were solved from PBs with bubble breakage and coalescence. Due to intense agitation, simulations were made with ideal mixing assumption. The comparison shows that both the present fitting and the model of Lehr et al. (2002) predict the bubble size fairly well under varying physical properties. In water and 1-propanol, the present fitting slightly overpredicts and the model of Lehr et al.

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M. Laakkonen et al. / Chemical Engineering Science 62 (2007) 721 – 740

Fig. 10. Local bubble size distributions in the CO2 –n-butanol system, 14 L tank, N = 700 rpm, Q = 0.7 vvm. The explanations are same as in Fig. 9.

(2002) underestimates bubble size. The adjusted constants of Eq. (28) agree closely with those given by Hu et al. (2005). The present model succeeds better in the case of diethylene glycol system than the model of Lehr et al. (2002). A more detailed analysis would require the comparison of BSDs. Qualitatively, the present fitting shows agreement with the observations of Hu et al. (2005), who noted that in all systems there exist a very wide range of bubble sizes. 5.8.3. The measurements of Calderbank (1958) Calderbank (1958) developed his widely used correlation (Eq. (29)) for the vessel-averaged bubble diameter based on local gas–liquid interfacial area measurements with a light scattering technique and gas holdup measurements with a sampling probe. The experiments were carried out with 10 liquids including water and n-butanol in the 5 and 100 L vessels: −0.4

a 32 /m = 4.15 · 



C

0.6 0.5 G + 0.0009/m,

(29)

where gas holdup is calculated from Eq. (27). Multiblock simulations with the adjusted and Lehr’s model are compared to Eq. (29) in Figs. 12a and b at varying gas feeds and mixing intensities for both air–water and CO2 –n-butanol systems. The

vessel-averaged Sauter mean diameters are obtained from the multiblock simulations as a 32 =

6G . a GL

(30)

The comparison shows that the present fitting is in good agreement with Calderbank’s correlation showing similar trend under varying agitation conditions. The model of Lehr et al. (2002) predicts notably smaller Sauter mean bubble diameters. 6. Conclusions Local bubble size distributions (BSDs) were measured from dense air–water and CO2 –n-butanol dispersions under hydrodynamic conditions in which many industrial gas–liquid reactors operate. The measured BSDs constitute a new data set for the validation of bubble breakage and coalescence closures. Parameters in phenomenological bubble breakage and coalescence models were adjusted comparing predicted and measured local BSDs. A multiblock stirred tank model was created to consider the inhomogeneity of dispersion in the fitting. The adjusted model predicts the complicated spatial variation of BSDs under varying vessel operating conditions and physical properties of dispersion realistically even though there

M. Laakkonen et al. / Chemical Engineering Science 62 (2007) 721 – 740

737

Fig. 11. Local volumetric bubble size distributions in the agitated vessels: (a) air–tap water ( = 0.069 N/m), 200 L vessel, (b) air–deionized water ( = 0.0716 N/m), 14 L vessel. The present fitting (solid line), the model of Lehr et al. (2002) (dashed line), capillary (circles) and photographing (squares) measurements of Laakkonen et al. (2005a,b).

Table 2 A comparison of population balance predictions to the empirical correlation of Hu et al. (2005) at gas fraction 0.01 System

a 32 ( = 35 W/kg) (mm)

Air–water, C = 1000 kg/m3 , Hu et al. This work Lehr et al.

= 72.28 mN/m, C = 1 mPa s

e

f

R2

0.113 0.211 0.535

−0.380 −0.436 −0.609

0.85 0.9999a 0.9999a

0.113 0.249 0.588

−0.380 −0.454 −0.619

0.85 0.9996a 0.9996a

Air–diethylene glycol, C = 1118 kg/m3 , = 44.77 mN/m, C = 30.2 mPa s Huet al. 0.598 2.72 This work 0.455 0.233 Lehr et al. 0.251 0.549

−0.7 −0.438 −0.612

0.78 0.9999a 0.9999a

0.447 0.545 0.359

Air–1-propanol, C = 804 kg/m3 , = 23.32 mN/m, C = 1.945 mPa s Hu et al. 0.316 This work 0.367 Lehr et al. 0.207

a The

fitting of Eq. (28) for the range  = 25–40 W/kg.

is still room for improvements. As a result of fitting local BSDs are described more accurately compared to the model of Lehr et al. (2002), which has been successful in bubble column studies (Lehr et al., 2002; Wang et al., 2005). This shows that phenomenological breakage and coalescence closures need

experimental validation for various flow environments. The present model was adjusted to describe both local number and volume BSDs. This issue has been neglected in many studies by comparing number BSDs or Sauter mean bubble diameters only.

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Fig. 12. The effects of (a) mixing intensity and (b) gas feed on the vessel-averaged Sauter mean bubble diameter.

Adjustable parameters have been often avoided in the model derivation by making numerous assumptions based on dimensional considerations, turbulence and probability theories. The fitting can produce reasonable estimates for parameter values. It can also guide in the model development. The present fitting indicates that majority of bubbles are born as a result of multibreakage events. This agrees with the experimental observations of Risso and Fabre (1998). In spite of this, binary breakage has been a commonly used assumption. Due to several superimposed phenomena and the complexity of gas–liquid mixing, the fitting alone is not sufficient for the validation PB closures. More basic research and isolated experiments under well-characterized, intense turbulent flow from both lean and dense dispersions are needed. So far, most validation studies have concentrated on the investigation of bubble columns. The experiments at such mild turbulence conditions may not be adequate for the validation of closures, in which turbulence is assumed to be the driving force. The rate of turbulent energy dissipation is an important parameter in the breakage and coalescence closures, but its accurate estimation is often difficult. In agitated vessels, turbulence energy dissipations can be related to the power consumption of impeller. In the CFD simulations local turbulent energy dissipations can be scaled with experimental values before passing them to the closure models. This can minimize the dependence of results on the simulation grid and turbulence model. A multiblock stirred tank model with varying flow fields and PBs for bubbles showed to be an efficient tool for the investigation of local gas–liquid hydrodynamics. It therefore seems that multiblock model is an optimal trade-off between the accuracy and CPU time for the testing and validation of PB closures.

Notation a aGL Aij B BSD c cij C1 C2 –C7, Cp CD Cflow DI e erfc() erf() Eo f F F∗ Fdrag Fl g g(aj ) h(ai , aj ) ni N Np NB NC

bubble diameter, m specific gas–liquid interfacial area, m2 /m3 the area over which liquid flows from subregion i to j, m2 blade width, m bubble size distribution acceleration, m/s2 adjustable parameter empirical constant, m−2/3 empirical constants bubble drag coefficient scaling factor for flow rates predicted by CFD impeller diameter, m adjustable parameter complementary error function error function Eötvös number (=a 2 c(C − D )/ ) adjustable parameter liquid flow rate, m3 /s liquid flow number (=F /N/DI3 ) drag force on bubble, kg m/s2 flow number, Q/N/DI3 acceleration due to gravity, m/s2 breakage frequency, 1/s coalescence rate, m3 /s dimensionless bubble number density agitation speed, 1/s power number of mixing (=P /(C N 3 DI5 )) number of observed bubbles number of discretization categories

M. Laakkonen et al. / Chemical Engineering Science 62 (2007) 721 – 740

∇p ∇p ∗ P Q Re s T u U Us vi vvm V Vb w(ai ) We Yk z zˆ

dynamic pressure gradient,kg/m2 /s2 scaled dynamic pressure gradient, kg/m2 power consumption of mixing, kg m2 /s gas feed, m3 /s Reynolds number (=C |U |a/) the depth of field, m diameter of vessel, m bubble approach velocity defined by Eq. (18) velocity, m/s superficial gas velocity, m/s dimensionless volume density of bubbles gas feed per vessel volume, m3 gas/m3 vessel min the volume of vessel or subregion, m3 volume of bubble, m3 weighting factor of measured BSDs in Eq. (1) Weber number, (=N 2 DI3 C / ) number of bubbles per unit volume, 1/m3 axial coordinate, m axial unit vector

Greek letters  (ai , aj ) v (ai , aj )  

(ai , aj )   D , C D , C



volume fraction probability that a bubble of size ai is formed when aj breaks, 1/m probability that a bubble of size ai is formed when aj breaks,1/m3 dissipation rate of turbulent energy, m2 /s3 rotational angle, radians coalescence efficiency molecular + turbulent viscosity, kg/m/s molecular viscosity of dispersed and continuous phase, kg/m/s densities of dispersed and continuous phase, kg/m3 interfacial tension, kg/s2 relative turbulence energy dissipation a fraction of the total liquid mass in a subregion

Subscripts g G i, j, k I L mix r u

gassed gas index impeller liquid mixing-induced radial ungassed

Acknowledgements Asta Nurmela, Elina Nauha and Altti Alastalo are acknowledged for their contribution to the experiments. Financial support from the Graduate School of Chemical Engineering (GSCE), ModCher (Modelling of Chemical Reactors) project

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