Simulation of bubbly flows Comparison between

ARTICLE IN PRESS Chemical Engineering Science 65 (2010) 1925–1941

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Simulation of bubbly flows: Comparison between direct quadrature method of moments (DQMOM) and method of classes (CM) B. Selma , R. Bannari, P. Proulx De´partement de ge´nie chimique et de ge´nie biotechnologique, Universite´ de sherbrooke, 2500 bd universite´, sherbrooke (QC), Canada J1K2R1

a r t i c l e in fo

abstract

Article history: Received 9 April 2009 Received in revised form 20 June 2009 Accepted 13 November 2009 Available online 24 November 2009

In typical bubbly flow applications, bubbles can break or coalesce due to bubble–bubble and bubble– fluid interactions in presence of turbulence. Under this assumption, a fixed bubble size model might not be suitable for predicting correct multiphase flow behaviour in the gas–liquid system. For example, breakage and coalescence events produce very different bubble size distribution and then affects the interfacial interactions between the phases as heat and mass transfer and exchange forces, for example the drag and lift forces. In the present work, a rectangular bubble column and stirred tank reactor are modelled using an open-source computational fluid dynamics CFD package OpenFOAM. A population balance equation is introduced in the mathematical model to account the effects of bubble size distribution taking account the effect of coalescence and break-up phenomena on the hydrodynamic behaviour of multiphase flow. For solving the population balance equation an efficient numerical technique is integrated to enable the simulation of more complex flows that are encountered in industrial applications. Furthermore, the direct quadrature method of moments (DQMOM) and the method of classes (CM) are implemented and compared using an open source CFD package (OpenFOAM). An Eulerian–Eulerian approach with a standard k2e model of turbulence is used. The momentum exchange between the bubbles and the continuous phase is taken into account with drag, lift and virtual mass forces. The predicted results are compared with measured data available in the scientific literature; they show that the gas volume fraction, velocity profiles and local bubble size are in good agreements. Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved.

Keywords: Bubble column CFD openFOAM Direct quadrature method of moments Classes method Size distribution Sauter mean diameter Coalescence Breakage

1. Introduction In the last decade, a rapidly increasing number of studies have been made on the modeling of bubble columns using advanced computational fluid dynamics methods (CFD). These studies are often made very closely to experimental work and aimed at identifying the hydrodynamic behaviour, heat and mass transfer, flow regime and mixing behaviour in this important type of chemical reactor. The importance of chemical processing technologies involving bubbly reacting flows brings sustained interest in fundamental studies of the hydrodynamics and flow regime for better reactor design and operation. A large number of the earlier fundamental studies have been realized on simple geometries of bubble columns or on simplified representations of the flow or bubble behaviour. For example, among the important models of gas–liquid flows in bubble columns, there are: Bel F’Dhila and Simonin (1992), Bakker and Van den Akker (1994), Pfleger and Becker (2001), Chen et al. (2004, 2005), Zhang et al. (2006). Most

 Corresponding author. Tel.: + 1 819 821 8000x65076; fax: + 1 819 821 7955.

E-mail address: [email protected] (B. Selma).

of them represent the bubbles with a single uniform size (one bubble model). However, recently Buwa and Ranade (2002) and Buwa et al. (2006) have shown different dynamic characteristics arise in these systems for different bubble sizes. It appears more and more that it is essential to represent the dispersed phase taking into account its bubble size distribution with consideration of the coalescence and break-up phenomena to model the bubble size evolution. Only a few authors have included a population balance equation to describe the bubbles behaviour. For example, the early work of Lo (1998), Carrica et al. (1999), and Lehr and Mewes (2001), and the more recent work Kerdouss et al. (2006, 2008). In these, the mass and momentum conservation equations for all bubble groups (multi-fluid model) are solved. The coalescence and break-up processes are accounted for by solving the continuity equation for all bubble classes. For some types of bubbly flows, Lehr and Mewes (2001) and Kerdouss et al. (2006) have assumed equilibrium between the coalescence and break-up processes transforming the population balance model to a single equation. However, in bubble columns this assumption may be not applicable because of the significant influence of convection. Several authors have used different models (multi-fluid model) to calculate coalescence and break-up rates and probabilities of

0009-2509/$ - see front matter Crown Copyright & 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2009.11.018

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bubble collisions (Prince and Blanch, 1990; Luo and Svendsen, 1996; Lehr and Mewes, 2001). The solution of population balance equations for multi-fluid model has been receiving, as a consequence of the interest in bubbly flows, increasing attention. At the root of the population balance of bubble sizes is the original work of von Smoluschowski which was published over 90 years ago (1917), it has the following form: X dnj 1 X ¼ K n n n K n dt 2 i þ k ¼ j ik i k j i ij j

ð1Þ

In the right-hand side of this equation, the first term represents the formation of j-mother particles due to binary collisions between i and k primary particles. The second term is the birth rate of j-mother particles due to aggregation. Since the early work of Ramkrishna in 1971, many solutions were proposed for solving the population balance equation using modern computational fluid mechanics methods. For example: Ramkrishna (1971), Bajpai et al. (1976), Singh and Ramkrishna (1977), Guimara~ es and Cruz-Pinto (1988), Aldis and Gidaspow (1989), Xiong and Pratsinis (1991), Erasmus et al. (1994), Chen et al. (1996), Kumar and Ramkrishna (1996a, 1996b), Liou et al. (1997), and Azizi and Al Taweel (2007). More recently, attempts have been made to solve the population balance equation using the socalled quadrature method of moments (QMOM) first proposed by McGraw (1997). It has proved an efficient and successful technique, for example, for describing aerosol dynamics under conditions that can include new particles formation, evaporation, growth and coagulation. In the case of bubble size distribution in bubble columns, the dispersed phase tends to break-up or coalesce due to presence of complex fluid mechanical mechanisms present in bubbly flows. The break-up occurs according to two mechanisms, the first is when turbulent eddies strike the bubble surface with sufficient energy, and the second is the break-up of large bubbles, due to their structural instability. Coalescence occurs when two bubbles or more collide because of velocity difference and/or because of turbulence. There are numerous challenges in the modeling of dispersed flows because the exchange of momentum between particles due to collision, and the changes of particles properties due to breakup and aggregation. The problem can be solved by coupling of the population balance equation (PBE) with the Computational Fluid Dynamics methods. The extra computational effort for the PBE solution, such as classical methods (Monte Carlo, sectional and method of classes) can, however, become very important. Therefore, the development of efficient and accurate numerical methods for solving the PBE has drawn a lot of attention. The direct quadrature method of moments (DQMOM) (Marchisio and Fox, 2007) is one of the most efficient methods developed recently; it was inspired by the quadrature method of moments (QMOM) (McGraw, 1997). The main advantage of DQMOM/ QMOM is that only a few abscissas are necessary to describe a particles distribution due to the numerical quadrature approximation closure used. In this work, the DQMOM is used for the modeling of polydispersed gas–liquid bubble columns and stirred tank reactors using the OpenFOAM CFD package. Besides the PBECFD coupling, other challenges in the modeling of multiphase flows are the breakage and coalescence phenomena. At the present, the current available breakup and aggregation models for gas–liquid flows are yet not completely adequate for many practical applications. For example, in a bubble column model one must consider several aspects such as turbulence, bubble induced turbulence, momentum interface exchange forces (drag, lift, virtual mass, gravity, drift velocity, etc.). The present work is aimed at the validation of the DQMOM with the method of classes (CM) and some experimental works available in the literature. Comparisons of the model and experimental results are carried out using breakup and

coalescence models available in the scientific literature. Some preliminary comparisons of the predicted Sauter mean diameter ðd32 Þ and experimental measurements are included.

2. Mathematical model In order to validate the model with the available results in the literature, a rectangular bubble column and a stirred tank are modelled using the two-phase flow approach. The flow model is based on solving the multiphase equivalent of Navier–Stokes equations along with the standard k2e model of turbulence. The mathematical model is solved by using an open-source CFD package OpenFOAM which is based on finite-volume-method (fvm) to solve mass conservation and momentum equations. The continuity for each phase j is written as @ ðr aj Þ þ rðrj aj uj Þ ¼ 0 @t j

ð2Þ

In the Eulerian multiphase transient modeling approach the following relationship is respected for each time step: X aj ¼ 1:0; j ¼ c; d ð3Þ The momentum equation is done by @ ! ! ! ðaj u j Þ þ rðaj Fj u j Þ þ rðaj nj r u j Þ @t ¼ ! ! ¼ aj rp þ rðtj Þ þ aj g þ F

ð4Þ

¼

where tj represent the Reynolds stress tensor for the continuous phase defined as ¼



T



2 2 tj ¼ neff j ðr uj þ r  uj 3 Ir  uj Þ þ 3Ik eff

ð5Þ

nj is the effective kinematic viscosity of phase j ðm s Þ. Fj and ! F are, respectively, the velocity field of phase j and the interphase momentum exchange term written as ! ! ! ! ð6Þ F ¼ F drag þ F lift þ F vm 1

The drag force is obtained as follows: ! 3 ad CD rc ! ! F drag ¼ j u rj u r ð7Þ 4 d32 ! ! ! where u r ¼ u c  u d is the relative velocity. The drag coefficient CD is a function of Reynolds number and the following correlation is used (Schiller and Naumann, 1935): 8 < 24 ð1þ 0:15Re0:687 Þ; Rer 1000 CD ¼ Re ð8Þ : 0:44 otherwise The Re number is based on the Sauter diameter d32 and the relative viscosity ur and is written as Re ¼

rc jur jd32 mc

ð9Þ

The lift force occurs due to interactions between bubbles and liquid velocity gradient and it is given by ! ! ! ð10Þ F lift ¼ ad Cl rc u r  ðr  u c Þ In this work, the expression of (Tomiyama et al., 2002) is used. It is given as ( min½0:288tanhð0:121ReÞ; f ðEod Þ; Eod o 4 Cl ¼ ð11Þ 4r Eod r 10:7 f ðEod Þ;

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

3.1. Solving PBE by the method of classes (CM)

where 3

2

f ðEod Þ ¼ 0:105ðEod Þ 0:0159ðEod Þ 0:0204Eod þ 0:474

ð12Þ

¨ os ¨ number based on the Sauter diameter defined as and Eod is Eotv Eod ¼

1927

g Drd232

s

ð13Þ

Here, g and s are, respectively, the gravity ðm s2 Þ and surface tension ðN m1 Þ. Virtual mass force can be physically described by considering the change in kinetic energy of fluid surrounding an accelerating bubble. In other term, if the bubble is accelerated relatively to the liquid, part of the surrounding liquid has to be accelerated as well. This additional force contribution is called the ‘‘virtual mass force’’ and naturally, it is much larger for a bubble dispersed in liquid than for a drop or solid particle in a gas due to the higher liquid density surrounding the bubble. The virtual mass coefficient is often set to 0.5 in the literature for rigid spherical particles (i.e. droplets). Based on previous studies (Bannari et al., 2008; Chen et al., 2004) the value of virtual mass coefficient is set to 0.25 for spherical bubbles (not rigid particles) and the virtual mass force is obtained by ! ! ! ! Dc u c Dd u d  ð14Þ F vm ¼ ac Cvm rc Dt Dt

To discretize the PBE in the size domain (internal coordinate), the fixed pivot approach of Kumar and Ramkrishna (1996a) is used. This technique is known as the method of classes (CM) and is today the most commonly used technique for solving PBE. The CM assumes that population bubbles are distributed on pivoted grid points xi where xi þ 1 ¼ sxi and s 4 1. Bubble break-up and coalescence may generate new bubbles with volume v such that xi o v oxi þ 1 . This bubble must be split by assigning, respectively, fraction gi and gi þ 1 to xi and xi þ 1 . In this method, the number density (zeroth moment) and mass conservation (first moment) are preserved by prescribing the following two constraints (Kerdouss et al., 2008): ( gi xi þ gi þ 1 xi þ 1 ¼ v ð20Þ gi þ gi þ 1 ¼ 1 Following Kumar and Ramkrishna (1996a, 1996b), the source term in this equation given by br Dcoal Þ þ ðBbr Si ¼ ðBcoal i i i Di Þ

where Bcoal ¼ i

where Dj =Dt indicates the substantive derivative which is defined as D @ ¼ þ uj  r Dt @t

ð15Þ

xi1 r xi þ xk M X

M X

Qi;k

aGk

ni;k Gk aGk

k¼i

ð16Þ

  1 x 1 dj;k ZQj;k aGk i x 2 j xk r

k¼1

3. Population balance modeling

@ ðr a f Þ þ rðuG rd ad fi Þ ¼ rd Si @t d d i

jX Zk

Dcoal ¼ aGi i

Bbr i ¼

The population balance equation (PBE) is the conservation equation of the number ni of the bubbles (per unit volume) of size i. Lo (1998) used computational fluid dynamics methods and added the population balance modelling. In this model, the population balance equation for the bubble volume fraction of class i is written as

ð21Þ

xk xi xk

Dbr i ¼ Gi aGi

ð22Þ

ð23Þ

ð24Þ

ð25Þ

in this work, Qj;k is defined as the coalescence frequency between bubbles of size j and k. Gi is the break-up frequency of bubble of size i. Here, ni;k and Z are defined as follows: Z xi þ 1 Z xi xi þ 1 v vxi1 bðv; xk Þ dv þ bðv; xk Þ dv ð26Þ ni;k ¼ xi þ 1 xi xi xi1 xi xi1 and 8 x v iþ1 > > < x x ;

xi r vr xi þ 1

vxi1 > > ; : xi xi1

xi1 rv rxi

fi is the bubble volume fraction of group of size i. Si is the source term due to coalescence and break-up. In terms of number density, ni , the population balance equation for a specific bubble of size, i, can be written as follows:

Z¼

@ ðr n Þ þ rðrd ud;i ni Þ ¼ rd Si @t d i

the variable bðv; xk Þ is the number of bubbles of volume v formed from the breakage of bubble of volume (or size) xk .

ð17Þ

The number density is related to the individual bubble volume through the gas volume fraction as follows: ni vi ¼ ai X

ai ¼ ad ¼ 1ac

ð18Þ ð19Þ

It is important to note the difference between Eqs. (16) and (17). In Eq. (17), the individual bubble velocity is used. On the other hand, in Eq. (16), the same gas velocity is used for all bubble sizes. For practical reasons, Eq. (16) is often preferred. This is mainly justified because the error in the spatial variation of number density is much more important than the error caused by the variation of individual size group volume fraction noted by scalar fi . In this work, the population balance equation chosen has the form of a transport equation of a scalar variable fi and is solved using OpenFOAM-1.4. The interfacial exchange forces coupling is done via the Sauter mean diameter ðd32 Þ.

iþ1

i

ð27Þ

3.2. Solving the PBE by the direct quadrature method of moments (DQMOM) The direct quadrature method of moments (DQMOM) is based on the direct solution of the transport equations for weights and abscissas of the quadrature approximation (Rong et al., 2004). The advantage of this method is that it is directly applicable to the population balance equation with more than one internal coordinate. Thus, the DQMOM offers a good approach for describing bubble size distribution undergoing break-up and coalescence processes in the context of CFD-bubble flow modeling. Since this method has been described extensively by many authors recently (Marchisio et al., 2003; Rong et al., 2004; Marchisio and Fox, 2007). The discussion is limited here to a brief review of the equations. The transport equations for calculating the weights and abscissas are written as (Rong et al.,

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and abscissas of the quadrature approximation rather than the product algorithm PD used in the simple quadrature method of moments.

2004) 8 @ > > < wi þ rðfwi Þ ¼ ai @t @ > > : Li þ rðfLi Þ ¼ bi @t

ð28Þ 3.3. Bubble break-up model

where ai and bi are found by the solution of the following non linear system in terms of the unknowns ai and bi (Stefano Bove, 2005): N X

k1

½ð1kÞLki ai þ kLi

bi  ¼ SðkÞ ;

k ¼ 0; . . . ; 2N1

ð29Þ

i¼1

This system is numerically solved using the Gauss–Seidel method. The source term SðkÞ is defined as follows: SðkÞ ¼ ðBðkÞ DðkÞ Þ þ ðBðkÞ DðkÞ Þ coal coal break break BðkÞ coal

ð30Þ

DðkÞ are the k coal and DðkÞ BðkÞ break break

and th birth and death rates due to where are the k th birth and death rates coalescence, and due to break-up of bubbles and its formulation are (Rong et al., 2004) ¼ BðkÞ coal

N X N X

ð31Þ

oi oj Lki bij

ð32Þ

i¼1j¼1

BðkÞ ¼ break

N X

ðkÞ

OB ðvj : vi Þ ¼ 0:923aL Ni

e

!1=3 Z

d2i

1

zmin

ð1 þ zÞ2

z11=3

exp 

!

12cf s 5=3 11=3

2rL e2=3 di

z

dz

ð37Þ

N X N 1X o o ðL3 þ L3j Þk=3 bij 2i¼1j¼1 i j i

DðkÞ ¼ coal

The bubble break-up can be related to the influence of small turbulent eddies on the bubble surface (Lee et al., 1989; Luo and Svendsen, 1996; Lehr and Mewes, 2001). The model of Luo and Svendsen (1996) has served as a basis for many of the studies published after 1996. It is derived from theories of isotropic turbulence. In the present study, the break-up model of Luo and Svendsen (1996) is used, which is based on the idea that break-up occurs when the eddy of higher energy hits the bubble surface. Hence, the break-up frequency is related to the frequency of collision of small eddies with the bubble and is given by (Bhole et al., 2008)

b i ai oi

ð33Þ

oi Lki ai

ð34Þ

%

Eq. (37) gives the break-up frequency for the breakage of bubble of volume i into a bubble of volume j. The increase energy in the surface area is given by 2=3

cf ¼ fBV þ ð1fFB Þ2=3 1

ð38Þ

where fFB ¼ vj =vi and z is the dimensionless eddy size (the ratio of eddy size to the bubble size). In this model no probability density

i¼1

¼ DðkÞ break

N X

%

i¼1

R1 ðkÞ where bij ¼ bðLi ; Lj Þ, ai ¼ aðLi Þ, and b i ¼ 0 Lk bðL=Li ÞdL. ðkÞ The above integrals of the daughter distribution function b i are approximated as %

ðkÞ

b i  Lki

mk=3 þ nk=3

i

0

1

2

wi ðÞ Li (m)

0.33 0.001

0.33 0.002

0.34 0.003

ð35Þ

ðm þ nÞk=3

where m and n represent the mass ratios between the two bubble breakage. For example, if m ¼ 1 and n ¼ 1, the two fragments have the same volume and thus, symmetric breakage is considered. In the present work, a symmetric breakage is considered ðm ¼ n ¼ 1Þ and if Eq. (35) is applied the following relation is obtained: ðkÞ bi

Table 2 Initial conditions of weights wi and abscissas Li used in Eq. (28).

 2ð3kÞ=3 Lki

ð36Þ

Here Li denotes the characteristic length of the bubble and is equivalent to the bubble diameter of class i in the present study. The coalescence rate bij can be approximated by using the Eq. (42), while the breakage kernel aðLi Þ can be formulated by using Eq. (37) addressed in the next section. In the present study, the direct quadrature method of moments is adopted, which is a computationally attractive alternative. It is based on the idea of directly tracking the weights

Table 3 Different closures used in the present model.

Technique for solving PBE

Closure A

Closure B

CM

DQMOM

Table 4 Values of s and the corresponding classes used with CM. Number of classes Value of r Value of s

7 3 2

15 7 1.3459

25 12 1.1892

Table 1 Numerical boundary conditions input. Parameters

Inlet

Outlet

Wall

Gas volume fraction a Gas velocity m s1 Liquid velocity m s1 Pressure (Pa) k m2 s2 e m2 s3

1 Calculated 0 0 Calculated Calculated

1 0 0 Atmospheric pressure Neuman condition Neuman condition

Neuman condition Dirichlet condition Dirichlet condition Dirichlet condition Dirichlet condition Dirichlet condition

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is needed and the break-up rate function can be calculated by using incomplete gamma functions in the following form (Kerdouss et al., 2008; Bannari et al., 2008): !1=3 18 0:923aL aG fj e OB ðvj ; vi Þ ¼  ðGð8=11; tm Þ 11 pb8=11 d3j d2j Gð8=11; bÞ þ 2b3=11 ðGð5=11; tm ÞGð5=11; bÞÞ

1929

where di and dj are the diameter of bubbles of class i and j with their number density ni and nj , respectively. In the present work, the coalescence probability Pc is expressed as follows (Hagesather et al., 2000): ! 2 3 ½0:75ð1 þ xij Þð1 þ xij Þ1=2 1=2 ð44Þ Weij Pc ¼ exp C ðrG =rL þ 0:5Þ1=2 ð1þ xij Þ3 1=2

þb6=11 ðGð2=11; tm ÞGð2=11; bÞÞÞ

ð39Þ

where Weij ¼ rL di u2ij =s; xij ¼ di =dj ; uij ¼ ðu2i þ u2j Þ1=2 ; ui ¼ b

ðedi Þ.

where b¼

12cf s 5=3

2rL e2=3 dj

ð40Þ

and 11=3

tm ¼ bðzmin =dj Þ

ð41Þ

At high Reynolds number, terms with tm in the incomplete Gamma function are taken equal to zero as tm C1 (Sanyal et al., 2005). 3.4. Bubble coalescence model Coalescence can occur when bubbles collide with each other, for example in a turbulent flow. There are various mechanisms of bubble coalescence, Prince and Blanch (1990) have considered three mechanisms of bubble collision. Bubbles can collide due to: (i) the random motion in a turbulent flow, (ii) to the different rise velocity and (iii) to the mean shear in the flow field (Bhole et al., 2008). Mathematically, the coalescence rates aðvi ; vj Þ are usually written as the product of the collision rate yi;j and the probability of collision or some times applied efficiency Pc (Hagesather et al., 2000): aðvi ; vj Þ ¼ yi;j Pc

ð42Þ

where the collision rate of bubble per unit volume is given by Saffman and Turner (1956) as follows:

yi;j ¼

9a2G fi fj 1=3 1=3 ðd þ d Þ2 e1=3 ðdi þ dj Þ1=2 pd3i d3j i j

ð43Þ

4. Numerical solution The numerical solution of the above equations is carried out by using the open source code CFD package OpenFOAM which is primarily destined to solve problems in mechanics of the continuous mediums. The Navier–Stokes conservation equations are discretized using finite volume method (FVM). The physical boundary conditions are summarized in Table 1. Typically, the OpenFOAM package is tailored to the specific CFD problem treated and a specific ‘solver’ is developed (compiled) for each specific application. In the present work for example, the PBE is implemented in OpenFOAM as a new dedicated ‘solver’ called momentsFoam using DQMOM in which the physical model of bubble coalescence and breakage is included. The coupled CFDPBE solver is based and extends an existing two-phase flow solver (twoPhaseEulerFoam). The developed solver takes in consideration the drag, lift and virtual mass interface forces and use the standard model of turbulence k2e.

4.1. Linear system solution The generated system (28) is solved using different numerical schemes implemented and compiled in OpenFOAM. For more details of these numerical methods refer you to the OpenFOAM documentation available in the http://www.opencfd.co.uk/open foam.

Fig. 1. Predicted Sauter mean diameter using DQMOM and the method of classes. In this case of H=W ¼ 2:25 and Ud;s ¼ 0:14 cm s1 .

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The solution of the nonlinear system generated by Eq. (29) is obtained using Gauss–Seidel method implemented in the present work as follows:

 solve the formed linear system using Gauss–Seidel method. As default, the Gauss–Seidel method is not our subject in the present study.

 construct

4.2. Example of initial conditions of wi and Li In this study, the initial conditions of weights wi and abscissas Li are summarized in Table 2.

Fig. 2. Predicted Sauter mean diameter using DQMOM and 7, 15, 25 different classes. In this case of H=W ¼ 4:5 and Ud;s ¼ 0:14 cm s1 .

0.2 Experimental (Pfleger et al. (1999)) 7 classes (this work) 15 classes (this work) 25 classes (this work) DQMOM (this work)

0.15

0.1 Axial liquid velocity [m/s]



a symmetric matrix A½i½j, i ¼ 0; . . . ; 2N1, j ¼ 0; . . . ; 2N1, where N is the node of the quadrature approximation used in DQMOM; filling the matrix A½i½j with coefficients ai and bi obtained by developing solving the Eq. (28);

0.05

0

-0.05

-0.1

-0.15

-0.2 0

0.05

0.1 Column width [m]

0.15

0.2

Fig. 3. Liquid velocity profile using E–E simulation with population balance. H=W ¼ 2:25; Ud;s ¼ 0:14 cm s1 for Y ¼ 0:37 m from the bottom.

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

5. Results and discussions 5.1. Test case I: bubble column The predicted results obtained with the present model are compared with the measurements available in the literature (Pfleger and Becker, 2001; Buwa et al., 2006; Bhole et al., 2008; Elena Da´z et al., 2008) both for qualitative and quantitative

1931

comparisons. The bubble column geometry is as used by Pfleger and Becker (2001), a rectangular bubble column 0:2  1:2  0:05 m ðW  H  DÞ. The domain is discretized into 16  96  4 control volumes, a total of 6144 cells. The time step is set to Dt ¼ 0:01 s resulting in a maximum Courant number between Co ¼ 0:40 and 0.70 depending on the gas flow rate. As in the experiments of Buwa and Ranade (2005), the air–water system is chosen and the superficial gas velocity is varied from 0.14 to

0.025 Experimental (Buwa and Ranade, 2005) 7 classes (this work) 15 classes (this work) 25 classes (this work) DQMOM (this work)

Time-averaged gas hold-up [-]

0.02

0.015

0.01

0.005

0 0

0.05

0.1 Column width [m]

0.15

0.2

Fig. 4. Gas volume profile using E–E simulation with population balance. H=W ¼ 2:25; Ud;s ¼ 0:14 cm s1 for Y ¼ 0:37 m from the bottom.

0.03 Experimental (Buwa and Ranade, 2005) 7 classes (this work) 15 classes (this work) 25 classes (this work) DQMOM (this work)

Time-averaged gas hold-up [-]

0.025

0.02

0.015

0.01

0.005

0 0

0.05

0.1 Column width [m]

0.15

0.2

Fig. 5. Gas volume profile using E–E simulation with population balance. H=W ¼ 4:5; Ud;s ¼ 0:14 cm s1 for Y ¼ 0:675 m from the bottom.

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0:73 cm s1 (corresponding to inlet velocities from 0.1296 to 0:67 m s1 in the model). Two closures (see A and B in Table 3) are used to model the bubble column and the predicted results from the models are compared with the available measurements. In the first closure used (A), the population balance equation is solved using the method of classes (CM), and in the second closure (B), the direct quadrature method of moments (DQMOM) is used.

In the method of classes, the bubbles are divided into n ¼ 2r þ 1 classes (Bannari et al., 2008), with n odd for symmetry. A distribution on pivoted grid point xi with xi þ 1 ¼ sxi and a value of s4 1 is used here (Bannari et al., 2008). With the assumption of spherical bubbles, the following formulation can be written as ð4=3Þpðdi þ 1 =2Þ3 ¼ ð4s=3Þpðdi =2Þ3 . In this relation, s is calculated to ensure that dn ¼ d2r þ 1 ¼ dmax and dr ¼ dmean . This gives the

0.01 7 Classes 15 classes 25 classes DQMOM

0.009

Sauter mean diameter d32 [m]

0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0

0.05

0.1

0.15

0.2

Column width [m] Fig. 6. Sauter mean diameter profile using E–E simulation. H=W ¼ 2:25; Ud;s ¼ 0:14 cm s1 for Y ¼ 0:37 m from the bottom.

0.01 7 Classes 15 Classes 25 Classes DQMOM

0.009

Sauter mean diameter d32 [m]

0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0

0.05

0.1

0.15

0.2

Column X [m] Fig. 7. Sauter mean diameter profile using E–E simulation. H=W ¼ 4:5; Ud;s ¼ 0:14 cm s1 for Y ¼ 0:675 m from the bottom.

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

following relation: di ¼ sðir1=3Þ dmean

5.2. Test case II: stirred-tank reactor ð45Þ

and s¼



dmax dmean

1933

3=r ð46Þ

In this study, the initial inlet bubble diameter is set to 5 mm as in Buwa and Ranade (2002) and Buwa et al. (2006). Table 4 gives the value of s used with the corresponding class. At the inlet, the mean diameter is used, and the same gas velocity is considered for all size groups. This assumption reduces the gas phase momentum equations to a single equation as described above. When using the class method (CM), Laakkonen et al. (2007) found that more than 80 classes should be used to minimize discretization errors. However, in the present case and for the given geometry, the number of classes was limited to 25 as shown in Table 4. By comparison with the experimental data and the DQMOM method, in the present case, with the present geometry and size of the columns it does not appear that increasing the number of classes further increases significantly the agreement and seems to be within experimental errors. Furthermore, increasing the number of classes to 80 the calculation becomes prohibitively costly when compared to the much faster DQMOM method. While approaching the walls (Figs. 1 and 2), the bubble size decreases because the breakage phenomena dominates coalescence, thus creating this observed radial bubble segregation. When approaching the center of bubble column the bubble size increases due the high volume fraction of gas, in this region the coalescence dominates and therefore larger bubble sizes are observed. As expected, population balance of the bubble size distribution gives better agreement than a single bubble size. For 7, 15 and 25 classes, respectively, only the last two cases give results within experimental errors when compared with the measurements of Buwa et al. (2006). In case of H=W ¼ 2:25 and 4:5 at two different gas flow rates, the predicted gas hold-up and liquid velocity profiles are shown in Figs. 3–5. In CM the bubble size range is set between 0 and 10 mm, the initial value of bubble diameter leaving the inlet is 5 mm as used by Buwa and Ranade (2005). Measured vertical liquid velocity by Pfleger and Becker (2001) and predicted results of the present model are shown in Fig. 3. In this case a superficial gas velocity of 0:14 cm s1 is used and y ¼ 0:37 m for data sampling. The predictions of the Sauter mean diameter using CM and DQMOM are shown in Figs. 6 and 7. Bannari et al. (2008) found that 25 classes give better results throughout the comparisons with experimental data under the conditions studied in the present work. Using more than 25 classes give also good results but the computational efforts quickly become prohibitive. Consequently, the use of other methods for solving the PBE using only a few moments to describe the distribution becomes very attractive. In the present study, an alternative is chosen in which the so-called direct quadrature method of moments DQMOM is used to close the equations. Until now, this technique has not been tested and validated on bubbles size distribution in presence of complex phenomena such as coalescence and breakage due to bubble–bubble and/or bubble–liquid interactions. In the present work the DQMOM technique is used and compared with the more complete PBE solution of the CM in order to show its potential as a fast and accurate alternative to the CM as illustrated in Figs. 6 and 7. The predicted results are obtained after the initial transient period which is, in the case of bubble column, of the order of 60 s.

In order to validate the predicted results, the experimental set-up of Alves et al. (2002a) is used, the solution domain is shown in Fig. 8. It consists of a flat bottom stirred cylindrical vessel with diameter T of 0.292 m and liquid height H ¼ 2T. A double six bladed standard Rushton impellers with diameter of D ¼ T=3 were located, respectively, at 0.146 and 0.438 m above the tank base. The impeller blade width, l, and the impeller blade height, w, are equal, respectively, to D=4 and D=5. The tank is equipped with four baffles of 0:1T width uniformly spaced around the periphery. Gas is supplied through a small sparger, which is located between the tank base and the lower impeller (Kerdouss et al., 2006). The model results (see

Fig. 8. Solution domain used in this work. The open-source CFD package OpenFOAM is used to solve the numerical model on the discretized geometry shown.

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B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

Table 5 Transport properties of different fluids used in CFD modelling. Fluids

Density kg m3

Dynamic viscosity (Pa s)

Surface tension N m1

Water ð203 CÞ

998.2

1:0  103

0.073

1:7894  105

–

Air ð203 CÞ

1.225

Fig. 10. Predicted Sauter mean diameter using 11 classes (a) and DQMOM (b).

Fig. 9. Unstructured mesh grid used in the present model.

and

Figs. 8 and 9) are compared to the experimental data (Alves et al., 2002a) of a stirred tank filled with tap water with a total height of 0.584 m, gas flow rate of 1:67  104 m3 s1 and an impeller rotation speed of 7:5 s1 corresponding to a turbulent Reynolds number, Re ¼ rc ND2 =mc ¼ 7:1  104 . The water and gas properties are set in Table 5. In the present work, a full tank model is used with a multiple reference of frames (MRF) techniques for the impeller regions. The modelling of stirred-tank is carried out with all interface forces (drag, lift, virtual mass) and the gravity as external force. In the case of CM as used by Kerdouss et al. (2008), the Sauter diameter d32 is calculated by combining Eqs. (47) ad (48) described as follows: n¼

6ad pd332

ð47Þ

! @n þ rðn Ud Þ ¼ Sbr Sco @t

ð48Þ

n, is the bubble number density. Sbr and Sco are, respectively, the bubble breakage and coalescence rates. Following Wu et al. (1998) bubble break-up rate can be written as     ðedÞ3 Wecrit 1=2 Wecrit Sbr ¼ Cbr n ð49Þ 1 exp  d32 We We and the coalescence rate is given by Sco ¼ Cco Zco d2 ðedÞ1=3 n2

1 1=3

ð1ad Þ

ð50Þ

where Cbr ¼ 0:075 and Cco ¼ 0:05 are adjustable parameters fitted to the experimental data of Alves et al. (2002a). Zco is the coalescence efficiency set to unity (Wu et al., 1998; Lane et al.,

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

1935

2002). We is the Weber number defined as We ¼

rc u2t d s

ð51Þ

where Wecrit is the critical value of the Weber number, and is set to 1.2 (Kerdouss et al., 2006). The velocity of eddies ut is given by ut ¼ 1:4ðedÞ1=3

ð52Þ

In the case of DQMOM, d32 is determined by the following formula: d32 ¼

Fig. 11. Identification of different loops used in the present model with the experimental points ðÞ from Alves et al. (2002a).

mð3Þ mð2Þ

ð53Þ

where mð3Þ and mð2Þ represent the first lower order moments. A qualitative comparison between CM and DQMOM is presented in Fig. 10(a) and (b). In both figures, the local bubble size decrease near impellers, because of high shear stresses due to impellers rotation. In this region, the breakage phenomena dominate coalescence. In the other regions (red color), the bubbles coalesce between them and form a large bubbles due to gas accumulation and liquid recirculation (Fig. 10). For a quantitative comparison Figs. 12–14 show the results obtained by the simulation of the stirred-tank reactor and they show good agreement in comparison with the experimental data of Alves et al. (2002a). In general, when increasing the radial position the gas hold-up increase due to the migration of bubbles near the reactor walls then the largest value of gas volume fraction is ad ¼ 8:4% founded in Fig. 14. Figs. 15–18 show the predicted local bubble mean diameter d32 in a vertical mid-plane between two baffles as these ‘loops’ (Fig. 11) were defined in the work of Alves et al. (2002a). These figures show quantitative comparison between the modelling results and measurements (Alves et al., 2002a). In the loops 2 and 4, the surface mean diameter increases because coalescence dominate breakage in these circulation zones. A Sauter mean diamter of 5.3 mm is predicted because the air is trapped in the circulation zone between impellers (coalescence dominate breakage). Also in the first loop, the bubble size increases due to the position of the gas sparger (inlet) and gas build-up. The

0.05 Experimental (Alves et al. (2002a)) this work (model with CM) this work (model with DQMOM)

Gas hold-up [-]

0.04

0.03

0.02

0.01

0 -0.146

-0.0925

-0.039

0.0145

0.068 0.1215 0.175 X-position [m]

0.2285

0.282

0.3355

0.389

Fig. 12. Gas hold-up profiles comparison between CM and DQMOM in the radial position r ¼ 0:024 m.

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B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

0.05 Experimental (Alves et al. (2002a)) this work (model with CM) this work (model with DQMOM)

Gas hold-up [-]

0.04

0.03

0.02

0.01

0 -0.146

-0.0925

-0.039

0.0145

0.068 0.1215 0.175 Y-position [m]

0.2285

0.282

0.3355

0.389

Fig. 13. Gas hold-up profiles comparison between CM and DQMOM in the radial position r ¼ 0:07775 m.

0.1 Experimental (Alves et al. (2002a)) this work (model with CM) this work (model with DQMOM)

0.09 0.08

Gas hold-up [-]

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 -0.146

-0.0925

-0.039

0.0145

0.068

0.1215

0.175

0.2285

0.282

0.3355

0.389

Y-position [m] Fig. 14. Gas hold-up profiles comparison between CM and DQMOM in the radial position r ¼ 0:1315 m.

predicted results are in good agreement as well with the measurements as other modeling results from the literature (Alves et al. 2002a, 2002b; Kerdouss et al., 2006).

6. Conclusion The model developed shows that in the closure A, a good agreement is obtained when the direct quadrature method of

moments (DQMOM) is used and the results are compared with the available experimental results as well as the method of classes (CM). It performs much better than using a constant bubble size while requiring only marginally higher computational effort. When using the method of classes, both the 15 classes and 25 classes are in agreement with the experimental results but as expected the 25 classes is more accurate, though it requires a significant computational effort. In a previous study (Bannari et al., 2008), the authors showed that for a given geometry of bubble column flow reactor a

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

1937

6 Alves et al. (2002a), Experimental Alves et al. (2002a), Model Kerdouss et al. (2006), Model this work, Model

Local bubble diameter [mm]

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

Loop 1 [m] Fig. 15. Predicted bubble diameter using DQMOM as a function of position along liquid circulation loop 1.

6 Alves et al. (2002a), Experimental Alves et al. (2002a), Model Kerdouss et al. (2006), Model this work, Model

Local bubble diameter [mm]

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

Loop 2 [m] Fig. 16. Predicted bubble diameter using DQMOM as a function of position along liquid circulation loop 2.

good compromise between computational effort and precision the 15 classes choice was a good choice for solving the population balance equation but other authors mention that up to 80 classes could be necessary to obtain sufficient accuracy. In order to have a method that represents adequately the population without the large computational effort associated with such a large number of additional equations, the DQMOM appears as a very interesting solution method.

In the closure B, the DQMOM gives again good results when compared to the experimental data and models published by Alves et al. (2002a) and Kerdouss et al. (2006). For quantitative comparison the Sauter mean diameter is chosen for its wide application for determining the mass transfer coefficient. Different loops are used to compare the present model with available measurements from the literature, agreement is again good. Drastic reduction in the computational requirements is obtained

ARTICLE IN PRESS 1938

B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

6 Alves et al. (2002a), Experimental Alves et al. (2002a), Model Kerdouss et al. (2006), Model this work, Model

Local bubble diameter [mm]

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

Loop 3 [m] Fig. 17. Predicted bubble diameter using DQMOM as a function of position along liquid circulation loop 3.

6 Alves et al. (2002a), Experimental Alves et al. (2002a), Model Kerdouss et al. (2006), Model this work, Model

Local bubble diameter [mm]

5

4

3

2

1

0 0

0.1

0.2

0.3

0.4

Loop 4 [m] Fig. 18. Predicted bubble diameter using DQMOM as a function of position along liquid circulation loop 4.

when using the DQMOM method when compared with the method of classes. Of course, as the number of classes used in the method of classes is increased (see Figs. 19 and 20), the advantage of the DQMOM becomes even more important, but even with a relatively low number of classes it is still much more efficient (computationally) to use DQMOM compared to CM.

Appendix A A.1. Product difference algorithm The PD algorithm proceeds in a sequence of steps beginning with setting up a triangular array of elements Pði; jÞ. Elements of

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

1939

30 Method of classes DQMOM (3 nodes)

Calculation time requirement [h]

25

20

15

10

5

0 5

10

15 20 Number of classes [-]

25

30

Fig. 19. Computational effort comparison between the method of classes and the DQMOM. Tested case I: bubble column (6144 cells, Dt ¼ 0:01 s).

440 Method of classes DQMOM (3 nodes)

400

Calculation time requirement [h]

360 320 280 240 200 160 120 80 40 0 5

10

15

20

25

30

Number of classes [-] Fig. 20. Computational effort comparison between the method of classes and the DQMOM. Tested case II: stirred-tank reactor (255000 cells, Dt ¼ 1e4 s).

the first column are Pði; 1Þ ¼ di;1

ð54Þ

where di;1 ¼ 0 for i a1 and di;1 ¼ 1 for i ¼ 1. The second column contains the moments with alternating sign: Pði; 2Þ ¼ ð1Þi1 mði1Þ

ai ¼

Pð1; i þ1Þ ; Pð1; iÞPð1; i1Þ

i A 2; . . . ; 2N

ð57Þ

ð55Þ

A symmetric tridiagonal matrix is obtained from sums and products of ai ,

ð56Þ

ai ¼ a2i þ a2i1 ;

The rest of elements of matrix Pði; jÞ are obtained from Pði; jÞ ¼ Pð1; j1ÞPði þ1; j2ÞPð1; j2ÞPði þ1; j1Þ

The coefficients of the continued fraction ðai Þ are generated by setting the first element equal to zero and computing the others according to the following relationship:

iA 1; . . . ; 2N1

ð58Þ

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B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

and

and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bi ¼ ða2i þ 1 a2i1 Þ;

i A 1; . . . ; 2N2

ð59Þ

where ai and bi are the diagonal and the co-diagonal of Jacobi matrix, respectively. When the tridiagonal matrix is determined, generation of the weights and abscissas is done by finding its eigenvalues and eigenvectors. In fact, the eigenvalues are the abscissas and the square of the first component of the j th eigenvectors vj are the weights (L½i ¼ d½i, w½i ¼ m0 ðv½i½1Þ2 ). Here, d½i and v½i½j are the obtained eigenvalues and eigenvectors from Jacobi matrix using TQL2 subroutine available in the literature.

A.2. Solving PBE by the method of moments (MOM) The MOM solves the problem by tracking the time evolution of first lower-order of moments (generally 0–5). These lower-order moments are often sufficient to describe the physical properties of particles (e.g. size distribution). The k th moments of particles (or bubbles) size distribution are defined for a continuous distribution as (McGraw, 1997) Z Lk f ðLÞ dL ð60Þ mðkÞ ¼ where f ðLÞ is the distribution function for the number density of particles of size L. k is the moment order. The key of the MOM is that the lower-order moments (generally the first 6 moments) can be tracked directly without requiring additional knowledge of the initial distribution. This method has been losing interest since its introduction because the closure requirements are very severe and depend upon the choice of the first lower order of moments.

A.3. Solving PBE by the quadrature method of moments (QMOM) The QMOM is a more recent approach of the method of moments introduced by McGraw (1997) for solving the population balance equation in the case of solid particles aggregation. In this method, the moments of the number density function f are tracked in time directly, and the analytic closure of MOM is replaced by an approximate closure condition which is based on the numerical quadrature (e.g. Gaussian, Laguerre, Lagrangian, etc.) and therefore is called the quadrature method of moments. Still more, the QMOM is based on the idea of solving the population balance equation in terms of the moments which are defined as follows (Marchisio and Fox, 2007): Z þ1 k nðx; tÞx dx ð61Þ mðkÞ ðtÞ ¼ 0

where x and nðx; tÞ are, respectively, a single internal coordinate and a number density function (NDF). Some of the moments of the NDF have specific physical meanings, for example the zeroth moment ðmð0Þ Þ represents the total number particles per unit volume. The second ðmð2Þ Þ and third ðmð3Þ Þ moments are, respectively, related to the total particles surface area and total particles volume through the following relationship: Atotal ¼ kA mð2Þ

ð62Þ

Vtotal ¼ kV mð3Þ

ð63Þ

d43 ¼ mð4Þ =mð3Þ

@mðkÞ ðtÞ þ rðuðkÞ mðkÞ Þ ¼ SðkÞ @t

ð66Þ

where SðkÞ is the source term of moment of order k and uðkÞ is the k th moment velocity defined as PN k i ¼ 1 uðLi Þwi Li ; k ¼ 0; . . . ; 2N1 ð67Þ uðkÞ ¼ P N k w L i i i¼1 The focus of the QMOM is to approximate the integral equation (61), typically by means of an n-point Gaussian quadrature as follows: Z þ1 N X k nðx; tÞx dx  Li ðtÞk wi ðtÞ ð68Þ mðkÞ ðtÞ ¼ 0

i¼1

where N is the node of the quadrature approximation (in general 3 nodes gives good results, Marchisio et al., 2003). Li ðtÞ and wi ðtÞ are the abscissas and weights of the quadrature. However, the direct solution of Eq. (68) is not recommended and requires a costly nonlinear search. A better approach is to use the moment sequence to construct a tridiagonal Jacobi matrix from which the quadrature abscissas and weights can be obtained using product-difference algorithm (PDA) described in this Appendix.

Notation CD Cl Ct Cvm Cm d D Eo f fi F~i

drag coefficient, dimensionless lift coefficient, dimensionless turbulence coefficient, dimensionless virtual mass coefficient, dimensionless k2e constant (0.09), dimensionless bubble diameter, m column depth, m Eotvos number, dimensionless friction coefficient for flow around bubble, dimensionless volume fraction of bubble of class i, dimensionless interphase forces exchange, N m3

g ~ g H

acceleration due to gravity, m s2 acceleration due to gravity, m s2 column height, m unit tensor

I k K L p Qi;k r Re t U Uj W

turbulent kinetic energy, m2 s2 exchange coefficient, kg m3 s1 length scale, m pressure, N m2 coalescence frequency between the bubbles of size group i and k, m3 s1 radial position, m Reynolds number, dimensionless time, s average velocity, m s1 velocity of phase j, m s1 column width, m

Greek letters

where kA and kV are the surface and volume shape factors. We can also define directly from the moments, different particles properties, for example the Sauter mean diameter d32 and the volume mean diameter d43 as follows:

a gi

d32 ¼ mð3Þ =mð2Þ

e

ð64Þ

ð65Þ

Now, if the moment transformation is applied to the population balance equation, the following equation is obtained:

Gi d

volume fraction, dimensionless fraction reassigned to nearby classes pivot, dimensionless break-up frequency of bubbles of size group i, s1 difference, dimensionless turbulent dissipation rate, m2 s3

ARTICLE IN PRESS B. Selma et al. / Chemical Engineering Science 65 (2010) 1925–1941

Z y

m r s t

test function angle between Ud and Ur , rad dynamic viscosity, kg m1 s1 density, kg m3 surface tension, N m1 stress tensor, kg m1 s2

Subscripts and superscripts c d dis D eff h i in l r s t vm

j

continuous phase dispersed phase dispersed drag effective hydraulic interface inlet lift relative superficial turbulent virtual mass phase index

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