Numerical simulation of gas liquid dynamics in cylindrical b.pdf

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*Corresponding author.

Chemical Engineering Science 54 (1999) 5071}5083

Numerical simulation of gas}liquid dynamics in cylindrical bubblecolumn reactors

Jayanta Sanyal*, Sergio VaH squez, Shantanu Roy, M. P. Dudukovic

Fluent Inc., Lebanon, NH 03766-1442, USA

Chemical Reaction Engineering Laboratory, Department of Chemical Engineering, Washington University, St. Louis, MO 63130, USA

Abstract

In this paper, we have attempted to validate a transient, two-dimensional axisymmetric simulation of a laboratory-scale cylindrical

bubble column, run under bubbly #ow and churn turbulent conditions. The experimental data was obtained via gamma-radiationbased non-invasive#ow monitoring methods, viz., computer automated radioactive particle tracking (CARPT) provided the data on

liquid velocity and turbulence, and computed tomography (CT) determined the gas holdup pro"les. The numerical simulation was

done using the FLUENT software and compares the results from the algebraic slip mixture model, and the two- #uid Euler}Euler

model. Reasonably, good quantitative agreement was obtained between the experimental data and simulations for the time-averaged

gas holdup and axial liquid velocity pro"les, as well as for the kinetic energy pro"les. The favorable results suggest that the simple

two-dimensional axisymmetric simulation can be used for reasonable engineering calculations of the overall #ow pattern and gas

holdup distributions. 1999 Elsevier Science Ltd. All rights reserved.

Keywords: Bubble columns; Axisymmetric simulation; FLUENT; Euler}Euler model; Algebraic slip mixture model

1. Introduction

Bubble columns are contactors in which a discontinu-

ous gas phase in the form of bubbles moves relative to the

continuous liquid phase. As reactors, they are used in

a variety of chemical processes, such as Fischer}Tropsch

synthesis (Kolbel & Ralek, 1980; Srivastava, Rao,

Cinquegrane & Stiegel, 1990), manufacture of"ne chem-

icals (Smidt, Hafner, Jira, Seiber, Sedimeier & Sabel,

1962), oxidation reactions (Hagberg & Krupa, 1976; Sit-

tig, 1967), alkylation reactions (Gehlawat & Sharma,

1970), e%uent treatment (Takahashi, Miyahara

& Nishizaki, 1979), coal liquefaction (Shah, 1981), fer-

mentation reactions, and more recently, in cell cultures

(Katinger, Scheirer, & Kromer, 1979) and production of

single cell proteins (Rosenzweig & Ushio, 1974). The

principal advantages of bubble columns are the absence

of moving parts, leading to easier maintenance, high

interfacial areas and transport rates between the gas and

liquid phases, good heat transfer characteristics, and

large liquid holdup which is favorable for slow liquid-

phase reactions (Shah, Kelkar, Godbole & Deckwer,

1982). Operation of bubble columns as reactors is a!ec-

ted by global operating parameters like gas-phase super-

"cial velocity (the liquid being processed as a batch in

many commercial applications), operating pressure and

temperature, and the liquid height. The actual variables

that in#uence bubble column performance as reactors

are gas holdup distribution, extent of liquid-phase back-

mixing, gas}liquid interfacial area, gas}liquid mass and

heat transfer coe$cients, bubble-size distributions,

bubble coalescence and redispersion rates, and bubble

rise velocities. In industrial operation, the complex two-

phase#ow and turbulence determines the transient and

time-averaged values of the above variables. The lack of

complete understanding of the #uid dynamics makes it

di$cult to improve the performance of a bubble column

reactor by judicious selection and control of the operat-

ing parameters.

The need to establish a rational basis for the inter-

pretation of the interaction of#uid dynamic variables has

been the primary motivation for active research in the

area of bubble column modeling based on computational

#uid dynamics (CFD) tools in the last decade (Kuipers

& van Swaaij, 1998). Various approaches have beensuggested for solving the same fundamental#ow problem

and modeling may be attempted at various levels

0009-2509/99/$- see front matter 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 0 09 - 2 5 0 9 (9 9 ) 0 0 2 35 - 3


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of sophistication. One may choose to treat both the

dispersed and continuous phases as interpenetrating

pseudo-continua (viz., the Euler}Euler approach, e.g.

Sokolichin & Eigenberger, 1994; Becker, Sokolichin

& Eigenberger, 1994; Ranade, 1992, 1995, 1997) or the

dispersed phase as discrete entities (viz., the Euler}Lag-

range approach: e.g. Lapin & Lubbert, 1994; De-

vanathan, Dudukovic, Lapin & Lubbert, 1995; Delnoij,Lammers, Kuipers & van Swaaij, 1997a; Delnoij,

Kuipers & van Swaaij, 1997b,c). The simulation may be

done fully transient (e.g. Becker, Sokolichin & Eigenber-

ger, 1994) or only for the steady-state time-averaged

results (e.g. Torvik & Svendsen, 1990; Svendsen, Jakob-

sen & Torvik, 1992; Jakobsen, Svendsen & Hjarbo, 1993,

Ranade, 1997). An appropriate mesh and a robust nu-

merical solver are crucial for getting accurate solutions

(e.g., Sokolichin, Eigenberger, Lapin & Lubbert, 1997).

Fundamental modeling of the "ne-scale phenomena

needed to predict the #ow pattern at the global scale is

also an issue that confronts the modeler (e.g. see Ranade,

1997; Kumar et al., 1995b).

Finally, it is highly imperative to validate the simula-

tion results against carefully designed non-invasive ex-

periments. Unfortunately, reliable data for the #ow

pattern and its dynamics in three-dimensional laborat-

ory-scale bubble columns is not common in the open

literature, so that most of the validation of the CFD

codes done in the past has only been attempted on simple

two-dimensional systems operating in the bubbly #ow

regime (e.g., Becker et al., 1994; Lin, Reese, Hong & Fan,

1996; Delnoij et al., 1997a}c).In the present work, the experiments were performed

in a 8 in diameter cylindrical air}water bubble column

using the computer automated radioactive particle track-

ing (CARPT) and the gamma-ray computed tomography

(CT) facilities. Both bubbly and churn-turbulent bubble

column#ow was simulated using FLUENT, and predic-

tions of the two-dimensional axisymmetric approxima-

tion are compared against the experimental results.

2. Experimental

At the Chemical Reaction Engineering Laboratory,

Washington University in St. Louis, USA the unique

CARPT-CT facilities allow non-invasive monitoring of

the#ow of two phases in opaque multiphase reactors on

a single platform (Devanathan, 1991; Yang, Devanathan

& Dudukovic, 1992; Kumar, 1994; Degaleesan, 1997).

Computer Automated Radioactive Particle Tracking

(CARPT) is the method employed for measuring the

time-averaged velocity and turbulence parameters of the

liquid phase in the bubble column. In CARPT, one

resorts to tagging the `typical #uid elementa

witha gamma ray source. For instance, in a bubble column

experiment in which the liquid-phase velocity pro"le is of

interest, the liquid phase is tagged with a 2.3 mm dia-

meter, `neutrally-buoyanta, hollow polypropylene

sphere "lled with radioactive Sc-46 (of 250}300Cistrength). Subsequently, the motion of this sphere is

monitored using an array of strategically positioned scin-

tillation detectors over a long time span in which the

particle (like any other typical #uid element) visits each

location in the bubble column a large number of times.Data is acquired over a very long time, typically 18}20 h,

in order to collect su$cient statistics (typically, two mil-

lion or more occurrences in the column) for the tracer

particle to sample instantaneous velocities at all points in

the column. A record of the gamma-ray photon counts at

each detector, and a pre-established calibration between

the detector counts and tracer particle location is used to

reconstruct the precise particle position at each time

instant. Time-di!erencing yields instantaneous velocities,

which when averaged at each spatial location over the

whole time span of the experiment, yield the ensemble-

averaged velocity #ow map. By the ergodic hypothesis,

this is also the time-averaged velocity "eld. The di!erence

between instantaneous and average velocity for each cell

yields the #uctuating component of velocity as a time

series. This time series is then used to construct the

cross-correlation matrix of #uctuating velocity compo-

nents. Trace of this matrix can be interpreted as the total

#uctuating kinetic energy of `turbulencea per unit vol-

ume, while the o!-diagonal components are directly re-

lated to the `turbulent Reynolds' shear stressesa

(Devanathan, 1991; Degaleesan, 1997). Error in recon-

struction of position using the CARPT technique hasbeen shown to be less than 0.5 cm, and error in spurious

root-mean-square velocities is less than 5 cm s in the

worst case (Degaleesan, 1997).

Computed tomography (CT) can be used to measure

time-averaged phase holdup pro"les in a multiphase re-

actor, and is used here to determine gas holdup pro"les in

the bubble column operated in bubbly and churn}turbu-

lent #ow. By placing a strong gamma-ray source in the

plane of interest and a planar array of scintillation de-

tectors in the same line on the other side of the reactor,

one measures attenuation of the beam of gamma radi-

ation. The attenuation is a function of the line-averaged

holdup distribution along the path of the beam. Many

such `projectionsa (e.g., 4851 for an 8 in. column) are

obtained at di!erent angular orientations around the

reactor. The complete set of projections is then used to

back-calculate the cross-sectional distribution of densit-

ies. Since the density at any point in the cross-section is

a sum of densities of individual phases weighted by their

volume fractions, the cross-sectional density distribution

of any particular phase can be uniquely recovered (if only

two phases are present).

The total scanning time in the CREL scanner is about2 h, thus the scanned image provides a time-averaged

cross-sectional distribution of mixture density. The CT

5072 J. Sanyal et al./ Chemical Engineering Science 54 (1999) 5071}5083


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setup at CREL provides a spatial resolution of better

than 5 mm, and a density resolution of about

0.04 g cm (Kumar, Moslemian & Dudukovic, 1995).

The results of the two-dimensional holdup distribution

can be subsequently averaged azimuthally for direct

comparison with the results of a suitable axisymmetric

simulation. For cylindrical bubble columns, the axisym-

metric pro"le of the gas holdup measured via CT istypically parabolic, with the highest gas holdup in the

center of the column. At lower super"cial gas velocity, the

pro"le is #atter across the cross-section and becomes

signi"cantly steeper with the increase in gas velocity.

This fact is used to discriminate between bubbly #ow

regime and churn-turbulent regime of operation of

a bubble column. The liquid circulation in a bubble

column is driven by buoyancy of the gas, thus the circula-

tion velocities are signi"cantly higher (with steeper mean

axial liquid velocity pro"les) in the churn}turbulent#ow

regime as compared to the bubbly #ow regime.

Details of the experimental setup and procedure

for the particle tracking and the tomographic techniques

may be found elsewhere (Devanathan, Moslemian

& Dudukovic, 1990; Devanathan, 1991; Moslemian,

Devanathan & Dudukovic, 1992; Yang et al.,

1992; Kumar, 1994; Kumar et al., 1995; Kumar, Moslem-

ian & Dudukovic, 1997; Degaleesan, 1997; Roy,

Chen, Degaleesan, Gupta, Al-Dahhan & Dudukovic,

1998).

3. Numerical Simulation

In the present work, the #ow in the bubble column

reactor was modeled using two di!erent approaches

incorporated in the FLUENT software *the Eulerian

multiphase model and the algebraic slip mixture

model (ASMM). Although both models are used

to predict multiphase #ows, there are fundamental

di!erences in their respective approaches which are

outlined here.

3.1. The Eulerian multiphase model

In the Eulerian two-#uid approach, the di!erent

phases are treated mathematically as interpenetrating

continua. The derivation of the conservation equations

for mass, momentum and energy for each of the

individual phases is done by ensemble averaging

the local instantaneous balances for each of the phases

(Anderson & Jackson, 1967). The basic assumptions of

this formulation used in the present computations are as

follows:

All phases are treated as interpenetrating continua andthe probability of occurrence of any one phase in

multiple realizations of the#ow is given by the instan-

taneous volume fraction of that phase at that point.

Sum total of all volume fractions at a point is identi-

cally unity.

Both#uids are treated as incompressible, and a single

pressure"eld is shared by all phases.

Continuity and momentum equations are solved for

each phase.

Momentum transfer between the phases is modeledthrough a drag term, which is a function of the local

slip velocity between the phases. A characteristic dia-

meter is assigned to the dispersed phase gas bubbles,

and a drag formulation based on a single sphere sett-

ling in an in"nite medium is used.

Turbulence in either phase is modeled separately.

The conservation equations can be written as follows:

Continuity (kth phase):

t()#) (u)"

m . (1)

Momentum (kth phase):

t(

u)#) (

uu

)"!

p#)

#F

#

(K

(u!u

)#m

u

), (2)

where is the kth phase stress}strain tensor, whose

components are given by

"

u

x

#u

x

!2

3

u

x

. (3)

The fourth term on the right-hand side of Eq. (2) repres-

ents the interphase drag term, with K

being the mo-

mentum exchange coe$cient between thepth and thekth

phases. The evaluation of the needed drag coe$cient

requires the bubble Reynolds number which is based on

the local slip velocity for a single sphere of constant

diameter sedimenting in stagnant #uid. In the present

computations, the drag coe$cient, K

is based on the

generalized correlations (Morsi & Alexander, 1972).

The turbulence in the continuous phase is modeled

through a set of modi"edk}equations with extra terms

that include interphase turbulent momentum transfer

(Launder & Spalding, 1974; Elghobashi & Abou-Arab,

1983), supplemented with extra terms that include the

interphase turbulent momentum transfer. This term can

be derived exactly from the instantaneous equation of the

continuous phase and involves the continuous-dispersed

velocity covariance. The turbulence quantities for the

dispersed phase in FLUENT are based on characteristic

particle relaxation time and Lagrangian time scales (Sim-onin & Viollet, 1990). For the dispersed gas phase, the

turbulence closure is e!ected through correlations from

J. Sanyal et al./ Chemical Engineering Science 54 (1999) 5071}5083 5073


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the theory of dispersion of discrete particles by homo-

geneous turbulence (Tchen, 1947).

The equations discussed above are solved using an

extension of the SIMPLE algorithm (Patankar, 1980).

The momentum equations are decoupled using the full

elimination algorithm (FEA). Using SIMPLE-FEA

(Spalding, 1980), the variables for each phase are elimi-

nated from the momentum equations for all other phases.The pressure correction equation is obtained by sum-

ming the continuity equations for each of the phases. The

equations are then solved in a segregated, iterative

fashion and are advanced in time. At each time step, with

an initial guess for the pressure "eld, the primary- and

secondary-phase velocities are computed. These are used

in the pressure correction equation (continuity), and

based on the discrepancy between the guessed pressure

"eld and the computed"eld, the velocities, holdups and

#uxes are suitably modi"ed to obtain convergence in an

iterative manner.

3.2. The Algebraic Slip Mixture Model

The Algebraic Slip Mixture Model (ASMM) has an

underlying philosophy that is quite distinct from the

Euler}Euler two-yuid model (Manninen, Taivassalo

& Kallio, 1996). The principal assumptions in this formu-

lation are as follows:

It models the phases as two interpenetrating continua,

with the probability of existence of each phase at

a point in the computational domain given by itsrespective volume fraction (holdup). In general, the

two phases move at di!erent velocities.

A single equation is solved for continuity of the mix-

ture and a single equation is solved for the momentum

of the mixture.

Motion of each phase relative to the center of mass of

the mixture in any control volume is viewed as a di!u-

sion of that phase; this introduces the concept of a dif-

fusion velocity of each phase (which is analogous and

directly related to the slip velocity and the drift velo-

city, as referred to in the classical drift #ux model for

a mixture, Wallis, 1969).

The Reynolds'-averaged mixture momentum equation

has a term, called the diwusion stress, which originates

because of the relative slip between the two phases.

This requires closure in terms of the di!usion velocity

of each phase (or, equivalently, the drift or the slip

velocity between the phases). In the ASMM, this is

supplied by assuming that the phases are in local

equilibriumovershort spatial length scales. This means

that the dispersed phase entity (bubble, particle) al-

ways slips with respect to the continuous phase at its

terminal Stokes' velocity in the local accelaration "eld. Thediwusion stressterm is also the only term in which

the phase volume fractions appear explicitly. In order

to back out the individual phase velocities and volume

fraction at the end of the computation at each time

step, it is necessary to solve a di!erential equation for

the volume fraction of the dispersed phase coupled

with the solution of the mixture equations. This equa-

tion is obtained from the equation of continuity for the

dispersed phase.

Finally, the turbulent stress term in the mixture equa-tion is closed by solving a k} model for the mixture

phase.

Based on these assumptions, the"nal equations of the

ASMM model are formulated as follows.

Equation of continuity for the mixture:

t(

)#

x

(

u

)"0. (4)

Equation for mixture momentum (jth component):

t(

u

)#

x

(

u

u

)"!p

x

#

x

u

x

#u

x

#g

#F#

x

u

u

. (5)

Volume fraction equation for the secondary phase:

t(

)#

x

(u

)"!

x

(u

). (6)

The above equations are formulated in terms of the

mixture density,

, mixture viscosity,

, and the mass-

averaged mixture velocity, u

, which are de"ned as fol-

lows:

"

,

"

, u

"

u

. (7)

u

is the drift velocity of the kth phase with respect to

the mixture center of mass, and is related to the slip

velocity with respect to the continuous phase in the

following manner:

u"u

!u

"v

!

1

v

, (8)

wherev

is the slip velocity of the kth phase with respect

to the continuous phase. In the ASMM, the slip velocityis calculated based on the assumption of local equilib-

rium between the phases over short spatial scales. The



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Table 1

Parameters used in the numerical simulation

Grid 300 (axial)19 (radial)

Cell size 0.66 cm (axial)0.5 cm (radial) (uniform grid)

Time step 0.01 s (Euler}Euler); 0.005 s (ASMM)

Iterations per time step for convergence around 25

Under-relaxation parameters Euler}Euler: 0.6 (pressure), 0.4 (velocities); ASMM: 0.3 (pressure), 0.7 (velocities)

Bubble size 0.5 cm

Drag formulation Single sphere drag correlation (Morsi & Alexander, 1972)

Turbulence model Standard k} model (Elghobashi & Abou-Arab, 1983)

convection terms of the dispersed phase are assumed to

be of similar magnitude as the convection terms of the

mixture (Manninen et al., 1996). Thus:

v"

(!

)d

18

f g!

Du

Dt, (9)where

f"1#0.05Re Re(1000,

0.018Re Re*1000.(10)

The complete set of the above equations are discretized

in an unstructureddomain, and solved in a segregated

fashion using an implicit time stepping algorithm. The

pressure correction is e!ected through the SIMPLEC

algorithm (van Doormal & Raithby, 1984). At each time

step and in each computational cell, the volume fraction

of the dispersed phase is used to back-calculate from the

mixture velocity the velocities of the individual phases.

4. Results and Discussion

The simulations were performed for a 8 in. (19.0 cm

i.d.) air}water bubble column. Experimental data using

the CARPT technique (Degaleesan, 1997) and the CT

scanner (Kumar, 1994) were available at the conditions of

interest. Gas super"cial velocities simulated were 2.0 and

12.0 cm s, characteristic of bubbly #ow and churn}tur-

bulent #ow regime, respectively. The column contained

a batch liquid with unexpanded height of 104.5 and95.0 cm, respectively. The distributor used in the experi-

ments was a perforated plate, with 0.33 mm diameter

holes in a square pitch, with an open area of 0.1%.

Distributor e!ects have been shown to a!ect (Kumar,

1994; Degaleesan, 1997) the overall #ow pattern in the

bubble column. However, in the present simulations,

distributor e!ects have been ignored and the perforated

plate has been modeled as a uniform surface source of the

gas phase.

The numerical simulation was performed on a 300

(axial)19 (radial) rectangular grid. Eqs. (1), (2), (4)}(6)

were solved in a transient fashion with a time step of

0.01 s for the Euler}Euler simulations and 0.005 s for the

ASMM. A number of sub-iterations were performed

within each time step to ensure continuity. It took ap-

proximately 25 iterations per time time step to converge

the residuals by three orders of magnitude or more for

both the Euler}Euler and the ASMM. The under-relax-

ation parameters were set to 0.6 for pressure and 0.4 for

velocities for the Euler}Euler case and 0.3 for pressure

and 0.7 for velocities for the ASMM. The power-lawdiscretization scheme was used for the Euler}Euler

model and a "rst-order upwinding scheme was used for

the ASMM. Inlet boundary conditions are assigned at

the distributor, and outlet conditions at the free surface.

No-slip conditions were applied at the wall, and sym-

metry conditions at the central axis of the column.

A bubble size of 5 mm, typical of air}water bubble col-

umns under atmospheric pressure operating at these

super"cial velocities with a perforated plate sparger, was

used in the simulations. Parameters used in the simula-

tion are summarized in Table 1.

In Fig. 1, a set of vector plots of the time-averaged

liquid velocity pro"le obtained via the CARPT technique

is shown (Degaleesan, 1997), at a gas super"cial velocity

of 12 cm s. The results are presented in four vertical

planes at four di!erent angular orientations. Fig. 2 shows

vector plots for the same experiment, at di!erent cross-

sectional planes. One notes that irrespective of angular

orientation (Fig. 1), in a time-averaged sense, the liquid

goes up at the center of the column and descends at the

walls in a single circulation loop. The circulation shows

a symmetric r}z dependence alone, with the instan-

taneous azimuthal component of velocity not being sig-ni"cant in determining the time-averaged #ow pro"le

(Degaleesan, 1997). Further, azimuthal}radial compo-

nents of time-averaged liquid velocity (Fig. 2) are signi"-

cantly smaller than the axial components (Fig. 1). Slight

asymmetries seen just above the distributor and below

the free surface are ascribed to asymmetric distributor

and gas disengagement e!ects, respectively. Data col-

lected using the CARPT technique under a variety of

operating conditions, distributors and column sizes, and

across various #ow regimes con"rms this picture (De-

galeesan, 1997); Figs. 1 and 2 being representative cases.

The message is that while the #ow in bubble columns is

highly turbulent and chaotic, and the transient #ow is



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Fig. 2. Velocity vector plots (cross-sectional views) for column diameter of 19 cm: ;"2.0 cm s(Degaleesan, 1997).

It is worth noting here that imposition of the symmetry

boundary condition at r"0 causes the liquid #ow to

develop very quickly (around "rst 5 s of real time) and

reach its long-time pattern. In contrast, the real experi-

ment is characterized by a highly dynamic #ow, with

three-dimensional vortical bubble swarms which cause

smaller bubbles and liquid to be trailed in their wakes.

Instantaneous #ow is never truly axisymmetric in reality,

while the simulation is tailored to be so. Thus, one isunable to capture realizations of the true instantaneous

velocities (which can have signi"cant azimuthal compo-

nents), and hence all the true time-scales in the problem.

Consequently, the true swirling bubble swarms motions,

as well as time scales of such phenomena, will be missed

by axisymmetric models.

In Fig. 4 the time-averaged liquid axial velocity pro-

"les are compared against the time-averaged velocity

pro"les obtained by CARPT. Fig. 5 shows the compari-

son for gas holdup pro"les obtained from the simulations

and from computed tomography (CT). Results presentedare at a height of 53 cm above the distributor, typical of

the fully developed region of the #ow. All the simulations



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Fig. 3. Velocity vector plots obtained from simulations: (a) ;"2.0 cm s (two #uid) (b) ;

"12.0 cm s (two #uid). (The r}z plane has been

`mirroredaalong the axis of symmetry to show the vector "eld across the diameter.)

showed that there is no signi"cant axial variation in any

of the variables, in the zone of developed #ow. Thecenterline axial velocity is overpredicted by both the

Euler}Euler two-#uid model, as well as the ASMM.

However, the general shape of the pro"le is well captured

and the discrepancy in the model predicted andCARPT-measured time-averaged axial liquid velocity di-

minishes as one moves radially outwards in the column.



9/13

Fig. 4. Comparison of axial liquid velocity pro"les.

One is also able to predict the cross-over point (i.e., the

radial location where the axial velocity component be-

comes zero) reasonably well.

In Fig. 5, one observes that in the bubbly #ow regime(;

"2 cm s), both the ASMM and the two-#uid

models agree in their prediction of the gas holdup pro"le

and seem to predict the mean holdup pro"le reasonably

well. (Unfortunately, the experimental data for gas hold-

up at this condition were not of the highest accuracy.) In

the churn}turbulent regime, however, there is still some

discrepancy between the predictions of the two models

and the data seems to be bracketed by the predictions.

The roughly parabolic pro"le observed in the experi-

mental data is seen in the simulation results as well. The

higher volume fraction of gas at the center of the column

drives the liquid#ow upwards at a high velocity, due to

gas buoyancy, and the liquid, being in batch mode, re-

turns downwards at the periphery.

Comparison of kinetic energy pro"les (obtained by

solution of the k} model in the simulations, and those

measured via CARPT, also shows good agreement in the

shape of the curves (Fig. 6). Kinetic energy pro"les typi-

cally exhibit a maximum around thecross-over point, due

to large gradients and large #uctuations in the liquid

velocity. In the bubbly#ow regime this e!ect is not very

signi"cant (because of suppressed turbulence) and the

turbulent kinetic energy is practically #at as a function ofradius. These e!ects are clearly captured in the simula-

tion results presented in Fig. 6. It may be noted that for

the ASMM, the kinetic energy plotted in Fig. 6 is the

mixture kinetic energy in contrast to the liquid phase

turbulent kinetic energy. However, it can be readily

shown that due to the signi"cantly lower gas-phase den-sity, liquid-phase inertia predominates and is the domi-

nant contributor to the mixture kinetic energy.

Quantitative agreement between the two-#uid model and

data is particularly good at churn}turbulent conditions.

Based on what we see in Fig. 6 one may argue that in

any bubble column simulation in which the mean velo-

city pro"le and the gas holdup pro"le is reasonably

predicted, oneshouldalso expect to observe a reasonable

comparison of the overall kinetic energy pro"les. The

total energy input to the system is through the gas in#ow,

and a simple overall energy balance reveals that this total

input energy is dissipated as work done against the

distributor, walls, and the #uid-phase viscosity and tur-

bulence. If the "rst two e!ects are small (distributors may

be properly modeled in a more sophisticated simulation),

then a reasonable prediction of the mean velocity and

holdup pro"les necessarily implies that the remaining

kinetic energy (both in the simulation as well as in the

real experiment)mustbe accounted for as the `turbulenta

kinetic energy. If the pertinent physics is properly

modeled, then this should result in good prediction of the

kinetic energy proxlesas well.

Turbulent kinetic energy referred to in models like thetwo-phase k} formulation arises from the turbulence

microscale, while that obtained from experiments like



10/13

Fig. 5. Comparison of gas holdup pro"les.

CARPT (Fig. 6) lumps energies fromallthe #uctuations

about the time-averaged (over 18}20 h) mean pro"le,

sampling more e$ciently the larger scales. (For example,

it is estimated that the `CARPT tracer particleacannot

respond to the turbulence #uctuations above 20}25 Hz in

frequency.) In the present type of axisymmetric simula-

tions (in which we attempt to predict the 18}20 h time-

averaged pro"les), the `turbulentakinetic energy (i.e.,all

the#ow energy that isnotdue to the mean#ow) is forced

to represent the large-scale turbulent kinetic energy (as

measured by CARPT). In other words, the 2D axisym-

metric simulation imposes allthe turbulence time scales

smaller than the total averaging time to contribute to the

turbulent kinetic energy at the `microscalea (hence, cap-

tured by the k} model). In general, however (e.g. from

other more sophisticated simulations), one would need to

compare the right scales between the simulation and the

experiment for a meaningful comparison of kinetic ener-

gies. Good comparison in Fig. 6 is not merely fortuitous,

but is actually very consistent with our understanding,expectation and development of the the two-dimensional

axisymmetric model, as well as the CARPT technique.

Similar assertions would hold for steady-state model

simulations (e.g., Ranade, 1997) as well.

It however, does notnecessarily follow that the entire

turbulence "eld is well captured, even if the mean

pro"les are. This is because turbulence is a multi-scale

phenomenon, with extremely complex energy cascading

in multiphase #ows. Models like the k} formulation

phenomenologically model turbulent kinetic energy pro-

duction at large scales and its dissipation as small scales

(Tennekes & Lumley, 1972). They do not "x the scale

information, neither spatial correlations between velocity

#uctuations between two directions, nor autocorrela-

tions (in time).

Thus, good prediction of kinetic energy pro"les does

not necessarilyimply good predictions of Reynolds' stres-

ses (i.e., quantities dependent on correlations between

various components of velocity #uctuations), for in-

stance, or of turbulent eddy di!usivities (i.e., quantities

dependent on autocorrelations in time). For either of

these quantities, our present models showed reasonableorder-of-magnitude comparisons and the radial trends

were also captured, but exact values were not. More



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Fig. 6. Comparison of turbulent kinetic energy pro"les.

sophisticated models with transport equations for the

entire second or higher-order turbulence velocity correla-

tions may be required for this purpose.

5. Conclusions

In summary, the usefulness of the two-dimensional

axisymmetric models has to be acknowledged. They pro-

vide good engineering descriptions, and can be used

reliably for approximately predicting the time-averaged

#ow and holdup patterns in bubble columns. We have

validated the models in di!erent#ow regimes, as well as

with di!erent fundamental modeling approaches to de-

scribe the#ow (i.e., ASMM versus Euler}Euler two#uid

models). We have also shown that a reasonable choice of

turbulence description is able to predict the kinetic en-

ergy pro"les, and must do so in a self-consistent model.

Further improvements in the turbulence model, or in

description of two-point correlations, is likely to improvethe description of the other turbulence parameters such

as shear stresses and turbulent di!usivities.

A fully transient three-dimensional model is necessary

to capture the transient #ow structures in the bubble

column, which are in general, not axisymmetric and have

a signi"cant azimuthal component. It would be interest-

ing to examine the relative importance of these phe-

nomena in determining the overall #ow pattern, and

more importantly, the overall reactor performance.

A satisfactory answer to these issues can only be deter-

mined through a detailed comparison of#ow, turbulence

and reactor performance variables between experiment,

two-dimensional codes and three-dimensional models.

Presently, work is in progress in this area and we hope to

present some of our results in this "eld in a subsequent

communication.

Notation

a local acceleration, cm s

C drag coe$cient, dimensionlessd particle (bubble) diameter, cm

f friction factor, dimensionless



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F external body force per unit volume, dyne cm

g acceleration due to gravity ("981 cm s)

K momentum exchange coe$cient, g cms

m mass transfer rate per unit volume between

phases, g cms

p pressure, dyne cm

Re Reynolds number, dimensionless

u velocity, cm s

v slip velocity (mixture model), cm s

t time, s

x spatial coordinate, cm

Greek letters

volume fraction (holdup) of phase

Kronecker's delta

density of phase, g cm

e!ective viscosity, g cms

stress tensor, dyne cm

Superscripts and subscripts

c continuous phase (mixture model)

D di!usion variable (mixture model)

i coordinate index

j coordinate index

k phase index

m mixture variable

n number of phases

p phase index (distinct fromk)

s secondary phase (mixture model)

6. For Further Reading

The following reference is also of interest to the reader:

Lin et al., 1996.

Acknowledgements

The authors would like to thank Dr. Sailesh B. Kumarand Dr. Sujatha Degaleesan for sharing their experi-

mental data, and Fluent, Inc. for their code. We express

our sincere appreciation to Dr. S. Subbiah of Fluent, Inc.

for o!ering constructive suggestions to improve the paper.

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