8 Lindborg Jakobsen

8/13/2019 8 Lindborg Jakobsen

1/10

Fourth International Conference on CFD in the Oil and Gas, Metallurgical & Process Industries

SINTEF / NTNU Trondheim, Norway

6-8 June 2005

1

2D SIMULATIONS OF BUBBLING GAS-SOLIDFLUIDIZED BED REACTORS

Hvard LINDBORG and Hugo A. JAKOBSEN

1NTNU Department of Chemical Engineering, 7491 Trondheim, NORWAY

ABSTRACT

This paper discusses the simulation of bubbling gas-solidflows by using an Eulerian modeling approach. Twoclosure models, the constant particle viscosity model(CPV) and a model based on kinetic theory of granularflow (KTGF), are compared in performance to describethe time-averaged velocity fields and bed expansion inboth a circular and rectangular column. The time averagedvelocity fields and bed expansion obtained in thesimulations are compared with experimental data obtainedby Lin et al. (1985) for validation.

In this work an in-house code has been developed basedon the finite volume and the fractional step approach usinga staggered grid arrangement. The velocities in bothphases are obtained by solving the two-dimensionalReynolds-averaged Navier/Stokes equations using apartial elimination algorithm (PEA) and a coupled solver.The k-epsilon turbulence model is used for the continuousphase.

Axi-symmetric simulations using the KTGF model givesresults in fairly well agreement with the experimentalresults, while the CPV model predicts unrealistically highsolids concentration along the central axis due to thesymmetry condition. Applying higher order discretizationsof the convective terms results in a higher solids voidfraction as long as the diffusion in the system issufficiently high.

The first step of including a chemical reactive systemseems to be successful, although validation of thechemical process still remains.

Keywords: Bubbling bed reactors; Fluid Mechanics;

Multiphase flow; Simulation; Viscous flow; CFD;

NOMENCLATURE

Dea dispersion coefficientMk interfacial momentum transfer terme restitution coefficientg gravity vectorg0 radial distribution functionp pressurevk velocityRj reaction rate

d conductivity of solid phase fluctuating energyk void fraction of phase k drag coefficient

collisional energy dissipationd solid phase bulk viscocity granular temperatured

dilutedilute viscosityk intrinsic density of phase k dynamic viscosityk Stress of phase ki species mass fraction

INTRODUCTION

The Eulerian modeling approach is probably the mostcommonly used approach for predicting the dynamicalbehavior in fluid-particle systems (for example Enwald etal., 1996, van Wachem et al., 1998). This approachdescribes both phases as interpenetrating continua and hasmore potential, compared to an Euler-Lagrangianapproach, in situations where the dynamics of the systemis of interest or the mass-loading of the dispersed phase isconsiderable. However, many Euler-Euler models used inthe literature suffer from uncertainties in prescribing theinternal momentum transfer in the solid phase (Ding andGidaspow, 1990). Various non-Newtonian models for theinternal stresses of the solid phase correlated withexperimental observations have been proposed. In recentyears, more fundamental closures for these stresses havebeen developed based on the application of the kinetictheory for dense gases to particulate assemblies. The workof Ding and Gidaspow (1990), Samuelsberg and Hjertager(1996) and Laux (1998) among others have shown theability of the Euler-Euler granular temperature approachto model, numerically, gas-solid fluidized beds. In thiswork two closure models, the classical constant particleviscosity model (CPV) and a more fundamental modelbased on kinetic theory of granular flow (KTGF), arecompared in performance to describe the time-averagedvelocity fields and bed expansion in both a circular andrectangular column. The former model can be consideredas a simplification of the latter, where a uniform andconstant granular temperature in the entire fluidized bed isassumed.

The motivation for this work is to describe reactive flowsin fluidized bed reactors and thereby couple the fluiddynamic models with a chemical reaction modelcontaining complex reaction kinetics and chemicalequilibriums. In a first step of this description, thechemical system chosen is the synthesis gas process forhydrogen production.


2/10


3/10


4/10

4

Chemical process

The chemical process implemented is the conversion ofnatural gas into synthesis gas. The kinetics is based onmethane steam reforming with Ni/MgAl2O3 as thecatalyst, where the most important reactions are:

CH4+ H2O = CO + 3H2 (19)

CO + H2O = CO2+ H2 (20)

CH4+ 2H2O = CO2+ 4H2 (21)

The kinetics and correlations needed are taken fromFroment and Bischoff (1990), De Groote and Froment(1995) and Xu and Froment (1989a,b). The operatingconditions and boundary conditions used in thesimulations as well as the data for the gas phase are givenby Jakobsen et al. (2002). The inlet temperature and thetemperature in the reactor are assumed constant and set to1100 K.

NUMERICAL SOLUTION ALGORITHM

The algorithm applied in this work is based on thealgorithm applied by Jakobsen et al. (2005). Thegoverning two-fluid equations are discretized by a finite-volume algorithm. A staggered grid arrangement is used.A fractional-step scheme similar to those used byTomiyama and Shimada (2001) and Lathouwers (1999) isdeveloped on the basis of the single-phase algorithmreported by Jakobsen et al. (2002), Lindborg et al. (2004)and Jakobsen et al. (2005). The fractional steps definingthe algorithm are sketched in the following discussion.

1. Velocity predictionTemporary velocity estimates are found by solvingthe momentum balances using implicit discretizationsof the velocity variables:

( )

*

*

*

*

n n n n n

k k k k k k

V

n n n

k k k k V

n nkk

V V

n n

k k k k V V

dvt

dv

p dv dv

dv dv

a r a r

a r

a

a r

- = D

-

- +

+ +

v v

v v

g M

(22)

The equations are solved by using a PartialElimination Algorithm (PEA) in a coupled manner,though separately for each velocity direction.

2. Void fraction prediction

Temporary void fractions are found by solving thedispersed phase continuity equation:

( )

*

* *

n n n

d d d d

V

n

d d dV

dvt

dv

a r a r

a r

- = D

-

v (23)

3. Granular temperature predictionIn intermediate granular temperature is computed byuse of temporary void fractions and velocity fields:

( )

( )

* *

* ** *

** **

*

** *

3

2

3

2

:

3

d d

n n n n

d d

V

n

d d dV

d dV

dV

V V

dvt

dv

dv

dv

dv dv

a r a r

a r

b g

Q - Q = D

- Q

+

+ G Q- Q -

v

v (24)

Since the source terms are all functions of thegranular temperature, negative temperatures can beavoided by using positive source terms only on theright hand side of the discretized equation, whilenegative source terms are added to the center-coefficient. The particle pressure is then calculatedbased on the intermediate values of the void fractionsand the granular temperature.

( )* * * * *01 2 1n

d d d d p e ga r a= Q + -

(25)

4. Particle pressure- and velocity correctionThe velocity of the dispersed phase is corrected dueto change in particle pressure:

* ** * *n nd d d d d d

V

d dV V

dvt

p dv dv

a r a r

d d

- = D

- +

v v

M

(26)

where* n

d d dp p pd = - and

** *d d dd = -M M M . The velocity of the

continuous phase is corrected due to velocity changeof the dispersed phase velocity through the phaseinteraction term:

* ** * *n nc c c c c c

V

cV

dvt

dv

a r a r

d

- = D

+

v v

M

(27)

Solved with respect to the corrected velocity of thecontinuous phase gives:

( )( )* * ** **

**

n n

c c c c d

c

t

D

a r b- D -= v M vv (28)


5/10

5

where* n nc cD ta r b= + D . Inserting equation

(28) into equation (26) solved with respect to thecorrected velocity of the dispersed phase gives:

( )

*

*

*** *

**

*** ** 1

n n

d d d

d c

n n

c c c c

d

n

d d

t p t

tt

D

tt

D

a r

d

ba r

ba r b

- D + D D + - D

= D + D -

v

M

v M

v (29)

Point 3 and 4 are performed in an iteration procedureuntil the change in void fraction is less than a certaincriterion.

5. Gas pressure- and velocity correctionA pressure update is then performed to update thepressure- and velocity-fields.

* 1 * **

*

n n n

k k k k k k

V

kV

dvt

p dv

a r a r

a d

+ - = D

-

v v

(30)

where1n np p pd += - . The equation for this

update is found by introducing a time-independentvolume fraction (and density) step.Summation of the normalized continuity equationsover both phases gives:

( )* 101

0n nk k kk k

a rr

+ = v (31)

The transient term in the continuity equations cancelsout as the sum of the volume fractions equals 1.Introducing the velocity correction

1 **nk k k

d += -v v v into equation (31) results in:

( )

( )

*

0

* **

0

1

1

n

k k k

k k

n

k k kk k

a r dr

a rr

=

v

v

(32)

The velocity correction can be written in terms of apressure correction. The pressure-velocitydependence can be obtained by subtracting aprediction step of the momentum equations from thesemi-implicit discretization:

( )*k k k d cT pd a d b d d = - -v v v (33)

where

* nk k

kT

t

a r=

D. Rearranging with respect to the

velocity correction gives for the continuous phase:*

d cc

c

pT

bd a d db

- =+

vv (34)

And for the dispersed phase:*

c dd

d

p

T

bd a d d

b

- =

+

vv (35)

Substituting the velocity correction in equation (35)with equation (34) gives the pressure-velocitydependence for the dispersed phase:

( )

*d c

d

c d c d

d

Tp

T T T T

G p

a bd d

b

d

+= -

+ +

= -

v (36)

For the continuous phase this gives:*

d cc

c

c

Gp

T

G p

b ad d

b

d

+= -

+

= -

v (37)

A Poisson equation is obtained after inserting thepressure velocity dependence into equation (32):

( )

( )

*

0

* **

0

1

1

n

k k k

k k

n

k k k

k k

G pa r dr

a rr

=

v (38)

6. Void fraction updateThe void fractions are updated by solving the

continuity equation for the dispersed phase:

( )

1

1 1

n n n n

d d d d

V

n n n

d d dV

dvt

dv

a r a r

a r

+

+ +

- = D

-

v (39)

7. Granular temperature updateThe granular temperature is updated based on the newvoid fraction and velocity-fields:

( )

( )

1 1

1 1 1

1 1 1

1

3

2

32

:

3

n n n n n n

d d d d

n n n n

d d d

nn n

d d d

n

t

a r a r

a r

b g

+ +

+ + +

+ + +

+

Q - Q = D

- Q

+ + G Q

- Q -

v

v

(40)

At the end a particle pressure-update is performed:

( )

1 1 1

1 101 2 1

n n n n

d d d

n n

d

p

e g

a r

a

+ + +

+ +

= Q

+ - (41)

8. Component mass fraction update


6/10

6

Integrate the transport equations for the componentsin the gas phase. This is split into two steps. Aconvection-diffusion step:

( )

( )

1 *

1 1 *

1 *

n n n n n

c c i c c i

V

n n nc c c i

V

n n

c c ea iV

dvt

dv

D dv

a r w a r w

a r w

a r w

+

+ +

+

- = D

-

+

v (42)

The final step is the reaction step:1 1 *n n n n n

c c i c c i

V

iV

dvt

R dv

a r w a r w+ + - = D

(43)

9. Density update

When including compressibility in the gas phase, adensity update is needed due to any eventual changesin temperature and fluid mixture composition. Thedensity is then updated by using a suitable equationof state (EOS). In the case of an ideal gas:

1*

n

mc

V V

p Mdv dv

RTr

+ = (44)

The density at time step n+1is obtained from the gasphase continuity equation:

( )

1 1 1 *

1 1 1

n n n

c c c c

V

n n nc c c

V

dvt

dv

a r a r

a r

+ + +

+ + +

- = D

-

v(45)

The fractional-step concept applied consists of successiveapplications of the predefined operators determining partsof the transport equation. The convective and diffusiveterms are further split into their components in the variouscoordinate directions. The time-truncation error in thesplitting scheme is of first order. The convective terms aresolved using a second-order TVD scheme in space and afirst-order explicit Euler scheme in time. The TVDscheme applied was constructed by combining the centraldifference scheme and the classical upwind scheme byadopting the "smoothness monitor" of van Leer (1974)and the monotonic centered limiter (van Leer; 1977).

All the linear equation systems are solved withpreconditioned Krylov methods. A bi-conjugate gradient-solver (BCG) is applied for the granular temperature andthe turbulent dissipation energy while a conjugategradient-solver (CG) is used for the other variables. Thesolvers are preconditioned with a Jacobi preconditionerwhich is computationally very cheap and effective. Inaddition, compared to the more robust ILU preconditioner,it does not introduce any numerical perturbations.

Overview of the algorithm

Velocity prediction. Solution of the momentum

equations (22). Void fraction prediction using equation (23).

Prediction of granular temperature usingequation (24).

Correction of particle pressure and velocitiesdue to particle pressure correction (equations(25), (28) and (29).

Correction of gas pressure and velocities usingequation (38).

Updating void fractions (equation (39)). Updating granular temperature (equation (40)

and particle pressure (equation (41)). Calculating species composition (equation (4)). Updating the gas phase density.

SAMPLE RESULTS AND DISCUSSION

Simulations described in this work were performed for 25s with a time step size of t=0.2 ms. Maximum Courantnumber is in the range 0.02 to 0.03. The results reportedare averaged over the last 20 seconds to avoid influence ofthe start-up phase.

To validate the models, the simulation results have beencompared with experimental results of Lin et al. (1985),who measured particle velocity in a cylindrical bed usinga radioactive particle tracking method. The bed, 13.8 cmin diameter, contained glass Ballotini beads withdiameters in the range of 0.42-0.60 mm, and a static bedheight of 11.3 cm. In this work a particle diameter of 0.50mm is assumed with a density of 2.5 g/cm3. To comparethe results with the work of Ding and Gidaspow (1990)and Pain et al. (2002) a uniform grid size of r=0.69 cmand z=1.13 cm was used. In addition, first order upwinddiscretization of convective terms is applied. Inletsuperficial velocity of air was set as 64.1 cm/s, and a

restitution coefficient value of 0.995 was used. Thesimulations are run in serial on an SGI Origin3800 withR14000 processors of 600MHz. A typical run on this gridsize takes approximately 3800 CPU seconds when usingthe CPV model while the KTGF model is morecomputationally expensive with 5100 CPU seconds.

Results by using the KTGF model are shown in Figure 1.On average, the particles ascend at the center and descendnear the wall. A vortex appeares in the upper portionbetween the center and the wall, as seen in theexperimental results. However, the opposite directedvortex of lower velocities in the lower portion between thecenter and the wall seen in the experiments does not

appear in the simulation results. Neither Ding andGidaspow (1990) nor Pain et al. (2002) obtained thesevortexes in their work (Figure 2).


7/10

7

0.06 0 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

25 cm/s

Vector scale

Figure 1Computed (left) and experimental (right) solidsvelocities in cylindrical bed.

Figure 2Solids velocities in cylindrical bed obtained byPain et al. (2001 (left) and ) Ding and Gidaspow (1990)(right).

On the other hand, the two oppositely directed vortexesare predicted by the CPV model, as shown in Figure 3.However, this model does not predict the correct particlemotion at the center, as the particles tend to compact anddescend quite fast in this region. The same motion is seenwhen applying two-dimensional Cartesian coordinates asshown in Figure 4. According to Sun and Gidaspow

(1999) and Pain et al. (2001) some unrealistic highconcentration of particles in the central region of thereactor can occur when applying axi-symmetricalboundary conditions, since the particles are prohibitedfrom crossing the central axis. To overcome this problem,the symmetrical boundary conditions have to be removedand the whole domain has to be simulated where thesymmetry is broken. It is common to insert some gaspockets asymmetrically into the bed initially or slightlytilt the gravity vector for a short period of time duringstart-up to break the symmetry. Special attention shouldbe paid to the calculation of the coefficients and sourceterms of the equation systems as well as the solversapplied for solving them to avoid that the symmetry is

broken by numerical perturbations originating fromaccuracy restrictions in the machine. The axi-symmetricboundary condition does however not always lead to

unphysical flows. Ding and Gidaspow (1990) show thatboth cases are possible depending on flow rate, bed heightand geometry.

0.06 0.04 0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.1 0.1 0.1

0.2 0.2 0.2

0.30.3 0.3

0.4

0.4

0.4

0.4

.

0.4

0.5

0.5

25 cm/s

Vector scale

Figure 3 Computed solids void fractions and velocities in

cylindrical bed obtained by using the CPV model (left)and the KTGF model (right).

The bed heights predicted by the two models arepractically equal, but somewhat lower than observed inthe experiments.

The solids motion obtained when simulating a rectangularaxi-symmetric column using Cartesian coordinates doesnot qualitatively differ much from the motion obtained inthe cylindrical column, although the highest velocities inthe center are approximately 30% less when using theCartesian coordinates. The ascending motion of theparticles at the center predicted by the CPV model is not

as dominant in the rectangular bed. The solids voidfraction profiles predicted by this model do not change asmuch when changing coordinate system as they do whenapplying the KTGF model.

0.06 0.04 0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.05 0.05 0.050.1 0.1 0.1

0.15 0.15 0.150.2 0.2 0.20.25 0.25 0.250.3 0.3 0.30.35 0.35 0.350.4

0.4 0.4

0.4

0.

4

0.4

0.4

0.45

0

.45

0.45

0.45

Vector scale

25 cm/s

Figure 4 Computed solidsvoid fractions and velocities intwo-dimensional axi-symmetric bed obtained by using theCPV model (left) and the KTGF model (right).

Letting the particles cross the central axis, by removingthe axi-symmetrical boundary conditions and introducingan asymmetric perturbation in the solids void fraction,

results in profiles shown in Figure 5. Compared to the axi-symmetric simulations, there are minor changes in solidsmotion predicted by the KTGF model. However, it is

0.06 0.04 0.02 0 0.02 0.04 0.06

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.10.1

0.10.2 0.2 0.2

0.3

0.30.3

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

Vector scale

25 cm/s

0.06 0.04 0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.1 0.1 0.1

0.2 0.2

0.20.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

Vector scale

25 cm/s


8/10

8

slightly asymmetric and the magnitude of the center-velocities is reduced by approximately 40%. There arealso minor changes in void fraction profiles.

0.06 0.04 0.02 0 0.02 0.04 0.06

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.05 0.05 0.050.1 0.1 0.1

0.15 0.15 0.150.2 0.2 0.20.25 0.25 0.25

0.3 0.3 0.30.35 0.35 0.350.4

0.4

0.4

0.4

0.4

0.4

0.4

5

0.4

5

Vectorscale

25cm/s

Figure 5 Computed solidsvoid fractions and velocities intwo-dimensional bed obtained by using the CPV model

(left) and the KTGF model (right).

The solids motion predicted by the CPV model with andwithout axi-symmetrical boundary conditions differ a lotin the way that it is more similar to the results obtainedwith the KTGF model when the particles are allowed tocross the centerline. There is no region of high particleconcentration at the center. The particles ascend at thecenter and descend near the wall creating two vortexes inthe upper portion of the bed. In addition, there are nolonger opposite directed vortexes in the upper and lowerparts as in the axi-symmetric simulations. The predictedparticle concentration in the upper half of the bed issomewhat lower than the concentration predicted by the

KTGF model causing a more expanded bed.

Increased order on convective terms

Applying the TVD scheme with the monotonic centeredlimiter when solving the convection terms in thecylindrical bed, results in profiles shown in Figure 6. Themajor differences from the results obtained when using 1.order differentiation is an increased bed height which is inbetter agreement with the experimental results of Lin et al.(1985). The solids motion remains the same although themagnitudes of the velocities are higher. The CPV modelpredicts a more compact region along the central axishalfway up in the bed.

0.06 0.04 0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.5

0.5

Vectorscale

25cm/s

Figure 6Computed solids void fractions and velocities incylindrical bed obtained by using the CPV model (left)and the KTGF model (right). TVD scheme is appliedwhen solving the convective terms.

Allowing the particles to cross the central axis gives theprofiles shown in Figure 7. The solids velocity- andconcentration profile obtained with the CPV model isfairly symmetric with up-flow in the center. Thecorresponding profiles obtained with the KTGF modelappear to be more chaotic. An increase in discretizationorder of convective terms, and thereby a reduction innumerical diffusion seems to destabilize the solution. Thiseffect is not seen with the CPV model as the shearviscosity is higher.

0.06 0.04 0.02 0 0.02 0.04 0.06

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.1

0.1

0.1

0.2

0.2

0.20.3

0.3

0.3

0.3

0.3

0.40.4

0.4

0.4

0.4

0.5

0.5

0.5

0.5

0.5

Vector scale

25 cm/s

Figure 7Computed solids void fractions and velocities innon-axi symmetric rectangular bed obtained by using theCPV model (left) and the KTGF model (right). TVDscheme is applied when solving the convective terms.

Synthesis gas production

The same physical data and fluid dynamic operatingconditions are applied as for the cold flow simulationsexcept for the outlet pressure which is 29 bars. Thechemical model is turned on after 3 seconds to avoidinfluence of the start-up phase. Simulations of axi-symmetric columns resulted in heavy down flow ofparticles in the center. However, simulations of the non-axi symmetric rectangular column predicted ascendingparticles in the center as shown in Figure 8. Both models

perform quite evenly both in predicting solids velocitiesand void fractions. The bed expansion is higher than in thecold flow simulations due to higher operating pressure and

0.06 0.04 0.02 0 0.02 0.04 0.06

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.050.05

0.05

0.1 0.1 0.10.15 0.15 0.150.2 0.2 0.2

0.25 0.25 0.25

0.3

0.3 0.3

0.35

0.35

0.35

0.35

0.4 0

.4

0.4

0.4

0.4

0.4

0.4

0.45

0.4

5 0.45

0.45

Vector scale

25 cm/s

0.06 0.04 0.02 0 0.02 0.04 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.1

0.1

0.1

0.2

0.2

0.2

0.3

0.3 0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

Vector scale

25 cm/s

0.06 0.04 0.02 0 0.02 0.04 0.06

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

z

0.1

0.1 0.10.2

0.20.2

0.3

0.3 0.3

0.3

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.4

0.5

0.5

Vector scale

25 cm/s


9/10

9

different gas mixture. The components mass fractionswere also predicted quite evenly by the two models. Thus,only the results from the KTGF model are shown inFigure 9. The mass fraction profile of the inert nitrogengas, which is supposed to be flat, indicates how well thegas continuity equation is fulfilled. The discrepanciesfrom the initial value of 0.0641 are small enough to avoid

the error growing.

0.06 0.04 0.02 0 0.02 0.04 0.06

0.05

0.1

0.15

0.2

0.25

0.3

z

0.05 0.05 0.05

0.1 0.1 0.1

0.15 0.15 0.150.2

0.2

0.2

0.25 0.25 0.25

0.25

0.25 0.2

50.25

0.25

0.25 0

.25

0.25

0.3 0.3

Vector scale

25 cm/s

Figure 8Computed solids void fractions and velocities innon-axi symmetric rectangular bed obtained by using theCPV model (left) and the KTGF model (right) whenrunning the synthesis gas process.

0.05 0

0.05

0

0.2

0.40

0.05

0.1

rz

CH

4

0.05 0

0.05

0

0.2

0.40

0.1

0.2

rz

CO

0.05 0

0.05

0

0.2

0.40.1

0.15

0.2

rz

CO

2

0.05 0

0.05

0

0.2

0.40.02

0.04

0.06

0.08

rz

H

2

0.05 0

0.05

0

0.2

0.40.4

0.5

0.6

rz

H

2O

0.05 0

0.05

0

0.2

0.40.064

0.0641

0.0642

rz

N

2

Figure 9 Mass fractions predicted when applying theKTGF model.

CONCLUSION

A numerical algorithm has been described for the solutionof particle flow adopting an Eulerian modeling approach.Two closure models, the constant particle viscosity model(CPV) and a model based on kinetic theory of granularflow (KTGF), are compared in performance to describethe time-averaged velocity fields and bed expansion inboth a circular and rectangular column. Axi-symmetricsimulations of a cylindrical column using the KTGFmodel gives solids velocity profiles that are in fairly wellagreement with experimental results obtained by Lin et al.

(1985), while results obtained by using the CPV modelgive unrealistic high void fraction in the central region dueto the axi-symmetric boundary conditions. Allowing theparticles to cross the centerline causes the solids motionspredicted by the two models to be more even. Thepredicted bed height is somewhat less than observed in theexperiments. However, applying higher order

discretizations of convective terms results in an increasedbed expansion closer to the experimental results as long asthe total diffusion is sufficiently high.The inclusion of a chemical reacting system seems to

perform satisfactorily. However, validation of thechemical process still remains.

ACKNOWLEDGEMENTS

This work has received support from the Research Councilof Norway (Program of Supercomputing) through a grantof computer time.

REFERENCES

CHAPMAN, S. and COWLING, T.G., (1970), Themathematical theory of non-uniform gases, CambridgeMathematical Library, Cambridge, 3rded.

CLIFT, R. and GRACE, J.R., (1985), In: J.F. Davidson,R. Clift, D. Harrison (Eds.), Fluidization, (Ch. 3, p. 77).London: Academic Press.

DE GROOTE, A.M. and FROMENT, G.F., (1995),Reactor modeling and simulation in synthesis gasproduction,Rev. Chem. Eng., 11(2), 145-183.

DING, J. and GIDASPOW, D. (1990), A bubblingfluidization model using kinetic theory of granular flow,

AlChE J., 42(4), 927-931.ENWALD, H, PEIRANO, E. and ALMSTEDT, A.E.,

(1996), Eulerian two-phase flow theory applied tofluidization,Int. J. Multiphase Flow, 22, 21-66.

ERGUN, S, (1952), Fluid flow through packedcolumns, Chem. Eng. Prog., 48, 89-94.

FROMENT, G.F. and BISCHOFF, K.B., (1990),Chemical reactor analysis and design, New York,Wiley.

GIDASPOW, D., (1994), Multiphase flow andfluidization, Academic Press Inc., Boston, 1sted.

JAKOBSEN, H.A., LINDBORG, H. andHANDELAND, V., (2002), A numerical study of theinteractions between viscous flow, transport and kinetics

in fixed bed reactors. Comput. Chem. Eng., 26, 333.JAKOBSEN, H.A., LINDBORG, H. and DORAO,

C.A., (2005), Modeling of Bubble Column Reactors:Progress and Limitations, Ind. Eng. Chem. Res., ASAPArticle.

LATHOUWERS, D. (1999), Modeling and simulationof turbulent bubbly flow. Ph.D. Thesis, The TechnicalUniversity of Delft, Delft, The Netherlands.

LAUX, H., (1998), Modeling of dilute and densedispersed fluid-particle flow, Ph.D. Thesis, NTNU,Trondheim, Norway.

LINDBORG, H., EIDE, V., UNGER, S., HENRIKSEN,S.T.; JAKOBSEN, H.A., (2004), Parallelization andperformance optimization of a dynamic PDE ractor model

for practical applications, Comput. Chem. Eng.28, 1585.

0.06 0.04 0.02 0 0.02 0.04 0.06

0.05

0.1

0.15

0.2

0.25

0.3

z

0.05 0.05

0.05

0.1 0.1 0.1

0.15 0.15 0.15

0.2

0.2

0.2

0.2

0.250.25

0.25 0.25

0.25

0.25 0.25

0.25

0.25

0.3

0.3

0.3

0.3

0.35

0.35 0

.35

0.4

Vector scale

25 cm/s


10/10

10

MASSOUDI, M., RAJAGOPAL, K.R., EKMANN, J.M.and MATHUR, M.P., (1992), Remarks on the modellingof fluidized systems,AlChE J., 38(3), 471-472.

PAIN, C.C., MANSOORZADEH, S., GOMES, J.L.M.and DE OLIVIERA, C.R.E., (2001), A study of bubblingand slugging fluidised beds using the two-fluid granulartemperature model,Int. J. Multiphase Flow, 27, 527-551.

PAIN, C.C., MANSOORZADEH, S., GOMES, J.L.M.and DE OLIVIERA, C.R.E., (2002), A numericalinvestigation of bubbling gas-solid fluidized bed dynamicsin 2-D geometries,Powder Technology, 128, 56-77

SAMUELSBERG, A. and HJERTAGER, B.H., (1996),An experimental and numerical study of flow patterns ina circulating fluidized bed reactor, Int. J. MultiphaseFlow,22, 575-591.

SUN, B. and GIDASPOW, D., (1999), Computation ofcirculating fluidized-bed riser flow for the fluidizationVIII benchmark test,Ind. Eng. Chem. Res., 38, 787-792.

TOMIYAMA, A. and SHIMADA, N. (2001), ANumerical Method for Bubbly Flow Simulation Based ona Multi-Fluid Model. J. Pressure Vessel Technol., 23,

510.VAN LEER, B., (1974), Towards the Ultimate

Conservation Difference Scheme 2. Monotonicity andConservation Combined in a Second-Order Scheme. J.Comput. Phys., 14, 361.

VAN LEER, B., (1977), Towards the UltimateConservation Difference Scheme 4. A New Approach toNumerical Convection., J. Comput. Phys.23, 276.

VAN WACHEM, B.G.M., SCHOUTEN, J.C.,KRISHNA, R. and VAN DEN BLEEK, C.M., (1998),Eulerian simulations of bubbling bed behaviour in gas-solid fluidised beds, Comput, Chem. Eng.22, 299-306.

WEN, Y.C. and YU, Y.H., (1966), Mechanics offluidization. Chem. Eng. Prog. Symp. Series, 62, 100-

111.XU, J. and FROMENT, G.F., (1989a), Methane steam

reforming, methanation and water-gas shift. I. Intrinsickinetics,AIChE J., 35(1), 88-96.

XU, J. and FROMENT, G.F., (1989b), Methane steamreforming, methanation and water-gas shift. II. Diffusionallimitations and reactor simulations, AIChE J., 35(1), 97-103.

Documents

8 Lindborg Jakobsen