Optimum Catalyst Size for Slurry Bubble Column Reactors - Industrial

Mar 16, 2005 - It was estimated from a relationship between the fundamental equations of the boundary layers and the turbulent kinetic energy of parti...
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Ind. Eng. Chem. Res. 2005, 44, 6393-6402

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Optimum Catalyst Size for Slurry Bubble Column Reactors Isaac K. Gamwo National Energy Technology Laboratory, U.S. Department of Energy, Pittsburgh, Pennsylvania 15236

Dimitri Gidaspow* and Jonghwun Jung Department of Chemical and Environmental Engineering, Illinois Institute of Technology, Chicago, Illinois 60616

A mathematical model to describe the hydrodynamics of the slurry bubble-column reactor (SBCR) for converting synthesis gas into liquid fuels has been developed. This model includes the complete granular temperature balance based on the kinetic theory of granular flow. The kinetic theory model and the computer code1 were extended to include the effect of the mass-transfer coefficient between the liquid and gas and the water gas shift reaction in the Air Products/DOE LaPorte SBCR. In this model, the mass-transfer coefficient is an input. It was estimated from a relationship between the fundamental equations of the boundary layers and the turbulent kinetic energy of particles (granular temperature) computed by the hydrodynamic model with no reaction. We have varied the particle size from 20 to 100 µm and discovered a maximum in the granular temperature. For the particles over this range, the mass-transfer coefficient has the highest values. With reaction, this model was used to predict the slurry height, gas holdup, and rate of methanol production of the Air Products/DOE LaPorte SBCR. The computed granular temperature was around 30 cm2/s2, and the computed catalyst viscosity was close to 1.0 cP, as shown by Wu and Gidaspow (Chem. Eng. Sci. 2000, 55, 573). The estimated volumetric masstransfer coefficient has a good agreement with experimental values shown in the literature. A critical issue in the SBCR that has not been addressed in the literature is that of optimum particle size. The optimum size was determined for maximum methanol production in a SBCR. The size was about 60-70 µm, which was found for maximum granular temperature in the model with no reaction. 1. Introduction Major oil companies are gearing up to build slurry bubble column reactors (SBCRs) to utilize natural gas located in remote areas of the world and to convert it to paraffin wax, which will be upgraded to gasoline and diesel fuels.2,3 SBCRs have recently become competitive with traditional fixed-bed reactors for converting synthesis gas into liquid fuels.4 Stiegel5 and Heydorn et al.6 published excellent reviews of DOE Research in Fischer-Tropsch (F-T) technology and liquid-phase methanol processes. They described the advantages of the slurry-phase reactor over the fixed-bed reactor. In the SBCRs, the fine powdered catalysts are suspended in the fluid and the gas bubbles provide the energy to keep the catalyst mixed. SBCRs have excellent heat- and mass-transfer characteristics for removal of the heat given off by exothermic reactions and the ability to replace catalysts easily. The design and scale-up of SBCRs require, among other things, precise knowledge of kinetics, hydrodynamics, and mass-transfer characteristics over a wide range of operating conditions for reactors with a diameter as large as 7 m and a height of 30 m being built by oil industries.7 Models were applied to the F-T conversion of synthesis gas in a SBCR.7-9 They require holdup correlations, diffusivity, mass-transfer coefficient, and bubble size as inputs. As inputs into such a model, eddy diffusivities were measured using computer-aided * To whom correpondence should be addressed. Tel.: 1-312567-3045. Fax: 1-312-567-8874. E-mail: [email protected].

radioactive particle tracking (CARPT).10 Mass-transfer coefficients in three-phase flows, including fine particles, were measured.11-13 Hold-up profiles were measured by Gandhi et al.14 However, they did not compute the coherent flow structures in bubble-column reactors in the churn-turbulent regime, as reviewed by Joshi et al.15 Computational fluid dynamics (CFD) is a recently developed tool that can help in the scale-up. The multiphase CFD codes with viscosity as an input computed hold-up and flow patterns for gas-liquid flow16 and gas-liquid-solid flow.17 Wu and Gidaspow,1 on the basis of a kinetic theory model, computed the hold-up and flow patterns and methanol production in an Air Products/DOE Laporte SBCR. Gamwo et al.18 used Wu and Gidaspow’s model for new reactor designs. However, most modeling studies did not address the effect of the catalyst size on the performance of the reactor. An issue of interest to the energy industries throughout the world is the size of the catalyst that they should make for SBCRs. The industries are gearing up to make catalysts for SBCRs.3,19 F-T catalysts, such as those used to produce methanol and other liquids from synthesis gas, normally come in powder form. Catalyst particles used in most fluidized-bed processes are small enough for external mass-transfer and internal diffusion resistance to be negligible. The size of the catalyst is typically in the range of 20-120 µm.20-22 Singleton et al.23 described a method of preparation of such catalysts with the preferred particle size between 20 and 80 µm. Small particle sizes are needed to have good effective-

10.1021/ie049205x CCC: $30.25 © 2005 American Chemical Society Published on Web 03/16/2005

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ness for reaction. However, small particles are entrained in the product gas stream and are known to cause liquid product filtration problems. Small particles also cause the formation of the clusters, which give large effective particle sizes and, hence, poor mass transfer. Here we have applied the mathematical model for gas-liquidsolid flows to determine the optimum particle size, which is the size that has the maximum granular temperature, similar to the experiments for gas-solid systems done at Exxon.24 For this particle size, the heatand mass-transfer coefficients have the highest values. In this model, the mass-transfer coefficient is an input. We have related the mass-transfer coefficient to the fluctuating velocities (granular temperature) computed by the hydrodynamic model. 2. Hydrodynamic Model A hydrodynamic model for the production of methanol from synthesis gas in the SBCR was developed. It uses the principles of mass, momentum, and energy conservation for each phase, as described in Table 1.25 This model includes the complete granular temperature balance based on the kinetic theory of granular flow.25,26 The kinetic theory model and the computer code1 were extended to include the effect of the mass-transfer coefficient between the liquid and gas and the water gas shift reaction in the Air Products/DOE LaPorte SBCR described in Figure 1. The reaction rates for methanol synthesis used by Wu and Gidaspow1 were all in the gas phase. The more realistic rates should be in the liquid phase. Here they were modified as given by Graaf et al.27,28 2.1. Reaction for Methanol in the Slurry Phase. A review of the literature showed that the following chemical reactions are accepted for production of methanol from synthesis gas. The reactions included are methanol production from hydrogen and CO, the water gas shift reaction, and the production of methanol from CO2 hydrogenation.

(1)

Graaf et al.27,28 developed the reaction rate for methanol synthesis in gas-catalyst phases and extended it to three-phase methanol synthesis using gas-liquid solubilities in thermodynamic equilibrium described by Henry’s law. The reaction of the jxth species of the liquid phase is IX

jx Rjx ∑ i M r′i i)1

(2)

The rate for the three reactions (IX ) 3) in the liquid phase is given as follows:

r′IX )

sFs

r′′IX (mol/cm3‚s) 1.0 × 103

/ kps,A3 k/CO(CCOCH23/2 - CCH3OH/CH21/2KC1) / 1/2 / / (1 + k/COCCO + kCO C )[CH21/2 + (kH /kH )CH2O] 2 CO2 2O 2

(4) r′′H2O,B2 ) / / kCO (CCO2CH2 - CH2OCCO/KC2) kps,B2 2 / / 1/2 / (1 + k/COCCO + kCO C )[CH21/2 + (kH /kH )CH2O] 2 CO2 2O 2

)

r′′CO,B2 (5) r′′CH3OH,C3 ) / / kps,C3 kCO (CCO2CH23/2 - CCH3OHCH2O/CH22/3KC3) 2 / / 1/2 / (1 + k/COCCO + kCO C )[CH21/2 + (kH /kH )CH2O] 2 CO2 2O 2

)

r′′H2O,C3 (6) where chemical equilibrium constants are

KCl ) 1.72 × 10-16e126011/RT, KC2 ) 5.81 × 10e-33760/RT, KC3 ) KC1KC2 (7) Reaction rate constants are / ) 1.66 × 105e93925/RT, kps,A3 / ) 7.21 × 1017e215130/RT kps,B2 / kps,C3 ) 8.52 × 10-1e43425/RT,

k/CO ) 9.01 × 10-12e92138/RT

/ / 1/2 kH /kH ) 2.71 × 10-12e103030/RT (8) 2O 2

CO2 + H2 T CO + H2O

rjx l )

r′′CH3OH,A3 )

/ kCO ) 3.15 × 10-5e34053/RT, 2

CO + 2H2 T CH3OH

CO2 + 3H2 T CH3OH + H2O

catalyst surface for three reactions.

(3)

where Rjx i represents the stoichiometric coefficient of the jxth species in the ith reaction of the liquid phase and Mjx represents the molecular weight of the jxth species. r′′IX (mol/kgcat‚s) is the rate of the reaction on a

Cjx (mol/m3) is the bulk concentration of the jxth species in the liquid phase, and R is the gas constant of 8.314 J/mol‚K. 2.2. Mass Transfer. The rate of mass transfer for each phase k is n

m ˘ k)

∑ m˘ jxk Mjx

(9)

jx)1

Mass transfer between the gas and liquid phases was only considered in this study. n

m ˘ l)

jx m ˘ jx ˘ g, ∑ l M ) m jx)1

m ˘ s)0

(10)

Mass-transfer rate between the gas and liquid phases can be expressed in terms of the volumetric masstransfer coefficient and the concentration difference between the gas-liquid interface and liquid phases

1 g-l (mol/cm3‚s) m ˘ jx l ) lkla(Cjx - Cjx) 106

(11)

Ind. Eng. Chem. Res., Vol. 44, No. 16, 2005 6395 Table 1. Hydrodynamic Kinetic Theory Model for Multiphase Flow (k ) l, g, s) Continuity Equations

∂(Fkk) ˘k + ∇‚(Fkkνk) ) m ∂t

(T1)

∂(Fkkyjx k) jx ˘ jx + ∇‚(Fkkνkyjx k) ) m k + rk ∂t

(T2) Momentum Equations

∂(Fkkνk) ∂t

∑β

+ ∇‚(Fkkνkνk) ) kFkFk + ∇‚τjk +

km(νm

- νk) + m ˘ kνk

(T3)

m)l,g,s m*k

Fluctuating Energy Equation for the Particle (θ ) 1/3〈C2〉)

3 ∂ ( F θ) + ∇‚(sFsνsθ) ) jτs:∇νs - ∇‚q - γs 2 ∂t s s

[

]

(T4) Constitutive Equations

(1) definitions p

∑

n

m ˘ k ) 0,

k)1

∑

n

rjx k ) 0,

jx)1

∑

p

yjx k ) 1.0,

jx)1

∑

k

) 1.0

(T5)

k)1

(2) equation of state

Fg,mixture )

PM h mixture zRT

(T6) 1

where M h mixture ) n

∑

(T7)

yjx g

jx jx)1M

(3) stress tensor

2 j τk ) -Pk + λk - µk tr(D hk) hI + 2µkD hk 3 1 where h Dk ) [∇νk + (∇νk)T] 2

[

(

]

)

(T8) (T9)

for particle

Ps ) Fssθ[1 + 2(1 + e)g0s] 2µsdil 4 4 µs ) 1 + (1 + e)g0s + s2Fsdsg0(1 + e) 5 5 (1 + e)g0

[

]

4 λs ) s2Fsdsg0(1 + e) 3

xπθ

(T10)

xπθ

(T11) (T12)

where g0 is the radial distribution function and µsdil is the particle-phase dilute viscosity

g0 ) [1 - (s/s,max)1/3]-1

(T13)

5xπ µsdil ) F d θ1/2 96 s s

(T14)

(4) granular conductivity of fluctuating energy (q ) -κ∇θ)

2 6 1 + (1 + e)g0s 5 (1 + e)g0 75 where κdil ) πF d θ1/2 384x s s

κ)

[

2

]κ

dil

+ 2s2Fsdsg0(1 + e)

xπθ

(T15) (T16)

(5) collisional energy dissipation

(

γs ) 3(1 - e2)s2Fsg0θ

4 ds

xπθ - ∇‚ν ) s

(T17)

(6) fluid-particle drag coefficient

s2µf Ffs|νf - νs| βB ) 150 2 2 + 1.75 f < 0.8 fds  f ds 3 Ffs|νf - νs| -2.65 βB ) Cd f f g 0.8 4 ds 24 where Cd ) [1 + 0.15Res0.697] for Res < 1000 Res Cd ) 0.44 for Res g 1000 fFfds|νf - νs| Res ) µf

(T18) (T19) (T20) (T21) (T22)

(7) particle-particle drag coefficient 2

slFslsmFsm(dsl + dsm) 3 β lm ) (1 + e) |νl - νm| 2 l*m  (F d 3 + F d 3) f

sl sl

sm sm

(T23)

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Table 1. (Continued) (8) external forces continuous phase

Ff ) g/f

(T24)

particulate phase

Fk )

g

(

f

1-

1

∑

Fk m)l,g,s

mF m

)

(T25)

(9) boundary conditions for a particle velocity

nτc )

x3θπΦFssg0Usl 6s,max

(T26)

granular temperature

-nq )

x3θπΦFssg0|Usl|2 x3πFssg0(1 - ew2)θ3/2 + 6s,max 4s,max

(T27)

where kla (s-1) is the volumetric mass-transfer coefficient and Cg-l jx is the concentration of the jxth species at the gas-liquid interface phase, which can be defined by Henry’s law. The fugacity of a very dilute species in a liquid phase is linearly proportional to its mole fraction at low mole fractions. 3 fjx ) HjxCg-l jx (mol/m )

(12)

In gas-liquid-phase equilibrium, the fugacity of the liquid phase can be the fugacity of the gas phase defined by the partial pressure of species in the gas phase

fjx ) y′jxP (bar)

(13)

the granular temperature shown in Figure 3. The computed granular temperature is about 202 (cm/s)2 for 20 µm. It rises to 356 (cm/s)2 for 60 µm and then decreases to 138 (cm/s)2 for 100 µm. The maximum granular-like temperature is at 60 µm, with a solid loading of about 10%. It agrees well with the experimental results of Cody’s study24,30 for a gas-solid bubbling bed. They showed that Geldart A glass spheres exhibit an order magnitude higher granular temperature than neighboring Geldart B glass spheres based on the experiments in a gas-solid bubbling bed. They showed a maximum for the gas-solid system to be about 34 (cm/s)2 at a particle size of about 75 µm. Our computed value of the granular-like temperature is almost an order of magnitude higher because of the fact

where fjx (bar) is the fugacity of the jxth species, y′jx is the mole fraction of the jxth species, and Hjx (bar‚m3/ mol) is Henry’s constant of the jxth species used by Graaf et al.27,28

HCO ) 0.175e638/RT, HCO2 ) 0.402e-6947/RT, HH2 ) 0.0782e4875/RT HH2O ) 0.330e-8633/RT, HCH3OH ) 1.49e-17235/RT (14) 3. Relationship between the Mass Transfer and Granular Temperature 3.1. Granular Temperature. The computer simulations were done for the experiment conducted at the IIT slurry bubble column (30.48 cm × 5.08 cm × 213.36 cm), as shown in Matonis et al.17 In the experiment, water was recirculated through the bed and air was injected through porous tubes. The particles were 800-µm lead glass beads with a density of 2.94 g/cm3. Many small bubbles act like another set of particles under the condition for Ug ) 3.37 cm/s and Ul ) 2.02 cm/s in the system. The complete mathematical model with particulate empirical viscosities and the method of analysis are described by Matonis et al.17 and Jung.29 Figure 2 shows typical large- and small-scale oscillations of particles obtained with the grid size of 1 cm. We computed the average random kinetic energy from such data. This gives us a granular temperature-like quantity similar to that computed in Matonis et al.17 The granular temperature, 2/3 the turbulence kinetic energy, can be introduced as a function of the particle fluctuation velocity. Here we have varied the particle sizes from 120 µm down to 20 µm and discovered a maximum in

Figure 1. Schematic of the LaPorte SBCR for the liquid-phase methanol synthesis process.

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Figure 4. Concentration and momentum boundary layers in a SBCR. It is assumed that the liquid layer surrounds the particles.

Figure 2. Large- and small-scale oscillations for the axial velocity of a solid as a function of time at x ) 15.5 cm and y ) 11.5 cm of the two-dimensional viscosity input model.

coefficient is beyond the present state of the art of the present CFD computations. The fluid dynamics of liquid-solid fluidization is being done by Daniel D. Joseph (http://www.aem.umn.edu/Solid-Liquid_Flows/ video.html) of the University of Minnesota in Minneapolis. So far, he has only been able to predict the bed expansion in such systems using solutions of the Navier-Stokes equations for the fluid and letting the particles oscillate. The computations with mass transfer will require resolution of very thin boundary layers. Hence, such computations are beyond the state of the art of computers. Instead, we have carried out an order of magnitude analysis to relate the mass-transfer coefficient to the granular temperature. By definition, the mass-transfer coefficient (kL) is related to the diffusivity (DL) of gas in liquid and the concentration boundary layer thickness (δc) (Figure 4) as follows:

kL ) DL/δc

Figure 3. Optimum particle size for mixing (maximum granularlike temperature) in the IIT slurry bubble column (VL ) 2.02 cm/ s; VG ) 3.37 cm/s).

that we have a flowing liquid stream. In the absence of liquid flow, our previous computations1 gave us a granular temperature of about 20 (cm/s)2. The turbulent kinetic energy for gas-liquid flow measured with a radioactive particle tracer was as high as 2000 (cm/s)2.31 This high value of turbulence is probably due to the fact that the actual gas velocity in the Chen et al.31 bubble column is an order of magnitude higher. An approximate relationship between the granular temperature and the particle velocity from the kinetic theory analysis is as follows:32

xθ/vs ) 0.5

(15)

where θ is the maximum granular temperature in the system and vs is the average particle velocity in the upstream portion of the fluidized-bed column. The relationship between the granular temperature and the mass-transfer coefficient in the SBCR can be obtained as follows. 3.2. Mass-Transfer Coefficient. In the fluidized bed, the catalyst particles have random and deterministic velocity components. Fluidized-bed dynamics for such systems without an input for the mass-transfer

(16)

The momentum boundary layer thickness (δm) can be related to the characteristic velocity, square root granular temperature (xθ), and the characteristic length, the particle diameter (dp), by the following relationship:

δm )

xνdp/xθ

(17)

where ν is the kinematic viscosity. The concentration boundary layer thickness is related to the momentum boundary layer thickness through the Schmidt number (Sc ) ν/DL) as follows:

δc )

1 δm Sc1/3

(18)

Substitution yields the following relationship between the mass-transfer coefficient and the granular temperature.

kL )

4 DLxθ 1/3 Sc xνdp

(19)

Equation 19 shows how the mass-transfer coefficient can be deduced from the computed granular temperature. Substitution for the optimum particle size of 60 µm in Figure 3 gives a reasonable value for masstransfer coefficients compared to literature values and rate constants for the reaction. The mass-transfer limitation for 60 µm will be negligible for the SBCR with

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Figure 5. Mass-transfer coefficient obtained from computed granular-like temperatures of Figure 3.

liquid recirculation. The mass-transfer limitation without liquid recirculation will probably be small but not negligible. 3.3. Volumetric Mass-Transfer Coefficient. Figure 5 shows the mass-transfer coefficient obtained from computed granular temperatures of Figure 3. The masstransfer coefficient generally increases with a decrease of the particle diameter. This has a good agreement with the literature.33 As the catalyst size is decreased, the combined resistance to internal diffusion, reaction, and external diffusion will be negligible. As a result, the absorption from the gas phase into the liquid phase will be the main resistance. The catalyst concentration should increase for the higher production rate. The process of coagulation due to random motion and subsequent collision of the particles may be considered in the SBCR. The effective particle diameter was obtained from coagulation theory34

de(t) ) d0(1 + N0Kt)1/3

(20)

where de is the particle size at time t, d0 is the initial particle size, N0 is the particle concentration, and K is the coagulation coefficient. The solid volume fractions for Geldart A particles were constant values of about 10% from the simulation. The particle concentrations were estimated from the particle diameters with the solid volume fraction of 10%. As a rule of thumb, if the particle concentration is less than 106/cm3, for particles larger than 120 µm in our simulation, the coagulation of particles is neglected.34 The time (t) was determined by such a principle. There is no coagulation for particles larger than 120 µm. The coagulation coefficients were calculated for particles at standard conditions. The effective particle diameters were calculated from the coagulation theory. The volumetric mass-transfer coefficient based on an effective particle diameter is given as

k La )

4 DLxθ 1/3 6 Sc  de s xνde

(21)

where DL is the diffusivity of 10-5 cm2/s and a is the interfacial area per unit volume estimated from the effective particle diameter. The volumetric mass-transfer coefficient based on an effective particle diameter, shown in Figure 6, has a maximum value near the particle diameter of 50 µm.

Figure 6. Volumetric mass-transfer coefficient based on the effective particle diameter using the coagulation theory.

This may be larger than the estimated value because of nonspherical particle effects or electrical forces. As pointed out previously, small particles cause the formation of the clusters, which give large effective particle sizes and, hence, poor mass transfer. Large particles, Geldart B particles, have very low mass-transfer coefficients and, hence, poor production rate. Our results show a good agreement with that of a SBCR in the Viking Systems International Report given by Prakash and Bendale.8 4. Numerical Considerations A search for an optimum catalyst size was carried out for methanol synthesis in the DOE Laporte SBCR shown in Figure 1. The powdered catalyst is suspended in an inert liquid to form slurry, and feed gas is introduced into the bottom of the reactor through a distributor. The upward-flowing gas bubbles provide the energy to keep the slurry highly mixed. The reactants from the gas phase dissolve in the liquid and diffuse to the catalyst surface, where they react. Heat is removed by generating steam in an internal tubular heat exchanger.6 The initial conditions and the configuration for the simulation are shown in Figure 7. The operating conditions are the same as those of LaPorte’s run E-8.1.35 To obtain the numerical solution of nonlinear-coupled partial differential equations, the IIT code was used. The simulations using the granular temperature model were carried out in a two-dimensional Cartesian coordinate with a total of 34 × 160 computational meshes. The Johnson and Jackson slip boundary condition described in Table 1 was employed for the solid phase.36 Neumann boundary conditions were applied to the three-phase flow with a constant pressure of 753.2 psig at the top wall. The boundary condition at the bottom wall was that the axial gas velocity is a parabolic profile with an average velocity of 15.24 cm/s, with zero velocity near the wall. The solid and liquid velocities are zero at the bottom wall. The restitution coefficient due to particle-particle collisions was 0.9995 for the simulation. The measurement of radial distribution functions of statistical mechanics showed that particles fluidized in water fly apart well before contact, at a radius of about 50% larger than the particle radius.37 In liquids, there exists a film between the particles that gives rise to a lubrication force. Thus, the restitution coefficient can be close to unity. The total volumetric mass-transfer coefficient was 0.5 s-1 for the reference condition. The synthesis gas composition fed into the bottom of the reactor is

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Figure 7. Initial and boundary conditions for methanol synthesis.

Figure 8. Computed gas (A) and liquid (B) holdups averaged from 15 to 30 s in the simulation of LaPorte’s methanol synthesis using the kinetic theory.

Table 2. Synthsis Gas Composition (CO-Rich) mol % wt %

CO

CO2

H2

CH3OH

H2O

N2

wax

51.00 68.07

13.00 27.26

35.00 3.34

0.0 0.0

0.0 0.0

1.00 1.33

0.0 0.0

shown in Table 2. In the initial condition of the simulation, the synthesis gas composition in the reactor was only nitrogen. The convergence criterion for the simulation was 10-4. The simulations were run for 40 s and then averaged from 15 to 30 s. 5. Computational Results and Discussion A search for an optimum catalyst size was carried out for methanol synthesis in the DOE Laporte SBCR using the kinetic theory model of granular flow. This model included the effect of the mass-transfer coefficient between the liquid and gas phases and the water gas shift reaction, which was not presented previously by Wu and Gidaspow.1 Figures 8-10 show the computed liquid and gas holdups, the flow patterns, the granular temperature, and the slurry viscosity averaged from 15 to 30 s for Laporte’s methanol synthesis. The computed average gas volume fraction is around 0.35. As expected, it is higher at the center of the reactor. The average liquid volume fraction is around 0.52, and the average catalyst volume fraction is about 0.13. The flow pattern is not significantly affected by the tubes in this configuration. The basic computed flow pattern is upflow in the center and downflow at the wall, as observed in the pilot plant. There are, however, many additional vortices around the heat-exchanger tubes not shown in Figure 9 because of time averaging. The flow pattern can be changed radically by rearranging the tube configuration.18 The average computed granular temperature is around 30 cm2/s2. It is close to that measured in the SBCR at IIT.1 The average computed catalyst viscosity is close to 1 cP. It is much higher around the heat-exchanger tubes because of the higher granular temperature at this location. This effect was not seen previously in the paper of Wu and Gidaspow.1 The new effect is due to the use

Figure 9. Computed catalyst (solids) volume fraction and catalyst (solids) flow pattern averaged from 15 to 30 s in the simulation of LaPorte’s methanol synthesis using the kinetic theory.

of more realistic Johnson-Jackson boundary conditions. Hence, there may be higher corrosion or collisions near the heat exchanger. The mole fractions of methanol and water in the gas and liquid phases are shown in Figures 11 and 12. The product water concentration is small because we assumed that the inlet synthesis gas was dry. Figures 11 and 12 show that the mixing in the reactor is very good. With the water gas shift reaction, the ratio of H2 to CO is 0.5 because of chemical equilibrium in the liquid phase of the simulation. Figure 13 shows the effect of the volumetric masstransfer coefficient estimated by the simulation in the SBCR without liquid circulation. Methanol production increases by 6.5 mol/kgcat‚h for the volumetric masstransfer coefficient of 0.75, and then methanol produc-

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Figure 10. Granular temperature (A) and catalyst shear viscosity (B) averaged from 15 to 30 s in the simulation of LaPorte’s methanol synthesis using the kinetic theory.

Figure 12. Computed mole fraction of methanol (A) and water (B) in the liquid phase averaged from 15 to 30 s in the simulation of LaPorte’s methanol synthesis using the kinetic theory.

Figure 11. Computed mole fraction of methanol (A) and water (B) in the gas phase averaged from 15 to 30 s in the simulation of LaPorte’s methanol synthesis using the kinetic theory.

tion is no longer limited by the volumetric mass-transfer coefficient. In this region, the granular temperature has a maximum of 33 (cm/s)2. This suggests that the maximum production can be at the maximum granular temperature shown in Figure 3. The estimated volumetric mass-transfer coefficient has a good agreement with experimental values shown in the literature.8,11,13 Figure 14 shows the effect of water in synthesis gas fed into the reactor. The methanol production has a maximum at a water mole fraction of 0.05 and then decreases slowly with increasing water. The enhancement of production due to the addition of water can be determined by a balance between the rates of the water gas shift reaction and CO2 hydrogenation as well as the composition of the synthesis gas fed into the reactor. However, the addition of lots of water decreases the rate of the production due to deactivation of catalyst as recognized by Parameswaran et al.38 Hence, the ratio of the fresh gas and recycle gas in the SBCR can be set near the water mole fraction of 0.05. The ratio of H2

Figure 13. Methanol production and granular temperature obtained from the different volumetric mass-transfer coefficients in the SBCR without liquid circulation.

and CO in the liquid phase increases linearly with the addition of water because of chemical equilibrium. Figure 15 shows the methanol production for five different catalyst sizes. For 75 µm, the production has increased and then decreases substantially for 100 µm. Hence, the optimum particle size is about 70 µm. This agrees with the previous simulations of about 60 µm for the case of no reaction. 6. Conclusions The SBCR for methanol and other hydrocarbon production from synthesis gas is an issue of interest to

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phases and the water gas shift reaction in the SBCR. The computed granular temperature was around 30 cm2/s2, and the computed catalyst viscosity was close to 1.0 cP. The volumetric mass-transfer coefficient estimated by the simulation has a good agreement with experimental values shown in the literature. The optimum particle size was determined for maximum methanol production in a SBCR. The size was about 60-70 µm, found for the maximum granular temperature. This particle size is similar to the fluidized catalytic cracking particle used in petroleum refining. Acknowledgment This study was supported by the University of Pittsburgh NETL Student Partnership Program, partially by a National Science Foundation grant to Professor Gidaspow, and by the ORISE program. Nomenclature

Figure 14. Methanol production and H2/CO mole ratio in the liquid phase due to the water feed effect in the SBCR without liquid circulation.

Figure 15. Methanol production for five different catalyst sizes obtained from the SBCR without liquid circulation.

the energy industries throughout the world. CFD is a recently developed tool that can help in the scale-up. A critical issue in the SBCR that has not been addressed in the literature is that of the optimum particle size. We have developed an algorithm for computing the optimum particle size for fluidized-bed reactors. The mathematical technique can be applied to gas-solid, liquid-solid, and gas-liquid-solid fluidized-bed reactors, as well as the LaPorte SBCR. Our computations show that there is a difference of about a factor of 2 between 20- and 60-µm sizes with the maximum granular temperature (turbulent kinetic energy) near the 60µm size particles. We also present a new method to theoretically estimate the mass-transfer coefficient in SBCRs and other fluidized beds, as well as bubble columns where the catalyst particle is replaced by an effective droplet or oscillating small bubble in a bubble column. The kinetic theory model1 was extended to include the effect of the mass-transfer coefficient between the liquid and gas

a ) interfacial area per unit volume Cd ) drag coefficient Cjx ) bulk concentration Cg-l jx ) concentration of the jxth species at the gas-liquid interface phase D ) diffusivity dk ) characteristic particulate phase diameter e ) coefficient of restitution fjx ) fugacity of the jxth species g ) gravity g0 ) radial distribution function Hjx ) Henry’s constant of the jxth species k ) mass-transfer coefficient K ) coagulation coefficient ka ) volumetric mass-transfer coefficient k* ) reaction rate constant Kc ) chemical equilibrium constant Mjx ) molecular weight of the jxth species m ˘ jx k ) mass-transfer rate of the jxth species in phase k N0 ) particle concentration P ) continuous phase pressure Pk ) dispersed (particulate) phase pressure R ) gas constant Rek ) Reynolds number for phase k r′IX ) reaction rate based on the unit volume r′′IX ) reaction rate based on the weight of the catalyst Sc ) Schmidt number t ) time T ) temperature Usl ) slip velocity v ) hydrodynamic velocity y′jx ) gas mole fraction of the jxth species in phase k Greek Letters Rjx i ) stoichiometric coefficient of the jxth species in the ith reaction βB ) interphase momentum transfer coefficient δc ) concentration boundary layer thickness δm ) momentum boundary layer thickness k ) volume fraction of phase k γs ) energy dissipation due to inelastic particle collisions κs ) granular conductivity λk ) bulk viscosity of phase k µk ) shear viscosity of phase k ν ) kinematic viscosity θ ) granular temperature Fk ) density of phase k τk ) stress of phase k Φ ) specularity coefficient

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Superscript jx ) species Subscripts g, l, s ) gas, liquid, solid, respectively jx ) species k ) phases

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Received for review August 28, 2004 Revised manuscript received February 2, 2005 Accepted February 7, 2005 IE049205X