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Ind. Eng. Chem. Res. 2008, 47, 4632–4640
Simulation of Performance of Cracking Reactions of Particle Clusters in FCC Risers Wang Shuyan,† He Yurong,† Lu Huilin,*,† Jianmian Ding,‡ Yin Lijie,† and Liu Wentie† School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China, and MSC Software, 2 MacArthur Place, Santa Ana, California 92707
The behavior of catalytic cracking reactions of particle cluster in fluid catalytic cracking risers was numerically analyzed using a four-lump mathematical model. The effects of the cluster porosity, inlet gas velocity, and cluster formation on cracking reactions were investigated. Distributions of temperature, gases, and gasoline from both the catalyst particle cluster and an isolated catalyst particle are presented. Simulated results show that the reactions from vacuum gas oil (VGO) to gasoline, gas, and coke of individual particle in the cluster are slower than those of the isolated particle, but faster for the reaction from gasoline to gas and coke. Particle clustering will reduce the reaction rates from VGO to gasoline, gas, and coke and increase the reaction rates from gasoline to gas and coke. Less gasoline is produced by particle clustering. More gas and gasoline are produced for the downward moving cluster than for the upward moving cluster. The cluster formation decreases the reaction rates and reduces gas and gasoline production. 1. Introduction Fluid catalytic cracking (FCC) is considered to be one of the most important refining processes. The fluid catalytic cracking involves the conversion of heavy oil feedstock into gasoline and other valuable products.1 The FCC unit is mainly comprised of a riser and a regenerator. The vacuum gas oil (VGO) is dispersed into the riser bottom in the form of drops through a feed nozzle system. The VGO drops contact hot regenerated catalyst particles from the regenerator and get vaporized. The vapor entrains the catalyst particles and the liquid drops getting cracked on the catalyst surface along the riser. In this process, the catalyst is progressively deactivated due to the deposition of coke formed during cracking reactions on its surface. The deactivated catalyst leaving the riser top is transferred to the regenerator where its activity is restored by burning the coke. Therefore, the hydrodynamics of the catalyst particle in the riser controls the performance of the FCC unit. It is generally agreed that the suspended particles may partially form denser clusters in the riser. These clusters move upward through the center of the riser and fall down along the riser wall.2,3 Flow characterization of the clusters has been experimentally investigated in the circulating fluidized beds (CFBs).4,5 Residence time, fraction, and size of the clusters were measured.6–9 A typical cluster diameter is 10–100 times the particle diameter.10,11 The cluster fluctuation energy is generally an order of magnitude smaller than that of dispersed particles.12 The particle clustering leads to inefficient gas–solids contact and, therefore, affects the CFB performance. Hence, in-depth knowledge of the cluster behavior is essential for designing and operating the FCC units. Numerical simulations have been performed to obtain the performance of the FCC unit based on catalyst reaction kinetic models. Most of them are from representations of oil in a few lumps, including the three-lump cracking model to study gasoline production in a FCC unit,13 the four-lump model to describe the catalytic cracking of gasoline,14–16 the five-lump * To whom correspondence should be addressed. Tel.: +010 0451 8641 2258. Fax: +010 0451 8622 1048. E-mail:
[email protected]. † Harbin Institute of Technology. ‡ MSC Software.
model to consider the heavy fraction with the gas oil splitting into paraffins, naphthenes, and aromatics,17,18 the six-lump model to predict the catalytic cracking of residual oil,19 and the ten-lump kinetic model to take into account for different feed properties in addition to the boiling point range.20 Flow and cracking reactions have been predicted by using threedimensional CFD models for FCC riser with Eulerian approach and reactions with lump models.21–23 Obviously, the number of lumps of the proposed models for catalytic cracking reactions has been increased to obtain more detailed descriptions of catalytic cracking reactions and product distributions. However, the effect of particle clustering on the cracking reactions mentioned above has not considered in the FCC riser. Experiments have suggested that the flow behavior of particle clusters be significantly different from the dispersed particles in a flow stream. The clusters play an important role on the performance of CFB risers. It is expected that the transport of particles within a cluster should be different from the dispersed particles. It is, therefore, desirable to understand the effect of particle clustering on catalytic reactions in a FCC riser. In present investigation, the effects of catalyst particle clusters on cracking reaction in a FCC riser are numerically analyzed. Reaction behavior of the catalyst particles in clusters is obtained. Chemical reactions involve heterogeneous and homogeneous reactions among gas, gasoline, and coke. Numerical results provide distributions of temperature, gas, and gasoline in the cluster. The effects of cluster porosity, gas velocity, flow direction, and the formation of cluster on the mass fluxes of gas and gasoline are presented. 2. Governing Equations For simplicity, the cluster is assumed to be spherical with a diameter of Dc containing 65 spherical particles as shown in Figure 1. The particles in the cluster are regularly arranged and assumed to be stationary during simulations. The cluster porosity is defined as εg ) (D3c - ndp3)/D3c . The cluster porosity can be changed by changing the cluster size for porosity variation simulations. With the particle diameter of 100 µm and the cluster size of 688 µm containing 65 particles in the cluster, the corresponding porosity is 0.8. The ratio of cluster diameter to
10.1021/ie071305q CCC: $40.75 2008 American Chemical Society Published on Web 06/14/2008
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4633
and the turbulent kinetic energy dissipation rate equation
( )
∂ ∂ ∂ µt ∂ε ε (Fgε) + (Fgujε) ) + [C1Gk - C2Fgε] + ∂t ∂xj ∂xj σε ∂xj k ε CG k 1 R where Gk is the production term
(
Gk ) µg
(4)
)
∂ui ∂uj ∂ui 2 ∂uk + - δ ∂xj ∂xi ∂xj 3 ij ∂xk
(5)
The empirical constants, Cµ, σε, C1, and C2, are listed in Table 1. 2.4. Energy Conservation of Gas Phase. The energy balance equation of the gas phase is written as follows:24,25
[(
) ]
Cgνt ∂Tg ∂ ∂ ∂ (C T ) + (ujCgTg) ) λ + + ∂t g g ∂xj ∂xj g σh ∂xj
∑ ∆H R
i i
i
(6) Figure 1. Cluster shapes studied in the simulations.
particle diameter is 6.9. Although the modeled three-dimensional cluster does not exactly match the cluster measured from a CFB riser, the simulated results will provide the general velocity, temperature, and concentration fields inside the cluster. Other assumptions are as follows: (1) The catalyst particle diameter and shape remains constant during the reactions. (2) Radiation heat transfer between gases and particles is neglected because of the small temperature difference between gases and particles; radiation heat transfer between particles and the riser wall is also not considered. (3) The instantaneous vaporization of feeding at the riser entry is specified, and the postcombustion reaction is not considered. The vaporized vacuum gas oil (VGO) diffuses into particulate phase and is cracked at the catalyst surface. The produced gases diffuse back to gas phase, and the produced coke is simultaneously deposited onto the catalyst surface. The four-lump model for cracking reaction kinetics is used in present analysis. The four lumps are feed (VGO), gasoline, gas, and coke. The primary cracking of gas-oil is assumed to be a first-order reaction. The turbulent flow field is computed using a k- turbulent model. The conservation equations of mass, momentum, gas species, and energy in Eulerian coordinates are listed in the following. 2.1. Gas Phase Continuity Equation. The mass conservation equation of gas can be written as follows:24,25 ∂ ∂ F + (F u ) ) 0 ∂t g ∂xj g j
(1)
2.2. Gas Phase Momentum Equation. The conservation of momentum for the gas phase is expressed as follows:24,25
( ) ( )
∂uj ∂ui ∂ ∂ ∂p ∂ ∂ (F u ) + (Fgujµi) ) + µ + µ + Fgg ∂t g i ∂xj ∂xi ∂xj g ∂xi ∂xj g ∂xj (2) where µg ) µl + µt is the effective gas viscosity, and the turbulent viscosity µt ) CµFgk2/ε is determined from a k- turbulent model. 2.3. Gas Phase Turbulent k-E Model. The standard k- turbulence model gives the following turbulent kinetic energy equation26
( )
∂ ∂ ∂ µt ∂k (Fgk) + (Fgujk) ) + Gk - Fgε + GR ∂t ∂xj ∂xj σk ∂xj
(3)
where Ri is the reaction rate of routes A, B, C, D, and E. The heat capacity and the thermal conductivity of the mixture are calculated by the mixing law Cg )
∑YC
(7)
∑Yλ
(8)
i gi
i
λg )
i gi
i
where Cgi and λgi are the heat capacity and the thermal conductivity of species i, respectively. The heat capacity and the thermal conductivity of gas and gasoline were estimated respectively using a modified Lee-Kesler correlation27 in which the pressure effect is neglected since the typical FCC processes are operated under a relatively low pressure.28,29 2.5. Conservation Equation of Gas Species. The mass balance for a gas species k (k ) VGO, gasoline, gas and coke) is:24,25
[(
) ]
µt ∂Yk ∂ ∂ ∂ (FgYk) + (FgujYk) ) FkDk + + ∂t ∂xj ∂xj σY ∂xj
∑MR
i i
i
(9) where σY is the turbulent Schmidt number with σY ) 0.7. The last term on the right-hand side of eq 9 is the mass source from the reactions. 2.6. Cracking Reactions of Single Catalyst Particle. A fourlump reaction kinetic model was considered in the present work to represent gas phase catalytic cracking reactions. The reaction schemes for these kinetic models are shown in Figure 2. The four lumps are VGO (feedstock), gasoline, gas, and coke. VGO is cracked to gasoline, gas, and coke. The rate of consumption of reactant k per unit catalyst volume can be expressed as29–32 -Rk ) Krk
( )
Ck n Cφ Cko k
(10)
where Ck is the concentration of component k and Cko is the initial concentration of pure component k. The value of n is set to be unity for VGO cracking. This implies that VGO evaporated at the beginning of the cracking reaction is easier to crack than VGO evaporated later since heavier components take longer to evaporate. For all other reactions, the value of n is set to be zero. The temperature dependence of kinetic parameters appearing in eq 10 is described by the Arhenius expression33,34 Krk ) Ak exp(-Ek/RTg)
(11)
4634 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 Table 1. Parameters Used in the Simulations inlet gas velocity inlet temperature inlet mass fraction of VGO inlet mass fraction of steam empirically determined constant C1 empirically determined constant Cµ empirically determined constant σ
1.0 m/s 990 K 0.27 0.73 1.44 0.09 1.3
100 µm 688 µm 65 0.8 1.92 1.0 1.0
diameter of particle size of cluster number of particles porosity of cluster empirically determined constant C2 empirically determined constant σk empirically determined constant σh
Table 2. Properties of Particles and Gas Phases Used in Simulations34,35 particle specific heat particle thermal conductivity molecular weight of VGO molecular weight of gasoline molecular weight of coke diffusion coefficient of gas diffusion coefficient of steam viscosity of gas viscosity of steam
1000 J/kg K 0.0454 W/m · K 400 kg/kmol 100 kg/kmol 400 kg/kmol 2.064 × 10-5 m2/s 2.178 × 10-5 m2/s 1.66 × 10-5 kg/m · s 2.0 × 10-5 kg/m · s
Using eq 10, the rates of production of the lumps per unit volume of catalyst can be expressed as 2 RVGO ) -(Kr1 + Kr2 + Kr3)φCVGO
(12a)
2 Rgasoline ) (Kr1CVGO - Kr4Cgasoline - Kr5Cgasoline)φ
(12b) 2 Rgas ) (Kr2CVGO + Kr4Cgasoline)φ
(12c)
2 Rcoke ) (Kr3CVGO + Kr5Cgasoline)φ
(12d)
The parameter φ in eq 10 is the activity factor. With nonselective deactivation of catalyst assumed, the activity factor φ is related to the coke deposition on the catalyst as:29–31 φ)
1 + Bc Bc + exp(AcCc)
(13)
where Cc is the coke concentration (weight percentage). The values for deactivation constants Ac and Bc were taken as 4.29 and 10.4, respectively.34 The kinetic constants for the four lump model proposed by Pitault et al. 34 and Han and Chung35 are used in present simulations. They are listed in Tables 2 and 3. Table 4 shows the comparison of the activation energies used in this work with those found in the literature.36,37 2.7. Initial and Boundary Conditions. Initially, both gas and particle temperatures were set to be 900 K. The composition, velocities, turbulent kinetic energy, and energy dissipation are specified by ug,y ) ug ;
ug,x ) ug,z ) 0;
Yi ) Yi,0 ;
Tg ) Tg,0 ;
k ) 0.004ug ;
ε ) k3/2/lm
(14)
where the turbulent mixing length lm is taken as lm ) 2Dc. At the outlet, the boundary conditions are ∂ug,y ∂ug,x ∂ug,z ) ) ) 0; ∂y ∂x ∂z
∂Yi ) 0; ∂y
∂Tg ) 0; ∂y ∂k ∂ε ) 0; ) 0 (15) ∂y ∂y On the particle surface, no-slip boundary condition for gas flow is assumed: u)k)ε)0 (16) The mass flux of each gas species is balanced with its rate of production/consumption. The wall surface reaction boundary condition of gas species can be expressed as26 -
FkDkSs ∂Yk Mk ∂xj
|
r)R
) Krk(Yk - Yk,R)
(17)
1730 kg/m3 991 K 50 kg/kmol 18 kg/kmol 8.8 × 10-6 m2/s 1.7 × 10-5 m2/s 5.0 × 10-5 kg/m · s 1.66 × 10-5 kg/m · s 1.66 × 10-5 kg/m · s
particle density temperature of particle molecular weight of gas molecular weight of steam diffusion coefficient of VGO diffusion coefficient of gasoline viscosity of VGO viscosity of gasoline viscosity of coke
The boundary condition of temperature at the catalyst particle surface is -λg
|
∂Tg ) ∂xj R
∑ R ∆H i
i
(18)
i
The governing eqs 1–9 coupled with reaction kinetics and boundary conditions are solved using a finite volume method.38 The modeled three-dimensional riser and its sizes are shown in Figure 1. The cluster is assumed to be located in the center of the riser. Particles in the cluster are labeled. Because the cluster size is so small compared with the riser size, the riser wall effect on the gas flow around the cluster can be ignored. The physical domain of the sphere boundary of each particle was mapped onto a cubic domain using the body-fitted coordinate to make the cubic grids for the boundary of each particle.39 Fresh gases are introduced to the cluster and react on the surface of catalyst particles. The sensitivity of the computed results to grid size was tested. The grid for all following computations is selected by increasing the number grids until the results are almost insensitive to the grid number. Figure 3 shows results from the medium grid sizes (162 × 162 × 328), the finer grid sizes (196 × 196 × 374), and the coarse grid sizes (122 × 122 × 260) (presented later in this paper). It is noticed that the medium grid sizes are appropriate for all following computations except otherwise stated. The finer grid size needs much more CPU time. A set of simulation using the medium grid sizes needs about 24 h of CPU time on a PC computer (80 GB hard disk, 128 Mb RAM, and 600 MHz CPU). 3. Computed Results and Analysis 3.1. Base Case Simulations. In the base case simulations, the parameters proposed by Pitault et al.34 and Han and Chung35 are used and listed in Tables 2 and 3. The inlet gas velocity, inlet temperature, and porosity were set to be 1.0 m/s, 990 K, and 0.8, respectively. Figure 4 shows the gas temperature distribution along the axes of the cluster. The cluster Reynolds number is 4144.6. The temperature reduction in the cluster is due to an endothermic reaction. The temperature reduction from the cluster located at the front of the cluster to the wake of the cluster is found. In the radial direction, the temperature increases radially from the center to the side of the cluster since unreacted gas flow out of the cluster. The relatively high temperature at the back of the cluster may be due to some hot gas back flow and relatively weak endothermic reactions there. In the base case simulation, the temperature drops about 3 K due to the cracking reaction.
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4635 Table 3. Kinetics Constants Reported by Pitault et al. (1995) and Heats of Reactions by Han and Chung (2000) pre-exponential factor Aj (mr3/mcal3 s) at 756 K (Pitaulat et al., 1995)
∆Hr (J/kg) (Han and Chung, 2000)
3.5890 × 106
195
Kr2
2.5410 × 107
670
Kr3
7
6.7910 × 10
745
Kr4
8.8572 × 102
530
Kr5
5.3198 × 107
690
cracking reaction Kr1
route A: VGO 98 gasoline route B: VGO 98 gas route C: VGO 98 coke route D: gasoline 98 gas route E: gasoline 98 coke
Table 4. Comparison of Activation Energies with Literature Values34,35 activation energy Ej (J/kmol) (present simulations)
activation energy Ej (J/kmol) (literature34,35)
Kr1
6.8 × 107
(4.187–15.072) × 107
Kr2
8.3 × 107
(5.024–8.795) × 107
cracking reaction route A: VGO 98 gasoline route B: VGO 98 gas
6.1 × 10
(2.798–6.280) × 107
Kr4
5.2 × 10
(5.443–6.280) × 107
Kr5
11.1 × 107
(6.699–11.304) × 107
7
Kr3
route C: 98 VGO coke route D: gasoline 98 gas route E: gasoline 98 coke
7
The effect of turbulent model of gas phase on gas temperature is also given in figure. The turbulent kinetic energy-specific dissipation rate model, k-ω, is a low-Reynolds-number model covering laminar, transitional and turbulent regions.40 The k-ω model includes two extra transport equations to represent the turbulent properties of the flow. The first transported variable is turbulent kinetic energy, and the second transported variable in this case is the specific dissipation. The turbulent viscosity is µt ) Fk/ω. We see that both models give a similar result of gas temperature. The k- model, capturing the physics of cracking reactions at reasonable computing resource requirements, has been suggested.
Figure 2. Four-lump kinetic model.
Figure 3. Ratio of gas flux through cluster to the total air flux as a function of time.
Figure 4. Profile of gas temperature inside of the cluster.
Figure 5. Concentrations of gas and gasoline inside of the cluster.
Figure 5 shows the axial and radial distributions of gas and gasoline concentrations along the centerline of the cluster. The gas concentration increases, reaches its maximum near the back, and then decreases due to the reactions and the back flow at the wake of cluster. The computed gasoline concentration increases along the flow direction. The highest gas concentration is found near the cluster center, while the highest gasoline concentration deviates from the cluster center. The lowest gas and gasoline concentrations are at the outside of the cluster, as shown in this figure. The produced gas and gasoline distributions vary spatially within the cluster. The gasoline concentration increases due to VGO conversion by reaction route A and decreases by reaction routes D and E. For gas, the molar concentration was increased due to reactions from VGO to gas and gasoline to gas. Various rates for particle in the cluster and for the isolated particle are shown in Figure 6. It is found that the reaction rates from VGO to gasoline, gas, and coke are higher in the front facing the incoming gas than those in the back of the cluster, while the reaction rates from gasoline to gas and coke are lower in the front than those in the back of the cluster. Figure 7 shows the produced gas and gasoline molar distributions from each particle in the cluster and from the isolated particle. There is a variation of the produced gas and gasoline from particle to particle in the cluster. For the isolated particle, the reactions from VGO to gasoline, gas, and coke are faster and from gasoline to gas and coke the reaction is slower compared with individual particles in the cluster. Hence, particle clustering will reduce the cracking reaction rates from VGO to gas and gasoline.
4636 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008
Figure 6. Distribution of molar fraction of gas and gasoline from particles in the cluster and isolated particle.
Figure 7. Profile of reaction rates of particles in the cluster and an isolated particle.
Figure 8. Profile of ratio of gas flux through cluster to inlet air flux and averaged gas temperature as a function of porosity.
3.2. Effect of Cluster Porosity. The ratio of gas flux through the cluster to inlet gas flux is shown in Figure 8 as a function of the cluster porosity. For cluster diameters of 638, 757, 866, and 1090 µm, the corresponding cluster porosities are 0.75, 0.85, 0.9, and 0.95, respectively. With the increase of particle concentration in the cluster, the ratio of gas flux is reduced due to the high flow resistance inside the cluster. The ratio of gas flux through the cluster increases rapidly as the porosity is greater than
Figure 9. Reaction rates as a function of cluster porosity.
Figure 10. Distributions of mass fluxes of gas and gasoline from isolated particle and particles in the cluster.
0.9. The averaged gas temperature inside the cluster will be reduced by endothermic reactions. On the other hand, the temperature will be increased as more gas passes through the cluster. As a result, the variation of gas temperature inside the cluster is not obvious with the change in the cluster porosity. With the increase of cluster porosity, more VGO can pass through the cluster and results in the cracking reaction rate increment, as shown in Figure 9. The reaction rates from VGO to gasoline, gas, and coke are increased, while the reaction rates from gasoline to gas and coke are decreased with the increase of the cluster porosity. The reaction rate of gasoline to gas is almost 1 magnitude order less than that of VGO to gasoline. These result in more gasoline and less gas produced, as shown in Figure 10. The produced gasoline increases with the increase of the cluster porosity. Hence, the effect of particle clustering on the cracking reactions is obvious in the FCC riser. 3.3. Effect of Inlet Gas Velocity. From simulations, the average reaction rate of the cluster, R ) Σni)1 Ri/65, is computed. Figure 11 shows the reaction rates of the cluster and the isolated particle as a function of inlet gas velocity. Roughly, the reaction rates from VGO to gasoline, gas, and coke increase, while the reaction rates from gasoline to gas and coke decrease with the increase of inlet gas velocity. For the isolated particle, the effect of inlet gas velocity on the reaction rates is not noticeable. The flow rates of gasoline and gas can be calculated by summing the product of density and mass fraction and velocity with the product of the facet area vector and the facet velocity vector at each grid. Figure 12 shows the yields of gas and
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4637
Figure 11. Profile of mass fluxes of gas and gasoline as a function of inlet velocity.
Figure 12. Profile of mass fluxes of gas and gasoline as a function of inlet gas velocity.
gasoline from the cluster and from the isolated particle as a function of inlet gas velocity. With the increase of inlet gas velocity, the mass flux of gasoline increases, while the mass flux of gas decreases. The cluster breakage may occur when the inlet gas velocity is increased to some point. Note that the cluster breakage is not considered in the present model. Experiments and simulations have shown that the gas velocity is high in the riser center and low near the riser wall. With uniform inlet gas velocity, the gas flow in the cluster is concavely distributed, as shown in Figure 13. If the inlet gas velocity is linearly increased from the left to the right of the riser with the same flow rate as that of the uniform inlet gas velocity, flow distribution in the riser is still close to the case with uniform inlet velocity, but the mass fraction of gas is increased, and the gasoline mass fraction is reduced for the case of the nonuniform inlet gas velocity compared with that of the uniform inlet velocity, as shown in Figure 14. Thus, the inlet gas velocity variations can change productions of gas and gasoline in the cluster. Experiments and simulations indicated that clusters move downward in the annulus and move upward in the core. The motion of cluster is somewhat stochastic, and the cluster shape is highly variable. For simplicity, we assumed that all particles in the cluster flow up or down with the same velocity along vertical direction, and the cluster remains its original size and shape. Figure 15 shows the contours of gasoline mass fraction at three instances using a dynamic mesh model of adaptively sampled distance fields.41 As the cluster moves up, the inlet
Figure 13. Influence of inlet gas velocity boundary conditions on axial gas velocity through the centerline of cluster.
Figure 14. Computed mass fluxes of gas and gasoline at two different inlet gas velocity boundary conditions.
Figure 15. Instantaneous mass fraction of gasoline as the cluster moves up at z ) 0 cross section (ug ) 1.5 m/s, uc ) 0.5 m/s, g ) 0.8, T ) 991, mass fraction of VGO ) 0.27, mass fraction of steam ) 0.73).
gas velocity and cluster velocity are 1.5 and 0.5 m/s, respectively. While as the cluster moves down, the inlet gas and cluster velocities are 0.5 and -0.5 m/s, respectively. For these two cases, the relative velocity, (ug - uc), equals 1.0 m/s. For case of the downward moving cluster, the cluster will meet more “fresh” incoming VGO and leads to high reaction rates and,
4638 Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008
Figure 16. Computed mass fractions of VGO, gas, and gasoline as the cluster moves up and down.
Figure 18. Variations of reaction rates of the moving particle during the formation of cluster.
Figure 17. Instantaneous mass fraction of gas during cluster formation at z ) 0 cross section (ug ) 1.0 m/s, g ) 0.8, T ) 991, mass fraction of VGO ) 0.27, mass fraction of steam ) 0.73).
therefore, higher mass fraction of gas and gasoline compared with the case of upward moving cluster, as shown in Figure 16. 3.4. Effect of the Cluster Formation. Figure 17 shows the cross-sectional view of a cluster with 64 particles in the cluster and one particle outside the cluster. Particles in the cluster remain stationary, and the particle outside the cluster moves toward the cluster by the drag force. The particle motion equation can be written as25 dup π ) CDFpd2p|ug - up|(ug - up) + mpg (19) dt 8 where ug and up are velocity vector of gas and particle, respectively. In the present work, the following correlation is used for estimating drag coefficient: mp
24 (1 + 0.15Re 0.687) Re Re ) Fdp|ug - up|/µg
CD )
(20) (21)
where Re is the particle Reynolds number. Local gas and particle velocities were obtained by solving the gas momentum equations and the particle motion equations. The trajectory of the particle motion can be calculated from dxi ) up (22) dt The movement of the moving particle is calculated by means of the dynamic mesh model of adaptively sampled distance fields.41 The instantaneous concentrations of gasoline are shown
Figure 19. Variations of mass fluxes of gas and gasoline of the cluster during the formation.
in Figure 17 with the inlet velocity and mass fraction of VGO of 1.0 m/s and 0.23, respectively. The initial position of the moving particle is located inside a vortex formed at the back of the cluster. As the particle is moving toward the cluster, the resistance between particles in the cluster and gases is varied, and the gas flux passing through the cluster is altered. From Figure 17, the instantaneous molar concentration of gasoline is higher in the center of the cluster compared with that in the back of the cluster. It is found that the molar fraction of gasoline is changed as the moving particle moves toward the cluster. The variation of the ratio of gas flux through the cluster to inlet gas flux as a function of time is shown in Figure 3. The gas flux through the cluster decreases when the moving particle moves toward the cluster. This causes a change of the cracking reaction of both particles in the cluster and the moving particle due to the concentration variations. Figure 18 shows the reaction rates of the moving particle as a function of time. The reaction rates from gasoline to gas and gasoline to coke of the moving particle decreased as it approaches the cluster, while the reaction rates from VGO to gasoline, gas, and coke are increased. It is found that the particle clustering delays cracking reactions. Figure 19 shows the produced mass fluxes of gas and gasoline of the cluster as a function of time. As the moving particle moves toward the cluster, the mass fluxes of gas and gasoline
Ind. Eng. Chem. Res., Vol. 47, No. 14, 2008 4639
decrease. Hence, the productions of gas and gasoline are changed during the cluster formation. 4. Conclusions A mathematical model for predicting gas and gasoline distributions in a cluster has been proposed by coupling a hydrodynamics model with four-lump catalyst cracking reactions. Numerical results indicate that gas temperature decreases in the cluster along the flow direction. The reaction rates from VGO to gasoline, gas, and coke of an individual particle in the cluster are slower than those of the isolated particle, while the reaction rates from gasoline to gas and coke of an individual particle in the cluster are faster than those of the isolated particle in the stream. More gas and gasoline are produced for the downward moving cluster compared with that of the upward moving cluster at the same relative velocity. Furthermore, the produced gasoline and gas amounts are decreased due to particle clustering. The effect of the cluster formation on the gas and gasoline productions is preliminary analyzed. Our model provides a good basis for the numerical modeling of cracking reactions of catalyst particle cluster in a riser. However, there are some limitations associated with the present model. First, the catalyst particle cluster maintains its mechanical structure throughout the simulations and thus the cluster breakup is not simulated. Certainly this model needs to be further refined to give an accurate and experimentally verified model for the cracking reaction in FCC risers. Acknowledgment This work was supported by Natural Science Foundation of China through Grant Nos. 50776023 and 20606006 and the National Key Project of Scientific and Technical Supporting Programs Funded by the Ministry of Science and Technology of China (NO.2006B-AA03B01) and by the Scientific Research Foundation of the Harbin Institute of Technology through Grant No. HIT. 2003.34. Nomenclature A ) cross-sectional area of cluster, m2 C1, C2 ) empirically determined constants Cc ) coke concentration Cg ) gas specific heat, (kJ/m3 K) Ck ) concentration of component k, kg/m3 d ) particle diameter, m D ) molecular diffusion coefficient, m2/s Dc ) diameter of cluster, m g ) gravity, m/s2 Gk ) turbulent kinetic energy production, kg/(m · s3) GR ) production due to reaction, kg/(m · s3) ∆Hk ) heat of reaction, kJ/kmol i, j, k ) coordinate direction k ) species, or turbulent kinetic energy, m2/s2 K ) reaction rates, kmol/(m3 · s) Mk ) molecular weight of species k, g/mol mP ) mass of single particle, kg M ) molar mass, kg/mol n ) particle number p ) gas pressure, Pa R ) gas constant, J/(kmol K) Rc ) radius of cluster, m Ri ) kinetic rate, mol/(m3 · s) r ) radial direction, m Ss ) specific surface area of particle, m2/m3
Tg ) temperature, K ug ) inlet gas velocity, m/s up ) particle velocity, m/s uyg ) axial velocity through the cluster, m/s x, y, z ) coordinates Yk ) mass fraction of species k Vg ) gas flux through the cluster, uyg × A, m3/s Vg,inlet ) gas flux from inlet, ug × A, m3/s Greek Symbols g ) porosity ) energy dissipation rate, m2/s3 λg ) thermal conductivity of gas, kJ/(m · s · K) µg ) gas viscosity, kg/(m · s) µl ) laminar gas viscosity, kg/(m · s) Fg ) gas density, kg/m3 Fs ) particle density, kg/m3 σ ) empirically determined constant ω ) specific dissipation rate, 1/s Subscripts g ) gas phase k ) gas species R ) at particle surface s ) solid phase
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ReceiVed for reView September 28, 2007 ReVised manuscript receiVed March 9, 2008 Accepted March 12, 2008 IE071305Q
