Chemical Engineering Science 59 (2004) 5225 – 5231 www.elsevier.com/locate/ces
DEM modeling and simulation of a catalytic gas–solid fluidized bed reactor: a spouted bed as a case study Sunun Limtrakul∗ , Asada Boonsrirat, Terdthai Vatanatham Department of Chemical Engineering, Faculty of Engineering, Kasetsart University, Bangkok, Thailand Received 28 February 2004 Available online 28 October 2004
Abstract A combined model of discrete element method (DEM) and mass transfer was developed for investigating the local mass transfer throughout a catalytic gas–solid fluidized bed reactor. The mathematical models are based on the equations of mass conservation coupling with the equations of gas and particle motion. The realistic model of DEM, which calculates the contact force acting on the individual particles, is used for monitoring the movement of individual particles in the bed. The contact force is calculated from the concept of spring, dash-pot and friction slider. The flow field of gas is predicted by the Navier–Stokes equation. This DEM-mass transfer model provides the information regarding the particle movement and distribution, gas velocity, gas holdup, and conversion profiles in the bed. A spouted bed reactor for decomposition of ozone on oxide catalyst was chosen as a case study. It was found that the ozone conversion is non-uniform in the bed with very low values near the center but high near the wall. The simulation results show very good agreement with the experimental results of Rovero and co-workers and are in better agreement than the results obtained from the one-dimensional model developed by Mathur and Lim. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Simulation; Discrete element method; Mass transfer; Spouted bed; Multiphase reactors; Fluidization
1. Introduction A catalytic gas–solid fluidized bed reactor has been important to various chemical industrial processes such as catalytic cracking, pyrolysis and combustion. The successful design and operation of a gas–solid fluidized bed reactor depends on an ability to predict the behavior in the system, especially the hydrodynamics, mixing of individual phases and the heat and mass transfer rates. Experimental approach to directly measure the behavior in the system is quite difficult technique and high cost of operation. Accordingly, a numerical simulation approach can provide a powerful tool to investigate the detail phenomena in the reactor. The performance of a fluidized bed reactor depends on gas and catalyst behavior which is difficult to access due to
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E-mail address:
[email protected] (S. Limtrakul). 0009-2509/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2004.09.020
the complex flow in the system. The flow behavior in a fluidized bed is usually predicted by the two-fluid model which regards the particle phase as a continuum (Gidaspow, 1994; Kuipers et al., 1992). Many researchers have investigated the performance of the fluidized reactor using a simple model, the two-region model (Davidson and Harrison, 1963; Kunii and Levenspiel, 1999). In this model, the reactor consists of two distinct zones, the bubble phase and the emulsion phase. The model is based on many simplifications. Nevertheless, the system containing a great number of large particles moving with complex behavior requires a model which is based on more realistic assumptions. Recently, many researchers (Limtrakul et al., 2003; Tsuji et al., 1992, 1993) presented the discrete particle simulation in a fluidized bed based on physical properties of particles in the bed by using the discrete element method (DEM). In the DEM model, the motion of individual particles is obtained from the calculation of the contact force acting on each particle. The contact force is calculated from analogy to a spring, dash-pot, and
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friction slider system which was first proposed by Cundall and Strack (1979). The flow field of gas is predicted by the Navier–Stokes equation. In addition, Kaneko et al. (1999) applied the DEM model combined with the energy balance to study the temperature distribution of gas and particles in a fluidized bed reactor for gas-phase olefin polymerization. The simple reaction rate form was used to account for the total heat of polymerization produced in the reactor. However, a combined model of DEM, gas momentum equation and complete mass transfer equation for investigating the local mass transfer throughout a catalytic gas–solid fluidized bed reactor has never been reported. A spouted bed is a special case of fluidization. It is an effective means of contacting gas with coarse solid particles. There is increasing application of spouted beds as chemical reactors, especially in the coal processing. In a spouted bed, the jet region extends up to the upper free surface of the bed leading to two distinct zones, i.e., spout and annulus, of different gas–solid contacts. The spout zone acts as a dilute transport system, while the annulus has similarities with a moving bed. However, the particles from the annulus can enter the spout at all levels resulting essentially in random behavior of the particles. The models for spouted bed reactors have been developed by Mathur and Lim (1974) and Piccinini et al. (1979). Both models are based on the concept of two regions consisting of spout and annulus. In the spout region, gas is simply assumed to enter the vessel vertically upwards without radial mixing. Mathur and Lim (1974) proposed that one-dimensional flow of gas can be assumed in the annulus without radial gradient, while 1979 et al. (1979) assumed that gas flow follows curved paths in the annulus without dispersion. The predicted profiles of reactor conversion by Mathur and Lim are too flat, but the Piccinini’s results are too steep. This research work extends the DEM model to combine with the mass conservation equations. This new DEM-mass transfer model can be used for investigating local conversion throughout a catalytic gas–solid fluidized bed reactor. The model is based on the mass balance equations coupling with the equations of gas and particle motion. The model can provide the information regarding the particle velocity and distribution, gas velocity, gas holdup, and concentration profiles. A simple first-order reaction, decomposition of ozone on iron oxide catalyst in a spouted bed is chosen as a case study. The simulation data were then compared with the experimental data reported by Rovero et al. (1983).
spring slider
dash-pot spring slider dash-pot
(a)
(b)
Fig. 1. Models of contact forces (a) normal force (b) tangential force.
2.1. Equations of particle motion A three-dimensional fluidized bed reactor containing single type of catalyst with uniform size is studied. Tracking of particle motion in the bed leads to the particle distribution and void fraction. Movement of individual particles is evaluated by the Newton’s equation of motion which includes the effects of gravitational force, contact force, and fluid force. Individual particles move with two types of motion, translational and rotational motions, given by a =
f + g, m
(1)
=
T , I
(2)
where the acceleration of particles ( a ) is a function of the sum of forces acting on the particle (f). The angular acceleration of particles () depends on the torque (T ) caused by the contact force and the moment of inertia of particles (I ). The force acting on the particles consists of the particle contact forces (fC ) and the force exerted by surrounding fluid (fD ). f = fC + fD .
(3)
The contact force between two spherical particles can be modeled by the simple concept of spring, dash-pot and friction slider as shown in Fig. 1. Thus the model depends on the parameters of stiffness, dissipation, and friction coefficients which can be obtained from the physical properties of the particles. The details of the applications of this model in gas–solid systems were illustrated in previous work (Chaleamwattanatai, 2000; Limtrakul et al., 2003; Moungrat, 2001; Tsuji et al., 1992, 1993). 2.2. Equations of gas motion
2. Mathematical modelling The mathematical model providing concentration profile in fluidized bed reactor consists of equations of particle motion, equations of gas motion, and equations of mass conservation.
The fluid flow field is considered in the axisymmetric cylindrical coordinates. The bed is divided into small fluid cells in a two-dimensional domain as shown in Fig. 2. Each fluid cell consists of gas phase contacting with catalyst particles. The pressure and gas velocity are assumed to be
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is the reaction rate based on volume of catalyst, k where RA r is reaction rate constant, b is the stoichiometric coefficient of reaction, and CA,s is the concentration of gaseous A in gas in the particle phase. The effectiveness factor, , is determined as a function of the Thiele modulus, MT (Levenspiel, 1999).
1 = MT
1 1 − tanh 3MT 3MT
.
(6)
The Thiele modulus is defined by dp MT = 6
uniform in each cell. Both values can be calculated from the momentum equation of gas phase as follows: (4)
where is the void fraction, u is the velocity vector of fluid, is the fluid density, p is the pressure of fluid, and fsi is the fluid drag force exerted on the particles. The fluid drag force can be calculated by the Ergun’s correlation (Ergun, 1952; Chaleamwattanatai, 2000; Limtrakul et al. (2003); Tsuji et al., 1992). 2.3. Equations of mass conservation kr
The first-order irreversible catalytic reaction, A(g) −→ bB (g) , was chosen to test this model. The reaction is assumed isothermal. The reactant gas mixture, consisting of A, B and inert I, is fed into the reactor through the injector. The equation of mass conservation, which explains the conversion of reactant in the reactor, is considered in each cell as shown in Fig. 2. In each cell, while gas is moving with the velocity of u, gaseous A diffuses through the gas film surrounding the particle to react at the surface of catalyst particles. The size of cell is small so that the reactant gas concentrations in gas phase and in gas bathing the particle phase are assumed uniform. The reaction rate is written as a function of effectiveness factor RA = −kr CA,s ,
kr , De
(7)
where dp is the diameter of particle and De is the effective diffusivity. The equations of mass conservation for gas and particle phases in a cylindrical fluidized bed reactor can be derived as follows. The mass conservation in the bed is assumed axisymmetric. Thus, the continuity equation for gas phase in twodimensional cylindrical coordinates for component i can be written as
Fig. 2. Fluid–particle flow field for two-dimensional model.
∇εp *(εu) + ( u.∇)εu = − + fsi , *t
(5)
*(εCi ) 1 *(rεur Ci ) *(εuz Ci ) + + r *t *r *z *(εDi,m Ci * *(εDi,m Ci 1 * r + = r *r *r *z *z (1 − ε) +6 kg (Ci,s − Ci ). dp
(8)
Considering the fluid cell containing gas and particle phases shown in Fig. 2, the equation of mass conservation for component i of gas in particle phase is written as follows: 6(1 − ε) *(1 − ε)Ci,s = kg (Ci − Ci,s ) + (1 − ε)Ri , dp *t
(9)
where Ci and Ci,s are the concentration of gaseous i in gas phase and in particle phase, respectively, ε is the void fraction in the unit cell, and Di,m is the diffusivity of gaseous i in a mixture. The gas to particle mass transfer coefficient in the fluid cell, kg , is assumed by the correlations for fixed and fluidized beds (Fogler, 1998): kg = 0.4548Re−0.4069 1/3 Di,m × Re dp Di,m 0.765 0.365 + Re0.82 Re0.386 1/3 Di,m × Re εdp Di,m
for Re < 10,
(10)
for Re 10.
(11)
kg =
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3. Numerical method
Table 1 Catalyst properties (Rovero et al., 1983)
The equations of mass conservation in gas and particle phases provide the concentration of unconverted reactant gas in both phases. The mass conservation equations combined with the equations of gas and particle motion were simultaneously solved for the profiles of particle and gas velocities, void fraction, and conversion of reactant gas in the system. The differential equations of gas motion and mass conservation were solved by finite difference method. The numerical method, semi-implicit method for pressurelinked equation (SIMPLE), developed by Patankar (1980), was used. The solid motion was solved algebraically. The differential two-dimensional equation of gas motion considered in an Eulerian framework and Lagrangian particle motion equations, with mutual interaction between gas and particles taken into account, were simultaneously solved to provide particle positions, particle velocities and gas velocities. In Eulerian–Lagrangian flow mapping, each fluid cell was considered to contain a group of particles interacting with the fluid. The position and velocity of individual particles were calculated using the Lagrangian equation of particle motion. The solids holdup in the cell was obtained from the information of particle positions. Concurrently, the mass transfer between gas phase in two-dimensional fluid cell and gas phase bathing the group of particles located in the fluid cell were simultaneously calculated to provide locally averaged concentrations in both gas and particle phases. In the finite difference method, the flow domain was divided into cells, the size of which was smaller than the macroscopic motion of particles in the system but larger than the particle size. The cell size for the calculation of gas flow and concentration in gas phase for this work was 9.5 × 16.4 mm(r × z). The time step should be set small enough to obtain the calculation stability. However, the time step should not be too small to save computation time. Tsuji et al. (1993) proposed that the time step should correspond to the following relation for stable calculation: t m/k, (12) 5
Constituent -Al2 O3 , 3Al2 O3 .2SiO2 , -Fe2 O3 , Al2 O3 .SiO2 Fe content 2.5% Particle density 2200 kg/m3 Particle diameter 4.4 mm
where t is the time step, m is the mass of a particle and k is the stiffness parameter. The simulation was carried out by using the time step of 2 × 10−4 s. 4. Simulation results and discussions Ozone decomposition on iron oxide catalyst with the rekr
action of O3 −→ 23 O2 was investigated in a spouted bed reactor, which is one type of the fluidized bed reactor. The cylindrical reactor has 0.152 m I.D., 0.60 m height with a conical gas plenum of 60◦ inclined angle. The inlet gas injector with the diameter of 19 mm is located centrally at the base. The properties of commercial catalyst, Girdler G47,
Table 2 Simulation conditions Gas phase: Inlet mole fraction: ozone(A):oxygen(B):nitrogen(I) Reaction rate constant, kr , s−1 Superficial gas velocity, m/s
0.001 : 0.001 : 0.998 5.4 2.28
Particle phase: Catalyst loading, kg Stiffness, N/m Friction coefficient Coefficient of restitution
3.93 (40,000 particles) 800 0.3 0.9
used in the simulation is shown in Table 1. The inlet gas is dilute gas of ozone in dry air with the ozone mole fraction of 0.001. This leads to an isothermal condition in the system in which the volume change and non-isothermal effects due to the heat of reaction could be neglected. The simulation conditions are shown in Table 2. As mentioned above, the fluid flow field and gas concentration distribution are calculated in two-dimensional coordinates by assuming axisymmetry while individual particles are monitored in three-dimensional coordinates. Therefore, the ozone concentration in gas phase are shown in two-dimensions, while the ozone concentration in gas bathing solid phase along with the particle distribution can be shown in three-dimensional coordinates. However, due to the difficulty of the visualization for solid phase information in a three-dimensional system, the axially sliced two-dimensional ozone fraction in particle phase and particle distribution are illustrated in this work. The central region of the bed was chosen in the illustration. Fig. 3 shows the dimensionless ozone concentration distributions in particle phase and gas phase in the bed as a function of time at the superficial gas velocity of 2.28 m/s. The dimensionless ozone concentration indicates the fraction of unconverted ozone, CA /CA0 . The color scales represent various values of dimensionless ozone concentrations. In addition, Fig. 3 provides the information regarding the distribution of particles in the bed. At the initial time, (0 < t < 0.16 s), slugs or large bubbles are generated particularly near the center of the bed where the strong stream of gas is upwards introduced. The solids holdup is very low in this area. The unconverted ozone fraction in gas phase is high (i.e., low conversion) in large bubble area due to less catalyst content. After 1 s, when the bed approaches the steady state, the effect of gas injection on the solid movement is reduced. The gas bubble size becomes smaller. The solid movement and
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Fig. 3. Particle distribution and unconverted ozone mole fraction distributions in gas and particle phases as a function of time, with a superficial gas velocity of 2.28 m/s. (Left: Particle phase, axially sliced two-dimensional information at the bed center, Right: Gas phase, two-dimensional axisymmetry).
the conversion are not varied with time as shown in Fig. 3. The catalyst content is still low in the center leading to low conversion or high unconverted ozone fraction. On the other hand, high catalyst holdup with low solid movement occurs near the wall. Thus the bed behavior approaches a packed bed resulting in high conversion. The conversion of reactant in the dense bed zone, where high loading catalyst is found, is much higher than that in the lean bed zone. In the dense bed zone, i.e., near the wall, the unconverted ozone fractions
in particle phase and gas phase are not significantly different at any position in the reactor, indicating that there is good mass exchange between gas and particle phases as shown in Fig. 4. The unconverted ozone fractions in gas phase in the lean bed zone are almost constant and approaches unity at any height because of less catalyst content in this region. In addition, due to the gas injection profile with extremely high velocity at the feed area, the ozone concentration is high in this region.
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were compared. It was found that the predictions from this DEM-mass transfer model are in better agreement with the experimental data than that obtained from the Mathur and Lim’s model. The one-dimensional model fails to account for the radial concentration gradient in both annulus and spout regions.
1
CA CA0
0.9
0.8
5. Conclusions z/Z
Gas
Solid
CA0
CAs
0.2 0.7
0.3 0.4 1.0
0.6 0
0.2 0.4 0.6 0.8 Dimensionless Radial Position (r/R)
1
Fig. 4. Radial profiles of unconverted ozone fraction at different heights for t = 4 s in gas and particle phases.
1
0.9
CA0
CA
0.8
A combined model of DEM and mass transfer equation was applied to investigate the conversion profile of gas–solid catalytic reaction in a fluidized bed reactor. The unconverted ozone fraction distributions in gas and particle phases as a function of time were obtained in a spouted bed using the DEM-mass transfer model. The unconverted ozone mole fractions in gas and particle phases rapidly increase with time and approaches steady state after 1 s. Solid distribution is much less in the lean bed region at the center than in the dense bed region near the wall and this limits the reactant conversion. Therefore, the unconverted ozone mole fractions in both phases at the center are always higher than that at the reactor wall. It was found that good mass transfer between gas and particle phases occurs. The simulation results for ozone conversion on iron oxide catalyst in a spouted bed reactor show good agreement with the experimental results of Rovero et al. (1983). In addition, the results from this DEM-mass transfer model are in better agreement than that obtained from the one-dimensional model of Mathur and Lim (1974).
0.7
Notation a b Ci
0.6
0.5
0
0.2 0.4 0.6 0.8 Dimensionless Radial Position (r/R)
1
Fig. 5. Comparison of the radial unconverted ozone fraction profiles in gas phase obtained from this model, the one-dimensional model developed by Mathur and Lim (1974), and the experiments of Rovero et al. (1983) for particle diameter = 4.4 mm, catalyst loading = 3.928 kg (40,000 particles) and superficial gas velocity = 2.28 m/s.
Fig. 5 shows the comparison of the simulations results obtained from this model and the one-dimensional model developed by Mathur and Lim (1974), and the experimental results measured by Rovero et al. (1983). The one-dimensional model assumes a vertically upwards flow of gas in both spout and annulus regions. Herein, radial gradient of flow is ignored. The radial profiles of unconverted ozone fractions in gas phase at the exit for the steady-state conditions
Ci,s dp De Di,m f fsi g I k kg kr m MT p RA
acceleration of particle, m/s2 stoichiometric coefficient of reaction concentration of component i in gas phase, kmol/m3 concentration of component i in gas in particle phase, kmol/m3 diameter of particle, m effective diffusivity, m2 /s diffusivity of gas i in a mixture, m2 /s sum of forces acting on the particle, N drag force, N gravitational acceleration vector, m/s2 moment of inertia of the particle, kg m2 stiffness, N/m mass transfer coefficient, m/s reaction rate constant, 1/s particle mass, kg Thiele modulus pressure, N/m2 reaction rate based on volume of catalyst, kmol/m3 s
S. Limtrakul et al. / Chemical Engineering Science 59 (2004) 5225 – 5231
T
Torque, Nm
u
fluid velocity vector, m/s
Greek letters
angular acceleration of particle, rads2 void fraction effectiveness factor fluid density, kg/m3
Subscripts i A B I
species of gas reactant gas, i.e., ozone product gas, i.e., oxygen inert gas, i.e., nitrogen
Acknowledgements This work was financially supported by the Thailand Research Fund (TRF) under Research Career Development Project and the Kasetsart University Research and Development Institute (KURDI). One of the authors (Asada Boonsrirat) gratefully acknowledges the scholarship from the National Science and Technology Development Agency (NSTDA). We are pleased to acknowledge Prof. Dr. Yutaka Tsuji and Assist. Prof. Dr. Toshihiro Kawaguchi for their collaboration. References Chaleamwattanatai, A., 2000. Modeling and simulation of particle mixing in a fluidized bed. M. Eng. Thesis, Kasetsart University, Thailand.
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