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Kinetic theory based computation of PSRI riser: Part II—Computation of mass transfer coefficient with chemical reaction
Chemical Engineering Science 64 (2009) 1212 -- 1222
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Kinetic theory based computation of PSRI riser: Part II—Computation of mass transfer coefficient with chemical reaction Benjapon Chalermsinsuwan a,b , Pornpote Piumsomboon a,b , Dimitri Gidaspow c,∗ a b c
Fuels Research Center, Department of Chemical Technology, Faculty of Science, Chulalongkorn University, Phayathai Road, Patumwan, Bangkok 10330, Thailand Center of Excellence for Petroleum, Petrochemicals, and Advanced Materials, Chulalongkorn University, Bangkok 10330, Thailand Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA
A R T I C L E
I N F O
Article history: Received 18 February 2008 Received in revised form 2 November 2008 Accepted 7 November 2008 Available online 19 November 2008 Keywords: CFD Clusters Fluidization Mass transfer Sherwood number Ozone decomposition
A B S T R A C T
The design of circulating fluidized bed systems requires the knowledge of mass transfer coefficients or Sherwood numbers. A literature review shows that these parameters in fluidized beds differ up to seven orders of magnitude. To understand the phenomena, a kinetic theory based computation was used to simulate the PSRI challenge problem I data for flow of FCC particles in a riser, with an addition of an ozone decomposition reaction. The mass transfer coefficients and the Sherwood numbers were computed using the concept of additive resistances. The Sherwood number is of the order of 4×10−3 and the mass transfer coefficient is of the order of 2×10−3 m/s, in agreement with the measured data for fluidization of small particles and the estimated values from the particle cluster diameter in part one of this paper. The Sherwood number is high near the inlet section, then decreases to a constant value with the height of the riser. The Sherwood number also varies slightly with the reaction rate constant. The conventionally computed Sherwood number measures the radial distribution of concentration caused by the fluidized bed hydrodynamics, not the diffusional resistance between the bulk and the particle surface concentration. Hence, the extremely low literature Sherwood numbers for fluidization of fine particles do not necessarily imply very poor mass transfer. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The design of circulating fluidized bed systems, such as gasifiers (Yoon et al., 1978; Gidaspow and Jiradilok, 2007), requires the knowledge of mass transfer coefficients or Sherwood numbers. Transport phenomena texts (e.g., Bird et al., 2002) show that the Sherwood number or the Nusselt number for related heat transfer is two, based on the diameter of particle. Indeed for large particles, Gunn (1978) has shown that the Sherwood number equals two, the diffusion limit, plus a contribution due to convection expressed in terms of Reynolds and Schmidt numbers. The mass transfer and heat transfer coefficients are known to be much lower for fine particles than those given by correlations for large particles (Zabrodsky, 1966; Turton and Levenspiel, 1989; Kunii and Levenspiel, 1991). Recently, Breault (2006) has reviewed the literature and has shown that the Sherwood numbers differ up to seven orders of magnitude ranging from a low value of 10−5 , as reported by Bolland and Nicolai (2001). One
possible explanation for this phenomenon is the formation of particle clusters. The second more plausible explanation is that the conventional method of computing mass transfer coefficients in fluidized beds shows the effect of radial distribution of concentration and has little to do with diffusion to an individual particle. Hence, the representation in terms of the Sherwood number is misleading. In the first part of the paper, the conventional methodology for estimation of the Sherwood numbers is described. In this second part, a different methodology for computation of the very low Sherwood numbers and mass transfer coefficients, with the chemical reaction is proposed. The computed Sherwood numbers agree with very small values reported in the literature. The variation of the Sherwood numbers with reactor length is similar to those found from the Leveque solution of laminar convection problems and experimental observations. The Sherwood numbers are high in the entrance region and reach a constant value, like in the developed flow mass transfer. 2. Ozone decomposition reaction
∗ Corresponding author. Tel.: +1 312 567 3045; fax: +1 312 567 8874. E-mail address: [email protected] (D. Gidaspow). 0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.11.006
The ozone decomposition or dissociation reaction is one of the frequently used chemical reactions for studying reactions in the circulating fluidized bed systems (Fryer and Potter, 1976; Samuelsberg
B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
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and Hjertager, 1995). The ozone reactant is converted to oxygen product. The reaction can be written as
compute the low mass transfer coefficients and Sherwood numbers observed experimentally in fluidized beds of fine particles.
2O3(g) → 3O2(g)
3. Computational fluid dynamics simulation
This decomposition reaction is a simple irreversible first-order catalytic reaction (Hansen et al., 2004). The rate of reaction (rO3 ) for ozone species is shown below:
3.1. Mathematical model
Ozone decompostion reaction rate = rO3 = −kreaction CO3 s where kreaction is the reaction rate constant, CO3 is the ozone molar concentration and s is the solids volume fraction or particle concentration of catalyst used in the reaction. The first-order reaction permits the total resistance to be expressed as the sum of reaction and diffusional resistances. This reaction requires only low concentrations of ozone reactant. Due to the low concentration of the reactant, the heat produced by the chemical reaction is negligible. The ozone decomposition reaction takes place at low temperature, ambient, and is therefore easy to carry out experimentally. The ozone concentration detection is rapid and accurate using fairly simple detection methods (Syamlal and O'Brien, 2003). Hence, the ozone decomposition reaction is selected as the base reaction for this study. 2.1. Previous experimental studies on ozone decomposition reaction There have been several attempts to analyze the experimental results from ozone decomposition in circulating fluidized bed systems. Jiang et al. (1991) investigated the baffle effects on the performance of ozone decomposition with FCC particles. It was found that, the riser with baffles gives a higher solid holdup and ozone conversion in the gas phase than that without baffles. This was because baffles could enhance the gas–solid contact efficiency by promoting radial gas and solid mixing. Ouyang et al. (1993, 1995) measured the ozone concentration profiles in a riser of a circulating fluidized bed with impregnated FCC as the catalyst. The axial ozone concentration profiles showed a significant deviation from plug flow profiles. The radial profiles showed that the ozone concentration was lower in the wall region than in the core region, consistent with the observations of near wall region flow of solid particles. Bolland and Nicolai (2001) studied the mass transfer between gas and solids in the riser section. The highest rate of reaction was found to be in the lower section, and the smallest in the upper zone of the riser. This phenomenon can be explained by means of mass transfer control in the upper zone. Zevenhoven and Ja¨ rvinen (2002) computed the slip velocity between the solid particles and the surrounding gas. They obtained improved gas–solid mass and heat transfer when the slip velocity was high. 2.2. Previous computational studies on ozone decomposition reaction A two-dimensional model for ozone decomposition in a circulating fluidized bed reactor was studied by Schoenfelder et al. (1996). They developed their model based on the empirical data from the experiment, including the height of bottom dense zone and top dilute zone, as well as axial and radial solids concentration profiles. Therdthianwong et al. (2003) modified a two-dimensional model of Schoenfelder et al. (1996) by using two different models for solid viscosity, a semi-empirical and a kinetic theory of granular flow model. Their model gave reasonable predictions of experimentally measured values. Syamlal and O'Brien (2003) also simulated the bubbling fluidized bed reactor with the ozone decomposition to test the capability of their computational fluid dynamics code, MFIX. The MFIX code quantitatively captured the effect of hydrodynamics on the chemical reaction in the bubbling bed system. This study is the first one to
In this study, a set of governing equations, mass, momentum, energy and gas species mass conservation equations, and constitutive equations are solved numerically. The equations used are based on the kinetic theory of granular flow, as reviewed by Gidaspow (1994). This theory has proved its validity by many researchers (Ding and Gidaspow, 1990; Neri and Gidaspow, 2000; Gidaspow et al., 2004; Jiradilok et al., 2006, 2007). Here the modified Wen and Yu drag law (Yang et al., 2003, 2004) has been replaced with that obtained from the energy minimization multi-scale (EMMS) approach. The EMMS drag model has proved to be an effective way for modeling a high solid mass flux system of FCC particles. This study uses commercial CFD program FLUENT 6.2.16 for modeling the system. There are several numerical models for gas–solid two-phase flow in the program such as, the Lagrangian model and the Eulerian model. In this case, the Eulerian model was selected. With this approach, governing equations in each phase are solved separately. A summary of the governing equations and constitutive equations is given in Table 1. Since the EMMS drag model is not available in the FLUENT 6.2.16 program, the new additional UDF (user-defined function) code (Fluent Inc., 2005a,b) in C programming language was added. According to the previous literature, there has not been enough information given on the values of operating parameters. Some optimization process of these values has been done to obtain the optimized parameter values, as summarized in Table 2. 3.2. System description and computational domain To validate the numerical results in this study, the PSRI challenge problem I, Knowlton et al. (1995), was chosen as the reference case. The system for Fluidization VII benchmark test is shown in Fig. 1. The validation and model verification were shown in the first part of this paper. In this second part, we use the same system, based on Knowlton et al. (1995) experiment, with the addition of the catalytic chemical reaction. The catalyst particles in the system were FCC particles, which are commonly used in circulating fluidized bed. The diameter (D) and height (h) of their riser were 0.2 and 14.2 m, respectively. Since a three-dimensional model requires long computation time, this study uses a two-dimensional model for the simulation. The schematic drawing of the riser is depicted in Fig. 2(a). This simplified schematic drawing is based on Benyahia et al.'s (2000, 2002) papers. It has a two inlet–outlet design, used to approximate the riser in two dimensions. The gas was fed to the system at the bottom of the riser. The solid particles were fed from the two sideinlets at 0.3 m above the bottom of the riser with a width of 0.1 m. The gas and solid exited through two symmetric side outlets at 0.3 m below the top of the riser with the same width as solid inlet ports. The other conditions for this simulation are listed in Table 2. The computational domain of the riser used in this study is illustrated in Fig. 2(b). It consists of 19 non-uniform grids in radial or horizontal direction and 285 uniform grids in axial or vertical direction, with a total of 5415 computational cells. The grid independence study is described in Appendix A. The models were solved by using a computer with Pentium 1.80 GHz CPU 2 GB RAM. It took approximately 10 days of computer time to obtain 40 s of simulation time. 3.3. Initial and boundary conditions At the inlet, the velocity, volume fraction, temperature and gas species composition were specified, as summarized in Table 2. On
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Table 1 A summary of the governing equations and constitutive equations.
(f) Solid phase shear viscosity
A. Governing equations (a) Conservation of mass (i) Gas phase
s =
j ( ) + ∇ · (g g vg ) = 0 jt g g
s =
j ( ) + ∇ · (s s vs ) = 0 jt s s
(2)
j ( v ) + ∇ · (g g vg vg ) = −g ∇P + ∇ · g + g g g − gs (vg − vs ) jt g g g
(3)
j ( v ) + ∇ · (s s vs vs ) = −s ∇P + ∇ · s − ∇Ps + s s g + gs (vg − vs ) jt s s s
(c) Conservation of energy (i) Gas phase
√ 2 6 150s dp 1 + s g0 (1 + e) + 2s 2s dp (1 + e)g0 384(1 + e)g0 5
(1 − g ) g
g d2p
+ 1.75
(1 − g )g |vg − vs |
when g 0.74 3 (1 − g )g g |vg − vs |CD0 () 4 dp
(18)
k
(5)
(16)
(17)
dp
24 (1+0.15Re0.687 ), Rek = with Re<1000, CD0 = Re k
j jpg ( h ) + ∇ · (g g vg hg ) = −g + g : ∇ · vg + Sg + Qsg jt g g g jt
(14)
(15)
2
(4)
2
(i) Gas–solid phase interphase exchange coefficient; when g < 0.74
gs =
4 s s dp g0 (1 + e) 3
gs = 150
(ii) Solid phase
(h) Conductivity of the fluctuating energy
s =
(b) Conservation of momentum (i) Gas phase
√
4 10s dp 1 + g0 s (1 + e) + 96(1 + e)g0 s 5
(g) Solid phase bulk viscosity
(1)
(ii) Solid phase
with hg =
4 s s dp g0 (1 + e) 5
when 0.74 g 0.82 () = −0.5760 + when 0.82 g 0.97 () = −0.0101 +
g g |vg −vs |dp , Re 1000, CD0 =0.44 g 0.0214
4(g − 0.7463) + 0.0044 0.0038 2
4(g − 0.7789) + 0.0040 when g > 0.97 () = −31.8295 + 32.8295g
cp,g dTg
2
(ii) Solid phase
j jps ( h ) + ∇ · (s s vs hs ) = −s + s : ∇ · vs + Ss + Qgs jt s s s jt with hs =
(6)
cp,s dTs
(d) Conservation of species (i = O3 , O2 and N2 (air))
j ( y ) + ∇ · (g g vg yi ) = ri jt g g i
(7)
(e) Conservation of solid phase fluctuating energy 3 2
j + ∇ · (s s )vs = (−∇ps I + s ) : ∇vs + ∇ · (s ∇ ) − s jt s s
(8)
Table 2 Parameters used for the simulation. Symbol
Description
Value
D h
Diameter of riser Height of riser Gas density Gas viscosity Particle density Diameter of particle Gas inlet velocity Gas inlet temperature Ozone species mass fraction inlet Air species mass fraction inlet Solid inlet velocity Solid inlet temperature Solid inlet volume fraction Restitution coefficient between particles Restitution coefficient between particle and wall Specularity coefficient Reaction rate constant
0.20 m 14.20 m 1.2 kg/m3 2×10−5 kg/m s 1712 kg/m3 76 m 5.200 m/s 298.15 K 0.00004 0.99996 0.476 m/s 298.15 K 0.60 0.95
g g s dp vg Tg yO3 yAir vs Ts
s
B. Constitutive equations (a) Gas phase stress
e
g = g g [∇vg + (∇vg )T ] − 23 g g (∇ · vg )I
(9)
eW
(b) Solid phase stress
kreaction
s = s s [∇vs + (∇vs )T ] − s ( s − 23 s )∇ · vs I
⎛
4 dp
⎞
⎠
(11)
(d) Radial distribution function g0 = 1 −
s
1/3 −1
s,max
(12)
the other hand, at the outlet, the system pressure was specified as atmospheric pressure. Initially, there were no gas and solid phases in the riser. At the wall, a no-slip condition was applied for all velocities, except for the tangential velocity of the solid phase and the granular temperature. Here, the boundary conditions of Johnson and Jackson (1987) were used. These conditions were first applied in the kinetic theory of granular flow modelling by Sinclair and Jackson (1989). They are vst,W = −
(e) Solid phase pressure ps = s s [1 + 2g0 s (1 + e)]
0.50 3.96, 39.60, 99.00 and 198.00 s−1
(10)
(c) Collisional dissipation of solid fluctuating energy
s = 3(1 − e2 )2s s g0 ⎝
0.90
(13)
6 s s,max jvst,W √ s s g0 3 jn
jW W = − s + W jn
√
(19) 3/2
3 s s v2s,slip g0 6s,max W
(20)
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3.4. Chemical reaction conditions The information needed for chemical reaction is the reaction rate constant. From the previous experimental data, the values of reaction rate constant for ozone decomposition reaction are varied by three orders of magnitude with the activity of catalyst particle. This decomposition reaction can be catalyzed with many types of catalyst compositions, such as FCC catalyst coated with ferric nitrate (Ouyang et al., 1993, 1995), FCC catalyst coated with ferric oxide (Jiang et al., 1991) and -alumina catalyst coated with ferric oxide (Pagliolico et al., 1992). In this study, four different values of reaction rate constants are used to study the effect of this parameter on the Sherwood number and mass transfer coefficient. The range of these values is selected from the literature experiments. The chosen reaction rate constants are summarized in Table 2. 4. Results and discussion 4.1. Calculation of Sherwood number and mass transfer coefficient From the conservation of species equation:
j ( y ) + ∇ · (g g vg yi ) = ri jt g g i
Fig. 1. Schematic drawing of a 20 cm diameter circulating fluidized bed test unit for the Fluidization VII benchmark test.
(7)
where t is the time, g is the volume fraction of gas phase, g is the density of gas phase, vg is the velocity of gas phase, yi is the mass fraction of specie “i” and ri is the reaction rate. Integration of Eq. (7) over time and over the riser radius (x- and zdirections) for ozone species gives the one-dimensional steady state balance: vy g
dCO3 = ri dY
(21)
where vy is the velocity of gas phase in axial or vertical direction and Y is the axial or vertical distance. The decomposition of ozone is a first-order reaction. Hence, the reaction rate constant is independent of the gas concentration. Substitution of the rate of reaction in Eq. (21) gives vy g
dCO3 = −KCO3 s dY
(22)
where K is the overall resistance. Solving Eq. (22) gives ln CO3 = ln CO3 ,0 −
Fig. 2. (a) Schematic drawing and (b) computational domain with their boundary conditions of a simplified riser used in this study.
where
W =
√ 3/2 3 (1 − e2W )s s g0 4s,max
K s Y vy g
(23)
where the subscript “0” is the initial molar concentration of ozone. In fluidized bed reactors, one measures the ozone concentration and obtains the overall resistance from Eqs. (22) and (23) or the finite difference form of Eq. (22). In this simulation, Eq. (23), the time averaged, area averaged result, gives the overall resistance. The linear plot of natural logarithm of ozone molar concentration versus the height of the riser gives the slope, which further gives the overall resistance. Then, the conventional additive resistance concept permits us to compute the mass transfer coefficient. At steady state, the external mass transfer in terms of global rate is equated to the mass transfer from the bulk gas to catalyst surface (Fogler, 1999; Levenspiel, 1999; Bolland and Nicolai, 2001; Welty et al., 2001): kmass transfer av (CO3 − CO3,surface ) = kreaction CO3,surface
(24)
where kmass transfer is the mass transfer coefficient, av is the external surface per volume of catalyst and CO3 ,surface is the surface molar concentration of ozone.
B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
1.20E-06
0 kReaction = 3.96 1/s kReaction = 39.60 1/s kReaction = 99.00 1/s kReaction = 198.00 1/s
1.00E-06
8.00E-07
6.00E-07
4.00E-07
2.00E-07
Natural logarithm of time-averaged ozone molar concentration
Time-averaged ozone molar concentration (kmol/m3)
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-5 -10 -15 -20 -25 -30 -35 -40
0.00E+00 0
2
4
6 8 10 Riser height (m)
12
0
14
Eliminating CO3 ,surface from Eq. (24) and expressing the global reaction rate per unit mass of catalyst (rp ) in terms of CO3 gives 1 1 kmass transfer av
+
1
CO3 s = KCO3 s
(25)
1 1 1 = + K kmass transfer av kreaction
(26)
No.
kreaction (s−1 )
kmass transfer dp D
(27)
where dp is the diameter of the catalyst particle and D is molecular diffusivity. Bolland and Nicolai (2001) have analyzed their ozone decomposition data in a fluidized bed using this exact method. 4.2. Computation and interpretation of Sherwood number and mass transfer coefficient Fig. 3 shows the variation of the computed time-averaged ozone molar concentration with the riser height for various reaction rate constants used in this study. Due to the reaction, all the ozone concentrations decrease with riser height. For the lowest reaction rate constant or kreaction = 3.96 s−1 case, the ozone reactant leaves the riser partially unreacted. For kreaction = 39.60, 99.00 and 198.00 s−1 cases, the ozone is almost used up due to the high reaction rate constant. For a first-order reaction, with or without mass transfer, and constant catalyst concentration, the natural logarithm of the ozone molar concentration varies linearly with height, as given by Eq. (23). Hence, in Fig. 4, the ozone concentrations are presented to test these assumptions and to enable the determination of the mass transfer coefficient. The natural logarithms of computed time-averaged ozone molar concentration with the riser height for various reaction
6 8 Riser height (m)
10
12
14
Riser height (m)
Slope
Intercept
1
3.96
3.5 7.0 10.5
−0.1680 −0.1375 −0.1236
−13.8290 −13.8290 −13.8290
2
39.60
3.5 7.0 10.5
−0.7584 −0.6630 −0.6319
−13.8290 −13.8290 −13.8290
3
99.00
3.5 7.0 10.5
−1.4370 −1.3074 −1.2216
−13.8290 −13.8290 −13.8290
4
198.00
3.5 7.0 10.5
−2.3236 −2.0776 −1.9759
−13.8290 −13.8290 −13.8290
The Sherwood number (Sh) is then given by Eq. (27) below: Sh =
4
Table 3 The fitted equation parameters with various reaction rate constants used in this study.
kreaction
The result from Eq. (25) and the overall resistance from Eq. (23) give the mass transfer coefficient as follows:
2
Fig. 4. Test of model for computation of mass transfer coefficients.
Fig. 3. Computed time-averaged ozone molar concentration with various reaction rate constants.
rp s s =
kReaction = 3.96 1/s kReaction = 39.60 1/s kReaction = 99.00 1/s kReaction = 198.00 1/s
Model equation: ln CO3 = Slope × Y + Intercept.
rate constants are displayed in Fig. 4. All the graph lines are approximately linear, which is consistent with the assumptions used in the ozone species mass balance. The fitted equation parameters, which are the slope and the intercept, for each graph are summarized in Table 3. These fitted slopes are consistent with the profiles of computed ozone molar concentration in Fig. 3. For the highest reaction rate constant or kreaction = 198.00 s−1 case, the highest slope value is obtained. The slopes decrease with decreasing reaction rate constants. The lowest slope is obtained for the lowest reaction rate constant or kreaction = 3.96 s−1 case. From the observation that the Sherwood numbers or mass transfer coefficients are not constant along the entire riser section, the slope values are separated into three regions for each graph. These three heights are at 3.5, 7.0 and 10.5 m, that is the middle section of the riser. From the first part of this paper, these three heights have the same flow structure. However, all the slopes with the same reaction rate constant are not significantly different. These results are also consistent with the assumption used in this study.
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Table 4 Computed information on the Sherwood numbers and mass transfer coefficients at three different heights of the riser. kreaction (s−1 )
No.
Riser height (m)
kmass transfer av (s−1 )
kmass transfer (m/s)
Sherwood number, dimensionless
1
3.96
3.5 7.0 10.5 Averaged
– – – Reaction controlled
– – – Reaction controlled
– – – Reaction controlled
2
39.60
3.5 7.0 10.5 Averaged
138.71 86.35 78.90 101.32
0.0018 0.0011 0.0010 0.0013
0.0046 0.0029 0.0026 0.0034
3
99.00
3.5 7.0 10.5 Averaged
132.18 110.07 109.51 117.25
0.0017 0.0014 0.0014 0.0015
0.0044 0.0037 0.0037 0.0039
4
198.00
3.5 7.0 10.5 Averaged
182.33 136.23 132.60 150.38
0.0023 0.0017 0.0017 0.0019
0.0061 0.0046 0.0044 0.0050
Sh =
kmass transfer dp D
=
6kmass transfer av D
(28)
The equation shows that for large av , small particles, the mass transfer coefficient must be extremely high to give a Sherwood number of two. The conventional method of computation presented here captures the small differences between the bulk and the surface concentration that are due to the hydrodynamics of the fluidized bed, not due to the diffusion between the bulk and the surface concentration. The mass transfer is not as poor as is implied by the extremely small values of the Sherwood numbers, although the particle cluster formation does decrease the mass transfer. Fig. 5 illustrates the effect of reaction rate constants on the computed Sherwood numbers and mass transfer coefficients for three different heights of the riser. The computed Sherwood numbers and mass transfer coefficients have a similar trend because the diameter of the catalyst particle and molecular diffusivity used in this study are constant. At the height of 3.5 m or at the lower section, the Sherwood number and mass transfer coefficient have a minimum value. For the low reaction rate constant or kreaction = 39.60 s−1 case, the ozone reactant is slowly converted. This makes the reaction resistance dominant for this system and results in the high values of Sherwood numbers and mass transfer coefficients. For the high reaction rate constant or kreaction = 198.00 s−1 case, the rate of reaction is also low due to the rapid conversion of ozone reactant. This gives high values of Sherwood numbers and mass transfer coefficients. Therefore, there exists an optimum condition at the suitable moderately
3.00E-03
8.00E-03
Sherwood number (-)
7.00E-03
2.50E-03
6.00E-03 2.00E-03
5.00E-03
1.50E-03
4.00E-03 3.00E-03
1.00E-03
2.00E-03
h = 3.5 m h = 7.0 m h = 10.5 m
1.00E-03 0.00E+00 0.00
kMass transfer (m/s)
Table 4 summarizes the computed information on the Sherwood numbers and mass transfer coefficients at three different heights of the riser for various reaction rate constants. For the lowest reaction rate constant or kreaction = 3.96 s−1 case, the reaction rate constant is too low and, thus, controls the system. Therefore, the mass transfer coefficient cannot be computed using the present method of additive resistances for mass transfer and reaction. In this case, the diffusional resistances are zero. Once the reaction rate constants become high or kreaction = 39.60, 99.00 and 198.00 s−1 cases, the diffusional resistances become large and can be computed using the concept of additive resistances. Appendix B shows a typical numerical example. Since the overall resistance (K) is close to reaction rate constant (kreaction ), the mass transfer is not bad. The reason for the low Sherwood number is its representation in terms of the particle diameter or surface area per unit volume, av . Eq. (27) rewritten as
5.00E-04
0.00E+00 50.00 100.00 150.00 200.00 250.00 kReaction (s-1)
Fig. 5. Effect of reaction rate constants on computed Sherwood numbers and mass transfer coefficients.
reaction rate constant, at kreaction = 99.00 s−1 in this simulation. At the height of 7.0 and 10.5 m or at the higher section, the ozone reactant is mostly converted. The conversion is higher, directly proportional to the reaction rate constant. Hence, the Sherwood numbers and mass transfer coefficients increase when the reaction rate constant increases. Fig. 6 shows the effect of reaction rate constants on the computed riser height-averaged Sherwood numbers and mass transfer coefficients. The average values are calculated at three different heights. The computed values of Sherwood numbers and mass transfer coefficients increase with the increasing reaction rate constant. The explanation of this phenomenon is similar to the reason for the case at the higher section or at the height of 7.0 and 10.5 m in the riser as shown in Fig. 5. Thus, we can state that the mass transfer resistance decreases when the reaction rate constants increases. These study results confirm the literature results that there is a dependence of the mass transfer coefficient or the Sherwood number on the reaction rate constant (Solbrig and Gidaspow, 1967; Kulacki and Gidaspow, 1967; Bird et al., 2002). The effect of riser height on the computed Sherwood numbers and mass transfer coefficients for various reaction rate constants is displayed in Fig. 7. The computed Sherwood numbers and mass transfer
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3.00E-03
8.00E-03
2.50E-03
6.00E-03 2.00E-03
5.00E-03
1.50E-03
4.00E-03 3.00E-03
1.00E-03
2.00E-03 5.00E-04
1.00E-03 0.00E+00 0.00
kMass transfer (m/s)
Sherwood number (-)
7.00E-03
0.00E+00 50.00 100.00 150.00 200.00 250.00 kReaction (s-1)
3.00E-03
8.00E-03
2.50E-03
6.00E-03 2.00E-03
5.00E-03
1.50E-03
4.00E-03 3.00E-03 2.00E-03 1.00E-03 0.00E+00 0.0
1.00E-03 kReaction = 39.60 1/s kReaction = 99.00 1/s kReaction = 198.00 1/s Kato et al., 1970
2.0
4.0 6.0 8.0 Riser height (m)
kMass transfer (m/s)
Sherwood number (-)
7.00E-03
5.00E-04 0.00E+00 10.0 12.0
Fig. 7. Effect of riser height on computed Sherwood numbers and mass transfer coefficients.
coefficients decrease with the riser height and reach constant values at the top section for all cases. This variation of the Sherwood numbers is similar to the typical behavior in convective mass transport. The computed trend from experimental data correlation (Kato et al., 1970) is also plotted in the figure. A reasonable agreement is obtained between this study and Kato's correlation results. As can be seen from Fig. 3, the highest and lowest conversions of the ozone are obtained at the bottom and the top sections of the riser, respectively. This confirms the low mass transfer resistance and results in the high values of Sherwood numbers and mass transfer coefficients near the inlet region. At the upper section, the mass transfer resistance has a large effect on the conversion of ozone reactant. Therefore, the values of Sherwood numbers and mass transfer coefficients decrease. The effect of Reynolds number on experimental and computed Sherwood numbers is displayed in Fig. 8. In this figure, the comparison of the computed Sherwood number from the part one of this paper is also illustrated. The computed Sherwood numbers obtained in this paper and those from the part I of the paper are lower than the experimental Sherwood numbers for large particles in fluidized and fixed bed systems. The values of Sherwood numbers and mass transfer coefficients are in reasonable agreement with the phenomena from the experimental data (Kunii and Levenspiel, 1991) and fall in the range as reviewed by Breault (2006). The computed Sherwood number and mass transfer coefficient from chemical reaction are of the same order of magnitude as the estimated values from the particle cluster diameter.
Fig. 8. Effect of Reynolds number on experimental and computed Sherwood numbers.
Time-averaged ozone molar concentration (kmol/m3)
Fig. 6. Effect of reaction rate constants on computed riser height-averaged Sherwood numbers and mass transfer coefficients.
1.40E-07 h = 3.5 m h = 7.0 m h = 10.5 m
1.20E-07 1.00E-07 8.00E-08 6.00E-08 4.00E-08 2.00E-08 0.00E+00 -0.10
-0.05
0.00 0.05 Radial distance (m)
0.10
Fig. 9. Radial distribution of computed time-averaged ozone molar concentration at three different heights with kreaction = 39.60 s−1 .
4.3. Confirmation of reduced Sherwood number and mass transfer coefficient The computed time-averaged ozone molar concentration versus radial distance at three different heights, with kreaction = 39.60 s−1 case, is illustrated in Fig. 9. The results obtained only for the case with kreaction = 39.60 s−1 are shown because similar trends were observed with other reaction rate constants. The results are similar to those shown by previous researchers (Ouyang et al., 1993, 1995; Bolland and Nicolai, 2001). There is a very little ozone concentration due to the high catalyst concentration near the wall. The highest ozone concentration is at the center region. The computed time-averaged and natural logarithm of computed time-averaged bulk and surface ozone molar concentrations with kreaction = 39.60 s−1 are displayed in Figs. 10(a) and (b), respectively. The surface ozone molar concentrations are calculated using Eq. (24). From the figures, the surface ozone molar concentration is slightly lower than the bulk ozone molar concentration due to the diluteness of the center ozone molar concentration at the catalyst or particle clusters. Fig. 11 shows plots of the computed instantaneous solids volume fraction and ozone molar concentrations with kreaction = 39.60 s−1 at
Time-averaged ozone molar concentration (kmol/m3)
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1.00E-06 Concentration (Bulk)
9.00E-07
Concentration (Surface)
8.00E-07 7.00E-07 6.00E-07 5.00E-07 4.00E-07 3.00E-07 2.00E-07 1.00E-07 0.00E+00
Natural logarithm of time-averaged ozone molar concentration
0
2
4
6 8 Riser height (m)
10
12
14
Fig. 11. Variation of instantaneous solid volume fraction and ozone molar concentration profiles with kreaction = 39.60 s−1 in the observed particle cluster.
-13 Concentration (Bulk) Concentration (Surface)
-15
Particle cluster
-17
-19
-21
-23
-25 0
2
4
6 8 10 Riser height (m)
12
14
Fig. 10. (a) Computed time-averaged and (b) natural logarithm of computed time-averaged of bulk and surface ozone molar concentrations with kreaction = 39.60 s−1 .
the observed particle cluster position in Fig. 12. The ozone concentration is not symmetric within the particle cluster due to the unequal surrounding reactant concentrations, as shown in Fig. 9. The ozone concentration decreases when the solid volume fraction increases or when it passes through the catalyst. The Sherwood number or the mass transfer coefficient, hence, measures the effect of the radial distribution of the ozone molar concentration and reflects this radial mal-distribution. The formation of particle cluster increases the mass transfer resistance in the system. This makes the Sherwood number and the mass transfer coefficient much lower than the diffusional limit of Sherwood number, which is equal to 2. Also, the reason for low Sherwood number for fine particles can be explained using arithmetic. From Eq. (27), a fine particle with high surface to volume ratio will have the Sherwood number lower than a large particle with low surface to volume ratio. To verify the explanation, the conventional and new computed Sherwood numbers with kreaction = 39.60 s−1 is displayed in Fig. 13. For the new computed Sherwood number, the particle diameter in Eq. (27) is changed to the particle cluster diameter. The results are plotted versus the riser height and compared with the computed trend from experimental data correlation (Kato
6.00e-1 5.70e-1 5.40e-1 5.10e-1 4.80e-1 4.50e-1 4.20e-1 3.90e-1 3.60e-1 3.30e-1 3.00e-1 2.70e-1 2.40e-1 2.10e-1 1.80e-1 1.50e-1 1.20e-1 9.00e-2 6.00e-2 3.00e-2 0.00e+00
Fig. 12. Contour of the computed instantaneous solid volume fractions in the riser at 30 s.
et al., 1970). The computed Sherwood numbers based on particle diameter are significantly lower than the computed Sherwood numbers based on particle cluster diameter. These results also infer that the convection to diffusion ratio of fine particles is lower than that of large particles. The reduction in the Sherwood number or the mass transfer coefficient has thus been confirmed. It can be explained by the particle cluster formation, as already proposed in the first part of the paper.
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Sherwood number (-)
10.000
h h I kmass transfer kreaction
This study - Based on particle diameter This study - Based on particle cluster diameter Kato et al., 1970
1.000
ShCluster = Sh ×
0.100
K n p Ps Qgs
dc dp
dc = 0.87 × 10−2 −1.01 × 10−2 m dp = 7.6 ×10−5 m
0.010
ShCluster = Sh × 133 (at the height of 3.5 m)
Qsg
0.001 0.0
2.0
4.0 6.0 8.0 Riser height (m)
10.0
12.0
Fig. 13. Comparison of the computed Sherwood numbers based on particle diameter (Sh) and particle cluster diameter (ShCluster ) with kreaction = 39.60 s−1 . For Sh = 0.0046, ShCluster = 0.62 at a height of 3.5 m.
r ri Rek S Sh Shcluster
5. Conclusions 1. From additive diffusional and chemical resistance concept, the computed Sherwood number is of the order of 4×10−3 and the mass transfer coefficient is of the order of 2×10−3 m/s, in agreement with measured literature experimental data for fluidization of small particles. 2. The Sherwood number or mass transfer coefficient is high near the inlet section, and decreases to a constant value with the height of the riser. This is similar to the normal behavior of the Sherwood number in convective mass transfer process (Kato et al., 1970). 3. The Sherwood number or mass transfer coefficient varies slightly with reaction rate constant. For the higher reaction rate constant case, the rate of reaction is lower due to the rapid conversion of ozone reactant. Therefore, the computed values of Sherwood numbers and mass transfer coefficients increase with the increasing of reaction rate constant. 4. In this study, two explanations are possible for the low Sherwood numbers measured in the fluidization of fine particles. One explanation is due to the particle cluster formation studied in part one of this study. The second more plausible explanation is that the conventional method of computing the mass transfer coefficients does not measure the diffusional resistance to the particle implied by the conventional Sherwood number representation. It shows the effect of radial distribution of concentration. Notation av cp C CD0 Csurface dc dp D D e eW g g0
external surface per volume of catalyst, m−1 heat capacity at constant pressure, J/kg K molar concentration, kg mol/m3 drag coefficient, dimensionless surface molar concentration, kg mol/m3 particle cluster diameter, m particle diameter, m diameter of riser, m molecular diffusivity, m2 /s restitution coefficient between particles, dimensionless restitution coefficient between particle and wall, dimensionless gravity force, kg/m3 radial distribution function, dimensionless
t T v vs, slip vst, W x y yi Y
height of riser, m specific enthalpy, J/kg unit tensor, dimensionless resistance due to mass transfer, m/s resistance due to reaction or reaction rate constant, s−1 overall resistance, m/s unit vector, dimensionless pressure, Pa solid pressure, Pa intensity of heat exchange from gas phase to solid phase, J/s m3 intensity of heat exchange from solid phase to gas phase, J/s m3 reaction rate, kg mol/s m3 reaction rate of specie “i”, kg mol/s m3 Reynolds number, dimensionless source term (e.g., due to chemical reaction), J/s m3 Sherwood number based on particle diameter, dimensionless Sherwood number based on particle cluster diameter, dimensionless time, s temperature, K velocity, m/s slip velocity of particle at the wall, m/s tangential velocity of particle at the wall, m/s radial distance from center of riser, m axial distance from bottom of riser, m mass fraction of specie “i”, dimensionless axial or vertical distance, m
Greek letters
gs s W s s,max W s
gas–particle interphase drag coefficient, kg/s m3 collisional dissipation of solid fluctuating energy, kg/m s3 collisional dissipation of solid fluctuating energy at the wall, kg/m s3 volume fraction, dimensionless solid volume fraction of particle, dimensionless solid volume fraction at maximum packing, dimensionless granular temperature, m2 /s2 granular temperature at the wall, m2 /s2 conductivity of the fluctuating energy, kg/m s viscosity, kg/m s bulk viscosity, kg/m s density, kg/m3 stress tensor, Pa specularity coefficient, dimensionless
Subscripts g O3 s y 0
gas phase ozone specie solid phase axial direction initial condition
Acknowledgments The authors gratefully acknowledge financial support for this study by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0021/2550). This study also was
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Table 5 The computed Sherwood numbers and mass transfer coefficients due to the effect of grid number. No.
kreaction (s−1 )
Grid number (radial direction×axial direction)
kmass transfer av (s−1 )
kmass transfer (m/s)
Sherwood number, dimensionless
– – –
– – –
1
3.96
19×285 29×428 38×570
– – –
2
39.60
19×285 29×428 38×570
101.32 99.02 96.74
0.0013 0.0013 0.0012
0.0034 0.0033 0.0032
3
99.00
19×285 29×428 38×570
117.25 115.30 116.92
0.0015 0.0015 0.0015
0.0039 0.0039 0.0039
4
198.00
19×285 29×428 38×570
150.38 153.23 151.77
0.0019 0.0019 0.0019
0.0050 0.0051 0.0051
partially supported by the Chulalongkorn University Graduate Scholarship to Commemorate the 72nd Anniversary of His Majesty King Bhumibol Adulyadej and the Centre of Excellence for Petroleum, Petrochemicals and Advanced Materials, as well as by US DOE university grant, DE-FG26-06NT42736.
The mass transfer coefficient is calculated from Eq. (26): 1 1 1 = + K kmass transfer av kreaction with dp = 76 × 10−6 m
Appendix A. The grid independence study To confirm that the results are grid independent of grid size, the increments of grid numbers in both the radial and axial directions were performed. The comparison of the Sherwood numbers and the mass transfer coefficients with the change in grid number is summarized in Table 5. Four different values of reaction rate constants were selected for study. For each reaction rate constant, the computed values were approximately same with all the grid number cases. The minimum and maximum mass transfer coefficient error percentages compared with the lowest grid number case were 0.29% and 4.52%, respectively. This indicates that all the grid number cases were sufficiently fine for providing reasonably good grid independence results. However, the computational power currently available is still a significant restriction for using large number of grids. Therefore, the computational domain which consists of 19 non-uniform grids in radial direction and 285 uniform grids in axial direction is chosen in this study. Appendix B. An example of Sherwood number and mass transfer coefficient calculation, explaining our argument for good mass transfer in fluidized beds An example of Sherwood number and mass transfer coefficient calculation is illustrated for the reaction rate constant equal to 39.60 s−1 case at a height of 3.5 m. The computed time-averaged and natural logarithm of computed time-averaged ozone molar concentration are displayed in Figs. 3 and 4, respectively. The obtained slope and the intercept are reported in Table 3. The slope for this case was −0.7584 and the intercept was −13.8290. From the simulation, the computed volume fraction of the solid phase was 0.1289, the computed volume fraction of the gas phase was 0.8711 and the velocity of gas phase in axial direction was 6.0115 m/s. Substitution of all the values into Eq. (23) to obtain the overall resistance gives −
K(0.1289) = −0.7584 (6.0115)(0.8711)
Overall resistance, K = 30.81 s−1 .
(26)
av = (3 × 4 (particle radius2 ))/(4 (particle radius3 )) = 3/particle radius = 3/(dp /2) = 3/((76 × 10−6 )/2) = 78947.37 m−1 kreaction = 39.60 s−1 Note that the overall resistance, 1/K, and the reaction resistance, 1/kreaction , are close to each other. This implies that the mass transfer resistance is small. Therefore, kmass transfer av = 138.71 s−1
and
kmass transfer = 0.0018 m/s The Sherwood number is calculated from Eq. (27): Sh =
kmass transfer dp D
(27)
with D = 2.88 × 10−5 m2 /s Therefore, Sh = 0.0046 Fig. 13 shows that the Sherwood number based on the particle cluster diameter, calculated to be 0.0087 to 0.0101 m in the first part of the paper, is 0.62, close to a typical value for diffusion to a particle. References Benyahia, S., Arastoopour, H., Knowlton, T.M., Massah, H., 2000. Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase. Powder Technology 112, 24–33. Benyahia, S., Arastoopour, H., Knowlton, T.M., 2002. Two dimensional transient numerical simulation of solids and gas flow in the riser section of a circulating fluidized bed. Chemical Engineering Communications 189 (4), 510–527. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 2002. Transport Phenomena. Wiley, New York.
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Bolland, O., Nicolai, R., 2001. Describing mass transfer in circulating fluidized beds by ozone decomposition. Chemical Engineering Communications 187, 1–21. Breault, R.W., 2006. A review of gas–solid dispersion and mass transfer coefficient correlations in circulating fluidized beds. Powder Technology 163, 9–17. Ding, J., Gidaspow, D., 1990. A bubbling fluidization model using kinetic theory of granular flow. A.I.Ch.E. Journal 36, 523–538. Fluent Inc., 2005a. Fluent 6.2 UDF Manual. Fluent Inc., Lebanon. Fluent Inc., 2005b. Fluent 6.2 User's Guide. Fluent Inc., Lebanon. Fogler, H.S., 1999. Elements of Chemical Reaction Engineering. Prentice-Hall, New Jersey. Fryer, C., Potter, O.E., 1976. Experimental investigation of models for fluidized bed catalytic reactors. A.I.Ch.E. Journal 22, 38–47. Gidaspow, D., 1994. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Description. Academic Press, Boston. Gidaspow, D., Jiradilok, V., 2007. Nanoparticle gasifier fuel cell for sustainable energy future. Journal of Power Sources 166 (2), 400–410. Gidaspow, D., Jung, J., Singh, R.K., 2004. Hydrodynamics of fluidization using kinetic theory: an emerging paradigm: 2002 Flour-Daniel lecture. Powder Technology 148 (2–3), 123–141. Gunn, D.J., 1978. Transfer of heat or mass to particles in fixed and fluidized beds. International Journal of Heat and Mass Transfer 21, 467–476. Hansen, K.G., Solberg, T., Hjertager, B.H., 2004. A three-dimensional simulation of gas/particle flow and ozone decomposition in the riser of a circulating fluidized bed. Chemical Engineering Science 59, 5217–5224. Jiang, P., Bi, H., Jean, R., Fan, L., 1991. Baffle effects on performance of catalytic circulating fluidized bed reactor. A.I.Ch.E. Journal 37, 1392–1400. Jiradilok, V., Gidaspow, D., Damronglerd, S., Koves, W.J., Mostofi, R., 2006. Kinetic theory based CFD simulation of turbulent fluidization of FCC particles in a riser. Chemical Engineering Science 61, 5544–5559. Jiradilok, V., Gidaspow, D., Breault, R.W., 2007. Computation of gas and solid dispersion coefficients in turbulent risers and bubbling beds. Chemical Engineering Science 62, 3397–3409. Johnson, P.C., Jackson, R., 1987. Frictional–collisional constitutive relations for granular materials, with application to plane shearing. Journal of Fluid Mechanics 176, 67–93. Kato, K., Kubota, H., Wen, C.Y., 1970. Mass transfer in fixed and fluidized beds. Chemical Engineering Progress Symposium Series 105 (66), 87–99. Knowlton, T., Geldart, D., Masten, J., King, D., 1995. Comparison of CFB hydrodynamic models. PSRI Challenge Problem Presented at the Eighth International Fluidization Conference, Tour, France. Kulacki, F.A., Gidaspow, D., 1967. Convective diffusion in a parallel plate duct with one catalytic wall-laminar flow-first order reaction. Part II—experimental. The Canadian Journal of Chemical Engineering 45, 72–78. Kunii, D., Levenspiel, O., 1991. Fluidization Engineering. Butterworth–Heinemann, Boston.
Levenspiel, O., 1999. Chemical Reaction Engineering. Wiley, New York. Neri, A., Gidaspow, D., 2000. Riser hydrodynamics: simulation using kinetic theory. A.I.Ch.E. Journal 46, 52–67. Ouyang, S., Lin, J., Potter, O.E., 1993. Ozone decomposition in a 0.254 m diameter circulating fluidized bed reactor. Powder Technology 75, 73–78. Ouyang, S., Lin, J., Potter, O.E., 1995. Circulating fluidized bed as a catalytic reactor: experimental study. A.I.Ch.E. Journal 41 (6), 1534–1542. Pagliolico, S., Tiprigan, M., Rovero, G., Gianetto, A., 1992. Pseudo-homogeneous approach to CFB reactor design. Chemical Engineering Science 47 (9–11), 2269–2274. Samuelsberg, A., Hjertager, B.H., 1995. Simulation of two-phase gas/particle flow and ozone decompositions in a 0.25 m I.D. riser. In: Serizawa, A., Fukano, T., Batalille, J. (Eds.), Advances in Multiphase Flow. Elsevier Science B.V., Amsterdam, p. 679. Schoenfelder, H., Kyuse, M., Werther, J., 1996. Two-dimensional model for circulating fluidized bed reactors. A.I.Ch.E. Journal 42, 1875–1888. Sinclair, J.L., Jackson, R., 1989. Gas–particle flow in a vertical pipe with particle–particle interaction. A.I.Ch.E. Journal 35, 1473–1486. Solbrig, C.W., Gidaspow, D., 1967. Convective diffusion in a parallel plate duct with one catalytic wall-laminar flow-first order reaction. Part I—analytical. The Canadian Journal of Chemical Engineering 45, 35–39. Syamlal, M., O'Brien, T.J., 2003. Fluid dynamic simulation of O3 decomposition in a bubbling fluidized bed. A.I.Ch.E. Journal 49 (11), 2793–2801. Therdthianwong, A., Pantaraks, P., Therdthianwong, S., 2003. Modeling and simulation of circulating fluidized bed reactor with catalytic ozone decomposition reaction. Powder Technology 133, 1–14. Turton, R., Levenspiel, O., 1989. An experiment investigation of gas–particle heat transfer coefficients in fluidized beds of fine particles. In: Grace, J.R., Schemildt, L.W., Bergougnou, M.A. (Eds.), Fluidization, vol. VI. Engineering Foundation, Alberta, p. 669. Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G., 2001. Fundamentals of Momentum, Heat and Mass Transfer. Wiley, New York. Yang, N., Wang, W., Ge, W., Li, J., 2003. CFD simulation of concurrent-up gas–solid flow in circulating fluidized beds with structure-dependent drag coefficient. Chemical Engineering Journal 96, 71–80. Yang, N., Wang, W., Ge, W., Wang, L., Li, J., 2004. Simulation of heterogeneous structure in a circulating fluidized-bed riser by combining the two-fluid model with the EMMS approach. Industrial and Engineering Chemistry Research 43, 5548–5561. Yoon, H., Wei, J., Denn, M.M., 1978. A model for moving-bed coal gasification reactors. A.I.Ch.E. Journal 24 (5), 885–903. Zabrodsky, S.S., 1966. Hydrodynamics and Heat Transfer in Fluidized Beds. MIT Press, Cambridge. Zevenhoven, R., J¨arvinen, M., 2002. CFB combustion, particle slip velocity and particle/turbulence interactions. Presented at the eighth International Conference on Circulating Fluidized Beds, Ontario, Canada.
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Kinetic theory based computation of PSRI riser: Part II—Computation of mass transfer coefficient with chemical reaction Benjapon Chalermsinsuwan a,b , Pornpote Piumsomboon a,b , Dimitri Gidaspow c,∗ a b c
Fuels Research Center, Department of Chemical Technology, Faculty of Science, Chulalongkorn University, Phayathai Road, Patumwan, Bangkok 10330, Thailand Center of Excellence for Petroleum, Petrochemicals, and Advanced Materials, Chulalongkorn University, Bangkok 10330, Thailand Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA
A R T I C L E
I N F O
Article history: Received 18 February 2008 Received in revised form 2 November 2008 Accepted 7 November 2008 Available online 19 November 2008 Keywords: CFD Clusters Fluidization Mass transfer Sherwood number Ozone decomposition
A B S T R A C T
The design of circulating fluidized bed systems requires the knowledge of mass transfer coefficients or Sherwood numbers. A literature review shows that these parameters in fluidized beds differ up to seven orders of magnitude. To understand the phenomena, a kinetic theory based computation was used to simulate the PSRI challenge problem I data for flow of FCC particles in a riser, with an addition of an ozone decomposition reaction. The mass transfer coefficients and the Sherwood numbers were computed using the concept of additive resistances. The Sherwood number is of the order of 4×10−3 and the mass transfer coefficient is of the order of 2×10−3 m/s, in agreement with the measured data for fluidization of small particles and the estimated values from the particle cluster diameter in part one of this paper. The Sherwood number is high near the inlet section, then decreases to a constant value with the height of the riser. The Sherwood number also varies slightly with the reaction rate constant. The conventionally computed Sherwood number measures the radial distribution of concentration caused by the fluidized bed hydrodynamics, not the diffusional resistance between the bulk and the particle surface concentration. Hence, the extremely low literature Sherwood numbers for fluidization of fine particles do not necessarily imply very poor mass transfer. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The design of circulating fluidized bed systems, such as gasifiers (Yoon et al., 1978; Gidaspow and Jiradilok, 2007), requires the knowledge of mass transfer coefficients or Sherwood numbers. Transport phenomena texts (e.g., Bird et al., 2002) show that the Sherwood number or the Nusselt number for related heat transfer is two, based on the diameter of particle. Indeed for large particles, Gunn (1978) has shown that the Sherwood number equals two, the diffusion limit, plus a contribution due to convection expressed in terms of Reynolds and Schmidt numbers. The mass transfer and heat transfer coefficients are known to be much lower for fine particles than those given by correlations for large particles (Zabrodsky, 1966; Turton and Levenspiel, 1989; Kunii and Levenspiel, 1991). Recently, Breault (2006) has reviewed the literature and has shown that the Sherwood numbers differ up to seven orders of magnitude ranging from a low value of 10−5 , as reported by Bolland and Nicolai (2001). One
possible explanation for this phenomenon is the formation of particle clusters. The second more plausible explanation is that the conventional method of computing mass transfer coefficients in fluidized beds shows the effect of radial distribution of concentration and has little to do with diffusion to an individual particle. Hence, the representation in terms of the Sherwood number is misleading. In the first part of the paper, the conventional methodology for estimation of the Sherwood numbers is described. In this second part, a different methodology for computation of the very low Sherwood numbers and mass transfer coefficients, with the chemical reaction is proposed. The computed Sherwood numbers agree with very small values reported in the literature. The variation of the Sherwood numbers with reactor length is similar to those found from the Leveque solution of laminar convection problems and experimental observations. The Sherwood numbers are high in the entrance region and reach a constant value, like in the developed flow mass transfer. 2. Ozone decomposition reaction
∗ Corresponding author. Tel.: +1 312 567 3045; fax: +1 312 567 8874. E-mail address: [email protected] (D. Gidaspow). 0009-2509/$ - see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2008.11.006
The ozone decomposition or dissociation reaction is one of the frequently used chemical reactions for studying reactions in the circulating fluidized bed systems (Fryer and Potter, 1976; Samuelsberg
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and Hjertager, 1995). The ozone reactant is converted to oxygen product. The reaction can be written as
compute the low mass transfer coefficients and Sherwood numbers observed experimentally in fluidized beds of fine particles.
2O3(g) → 3O2(g)
3. Computational fluid dynamics simulation
This decomposition reaction is a simple irreversible first-order catalytic reaction (Hansen et al., 2004). The rate of reaction (rO3 ) for ozone species is shown below:
3.1. Mathematical model
Ozone decompostion reaction rate = rO3 = −kreaction CO3 s where kreaction is the reaction rate constant, CO3 is the ozone molar concentration and s is the solids volume fraction or particle concentration of catalyst used in the reaction. The first-order reaction permits the total resistance to be expressed as the sum of reaction and diffusional resistances. This reaction requires only low concentrations of ozone reactant. Due to the low concentration of the reactant, the heat produced by the chemical reaction is negligible. The ozone decomposition reaction takes place at low temperature, ambient, and is therefore easy to carry out experimentally. The ozone concentration detection is rapid and accurate using fairly simple detection methods (Syamlal and O'Brien, 2003). Hence, the ozone decomposition reaction is selected as the base reaction for this study. 2.1. Previous experimental studies on ozone decomposition reaction There have been several attempts to analyze the experimental results from ozone decomposition in circulating fluidized bed systems. Jiang et al. (1991) investigated the baffle effects on the performance of ozone decomposition with FCC particles. It was found that, the riser with baffles gives a higher solid holdup and ozone conversion in the gas phase than that without baffles. This was because baffles could enhance the gas–solid contact efficiency by promoting radial gas and solid mixing. Ouyang et al. (1993, 1995) measured the ozone concentration profiles in a riser of a circulating fluidized bed with impregnated FCC as the catalyst. The axial ozone concentration profiles showed a significant deviation from plug flow profiles. The radial profiles showed that the ozone concentration was lower in the wall region than in the core region, consistent with the observations of near wall region flow of solid particles. Bolland and Nicolai (2001) studied the mass transfer between gas and solids in the riser section. The highest rate of reaction was found to be in the lower section, and the smallest in the upper zone of the riser. This phenomenon can be explained by means of mass transfer control in the upper zone. Zevenhoven and Ja¨ rvinen (2002) computed the slip velocity between the solid particles and the surrounding gas. They obtained improved gas–solid mass and heat transfer when the slip velocity was high. 2.2. Previous computational studies on ozone decomposition reaction A two-dimensional model for ozone decomposition in a circulating fluidized bed reactor was studied by Schoenfelder et al. (1996). They developed their model based on the empirical data from the experiment, including the height of bottom dense zone and top dilute zone, as well as axial and radial solids concentration profiles. Therdthianwong et al. (2003) modified a two-dimensional model of Schoenfelder et al. (1996) by using two different models for solid viscosity, a semi-empirical and a kinetic theory of granular flow model. Their model gave reasonable predictions of experimentally measured values. Syamlal and O'Brien (2003) also simulated the bubbling fluidized bed reactor with the ozone decomposition to test the capability of their computational fluid dynamics code, MFIX. The MFIX code quantitatively captured the effect of hydrodynamics on the chemical reaction in the bubbling bed system. This study is the first one to
In this study, a set of governing equations, mass, momentum, energy and gas species mass conservation equations, and constitutive equations are solved numerically. The equations used are based on the kinetic theory of granular flow, as reviewed by Gidaspow (1994). This theory has proved its validity by many researchers (Ding and Gidaspow, 1990; Neri and Gidaspow, 2000; Gidaspow et al., 2004; Jiradilok et al., 2006, 2007). Here the modified Wen and Yu drag law (Yang et al., 2003, 2004) has been replaced with that obtained from the energy minimization multi-scale (EMMS) approach. The EMMS drag model has proved to be an effective way for modeling a high solid mass flux system of FCC particles. This study uses commercial CFD program FLUENT 6.2.16 for modeling the system. There are several numerical models for gas–solid two-phase flow in the program such as, the Lagrangian model and the Eulerian model. In this case, the Eulerian model was selected. With this approach, governing equations in each phase are solved separately. A summary of the governing equations and constitutive equations is given in Table 1. Since the EMMS drag model is not available in the FLUENT 6.2.16 program, the new additional UDF (user-defined function) code (Fluent Inc., 2005a,b) in C programming language was added. According to the previous literature, there has not been enough information given on the values of operating parameters. Some optimization process of these values has been done to obtain the optimized parameter values, as summarized in Table 2. 3.2. System description and computational domain To validate the numerical results in this study, the PSRI challenge problem I, Knowlton et al. (1995), was chosen as the reference case. The system for Fluidization VII benchmark test is shown in Fig. 1. The validation and model verification were shown in the first part of this paper. In this second part, we use the same system, based on Knowlton et al. (1995) experiment, with the addition of the catalytic chemical reaction. The catalyst particles in the system were FCC particles, which are commonly used in circulating fluidized bed. The diameter (D) and height (h) of their riser were 0.2 and 14.2 m, respectively. Since a three-dimensional model requires long computation time, this study uses a two-dimensional model for the simulation. The schematic drawing of the riser is depicted in Fig. 2(a). This simplified schematic drawing is based on Benyahia et al.'s (2000, 2002) papers. It has a two inlet–outlet design, used to approximate the riser in two dimensions. The gas was fed to the system at the bottom of the riser. The solid particles were fed from the two sideinlets at 0.3 m above the bottom of the riser with a width of 0.1 m. The gas and solid exited through two symmetric side outlets at 0.3 m below the top of the riser with the same width as solid inlet ports. The other conditions for this simulation are listed in Table 2. The computational domain of the riser used in this study is illustrated in Fig. 2(b). It consists of 19 non-uniform grids in radial or horizontal direction and 285 uniform grids in axial or vertical direction, with a total of 5415 computational cells. The grid independence study is described in Appendix A. The models were solved by using a computer with Pentium 1.80 GHz CPU 2 GB RAM. It took approximately 10 days of computer time to obtain 40 s of simulation time. 3.3. Initial and boundary conditions At the inlet, the velocity, volume fraction, temperature and gas species composition were specified, as summarized in Table 2. On
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Table 1 A summary of the governing equations and constitutive equations.
(f) Solid phase shear viscosity
A. Governing equations (a) Conservation of mass (i) Gas phase
s =
j ( ) + ∇ · (g g vg ) = 0 jt g g
s =
j ( ) + ∇ · (s s vs ) = 0 jt s s
(2)
j ( v ) + ∇ · (g g vg vg ) = −g ∇P + ∇ · g + g g g − gs (vg − vs ) jt g g g
(3)
j ( v ) + ∇ · (s s vs vs ) = −s ∇P + ∇ · s − ∇Ps + s s g + gs (vg − vs ) jt s s s
(c) Conservation of energy (i) Gas phase
√ 2 6 150s dp 1 + s g0 (1 + e) + 2s 2s dp (1 + e)g0 384(1 + e)g0 5
(1 − g ) g
g d2p
+ 1.75
(1 − g )g |vg − vs |
when g 0.74 3 (1 − g )g g |vg − vs |CD0 () 4 dp
(18)
k
(5)
(16)
(17)
dp
24 (1+0.15Re0.687 ), Rek = with Re<1000, CD0 = Re k
j jpg ( h ) + ∇ · (g g vg hg ) = −g + g : ∇ · vg + Sg + Qsg jt g g g jt
(14)
(15)
2
(4)
2
(i) Gas–solid phase interphase exchange coefficient; when g < 0.74
gs =
4 s s dp g0 (1 + e) 3
gs = 150
(ii) Solid phase
(h) Conductivity of the fluctuating energy
s =
(b) Conservation of momentum (i) Gas phase
√
4 10s dp 1 + g0 s (1 + e) + 96(1 + e)g0 s 5
(g) Solid phase bulk viscosity
(1)
(ii) Solid phase
with hg =
4 s s dp g0 (1 + e) 5
when 0.74 g 0.82 () = −0.5760 + when 0.82 g 0.97 () = −0.0101 +
g g |vg −vs |dp , Re 1000, CD0 =0.44 g 0.0214
4(g − 0.7463) + 0.0044 0.0038 2
4(g − 0.7789) + 0.0040 when g > 0.97 () = −31.8295 + 32.8295g
cp,g dTg
2
(ii) Solid phase
j jps ( h ) + ∇ · (s s vs hs ) = −s + s : ∇ · vs + Ss + Qgs jt s s s jt with hs =
(6)
cp,s dTs
(d) Conservation of species (i = O3 , O2 and N2 (air))
j ( y ) + ∇ · (g g vg yi ) = ri jt g g i
(7)
(e) Conservation of solid phase fluctuating energy 3 2
j + ∇ · (s s )vs = (−∇ps I + s ) : ∇vs + ∇ · (s ∇ ) − s jt s s
(8)
Table 2 Parameters used for the simulation. Symbol
Description
Value
D h
Diameter of riser Height of riser Gas density Gas viscosity Particle density Diameter of particle Gas inlet velocity Gas inlet temperature Ozone species mass fraction inlet Air species mass fraction inlet Solid inlet velocity Solid inlet temperature Solid inlet volume fraction Restitution coefficient between particles Restitution coefficient between particle and wall Specularity coefficient Reaction rate constant
0.20 m 14.20 m 1.2 kg/m3 2×10−5 kg/m s 1712 kg/m3 76 m 5.200 m/s 298.15 K 0.00004 0.99996 0.476 m/s 298.15 K 0.60 0.95
g g s dp vg Tg yO3 yAir vs Ts
s
B. Constitutive equations (a) Gas phase stress
e
g = g g [∇vg + (∇vg )T ] − 23 g g (∇ · vg )I
(9)
eW
(b) Solid phase stress
kreaction
s = s s [∇vs + (∇vs )T ] − s ( s − 23 s )∇ · vs I
⎛
4 dp
⎞
⎠
(11)
(d) Radial distribution function g0 = 1 −
s
1/3 −1
s,max
(12)
the other hand, at the outlet, the system pressure was specified as atmospheric pressure. Initially, there were no gas and solid phases in the riser. At the wall, a no-slip condition was applied for all velocities, except for the tangential velocity of the solid phase and the granular temperature. Here, the boundary conditions of Johnson and Jackson (1987) were used. These conditions were first applied in the kinetic theory of granular flow modelling by Sinclair and Jackson (1989). They are vst,W = −
(e) Solid phase pressure ps = s s [1 + 2g0 s (1 + e)]
0.50 3.96, 39.60, 99.00 and 198.00 s−1
(10)
(c) Collisional dissipation of solid fluctuating energy
s = 3(1 − e2 )2s s g0 ⎝
0.90
(13)
6 s s,max jvst,W √ s s g0 3 jn
jW W = − s + W jn
√
(19) 3/2
3 s s v2s,slip g0 6s,max W
(20)
B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
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3.4. Chemical reaction conditions The information needed for chemical reaction is the reaction rate constant. From the previous experimental data, the values of reaction rate constant for ozone decomposition reaction are varied by three orders of magnitude with the activity of catalyst particle. This decomposition reaction can be catalyzed with many types of catalyst compositions, such as FCC catalyst coated with ferric nitrate (Ouyang et al., 1993, 1995), FCC catalyst coated with ferric oxide (Jiang et al., 1991) and -alumina catalyst coated with ferric oxide (Pagliolico et al., 1992). In this study, four different values of reaction rate constants are used to study the effect of this parameter on the Sherwood number and mass transfer coefficient. The range of these values is selected from the literature experiments. The chosen reaction rate constants are summarized in Table 2. 4. Results and discussion 4.1. Calculation of Sherwood number and mass transfer coefficient From the conservation of species equation:
j ( y ) + ∇ · (g g vg yi ) = ri jt g g i
Fig. 1. Schematic drawing of a 20 cm diameter circulating fluidized bed test unit for the Fluidization VII benchmark test.
(7)
where t is the time, g is the volume fraction of gas phase, g is the density of gas phase, vg is the velocity of gas phase, yi is the mass fraction of specie “i” and ri is the reaction rate. Integration of Eq. (7) over time and over the riser radius (x- and zdirections) for ozone species gives the one-dimensional steady state balance: vy g
dCO3 = ri dY
(21)
where vy is the velocity of gas phase in axial or vertical direction and Y is the axial or vertical distance. The decomposition of ozone is a first-order reaction. Hence, the reaction rate constant is independent of the gas concentration. Substitution of the rate of reaction in Eq. (21) gives vy g
dCO3 = −KCO3 s dY
(22)
where K is the overall resistance. Solving Eq. (22) gives ln CO3 = ln CO3 ,0 −
Fig. 2. (a) Schematic drawing and (b) computational domain with their boundary conditions of a simplified riser used in this study.
where
W =
√ 3/2 3 (1 − e2W )s s g0 4s,max
K s Y vy g
(23)
where the subscript “0” is the initial molar concentration of ozone. In fluidized bed reactors, one measures the ozone concentration and obtains the overall resistance from Eqs. (22) and (23) or the finite difference form of Eq. (22). In this simulation, Eq. (23), the time averaged, area averaged result, gives the overall resistance. The linear plot of natural logarithm of ozone molar concentration versus the height of the riser gives the slope, which further gives the overall resistance. Then, the conventional additive resistance concept permits us to compute the mass transfer coefficient. At steady state, the external mass transfer in terms of global rate is equated to the mass transfer from the bulk gas to catalyst surface (Fogler, 1999; Levenspiel, 1999; Bolland and Nicolai, 2001; Welty et al., 2001): kmass transfer av (CO3 − CO3,surface ) = kreaction CO3,surface
(24)
where kmass transfer is the mass transfer coefficient, av is the external surface per volume of catalyst and CO3 ,surface is the surface molar concentration of ozone.
B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
1.20E-06
0 kReaction = 3.96 1/s kReaction = 39.60 1/s kReaction = 99.00 1/s kReaction = 198.00 1/s
1.00E-06
8.00E-07
6.00E-07
4.00E-07
2.00E-07
Natural logarithm of time-averaged ozone molar concentration
Time-averaged ozone molar concentration (kmol/m3)
1216
-5 -10 -15 -20 -25 -30 -35 -40
0.00E+00 0
2
4
6 8 10 Riser height (m)
12
0
14
Eliminating CO3 ,surface from Eq. (24) and expressing the global reaction rate per unit mass of catalyst (rp ) in terms of CO3 gives 1 1 kmass transfer av
+
1
CO3 s = KCO3 s
(25)
1 1 1 = + K kmass transfer av kreaction
(26)
No.
kreaction (s−1 )
kmass transfer dp D
(27)
where dp is the diameter of the catalyst particle and D is molecular diffusivity. Bolland and Nicolai (2001) have analyzed their ozone decomposition data in a fluidized bed using this exact method. 4.2. Computation and interpretation of Sherwood number and mass transfer coefficient Fig. 3 shows the variation of the computed time-averaged ozone molar concentration with the riser height for various reaction rate constants used in this study. Due to the reaction, all the ozone concentrations decrease with riser height. For the lowest reaction rate constant or kreaction = 3.96 s−1 case, the ozone reactant leaves the riser partially unreacted. For kreaction = 39.60, 99.00 and 198.00 s−1 cases, the ozone is almost used up due to the high reaction rate constant. For a first-order reaction, with or without mass transfer, and constant catalyst concentration, the natural logarithm of the ozone molar concentration varies linearly with height, as given by Eq. (23). Hence, in Fig. 4, the ozone concentrations are presented to test these assumptions and to enable the determination of the mass transfer coefficient. The natural logarithms of computed time-averaged ozone molar concentration with the riser height for various reaction
6 8 Riser height (m)
10
12
14
Riser height (m)
Slope
Intercept
1
3.96
3.5 7.0 10.5
−0.1680 −0.1375 −0.1236
−13.8290 −13.8290 −13.8290
2
39.60
3.5 7.0 10.5
−0.7584 −0.6630 −0.6319
−13.8290 −13.8290 −13.8290
3
99.00
3.5 7.0 10.5
−1.4370 −1.3074 −1.2216
−13.8290 −13.8290 −13.8290
4
198.00
3.5 7.0 10.5
−2.3236 −2.0776 −1.9759
−13.8290 −13.8290 −13.8290
The Sherwood number (Sh) is then given by Eq. (27) below: Sh =
4
Table 3 The fitted equation parameters with various reaction rate constants used in this study.
kreaction
The result from Eq. (25) and the overall resistance from Eq. (23) give the mass transfer coefficient as follows:
2
Fig. 4. Test of model for computation of mass transfer coefficients.
Fig. 3. Computed time-averaged ozone molar concentration with various reaction rate constants.
rp s s =
kReaction = 3.96 1/s kReaction = 39.60 1/s kReaction = 99.00 1/s kReaction = 198.00 1/s
Model equation: ln CO3 = Slope × Y + Intercept.
rate constants are displayed in Fig. 4. All the graph lines are approximately linear, which is consistent with the assumptions used in the ozone species mass balance. The fitted equation parameters, which are the slope and the intercept, for each graph are summarized in Table 3. These fitted slopes are consistent with the profiles of computed ozone molar concentration in Fig. 3. For the highest reaction rate constant or kreaction = 198.00 s−1 case, the highest slope value is obtained. The slopes decrease with decreasing reaction rate constants. The lowest slope is obtained for the lowest reaction rate constant or kreaction = 3.96 s−1 case. From the observation that the Sherwood numbers or mass transfer coefficients are not constant along the entire riser section, the slope values are separated into three regions for each graph. These three heights are at 3.5, 7.0 and 10.5 m, that is the middle section of the riser. From the first part of this paper, these three heights have the same flow structure. However, all the slopes with the same reaction rate constant are not significantly different. These results are also consistent with the assumption used in this study.
B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
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Table 4 Computed information on the Sherwood numbers and mass transfer coefficients at three different heights of the riser. kreaction (s−1 )
No.
Riser height (m)
kmass transfer av (s−1 )
kmass transfer (m/s)
Sherwood number, dimensionless
1
3.96
3.5 7.0 10.5 Averaged
– – – Reaction controlled
– – – Reaction controlled
– – – Reaction controlled
2
39.60
3.5 7.0 10.5 Averaged
138.71 86.35 78.90 101.32
0.0018 0.0011 0.0010 0.0013
0.0046 0.0029 0.0026 0.0034
3
99.00
3.5 7.0 10.5 Averaged
132.18 110.07 109.51 117.25
0.0017 0.0014 0.0014 0.0015
0.0044 0.0037 0.0037 0.0039
4
198.00
3.5 7.0 10.5 Averaged
182.33 136.23 132.60 150.38
0.0023 0.0017 0.0017 0.0019
0.0061 0.0046 0.0044 0.0050
Sh =
kmass transfer dp D
=
6kmass transfer av D
(28)
The equation shows that for large av , small particles, the mass transfer coefficient must be extremely high to give a Sherwood number of two. The conventional method of computation presented here captures the small differences between the bulk and the surface concentration that are due to the hydrodynamics of the fluidized bed, not due to the diffusion between the bulk and the surface concentration. The mass transfer is not as poor as is implied by the extremely small values of the Sherwood numbers, although the particle cluster formation does decrease the mass transfer. Fig. 5 illustrates the effect of reaction rate constants on the computed Sherwood numbers and mass transfer coefficients for three different heights of the riser. The computed Sherwood numbers and mass transfer coefficients have a similar trend because the diameter of the catalyst particle and molecular diffusivity used in this study are constant. At the height of 3.5 m or at the lower section, the Sherwood number and mass transfer coefficient have a minimum value. For the low reaction rate constant or kreaction = 39.60 s−1 case, the ozone reactant is slowly converted. This makes the reaction resistance dominant for this system and results in the high values of Sherwood numbers and mass transfer coefficients. For the high reaction rate constant or kreaction = 198.00 s−1 case, the rate of reaction is also low due to the rapid conversion of ozone reactant. This gives high values of Sherwood numbers and mass transfer coefficients. Therefore, there exists an optimum condition at the suitable moderately
3.00E-03
8.00E-03
Sherwood number (-)
7.00E-03
2.50E-03
6.00E-03 2.00E-03
5.00E-03
1.50E-03
4.00E-03 3.00E-03
1.00E-03
2.00E-03
h = 3.5 m h = 7.0 m h = 10.5 m
1.00E-03 0.00E+00 0.00
kMass transfer (m/s)
Table 4 summarizes the computed information on the Sherwood numbers and mass transfer coefficients at three different heights of the riser for various reaction rate constants. For the lowest reaction rate constant or kreaction = 3.96 s−1 case, the reaction rate constant is too low and, thus, controls the system. Therefore, the mass transfer coefficient cannot be computed using the present method of additive resistances for mass transfer and reaction. In this case, the diffusional resistances are zero. Once the reaction rate constants become high or kreaction = 39.60, 99.00 and 198.00 s−1 cases, the diffusional resistances become large and can be computed using the concept of additive resistances. Appendix B shows a typical numerical example. Since the overall resistance (K) is close to reaction rate constant (kreaction ), the mass transfer is not bad. The reason for the low Sherwood number is its representation in terms of the particle diameter or surface area per unit volume, av . Eq. (27) rewritten as
5.00E-04
0.00E+00 50.00 100.00 150.00 200.00 250.00 kReaction (s-1)
Fig. 5. Effect of reaction rate constants on computed Sherwood numbers and mass transfer coefficients.
reaction rate constant, at kreaction = 99.00 s−1 in this simulation. At the height of 7.0 and 10.5 m or at the higher section, the ozone reactant is mostly converted. The conversion is higher, directly proportional to the reaction rate constant. Hence, the Sherwood numbers and mass transfer coefficients increase when the reaction rate constant increases. Fig. 6 shows the effect of reaction rate constants on the computed riser height-averaged Sherwood numbers and mass transfer coefficients. The average values are calculated at three different heights. The computed values of Sherwood numbers and mass transfer coefficients increase with the increasing reaction rate constant. The explanation of this phenomenon is similar to the reason for the case at the higher section or at the height of 7.0 and 10.5 m in the riser as shown in Fig. 5. Thus, we can state that the mass transfer resistance decreases when the reaction rate constants increases. These study results confirm the literature results that there is a dependence of the mass transfer coefficient or the Sherwood number on the reaction rate constant (Solbrig and Gidaspow, 1967; Kulacki and Gidaspow, 1967; Bird et al., 2002). The effect of riser height on the computed Sherwood numbers and mass transfer coefficients for various reaction rate constants is displayed in Fig. 7. The computed Sherwood numbers and mass transfer
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B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
3.00E-03
8.00E-03
2.50E-03
6.00E-03 2.00E-03
5.00E-03
1.50E-03
4.00E-03 3.00E-03
1.00E-03
2.00E-03 5.00E-04
1.00E-03 0.00E+00 0.00
kMass transfer (m/s)
Sherwood number (-)
7.00E-03
0.00E+00 50.00 100.00 150.00 200.00 250.00 kReaction (s-1)
3.00E-03
8.00E-03
2.50E-03
6.00E-03 2.00E-03
5.00E-03
1.50E-03
4.00E-03 3.00E-03 2.00E-03 1.00E-03 0.00E+00 0.0
1.00E-03 kReaction = 39.60 1/s kReaction = 99.00 1/s kReaction = 198.00 1/s Kato et al., 1970
2.0
4.0 6.0 8.0 Riser height (m)
kMass transfer (m/s)
Sherwood number (-)
7.00E-03
5.00E-04 0.00E+00 10.0 12.0
Fig. 7. Effect of riser height on computed Sherwood numbers and mass transfer coefficients.
coefficients decrease with the riser height and reach constant values at the top section for all cases. This variation of the Sherwood numbers is similar to the typical behavior in convective mass transport. The computed trend from experimental data correlation (Kato et al., 1970) is also plotted in the figure. A reasonable agreement is obtained between this study and Kato's correlation results. As can be seen from Fig. 3, the highest and lowest conversions of the ozone are obtained at the bottom and the top sections of the riser, respectively. This confirms the low mass transfer resistance and results in the high values of Sherwood numbers and mass transfer coefficients near the inlet region. At the upper section, the mass transfer resistance has a large effect on the conversion of ozone reactant. Therefore, the values of Sherwood numbers and mass transfer coefficients decrease. The effect of Reynolds number on experimental and computed Sherwood numbers is displayed in Fig. 8. In this figure, the comparison of the computed Sherwood number from the part one of this paper is also illustrated. The computed Sherwood numbers obtained in this paper and those from the part I of the paper are lower than the experimental Sherwood numbers for large particles in fluidized and fixed bed systems. The values of Sherwood numbers and mass transfer coefficients are in reasonable agreement with the phenomena from the experimental data (Kunii and Levenspiel, 1991) and fall in the range as reviewed by Breault (2006). The computed Sherwood number and mass transfer coefficient from chemical reaction are of the same order of magnitude as the estimated values from the particle cluster diameter.
Fig. 8. Effect of Reynolds number on experimental and computed Sherwood numbers.
Time-averaged ozone molar concentration (kmol/m3)
Fig. 6. Effect of reaction rate constants on computed riser height-averaged Sherwood numbers and mass transfer coefficients.
1.40E-07 h = 3.5 m h = 7.0 m h = 10.5 m
1.20E-07 1.00E-07 8.00E-08 6.00E-08 4.00E-08 2.00E-08 0.00E+00 -0.10
-0.05
0.00 0.05 Radial distance (m)
0.10
Fig. 9. Radial distribution of computed time-averaged ozone molar concentration at three different heights with kreaction = 39.60 s−1 .
4.3. Confirmation of reduced Sherwood number and mass transfer coefficient The computed time-averaged ozone molar concentration versus radial distance at three different heights, with kreaction = 39.60 s−1 case, is illustrated in Fig. 9. The results obtained only for the case with kreaction = 39.60 s−1 are shown because similar trends were observed with other reaction rate constants. The results are similar to those shown by previous researchers (Ouyang et al., 1993, 1995; Bolland and Nicolai, 2001). There is a very little ozone concentration due to the high catalyst concentration near the wall. The highest ozone concentration is at the center region. The computed time-averaged and natural logarithm of computed time-averaged bulk and surface ozone molar concentrations with kreaction = 39.60 s−1 are displayed in Figs. 10(a) and (b), respectively. The surface ozone molar concentrations are calculated using Eq. (24). From the figures, the surface ozone molar concentration is slightly lower than the bulk ozone molar concentration due to the diluteness of the center ozone molar concentration at the catalyst or particle clusters. Fig. 11 shows plots of the computed instantaneous solids volume fraction and ozone molar concentrations with kreaction = 39.60 s−1 at
Time-averaged ozone molar concentration (kmol/m3)
B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
1219
1.00E-06 Concentration (Bulk)
9.00E-07
Concentration (Surface)
8.00E-07 7.00E-07 6.00E-07 5.00E-07 4.00E-07 3.00E-07 2.00E-07 1.00E-07 0.00E+00
Natural logarithm of time-averaged ozone molar concentration
0
2
4
6 8 Riser height (m)
10
12
14
Fig. 11. Variation of instantaneous solid volume fraction and ozone molar concentration profiles with kreaction = 39.60 s−1 in the observed particle cluster.
-13 Concentration (Bulk) Concentration (Surface)
-15
Particle cluster
-17
-19
-21
-23
-25 0
2
4
6 8 10 Riser height (m)
12
14
Fig. 10. (a) Computed time-averaged and (b) natural logarithm of computed time-averaged of bulk and surface ozone molar concentrations with kreaction = 39.60 s−1 .
the observed particle cluster position in Fig. 12. The ozone concentration is not symmetric within the particle cluster due to the unequal surrounding reactant concentrations, as shown in Fig. 9. The ozone concentration decreases when the solid volume fraction increases or when it passes through the catalyst. The Sherwood number or the mass transfer coefficient, hence, measures the effect of the radial distribution of the ozone molar concentration and reflects this radial mal-distribution. The formation of particle cluster increases the mass transfer resistance in the system. This makes the Sherwood number and the mass transfer coefficient much lower than the diffusional limit of Sherwood number, which is equal to 2. Also, the reason for low Sherwood number for fine particles can be explained using arithmetic. From Eq. (27), a fine particle with high surface to volume ratio will have the Sherwood number lower than a large particle with low surface to volume ratio. To verify the explanation, the conventional and new computed Sherwood numbers with kreaction = 39.60 s−1 is displayed in Fig. 13. For the new computed Sherwood number, the particle diameter in Eq. (27) is changed to the particle cluster diameter. The results are plotted versus the riser height and compared with the computed trend from experimental data correlation (Kato
6.00e-1 5.70e-1 5.40e-1 5.10e-1 4.80e-1 4.50e-1 4.20e-1 3.90e-1 3.60e-1 3.30e-1 3.00e-1 2.70e-1 2.40e-1 2.10e-1 1.80e-1 1.50e-1 1.20e-1 9.00e-2 6.00e-2 3.00e-2 0.00e+00
Fig. 12. Contour of the computed instantaneous solid volume fractions in the riser at 30 s.
et al., 1970). The computed Sherwood numbers based on particle diameter are significantly lower than the computed Sherwood numbers based on particle cluster diameter. These results also infer that the convection to diffusion ratio of fine particles is lower than that of large particles. The reduction in the Sherwood number or the mass transfer coefficient has thus been confirmed. It can be explained by the particle cluster formation, as already proposed in the first part of the paper.
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Sherwood number (-)
10.000
h h I kmass transfer kreaction
This study - Based on particle diameter This study - Based on particle cluster diameter Kato et al., 1970
1.000
ShCluster = Sh ×
0.100
K n p Ps Qgs
dc dp
dc = 0.87 × 10−2 −1.01 × 10−2 m dp = 7.6 ×10−5 m
0.010
ShCluster = Sh × 133 (at the height of 3.5 m)
Qsg
0.001 0.0
2.0
4.0 6.0 8.0 Riser height (m)
10.0
12.0
Fig. 13. Comparison of the computed Sherwood numbers based on particle diameter (Sh) and particle cluster diameter (ShCluster ) with kreaction = 39.60 s−1 . For Sh = 0.0046, ShCluster = 0.62 at a height of 3.5 m.
r ri Rek S Sh Shcluster
5. Conclusions 1. From additive diffusional and chemical resistance concept, the computed Sherwood number is of the order of 4×10−3 and the mass transfer coefficient is of the order of 2×10−3 m/s, in agreement with measured literature experimental data for fluidization of small particles. 2. The Sherwood number or mass transfer coefficient is high near the inlet section, and decreases to a constant value with the height of the riser. This is similar to the normal behavior of the Sherwood number in convective mass transfer process (Kato et al., 1970). 3. The Sherwood number or mass transfer coefficient varies slightly with reaction rate constant. For the higher reaction rate constant case, the rate of reaction is lower due to the rapid conversion of ozone reactant. Therefore, the computed values of Sherwood numbers and mass transfer coefficients increase with the increasing of reaction rate constant. 4. In this study, two explanations are possible for the low Sherwood numbers measured in the fluidization of fine particles. One explanation is due to the particle cluster formation studied in part one of this study. The second more plausible explanation is that the conventional method of computing the mass transfer coefficients does not measure the diffusional resistance to the particle implied by the conventional Sherwood number representation. It shows the effect of radial distribution of concentration. Notation av cp C CD0 Csurface dc dp D D e eW g g0
external surface per volume of catalyst, m−1 heat capacity at constant pressure, J/kg K molar concentration, kg mol/m3 drag coefficient, dimensionless surface molar concentration, kg mol/m3 particle cluster diameter, m particle diameter, m diameter of riser, m molecular diffusivity, m2 /s restitution coefficient between particles, dimensionless restitution coefficient between particle and wall, dimensionless gravity force, kg/m3 radial distribution function, dimensionless
t T v vs, slip vst, W x y yi Y
height of riser, m specific enthalpy, J/kg unit tensor, dimensionless resistance due to mass transfer, m/s resistance due to reaction or reaction rate constant, s−1 overall resistance, m/s unit vector, dimensionless pressure, Pa solid pressure, Pa intensity of heat exchange from gas phase to solid phase, J/s m3 intensity of heat exchange from solid phase to gas phase, J/s m3 reaction rate, kg mol/s m3 reaction rate of specie “i”, kg mol/s m3 Reynolds number, dimensionless source term (e.g., due to chemical reaction), J/s m3 Sherwood number based on particle diameter, dimensionless Sherwood number based on particle cluster diameter, dimensionless time, s temperature, K velocity, m/s slip velocity of particle at the wall, m/s tangential velocity of particle at the wall, m/s radial distance from center of riser, m axial distance from bottom of riser, m mass fraction of specie “i”, dimensionless axial or vertical distance, m
Greek letters
gs s W s s,max W s
gas–particle interphase drag coefficient, kg/s m3 collisional dissipation of solid fluctuating energy, kg/m s3 collisional dissipation of solid fluctuating energy at the wall, kg/m s3 volume fraction, dimensionless solid volume fraction of particle, dimensionless solid volume fraction at maximum packing, dimensionless granular temperature, m2 /s2 granular temperature at the wall, m2 /s2 conductivity of the fluctuating energy, kg/m s viscosity, kg/m s bulk viscosity, kg/m s density, kg/m3 stress tensor, Pa specularity coefficient, dimensionless
Subscripts g O3 s y 0
gas phase ozone specie solid phase axial direction initial condition
Acknowledgments The authors gratefully acknowledge financial support for this study by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant no. PHD/0021/2550). This study also was
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Table 5 The computed Sherwood numbers and mass transfer coefficients due to the effect of grid number. No.
kreaction (s−1 )
Grid number (radial direction×axial direction)
kmass transfer av (s−1 )
kmass transfer (m/s)
Sherwood number, dimensionless
– – –
– – –
1
3.96
19×285 29×428 38×570
– – –
2
39.60
19×285 29×428 38×570
101.32 99.02 96.74
0.0013 0.0013 0.0012
0.0034 0.0033 0.0032
3
99.00
19×285 29×428 38×570
117.25 115.30 116.92
0.0015 0.0015 0.0015
0.0039 0.0039 0.0039
4
198.00
19×285 29×428 38×570
150.38 153.23 151.77
0.0019 0.0019 0.0019
0.0050 0.0051 0.0051
partially supported by the Chulalongkorn University Graduate Scholarship to Commemorate the 72nd Anniversary of His Majesty King Bhumibol Adulyadej and the Centre of Excellence for Petroleum, Petrochemicals and Advanced Materials, as well as by US DOE university grant, DE-FG26-06NT42736.
The mass transfer coefficient is calculated from Eq. (26): 1 1 1 = + K kmass transfer av kreaction with dp = 76 × 10−6 m
Appendix A. The grid independence study To confirm that the results are grid independent of grid size, the increments of grid numbers in both the radial and axial directions were performed. The comparison of the Sherwood numbers and the mass transfer coefficients with the change in grid number is summarized in Table 5. Four different values of reaction rate constants were selected for study. For each reaction rate constant, the computed values were approximately same with all the grid number cases. The minimum and maximum mass transfer coefficient error percentages compared with the lowest grid number case were 0.29% and 4.52%, respectively. This indicates that all the grid number cases were sufficiently fine for providing reasonably good grid independence results. However, the computational power currently available is still a significant restriction for using large number of grids. Therefore, the computational domain which consists of 19 non-uniform grids in radial direction and 285 uniform grids in axial direction is chosen in this study. Appendix B. An example of Sherwood number and mass transfer coefficient calculation, explaining our argument for good mass transfer in fluidized beds An example of Sherwood number and mass transfer coefficient calculation is illustrated for the reaction rate constant equal to 39.60 s−1 case at a height of 3.5 m. The computed time-averaged and natural logarithm of computed time-averaged ozone molar concentration are displayed in Figs. 3 and 4, respectively. The obtained slope and the intercept are reported in Table 3. The slope for this case was −0.7584 and the intercept was −13.8290. From the simulation, the computed volume fraction of the solid phase was 0.1289, the computed volume fraction of the gas phase was 0.8711 and the velocity of gas phase in axial direction was 6.0115 m/s. Substitution of all the values into Eq. (23) to obtain the overall resistance gives −
K(0.1289) = −0.7584 (6.0115)(0.8711)
Overall resistance, K = 30.81 s−1 .
(26)
av = (3 × 4 (particle radius2 ))/(4 (particle radius3 )) = 3/particle radius = 3/(dp /2) = 3/((76 × 10−6 )/2) = 78947.37 m−1 kreaction = 39.60 s−1 Note that the overall resistance, 1/K, and the reaction resistance, 1/kreaction , are close to each other. This implies that the mass transfer resistance is small. Therefore, kmass transfer av = 138.71 s−1
and
kmass transfer = 0.0018 m/s The Sherwood number is calculated from Eq. (27): Sh =
kmass transfer dp D
(27)
with D = 2.88 × 10−5 m2 /s Therefore, Sh = 0.0046 Fig. 13 shows that the Sherwood number based on the particle cluster diameter, calculated to be 0.0087 to 0.0101 m in the first part of the paper, is 0.62, close to a typical value for diffusion to a particle. References Benyahia, S., Arastoopour, H., Knowlton, T.M., Massah, H., 2000. Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase. Powder Technology 112, 24–33. Benyahia, S., Arastoopour, H., Knowlton, T.M., 2002. Two dimensional transient numerical simulation of solids and gas flow in the riser section of a circulating fluidized bed. Chemical Engineering Communications 189 (4), 510–527. Bird, R.B., Stewart, W.E., Lightfoot, E.N., 2002. Transport Phenomena. Wiley, New York.
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B. Chalermsinsuwan et al. / Chemical Engineering Science 64 (2009) 1212 -- 1222
Bolland, O., Nicolai, R., 2001. Describing mass transfer in circulating fluidized beds by ozone decomposition. Chemical Engineering Communications 187, 1–21. Breault, R.W., 2006. A review of gas–solid dispersion and mass transfer coefficient correlations in circulating fluidized beds. Powder Technology 163, 9–17. Ding, J., Gidaspow, D., 1990. A bubbling fluidization model using kinetic theory of granular flow. A.I.Ch.E. Journal 36, 523–538. Fluent Inc., 2005a. Fluent 6.2 UDF Manual. Fluent Inc., Lebanon. Fluent Inc., 2005b. Fluent 6.2 User's Guide. Fluent Inc., Lebanon. Fogler, H.S., 1999. Elements of Chemical Reaction Engineering. Prentice-Hall, New Jersey. Fryer, C., Potter, O.E., 1976. Experimental investigation of models for fluidized bed catalytic reactors. A.I.Ch.E. Journal 22, 38–47. Gidaspow, D., 1994. Multiphase Flow and Fluidization: Continuum and Kinetic Theory Description. Academic Press, Boston. Gidaspow, D., Jiradilok, V., 2007. Nanoparticle gasifier fuel cell for sustainable energy future. Journal of Power Sources 166 (2), 400–410. Gidaspow, D., Jung, J., Singh, R.K., 2004. Hydrodynamics of fluidization using kinetic theory: an emerging paradigm: 2002 Flour-Daniel lecture. Powder Technology 148 (2–3), 123–141. Gunn, D.J., 1978. Transfer of heat or mass to particles in fixed and fluidized beds. International Journal of Heat and Mass Transfer 21, 467–476. Hansen, K.G., Solberg, T., Hjertager, B.H., 2004. A three-dimensional simulation of gas/particle flow and ozone decomposition in the riser of a circulating fluidized bed. Chemical Engineering Science 59, 5217–5224. Jiang, P., Bi, H., Jean, R., Fan, L., 1991. Baffle effects on performance of catalytic circulating fluidized bed reactor. A.I.Ch.E. Journal 37, 1392–1400. Jiradilok, V., Gidaspow, D., Damronglerd, S., Koves, W.J., Mostofi, R., 2006. Kinetic theory based CFD simulation of turbulent fluidization of FCC particles in a riser. Chemical Engineering Science 61, 5544–5559. Jiradilok, V., Gidaspow, D., Breault, R.W., 2007. Computation of gas and solid dispersion coefficients in turbulent risers and bubbling beds. Chemical Engineering Science 62, 3397–3409. Johnson, P.C., Jackson, R., 1987. Frictional–collisional constitutive relations for granular materials, with application to plane shearing. Journal of Fluid Mechanics 176, 67–93. Kato, K., Kubota, H., Wen, C.Y., 1970. Mass transfer in fixed and fluidized beds. Chemical Engineering Progress Symposium Series 105 (66), 87–99. Knowlton, T., Geldart, D., Masten, J., King, D., 1995. Comparison of CFB hydrodynamic models. PSRI Challenge Problem Presented at the Eighth International Fluidization Conference, Tour, France. Kulacki, F.A., Gidaspow, D., 1967. Convective diffusion in a parallel plate duct with one catalytic wall-laminar flow-first order reaction. Part II—experimental. The Canadian Journal of Chemical Engineering 45, 72–78. Kunii, D., Levenspiel, O., 1991. Fluidization Engineering. Butterworth–Heinemann, Boston.
Levenspiel, O., 1999. Chemical Reaction Engineering. Wiley, New York. Neri, A., Gidaspow, D., 2000. Riser hydrodynamics: simulation using kinetic theory. A.I.Ch.E. Journal 46, 52–67. Ouyang, S., Lin, J., Potter, O.E., 1993. Ozone decomposition in a 0.254 m diameter circulating fluidized bed reactor. Powder Technology 75, 73–78. Ouyang, S., Lin, J., Potter, O.E., 1995. Circulating fluidized bed as a catalytic reactor: experimental study. A.I.Ch.E. Journal 41 (6), 1534–1542. Pagliolico, S., Tiprigan, M., Rovero, G., Gianetto, A., 1992. Pseudo-homogeneous approach to CFB reactor design. Chemical Engineering Science 47 (9–11), 2269–2274. Samuelsberg, A., Hjertager, B.H., 1995. Simulation of two-phase gas/particle flow and ozone decompositions in a 0.25 m I.D. riser. In: Serizawa, A., Fukano, T., Batalille, J. (Eds.), Advances in Multiphase Flow. Elsevier Science B.V., Amsterdam, p. 679. Schoenfelder, H., Kyuse, M., Werther, J., 1996. Two-dimensional model for circulating fluidized bed reactors. A.I.Ch.E. Journal 42, 1875–1888. Sinclair, J.L., Jackson, R., 1989. Gas–particle flow in a vertical pipe with particle–particle interaction. A.I.Ch.E. Journal 35, 1473–1486. Solbrig, C.W., Gidaspow, D., 1967. Convective diffusion in a parallel plate duct with one catalytic wall-laminar flow-first order reaction. Part I—analytical. The Canadian Journal of Chemical Engineering 45, 35–39. Syamlal, M., O'Brien, T.J., 2003. Fluid dynamic simulation of O3 decomposition in a bubbling fluidized bed. A.I.Ch.E. Journal 49 (11), 2793–2801. Therdthianwong, A., Pantaraks, P., Therdthianwong, S., 2003. Modeling and simulation of circulating fluidized bed reactor with catalytic ozone decomposition reaction. Powder Technology 133, 1–14. Turton, R., Levenspiel, O., 1989. An experiment investigation of gas–particle heat transfer coefficients in fluidized beds of fine particles. In: Grace, J.R., Schemildt, L.W., Bergougnou, M.A. (Eds.), Fluidization, vol. VI. Engineering Foundation, Alberta, p. 669. Welty, J.R., Wicks, C.E., Wilson, R.E., Rorrer, G., 2001. Fundamentals of Momentum, Heat and Mass Transfer. Wiley, New York. Yang, N., Wang, W., Ge, W., Li, J., 2003. CFD simulation of concurrent-up gas–solid flow in circulating fluidized beds with structure-dependent drag coefficient. Chemical Engineering Journal 96, 71–80. Yang, N., Wang, W., Ge, W., Wang, L., Li, J., 2004. Simulation of heterogeneous structure in a circulating fluidized-bed riser by combining the two-fluid model with the EMMS approach. Industrial and Engineering Chemistry Research 43, 5548–5561. Yoon, H., Wei, J., Denn, M.M., 1978. A model for moving-bed coal gasification reactors. A.I.Ch.E. Journal 24 (5), 885–903. Zabrodsky, S.S., 1966. Hydrodynamics and Heat Transfer in Fluidized Beds. MIT Press, Cambridge. Zevenhoven, R., J¨arvinen, M., 2002. CFB combustion, particle slip velocity and particle/turbulence interactions. Presented at the eighth International Conference on Circulating Fluidized Beds, Ontario, Canada.