Hydrodynamic Model for the Gas Flow in Circulating Fluidized Bed

Oct 26, 2002 - The circulating fluidized bed (CFB) riser may be regarded as composed ...... P., Large, J. F., Eds.; Pergamon Press: Toronto, 1988; pp ...
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Ind. Eng. Chem. Res. 2002, 41, 5983-5989

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Hydrodynamic Model for the Gas Flow in Circulating Fluidized Bed Reactors Charlotte Vandewalle,* Jan Baeyens, and Anna Claerbout Department of Chemical Engineering, Catholic University Leuven, de Croylaan 46, 3001 Heverlee (Leuven), Belgium

The circulating fluidized bed (CFB) riser may be regarded as composed of two hydrodynamically distinct regions: in the central part, the “core”, particles, and gas are transported upward in a cocurrent dilute flow with a fairly high voidage. In the wall zone or “annulus”, particles are, on average, transported downward, countercurrently to the gas flow: the voidage in this region is significantly lower than that in the central region. The hydrodynamics of the gas phase in CFBs are of dominant importance for predicting the conversion of CFB chemical reactions. Both the core plug-flow gas and the annulus backmixing determine the average residence time, the residence time distribution, and hence the reaction yield. The scope of this paper is to develop a model to predict the residence time and the residence time distribution of the gas phase. Values of underlying parameters are selected from fitting experimental data and model predictions. The model contains two dominant parameters: the exchange coefficient for the gas phase, kg, and the exchange coefficient for the solids between the annulus and core, KAfC. The model is evaluated for different values of these coefficients so that the predicted response curve for the concentration in the gas phase is in best possible agreement with the measured one. The values of both parameters are given and compared with literature data. To complete the assessment, a sensitivity analysis was performed. Introduction The hydrodynamics of the gas phase in circulating fluidized beds (CFBs) are of dominant importance for predicting the conversion of CFB chemical reactions. Both the core plug-flow gas and the gas backmixing determine the average residence time and the residence time distribution (RTD). If the contact time between the gas phase and the catalyst is too short, the degree of conversion might be too low. If the gas is in contact with the solids for too long a period, unwanted side reactions might occur. The scope of this paper is to develop a model to predict the residence time and the RTD of the gas phase. Values of underlying parameters are selected from fitting experimental data and model predictions. Background The CFB riser may be regarded as composed of two hydrodynamically distinct regions: in the central part, the “core”, particles and gas are, as a whole, transported upward in a cocurrent-type flow with a fairly high voidage. In the wall zone or “annulus”, particles are, on average, transported downward, countercurrently to the gas flow: the voidage in this region is significantly lower than that in the central region. This so-called core-annulus structure of the CFB is used in the present paper: the riser is divided into a core region characterized by an upflow of a dilute gasparticle suspension, surrounded by an annular region * To whom correspondence should be addressed. E-mail: [email protected]. Tel: 0032/16/32.23.66. Fax: 0032/16/32.29.91.

characterized by a relatively dense particle phase moving, on average, downward.1-4 As a first approach, the radius of the core can be considered of constant size over the whole height of the riser, although it is probably more acceptable to assume that the core zone gets larger toward the exit of the CFB riser. Different approaches are cited in the literature.1-4 The voidage can, on average, be assumed to have a constant value in the core region and a much lower but constant value in the annulus. This assumption has been made by previous authors.2,4 Experimental results of RTD experiments using sand as the solid phase (a nonactive system without adsorption of the gas component on the solid phase), where gas backmixing is practically nonexistent, will demonstrate that the gas velocity profile is of a plug-flow nature in the core region, characterized by a constant gas velocity in the core. This is acceptable, especially when operating at high gas velocities, because higher gas velocities promote plug-flow conditions. Some investigators stated that gas backmixing is negligible;5 others stated that it is only negligible in the core but very significant in the annulus.6 It has also been shown that the working conditions determine the amount of gas backmixing.7,8 Using an “active” particle, such as a catalyst, adsorption/desorption of the active gas component on the catalyst is a very important phenomenon. This phenomenon largely influences the flow mode of the suspensions in both the core and annulus and is the main reason for gas backmixing in CFB systems. Backmixing of solids or internal refluxing of solids in a CFB system is well-known,1-4 and with the gas component adsorbed on the catalyst, the gas will also backmix. This adsorp-

10.1021/ie020356e CCC: $22.00 © 2002 American Chemical Society Published on Web 10/26/2002

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Figure 1. Experimental setup.

Figure 3. Influence of the solids loading on the response tailing (CFB riser using a CFB catalyst).

Figure 2. Typical response peak for a CFB riser using sand.

tion/desorption phenomenon is hence important and needs to be included in the model. It must be based upon the adsorption characteristics of the gas on the catalyst. In this model, a linear adsorption isotherm is assumed (certainly a valid assumption at low tracer gas concentrations). This assumption is translated in the equations of the model in the following way:

concentration adsorbed on the catalyst ) Kads(concentration in the bulk)n (1) with Kads a constant, derived from Perry’s Chemical Engineers’ Handbook (6th ed., 1985, Chapter 16) and n the power (here n ) 1, in view of the linear relationship). As a result of gas backmixing, the residence time of the gas phase will increase and will be widely spread. This has an obvious effect on the conversion and on the possible occurrence of side reactions. Axial dispersion of the gas phase is minimal and therefore neglected. Radial gas transport in the riser is mainly due to convective mass transfer between the core and annulus of both the gas and solids. This mass transfer is characterized by exchange coefficients, for both the gas and solids.1,2,6,9,10 These coefficients appear to be the most important parameters of the model. All model parameters will be fitted to literature data and to results of our own experiments conducted in a 100 mm i.d. CFB riser.

Figure 4. Schematic representation of the process for calculating the stationary mass balances of solids.

Experimental Procedure and Results The experimental setup is shown in Figure 1. The rig consists of a 6.5 m high riser with an internal diameter of 0.1 m. Air is supplied through a distributor plate and leaves the system through a cyclone after the riser exit. The particle circulation rate is controlled by an L valve and measured in a measuring bed. Pressure taps are located along the height of the riser and connected to a data acquisition system. Flow rates and pressure drops were monitored. The solids used for the experiments are sand and fresh fluidized catalytic cracking (FCC) catalyst with an average particle size of respectively 120 and 70 µm and a bulk density of 1500 kg/m3 for the sand and 1000 kg/m3 for the catalyst. Gas flow rates and solids circulating rates were varied within the experiments. In all experiments, a tracer gas (propane) was injected into the bed, 0.5 m above the distributor plate, and the concentration profile at the exit was measured by means of a gas chromatograph. In a first set of experiments, “nonactive” sand was used as the bed material. Figure 2 shows a typical

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response curve. As expected the response curve shows a peak, indicating the part of the injected gas that travels through the riser in plug flow, and a tailing signal, indicating the part that has been recycled or backmixed in the riser. For all operating conditions of the gas velocity and solids circulation rate, the response curves all have the shape of Figure 2: almost all of the gas travels through the riser in plug flow, and there is nearly no gas backmixing. In a second set of experiments, with the catalyst as the solid phase, propane is adsorbed on the catalyst and this provokes a different picture where the tailing effect is outspoken: a second peak occurs which represents the propane that is progressively desorbed, after it has been adsorbed. Evidently, this second peak becomes more relevant at higher solids loading. This effect is shown in Figure 3, where a comparison is given for experiments with almost equal gas velocities but different solid fluxes. Modeling The model is based on mass balances for the gas phase in the core and annulus, respectively. Because the adsorption/desorption phenomenon is of utmost importance, the mass balances of the solids need to be included. Figure 4 gives a schematic presentation of the model for calculating the stationary mass balances of the solid phase. The symbols correspond to the parameters used and include R ) radius of the riser of the CFB, rC ) radius of the core region, H ) height of the riser, Gs ) total (external) solid flux, Gs,C ) (internal) solid flux in the core region, Gs,A ) (internal) solid flux in the annulus region, Cs,C ) solid concentration in the core region, Cs,A ) solid concentration in the annulus region, KCfA ) exchange coefficient for the solids from the core to the annulus, KAfC ) exchange coefficient for the solids from the annulus to the core, KCfA(top) ) exchange coefficient for the solids at the top, and KAfC(bottom) ) exchange coefficient for the solids at the bottom. For an infinitesimal control volume of the riser (neither at the top nor at the bottom), the balances for the tracer (propane) can be written as

In the core: (accumulation of propane in the bulk) + (accumulation of adsorbed propane on the catalyst) ) (net convective axial transport of propane in the bulk) + (mass transfer of propane between core and annulus) + (net convective axial transport of adsorbed propane) + (exchange between the core and annulus of adsorbed propane) 2

2

[C πrC δz δCg,C] + [(1 - C)πrC δz FpKads δCg,C] ) {[πrC Ug,CCg,C|z δt - πrC Ug,C(Cg,C|z + δCg,C) δt} + 2πrC δz kgCg,A|z δt} - [2πrC δz kgCg,C|z δt + 2

2

πrC2Us,CKadsCs,CCg,C|z δt] - [πrC2Us,CKadsCs,C(Cg,C|z + δCg,C) δt + 2πrC δz KAfCKadsCs,ACg,A|z δt] [2πrC δzKCfAKadsCs,CCg,C|z δt] (2)

In the annulus: (accumulation of propane in the bulk) + (accumulation of adsorbed propane on the catalyst) ) (net convective axial transport of absorbed propane) + (mass transfer of propane between the core and annulus) + (exchange between the core and annulus of adsorbed propane) [Aπ(R2 - rC2) δz δCg,A] + [(1 - A)π(R2 - rC2) δz FpKads δCg,A] ) {π(R2 - rC2)Us,AKads[(Cs,ACg,A)z + δ(Cs,ACg,A)] δt π(R2 - rC2)Us,AKads(Cs,ACg,A|z) δt} + (2πrC δz kgCg,C|z δt - 2πrC δz kgCg,A|z δt) + (2πrC δz KCfAKadsCs,CCg,C|z δt 2πrC δz KAfCKadsCs,ACg,A|z δt) (3) For the last control volume at the top of the riser, with δz ) (1/20)H, the balances for the tracer (propane) can be written as

In the core: (accumulation of propane in the bulk) + (accumulation of adsorbed propane on the catalyst) ) (net convective axial transport of propane in the bulk) + (mass transfer of propane between the core and annulus) + (net convective axial transport of adsorbed propane) + (exchange between the core and annulus of adsorbed propane) [C πrC2 δz δCg,C] + [(1 - C)πrC2 δz FpKads δCg,C] ) [πrC2Ug,CCg,C|z δt - πrC2Ug,C(Cg,C|z + δCg,C) δt] + (2πrC δz kgCg,A|z δt - 2πrC δz kgCg,C|z δt) + [πrC2Us,CKadsCs,CCg,C|z δt - πrC2Us,CKadsCs,C(Cg,C|z + δCg,C) δt] - [2πrC δt KCfA(top)KadsCs,CCg,C|z δt] (4) In the annulus: (accumulation of propane in the bulk) + (accumulation of adsorbed propane on the catalyst) ) (outgoing convective axial transport of absorbed propane) + (mass transfer of propane between the core and annulus) + (exchange between the core and annulus of adsorbed propane) [Aπ(R2 - rC2) δz δCg,A] + [(1 - A)π(R2 - rC2) δz FpKads δCg,A] ) [-π(R2 - rC2)Us,AKads(Cs,ACg,A|z) δt] + (2πrC δz kgCg,C|z δt - 2πrC δz kgCg,A|z δt) + (2πrC δz KCfA(top)KadsCs,CCg,C|z δt) (5) For the first control volume at the bottom of the riser, again with δz ) (1/20)H, the balances for the tracer (propane) can be written as

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In the core: (accumulation of propane in the bulk) + (accumulation of adsorbed propane on the catalyst) ) (net convective axial transport of propane in the bulk) + (mass transfer of propane between the core and annulus) + (outgoing convective axial transport of adsorbed propane) + (exchange from the annulus to the core of adsorbed propane)

The solids velocity in the core and the internal upward solid flux are related by the following expression:

Gs,C ) Us,CFp(1 - C)(rC/R)2

In the same manner, the internal downward solid flux can be written as

Gs,A ) Us,AFp(1 - A)[1 - (rC/R)2]

[πrC2Us,CKadsCs,C(Cg,C|z + δCg,C) δt] + [2πrC δz KAfC(bottom)KadsCs,ACg,A|z δt] (6) In the annulus: (accumulation of propane in the bulk) + (accumulation of adsorbed propane on the catalyst) ) (incoming convective axial transport of absorbed propane) + (mass transfer of propane between the core and annulus) + (exchange from the annulus to the core of adsorbed propane)

Gs ) Gs,C - Gs,A

UT ) 0.226 ) Ug,C - Us,C )

rC2) δz FpKads δCg,A] ) [π(R2 - rC2)Us,AKads[ (Cs,ACg,A)z + δ(Cs,ACg,A)] δt] + [2πrC δz kgCg,C|z δt] [2πrC δz kgCg,A|z δt] [2πrC δz KAfC(bottom)KadsCs,ACg,A|z δt] (7)

Ug,C )

()

Usup R C rC

2

(8)

The velocity of the solids in the core, Us,C, is determined as the particles’ slip velocity, i.e., the difference of the gas velocity and terminal velocity of the catalyst:

Us,C ) Ug,C - UT

(9)

This terminal velocity is calculated by the method of Geldart,14 and its value for the used catalyst is 0.226 m/s.

(12)

The voidages in both the core and annulus, C and A, are taken to be constant over the whole height of the riser, again depending on the gas velocity and the solids loading. Provided the radius of the core and the voidage in the annulus are fixed (these values will be chosen, based on the literature data2,4) and both the superficial gas velocity and the total solid flux are known for each experiment, then the voidage in the core can be calculated from the previous equations:

[Aπ(R2 - rC2) δz δCg,A] + [(1 - A)π(R2 -

The determination of the parameters necessary for the model will be briefly explained. The radius of the core, rC, is taken to be constant over the whole height of the riser, depending on the gas velocity and the solids loading. The final determination of the used value is based on previous experiments4 and is included in Table 1. The downward velocity of the solids in the annulus is taken to be constant over the complete riser height and the gas velocity range. On the basis of literature data,4,6,11-13 Us,A is fixed as 1 m/s. The gas velocity in the core, Ug,C, can be calculated when the voidage in the core, the radius of the core, and the superficial gas velocity, Usup, are known:

(11)

Both values are linked to the total (external) solid flux:

[C πrC2 δz δCg,C] + [(1 - C)πrC2 δz FpKads δCg,C] ) [πrC2Ug,CCg,C|z δt - πrC2Ug,C(C0 + δCg,C) δt] + (2πrC δz kgCg,A|z δt - 2πrC δz kgCg,C|z δt) -

(10)

)

)

()

Usup R ‚ C rC

( )[ R rC

2

() R rC

2

2

-

Gs,C Fp(1 - C)

() R rC

2

]

Usup Gs,C + Gs,A C Fp(1 - C)

[

[ ( )]

rC Usup Gs + Us,AFp(1 - A) 1 - R C Fp(1 - C)

2

]

(13)

The voidage of the core is the only unknown parameter in this equation and can hence be determined. The solids concentration in both the core and annulus, Cs,C and Cs,A, can be calculated as well:

Cs,C ) Gs,C/Us,C

(14)

Cs,A ) Gs,A/Us,A

(15)

and

The exchange coefficient of gas between the core and annulus, kg, is one of the parameters that will be varied to evaluate the model, although its value is chosen in agreement with the literature data.1,2,10 The exchange coefficients of solids between the core and annulus, KCfA and KAfC, are both taken to be constant over the whole height of the riser but are kept as variable in the model. The dependency between both parameters is determined from the solids mass balance in an infinitesimal control volume of the riser. If one or both exchange coefficients is fixed at a certain value (based on the literature data4,10), the other one will easily be calculated because there is no net exchange of solids between the core and annulus:

KCfA ) KAfC(Cs,A/Cs,C)

(16)

Ind. Eng. Chem. Res., Vol. 41, No. 24, 2002 5987 Table 1. Review of the Model Parameters for a Particular Experiment parameter

value

ref

radius of the riser height of the riser adsorption constant particle density injected concentration terminal velocity of the solids solids velocity in the annulus total solid flux superficial gas velocity radius of the core voidage in the annulus voidage in the core gas velocity in the core solids velocity in the core internal upward solid flux internal downward solid flux solids concentration in the core solids concentration in the annulus exchange coefficient (top) exchange coefficient (bottom)

R ) 0.05 m H ) 5.97 m Kads ) 0.0579 mgas3/kgsolids Fp ) 1700 kg/m3 C0 ) 0.9 molpropane/mgas3 UT ) 0.226 m/s Us,A ) 1 m/s Gs ) 25 kg/m3‚s Usup ) 4 m/s rC ) 0.04 m A ) 0.8 C ) 0.978 Ug,C ) 6.39 m/s Us,C ) 6.164 m/s Gs,C ) 147.4 kg/m2‚s Gs,A ) 122.4 kg/m2‚s Cs,C ) 23.9 kg/m3 Cs,A ) 122.4 kg/m3 KCfA(top) ) 0.75 m3/m2‚s KAfC(bottom) ) 0.105 m3/m2‚s

15 FCC catalyst experiment calculated fixed experiment experiment 4 2 mass balance calculated calculated calculated calculated calculated calculated calculated calculated

To make sure that there is no accumulation of solids at the bottom or at the top of the riser, the exchange coefficients for the solids are predicted in the bottom zone and in the top zone. A major exchange of solids is assumed from the core to the annulus at the top of the column, and a similar exchange of solids from the annulus to the core occurs at the bottom of the column. The coefficients are determined from the stationary mass balances in the top zone and the bottom zone, respectively. The height for these zones is assumed to correspond to 1/20th of the total height of the column. The coefficients are defined as

At the top: KCfA(top) )

Us,AR2Cs,A 1 2rC HCs,C 20

(17)

2

Us,AR 1 2rC H 20

Gs (kg/m3‚s) Usup (m/s) rC (m) A C Ug,C (m/s) Us,C (m/s) Gs,C (kg/m3‚s) Gs,A (kg/m3‚s) Cs,C (kg/m2) Cs,A (kg/m2) KCfA(top) (m2/m3‚s) KAfC(bottom) (m2/m3‚s)

6 4 0.045 0.9 0.994 4.97 4.74 38.3 32.3 8.1 32.3 0.37 0.093

15 4 0.0425 0.85 0.987 5.61 5.38 85.8 70.8 15.9 70.8 0.44 0.100

25 4 0.04 0.8 0.978 6.39 6.16 147.4 122.4 23.9 122.4 0.75 0.105

ment, i.e., at a gas velocity of 4.0 m/s and a solid flux of 25 kg/m3‚s. Modeling Results

at the bottom: KAfC(bottom) )

Table 2. Parameters for Different Working Conditions at Usup ) 4 m/s

(18)

To illustrate the use of the previous calculations, Table 1 reviews all parameters for a particular experi-

The model contains two unknown parameters: the exchange coefficient for the gas phase, kg, and the exchange coefficient for the solids between the annulus and core, KAfC. The model is evaluated for different values of these coefficients so that the predicted response curve for the concentration in the gas phase is in best possible agreement with the measured one. Figure 5 illustrates the results for the experiment of Table 1. The values of both parameters, for this particular experiment, are given and compared with the literature data: kg [mgas3/mexchange surface2‚s] by fitting from the literature

0.012 0.03-0.1 0.08-0.11

2 1

KAfC [mcatalyst3/mexchange surface2‚s] by fitting from the literature

Figure 5. Comparison of the experimental (on the left) and predicted (on the right) response curves for Usup ) 4 m/s and Gs ) 25 kg/m2‚s.

10-5 1.47 × 10-5-1.47 × 10-4

4

The modeling has been done for the different sets of experiments. Table 2 summarizes the values of the different parameters when working at different conditions. The experimental response curve shows a second peak at approximately 13-15 s. This second peak is predicted

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at approximately 20 s. This difference is due to an insufficient annular solids flux, as predicted by the model: this too low recycle flow delays the presentation of tracer-loaded catalyst to the incoming air flow, thus leading to a delayed adsorption. An increase in the internal recycle flux can be achieved through an increasing thickness of the annulus, through an increased downward velocity in the annulus, and/or through a reduced voidage in the annulus. A higher internal reflux will shift the second RTD peak to a lower time value. To assess these tentative assumptions, a sensitivity analysis was performed. First of all, the downward particle velocity in the annulus was examined. A typical value between 0.5 and 1 m/s can be found in the literature, and therefore the velocity has been varied between these two values. Some parameters will significantly change when the particle velocity in the annulus is changed because of the relations between them (mentioned above). Especially, the particle fluxes in both the core and annulus will decrease significantly with decreasing downward velocity. This leads to a reduced and slower internal circulation of solids. Translated to the RTD response curve, the first peak will be higher, leaving the tailing signal smaller, and the tailing occurs over an extended period of time with a less pronounced second peak. The second investigated parameter is the radius of the core. Again, it is easy to see that several parameters will change when the radius of the core is changed. The effect of decreasing the size of the core (decreasing the radius of the core) is similar to the effect of increasing the particle velocity in the annulus. The first peak becomes less important, and the second one appears sooner. The porosity in the annulus was varied between 0.75 and 0.85. Increasing the porosity in the annulus will increase the porosity in the core and will decrease the particle fluxes in the riser. Again, this leads to a greater first peak and a less important tailing signal. In this case, the second peak will appear at exactly the same time, because the velocities in the core and annulus are invariant. The assumption made for the height of both the top and bottom zones is examined. The height is varied between 1/10th and 1/40th of the total height of the riser. Most other parameters remain unchanged, but in the RTD response curve, a movement of the tailing signal can be seen. When the exchange surfaces are reduced by decreasing the height of the bottom and top zones, the second peak will appear later, because the particles and the adsorbed propane have to travel longer before they leave the column. The sensitivity analysis demonstrates that it is very important to choose appropriate values of the parameters because they have an important effect on the RTD response curve. A better fit of experimental and predicted RTD curves is obtained when decreasing the size of the core. Current investigations, using electrical capacitance tomography and positron imaging, will probably enable us to better define the core and annulus characteristics. These results will be published as soon as available, and the underlying model parameters will be adapted if so required. As was already mentioned in the introduction of the paper, the practical importance of the experimental

results and of the model predictions relates to predicting the gas-catalytic conversion in a CFB reactor. The residence time of the gas is determined by two contributions, i.e., (i) by a core plug flow, for about 50% (area of the first peak), at an average time of the first peak and (ii) for about 50% (area of the tail) with a weighed average time of approximately the second peak (or slightly higher). The overall conversion will hence be the result of the short core contact and the prolonged annular contact. To illustrate the effect, the findings are tentatively applied for a first-order reaction such as butane to maleic anhydride, with a reaction rate constant of 0.1 s-1.16 For a representative core contact of 2 s, the conversion in the core would be 18%, whereas the annulus contact of 20 s would lead to a conversion of 86%. The average conversion will hence exceed 52%. Conclusions The hydrodynamics of the gas phase in CFBs are of dominant importance for predicting the conversion of CFB chemical reactions. Both the core plug-flow gas and the gas backmixing determine the average residence time and the RTD. Too long or too short contact times between the gas phase and the catalyst will affect the degree of conversion. Experimental results of RTD experiments using sand as the solid phase (a nonactive system without adsorption of the gas component on the solid phase), where gas back mixing is practically nonexistent, demonstrated that the gas velocity profile is of plug-flow nature in the core region. This is acceptable, especially when operating at high gas velocities, because higher gas velocities promote plug-flow conditions. Using an “active” particle, such as a catalyst, adsorption/desorption of the active gas component on the catalyst is a very important phenomenon. This phenomenon largely influences the behavior of the suspension in both the core and annulus and is the main reason for gas backmixing in CFB systems. Backmixing of solids or internal refluxing of solids in a CFB system is well-known, and with the gas component adsorbed on the catalyst, the gas will also backmix. This adsorption/ desorption phenomenon of gas and catalyst is hence important and needs to be included in the model. In this model, a linear adsorption isotherm is assumed. (certainly a valid assumption at low tracer gas concentrations). As a result of gas backmixing, the residence time of the gas phase will increase and will be widely spread. The model contains two dominant parameters: the exchange coefficient for the gas phase, kg, and the exchange coefficient for the solids between the annulus and core, KAfC. The model is evaluated for different values of these coefficients so that the predicted response curve for the concentration in the gas phase is in best possible agreement with the measured one. Figure 5 illustrates the results for the typical experiment of Table 1, and the corresponding values of both parameters are defined and compared with the literature data. To complete the assessment, a sensitivity analysis was performed and demonstrates that the RTD response curve is highly sensitive to the values of the parameters.

Ind. Eng. Chem. Res., Vol. 41, No. 24, 2002 5989

Additional research is ongoing to narrow the range of parameter values. List of Symbols C0 ) injected concentration of propane, molpropane/mgas3 Cg,A ) concentration of the gas phase in the annulus, molpropane/mgas3 Cg,C ) concentration of the gas phase in the core, molpropane/ mgas3 Cs,A ) concentration of the solid phase in the annulus, kgsolids/mriser3 Cs,C ) concentration of the solid phase in the core, kgsolids/ mriser3 Gs ) total solid flux, kgsolids/mriser2‚s Gs,A ) internal downward solid flux (annulus), kgsolids/ mriser2‚s Gs,C ) internal upward solids flux (core), kgsolids/mriser2‚s H ) height of the riser, m Kads ) adsorption-constant, mgas3/kgsolids KAfC ) solids exchange coefficient (from the annulus to the core), mcatalyst3/mexchange surface2‚s KCfA ) solids exchange coefficient (from the core to the annulus), mcatalyst3/mexchange surface2‚s kg ) gas exchange coefficient, mgas3/mexchange surface2‚s R ) radius of the riser, m rC ) radius of the core, m t ) time variable, s Ug,C ) gas velocity in the core, mgas3/mcore2‚s Us,A ) solids velocity in the annulus, msolids3/mannulus2‚s Us,C ) solids velocity in the core, msolids3/mcore2‚s Usup ) superficial gas velocity, mgas3/mriser2‚s UT ) terminal velocity of the solids, m/s z ) height variable, m A ) voidage in the annulus, mgas3/mriser3 C ) voidage in the core, mgas3/mriser3 Fp ) particle density, kgsolids/msolids3

Literature Cited (1) Brereton, C. M. H.; Grace, J. R.; Yu, J. Axial gas mixing in a circulating fluidised bed. In Circulating fluidised bed technology II; Basu, P., Large, J. F., Eds.; Pergamon Press: Oxford, 1988; pp 209-212. (2) Patience, G. S.; Chaouki, J. Gas-phase hydrodynamics in the riser of a circulating fluidised bed. Chem. Eng. Sci. 1993, 48, 3195-3205.

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Received for review May 15, 2002 Revised manuscript received August 29, 2002 Accepted September 6, 2002 IE020356E