CFD simulation study of the effect of baffles on the fluidized bed for hydrogenation of silicon tetrachloride

Ning Liu, Xingping Liu, Fumin Wang, Feng Xin, Mingshuai Sun, Yi Zhai, Xubin Zhang,*

1 School of Chemical Engineering and Technology, Tianjin University, Tianjin 300350, China

2 Xinte Energy Co., Ltd., Urumchi, Xinjiang 830011, China

Keywords:Fluidized-bed Baffle Gas-solids flow Hydrogenation Computational fluid dynamics

ABSTRACT In this study,the effect of channel baffles and louver baffles on the flow pattern in the large-scale industrial fluidized beds was studied by computational fluid dynamics(CFD)methods.Then,the effect of flow pattern on the chemical reaction performance was studied for the first time.Simulation results showed that the gas velocity distributed more uniformly, solid particles dispersed more homogeneously and aggregation scarcely occurred in the fluidized bed with louver baffles than that with channel baffles.The residence time distribution indicated that louver baffles remarkably suppressed gas back-mixing in comparison with channel baffles.The reasonable agreements of pressure distribution and reaction results between the simulation in the bed with channel baffles and the data on a large-scale industrial apparatus demonstrated the accuracy of the CFD model.The predicted conversion of SiCl4 in the bed with louver baffles (27.44%) was higher than that with channel baffles (22.69%), indicating that louver baffles markedly improved the performance of the fluidized bed.This study could provide useful information for future structural improvements of baffles in large-scale fluidized beds.

1.Introduction

Silicon-based photovoltaic (PV) cells play a critical part in converting solar energy into electricity,and the demand of polysilicon is expected to increase rapidly in the next few decades.The Siemens process and monosilane pyrolysis are commonly considered as two main methods to produce polysilicon at present.Unfortunately,enormous amount of silicon tetrachloride (SiCl4) is generated as a major byproduct through both two methods.And for both environmental and economic reasons, SiCl4needs to be recycled and converted to the valuable trichlorosilane(SiHCl3)which is the starting material in a closed-loop polysilicon production process[1-3].

Up to now, there have been two basic chemical routes to convert SiCl4back to SiHCl3,the thermal hydrogenation of SiCl4(Eq.(1)) and the catalytic hydrogenation of SiCl4in the presence of metallurgical-grade silicon(MG silicon)(Eq.(2)).The former route(Eq.(1)) involves the homogeneous thermal reaction of SiCl4and H2at temperatures higher than 1100°C in a bell-jar furnace.This route has been widely applied in polysilicon industry, but large quantities of energy consumption and high cost of reactor materials used for high-temperature reaction also present severe restrictions to this process [4,5].The latter route (Eq.(2)) involves the heterogeneous reaction with the use of MG silicon.In contrast with the former route, the latter route is conducted at significantly lower operating temperatures of 400-600°C and can obtain higher SiCl4conversion and SiHCl3selectivity,which has sparked tremendous interests of global researchers.As for the latter route, a fluidized bed reactor is always used [6,7].

Nowadays, the fluidized beds have been diffusely used in the chemical industry, including catalytic cracking, methanol to olefins, polysilicon production, and biomass gasification [3,8,9],because of their excellent gas-solids mixing, heat and mass transfer performance.Various types of baffles are often installed in industrial fluidized beds with the purpose of improving the contact between gas and particles to further increase the conversion of reactant and the selectivity of the desired product.In the past few decades, lots of experimental investigations have been conducted focusing on the effects of baffles on gas-solids hydrodynamics in the fluidized beds.Ramamoorthyet al.[10]investigated the axial solids dispersion in a lab-scale fluidized bed with vertical springs and rods internals and a conclusion was come to that axial dispersion coefficient decreased when internals were installed.Hartholtet al.[11]and van Dijket al.[12]introduced horizontal sieve-like baffles into fluidized beds to investigate the influences on the axial mixing of particles, and the results showed that such baffles abated the vertical mixing of particles remarkably.The effect of a single layer of louver baffles in a 2-D fluidized bed was studied by Zhanget al.[13]and they proved that louver baffles could break bubbles and suppressed solids backmixing while internal emulsion circulation was promoted above the louver baffle.Their another work [14]showed that the extent of solids back-mixing was further reduced when multilayer louver baffles were inserted in the bed.But, the data from all above researches in laboratories using a 2D column or a small-scale apparatus are still far from abundant for the applications of baffles in industrial fluidized beds.

The design and scale-up of large-scale industrial apparatus require more accurate information of the flow hydrodynamics in the fluidized beds with various internal structures.Computational fluid dynamics (CFD), which can obtain the local and transient hydrodynamics details conveniently,has become an efficient technique for extensive researches.There are two approaches for the simulation of the gas-solids flow in fluidized bed, Eulerian-Eulerian approach and Eulerian-Lagrangian approach.Eulerian-Lagrangian approach was proper for a system with less than 106particles,proved by Goldschmidtet al.[15].Therefore,simulations of the industrial fluidized beds with extremely large number of particles always use the Eulerian-Eulerian CFD model, which significantly save both the calculating resources and time.

Huaet al.[16]used the Eulerian-Eulerian CFD methods to investigate a gas-solids dense fluidized bed with vertical baffles,and they found that the residence time distribution of particles transformed from a complete mixing flow to a non-ideal plug flow when the baffles were applied.Rossbachet al.[17]improved the radial gas-solids distribution in the circulating fluidized bed by inserting ring baffles that increased the solids velocity near the wall and redirect the particles to the riser center.Zhaoet al.[18]conducted 3D CFD simulations to study the gas-solids flow hydrodynamics and bubble behaviors in the bubbling fluidized beds.The bubble diameter in the bed with multilayer structured packing was much smaller and the amount of bubbles was much more compared with those in the empty bed.The investigations of Yanget al.[19]demonstrated the ability of louver baffles to break up bubbles as well as suppress the gas-solid back-mixing in fluidized beds by means of CFD simulation.Klenovet al.[20]experimentally and numerically studied a large-scale fluidized bed separated by horizontal baffles made of angle bars based on CFD methods,demonstrating that that the amount of solids was almost evenly distributed in the regions between the baffles.Meanwhile, they confirmed that the accuracy of simulation for industrial applications using a coarse computational grid was sufficient.

From the above analysis, it can be seen that all kinds of baffles can change the flow behaviors of gas and solids compared with the empty bed.However,the investigations of practical applications of baffles in industrial fluidized beds were insufficient in previous studies.Meanwhile, the effect of baffles on a specific chemical reaction was scarcely involved.

The channel baffles have been used in a large-scale industrial fluidized bed for the hydrogenation of SiCl4over the last several years.Nevertheless, the desired value of the conversion of SiCl4can’t be attained,which may be attributed to the insufficiency of the channel baffles in the radial gas-solids mixing and axial gas back-mixing suppression.Therefore,the multi-turn louver baffles that could generate multi-turn annular gas flow are supposed to be better choices and needed to be evaluated by CFD methods at first.

In the present paper, the flow hydrodynamics in the fluidized beds with channel baffles or louver baffles were studied respectively by CFD methods.Then, the chemical reaction simulation results of the hydrogenation of SiCl4in a fluidized bed with channel baffles were compared with the results of experiments on a largescale industrial apparatus.The validated CFD model was then applied to the reaction simulation in a fluidized bed with louver baffles.And it was expected to provide a prediction for future structural improvements of baffles of the fluidized beds for the catalytic hydrogenation of SiCl4.

2.Industrial Apparatus

Experimental investigations have been conducted on a largescale industrial fluidized bed with channel baffles.The schematic of the experimental setup and its picture are shown in Fig.1.The inner diameter of the industrial fluidized bed was 3.9 m and the total height was 18 m.Gas was introduced from the bottom of the fluidized bed and then passed through the gas distributor nozzles.MG silicon particles were introduced from the solids inlet,then mixed and reacted with the gas in the baffled region.The particles entrained by gas were separated by cyclone and moved downward to the bed.Gas flowed out of the system from the gas outlets at the top of the fluidized bed.

Fig.1. Schematic and optical image of the industrial fluidized bed.

3.Simulation

3.1.CFD model

In this work,the Eulerian-Eulerian two-fluid model was used to simulate gas-solids flow in the fluidized bed.The main governing equations and constitutive relations are listed in Table 1 [21].The widely used Gidaspow drag model[22],which combined the Ergun drag model [23](for dense phase) and the Wen-Yu drag model[24](for dilute phase), was applied to describe the interaction between the gas and solids.

Table 1Governing equations and constitutive relations

Table 2Reaction kinetics parameters

3.2.Chemical reaction kinetic model

The CuCl-catalyzed heterogeneous hydrogenation of SiCl4(Eq.(2)) has been implemented industrially for decades.The kinetic model of Dinget al.[6]was modified to apply to the operating conditions in this simulation.The modified kinetic model(Eq.(3))was used to calculate the chemical reaction raterc(mol·gSi-1·s-1) of the hydrogenation of SiCl4.

The apparent rate constantk, equilibrium constantKpand the adsorption equilibrium constantsK1,K2in Eq.(3)can be expressed as

where the relevant parameter values are listed in Table 2.

3.3.Geometrical configuration

It was unaffordable to calculate the industrial fluidized bed with 3.9 m in diameter mainly because that lots of baffles with complicated configuration will result in a large number of grids.Hence,the industrial fluidized bed was scaled down appropriately in the radial direction.And the solids inlet was not taken into consideration since the consumption of particles over the calculating time could be ignored.Meanwhile, the cyclones were omitted.The geometry of the fluidized bed model used in our simulations is shown in Fig.2.The fluidized bed has a height of around 18 m and an inner diameter of 1.1 m.The channel baffles are set at the height of 1.8-8.6 m in model A.11 rows of louver baffles are arranged at the height of 1.8-7.8 m with equal spacing of 0.6 min model B.Most of the previous studies on the effect of baffles on gas-solids hydrodynamics in the fluidized beds assumed uniform gas injection at inlet,which ignored the influence of gas distributor configuration on the flow field[25].Hence,gas distributor nozzles including 21 inlets with 0.0325 m in diameter were taken into consideration to let the simulation be much closer to the actual situation.Two gas outlets with 0.066 m in diameter are at the top of the fluidized bed.

Fig.2. Schematic of the fluidized beds used in our simulations.

Fig.3 exhibits the top view of the configuration of channel baffles and louver baffles.Fig.3(a) shows 4 rows of vertical channel baffles.Each side of one channel baffle measures 0.1 m and the interval between two channel baffles is also 0.1 m.Fig.3(b)shows the arrangement of multi-turn louver baffles with a diameter of 1 m.Each straight line symbolizes an inclined vane and the arrows symbolize the incline direction of vanes.The vanes have an inclination of 50° and the distance between two vanes is 0.028 m.The shape of a bundle of vanes with same orientations is a square with a side of length 0.3 m.The louver baffle has a height of 0.03 m.

3.4.Simulation setting

Fig.3. Configuration of (a) channel baffles and (b) louver baffles.

The commercial CFD code ANSYS Fluent was used to obtain the results of this simulation.The Phase Coupled SIMPLE (PC-SIMPLE)algorithm was employed for pressure-velocity coupling and correction.The first order upwind interpolation scheme was applied for the equations of momentum, turbulent kinetic energy, turbulent dissipation rate and volume fraction.The details of gas and solids properties, operating conditions and main simulation parameters are shown in Table 3.Operating pressure was set at 1.7,1.9, 2.1, 2.3 MPa in different cases respectively with same mass flow of inlet gas of 2.9 kg·s-1and operating temperature of 813 K.The inlet of gas was defined as velocity inlet and the molar composition of inlet gas was 64%H2and 36%SiCl4.The outlet of gas was defined as pressure outlet.On the wall, a non-slip boundary condition was set for both gaseous and solid phases.

Table 3Basic parameters of modeling

The simulation was performed using a very small time step equal to 0.001 s with 20 iterations per time step.The solids were filled at the bottom of the bed up to a height of 6 m at the solids volume fraction of 0.63 initially.CFD simulations of flow hydrodynamics were carried out for a total of 70 s of physical time to ensure that the gas-solids flow was completely developed.Then,chemical reaction simulations were also carried out for 70 s to ensure the reversible reaction had reached steady state.One transient simulated case of channel baffles and louver baffles needed about 10 and 15 days to calculate 70 s, respectively, using 44 processors (Intel Xeon(R)CPU E5-2699 v4@ 2.2 GHz).The timeaveraged data were obtained through the later calculations for 20 s after a steady state was reached.

4.Results and Discussion

4.1.Validation of CFD model

The accuracy of the CFD model was validated through comparison of between the predicted axial pressure distribution and the data on an industrial apparatus at an operating pressure of 2.3 MPa, as shown in Fig.4.The experimental pressure data were measured at positions of the bottom of the bed, solids inlets and gas outlets, respectively.The predicted pressure drop mainly occurs between the bottom and middle of the fluidized bed,which is in good agreement with the experimental data.It can be seen that the CFD model used in this work can provide reasonable results to investigate the hydrodynamics in the fluidized bed.In this section, the hydrodynamic behaviors at 2.3 MPa is evaluated as an example, because the phenomena at other operating pressures are similar.

Fig.4. Pressure distribution along the height of the fluidized bed with channel baffles at an operating pressure of 2.3 MPa.

In order to confirm the grid independence,the simulations with three different grid resolutions were conducted.Considering both simulation accuracy and computational effort, the grid numbers of 789,909 and 1,576,789 were chosen for the beds with channel baffles and louver baffles respectively in the following simulations.

4.2.Flow pattern and solids distribution

The instantaneous gas velocity vectors of the fluidized beds with channel baffles or louver baffles are shown in Fig.5.And the flow pattern of particles is similar to that of gas in the gassolids mixing region.The flow field in the fluidized bed with channel baffles is separated into a few regions by channel baffles, as shown in Fig.5(a).The macro gas flow directions are upward in the channels near the center of the bed and downward in the channels near the wall region, respectively.In each channel, the gas flow behavior is close to an ideal plug flow mode while there are almost no vortexes because of the special structure and spatial size of channel baffles.However, gas back-mixing occurs obviously in the whole bed at the macro level.The marked velocity vectors painted by red indicate that more gas rapidly passes through the channels near the center of the bed,which may lead to undesirable gas short-cut.From Fig.5(b)it is seen that the louver baffles divide the flow field along the bed height into a dozen regions.On balance, gas tends to flow upward in the most area of the bed.The near wall area in which the gas flows downward is much narrower than that of the bed with channel baffles, and the radial distribution of gas velocity is more uniform.All above characteristics of gas velocity radial distribution (in the baffled region,Y=1.8-8.4 m) can also be found in Fig.6.The radial distanceris nondimensionalized by being divided by the fluidized bed diameterRf.The velocity directions of gas are changed by the inclined vanes in the multi-turn louver baffles, creating multi-turn annular gas flow in the bed.Small-scale vortex motions exist in the space between two layers of baffles, resulting in the radial mixing to some extent.In whole it can be said that the gas back-mixing in the bed with louver baffles is weaker than that in the bed with channel baffles in some degree.

Fig.5. Instantaneous gas velocity vectors of the fluidized bed with(a)channel baffles and(b)louver baffles(I and II are the local enlarged figures of(a),III and IV are the local enlarged figures of (b)).

Fig.6. Radial distribution of gas velocity invertical direction.

Fig.7 depicts the instantaneous distribution of solid volume fraction in vertical and cross sections in the fluidized beds with channel baffles or louver baffles.The separation phenomenon caused by channel baffles can also be seen in the instantaneous contours of solids holdup in both vertical and cross sections, as shown in Fig.7(a).The channel baffles considerably restrict the generation of large bubbles, which is beneficial for the mixing of gas and solids to a degree.However,some regions of high concentration of particles exist at the channel baffles walls and the internal wall of the bed, which may be attributed to the complicated geometry of channel baffles and that there is no lateral turbulence in channels.Meanwhile, the aggregation and deposition of particles occur at the bottom of the bed with channel baffles, as well as in the model with louver baffles, forming another portion of the stagnant zones.In Fig.7(b),much less aggregation and deposition of particles occur on the surfaces of vanes of the louver baffles while no particles clustering on the internal wall of the bed.A few bubbles are generated mostly in the center of the bed.In addition,the louver baffles lead to a higher bed expansion ratio.From the contours of solid volume fraction in cross sections, it can be seen that the solids distribute less non-uniformly in the fluidized bed with louver baffles, due to the multi-turn annular gas flow and a greater radial mixing caused by louver baffles.

The radial distribution of solids at a height of 2.6 and 5.8 m is further compared,as shown in Fig.8.In both the fluidized bed with channel baffles and the bed with louver baffles, solid volume fraction is higher near the wall and lower in the bed center, because the gas velocity is higher in the bed center and lower near the wall.And the in homogeneity of solid volume fraction between the bed center and the wall area is moderated in the bed with louver baffles.The variance of the radial distribution of solid volume fraction also certifies that the solids distribute more homogeneously in the fluidized bed with louver baffles.Moreover, with the increasing of the bed height, solid volume fraction becomes lower in the bed with channel baffles and higher in the bed with louver baffles.

Fig.7. Instantaneous distribution of solid volume fraction of the fluidized bed with (a) channel baffles and (b) louver baffles.

Fig.8. Radial distribution of solid volume fraction at a height of (a) 2.6 m and (b)5.8 m.

4.3.The residence time distribution of gas

The residence time distribution (RTD) of gas in fluidized bed reactors, a quantitative criterion of flow pattern and back-mixing degree, is always the crucial topic in both the investigations and industrial fields [26-29].Measuring gas RTD curves in the largescale industrial apparatus through experimental methods suffer from high cost and complex operation.In this paper,the numerical simulation tracing method was applied to obtain the RTD of gas in the fluidized beds with channel baffles or louver baffles.The gas tracer with the identical characteristics to the main gas was injected into the bed through gas inlets within 5 s after steady flow field was reached.And to reduce the computational effort, the molar concentration of gas tracer was monitored at the cross section at the height of 10.6 m where the gas velocity vectors were totally straight upward.The mass ratio of gas tracer to the all inlet gas was controlled to be 5% to weaken the influence of the tracer on the main flow of the bulk bed materials.

Fig.9 illustrates the predicted RTD curves of gas for the fluidized beds with channel baffles or louver baffles.It can be seen that two RTD curves of gas both display long tails, indicating that some extents of gas back-mixing both exist in the bed with channel baffles and the bed with louver baffles.The RTD curve of gas in bed with channel baffles exhibits multiple peaks:the former peaks are assigned to the gas rapidly flowing through the channels near the center of the bed, and the latter peaks are assigned to the gas circulating in the channels near the wall region.The RTD curve of gas in bed with louver baffles exhibits single peak that is narrower and sharper than that in the bed with channel baffles, indicating that the louver baffles weaken the gas back-mixing,which is consistent with the gas flow pattern in Fig.5.

The average residence timetmand dimensionless varianceare key parameters reflecting the characteristics of RTD curves.The average residence timetmcan indicate the recirculation extent of gas flow, and the dimensionless varianceis able to quantitatively demonstrate the degree of global gas back-mixing.The space time of the gas was calculated by the equation τ=V/Q, whereVis the volume of the region used for the measurement of gas RTD curve, andQis the volume flow rate of inlet gas [30].The average residence timetm, space time τ and dimensionless variancefrom Fig.9 are listed in Table 4.

Table 4The characteristic parameters of gas RTD curves

Theoretically, the average residence timetmshould be equal to space time τ when no internal stagnant zones and short circuits exist in the fluidized bed.As for the results in this work,the general tendency oftm＞τ gives evidence of that there is some stagnant zones in the bed with channel baffles and the bed with louver baffles [31].The stagnant space can occur at the bottom of the bed(deposition of particles in Fig.7),the junctions between the baffles and the internal wall and between two adjacent baffles,where the gas flows slowly or scarcely moves.The dimensionless variance σ2θin the bed with louver baffles is markedly smaller than that in the bed with channel baffles, manifesting that the gas has a relatively weaker back-mixing and the gas flow is much closer to a plug flow in the bed with louver baffles.

Fig.9. Residence time distribution (RTD) of gas of the fluidized beds with channel baffles or louver baffles.

4.4.The simulations of chemical reaction

The simulation results of the conversion of SiCl4in the fluidized bed with channel baffles are compared with a series of experimental data on a large-scale industrial apparatus under operating pressures range from 1.7 to 2.3 MPa to validate the accuracy of the CFD model used in this work, as shown in Fig.10.It can be observed that, the simulation results of the fluidized bed with channel baffles have good agreements with the experimental data as a whole,which indicates that the CFD model can be used to precisely predict the catalytic hydrogenation of SiCl4in the fluidized bed reactors.As the operating pressures increase, which is beneficial to the proceeding of forward reaction, the conversion of SiCl4increases distinctly.

Fig.10. Comparison of the conversion of SiCl4between experimental data and simulation results at different operating pressures of the fluidized beds with channel baffles.

Fig.11. Instantaneous distribution of mass fraction of gaseous product SiHCl3 of the fluidized bed with channel baffles(a)and louver baffles(b)at an operating pressure of 2.3 MPa.

Fig.11 displays the instantaneous mass fraction distribution of gaseous product SiHCl3in the fluidized bed with channel baffles or louver baffles at an operating pressure of 2.3 MPa.As shown in Fig.11(a), the mass fraction of SiHCl3in channels near the center of the bed is much lower than that in channels near the wall region,and the mass fraction of SiHCl3in channels near the wall region is equal to or even higher than that in the area above the channel baffles.This can be attributed to that a portion of gaseous reactants pass fast through central channels and others circulate in the bed.In contrast with the product distribution in the bed with channel baffles, an axial gradient distribution of the mass fraction of gaseous product SiHCl3is formed along the height the bed with louver baffles, as shown in Fig.11(b), due to the gas flow pattern is close to a plug flow in the whole bed and its weaker backmixing mostly occurs in the space between two layers of baffles.This also can be seen in Fig.12, the axial distribution of gaseous product SiHCl3appears to be uniform along the height of the fluidized bed with channel baffles.The mass fraction of SiHCl3is relatively higher at the bottom of the bed with channel baffles or louver baffles, since gaseous reactants are trapped in the particles deposition zone and have more time to react with particles.

Fig.12. Axial distribution of gaseous product SiHCl3 along the height of the fluidized bed with channel baffles or louver baffles at an operating pressure of 2.3 MPa.

The conversion of SiCl4in the bed with louver baffles(27.44%)is remarkably higher than that in the bed with channel baffles(22.69%)at an operating pressure of 2.3 MPa.Because the improvement of solids distribution in the bed with louver baffles could enhance the contact between gas and particles.Furthermore, the gas flow closer to a plug flow is more beneficial to the reversible reaction of catalytic hydrogenation of SiCl4.

5.Conclusions

The industrial fluidized bed,separated by channel baffles or louver baffles, was studied by CFD methods.The flow pattern and chemical reaction performance were analysed to investigate the function of baffles.The following conclusions were obtained,

(1) The CFD model was validated based on a comparison of pressure distribution between simulation results and the data on an industrial apparatus.

(2) Generally, gas flowed upward in the center of the fluidized bed and downward near the wall region.In the bed with louver baffles,the radial range in which gas flowed upward was wider than that with channel baffles,and the radial distribution of gas velocity was more uniform.

(3) The solids distributed more homogeneously and much less aggregation of particles occurred in the fluidized bed with louver baffles compared to the bed with channel baffles.

(4) The degree of back-mixing for gas flow was reduced by the inserting of louver baffles and the flow behavior was closer to the plug flow mode in comparison with the applying of channel baffles.

(5) The simulation results of chemical reaction in the fluidized bed with channel baffles had good agreements with the experimental data on a large-scale industrial apparatus.The conversion of gaseous reactant SiCl4was increased to 27.44%in the bed with louver baffles,which was higher than that in the bed with channel baffles(22.69%)at an operating pressure of 2.3 MPa.

A basic study of the effect of louver baffles on the large-scale fluidized bed was provided in this paper.And future works are still needed to evaluate the effect of the geometric parameters (e.g.,height of baffle, pitch of inclined vanes and inclination angle of vanes) of the louver baffles to promote the industrial application.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported by the National Key Research and Development Program ofChina (2018YFB0604900), the National Natural Science Foundation of China(21978198),and the National Key Research and Development Program of China(2016YFF0102503).

Nomenclature

CDdrag coefficient

dpdiameter of particle, m

Eactivation energy, kJ·mol-1

erestitution coefficient

gigravity acceleration, m·s-2

g0radial distribution function

Hspecific enthalpy, kJ·kg-1

ΔH enthalpy change, kJ

hinterphase heat transfer coefficient, kJ·m-3·K-1·s-1

Jidiffusion flux of speciesi,kg·m-3·s-1

K1adsorption equilibrium constants, atm-1

K2adsorption equilibrium constants, atm2/3

Kpequilibrium constant, atm-1/3

kapparent rate constant, mol·g-1·s-1·atm-2

mgpsource terms of mass generation for the gas phase, kg·m-3·s-1

mpgsource terms of mass consumption for the solids phase,kg·m-3·s-1

ppressure, Pa

Qgas volume flow rate, m3·s-1

Rgas constant, J·mol-1·K-1

Rffluidized bed diameter, m

Riproduction rate of speciesidue to chemical reactions,kg·m-3·s-1

RepReynolds number

rradial distance, m

rcchemical reaction rate,

Sheat source due to chemical reactions, kJ·m-3·s-1

Ttemperature, K

ttime, s

tmaverage residence time, s

uvelocity vector, m·s-1

Vreactor volume, m3

xdirection coordinate, m

yimass fraction of speciesi

Yycoordinate

α volume fraction

β inter-phase momentum exchange coefficient, kg·m-3·s-1

ΓΘdiffusion coefficient for the energy fluctuation, kg·m-1·s-1

γ collisional dissipation of energy fluctuation, kg·m-1·s-3

δijunit tensor, m2·kg-1

Θ granular temperature, m2·s-2

κ thermal conductivity, kJ·m-1·K-1·s-1

λpsolids bulk viscosity, kg·m-1·s-1

μpsolids phase shear viscosity, kg·m-1·s-1

ρ density, kg·m-3

τ space time, s

τggas phase stress tensor, kg·m-1·s-2

τpsolids phase stress tensor, kg·m-1·s-2

Subscripts

i, j, kdirection coordinate

ggas phase

psolids phase