ARTICLE pubs.acs.org/IECR
Effects of GasSolid Hydrodynamic Behavior on the Reactions of the Sorption Enhanced Steam Methane Reforming Process in Bubbling Fluidized Bed Reactors Yuefa Wang, Zhongxi Chao, and Hugo A. Jakobsen* Department of Chemical Engineering, Norwegian University of Science and Technology (NTNU), Sem Sælands vei 4, NO-7491 Trondheim, Norway ABSTRACT: A three-dimensional two-fluid modeling approach with the kinetic theory of granular flow was used to study the process of sorption enhanced steam methane reforming (SE-SMR) carried out in a bubbling fluidized bed reactor. The effects of the superficial gas velocity and restitution coefficient on the solid flow pattern and reactions were investigated. The hydrodynamic properties of the reactive flow showed different behaviors compared to the cold flow. The superficial gas velocity and restitution coefficient influence the bed expansion and bed uniformity in different ways. The reactions of SE-SMR were affected mainly by the superficial gas velocity, while the restitution coefficient has a smaller influence on the reactions. The CO2 adsorption is a faster reaction than the SMR reactions. A uniform bed state will be of more benefit to slow reactions than to fast reactions.
1. INTRODUCTION The bubbling fluidized bed (BFB) is the most common densephase fluidization system for particular applications. Lin et al.1 used the computer automated radioactive particle tracking technique to map the velocity field of the whole system by tracking the movement of a tracer particle in a bubbling fluidized bed. In the experiments they found that there exist different solid flow patterns, even with opposite flow directions in the bubbling fluidized bed reactors. Their work was the first successful attempt to obtain detailed quantitative information on solid velocities, though several researchers had made measurements on the circulating pattern of solid particles and found some particular types of solid circulation patterns before them (e.g., Marsheck and Gomezplata,2 Werther and Molerus,3 and Whitehead et al.4). The two-fluid model containing closure relations for the particle stresses derived on the basis of the kinetic theory of granular flow (KTGF) is commonly used to describe the gassolid fluidization system. The KTGF postulates that the particulate system can be represented considering a collection of identical, smooth, and rigid spheres. The macroscopic transport equations are derived starting out from a Boltzmann type of equation. Savage and Jeffrey5 were the first to apply the kinetic theory for dense gases with the assumption of elastic particle collisions to determine the stress tensor in a granular flow. Jenkins and Savage6 and Jenkins and Richman7 extended the theory to the dense systems of inelastic particles. Lun et al.8 established a generalized model valid for both dilute and dense flows by incorporating the kinetic stresses with the collisional stresses. Ding and Gidaspow9 and Gidaspow10 derived a two-phase flow model by incorporating the fluidparticle interactions (drag force) into the governing equations for granular flow. The closure parameters and gas flow rate have strong influences on the flow pattern of the solid particles.11,12 Many authors have studied the influence of closure parameters on the hydrodynamic behavior of bubbling fluidized beds to improve the KTGF and twofluid models. The drag coefficient (β) representing the main gasparticle interacting force1318 and the coefficient of restitution r 2011 American Chemical Society
(e) representing a measure of elasticity for particleparticle collisions11,1927 are two very important parameters in the KTGF and two-fluid models for gasparticle systems. The gas flow rate is also an important factor to affect the solid flow pattern. However, the influence of the solid flow pattern on the reactions is less investigated by most researchers. The process of sorption enhanced steam methane reforming (SE-SMR) is becoming an important topic due to its integration of hydrogen production and CO2 separation. It can be carried out in a fixed bed reactor2839 or in a bubbling fluidized bed.4047 In this paper the performance of the SE-SMR process in a bubbling fluidized bed reactor under different hydrodynamic states is studied with a three-dimensional two-fluid model.
2. MATHEMATICAL MODEL 2.1. Hydrodynamic Model. An Eulerian two-fluid model implemented in an in-house code is used to simulate the fluidized bed performance. The standard kε model is used to describe the turbulent flow of gas phase. The kinetic theory of granular flow (KTGF) is used to treat the particle phase as a fluid in the model. More detailed descriptions of the model and solution methods can be found in Lindborg et al.26 and Jakobsen.48 The governing equations are given in Table 1. The constitutive equations are listed in Tables 2 and 3 for internal phases and Table 4 for the interface, respectively. The drag model presented by Benyahia et al.18 is used in this work to account for the gassolid interactions. 2.2. Reaction Kinetics. The chemical system of the SE-SMR includes the steam methane reforming reactions with Ni-based catalysts and the CO2 adsorption by CaO sorbent. The reaction Received: November 18, 2010 Accepted: April 28, 2011 Revised: March 30, 2011 Published: April 28, 2011 8430
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Table 1. Governing Equations continuity equation for phase k (=c, d): D ðRk Fk Þ þ r 3 ðRk Fk v k Þ ¼ 0 Dt momentum equation for phase k (=c,d): D ðRk Fk vk Þ þ r 3 ðRk Fk v k v k Þ ¼ Rk rp Dt þ r 3 τ k þ Rk Fk g þ Mk
molecular temperature equation for phase c: DTc ¼ r 3 ðRc keff Rc Fc Cp, c ð ΔHiSMR ÞRiSMR þ Qci c rTc Þ þ Dt
∑i
ð1Þ
molecular temperature equation for phase d: DTd SP SP i ¼ r 3 ðRd keff Rd Fd Cp, d d rTd Þ þ ð ΔH ÞR Qc Dt
ð2Þ
ð4Þ
ð5Þ
species composition for phase k (=c, d): D i ðRk Fk ωk, j Þ þ r 3 ðRk Fk v k ωk, j Þ ¼ r 3 ðRk Fk Deff k, j rωk, j Þ þ Mj Rk, j ð6Þ Dt
granular temperature equation: 3 D ðRd Fd ΘÞ þ r 3 ðRd Fd v d ΘÞ ¼ τ d :rvd 2 Dt ð3Þ 3 þ r 3 ðkd rΘÞ γ 3βΘ þ βÆ~v0c 3 Cd æ þ Γk Θ 2
Table 2. Constitutive Equations for Internal Momentum Transfer
Table 3. Constitutive Equations for Internal Heat and Mass Transfer
gas phase stress:
effective conductivity: τ c ¼ 2Rc μc Sc
ð7Þ
m keff k ¼ kk þ
solid phase stress: _ τ d ¼ ð pd þ Rd ζd r 3 vk ÞI þ 2Rd μd Sd
ð8Þ
_ 1 1 ðrvk þ ðrvk ÞT Þ ðr 3 vk ÞI 2 3
Λ ¼ ð10Þ
8
ð11Þ
10=9 Rd B ¼ 1:25 , Rc
k0d , k0c
m Deff k, j ¼ Dk, j þ
solid phase shear viscosity (Gidaspow ): rffiffiffiffi 2 2μdilute 4 4 Θ d R R ð12Þ 1 þ ¼ g ð1 þ eÞ þ F g ð1 þ eÞ μkc d 0 d d 0 d 5 5 π Rd g0 ð1 þ eÞ
μdilute ¼ d
pffiffiffiffiffiffiffi 5 F dd Θπ 96 d
ð23Þ
ð24Þ
∑
j6¼ i
binary diffusion coefficient (Fuller et al.52): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Tc 1:75 1=Mj þ 1=Mi Dij ¼ 101:325P½ð∑V Þj 1=3 þ ð∑V Þi 1=3 2
ð13Þ
dilute viscosity (Gidaspow10):
μtk Fk Sct
molecular diffusion coefficient (Wilke51): 1 ω ωj Dm c, j ¼ Mm Mj Dij
conductivity of the granular temperature: rffiffiffiffi 2 15 μdilute 6 Θ 2 d kkc R 1 þ ¼ g ð1 þ eÞ þ 2R F d g ð1 þ eÞ d 0 d d d 0 d 2 g0 ð1 þ eÞ 5 π
radial distribution function (Ma and Ahmadi49): 1 þ 2:5Rd þ 4:5904Rd 2 þ 4:515439Rd 3 g0 ¼ 2 !3 30:67802 R d 41 5 Rmax d
φ ¼ 7:26 10 3
effective diffusivity:
10
collisional energy dissipation (Jenkins and Savage6): ! rffiffiffiffi 4 Θ 2 2 r 3 vd γ ¼ 3Rd Fd g0 Θð1 e Þ dd π
ð21Þ
" # 2 A1 B A B1 1 ln ðB þ 1Þ ð22Þ 1 B=A ð1 B=AÞ2 A B 1 B=A 2
A ¼
solid phase bulk viscosity (Lun et al. ):
rffiffiffiffi 4 Θ ζd ¼ Rd Fd dd g0 ð1 þ eÞ 3 π
ð20Þ
k0c km d ¼ pffiffiffiffiffi ðφA þ ð1 φΛÞÞ Rd
ð9Þ
solid phase pressure (Lun et al.8): pkc d ¼ Rd Fd Θ½1 þ 2ð1 eÞRd g0
ð19Þ
molecular conductivity (Bauer and Schl€under50): pffiffiffiffiffi k0c ð1 Rd Þ km c ¼ Rc
deformation rate: Sk ¼
μtk Fk Pr t
ð25Þ
ð14Þ
Table 4. Interfacial Momentum and Heat Transfer Equations interfacial force:
ð15Þ
Mc ¼ Md ¼ FD ¼ βðvd vc Þ interfacial heat transfer (spherical particles): 6Rd hcd ðTd Tc Þ Qc ¼ dd
dissipation of granular energy due to fluid turbulence (Lindborg et al.26): β2 dp jvd Ævc æj2 Rs pffiffiffiffiffiffiffi βÆ~v0c 3 Cd æ ¼ 4Rc 4 Rd Fd πΘ
ð17Þ
1 Rs ¼ pffiffiffiffiffi g0 ð1 þ 3:5 Rd þ 5:9Rd Þ
ð18Þ
ð27Þ
interfacial heat transfer coefficient (Gunn53): kc hcd ¼ ½ð7 10Rc þ 5Rc 2 Þð1 þ 0:7Rep 0:2 Pr 1=3 Þ dp
ð16Þ
ð26Þ
ð28Þ
þ ð1:33 2:4Rc þ 1:2Rc 2 ÞRep 0:7 Pr 1=3 particle Reynolds number and Prandtl number: Rep ¼ 8431
Rc Fc jv d vc jdd , μc
Prp ¼
μc Cp, c kc
ð29Þ
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Table 5. Physical Properties and Model Parameters particle diameter
500 μm
particle density
2400 kg/m3
sorbent-to-catalyst ratio
1:4 kg/kg
reactor size
dt = 0.14 m, Ht = 0.34 m
initial bed height grid cell number
0.113 m 12 12 80
time step
1 104 s ) )
) )
rm < ε b ; ε = 1010
convergence criterion reaction temperature
848 K
pressure
1 bar
steam-to-carbon ratio
5 kg/kg
rate equations and the correlations for these reactions are taken from the literature. Steam methane reforming (SMR) includes the following three main reactions: CH4 þ H2 O S CO þ 3H2
ΔH298 ¼ 206 kJ=mol ð30Þ
CO þ H2 O S CO2 þ H2
ΔH298 ¼ 41 kJ=mol ð31Þ
CH4 þ 2H2 O S CO2 þ 4H2 ΔH298 ¼ 165 kJ=mol
ð32Þ
The kinetic expressions of SMR reactions from Xu and Froment54 are used in our simulations: " # k1 pCH4 pH2 O pH2 3 pCO =K1 R1 ¼ ð33Þ pH2 2:5 DEN2 k2 pCO pH2 O pH2 pCO2 =K2 R2 ¼ p H2 DEN2 R3 ¼
k1 pH2 3:5
"
pCH4 pH2 O 2 pH2 4 pCO2 =K3 DEN2
ð34Þ # ð35Þ
where DEN ¼ 1 þ KCO pCO þ KH2 pH2 þ KCH4 pCH4 þ KH2 O pH2 O =pH2 ð36Þ The rate equation for CO2 adsorption by the CaO sorbent is taken from Sun et al.55 CaO þ CO2 S CaCO3 Ra ¼
ΔH298 ¼ 178 kJ=mol ð37Þ
dX ¼ 56ks ð1 XÞðpCO2 pCO2 , eq Þn S dt
ð38Þ
where the equilibrium pressure of CO2 is calculated from the equilibrium concentration of CO2 presented by Abanades et al.40 1:462 1011 19130 exp CCO2 , eq ¼ ð39Þ T T 2.3. Initial and Boundary Conditions. At initial time (t = 0), there is no gas flow in the reactor, and the bed is at rest with a particle volume fraction slightly below that of the minimum fluidization state. The temperature of the bed was set to the reaction temperature.
For time t > 0, the wall boundary condition for the gas phase is based on the wall function approach consistent with the single-phase kε model. The particles were allowed to slip along the wall, following the boundary conditions used by Ding and Giaspow.9 The radial and azimuthal velocity components for both phases are set to zero at both inlet and outlet boundaries. Uniform plug flow is assumed at the inlet. A prescribed pressure is specified at the outlet. The inlet gas composition is the pure steam at the beginning step (about 510 s). When the hydrodynamic state of the fluid flow in the reactor became stable, the inlet gas was switched to the mixture of steam and methane with the desired ratio. The temperature at the wall was set to 10 K higher than the reaction temperature.
3. RESULTS AND DISCUSSION The simulated reactor is a cylinder of 34 cm in length and 14 cm in diameter. The solid particles are the mixture of reforming catalyst and sorbent with the sorbent-to-catalyst mass ratio of 1:4. Superficial gas velocity is from 0.32 to 0.89 m/s. These conditions are close to those used by Lin et al.1 The physical properties of the reactor and particles as well as simulation parameters are listed in Table 5. The model was solved in a three-dimensional cylindrical coordinate system with uniform grids in all three directions. The governing equations are discretized on a staggered grid arrangement with the finite-volume algorithm. The linear equations are solved with biconjugated gradient (BCG) algorithms. The restitution coefficient and superficial gas velocity have strong influences on the solid flow patterns in the bubbling fluidized bed.11,12 The hydrodynamic parameters may also affect the reactions, and reversely the reactive flow may affect the solid flow patterns. The drag model is another important factor to affect the flow pattern in the bubbling fluidized bed. In our previous paper12 the drag model presented by Benyahia et al.18 was shown to be suitable for simulation of the experimental conditions similar to those used by Lin et al.1 In this paper the simulated hydrodynamic conditions are also the same as those mentioned above. Therefore, the Benyahia drag model is used in the simulation, and only the effects of restitution coefficient and superficial gas velocity were investigated in this work. 3.1. Hydrodynamic Behavior of Reactive Flow. The experimental data (Lin et al.1) and simulation results (Wang et al.12) indicated that there exist two counter-rotating toroidal vortices for solid particles under certain operating conditions in the bubbling fluidized bed. The simulation also showed that different values of the restitution coefficient can change the rotating direction of the vortices while keeping the other conditions the same. However, in the reactive flow, the temperature and pressure are usually much higher than in cold flow, and the compressibility of the gas phase is more important. These differences may change the solid flow patterns. Figure 1 is the simulated solid phase velocity profiles for bubbling fluidized bed reactors with the reactions of the SE-SMR process under conditions of 848 K and compressible gas. There is no clear dominant vortex with either smaller e value or larger e value. The simulation results of the solid flow patterns under other superficial gas velocities (from 0.32 to 0.89 m/s as used by Lin et al.1) have the same behavior. The simulation results in Figure 1 also showed that the bed expansion increased largely under the conditions of reactive flow, and increased more with large e values. Figure 2 is the relationship between bed expansion and superficial gas flow rate. It can be 8432
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Figure 3. Variation of solid fraction (averaged over the whole bed) with time under different gas flow rates (þ, 0.32 m/s; 0, 0.45 m/s; , 0.64 m/s; O, 0.89 m/s) and restitution coefficients for reactive flow (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 1. Velocity profiles of solid particles (u0 = 0.64 m/s, T = 848 K, rsc = 5:1, p = 1 bar).
Figure 4. Time-averaged solid fraction under different gas flow rates and restitution coefficients for reactive flow (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 2. Relationship between bed expansion and gas flow rate for reactive and nonreactive flows (T = 848 K, rsc = 5:1, p = 1 bar).
concluded from Figures 1 and 2 that the solid flow pattern is not sensitive to the e value and the bed expansion is sensitive to the e value in the reactive flows. Such a conclusion is opposite that for nonreactive flows as indicated by Lindborg et al.26 and Wang et al.12 3.2. Solid Fraction and Its Mean Square Root. The solid fraction in the bed is related directly to the bed expansion. Usually the solid fraction will decrease with the increasing bed expansion. Figure 3 shows such property of the solid fraction. The larger the gas flow rate and restitution coefficient are, the smaller the solid fraction is. After the startup period of about 10 s, the bed will become more stable in spite of a small fluctuation of the solid fraction as shown in Figure 3. The time-averaged solid fraction started from 10 s was calculated for different superficial gas velocities and restitution coefficients and is shown in Figure 4. The mean square root (MSR) of the solid fraction over the whole bed can express the uniformity of distribution of solid particles in the bed. A smaller MSR of solid fraction means the bed is more uniform. Figure 5 shows the influence of gas flow rate and restitution
coefficient on the MSR of solid fraction. The values of MSR of solid fraction are distinctly divided into two groups: one with larger values for e = 0.95 and another with smaller values for e = 0.9995. That means a smaller e value corresponds to the less uniform solid distribution, and larger e values correspond to the more uniform solid distribution. The time-averaged MSR of solid fraction started from 10 s (Figure 6) manifests two approximately horizontal lines for two e values as the gas flow rate changes. The gas flow rate has little influence on the MSR of the solid fraction. The bed uniformity changed mainly with the restitution coefficient, and kept similar values for different gas flow rates simulated for the same e value. Some conclusions were obtained from the relationships between the solid fraction or MSR values and the gas flow rate or restitution coefficient for reactive flows. A higher gas flow rate made a higher bed expansion. A larger e value also made a higher bed expansion and smaller solid fraction, which is different from its influence in a circulating fluidized bed.12 However, the gas flow rate has stronger effects on the bed expansion than the restitution coefficient has. Contrarily, the restitution coefficient has stronger effects on the bed uniformity than the gas flow rate has. 3.3. Concentrations of Methane and Hydrogen. The variations of outlet concentrations of methane and hydrogen in the 8433
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Figure 5. Variation of mean square root of solid fraction with time under different gas flow rates (þ, 0.32 m/s; 0, 0.45 m/s; , 0.64 m/s; O, 0.89 m/s) and restitution coefficients for reactive flow (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 6. Time-averaged mean square root of solid fraction over whole bed under different gas flow rates and restitution coefficients for reactive flow (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 7. Outlet methane concentrations in gas phase under different gas flow rates and e values (T = 848 K, rsc = 5:1, p = 1 bar).
gas phase with time for different gas flow rates and restitution coefficient values are shown in Figures 7 and 8. In order to make the figures look more clear, only the data for the smallest and the largest values of the gas flow rate are drawn out. The data for the
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Figure 8. Outlet hydrogen concentrations in gas phase under different gas flow rates and e values (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 9. Averaged outlet methane concentrations in gas phase under different gas flow rates and e values (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 10. Averaged outlet hydrogen concentrations in gas phase under different gas flow rates and e values (T = 848 K, rsc = 5:1, p = 1 bar).
gas flow rates of 0.45 and 0.64 m/s would actually lie in between. These situations can be seen from Figures 9 and 10. The gas flow rate and restitution coefficient affect the outlet methane and hydrogen concentrations in a similar way as they affect the bed expansion; i.e., the gas flow rate has stronger effects than the restitution coefficient has. However, the difference is that 8434
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Figure 11. Outlet CO2 concentrations in gas phase under different gas flow rates and e values (T = 848 K, rsc = 5:1, p = 1 bar).
Figure 12. Hydrogen concentrations in sorbents under different gas flow rates and e values (T = 848 K, rsc = 5:1, p = 1 bar).
the effect of the restitution coefficient increased with increasing gas flow rate. Therefore, the two lines for different e values in Figures 9 and 10 are emanative from smaller gas flow rate to larger gas flow rate, while the two lines in Figure 4 for the solid fraction are parallel, which means the effect of the restitution coefficient on the solid fraction is around the same for different values of the gas flow rate. Increasing the gas flow rate, the outlet methane concentration increased and the outlet hydrogen concentration decreased. This is probably due to the short bed height (less than 40 cm), which cannot provide a long enough residence time for reactants to complete the reactions. When the residence time for reactants is long enough, the reactions of SE-SMR are very close to equilibrium. The reactions will be independent of the gas flow rate, such as the situations simulated by Wang et al.46 Lower methane concentration and higher hydrogen concentration were obtained with the larger e value. The reason may be that the solid distribution in the bed is more uniform with the larger e value, which is in favor of the reactions. 3.4. CO2 Concentration in Gas Phase and Solid Phase. Figure 11 is the time variation of outlet concentration of CO2 in the gas phase. The effect of the gas flow rate was the same as that on the methane concentration as shown in Figure 7. However, the effect of the restitution coefficient is uncertain. Actually, though the CO2 concentration varied with the gas flow rate, its absolute value for all gas flow rates involved is in the range (36) 103, which is near the inlet CO2 concentration of
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1 104. The inlet concentration for all the components except for the reactants is given a small value to make the simulation go on smoothly. This concentration is negligible compared with the reactant and product concentrations. In this case, the CO2 produced by the SMR reactions is considered to be adsorbed completely by the sorbents. The reaction rate of CO2 adsorption is fast enough for the simulated conditions. Figure 12 shows that the CO2 adsorption rate is constant for all values of the gas flow rate and restitution coefficient. The effect of the restitution coefficient on the CO2 concentration in the sorbent is the same as that on methane and hydrogen concentration (Figure 12). The effect increased with an increasing gas flow rate. At lower gas flow rates such as 0.32 m/s, the restitution coefficient has no effect on the CO2 adsorption by sorbent. As a whole, the effect of the restitution coefficient on the CO2 adsorption in sorbent is small since the adsorption rate is sufficiently fast. Larger e values correspond to the more uniform solid distributions. Such a bed state will benefit the reactions. However, the effect on the faster reactions will be smaller than that on the slower reactions.
4. CONCLUSIONS The solid flow patterns of reactive flows in the simulation of the bubbling fluidized bed are different from those of the cold flow. The gas flow rate and restitution coefficient will affect the solid flow in different ways. The gas flow rate mainly affects the bed expansion of the bubbling fluidized bed. Higher gas flow rates make the bed expansion higher. A larger restitution coefficient makes the bed expansion higher, but to a smaller degree. The restitution coefficient mainly affects the uniformity of the bed. A larger restitution coefficient corresponds to a more uniform state of the bed. The reactions are also affected by the gas flow rate and restitution coefficient. Under the simulated conditions, the CO2 adsorption is fast enough to be completed, and the SMR reactions are kept away from equilibrium. When the reaction is not close to equilibrium, a higher gas flow rate will decrease the degree of completeness of the reaction. A larger restitution coefficient will increase the progress of the reaction. A uniform state of the solid bed is in favor of the reactions, and the resulting effects on the slower reactions are stronger than that on the faster reactions. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The postdoctoral fellowship (Y.W.) financed through the RENERGI program (Hydrogen Production by Sorbent Enhanced Reforming) of the Norwegian Research Council and the Ph.D. fellowship (Z.C.) financed through the GASSMAKS program (Advanced Reactor Modeling and Simulation) are greatly appreciated. ’ NOMENCLATURE Latin Symbols
CD = drag coefficient for a particle Cp,k = heat capacity of phase k CsCO2 = CO2 fraction in sorbent dd, dp = particle diameter dt = reactor diameter 8435
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Industrial & Engineering Chemistry Research Dji = binary diffusion coefficient Dk,j = diffusion coefficient for component j in phase k e = coefficient of restitution F = dimensionless drag coefficient g = gravity g0 = radial distribution function hcd = interfacial heat transfer coefficient HRi = reaction enthalpy for reaction i Ht = height of reactor k = rate coefficient kk = thermal conductivity of phase k K1, K2, K3 = equilibrium constants KY = adsorption constants for component Y (Y = CH4, CO2, H2, H2O) L = bed height M = mole mass Qik = interfacial heat transfer to phase k n = reaction order of CO2 adsorption p = pressure pd = particle pressure Pr = Prandtl number r = radial coordinate rsc = steam-to-carbon ratio Re = Reynolds number Rep = particle Reynolds number Ri = reaction rate of reaction i Rj = formation rate of component j Rs = energy source S = specific surface area of sorbent Sh = Sherwod number t = time T = temperature u0 = superficial inlet gas velocity umf = minimum fluidization velocity X = conversion ydry = dry mole fraction in gas phase z = axial coordinate Cd = peculiar velocity _FD = drag force I = unit tensor Mk = interfacial momentum transfer of phase k S = deformation rate tensor of phase k vc0 = fluid velocity fluctuation vk = velocity of phase k BFB = bubbling fluidized bed CFB = circulating fluidized bed Greek Symbols
rk = volume fraction of phase k β = interfacial drag coefficient γ = collisional energy dissipation φ = angle of internal friction ε = convergence criterion ζd = solid phase bulk viscosity θ = azimuthal angle kd = conductivity of granular temperature μddilute = dilute viscosity μk = viscosity of phase k Fk = density of phase k σ = mean square error hτk = stress tensor of phase k ω = mass fraction
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Γ = averaged interfacial mass flux Θ = granular temperature Superscripts
dilute = dilute c = collisional eff = effective f = frictional i = interfacial k = kinetic m = molecular max = maximum min = minimum SMR = reactions of steam methane reforming SP = reaction of CO2 sorption by sorbent Subscripts
0 = initial a = adsorption c = continuous phase d = dispersed phase eq = equilibrium i = reaction number j = component number k = phase (k = c, d) mf = minimum fluidization p = particle
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