Simulation of the Chemical Looping Reforming Process in the Fuel

Article pubs.acs.org/EF

Simulation of the Chemical Looping Reforming Process in the Fuel Reactor with a Bubble-Based Energy Minimization Multiscale Model Shuai Wang, Huilin Lu,* Dan Li, and Yanjia Tang School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China ABSTRACT: Chemical looping reforming (CLR) is a novel hydrogen production technology, which has attracted great attention, owing to the benefits of lower cost and better concept of environmental protection. To investigate the CLR performance in the fuel reactor (FR), this study develops a multiphase computational fluid dynamics (CFD) reactive model to obtain a better understanding of complex gas−solid flow behavior and reactive characteristic during the CLR process. A bubblebased energy minimization multiscale (EMMS) approach is applied to account for the bubble effect on the gas−solid interaction. A two-fluid model is adopted with a kinetic theory of granular flow for closure. The distributions of concentrations of particles and gas species are obtained in the FR with Ni-based oxygen carriers by means of numerical simulation. This bubble-based EMMS model gives a more reasonable agreement with experimental results by a comparison of gas compositions at the outlet of the reactor.

1. INTRODUCTION Hydrogen has been widely used as an environmental clean energy for power generation because of no emission besides water during the combustion process. The current method for hydrogen production is mainly through catalytic reforming of methane or natural gas, although this approach leads to large amounts of CO2 emissions.1 Chemical looping reforming (CLR), as an extension of chemical looping combustion (CLC), plays an extremely important role in hydrogen production from natural gas, which has attracted increasing attention. This concept of CLR eliminates the expensive cost for air separation because of the fact that metal oxide carriers are used for the oxygen transfer between two reactors. For the CLR, the air/fuel ratio is controlled at a lower level to avoid fuel complete conversion. The required heat for reactions can be supplied without additional oxygen production, which prevents direct contact of air with fuel gas and controls CO2 emissions during the heat production process.2 The basic principle for the CLR process is illustrated in Figure 1.3 In the fuel reactor (FR), fuel gases are converted to synthesis gas via

partial oxidation of metal oxygen carriers and then reduced oxygen carriers are delivered to the air reactor (AR), where the reoxidized process is completed. In this way, the products are prevented from the N2 dilution. There are a great amount of studies on the performance of oxygen carriers for CLR.4,5 Zafar et al.6 examined different oxygen carriers in a fluidized bed and concluded that NiO appeared the most promising active metal for CLR, owing to its high reactivity and strong catalytic properties. Two metal oxides based on ilmenite and nickel were tested for raw gas derived from biomass gasification in the CLR reactor, and the result showed that the ilmenite catalyst was less active than the NiO/ Al2O3 catalyst.7 Ryden et al.8 preformed the CLR process in the circulating fluidized bed and found that there was a significant solid carbon formation in the FR when fuel was only natural gas. He et al.9 evaluated the reactivity of the perovskite-type oxygen carrier during the CLR process, and the results indicated that the synthesized oxygen carriers had a good regenerability for CLR. With the development of computer technology, numerical simulation has become one of the most promising means for understanding and evaluating the performance of complex gas− solid flow and chemical reactions.10,11 Xiaojia et al.12 developed a multiphase computational fluid dynamics (CFD) model with heterogeneous reactions to investigate the CLC process in the FR and analyzed effects of operating parameters on the fuel conversion. Brahimi et al.13 performed a simulation on the performance of CLC in a continuous bubbling fluidized bed and discussed proper operating conditions for the fuel complete combustion. A one-dimensional heterogeneous dynamic mathematical model in the fixed bed was applied to investigate the sorption-enhanced methane steam reforming performReceived: May 12, 2013 Revised: July 2, 2013 Published: July 2, 2013

Figure 1. Schematic diagram of CLR. © 2013 American Chemical Society

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dx.doi.org/10.1021/ef401101p | Energy Fuels 2013, 27, 5008−5015

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Table 1. Governing Equations and Constitutive Correlations in the Model 1. continuity equations

∂ (εgρg ) + ∇(εgρg ug) = ṁ g ∂t

(T1-1)

∂ (εsρs ) + ∇(εsρs us) = ṁ s ∂t 2. momentum conservation equations

(T1-2)

∂ (εgρg ug) + ∇(εgρg ugug) = − εg∇p + εg∇τg + εgρg g − β(ug − us) + ṁ g ug ∂t

(T1-3)

∂ (εsρs us) + ∇(εsρs usus) ∂t = − εs∇p − ∇ps + εs∇τs + εsρs g + β(ug − us) + ṁ sus

(T1-4)

3. species transport equations

⎤ ⎡ ⎛ μ⎞ ∂ (εgρg Yg, j) + ∇(εgρg ugYg, j) = ∇⎢εg ⎜ρg Dj + t ⎟∇Yg, j ⎥ + ṁ g, j ⎥⎦ ⎢⎣ ⎝ Sc ⎠ ∂t

4. energy conservation equations

∂ (εgρg Hg) + ∇(εgρg ugHg) = ∇(λg ∇Tg) + Q gs + ṁ g Hg ∂t ∂ (εsρs Hs) + ∇(εsρs usHs) = ∇(λs∇Ts) + Q sg + ṁ sHs ∂t

5. conservation equation of granular temperature

(T1-6) (T1-7)

⎤ 3⎡ ∂ ) + ∇(εsρθ )us⎥ ⎢ (εsρθ s s ⎦ 2 ⎣ ∂t = (−∇ps I + τs): ∇us + ∇(ks∇θ) − γs − 3βθ + Dgs

6. stress tensor

(T1-5)

(T1-8)

2 τg = μg [∇ug + (∇ug)T ] − (∇ug)I 3

{ } (T1-9) 2 τ = μ {[∇u + (∇u ) ] − (∇u )I} + ξ ∇u I (T1-10) 3 T

s

s

s

s

s

s

s

7. solid pressure

ps = εsρθ + 2ρs (1 + e)εs 2g0θ s

8. solid shear viscosity and bulk viscosity

10ρs ds πθ ⎡ ⎤2 4 4 θ μs = εs 2ρs dsg0(1 + e) + ⎢⎣1 + g0εs(1 + e)⎥⎦ 5 π 96(1 + e)εsg0 5

(T1-11)

ξs =

⎛θ⎞ 4 2 εs ρ dsg (1 + e)⎜ ⎟ ⎝π ⎠ 3 s 0

ks =

25ρs ds πθ ⎡ ⎤2 ⎛ θ ⎞1/2 6 2 ⎢⎣1 + (1 + e)g0εs⎥⎦ + 2εs ρs dsg0(1 + e)⎝⎜ ⎠⎟ π 64(1 + e)g0 5

(T1-12)

1/2

9. thermal conductivity of particles

10. dissipation of fluctuation kinetic energy

⎛4 γs = 3(1 − e 2)εs 2ρs g0θ ⎜ ⎝ ds

11. rate of energy dissipation per unit volume

(T1-13)

⎞ θ − ∇us⎟ π ⎠

dsρs ⎛ 18μg ⎞ ⎜⎜ ⎟⎟ |ug − us|2 4 πθ g0 ⎝ ds 2ρs ⎠

(T1-14)

(T1-15)

2

Dgs =

ance.14 However, few studies focus on the oxygen carrier behaviors for the CLR process by means of the CFD method. For the FR in the form of a bubbling fluidized bed, the bubble phase is regarded as a mesoscale structure and similar to clusters in circulating fluidized beds. The effect of this mesoscale structure is of great importance on CFD simulations, which need to be considered in the model.15,16 Shi et al.17 extended the original energy minimization multiscale (EMMS) method, which was limited in the circulating fluidized-bed field18 and developed a bubble-based EMMS model to consider the mesoscale structure. This model was evaluated by comparisons to experimental data and empirical relations and agreed reasonably with the available data. The goal of this study is to investigate the behavior of Nibased oxygen carriers in the FR in the form of a bubble fluidized bed during the CLR process. The distributions of flow variables and concentrations of gas species are obtained by means of CFD simulation. The bubble-based EMMS drag model is used to account for the effect of bubbles in the FR.

(T1-16)

2. MATHEMATICAL MODEL The model in this study is mainly assumed as follows: (1) The particle size is uniform, and the particle shape is spherical. (2) The overall reaction rate is solely controlled by a chemical reaction. A two-fluid model is adopted, where solid phases are modeled as a continuum fluid and the kinetic theory of granular flow is used to close this model.19 2.1. Hydrodynamic Model. Mass conservation equations of gas and particle phases are described by eqs T1-1 and T1-2, which are listed in Table 1. Momentum balance equations of both phases are given by eqs T1-3 and T1-4, where τ represents the stress tensor and β is the gas−solid drag coefficient. The transport equation for gas species is expressed as eq T1-5, where Dj represents the jth species diffusion coefficient. The energy balance equations are given by eqs T1-6 and T1-7, where the gas−solid heat exchange Q takes the following term: Q sg = − Q gs = hsg (Ts − Tg)

(1)

The heat-transfer coefficient hsg is expressed as follows:

hsg = 5009

6kgεsεgNu ds 2

(2) dx.doi.org/10.1021/ef401101p | Energy Fuels 2013, 27, 5008−5015

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Table 2. Model for the Bubble-Based EMMS Drag Coefficient 1. balance equations

εg = δ b + (1 − δ b)εe

(T2-1)

Ug = δ bUb + (1 − δ b)Uge

Us = (1 − δ b)Upe

(T2-3)

ε Use = Uge − Upe e 1 − εe

2. superficial slip velocity

(T2-2)

(T2-4)

Usb = (Ub − Ue)(1 − δ b)

(T2-5)

⎡ ⎤ (1 − εe)2 μg 7 (1 − εe)ρg Use ⎥ Use ⎢150 + = (1 − εe)(ρs − ρg )(g + ae) 2 ⎢⎣ ⎥⎦ εe 2 4 εeds εeds

3. equation for particles force balance in the emulsion phase

4. equation for force balance on the bubble phase

(T2-6)

2

πd b 1 π Cdb ρe Usb2 = db3(ρe − ρg )(g + ab) 4 2 6

5. accelerations

ab − ae =

C b(1 − εe)δ bρe

⎛ 1 + 2δ b ⎞ C b = 0.5⎜ ⎟ ⎝ 1 − δb ⎠ σ2 = 6. stability criterion by minimization of the energy dissipation by drag force

(T2-7)

σ 2(ρs − ρg )g

(T2-8)

(T2-9) (1 − εg)2 εg 4

1 + 4(1 − εg) + 4(1 − εg)2 − 4(1 − εg)3 + (1 − εg)4 ρg U 3 se Cde Uge + fb Ug(g + ab) → min ρs ds 4

(T2-10)

2

Ns =

(T2-11)

where Nu is calculated as eq 3.20

Nu = (7 − 10εg + 5εg 2)(1 + 0.7Re 0.2Pr1/3) + (1.33 − 2.4εg + 1.2εg 2)Re 0.7Pr1/3

(3)

Analogous to the thermodynamic temperature, the granular temperature is introduced to describe the solid fluctuating energy, which is defined as θ = ⟨C2⟩/3. The granular temperature transport equation is expressed as eq T1-8. To close the above governing equations, constitutive equations are also listed in Table 1. The fluid- and solidphase stress tensors are calculated as eqs T1-9 and T1-10, respectively. The pressure and viscosity of the solid phase are given by eqs T1-11 and T1-12, respectively.21 The thermal conductivity of particles is expressed as eq T1-14. The fluctuation kinetic energy dissipation rates resulting from collisions of particles and because of the transfer of the interphase fluctuations are predicted by eqs T1-15 and T1-16, respectively.22 2.2. Bubble-Based Drag Model. In the momentum conservation equations, the gas−solid drag interaction is the most important component. The EMMS method is widely applied to the calculation of the heterogeneous drag coefficient.23,24 In this study, the drag coefficient model with consideration of bubble effects is adopted17 and expressed as

βbubble − EMMS =

εg 2Fgs Uslip

=

εg 2 Uslip

[(1 − δ b)neFde + δ bnbFdb]

(4)

These structure parameters are solved through a set of nonlinear equations under the specified operating conditions and are shown in Table 2. Detailed formulas and parameters in this model can be found in the study by Shi et al.17 The code is implemented on the basis of the Visual C++6.0. The process of the program is displayed in Figure 2. To solve the nonlinear equations in this model, a revised Aitken iteration method is selected because of fast convergence and simple implementation.25 With consideration of the heterogeneity in the FR, the heterogeneous index Hd takes the following form:

Figure 2. Procedure of the bubble-based EMMS drag force model.

Hd =

βbubble − EMMS βWen − Yu

(5)

where βWen−Yu is expressed as follows: βWen − Yu = 5010

26

3 ρg (1 − εg)εg|ug − us| −2.65 Cd εg 4 ds

(6)


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The profile of heterogeneous index Hd as a function of voidage at different gas velocities is displayed in Figure 3. The particle is 2500 kg/

Table 3. Chemical Kinetic Parameters for the Ni-Based Oxygen Carrier CH4

H2

CO

0.2 5 0.2

0.15 5 0.4

0.059 5 0.6

k0 (mol1−n m3n−2 s−1) E0 (kJ/mol) n

With regard to the methane steam reforming reaction during the CLR process, the decompositions of hydrocarbons may form carbon and the global reaction rate will be affected. However, according to Ortiz et al.,32 the following three reactions are only considered to represent methane steam reforming:

CH4 + H 2O → 3H 2 + CO CO + H 2O → H 2 + CO2

ΔHr4 = 206.0 kJ/mol

(R4)

ΔHr5 = − 41.0 kJ/mol

(R5)

CH4 + 2H 2O → 4H 2 + CO2

(R6) Hence, in this simulation, the above three reactions are also taken into account and the corresponding reaction rate equations are given by33

Figure 3. Variation of the heterogeneous index with voidage at different gas velocities.

r1 = k1

m3 in density and 0.2 mm in diameter, belonging to Geldart B. It can be observed that Hd decreases with the increasing gas velocity, which reflects the increasing heterogeneity. Under the same operating velocity, the lowest value of Hd occurs at the minimum fluidization state. With the gas volume fraction increased, Hd shows a nearly logarithmic growth trend and tends toward homogeneity. The Wen− Yu formula is applied to the region where the homogeneity state is reached (Hd = 1). The drag coefficient model proposed by Ergun is adopted for εg < εmf.27 2

βErgun = 150

(1 − εg) μg εgds

2

+ 1.75

r3 = k 3

ds

CH4(g) + NiO(s) → Ni(s) + 2H 2(g) + CO(g) (R1)

ΔHr2 = −2.1 kJ/mol (R2)

CO(g) + NiO(s) → Ni(s) + CO2 (g) ΔHr3 = − 43.3 kJ/mol

(R3)

From the previous research, the chemical reaction seemed to be the main resistance controlling the reduction reaction rate when the particle diameter was located in the region of 0.09−0.5 mm.29 The average particle diameter of 0.2 mm is used in this simulation. Therefore, the shrinking-core model is adopted to determinate the reduction reaction rate, which is expressed as follows:30,31

dXs,r dt

= k 0 exp(− E0 /RT )Cg nfNiO−1/3

(− rNiO)r =

(8)

ρm,NiO εsYNiO dXs,r b

dt

(10)

(PCOPH2O/PH2) − (PCO2/K 2) DEN2

(11)

(PCH4PH2O2/PH2 3.5) − (PCO2PH2 0.5/K1K 2) DEN2

(12)

(13) where the detailed correlations and constants can be found in the study by Xu and Froment.33 2.4. Initial and Boundary Conditions. The FR with a bed diameter of 0.052 m and reactor height of 0.25 m is selected in this work, which is shown in Figure 4. The main physical properties and operating parameters for the simulation can be found in Table 4. The above equations are solved using a higher order scheme, which is called the total variation diminishing (TVD) method. The simulation is performed using a K-FIX CFD code, which was applied to the simulation of the circulating fluidized bed in advance and allowed a modification and implement of governing equations and boundary conditions. The maximum convergence residual of 10−3 for the simulation is set, and the time step varies between 10−4 and 10−6 s. The distributions of flow variables are time-averaged from 10 to 15 s when the steady state is reached. The dependency of mesh is known to be extremely significant to eliminate numerical errors. Hence, a mesh sensitivity was previously evaluated with domains containing 2600, 5200, and 10 400 cells, respectively. The less refined mesh leads to a slight difference in the mean axial concentration of particles, which is not obvious. Therefore, a domain containing 5200 grids is selected, where the grid size is 2 × 1.25 mm. Initially, the metal oxygen carriers are loaded in the reactor at the bed height of 0.1 m and solid concentration of 0.5. The inlet velocity of fuel gas is set at the bottom of the reactor. Solid oxygen carriers are fed to the reactor from the bottom of the right wall with a constant mass flow rate. The solid mass outlet is set at the height of 0.08 m from the bottom. The atmospheric pressure outlet is set at the top of the FR. An adiabatic and no slip wall condition is applied.

2.3. Reaction Kinetics Models. According to the experiment by de Diego et al.,28 the reduction reactions using a Ni-based oxygen carrier were carried out in the FR for CLR. The reducing reactions in the FR are considered as follows:

H 2(g) + NiO(s) → Ni(s) + H 2O(g)

DEN2

DEN = 1 + K CH4PCH4 + K H2PH2 + K COPCO + K H2OPH2O/PH2

(7)


(PCH4PH2O/PH2 2.5) − (PCOPH2 0.5/K1)

r2 = k 2

ρg (1 − εg)|ug − us|

εg < εmf


3. RESULTS AND DISCUSSION To evaluate the applicability of this model, a comparison of the concentration of particles along the radial direction with the measured data is shown in Figure 5. The detailed experimental operation parameters can be found in the study by Taghipour

(9)

where C, n, and f NiO represent the concentration of the gas reactant, the reaction order, and the fraction of NiO, respectively, k0 is the preexponential factor, E0 is the activation energy, and ρm is the molar density. The kinetic parameters in detail are given in Table 3. 5011


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et al.34 It can be distinguished that the prediction by the bubble-based EMMS model is in more reasonable agreement with the experimental result. With consideration of the mesoscale bubble phase effect on the drag force, the bubblebased EMMS model makes a reduction of the drag force, which results in an increase of the concentration of particles. While the Ergun/Wen−Yu drag model agrees with the measured value near the center region, this model underestimated the overall solid concentration. To further verify the reaction model, the outlet gas concentrations predicted by the simulation are compared to measured values by de Diego et al.,28 which are displayed in Figure 6. It can be found that the model can capture a

Figure 4. Schematic and grids of the FR.

Table 4. System Properties and Parameters Used in the Simulation parameter

unit

value

reactor height reactor diameter NiO content particle diameter particle density restitution coefficient wall restitution coefficient specularity coefficient initial static bed height initial concentration of particles initial temperature inlet gas velocity minimum fluidizing velocity voidage at minimum fluidization

m m % μm kg/m3

0.25 0.052 18.0 200 2500 0.9 0.9 0.5 0.1 0.5 1173 0.1 0.011 0.4

m K m/s m/s

Figure 6. Comparisons of gas concentrations at the outlet between predictions and experimental data.

reasonable gas concentration close to the experiment data. The result obtained by means of simulation gives a lower value on the steam concentration than the experimental data. Other gas concentrations are a bit overpredicted. This may be due to the fact that reactions for carbon formation are neglected. However, the maximum relative error is in the acceptable range. The predictions by the Ergun/Wen−Yu drag model are also shown in Figure 6. We can observe that the results from the bubble-based EMMS model to take the mesoscale effect on the drag force into account are closer to the measured results than that obtained by the Ergun/Wen−Yu drag model. Figure 7 displays the contour plots of solid volume fractions in the FR. The low concentration of particles reflects a high gas volume fraction. The coexistence and mixing of the bubble and

Figure 5. Comparisons of the concentration of particles experimentally obtained and predicted by the model.

Figure 7. Instantaneous concentration of particles in the FR. 5012


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emulsion phases can be clearly distinguished in the reactor. With the bubble formation at the bottom distributor, coalescence, passing through the bed and bursting at the bed surface, an intensive interaction and oscillation between particles and the gas phase is formed. The instantaneous solid velocity is also exhibited in Figure 7. There is a downward flow of particles close to the wall because of the wall effects. At the bed surface, the particles are driven by the gravity and fall down. We can find that there is a vortex field formation as a result of the combined effects of gravity and drag force, which offers an energy to maintain a mixing of fuel gas and oxygen carriers. Figure 8 shows the instantaneous gas concentrations in the quasi-steady state at 12 s. At the bottom zone of the FR, the Figure 9. Profile of solid velocity and concentration of particles at different heights of the FR.

Figure 8. Instantaneous molar fractions of gas species at 12.0 s. Figure 10. Radial distribution of gas components at the height of 0.075 m.

CH4 concentration displays a higher value. With the reaction in progress, the concentrations of H2 and CO become increased along the height, which are consumed by the oxygen carriers simultaneously and produce CO2 and H2O. During the reactions, the upward flowing gas in the form of bubble phases leads to fuel gas within the bubbles not mixing with the solid oxygen carrier, which decreases fuel conversion. Hence, the local higher concentration of CO2 can be observed near the wall. Above the bed surface, concentrations of particles approach zero and the change of the gas concentration is not obvious. The lateral profiles of time-averaged solid velocity and concentration of particles are outlined in Figure 9. It can be observed that the concentration of particles shows a dilute zone in the middle of the reactor and a denser region close to the wall. The profile of the solid velocity at three different heights shows a similar trend. However, the difference among them can be distinguished. At the lower height of 0.05 m, the distribution of the solid velocity displays asymmetry. Accordingly, this phenomenon occurs at the profile of the solid volume fraction. This may be attributed to the location of the mass inlet of particles. Because of the existing bubble phase, the profile of the solid volume fraction along the lateral direction is not uniform. This is consistent with that shown in Figure 7. With the height increased, the concentration of the particle becomes flatter in the center. Figure 10 demonstrates the time-averaged gas concentrations along the lateral direction. As seen, the concentrations of H2O and CO2 show a low value in the center zone and an increase

toward the wall, which is similar to the distribution of the solid volume fraction along the lateral direction. Because of the fact that a higher concentration of particles near the wall region enhances the reaction rate, there is an increase in the amount of gas products. The lateral distribution of other gas species shows a reverse trend. The non-uniform gas composition has a great influence on fuel conversion. Therefore, to control the residence time reasonably is a key to improve fuel conversion and operate in the CLR process. The axial profile of gas composition in the FR is shown in Figure 11. There is an obvious difference among different gas compositions at the bottom dense region. The profile of reactant CH4 decreases with the bed height increased in the dense region, while the reverse trend is observed for CO and H2. About at the height of 0.05 m from the bottom of the bed, CO and H2 concentrations decrease slightly because of reactions R2 and R3. The concentration of H2O decreases sharply at the bottom because of methane steam reforming and increases because of reaction R2, with the bed height increased and a great amount of CH4 consumed. For the gas concentration of the freeboard region, there is no obvious variation.

4. CONCLUSION A multiphase CFD model with consideration of chemical reactions is developed for the CLR process. Simulations are carried out with Ni-based oxygen carriers in the FR to 5013


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ṁ = mass source term (kg m−3 s−1) n = reaction order ne = number of particles in the emulsion phase per unit volume nb = number of bubbles per unit volume Nu = Nusselt number Ns = energy dissipation (W kg−1) p = fluid pressure (Pa) ps = particle pressure (Pa) r = reaction rate (mol m−3 s−1) R = universal gas constant (J mol−1 K−1) Re = Reynolds number Sc = turbulent Schmidt number T = temperature (K) u = velocity (m s−1) U = superficial velocity (m s−1) Umf = minimum fluidizing gas velocity (m s−1) Use = superficial slip velocity in emulsion phase (m s−1) Usb = superficial slip velocity between bubble and emulsion (m s−1) X = conversion rate Yi = mole fractions of gas species

Figure 11. Axial distribution of molar fractions of gas species in the FR.

investigate gas−solid flow characteristics and gas species distribution in the reactor during the CLR process. A multiscale drag coefficient model is incorporated to account for dynamic effects of bubble phases, which is based on the EMMS model. The predictions show that a heterogeneous drag model gives a more reasonable agreement with experimental data by a comparison of concentrations of gas species at the outlet. Multiscale mass and heat transfers during the reactions because of the coexistence of the emulsion and bubble phases are required to be considered in the future.

■

Greek Letters

β = drag coefficient (kg m−3 s−1) γ = collisional energy dissipation (J m−3 s−1) ε = volume fraction θ = granular temperature (m2 s−2) λ = thermal conductivity (W m−2 K−1) μ = viscosity (Pa s) ξ = bulk viscosity (Pa s) ρ = density (kg m−3) ρm = molar density of the reacting material (mol m−3) τ = stress tensor (Pa) δ = bubble holdup

AUTHOR INFORMATION

Corresponding Author

*Telephone: +0451-8641-2258. Fax: +0451-8622-1048. E-mail: [email protected].

Subscripts

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was financially supported by the project supported by the National Natural Science Foundation of China (51176042) and (21276056).

■

■

NOMENCLATURE a = acceleration (m s−2) b = stoichiometric factor CD = drag coefficient of a single particle d = particle diameter (m) db = bubble diameter (m) D = diffusivity (m2 s−1) e = restitution coefficient f b = ratio of gas in the bubble phase to that in total E = activation energy (kJ mol−1) F = force acting on each particle or cluster (N) Fde = drag force in the emulsion phase per unit volume Fdb = drag force acting on the bubble per unit volume g0 = radial distribution function g = gravity (m s−2) h = heat-transfer coefficient (W m−2 K−1) H = specific enthalpy (J kg−1) Ji = diffusion flux of species i (kg m−2 s−1) k0 = pre-exponential factor (mol1−n m3n−2 s−1) ks = conductivity of fluctuating energy (kg m−1 s−1)

b = bubble phase e = emulsion phase g = gas phase p = particle s = solid phase w = wall

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