Comparative CFD Analysis of Heterogeneous Gas–Solid Flow in a

Article pubs.acs.org/IECR

Comparative CFD Analysis of Heterogeneous Gas−Solid Flow in a Countercurrent Downer Reactor Zhan Shu,†,‡,§ Guo Peng,†,‡,§ Junwu Wang,*,† Nan Zhang,† Songgeng Li,† and Weigang Lin† †

State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, P. R. China ‡ University of Chinese Academy of Sciences, Beijing, 100490, P. R. China ABSTRACT: It has been widely recognized that input parameters affected the results of Eulerian simulation of gas−solid flow. In this study, a comparative study was performed to study the effect of input parameters on the hydrodynamics of gas−solid flow in a countercurrent downer reactor. It was concluded that the input parameters of the Eulerian model, including restitution coefficients, specularity coefficient, and drag correlation, have a major effect on the bed hydrodynamics, in accordance with previous studies on other types of gas-fluidized beds. More importantly, we show that it is possible to match the experimental data either by changing the wall boundary conditions or by correcting the interphase drag force. Therefore, there is no unique way to tune the input parameters to fit the experimental data. This study indicated that researchers have to pay great caution when selecting input parameters and highlighted the requirement of stringent experimental tests of the state-of-the-art Eulerian models.

1. INTRODUCTION

values of which are unfortunately very difficult to determine, especially the specularity coefficient.30 In this study, we show that the hydrodynamics of the countercurrent downer are also sensitive to the restitution coefficient, the wall boundary condition, and the drag correlations used. Specifically, we show that it is possible to match the experimental data either by changing the wall boundary conditions or by correcting the interphase drag force. Therefore, there is no unique way of tuning the input parameters to fit the experimental data, highlighting the requirement of stringent experimental tests of the state-ofthe-art Eulerian models

Gas-fluidized beds are widely used in industry, which can be operated in different modes, such as cocurrent up-flow (riser), cocurrent down-flow (downer), and countercurrent down-flow (downer).1 Compared to cocurrent up-flow and cocurrent down-flow modes, studies on the countercurrent gas−solid downer are sparse, although it has the potential to combine the merits of cocurrent riser and cocurrent downer, that is, high solid holdup in the bed with small solid backmixing and short contact time,2 and has been used in a number of applications, such as the high efficient utilization of coal and biomass.3,4 In addition to the experimental studies on the hydrodynamics of countercurrent gas−solid flow in downers,2,3,5−7 Peng et al.8 have recently performed a preliminary computational fluid dynamics (CFD) study on the hydrodynamics of countercurrent gas−solid flow in a two-dimensional downer, although there are extensive CFD studies on cocurrent downer.9−18 It was shown that a Eulerian−Eulerian model with the proposed empirical interphase drag force correlation has the ability to correctly capture the main features of countercurrent gas−solid flow. It is well-known that input parameters have an important impact on the results of CFD simulations of gas−solid flow, such as the drag correlations, wall boundary condition, and parameters for particle−particle collisions (restitution coefficient). It has long been recognized that proper determination of interphase drag force is crucial for a successful simulation of heterogeneous gas−solid flow19−21 and that the restitution coefficient in kinetic theories has a significant effect on the predicted hydrodynamics of CFB risers.22−25 Furthermore, extensive numerical simulations have consistently shown that the hydrodynamics of gas−solid flow are sensitive to wall boundary conditions,22,26−28 where the Johnson and Jackson’s model is used.29 The model requires two input parameters, the © 2014 American Chemical Society

2. MATHEMATICAL MODEL AND SIMULATION LAYOUT The numerical method used here is a state-of-the-art Eulerian model with standard kinetic theory of granular flow31,32 and the frictional model of Schaeffer33 for particle phase and the Johnson and Jackson’s model for wall boundary conditions of the solid phase.29 Furthermore, an empirical drag correlation for interphase drag force proposed in our previous study,8 which was formulated following the idea of previous studies,34,35 is used to include the effects of particle clustering structures on the effective interphase drag force. Table 1 gives a summary of governing equations, constitutive relations, and wall boundary conditions of the solid phase. The mathematical model was solved using commercial software ANSYS Fluent. Figure 1 shows the schematic of geometry of a 3D simulated downer; a more detailed description can be found in our previous publication.8 The solid particles are fed into the Received: Revised: Accepted: Published: 3378

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Table 1. Governing Equations for Gas−Solid Flow and Constitutive Law continuity equations for gas and solid phases: ∂ (εgρg ) + ∇·(εgρg ug⃗ ) = 0 ∂t

∂ (εsρs ) + ∇·(εsρs us⃗ ) = 0 ∂t momentum equations for gas and solid phases: ∂ (εgρg ug⃗ ) + ∇·(εgρg ug⃗ ug⃗ ) = − εg∇p + ∇·(εgτg ) + εgρg g ⃗ + β(us⃗ − ug⃗ ) ∂t ∂ (εsρs us⃗ ) + ∇·(εsρs us⃗ us⃗ ) = − εs∇p − ∇ps + ∇·(εsτs ) + εsρs g ⃗ + β(ug⃗ − us⃗ ) ∂t energy conservation equation for granular temperature: ⎤ 3 ⎡ ∂(εsρs Θs) ⎢ + ∇·(εsρs us⃗ Θs)⎥ = (− ps I + εsτs ): ∇us⃗ − ∇·(εsqs) − γ − 3β Θs 2 ⎣ ∂t ⎦ stress−strain tensor for gas and solid phases 2 τg = μg (∇ug⃗ + ∇ug⃗ T ) − μg (∇·ug⃗ )I 3

⎛ 2 ⎞ τs = μs (∇us⃗ + ∇us⃗ T ) + ⎜λs − μs ⎟(∇·us⃗ )I ⎝ 3 ⎠ solid pressure

ps = εsρs Θs + 2ρs (1 + e)εs2g0Θs + Fr

(εs − εs,min)n (εs,max − εs) p

with Fr = 0.1εs, n = 2, p = 3, εs,min = 0.5, εs,max = 0.63 solid bulk viscosity

Θs 4 εsρ dsg (1 + e) 3 s 0 π

λs =

solid shear viscosity

5ρs d s π Θs ⎡ ⎤2 Θs 4 4 ⎢1 + g0εs(1 + e)⎥⎦ + εsρs dsg0(1 + e) 48εs(1 + e)g0 ⎣ 5 5 π

μs =

ps,friction sin φ

+

2 I2D

radial distribution function −1 ⎡ ⎛ ε ⎞1/3⎤ s ⎥ ⎢ ⎟⎟ g0 = 1 − ⎜⎜ ⎢ ⎝ εs,max ⎠ ⎥⎦ ⎣

pseudo-Fourier flux of kinetic fluctuation energy: qs = − ks∇Θs

k Θs =

150ρs d p Θsπ ⎡ ⎤2 Θs 6 2 d (1 + e)g0 ⎢⎣1 + εsg0(1 + e)⎥⎦ + 2ρε s s p π 384(1 + e)g0 5

collisional dissipation of energy

γΘ =

12(1 − e 2)g0 dp π

s

2 3/2 ρε Θ s s s

drag coefficient 3 ρg εgεs|ug⃗ − us⃗ | −2.65 εg β = CD 4 dcl where

⎧(24/Re)(1 + 0.15Re 0.687), Re < 1000 εgρg dcl|ug⃗ − us⃗ | CD = ⎨ , Re = μg Re ≥ 1000 ⎩ 0.44, ⎪

⎪

Uslip dcl = 0.5987 + 5.1128εs + 0.4573 dp Ut

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Table 1. continued wall boundary conditions of the solid phase ε π τs = − 3π ϕ s ρs g0 Θs Us,⃗ || 6 εs ,max

qs =

ε ε π π 3π ϕ s ρs g0 Θs Us,⃗ ||·Us,⃗ || − 3 s (1 − e w2)ρs g0Θ1.5 s 6 4 εs,max εs,max

interested in the solid holdup (or pressure drop) along the downer, only the pressure drop is reported. (iii) In the base case, the secularity coefficient is 0.01, the restitution coefficients for particle−particle and particle−wall interactions are 0.995 and 0.2, respectively, and the recently proposed drag coefficient is used.8 (iv) All the experimental data are obtained from the thesis of Li.7

3. SIMULATION RESULTS 3.1. 2D vs 3D. CFD simulations of gas−solid flows are computationally expensive. In order to save computational cost, early studies have used two-dimensional (2D) rectangular geometry to represent the real three-dimensional (3D) cylindrical geometry, although the real structure of gas−solid flow in fluidized beds is always three-dimensional. Many studies have been carried out to assess the reasonability of 2D approximation for real 3D structures. It was concluded that 2D simulations should be used with great caution and only for sensitivity analysis. 3D simulations are highly preferable for validating CFD models with available empirical correlations and experimental data.36 2D simulations can be successfully applied to study the hydrodynamics of fluidized beds only in cases where the spatiotemporal variations in a given direction of the physical space are negligible compared to those in other directions,36−38 such as low-velocity fluidization in bubbling fluidized beds. A later detailed study39 concluded that when mass transfer has been included, 3D simulations should be performed even in cases of bubbling fluidized beds. The difference between 2D and 3D increases with increasing inlet superficial gas velocity;37,38 therefore, for high-velocity fluidization, such as gas−solid flow in risers, it is mandatory to perform 3D simulations when quantitative comparisons with experimental data are carried out. For example, it was shown that 2D simulations significantly underpredicted the solid inventory in a square cross-sectional riser; however, much better agreement with experimental data can be achieved, simply by including the third dimension in CFD simulations.40 Figure 2 shows the CFD results of gas−solid flow in a countercurrent downer. It can be seen that there is only a minor difference between 2D and 3D results, and both 2D and 3D results are in reasonable agreement with experimental measurements. This is possibly due to the fact that the hydrodynamics in the countercurrent downer are essentially 2D, as has been discussed in the preceding paragraph. However, although the time-averaged axial pressure distribution and pressure drop profile are very similar, the transient hydrodynamics near the solid inlet are quite different between 2D and 3D simulations, as shown in Figure 3: Our previous 2D simulations8 indicated that although the geometry is fully symmetric, the granular jet at the top inlet is nonsymmetric; it shifts from left to right and vice versa, while present 3D simulations do not find this phenomenon. Note that even the gas velocity in the main gas inlet is zero; the superficial gas velocity in the downer is nonzero, because there is a superficial

Figure 1. Schematic geometry of simulated 3D downer

downer with a given solid flux; the particles are collected at the bottom of the downer and then go out of the simulated domain via the gas inlet and solid outlet at the bottom (Ug = 0.048 m/s which is slight larger than Umf), where the solid outlet is achieved by setting the inlet velocity of solid particle as a negative value. No slip-wall condition is applied to gas phase, and the gas velocity at the gas inlet is specified to make sure that the superficial gas velocity within the downer matches experimental values. The physical properties of the gas and solid and parameters used in simulations are summarized in Table 2. Table 2. Summary of Parameters Used in Numerical Simulations dp ρs μg ρg

2.17 × 10−4 m 2030 kg/m3 1.8 × 10−5 kg/(m·s) 1.225 kg/m3

φ εs,max solid flux Ug

30° 0.63 63.61 kg/(m2·s) 0, 0.437 m/s

Note the following: (i) We have ensured that all the reported results are grid-size-independent by checking the axial solid concentration profiles, although the results are not reported. Furthermore, in order to reduce numerical diffusion, a second order upwind scheme is used to discretize the convection terms of momentum and granular temperature equations, and a QUICK scheme is used for mass conservation equations. (ii) For brevity of the article and the reason that we are primarily 3380

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Figure 2. Effect of two-dimensional and three-dimensional simulations on the axial pressure distribution and pressure gradient profiles and their comparison with experimental data

Figure 3. Snapshots of solid concentration distribution at the center slice of the downer obtained from 3D simulation

gas velocity of 0.048 m/s which is fed into the downer from the bottom inlet. 3.2. Restitution Coefficient. In the state-of-the-art twofluid model, kinetic theory of granular flow is used to close the particulate phase stress, where the restitution coefficient for particle−particle interaction (e) is a necessary input parameter. In principle, the restitution coefficient for a specific type of particles can be measured by independent experiments; however, a constant restitution coefficient is usually introduced in kinetic theories to characterize all the energy dissipation during direct particle−particle interactions, due to both normal inelastic collisions and tangential frictions and also to the lubrication effect in close contact. Therefore, it is more rigorous to say that the restitution coefficient in kinetic theories is an effective restitution coefficient, the exact value of which is hard to determine. Figure 4 shows the effects of the restitution coefficient on the pressure drop along the downer. It can be seen that the model prediction is affected by the used restitution coefficient as in the studies of gas−solid flow in risers, with a value of about 0.99 fitting best with experimental data. The results might be explained as follows: In the dilute gas−solid flow as in CFB risers and downers, the gas−particle interaction is less important than that of dense gas−solid flow as in bubbling beds. Furthermore, the kinetic theory of granular flow usually

Figure 4. Effect of restitution coefficient for particle−particle interaction on the pressure distribution along the downer.

predicts a higher granular temperature at dilute flow as compared to dense flow, which means that the importance of particulate phase stresses in determining the hydrodynamics of gas−solid flow is increased, and therefore, sensitivity to the restitution coefficient is expected. It is interesting to note that the predicted high granular temperature is partially due to the fact that the effect of the presence of fluid is not considered in the used kinetic theory of granular flow.41 3381

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3.3. Wall Boundary Conditions. In addition to the governing equations, initial and boundary conditions are necessary for a complete description of gas−solid flow. In gas fluidized beds, the solid particles are confined by solid walls; therefore, wall boundary conditions have to be specified. Except the no-slip and free-slip wall boundary conditions, the wall conditions proposed by Johnson and Jackson29 are the most popular in the fluidization community. In the model, as presented in Table 1, two input parameters are necessary, that is, the restitution coefficient for particle−wall interaction (ew), which characterizes the energy dissipation due to particle−wall interaction, and specularity coefficient (ϕ), which characterizes the smoothness of the solid wall. Those parameters are very difficult to determine,30 which are usually treated as adjustable parameters in CFD simulations. Figures 5 and 6 show the effect of the restitution coefficient for particle−wall interaction and the specularity coefficient on

From the present study and previous studies, it is safe to conclude that two-fluid modeling of gas−solid flow is affected significantly by the input parameters; unfortunately, those parameters normally play the role of tuning parameters. This is one of the main drawbacks of state-of-the-art two-fluid model, although effort has been put forward.30 It should be emphasized that in our viewpoint, it is still unclear whether the sensitivity to those input parameters (e, ew, and ϕ) is physical or unphysical; therefore, stringent one-to-one experimental tests are highly appreciated in the validation of the two-fluid model. 3.4. Drag Correlation. Figure 7 shows the effect of drag correlations on the simulated pressure distribution along the

Figure 7. Effect of drag correlations on the pressure distribution along the downer

downer (more comparison with experimental data can be found in previous publications8). It can be seen that with the given values for e, ew, and ϕ, the drag correlation proposed by Peng et al.8 fits much better with experimental data than the one of Gidaspow;31 this is in agreement with previous studies.8 However, the results presented in sections 3.2 and 3.3 have shown that e, ew, and ϕ have significant effects on the simulation results, the unsatisfactory agreement with experimental data, therefore, may be due to the values we selected. Figure 8 shows that if another group of values has been selected, a two-fluid model with Gidaspow’s drag correlation can also fit with experimental data reasonably well. Note that the selected values are not the best fitting with experimental data, if those values are adjusted further, better fits with experimental data are possible. A dilemma arises whether we have to correct the drag correlation to consider the effect of particle clustering structures or simply use Gidaspow’s drag correlation with proper selection of e, ew, and ϕ. In view of the fact that experimental studies have shown that there are particle clustering structures in countercurrent downers2,5,7 and particle clustering structures have a pronounced effect on the interphase drag force, see Wang42 for a review, we believe that the drag force should be corrected to consider the effect of particle clustering structures, instead of unphysical adjustment of e, ew, and ϕ, as has been concluded in a previous study on CFB risers.20 This study indicates that researchers have to pay great caution to the selection of input parameters and highlights the requirement of stringent experimental tests of Eulerian models.

Figure 5. Effect of restitution coefficient for particle−wall interaction on the pressure distribution along the downer.

Figure 6. Effect of the specularity coefficient on the pressure distribution along the downer.

the pressure distribution along the downer, respectively. It can be seen that the results are sensitive to the restitution coefficient when the value of it is close to unity; however, the sensitivity decreased with decreasing value (for the specific operation conditions we studied, we can see that the values from 0 to 0.5 do not give out too much difference). Similarly, the results are very sensitive to the specularity coefficient when its value is close to zero, but the values from 0.2 to 1.0 predict vary similar pressure distribution along the downer. 3382

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Figure 8. Comparison of two-fluid simulation with Gidaspow (1994) drag correlation to experimental data

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4. CONCLUSION AND DISCUSSION

ACKNOWLEDGMENTS We thank the anonymous reviewers for suggestions and comments, which significantly improved the quality of this article. This study was financially supported by the National Nature Science Foundation of China under grant no. 21206170 and by the “Strategic Priority Research Program” of the Chinese Academy of Sciences, grant nos. XDA07080200 and XDA07010200.

A comparative study has been carried out to assess the effects of various input parameters related to Eulerian simulation of gas− solid flow. We show that there is no unique way of tuning the input parameters to match the experimental data, due to the fact that the simulations are sensitive to the input parameters. Either correcting the drag force or adjusting the wall boundary conditions can result in a nice fitting to experimental data, although the underlying mechanics are completely different. The present study highlights the requirement of stringent experimental tests of Eulerian models. An EMMS-based two-fluid model has recently been proposed for heterogeneous gas−solid flow in CFB risers,43,44 where it was shown that by alternatively treating the dilute and dense phase as the two interpenetrating continua according to the physical principle proposed in the EMMS model,45 the resulted two-fluid model was able to capture the main hydrodynamics of both low-density and high-density CFB risers without the needs of input parameters like a specularity coefficient and restitution coefficient. Furthermore, the model was also insensitive to the wall boundary conditions. Therefore, the EMMS-based two-fluid model eliminates one of the main drawbacks of state-of-the-art two-fluid model, while maintaining the comparative accuracy and computational efficiency of two-fluid modeling with suitable mesoscale (or subgrid scale) models for constitutive relations.

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SYMBOLS USED dcl = cluster diameter, m dp = particle diameter, m e = coefficient of restitution in particle−particle interaction ew = coefficient of restitution in particle and wall interaction g⃗ = gravitational acceleration, m/s2 g0 = radial distribution function p = gas pressure, Pa ps = particle pressure, Pa Ug = superficial gas velocity, m/s Uslip = slip velocity, m/s Ut = particle terminal velocity, m/s u⃗g, u⃗s = gas and solid velocity vectors, m/s

Greek Symbols

AUTHOR INFORMATION

Corresponding Author

■

*Tel.: +86-10-82544842. Fax: +86-10-62558065. E-mail: [email protected]. Author Contributions

β = drag coefficient for a control volume, kg/(m3·s) εg = voidage εs = solid volume fraction εs,max = solid volume fraction at packed condition φ = angle of internal friction, deg ϕ = specularity coefficient μg, μs = fluid and solid viscosity, Pa·s Θs = granular temperature, m2/s2 ρg, ρs = fluid density and solid density, kg/m3

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§

These authors have contributed equally.

Notes

The authors declare no competing financial interest. 3383

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