Anisotropic Turbulent Mass Transfer Model and Its Application to a

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Anisotropic Turbulent Mass Transfer Model and its Application to a Gas-Particle Bubbling Fluidized Bed Wenbin Li, Yuanyuan Shao, and Jesse Zhu Ind. Eng. Chem. Res., Just Accepted Manuscript • DOI: 10.1021/acs.iecr.7b03715 • Publication Date (Web): 11 Jan 2018 Downloaded from http://pubs.acs.org on January 11, 2018

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Industrial & Engineering Chemistry Research

Anisotropic Turbulent Mass Transfer Model and its Application to a Gas-Particle Bubbling Fluidized Bed Wenbin Li†, Yuanyuan Shao∗,†, Jesse Zhu†,‡ †

Collaborative Innovation Center of Chemical Science and Engineering (Tianjin), School of

Chemical Engineering and Technology, Tianjin University, Tianjin 300072, China ‡

Particle Technology Research Center, Department of Chemical & Biochemical Engineering,

The University of Western Ontario, London, Ontario, Canada Abstract: The recently developed Reynolds Mass Flux (RMF) model is applied to simulating the reactive flow in gas-particle bubbling fluidized bed (BFB). By using this model, the profiles of species/particles concentration and phase velocities are able to be predicted. The proposed model avoids the generalized Boussinesq’s postulation, thereby realizing the simulation of anisotropic mass transfer. The simulations are validated by experiments for ozone decomposition in a gas-particle bubbling fluidized bed and satisfactory agreement is found between them. Furthermore, the proposed model successfully characterizes the anisotropy of turbulent mass diffusivity in gas-particle BFB. Keyword: Mathematical modeling; Bubbling fluidized bed (BFB); Chemical reaction; computational fluid dynamics (CFD); Anisotropic turbulent mass diffusion

1. INTRODUCTION Mass diffusivity is a measure of the quality of mixing. Generally, two types of mass diffusion exists in mass transfer process occurs in fluidized bed: the molecular mass diffusion, Dg, and the turbulent mass diffusion, Dgt, that induced by turbulent flow. Experimental studies ∗

Corresponding author. Tel.: +862223497607. E-mail address: [email protected] (Y. Shao).

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Breault et al. 12 show the axial gas turbulent mass diffusivity, Dgt,x, in the fluidized bed is larger than the radial one, Dgt,r. Further studies

13

reveal that in circulating fluidized bed (CFB) riser Dgt,x is

often 2 to 3 orders of magnitude larger than Dgt,r, but less than an order of magnitude for the bubbling fluidized bed. All of the previous works mentioned above indicate a fact that the turbulent mass diffusion in fluidized beds shows anisotropic character. According to the works by Breault and Guenther

14-16

, Dgt play an important role on the mass transfer behavior in fluidized bed, thereby

many researchers have devoted their attention to the investigation of anisotropic turbulent mass diffusion through experiments. There are many attempts 3, 5, 6, 17-19 for correlating the Dgt,i in fluidized bed from experimental data. However, a wide variation (more than five orders of magnitude) has been found for the Dgt,i obtained by different experimental correlations. Breault

12

has noted that the significant difference between

them can be attributed to the extreme differences in the experimental conditions and facilities. No doubt, the industrial application of such correlations for design, scale-up and optimize the fluidized bed is limited. By using combined computational fluid dynamics (CFD), Li et al.

20

originally

introduced a model that enables simulation of anisotropic mass transfer in distillation column. And recently, Li et al. have applied this model to the simulations of various chemical engineering equipments, including those of packed beds 21-24 and circulating fluidized beds risers 25. This model could be called as Reynolds Mass Flux (RMF) model because the key point of the model is to close the turbulent mass transfer equation by the newly deduced Reynolds mass flux equation. And with the RMF model, the anisotropy of turbulent mass diffusion in different chemical engineering processes is characterized. The objective of this study is to extend the newly developed RMF model application to the ozone decomposition simulation in BFB. The simulations on hydrodynamics and mass transfer process are


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provided.

And the predictions are testified by comparison against the published experimental

measurements 26. The anisotropic character of Dgt,i in the BFB is discussed.

2. SIMULATION TARGET In this study, the gas-particle BFB with I.D. = 0.229 m and H = 2 m, as reported by Fryer and Potter 26, is simulated. The ozone decomposition is conducted in the gas-particle BFB system and the sand impregnated with iron oxide is taken as catalyst. More detailed information about the BFB system and the relevant experiments are available in literature 26.

3. MODELING The presented RMF model is composed of the turbulent mass transfer equations with the modeled Reynolds mass flux equations as its closure to describe the species mass transfer process, and the accompanied CFD approach based on Eulerian-Eulerian method for the multiphase flow.

3.1. Model formulations. (1) The gas phase turbulent mass transfer equation.

∂ ∂ ∂  ρ α ρ g α g Cn + ρ gα g u gi Cn = ∂t ∂xi ∂xi  g g 

(

)

(

)

  ∂ Cn − u ′gi cn′   + SC n  Dgn   ∂xi   

(1)

CO + CO + Cair = 1 3

(2)

2

In which Cn represents the species (O3, O2 and air) mass concentration; Dgn denotes the species molecular diffusivity; − ρ gα g ugi′ cn′ is an unknown term, namely the Reynolds mass flux, and is

(

)

conventionally solved by the generalized Boussinesq’s postulation − ρ gα g u gi′ cn′ = ρ g α g Dgn, t ∂Cn ∂xi , where the assumed isotropic turbulent mass diffusivity, Dgn, t , is calculated by using a experimental correlation or a constant turbulent Schmidt number Sct 27-31. Although such postulation has a simple ACS Paragon Plus Environment


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expression and would be convenient to be applied, the use of isotropic Dgn, t is unjustified since many researchers have revealed the anisotropic character of turbulent mass diffusion in FB 9, 13, 32. In this light, the newly deduced Reynolds mass flux equation by Li et al. 20, 25 is adopted in this study as a closure equation for the turbulent mass transfer equation.

(

∂ ρ g α g u gi′ cn′ ∂t

) + ∂(ρ α g

g

u gi u gi′ cn′

∂x j

)=

(

)

(

 ∂ ρ g α g u gi′ cn′ ∂ ρ g α g u ′gi cn′ kg ∂  ′ δ lmjk CC1 u gl′ u gm + Dgn ∂x j  ∂xk ∂x j εg 

)   

(3)

εg ∂ u gi ∂ Cn − ρ g α g u gi′ u gj′ − Cc 2 ρ g α g u gi′ cn′ − Cc 3 ρ g α g u gj′ cn′ kg ∂x j ∂x j

The constants in the above equation are

20

: CC1 = 0.09 , CC 2 = 3.2 , CC 3 = 0.55 . The unknown

terms of Reynolds stress ( − ρ g α g ui′u ′j ), turbulent kinetic ( k g ) and the turbulent kinetic dissipation ( ε g ) are solved by the modeled Reynolds stress equation

33

and the k g − ε g equations which are

summarized in the formulations of the accompanied CFD. In the course of simulation, the following numerical procedures are carried out for solving the model differential equations: firstly, the simulation domain is discretized into numerous grids, and the model differential equations are simultaneously changed to a algebraic form; then, the source terms are referred to the local averaged conditions within each small finite grids. In this light, the source term SC i in eq. (1) can be determined by the following formulation with local averaged concentration: SC i = ± r

Pg RTg

Mi

(4)

here the plus/negative sign refers to the product/reactant component of the reaction; Tg Pg,are the gas phase temperature and pressure; R is the universal gas constant; Mi is the local species molar mass; r denotes the reaction rate and is calculated by:

r = kr X O α s 3

(5)

where X O is the local ozone mole concentration; α s is the volume fraction of particle phase; kr 3


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is the constant of reaction rate. (2) Formulations of the accompanied CFD approach. The formulations of the accompanied CFD for describing the gas-particle two-phase flow is composed by a set of governing equations, including the continuity and momentum equations, and the constitutive equations, including the KTGF integrated by the modeled Reynolds stress equations and the EMMS drag model equations. The accompanied CFD formulations are given in the Supporting Information. 3.2. Arrangement of Boundary Conditions. Figure 1 shows the computational domain and boundary arrangement.

Figure 1. Computational domain and boundary condition arrangements. (1) Inlet conditions Gas phase: The gas are induced to the BFB from the bottom, and the “velocity inlet” condition 20, 25, 34

is chosen for the gas phase where the velocity and ozone concentration is specified as

U g ,in = U g and Cg ,in = Cg ,in . The inlet gas phase turbulent energies, k g , in , and dissipation rates, ε g ,in ,

are set as those reported by Zheng et al. 34. And the inlet Reynolds stress is taken as 35: ACS Paragon Plus Environment


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2  k , i = j ui′u ′j =  3 in 0, i ≠ j 

As for the inlet Reynolds mass flux, −u′gi cn′ , the generalized Boussnesq’s postulation is used:

( −u ′ c ′ ) gi n

in

(

= Dt , gn ∂ Cn ∂xi

)

in

where Dgn, t is estimated as Dgn, t =

ν gt , in Sc gt

by assuming turbulent Schmidt number as 0.7 36, and the

inlet turbulent viscosity ν t , in is equal to ν gt , in = Cµ k g2, in ε g , in with constant Cµ = 0. 09

34

.

Solid phase: No solids are entered to the BFB, thus the inlet velocity of solids is set as U s ,in = 0 ; the granular temperature is taken as 34 Θ = 0.0001 . (2) Outlet conditions The "pressure outlet" condition is specified at the top of the BFB. (3) Wall conditions The no-slip and partial-slip37 conditions are defined for the gas and particles phase, respectively. 3.3. Numerical procedures. The simulation is conducted by the software Fluent 6.3.26 under 2D space. The computation domain of the BFB is 150×245 (radial×axial) quadrilateral cells that are created by the software Gambit 2.4.6. The SIMPLE algorithm is used for solving the pressure-velocity coupling problem. The first order discretization scheme is adopted for the convection and viscous terms of the volume fraction equation, whereas the second order discretization scheme is applied for all of the other model equations. 4. SIMULATIONS AND DISCUSSIONS In this section, comparisons of simulated results against experimental measurements reported in literature

26

on species concentration distributions and solid concentration distributions are made to

validate the RMF model. Then, the anisotropic turbulent mass diffusion in the BFB is characterized ACS Paragon Plus Environment

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and analyzed.

4.1. Profiles of species and solid concentration As shown in Figure 2, the outlet ozone concentration increases sharply from the flow time t = 12 to 25 s, and maintains almost constant after t = 30 s (the standard deviation is within 1%). And all of the simulated results provided in the following are those transient values of 30 s or time-averaged values from 30-50 s. Figure 3 shows the comparisons of ozone concentration (C/C0) profiles between experimental measurements and model predictions. It is seen that the present model predictions (solid line) show reasonable agreement with the experimental measurements (circle symbols), which confirms that the present RMF model is suitable for simulating ozone decomposition process undertaken in BFB. Figure 3 also provides the predictions by using the empirical Sct model (dot line). As mentioned before the Sct model is based on the assumption of isotropic turbulent mass diffusivity. Difference of the simulated results is clear seen between using the two models, and the present model performs better than the Sct model. It indicates that the Sct model with isotropic turbulent mass diffusivity is not precise enough for the simulation in this case. It is interesting to find that the experimental data in Figure 3 pass through a minimum value within the bed. As stated by Fryer and Potter 26, this is characteristic of gas backmixing behavior. In other word, the dramatic inversed flow of gas would lead to transportation of ozone from lower concentration region (upper part of the BFB) to higher concentration region (bottom part of the BFB). Further explanation should be that there is a trade-off between the gas inverse flow and the turbulent mass diffusion (representing the quality of mixing or the ability of ozone transferred from higher concentration to lower concentration). Since the performance of the present model and the simplified Sct model on simulation of gas-solid phase flow



are the same, the predictions of more flat concentration profiles should be attributed to the overestimation of turbulent mass diffusivity for the Sct model. As mentioned in the introduction section, it has been observed that radial gas turbulent mass diffusivity in the fluidized bed is much smaller than the axial one 1-11. However, in the simplified model isotropic turbulent mass diffusivity is used which means Dgt,r = Dgt,x = Dgt. Further explanation would be made in the section 4.3. It is noted from Figure 3 that deviations does exist between the present model predictions and experimental data. This may be induced by the following reason: the gas-solids two phase flow in the BFB is not two dimensional especially in the region near the gas distributor at the bottom part. However, the simulations are conducted using 2D mesh to reduce the need of computational demand. The contour of simulated ozone concentration in the BFB at t = 30 s are shown in Figure 4. In combination with Figure 5, it is found that values of ozone concentration are relatively lower in the dense region than in the bubble region. Such phenomena can be explained that ozone decomposition rate is related with the particles concentration (see eq. (5)). Maldistributions of ozone concentration are seen both in Figure 4(a) and (b), whereas the prediction of the simplified Sct model shows less significant non-uniformity. As mentioned before, this might be attributed to the overestimation of turbulent mass diffusivity for the simplified Sct model.


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Figure 2. Simulated outlet radial averaged ozone concentration for experiment: (Hmf = 0.231 m, Ug = 0.104 m s-1, kr = 0.33 s-1)

Figure 3. Comparisons of ozone concentration profiles between model predictions and experimental data for: (Hmf = 0.231 m, Ug = 0.104 m s-1, kr = 0.33 s-1)

(a)

(b)



Figure 4. Contours of simulated ozone concentration in BFB at t = 30s: (a) present model prediction; (b) simplified Sct model prediction.

(a)

(b) Figure 5. Solid volume fraction in BFB at t = 30s: (a) contour of α s ; (b) axial profile of α s .

4.2. Profiles of gas and solid velocities Velocity vectors of gas and particles in the BFB are shown in Figure 6. Several gas phase vortex can be found in (or near) the bubble region (see Figure 6(a)). And the solid phase vortex (see Figure 6(b)) are found to be accompanied with that of gas phase, due to the dramatic gas-particle momentum interaction. Figure 7 and 8 provide the radial distributions of the simulated gas and particles velocities. significant variations of the fluid velocity are seen at a fixed bed height, which


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implies a dramatic turbulence of both the gas and solid phase.

(a)

(b) Figure 6. Simulated gas and particles velocity vectors in BFB for experiment: OC-b-(Hmf = 0.231 m, Ug = 0.104 m s-1, kr = 0.33 s-1)

Figure 7. Simulated gas velocities profiles in BFB at different heights: (a) u g , x ; (b) u g , y

Figure 8. Simulated particles velocities profiles in BFB at different heights: ACS Paragon Plus Environment


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(a) us , x ; (b) us , y

4.3. Profiles of the anisotropic turbulent mass diffusivity Figure 9 and Figure 10 give the simulated Reynolds mass flux and ozone concentration gradient, respectively. And the anisotropic turbulent mass diffusivity can be characterized by transforming the generalized Boussinesq’s postulation as

Dgt ,x = u ′gx c′ −

∂C ∂C and Dgt ,y = u ′gy c′ − ∂x ∂r

(6)

It is shown from Figure 11 that the shapes of the Dgt,x and Dgt,y profiles are found to be related to the gas velocity field, as demonstrated in Figure 7. At a fixed axial position, more significant variation of Dgt,i is found at the region where velocity oscillates dramatically. According to the present anisotropic RMF model, by volume-averaging the mass turbulent diffusivity in the whole BFB (bed bottom to the surface), it is found that the axial turbulent mass diffusivity ( Dgt ,x = 0.0024 ) is about five times larger than the radial one ( Dgt ,x = 0.00051 ), while the computed isotropic turbulent mass diffusivity from the Sct model is Dgt ,x = Dgt ,y = 0.0061 m 2s -1 . One can observe that both the radial and axial turbulent mass diffusivities are overestimated by the Sct model, especially for the radial one. This also explains the simulation error induced by the empirical model with isotropic Dgt,, as shown in Figure 3.


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Figure 9. Simulated − ρ g u gi′ c′ profiles at different heights:

′ c′ ; (b) − ρ g u ′gy c′ . (a) − ρ g u gx

Figure 10. Simulated ozone concentration gradient at different heights: (a) ∂C ∂x ; (b) ∂C ∂y .

Figure 11. Simulated Dgt,i at different heights: (a) Dgt ,x ; (b) Dgt ,y .



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4.4. Study on the influence of grid number on the accuracy of simulation Generally, larger number of grids is preferred for more precise numerical simulation. However, the growth of grid number will increase the computation effort, therefore it is necessary to find a proper number of grids that on one hand it ensures a reliable simulation and on the other hand finer grid would not result in significant different simulation. For this purpose, the BFB is meshed with different grids of 80×120, 150×245 and 200×360 respectively. The simulation on solid phase volume fraction contour under different grid sizes are given in Figure 12. Although all simulations predict the correct qualitative bed expansion, the finer grid cases (150×245 and 200×360) obtain similar solid volume fraction contour.

(80×120)

(150×245)

(200×360)

Figure 12. Simulated contours of α s at t = 1.6 s with different mesh schemes.

The simulated bed expansion and outlet ozone concentration are illustrated in Figure 13 for different grid sizes. It is observed that the changing of the predictions is not significant if grids number is more than 150×245, which indicates that the simulation with 150×245 grids is sufficient for the present study.


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Figure 13. Comparisons of simulated bed expansion and outlet ozone concentration with different grid number

Table 1 provides the anisotropic turbulent mass diffusivities under different grid numbers. From this table, the values of Dgt,i demonstrate obvious deviation when grid number is lower than 150×245; whereas difference is small when grid number is beyond 150×245. It further supports that the simulations in present case become independent of grid number when the number is 150×245 or beyond.

Table 1. Simulated Dgt,i with different meshes. Grid number

Dgt,x×10-4 /m2 s-1

Dgt,y×10-4 /m2 s-1

80×120

29.4(av.)

6.35(av.)

150×245

24.1(av.)

5.13(av.)

200×360

23.8(av.)

5.06(av.)

5. CONCLUSION (1) A CFD-based model is presented for simulating the reactive flow in bubbling fluidized bed. The proposed model avoids the generalized Boussinesq’s postulation, thereby realizing the simulation of anisotropic mass transfer. To test the validity of the model, simulation of a gas-particle ACS Paragon Plus Environment


bubbling fluidized bed for ozone decomposition is conducted. By comparing the model predictions with the experimental data, the present model shows its advantage over the empirical Sct model with isotropic turbulent mass diffusivity for the BFB simulation. (2) The anisotropic character of the mass transfer in BFB is characterized and discussed. The distribution pattern of Dgt,x is rather different from that of Dgt,y . By volume-averaging the mass turbulent diffusivity in the whole BFB, the Dgt,x is about five times larger than the Dgt,y.

ACKNOWLEDGEMENT This work has been supported by the National Natural Science Foundation of China (Project Nos. 21506146 and 21606170).

SUPPORTING INFORMATION The formulations of the accompanied CFD for the presented Reynolds Mass Flux (RMF) model are supplied as Supporting Information. This information is available free of charge via internet at http://pubs.acs.org/.

NOMENCLATURE

Cn

mass concentration of O3, O2 and air

Cc1, Cc2, Cc3

RMF model constants

Dg, Dgt, Dgt,i

species molecular diffusivity, isotropic turbulent diffusivity and anisotropic turbulent mass diffusivity, m2/s

H

total height of the BFB reactor, m

h

bed height, m

kg

gas phase turbulent kinetic energy, m2/s2

kr

apparent reaction rate constant, 1/s

Pg

gas phase pressure, kPa

Sci

source term, kg/(m3·s)

Sct

turbulent Schmidt number


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t

time, s

Tg

gas phase operation temperature, K

u g , i , us , i

gas and particles phase velocities, m/s

XO

mole concentration of O3 in gas phase, kmol/m3

αg, αs

gas and particles volume fractions

εg

dissipation rate of turbulent kinetic energy, m2/s3

ρg

gas phase density, kg/m3

3

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Anisotropic Turbulent Mass Transfer Model and Its Application to a

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