Effect of particle clusters on mass transfer between gas and particles in gas-solid flows

    Effect of particle clusters on mass transfer between gas and particles in gas-solid flows Limin Wang, Chengyou Wu, Wei Ge PII: DOI: Reference:

S0032-5910(17)30505-3 doi:10.1016/j.powtec.2017.06.046 PTEC 12622

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Powder Technology

Received date: Revised date: Accepted date:

8 January 2017 3 May 2017 17 June 2017

Please cite this article as: Limin Wang, Chengyou Wu, Wei Ge, Effect of particle clusters on mass transfer between gas and particles in gas-solid flows, Powder Technology (2017), doi:10.1016/j.powtec.2017.06.046

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ACCEPTED MANUSCRIPT Effect of particle clusters on mass transfer between gas and particles in gas–solid flows State Key Laboratory of Multiphase Complex Systems, Institute of Process

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Limin Wanga*, Chengyou Wub, Wei Gea

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Engineering, Chinese Academy of Sciences, Beijing 100190, China; b

Changsha Research Institute of Mining and Metallurgy Co., Ltd, Changsha 410012, Hunan, China

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*Corresponding author; email addresses: [email protected]

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Tel: +86 10 8254 4942; Fax: +86 10 6255 8065

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Abstract Gas–solid riser flows tend to be characterized by particle clusters that

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significantly affect the flow, the mass/heat transfer, and the reaction behavior. To

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account for the effect of such particle clustering on the mass transfer between gas and particles, we use a lattice Boltzmann model with coupled mass transfer to conduct fully resolved simulations for flow past arrays of particles with both homogeneous

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and heterogeneous structures. We find that the distributions of velocity and concentration among the particle arrays are both affected significantly by particle clustering, and that the computed Sherwood number for either the homogeneous or heterogeneous particle structure increases exponentially with Reynolds number. However, the Sherwood number for homogeneously distributed particles is 3–5 times greater than that for heterogeneously distributed particles. This further supports the case for particle clustering having a serious effect on the mass-transfer efficiency between gas and solid in processes that involve flow past arrays of particles. Finally, the feasibility of using the lattice Boltzmann method with our mass-transfer model to

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ACCEPTED MANUSCRIPT describe the mass transfer in a heterogeneous two-phase gas–solid flow is also discussed.

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Keywords gas–solid flows; particle cluster; lattice Boltzmann method; mass transfer;

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fully resolved simulation

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ACCEPTED MANUSCRIPT 1 Introduction The interactions between a gas and an ensemble of particles are completely

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different from those between a gas and a single particle because of inter-particle

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collisions and the effects of particle wakes. In most cases, the correlations of the drag,

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mass-transfer and heat-transfer coefficients of gas–particle-ensemble systems are usually obtained under the assumption of an ensemble of spherical particles

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distributed homogeneously within the flow. Typically, flow through a packed bed is described using the Ergun equation, which was derived empirically and relates the

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drag to the pressure drop across the packed column [1]. However, the Ergun equation does not take into account the effect of a heterogeneous particle structure. In gas–solid

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flows, particles aggregate into clusters that are surrounded by a dilute phase, which

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greatly affects the momentum, mass, and heat transfer. In such situations, there are

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considerable differences between experimental measurements and the predictions of the traditional averaging approach without considering particle clusters. Therefore, in the context of gas–solid flows, there is much interest in the effect of mesoscale

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structures (i.e., particle clusters) on the flow, the mass and heat transfer, and the reaction behavior.

In the 1980s, Li and Kwauk [2] pioneered the exploration of mesoscale phenomena in gas–solid fluidization. Starting with particle clustering, they viewed mesoscale phenomena as originating from a balance between gaseous and particle kinetics, which led to the development of the energy-minimization multi-scale (EMMS) model [2–4]. From the drag values calculated using the EMMS model, differences of several orders of magnitude are found among the drag coefficients for different flow structures, such as a dilute phase, a dense phase, and the interphase between them [5]. However, in traditional two-fluid models (TFMs) [6, 7], the 3

ACCEPTED MANUSCRIPT constitutive laws are based on the assumption of a homogeneous structure on the computational grid, which is regarded as a source of modeling inaccuracies in relation

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to gas–solid riser flows.

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Various sub-grid drag models have been proposed to account for the effect of

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particle clusters [8–14]. When coupled with a TFM, some EMMS-based drag models [10–14] are more accurate at predicting the hydrodynamics of heterogeneous gas– solid flows than is a traditional TFM with homogeneous drag [15–20]. O’Brien and

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Syamlal [21] developed an empirical heterogeneous drag model. Zhang and

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Vanderheyden [22] used a scaling factor to correct the interphase drag force while considering the effects of mesoscale structures. Mehrabadi et al. [23] developed a

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gas–solid drag law for clustered particles by using direct numerical simulation (DNS).

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However, relatively few studies have been extended to consider the influence of mesoscale structures on mass transfer. Most numerical studies have been limited to

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the influence of mesoscale structures on momentum transfer (e.g., the drag coefficient) in gas–solid flows [8-23] and conventional Sherwood equations for reacting flows in

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fluidized beds [24, 25]. Wang and Li [26] developed a steady multi-scale mass-transfer model based on the structure analysis in the EMMS model. Dong et al. [27, 28] coupled the EMMS/mass model with a TFM to realize a dynamic multi-scale mass-transfer model. From three-dimensional plots for different combinations of structural parameters, Dong et al. [27, 28] were able to account for the differences of several orders of magnitude among the mass-transfer coefficients reported in the literature. Li and Hou [29, 30] carried out a theoretical analysis of the quantitative relationship

between

the

heterogeneous

structure

parameters

and

the

heat/mass-transfer coefficients of a fast-fluidized bed. Wang et al. [31, 32] established a mathematical model of the heat/mass-transfer processes associated with particle

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ACCEPTED MANUSCRIPT clustering, and simulated these processes numerically for the sublimation of Ne particles. Regarding the simulation of mass transfer by means of DNS for gas-solid

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systems, limited literature is available. Recently, Deen and Kuipers [33, 34]

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performed DNS to investigate fluid–particle mass transfer involving random arrays of

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particles. Deen et al. [35, 36] also reviewed the DNS of fluid–particle mass, momentum and heat transfer in dense gas–solid flows. Feng and Musong [37] developed a DNS approach combined with the immersed boundary (DNS–IB) method

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for studying heat and mass transfer in particulate flows but with specific emphasis on

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the heat transfer of 225 heated spheres in a fluidized bed. Sahu et al. [38] directly calculated the heat and mass transfer for a structured packed bed using

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particle-resolved simulations taking into account heterogeneous chemical reactions on

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the particle surface. Bale et al. [39] performed DNS to understand the effects of

flows.

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confining walls on mass transfer through a packed bed for moderate Reynolds number

The lattice Boltzmann method (LBM) [40] operates at a more fundamentally

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kinetic level via the discrete Boltzmann equation. Because of its advantageous speed due to its natural parallelism, algorithmic simplicity, and explicit methods, the LBM has been developed into a powerful alternative tool for computational fluid dynamics and beyond [41]. Hill et al. [42] were the first to apply the LBM to study flow past ordered and random arrays of particles, and succeeded in correlating the gas–solid drag coefficients with the particle Reynolds numbers and solid volume fraction. Beetstra et al. [43] used the LBM to study gas–solid drag coefficients for particle clustering in different geometries. Zhang et al. [44] used the LBM to verify the rationality of heterogeneous drag distribution in the EMMS model. Shah et al. [45, 46] investigated the influence of particle clustering on drag force, and verified the drag

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ACCEPTED MANUSCRIPT from the EMMS formulation with high-resolution three-dimensional lattice Boltzmann (LB) simulations. Rong et al. [47, 48] quantified the drag force on

on

the

internal

fluid

flow.

Zhou

and

Fan

[49, 50]

used

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distribution

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particles in packed beds, as well as the effects of porosity and particle-size

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immersed-boundary LB simulations to examine the effects of rotation on the drag force, Magnus lift force, and torque on spherical particles. Zhou et al. [51] and Liu et al. [52] used large-scale gas-solid DNS to study the effects of mesoscale structures on

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the drag and statistical properties of particles, respectively.

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However, although previous studies have conducted fully resolved DNS of flows past monodisperse arrangements of particles, the use of DNS for mass transfer in gas–

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solid systems is much less common. In this paper, we use a LB model coupled with

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mass transfer to simulate the flow past arrays of particles with both homogeneous and heterogeneous structures. We calculate the mass-transfer coefficient directly from the

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local flow field and concentration. Hence, we are able to examine in detail the hydrodynamic behavior in particle clusters, the fluid velocity distribution, and the

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mass-transfer coefficients for flows past arrays of particles. This paper is organized as follows. Sections 2.1 and 2.2 present the LBM used to model the fluid flow and the mass transfer, respectively; the quantification of the overall mass-transfer coefficients is

illustrated

in

Section 2.3.

The

initial/boundary

conditions

and

the

simulation/physical parameters are described in Section 3. The validity of the LB model with coupled mass transfer is examined in Section 4.1. Then, in Sections 4.2 and 4.3, the effects of particle configuration on the fluid flow distribution, concentration field, and mass-transfer behavior are investigated for different flow velocities. Finally, we summarize the present study in Section 5. 2 Numerical approach 6

ACCEPTED MANUSCRIPT The drag between gas and particles has been studied using various approaches [8– 14, 42–51], but the mass transfer between gas and particles has been less considered.

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Quantifying particle clustering and its effect on mass transfer is critical for the

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realistic simulation of heterogeneous gas–solid flows. In the present work, we

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consider the effect of mesoscale structures on the mass transfer in gas–solid flows by means of fully resolved simulations of flow past arrays of particles. As a smoothed alternative to lattice gas automata (LGA), the LBM is an efficient second-order

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Navier–Stokes solver that is capable of resolving the hydrodynamics of various

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systems because of its algorithmic simplicity, natural parallelism, and ability to yield particle distribution functions explicitly. Therefore, the LBM is an alternative

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arrangements of particles.

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approach that is well suited to fully resolved simulations of flows past monodisperse

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2.1. Lattice Boltzmann model for fluid flow We use the D2Q9 (two dimensions and nine velocities) LB model to solve for the flow field. Here, flow past arrays of particles can be simulated by the following

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Bhatnagar–Gross–Krook(BGK) lattice model [41]: f ( x  e  t , t   t )  f ( x, t ) 

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 feq ( x, t )  f ( x, t )  , 

(1)

where f ( x, t ) is the particle distribution function in the αth direction,  t is the time step,  is the dimensionless relaxation time, and e is the discrete velocity vector related to the D2Q9 model as follows:

 (0, 0)     e  (c cos[( -1) ],sin[( -1) ]) 2 2     ( 2c cos[(2 -1) 4 ],sin[(2 -1) 4 ]) 7

 0   1, 2,3, 4   5, 6, 7,8

(2)

ACCEPTED MANUSCRIPT where the lattice velocity is defined as c= x /  t with the lattice length  x .

f

 e  u (e  u)2 u 2    1  2   2 , cs 2cs4 2cs  

(3)

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eq

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eq The corresponding equilibrium distribution function f ( x, t ) can be expressed as

where the lattice speed of sound is c =c / 3 and  is a weighting factor (  =4 / 9

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s

for α = 0,  =1 / 9 forα = 1–4, and  =1 / 36 forα = 5–8). The fluid density  and u

are calculated respectively as 8

 =  f ,

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 =0

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velocity

(4)

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 u=  e f .

(5)

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 =0

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In the limit of incompressible flow and with the Chapman–Enskog expansion, we

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can recover the Navier–Stokes equations (6) and (7) from the above LB model [53]:       u  0, t

(7)

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   u      uu  p  [  (u  u)], t

(6)

with the kinematic viscosity given by   c2 t (  0.5) / 3 and pressure by p= cs2 . 2.2. Lattice Boltzmann model for mass transfer We use the following BGK lattice model [54] to simulate the mass transfer in gas– solid flow:

g ( x  e t , t   t )  g ( x, t ) 

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 geq ( x, t )  g ( x, t )  , c

(8)

where g ( x, t ) is the gas concentration distribution function in the αth direction and

 c is the dimensionless relaxation time.

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ACCEPTED MANUSCRIPT The corresponding equilibrium distribution function geq ( x, t ) can be expressed as [54]

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C =  g .  =0

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where the gas concentration C is calculated from

(9)

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 3(e  u)  geq  Cω 1  , cs2  

(10)

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Similarly, according to the Chapman–Enskog expansion, the following macroscopic advection–diffusion equation [54] can be recovered from the above LB model

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describing the mass transfer:

(11)

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C     Cu     DC  , t

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with the diffusion coefficient given by D  c2 t ( c  0.5) / 3 .

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2.3. Quantitation of overall mass-transfer coefficients Based on the overall mass balance in the fluidized bed, the governing equation can

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be formulated as [55]

 2C C a Day 2  u j  ks (C  C0 )  0, y y 

(12)

where Day is the axial dispersion coefficient, a is the effective surface area for mass transfer,  is the porosity of the fluidized bed, and u j is the fluid velocity in the pore spaces when a homogeneous cross-sectional flow distribution is assumed. Because axial dispersion can be neglected in relation to gas–solid mass transfer under specific circumstances [56], namely for the case of Day =0 , Eq. (12) can be simplified to uj

C a = ks (C0  C ). y  9

(13)

ACCEPTED MANUSCRIPT Then, we integrate Eq. (13) over the packed-core height from y=0 to y=H with inlet concentration C=Cin and outlet concentration C=Cout , respectively. Therefore, the

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u C  Cin ln 0 , a  H C0  Cout

(14)

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ks =

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mass-transfer coefficient k s can be calculated as

with the superficial fluid velocity u=u j   .

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Correspondingly, the Sherwood number Sh, which describes the overall mass transfer in dimensionless form, is defined as

ks  d p

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Sh =

D

.

(15)

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Hence, using the known average particle diameter d p , the solute diffusion coefficient

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D, and the aforementioned mass-transfer coefficient k s , we calculate the Sherwood number and use it as a reference for further analyzing and comparing the

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mass-transfer data for gas–solid flows.

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3 Simulation setup

3.1. Initial and boundary conditions The simulation objective is the gas flow around particles in a two-dimensional rectangular domain of height H= 6 mm and width W=3 mm. As shown in Fig. 1, two different initial particle distributions are used in the homogeneous domain. Specifically, each structure contains 162 equal-diameter particles with an overall average porosity of 0.964. In the flow process, the gas is injected into the domain through the lower wall with constant inflow velocity u0 , and flows out through the upper wall. Meanwhile, the particles are regarded as being stationary.

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ACCEPTED MANUSCRIPT In the simulation, we assume that the gas flow began from a saturated situation and that both the initial velocity and concentration in each node of the mobile phase

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are zero. Subsequently, both the left- and right-hand walls are subject to no-slip

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boundary conditions in which both components of the flow velocity are zero with a

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bounce-back scheme. In addition, we apply the boundary conditions of constant inflow velocity at the flow inlet and constant concentration at the solid/gas interface.

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3.2. Simulation conditions

The simulation and physical parameters for flow around the particles at an

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4.1. Model validation

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4 Results and discussion

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inflow velocities that were used.

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ambient temperature of 25℃ are listed in Table 1, and Table 2 lists the different gas

To evaluate the validity of the computational model, we simulate a steady two-component diffusion process between two parallel walls with the same simulation

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parameters and conditions as those used by Inamuro et al. [57]. We divide a square domain into a square lattice of spacing  x . The lower and upper porous walls are located at y = 0 and 1, respectively. Moreover, a constant normal flow u0 of component A is injected through the lower wall and flows out through the upper wall. The concentrations of component B at the lower and upper walls are maintained at CL and CU , respectively, with CL ≪ CU ; therefore, the diffusion direction of

component B is opposite to the flow direction of component A. The governing equation is formulated as

u0

  2 D 2 , y y 11

(16)

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C  CL . CU  CL

(17)

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ψ

exp(u0 y / D)  1 . exp(u0 / D)  1

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ψ* 

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Thus, the analytical solution ψ* becomes [57]

(18)

In this problem, we apply the boundary conditions of constant concentrations of

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component Bon the lower and upper walls, and periodic boundary conditions in the x

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direction. Under the conditions of  x =0.05 ,  c =1.1 , and u0 / D  4 , the numerical simulation gives the results shown in Fig. 2, where the results of using the

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aforementioned LBM are seen to be in good agreement with the analytical solution.

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4.2. Analysis of velocity and concentration fields

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Using the LB model to simulate gas flow around particles with the parameters provided in Table 1, we obtain the two different gas velocity fields shown in Fig. 3. For the homogeneous particle distribution in Fig. 3(a), the velocity field is relatively

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homogeneous, with a magnitude close to 0.25 ms-1 at each node; only in small areas surrounding the particles does the gas flow any slower. However, for the clustered particle distribution in Fig. 3(b), the velocity field is much more node-dependent. The gas flows rapidly around the particles, usually at speeds in excess of 0.25 ms-1. Meanwhile, because it is difficult for gas to flow into a particle cluster, the gas flows more slowly within the clusters, generally at speeds less than 0.025 ms-1. It follows that particle clustering has a significant effect on the gas velocity field for flow around particles. Similarly, by using the LB model coupled with mass transfer, we obtain the gas-concentration distributions shown in Fig. 4. For the homogeneous particle 12

ACCEPTED MANUSCRIPT distribution in Fig. 4(a), the gas is concentrated relatively homogeneously in the longitudinal direction, while increasing gradually to 4.0 kgm−3 in the axial direction.

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However, for the clustered particle distribution in Fig. 4(b), the gas is concentrated

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quite differently in the longitudinal direction. The gas concentration is relatively high

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(≳4.0 kgm−3) within the particle clusters (where the gas flows relatively slowly), but is far less concentrated (≲0.5 kgm−3) around the particles (where it flows much faster). Consider the transfer of mass from particles' surface to a fluid flowing through

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arrays of particles, nearly all of the resistance to mass transfer is in the particle’s

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surface boundary layer. The mean thickness of particles' surface boundary layer becomes large due to slow gas flows more slowly within the clusters. Therefore, the

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particle clustering reduces the mass transfer between gas and particles and also has a

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significant effect on the distribution of gas concentration for flow around particles.

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4.3. Analysis of overall mass transfer From the numerical calculations of gas flow around particles with the parameters

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provided in Table 2, we obtained values of the mass-transfer Sherwood number as plotted in Fig. 5. It can be seen that the Sherwood number increases rapidly with time at first and then reaches a steady state. Moreover, for increasing flow speed through the same particle structure, the time required to reach this steady state decreases. We calculated the Sherwood number for gas flow around particles for inflow speeds of 0.05–0.5 ms-1. Therefore, the relationship between Sherwood number and Reynolds number can be investigated for either particle structure, as shown in Fig. 6. Regardless of the particle structure, the calculated Sherwood number increases with Reynolds number. Furthermore, we find from the curves fitted to the obtained data that this increase is exponential in both cases. However, the Sherwood numbers for

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ACCEPTED MANUSCRIPT the homogeneous structure are 3–5 times larger than those for the heterogeneous structure. This illustrates that the emergence of particle clustering would have a

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seriously detrimental influence on the gas–solid mass-transfer effectiveness of process

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involving flow around particles.

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Traditionally, the mass-transfer coefficients [58–60] for gas–solid flows are expressed as empirical functions of particle Reynolds number, Schmidt number, and solid volume fraction. Structural heterogeneity is often neglected within

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computational grids, which leads to incorrect predictions of the transport processes of

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practical gas–solid flows. Therefore, for a heterogeneous gas–solid fluidization system, the development of new processes and the improvement of existing processes

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(e.g., vessel configuration, internal design, particle size distribution, extraneous fields)

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should be directed as much as possible toward reducing the formation of particle clusters, to maintaining a state of homogeneous distribution, and to enhancing the

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gas–solid mass transfer.

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5 Conclusions

In this paper, we presented a LB method with mass transfer for fully resolved simulations of flow past arrays of particles with both homogeneous and heterogeneous structures. A steady diffusion process between two parallel walls was simulated successfully with the proposed approach. In general, the agreement between the simulated results and the analytical solution was reasonable. Based on this validation of our model, we performed fully resolved simulations of the flow around two different particle configurations by using the LBM coupled with mass transfer. From this, we obtained not only the gas flow field around the particles but also the distribution of gas concentration in the flow. We found that particle

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ACCEPTED MANUSCRIPT clustering had a significant influence on the velocity and concentration distributions in the flow process. In addition, through quantitative analysis of the mass-transfer

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process for flow around particles, we found that for both homogeneous and

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heterogeneous particle distributions, the obtained Sherwood number increased

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exponentially with Reynolds number. In general, the Sherwood number for the homogeneous configuration was 3–5 times larger than that for the heterogeneous configuration, illustrating that the emergence of particle clustering would seriously

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influence the gas–solid mass-transfer effectiveness of processes involving flow

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around particles.

In closing, we have shown by our numerical results that particle clustering greatly

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affects the mass transfer between gas and particles in gas–solid flows and that the

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LBM is an efficient way to simulate flow past arrays of particles that are distributed either homogeneously or heterogeneously. Further investigation should be made of the

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use of lattice Boltzmann simulation for practical gas–solid fluidization involving chemical reactions and the transfer of heat and mass.

A

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Notation

effective surface area for mass transfer, m2

C

lattice velocity, ms-1

cs

speed of sound, ms-1

C

gas concentration, kgm-3

Cin

inlet concentration, kgm-3

Cout

outlet concentration, kgm-3

D

diffusion coefficient, m2s-1

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ACCEPTED MANUSCRIPT axial dispersion coefficient, m2s-1

dp

solid particle diameter, m

De

volume averaged equivalent bubble diameter, m

ea

discrete velocity vector

F

distribution functions

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feq

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Day

particle equilibrium distribution function concentration distribution function

geq

particle concentration equilibrium distribution function

H

domain height, m

ks

mass-transfer coefficient, g ms-1

m

lattice mass, kg

P

pressure, pa

Rep

particle Reynolds number

Sh

Sherwood number

U

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D

TE

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AC

T

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G

time, s

gas velocity, ms-1

u0

gas inflow velocity, ms-1

wa

weighting factors

W

domain width, m

x

position of lattice, m

Greek letters

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ACCEPTED MANUSCRIPT time step, s

x

lattice length, m



porosity of fluidized bed



kinematic viscosity, m2s-1



gas density, kgm-3

Τ

dimensionless relaxation time for flow

τc

dimensionless relaxation time for mass transfer

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t

Subscripts solid particle k



the  th direction of lattice discrete velocity

p

Particle

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Abbreviations

TE

D

K

Bhatnagar–Gross–Krook

D2Q9

Two-dimensional nine-velocity

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BGK

EMMS

Energy-minimization multi-scale

LBM

Lattice Boltzmann method

LGA

Lattice gas automata

TFM

Two-fluid model

Acknowledgements This work was supported financially by the National Natural Science Foundation of China (Nos. 91434113 and 21106155), the National Key Basic Research Program

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ACCEPTED MANUSCRIPT of China (No. 2015CB251402) and the Chinese Academy of Sciences (Nos. QYZDB-SSW-SYS029, CXJJ-14-Z72, and XDA07080303).

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ACCEPTED MANUSCRIPT List of Figures

Fig. 1. Schematic diagram of two different particle distributions: (a)

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Fig. 3. Detailed flow fields for gas flow around particles: (a) homogeneous

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Fig. 4. Concentration distributions for gas flow around particles: (a)

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homogeneous distribution, (b) clustered distribution Fig. 5. Evolutions of Sherwood number: (a) homogeneous distribution, (b)

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List of Tables

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1. Table 1 Simulation and physical parameters for flow around particles

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2. Table 2 Gas inflow velocities for flow around particles

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Table 1 Simulation and physical parameters for flow around particles Dimensionless

310-6 m

Time step  t

1.210-7 s

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Lattice mass m

9.010-9 kg

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25 ms-1

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1.209 kgm-3

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Gas inflow velocity u0

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Domain height H

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Domain length W

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Kinematic viscosity

1.210-5 m2s-1

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Diffusion coefficient D

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Flow relaxation time τ

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Mass-transfer relaxation time τc

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Lattice velocity c

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Table 2 Gas inflow velocities for flow around particles Gas inflow velocityu0

Dimensional

0.05 ms-1 0.10 ms-1 0.15 ms-1 0.20 ms-1 0.25 ms-1 0.30 ms-1 0.35 ms-1 0.40 ms-1 0.45 ms-1 0.50 ms-1

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0.014

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0.018

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Graphical abstract

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ACCEPTED MANUSCRIPT Highlights A lattice Boltzmann model coupled mass transfer is proposed and verified.

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Fully resolved simulations for flow past arrays of particles are conducted.

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Macroscopic mass balance is used to quantify the mass transfer coefficients.

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Particle clusters seriously affect mass transfer efficiency between gas and solid.

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