A cluster structure-dependent drag coefficient model applied to risers

Powder Technology 225 (2012) 176–189

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A cluster structure-dependent drag coefficient model applied to risers Wang Shuai a, Liu Guodong a, Lu Huilin a,⁎, Xu Pengfei a, Yang Yunchao a, Dimitri Gidaspow b a b

School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, 150001, China Department of Chemical and Biological Engineering, Illinois Institute of Technology, Chicago, IL 60616, USA

a r t i c l e

i n f o

Article history: Received 29 May 2011 Received in revised form 14 March 2012 Accepted 2 April 2012 Available online 10 April 2012 Keywords: Fluidization Computational fluid dynamics Hydrodynamics Kinetic theory of granular flow Structure-dependent drag coefficient Wall friction

a b s t r a c t Cluster structures affect macroscopic hydrodynamic behavior in gas–solid risers. The moment and energy balances for the dense phase and dilute phase are presented by the multi-scale resolution approach to investigate the dependence of drag coefficient on structure parameters. The modified model of cluster structuredependent (CSD) drag coefficient is proposed on the basis of the minimization of energy dissipation by heterogeneous drag (MEDHD). Unlike previous works on CSD drag coefficient model, the modified CSD drag model takes wall friction into account. The closure for the drag coefficient depends not only on flow behavior of gas and particles but also on the wall friction. The structure-dependent drag coefficients calculated from the approach of the minimization of energy dissipation by drag force are then incorporated into the twofluid model to simulate the behavior of gas–solid flow in a riser. The distributions of concentration and velocity of particles are predicted. Simulated results are in agreement with experimental data published in the literature. The effect of the wall friction on flow behavior of particles is analyzed. © 2012 Elsevier B.V. All rights reserved.

1. Introduction Computational fluid dynamics (CFD) have been extensively applied to understand the hydrodynamics of the riser flow. Most studies apply the Eulerian–Eulerian two-fluid model (TFM) which assumes the gas–solid phases as continuous and fully interpenetrating within each control volume. Among the various attempts to formulate the particulate flow, the kinetic theory of granular flow (KTGF) is widely used in fluidization [1]. This treatment of the particulate phase uses classical results from the kinetic theory of dense gases [2]. This approach uses a one equation model to describe the turbulent kinetic energy of particles by introducing the concept of granular temperature of particles. The granular temperature equation can be expressed in terms of production of fluctuations by shear, dissipation by kinetic and collisional heat flow, dissipation due to inelastic collisions, production due to fluid turbulence, and dissipation due to interaction with the fluid. The kinetic theory approach for granular flow allows the determination of the solid pressure, shear viscosity and bulk viscosity of particles. The empirical models for solid pressure and viscosity are avoided completely. Therefore, this theory makes it possible to use a more fundamental approach in the fluidization. Thus, the flow behavior of particles can be predicted in combination with the kinetic theory of granular flow. Eulerian–Eulerian models require closure laws for gas–particle interactions. The interphase momentum transfer between the gas phase

⁎ Corresponding author. Tel.: +86 451 8641 2258; fax: +86 451 8622 1048. E-mail address: [email protected] (L. Huilin). 0032-5910/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2012.04.006

and solid phase is expressed by the drag coefficient multiplied by the relative velocity. The available drag models can be briefly classified into two categories; (i) the conventional drag models [1,3,4] and (ii) the structure-based drag models [5,6]. The conventional drag models, such as those of Gidaspow [1] and Syamlal–O'Brien [7], are derived using the terminal velocity data for a single particle and pressure drop data from a dense packed bed. The Gidaspow drag model is a combination of the Wen–Yu [3] and Ergun [8] equations. It approaches to the Wen–Yu equation for the porosity greater than 0.8, and to the Ergun equation for the porosity less than 0.8. The Syamlal–O'Brien drag model [7] is derived for a single spherical particle in a fluid, and modified with a relative velocity correlation. The relative velocity correlation is the terminal settling velocity of a particle in a system divided by the terminal settling velocity of a single sphere. The main idea about this model is the assumption that the Archimedes number is the same in a single particle and a multiparticle system. The drag correlation has been developed from latticeBoltzmann method (LBM) simulations [9]. For the case of LBM, particles are tracked individually and a fine grid (compared to the particle size) is used to solve for the fluid phase. A no-slip boundary condition is applied over the particle surface. Thus, the flow field around the each particle is resolved in detail. The drag force on each particle is calculated from the integration of the pressure and stress fields of the fluid over the particle surface. Gas–particle flows in risers are inherently unstable, and they manifest fluctuations in velocities and local suspension density over a wide range of length and time scales. These fluctuations are associated with the random motion of the individual particles (typically characterized through the granular temperature) and with the chaotic

W. Shuai et al. / Powder Technology 225 (2012) 176–189

motion of particle clusters, which effect on gas–solid interactions. The conventional drag models do not represent this aspect appropriately, and a direct application of these models leads to unrealistic predictions. Zhang and VanderHeyden [6] have shown that the drag of suspensions depends strongly on the arrangement of particles and therefore the structure of the suspension should be taken into account in the calculation of the momentum exchange. They attributed the most important effects of cluster structures to drag reduction. Since the surrounding mixture is much denser than pure gas, they stated that the added-mass force between the cluster and the surrounding mixture of solid and gas is important. A major limitation of these approaches is the assumption that the clusters are dense with a solid fraction close to maximum packing. A scaled drag model is implemented into the simulation of a fluidized bed of FCC particles [10,11]. A value of the scale factor between 0.2 and 0.3 gives reasonable results for the bubbling fluidized bed regime investigated. Andrews et al. [12] showed that the effective drag law, obtained by averaging (the results gathered in highly resolved simulations of a set of microscopic two-fluid model equations) over the whole domain, is very different from those used in the microscopic two-fluid model and depends on size of the periodic domain. Using these numerical results, they constructed ad hoc subgrid models for the effects of the fine-scale flow structures on the drag force. They demonstrated that this subgrid scale drag force correction affects the predicted large scale flow patterns profoundly. This filtered drag force model is necessary to simulate large-scale risers due to its effectiveness on coarse-sized grids. The structure-dependent drag coefficient model is used in the numerical simulations of risers using an energy minimization multiscale approach (EMMS) to take the heterogeneity of flow into account [13,14]. A structure-based drag coefficient model was proposed, and used to carry out computational fluid dynamics simulations for low solid flux fluid catalytic cracking (FCC) risers [15,16]. In their model, the force balances were derived from the momentum conservation of the individual phases assuming that gas pressure only acts on the gas phase. While a gas–solid twophase model assumed that gas pressure acts on both gas phase and solid phase. The equation for the pressure balance between cluster and dilute phase is also not reasonable. Therefore, such models are unable to accurately predict the momentum exchange between the individual particles in the dilute phase and the clusters in the dense phase. Recently, we proposed a cluster structure-dependent (CSD) drag coefficient model from the local structure parameters of the dense and dilute phases based on the minimization of the energy consumed by heterogeneous drag [17]. The heterogeneous gas–solid flow structure is resolved into the dense phase and the dilute phase. The forces of the dense phase, dilute phase and the interface between the two phases are determine from the gas and solid momentum conservation equations of the dense and dilute phases. However, in the CSD drag coefficient model, the effect of the wall friction force is neglected in the momentum balance. This results in the under-predicted the concentration of particles near the walls. Reviewing the literature there is strong evidence that the many phenomena, such as agglomeration of particles in the vicinity of the wall, are greatly affected by wall friction. In the present work, the modified cluster structuredependent drag coefficient (modified CSD drag) model is proposed with consideration of wall friction. The gas–solid flow behavior in the riser is simulated and compared with experimental results published in the literature. Simulated results with and without wall friction are analyzed. 2. Wall friction factor Many experimental results have shown that the contributions of friction and acceleration to the total pressure drop cannot be neglected under certain operating conditions in risers [18,19]. Swaaij

177

et al. [18] found the pressure drop due to friction to be 20–40% of the measured total pressure drops in dilute flows. Wirth et al. [19] found the deviation of the apparent solid concentrations from the actual ones to be about 20%. Rautiainen and Sarkomaa [20] found that when the solids near the riser wall moved downward, the particle friction factor became negative. They also found that particle diameter had great influences on the particle friction factor. The friction pressure loss is often separated into two parts due to gas alone and to the effect of solid particles


 dp dp dp ¼ þ dz f dz gw dz pw

ð1Þ

where the first term on the right hand side of the equation stands for the pressure drop due to gas–wall friction, and the last term is the pressure drop due to particle–wall friction. The pressure drop due to gas–wall friction is assumed to be the same when only gas is flowing in the same riser, which can be expressed by the Fanning equation:
 2f g ε g ρg u2g dp ¼ dz gw D

ð2Þ

where D is the internal diameter of riser. The gas friction factor, fg, has been calculated using the Blasius correlation:  fg ¼

−1

4Re 0:0791Re−0:25

Re≤2300 Re > 2300

ð3Þ

where the Reynolds number, Re, is defined as (Dρgug) / μg. Similar to the gas–wall friction factor, most investigators (e.g., Stemerding [21]; Capes and Nakamura [22]) have defined the frictional pressure drop due to particle–wall friction following the Fanning equation as a function of particle–wall friction factor.
 2f p εs ρs u2s dp ¼ dz pw D

ð4Þ

where fp is the particle–wall friction factor that is traditionally determined by using planar shear cell equipment [23,24] and fitting the measured pressure drops. A large number of correlations have been proposed in the literature in order to predict the particle–wall friction factor, which can be defined as: a constant; a function of particle velocity; a function of dimensionless numbers; or a function of both solid concentration and particle velocity. This factor has been experimental studied by many researchers for pneumatic transport lines. However, the particle–wall friction factor, fp, is the term in dispute. Some of the expressions for particle–wall friction factors found in the literature are summarized in Table 1. Since the existence of dispersed particles and clusters has significant impacts on the gas flow field in gas–solid two-phase flow, it is difficult to measure the gas–wall friction and the particle–wall friction separately. Usually, it is the combined friction between gas– solid suspension and wall that most investigators would measure. As a result, it is different from the common approach to separately evaluate the gas–wall and particle–wall frictional pressure loss. In this study, the Fanning friction equation for single fluid flow is used to define a friction factor of gas phase and solid phase between suspension and riser wall. 3. Gas–solid two-fluid model with CSD drag model In the present work, the fluid dynamic model is based on the Eulerian–Eulerian approach and treats the gas and solid phases as interpenetrating continua. Separate conservation equations of mass and momentum are formulated for both the phases. Since the solid phase is considered as a fluid, constitutive equations for the solid phase

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Table 1 Particle–wall friction factor correlations reported in the literature. Authors

Correlation

Capes et al. [22] Breault et al. [42] Kmiec et al. [43] Stemerding [44] Yousfi et al. [45] Garic et al. [46] Mabrouk et al. [47]

fp fp fp fp fp fp fp fp

¼ 0:048u−1:22 −1 s ¼ 12:2εs ε3g us −0:75 ¼ 0:074us ¼ 0:003 ¼ 0:0015−0:003  −1:5 ut ¼ 0:0017 εε4s uut ugεs−u s g g ¼ 0:022u−1:0 for smooth wall s ¼ 0:051u−1:0 for rough wall s

ds (μm)

ρs (kg/m3)

D (m)

H (m)

470–3400 296–452 683–2240 65 20–290 1200–2900 170–250

911–7850 2180–2560 802–1154 1600 868–2740 2507–2640 2500–3400

0.0762 0.038 0.04 0.051 0.038–0.05 0.03 0.052

4.87 2 16.2 10 6 1.2 1.0

viscosity and pressure are required. In the present work, the constitutive equations are obtained from the kinetic theory of granular flows (KTGF). In KTGF, the solid viscosity, pressure and thermal conductivity are expressed in terms of a granular temperature which characterizes the random motion of particles [1]. The governing equations are given below.

each phase and the constitutive relations are given in Table 2. The continuity for gas phase and solid phase is expressed by Eqs. (T2-1) and (T2-2). Mass exchanges between the phases, e.g. due to reaction, is not considered. The momentum balance for the gas phase is given by the Navier– Stokes equation, modified to include an interphase momentum transfer term.

3.1. Governing equations

   ∂ ε g ρg ug þ ∇⋅ εg ρg ug ug ¼ −εg ∇pg þ εg ∇⋅τg þ εg ρg g ∂t   þ βgs us −ug

For simplicity, the following hypotheses are considered: (1) both phases are assumed to be isothermal, and no interface mass transfer is assumed; and (2) the solid phase is characterized by a mean particle diameter and density. Both phases are continuous assuming a single gas phase and a single solid phase. The governing equations for Table 2 Mathematical model of gas–solid flow in fluidized beds. A. Conservation equations (1) Continuity equations (a) Gas  phase   ∂ ε g ρg þ ∇⋅ εg ρg ug ¼ 0 ∂t (b) Solid phase ∂ ðεs ρs Þ þ ∇⋅ðεs ρs us Þ ¼ 0 ∂t (2) Momentum equations (a) Gas phase    ∂ εg ρg ug þ ∇⋅ ε g ρg ug ug ∂t

¼ −εg ∇pg þ ε g ∇⋅τ g þ εg ρg g þ β gs us −ug (b) Solid phase ∂ ðεs ρs us Þ þ ∇⋅ðεs ρs us us Þ ∂t

¼ −εs ∇pg −∇ps þ εs ∇⋅τ s þ ε s ρs g þ β gs ug −us (3) Equation of conservation of solid fluctuating energy   3 ∂ ðεs ρs θÞ þ ∇⋅ðεs ρs θÞus 2 ∂t ¼ ð−∇ps I þ τ s Þ : ∇us þ ∇⋅ðks ∇θÞ−γs þ ϕs þ Dgs B. Constitutive equations (a) Gasnphase stress h

T i

o τ g ¼ μ g ∇ug þ ∇ug − 13 ∇⋅ug I (b) Solid nh phase stress i o τ s ¼ μ s ∇us þ ð∇us ÞT − 13 ð∇⋅us ÞI þ ξs ∇⋅us (c) Solid pressure ps = εsρsθ[1 + 2goεs(1 + e)] (d) Shear viscosity ofq solids ffiffi pffiffiffiffi

2 10ρs ds πθ 4 μ s ¼ 45 ε2s ρs ds g o ð1 þ eÞ πθ þ 96 ð1þeÞε s g o 1 þ 5 g o ε s ð1 þ eÞ (e) Bulk solid viscosity qffiffi ξs ¼ 43 ε2s ρs ds g o ð1 þ eÞ πθ (f) Thermal of particles pffiffiffifficonductivity

2 1=2 6 s ds πθ þ 2ε2s ρs ds g o ð1 þ eÞ πθ ks ¼ 25ρ 64ð1þeÞgo 1 þ 5 ð1 þ eÞg o ε s (g) Dissipation fluctuating energy  qffiffi 

γs ¼ 3 1−e2 ε2s ρs g o θ d4s πθ −∇⋅us (h) Radial distribution function at contact   1=3 −1 εs g o ¼ 1− εs;max

(T2-1) (T2-2)

(T2-3)

(T2-4)

(T2-5)

(T2-6) (T2-7) (T2-8) (T2-9) (T2-10) (T2-11) (T2-12) (T2-13)

(i) Rate of energy dissipation per unit volume
2   18μ g ug −us 2 2

(T2-14)

(j) Exchange of fluctuating energy between gas and particles ϕs = − 3βgsθ

(T2-15)

ρs Dgs ¼ 4pdsffiffiffiffi πθg

o

d s ρs

ð5Þ

where g is the gravity acceleration, pg the thermodynamic pressure, βgs the gas–solid drag coefficient and τg the viscous stress tensor. The stress tensor of gas phase is represented by Eq. (T2-6), where μg is the viscosity of gas phase. For simplicity, a constant viscosity of gas phase is used in present simulations. Note that a variety of turbulence models are available to calculate gas viscosity; however, these models are commonly used due to their simplicity and reasonable accuracy for a wide range of turbulent flows [25]. Thus, the effect of gas turbulent model is further investigated in future. The solid phase momentum balance is given by [1]: ∂ ðε ρ u Þ þ ∇⋅ðε s ρs us us Þ ¼ −εs ∇pg −∇ps þ εs ∇⋅τ s þ εs ρs g ∂t s s s   þ βgs ug −us

ð6Þ

where τs is the solid stress tensor and expressed by Eq. (T2-7). Note that the momentum equations of gas and solid phases treat the gas pressure drop in both the gas phase and solid phase. This model is referred to as model A [1] in fluidized beds. The modeling of the kinetic stress of particles is used by means of the kinetic theory of granular flow (KTGF). This theory assumes dry granular nearly elastic particles where particles behavior is the major source of momentum transfer. A granular temperature is introduced in order to quantify the energy contained within these random particle motions. The granular temperature, θ, is defined as: θ = C 2 / 3, where C is the particle fluctuating velocity. The equation of conservation of solid fluctuating energy is [1]   3 ∂ ðεs ρs θÞ þ ∇⋅ðε s ρs θÞus ¼ ð−∇ps I þ τs Þ 2 ∂t : ∇us þ ∇⋅ðks ∇θÞ−γs þ ϕs þ Dgs :

ð7Þ

The two terms on left hand side represent accumulation and convection, while the terms on the right hand side represent production due to shear, diffusive transport, dissipation due to inelastic collisions, exchange of fluctuating energy between gas and solid phase and dissipation due to fluid friction. The particle pressure represents the particle normal forces due to particle–particle interaction. Its description based on the kinetic theory of granular flow was developed. In this approach, both the kinetic and the collisional influences are taken into account. The kinetic

W. Shuai et al. / Powder Technology 225 (2012) 176–189

179

Fig. 1. Local heterogeneous flow with three phases in a cell.

portion describes the influence of particle translations, whereas the collisional term accounts for the momentum transfer by direct collisions [1]. The particle pressure is calculated by Eq. (T2-8). Granular temperature as well as the radial distribution function (measure of the average distance between particles) are used to determine the solid pressure. The shear viscosity accounts for the tangential forces. It was shown by Gidaspow [1] that it is possible to combine different inter-particle forces and to use a momentum balance similar to that of a true continuous fluid. The shear viscosity of particles is calculated by Eq. (T2-9). The bulk viscosity formulates the resistance of solid particles to compression and expansion and is expressed by Eq. (T2-10). For the conductivity of granular energy, the correlation proposed by Gidaspow [1] is used, and expressed by Eq. (T2-11). The rate of dissipation of fluctuation kinetic energy due to particle collisions and the rate of energy dissipation per unit volume resulting from the transfer of gas phase fluctuations to the particle phase fluctuations are calculated by Eqs. (T2-12) and (T2-14). 3.2. Cluster structure-dependent (CSD) drag model In the CFB riser, gas and particles are considered to be either in the clusters of the dense phase or in the gas-rich dilute phase. This means that particle movements are in the form of clusters in the dense phase or in the form of a dispersed particle in the dilute phase in a grid cell, seeing Fig. 1. The local flow divided into three kinds of flow, the dense phase in (b) that characterizes the clusters, the dilute phase in (c) which is outside the clusters and the interface in (d) between the dense phase and the dilute phase. The dense phase and the dilute phase are not in equilibrium, their gas and particles being accelerated or decelerated by complex interactions. The disparity of gas–solid interaction mechanisms inside the two phases gives rise to the interaction between the dense phase and the dilute phase. Thus, the occurrence of interaction between the dense phase and the dilute phase arises from the appearance of the heterogeneous two-phase structure, as shown in Fig. 1d. Therefore, when the clusters are analyzed as large particles, the drag from the dilute phase will act on the dense phase. The volume fraction of dense phase is defined as f = Vden / V, where V is the control volume of a grid cell, Vden is the volume of dense phase in a grid cell. The porosities of dense phase and dilute phase are εden = Vg,den / Vden and εdil = Vg,dil / Vdil, where Vdil is the volume of dilute phase in a grid cell. Vg,den and Vg,dil are the gas volumes of dense and dilute phases in the control volume of a grid cell. The correlations for these parameters are given in Table 3. In order to establish a mathematical model for both dense phase and dilute phase, we make the following assumptions: (1) The dense phase exists as spherical clusters with diameter dc. (2) Particles in the dilute phase are uniform. (3) Particles within the clusters and the clusters in a control volume are homogeneously dispersed. (4) The mass and momentum exchanges between the dilute phase and the dense phase are neglected in the control volume of a grid cell. (5) The stresses of gas phase and particles in the dense and dilute

phases are neglected. Considering these assumptions, the conservation equations of mass and momentum of gas phase and particles are given as follows. 3.2.1. Correlations for dense phase and dilute phase in the cell Flow of gas phase and solid phase in the grid cell is characterized by two-phase structures consisting of particle-rich dense phase and fluid-rich dilute phase. The dense and the dilute phases pass through an arbitrary microspace intermittently with varying properties. The correlations of the dense phase and dilute phase are listed in Table 4. From the mass balance for the gas and the solid phases, the mean velocities of gas phase and solid phase in the cell are shown in Eqs. (T4-6) and (T4-7), where Ug,den and Ug,dil are the gas superficial velocities through the dense phase and the dilute phase in the cell along vertical (z) direction. Us,den and Us,dil are the superficial velocities of particles in the dense phase and the dilute phase in the cell along vertical (z) direction. The parameters for the dense and dilute phases are related to the local average parameters. The overall porosity is related to the dense-phase and the dilute-phase porosities, and expressed by Eq. (T4-5). 3.2.2. Momentum conservation of gas phase in the dense and dilute phases in the cell When the clusters are analyzed as large particles, the drag from the dilute phase will act on the clusters, i.e., the dense phase. Both phases are combined through the drag term to form the interface. The gas tends to flow through the path between clusters because of the larger flow resistance within the clusters. The drag from the dilute phase will act on the dense phase. Thus, the drag forces include the drag force components on dense phase Fden, dilute phase Fdil, and interfacial force between the dense and the dilute phases, Fint. The momentum

Table 3 Correlations and parameter used in the model. 1. Superficial slip velocity, gas velocity through dense phase and number of particles in dense phase f U g;den ε U s;den den Þ U g;den ¼ U den þ den and nden ¼ f ð1−ε 3 1−εden ; ug;den ¼ εden πds =6 2. Superficial slip velocity, gas velocity through dilute phase and number of particles in dilute phase ð1−f ÞU ε dil U s;dil U g;dil ¼ U dil þ 1−ε ; ug;dil ¼ ð1−εdeng;dilÞ and ndil ¼ ð1−f Þð31−εdil Þ dil πds =6 3. Superficial slip velocity   and number of clusters εdil U s;den f U int ¼ U g;dil − 1−εden ð1−f Þand nint ¼ 3 πdc =6 4. Superficial velocity hof particles in the dilute phase i

εden dil Þð1−ε den Þ us −ð1−f ÞU dil −f U den U s;dil ¼ ðð1−ε ε u − 1−ε g g g 1−εden ε dil −ε den Þð1−f Þ 5. Superficial velocity of particles in the dense phase ð1−εg Þus −ð1−f ÞU s;dil U s;den ¼ f 6. Drag coefficients    −0:313 ρg ds U den −1 ρ d U þ 3:6 g μs den C den ¼ 24 μg  −1  g−0:313 ρ d U ρ d U C dil ¼ 24 g μs dil þ 3:6 g μs dil  g −1  g −0:313 ρg dc U int ρ d U C int ¼ 24 þ 3:6 g μc int μg g 7. Particle–wall friction  2f p ð1−εg Þρs us2 dp ¼ D dz pw f p ¼ 0:048u−1:22 s

(T3-1)

(T3-2)

(T3-3) (T3-4) (T3-5) (T3-6) (T3-7) (T3-8) (T3-9)

(T3-10)

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Table 4 Model for cluster structure-dependent drag coefficient. 1. Interface momentum transfer coefficient h i ε2 ρ f ð1−εden ÞC den U 2den ð1−f Þð1−εdil ÞC dil U 2dil f C U2 β CSD ¼ 34 u g−ug þ þ intdc int ε g ≥0:8 ds ds j g sj 2 ρg ð1−εg Þjug −us j ð1−εg Þ μ g þ 1:75 εg b0:8 β E ¼ 150 2 ds

(T4-1) (T4-2)

ε g ds

βgs = φgsβE + (1 − φgs)βCSD

(T4-3)

ð0:2−ε s Þ þ 0:5 ϕgs ¼ arctan½1501:75 π 2. Balance equations εg = fεden + (1 − f)εdil

ug ¼ us ¼

1 εg



(T4-4) (T4-5)

f U g;den þ ð1−f ÞU g;dil andU g ¼ f U g;den þ ð1−f ÞU g;dil

1

ð1−εg Þ

(T4-6)



f U s;den þ ð1−f ÞU s;dil andU s ¼ f U s;den þ ð1−f ÞU s;dil

(T4-7)

3. Equations for drag force of dense and dilute phases   2 24μ 3:6μ 0:313 πd ρ ε−4:7 g Fden ¼ s g8 den ρ ds Ugden þ 0:313 jU den jU den g ρ d U ð g s den Þ   πd2 ρ ε−4:7 24μ 3:6μ 0:313 g Fdil ¼ s 8g dil ρ ds Ug þ 0:313 jU dil jU dil dil g ðρg ds U dil Þ   2 −4:7 24μ g 3:6μ 0:313 πd ρ ð1−f Þ g þ Fint ¼ c g 8 0:313 jU int jU int ρg dc U int ðρg dc U int Þ

(T4-8) (T4-9) (T4-10)

4. Equation for superficial slip velocity of the dense phase " # !0:687 (
 2

∂pg ρg ds ε5:7 dp δ den ds iw

U 1:687 þ 1−εg 0:15 den þ U den ¼ dz μg 18εg ð1−εden Þμ g ∂z pw 1−ε g " #)
  





dp δiw þ ρs −ρg ð1−f Þð1−εdil Þ g þ as;dil þ f ð1−εden Þ g þ as;den þ ð1−f Þ εdil ρg ag;dil −ag;den þ ðεden −εdil Þ dz gw ε den 5. Equation for superficial slip velocity of the dilute phase     0:687    2

ε4:7 d ρ d ∂p δiw dil s U 1:687 þ U dil ¼ 18μ ρs −ρg g þ as;dil þ ∂zg þ dp 0:15 μg s dil 1−εdil dz g

g

pw

6. Equation for superficial slip velocity of the meso-scale interface !0:687 ( " #
 2 5:7 ρg dc ∂pg d ð1−f Þ dp 1:687 U int þ U int ¼ c ðε dil −εden Þ δiw þ 0:15 μg 18ε g μ g dz pw ∂z


  





dp δ þ ρs −ρg εdil ð1−εden Þ g þ as;den −εden ð1−εdil Þ g þ as;dil þ εden εdil ρg ag;den −ag;dil þ ðεdil −εden Þ dz gw iw 7. Hydrodynamic equivalent diameter of the cluster n  o f U s;den þð1−f ÞU s;dil ε mf ds − U mf þ1−ε ½f U s;den þð1−f ÞU s;dil  g 1−ε max mf n o dc ¼ ε ρs mf Ndf;min ρ −ρ − U mf þ1−ε f U s;den þð1−f ÞU s;dil  g ½ s g mf

)

8. Stability criterion by minimization of the energy dissipation by drag force N df ;min ¼

(T4-11)

(T4-12)

(T4-13) (T4-14)

(T4-15)

nden F den U g;den þndil F dil U g;dil þnint F int U g;dil ð1−f Þ →minimum ð1−εg Þρs

equation of gas phase in the dense phase at the steady state along vertical (z) direction in the absence of gas stress term is [1,17]  ∂   ∂  f ε den ρg ug;den ug;den þ f ε den ρg ug;den vg;den ¼ ∂z ∂x
 ∂pg dp −f εden δ −nden Fden −f εden ρg g−f dz gw iw ∂z

ð8Þ

where ug,den and vg,den are the gas velocities through the dense phase along vertical (z) and lateral (x) directions. δiw is the Kronecker delta. The term on the left hand side of the equation represents the momentum change of gas in the dense phase. The first term on the right hand side of the equation stands for the pressure gradient component of gas in the dense phase, the second term is the drag force in the dense phase and the third term is the gravitational force. The last term is the gas friction force of the dense phase, which is zero away from the wall. Without the gas wall friction term, Eq. (8) is the same as the gas momentum balance equation used in Shuai et al. [17]. For the dilute phase, the momentum equation of gas phase at the steady state along vertical (z) direction in the absence of gas viscous effects is i ∂ h i ∂ h ð1−f Þεdil ρg ug;dil ug;dil þ ð1−f Þεdil ρg ug;dil vg;dil ¼ ∂z ∂x
 ∂pg dp −ð1−f Þεdil δ −ndil Fdil −n int F int −ð1−f Þεdil ρg g−ð1−f Þ dz gw iw ∂z ð9Þ

where ug,dil and vg,dil are the gas velocities through the dilute phase along vertical (z) and lateral (x) directions. The term on the left hand side of the equation represents the momentum change of gas in the dilute phase. The first term on the right hand side of the equation stands for the pressure gradient, the second and third terms are the drag force in the dilute phase and the force in the interface between the dilute phase and dense phase, the fourth term is the gravitational force and the last term is the gas friction force of the dilute phase which is zero away from the wall. The occurrence of interfacial interaction between the dense phase and the dilute phase arises from the appearance of clusters. Fint represents the drag force acting on cluster. For simplicity, the second term on the left hand side of Eqs. (8) and (9) is neglected. Eliminating the pressure drop from Eqs. (8) and (9) yields   nden Fden n F n F ¼ dil dil þ int int þ ρg ag;dil −ag;den f εden ð1−f Þεdil
ð1−fÞεdil ε −εdil dp þ den δ εdil εden dz gw iw

ð10Þ

  where  ag;dil ¼ ∂ ð1−f Þεdil ug;dil ug;dil =ðð1−f Þεdil ∂zÞ and ag;den ¼ ∂ f εden ug;den ug;den =ðf εden ∂zÞ, and are known as the accelerations of gas in the dilute and dense phases. The correlations for the number densities of particles and the superficial slip velocities are listed in Table 3. Without the gas–wall friction term, Eq. (10) is the same as the model proposed by Shuai et al. [17]. The third term on

W. Shuai et al. / Powder Technology 225 (2012) 176–189

the right hand side of the equation is contributed by the accelerations of gas in the dilute phase and the dense phase. 3.2.3. Momentum equations for solid phase in the dense and dilute phases in the cell As stated in assumptions, the mass and momentum exchanges of particles between the dilute phase and the dense phase are not accounted, and the solid stress is neglected. For flow of particles in the dilute phase, the momentum equation of particles at the steady state along vertical (z) direction is: i ∂ h i ∂ h ð1−f Þð1−εdil Þρs us;dil us;dil þ ð1−f Þð1−εdil Þρs us;dil vs;dil ¼ ∂z ∂x
 ∂pg dp −ð1−f Þð1−εdil Þ δ þ ndil Fdil −ð1−f Þð1−εdil Þρs g−ð1−f Þ dz pw iw ∂z ð11Þ where us,dil and vs,dil are the particles velocities through the dilute phase along vertical (z) and lateral (x) directions. As mentioned above, the solid pressure and viscosity due to the collisions of particles are not accounted in Eq. (11). For flow of gas–solid system, the density of particles is greater than that of the gas phase, and (ρs −ρg)≈ρs. For simplicity, the second term on the left hand side of Eq. (11) is neglected. Eq. (11) can be rewritten to    ndil Fdil ¼ ð1−f Þð1−εdil Þ ρs −ρg g þ as;dil " #
 ∂pg dp δiw þ ð1−f Þð1−ε dil Þ þ dz pw ð1−εdil Þ ∂z

ð12Þ

  where as;dil ¼ ∂ ð1−f Þð1−εdil Þus;dil us;dil =ðð1−f Þð1−ε dil Þ∂zÞ, and is called as the acceleration of particles in the dilute phase. The correlations of ndil and Fdil are given in Table 3. Thus, we obtain the equation for the superficial slip velocity of the dilute phase Udil, and is given in Eq. (T4-12). The superficial slip velocity Udil is computed at the specified ∂pg =∂z which is obtained from numerical simulations in the computational cells. Both the gas pressure gradient and the frictional pressure drop due to particle–wall friction affect the superficial slip velocity in the dilute phase at the walls. Without the particle– wall friction, Eq. (T4-12) is the same as the model proposed by Shuai et al. [17] in the determination of superficial slip velocity of the dilute phase. For flow of particles in the dense phase, the momentum equation at the steady state along vertical (z) direction can be expressed by: i ∂ h i ∂ h f ð1−εden Þρs us;den us;den þ f ð1−εden Þρs us;den vs;den ¼ ∂z ∂x
 ∂pg dp −f ð1−εden Þ δ þ nden Fden þ n int F int −f ð1−εden Þρs g−f dz pw iw ∂z ð13Þ where us,den and vs,den are the particles velocities through the dense phase along vertical (z) and lateral (x) directions. For simplicity, the second term on the left hand side of Eq. (13) is neglected. It can be rewritten to    nden Fden þ n int F int ¼ f ð1−εden Þ ρs −ρg g þ as;den " #
 ∂pg dp δiw þ f ð1−ε den Þ þ dz pw 1−εden ∂z

ð14Þ

  where as;den ¼ ∂ f ð1−εden Þus;den us;den =ðf ð1−εden Þ∂zÞ, and is known as the acceleration of particles in the dense phase. The correlations of nden, nint, Fden and Fint are given in Table 3. Combining Eqs. (10), (12) and (14) yields the expression for the superficial slip velocity in the dense phase Uden given in Eq. (T4-11).  Thesuperficial slip velocity Uden is predicted at the specified 1−εg ∂pg =∂z which the

181

porosity and gas pressure gradient are obtained from numerical simulations in the computational cells. Both the pressure drop due to gas–wall friction and the frictional pressure drop due to particle– wall friction influence on the superficial slip velocity in the dense phase. Substituting Eqs. (10) and (12) into Eq. (14), the expression for the superficial slip velocity Uint is expressed by Eq. (T4-13). The super

ficial slip velocity Uint relates to εdil −εg ∂pg =∂z in which the porosity of the dilute phase is obtained by solving the above non-linear equations and the porosity is obtained from numerical simulations in the computational cells. In the calculation of superficial slip velocities Udil, Uden and Uint, the gas pressure gradient ∂pg =∂z in the computational cell is considered. This term effects significantly on concentration of particles in the cell. Here, the gas pressure gradient is imposed to the computational cell to keep the pressure gradient components of the dense phase and the dilute phase constant in the computational cell. We also note that the importance of the pressure drop due to particle– wall friction is involved in. It affects the superficial solid velocities in the cluster and dilute phases, and the interface between two phases at the walls. In Eq. (T4-13), dc is the equivalent diameter of clusters. The occurrence of particle clusters is well accepted with upward moving particle clusters in the center and downward moving particle clusters near the walls in risers. Experimental (e.g. Gu and Chen [26]; Harris et al. [27]) or theoretical work yields expressions for the calculation of clusters diameter. Harris et al. [27] presented correlations for predicting the properties of cluster of particles traveling near the riser wall. The cluster diameter used in the above momentum balance equation is given by (e.g., Chavan [28]; Li et al. [29])

dc ¼

ds

h

Us 1−εmax

ρs Nst ρ −ρ s

g

 i εmf Us g − Umf þ 1−ε mf   εmf Us − Umf þ 1−ε g

ð15Þ

mf

where Nst is the energy consumption for suspension and transport of particles. Us is the superficial velocity of particles which given in Eq. (T4-7). εmax represents the maximum porosity beyond which clusters do not exist. The value of εmax depends on operating conditions and material properties. For fine powders fluidized by air, it could be taken as 0.9997 [30–32]. When the porosity is greater than the maximum porosity, the slip velocity is very nearly the terminal velocity of a single particle [30,31]. The value for εmax of 0.9997 is used as a first approximation in present simulations. 3.2.4. Minimization of energy consumed by drag force Gas and particles have their respective movement tendencies in the dense and the dilute phases. Both phases are usually not in equilibrium by complex interactions. Interfacial particles experience a locally interstitial force difference between the dense and the dilute phases, featuring deformable interface. The drag force times the gas velocity equals to the energy dissipation rate [33]. Hence, with respect to unit mass of particles in unit cross-sectional area, the energy consumed by drag force per unit mass of particles defines h i 1  Ndf ¼  nden Fden U g;den þ ndil Fdil U g;dil þ n int F int U g;dil ð1−f Þ : 1−εg ρs ð16Þ Characteristically, the particles tend to aggregate into clusters to reduce the gas flow resistance, while the gas phase tends to bypass the clusters instead of passing through them. Both the particles and the gas phases have their respective movement tendencies. That is, particles tend to array themselves with minimal energy loss by drag, while the gas phase tends to choose an upward path with minimal resistance.

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From Eq. (16), it shows that the resistance includes the resistance between the gas phase and dispersed particles in the dilute phase, the resistance between the gas phase and particles in the dense phase and the interfacial interaction between the dilute phase and the dense phase. The energy Ndf is given as sum of the respective drag force times slip velocity in the cluster, dilute phase and in the interphase. The term Ndf characterizes the intrinsic tendency of particles toward formation of an array which offers the lowest interaction with the fluid. Therefore, the system is postulated by minimum energy loss by drag force per unit mass of the particles, Ndf, by which the particles aggregate into clusters and the gas flows with relatively low resistance. Hence, the stability condition is needed to the extremum of energy dissipation by drag force, for which the minimization of the energy dissipation by heterogeneous drag (MEDHD) is

Ndf

h i nden Fden U g;den þ ndil Fdil U g;dil þ n int F int U g;int ð1−f Þ   ¼ →minimum: 1−εg ρs

3.3. Boundary conditions The governing equations mentioned above are numerically solved with appropriate boundary and initial conditions. Initially, there are no motions for both the gas phases and the particles in the riser. At the inlet, all velocities and volume fractions of gas phase and particles are specified. At the top, Neumann boundary conditions are applied to the gas–particle flow. The gas pressure is set to be 1 atm. At the inlet, the gas velocity is constant with a specified porosity. The velocities and granular temperature of particles are specified. At the wall, the tangential and normal velocities of gas phase are set to zero (no slip condition). The normal velocity of the particles is also set at zero. The following boundary equations apply to the tangential velocity us,w and granular temperature of particles θw at the wall [35]:

us;w ¼ −

6μ s εs;max ∂u pffiffiffiffiffiffiffiffiffi s;w 3θw ∂n

πϕρs εs g o

ð17Þ θw ¼ − The prediction of the equivalent diameter of clusters is taken at Nst ¼ Ndf . Thus, from Eq. (15), the correlation for the cluster equivalent diameter is given by Eq. (T4-14). There are 8 independent variables (Ug,den, Us,den, Ug,dil, Us,dil, εg,dil, εg,den, f and dc). With the computed gas pressure gradient ∂pg =∂z, gas velocity ug, solid velocity us and porosity εg in the grid cell from continuity and momentum equations, these eight variables are solved to define the gas-solid behavior with seven equations (Eqs. (T4-5), (T4-6), (T4-7), (T4-11), (T4-12), (T4-13) and (T4-14)) and the stability criterion by MEDHD (Eq. (T4-15)). These equations are solved in the cell employing bisection search algorithm. The solution is initiated with guess value for the cluster fraction and iterations are continued until cluster fraction converges satisfying the criteria given by | fi − fi − 1| ≤ 10− 6, where i is the iteration number. Thus, these 8 variables in the model describing flow structure can be determined. 3.2.5. Modified cluster structures-dependent drag coefficient (modified CSD drag model) The drag force is represented by the term βgs(ug − us), the product of the interphase momentum exchange coefficient βgs and the slip velocity. From Eqs. (T4-8)–(T4-10), it can be observed that the drag forces are correlated with the structure parameters by the non-linear equations, implying that it can be regarded as an implicit function of structure parameters. In this manner, the relationship between structure parameters and drag coefficient can be written as follows

βCSD ¼

εg F gs ε2g  ½nden Fden þ ndil Fdil þ nint Fint  εg ≥0:8 : ¼   U slip ug −us 

ð18Þ

Substituting Eqs. (T3-6), (T3-7) and (T3-8) into Eq. (18), the modified cluster structure-dependent drag coefficient βCSD (modified CSD drag model A) is expressed by Eq. (T4-1). Without the wall frictions, the modified CSD drag model A will be recovered to the CSD drag model proposed by Shuai et al. [17]. For flow of particles with high concentrations, the correlations given by Gidaspow [1] are often used in the numerical simulations of fluidized beds, which is based on Ergun equation [8]. The Ergun expression, Eq. (T4-2), for drag coefficient of packed-bed flows, is used when the porosity εg is below 0.8. To avoid discontinuity of these two correlations Eq. (T4-1) and (T4-2), a switch function φgs is introduced to give a smooth from the dilute regime to the dense regime [34], and expressed by Eq. (T4-4).

ks θ ∂θw þ χ w ∂n

pffiffiffi 3πϕρs εs u2s g o θ3=2 w 6ε s;max χ w

ð19Þ

ð20Þ

where ϕ is a specularity coefficient. n is the normal component to the wall. χw is the energy dissipation due to inelastic collisions between particles and the wall, and is given by

χw ¼

 pffiffiffi 2 3=2 3 1−ew πεs ρs g o θw 4εs;max

ð21Þ

where ew is the coefficient of restitution of the wall. The simulations are carried out with the CFD code which previously used to model gas–solid flow in a bubbling fluidized bed [36], and incorporates with the modified CSD drag model A in this work. This software is based on the CFD K-FIX code [1,36]. It allows free implementation of extra equations, boundary conditions, and differencing schemes. The granular kinetic theory and the granular equations described in the previous section are implemented into this code. The numerical scheme used in the K-FIX code is the implicit continuous Eulerian (ICE) approach. The model uses donor cell differencing. The conservation of the momentum equations is in explicit form. The continuity equations are in implicit form. The set of non-linear equations is linearized using a modified version of the SIMPLE algorithm using the void fraction and gas pressure correction equations. The solution of the pressure from the momentum equations requires a pressure correction equation, correcting the pressure and the velocities after each iteration of the discretized momentum equations. A sequential iterative solver is used to calculate the field variables at each time step. The adaptive time step in the range of 0.00001–0.0005 is used. The time step automatically decreases when the solution is changing more rapidly, and it increases when fast transients subside in order to minimize computation time. The fluid dynamics parameters of the inhomogeneous structure, f, εdil, εden, dc, Ug,den, Ug,dil, Us,den, and Us,dil, are obtained from the Eqs. (T4-5), (T4-6), (T4-7), (T4-11), (T4-12), (T4-13) and (T4-14) and the stability criterion from Eq. (T4-15). Then, the three drag components in the three phases can be obtained from Eqs. (T4-8), (T4-9) and (T4-10), and the drag of the overall heterogeneous system from Eq. (T4-3). During each iteration of the CFD computation, the flow parameters obtained from the previous iteration are used in the new drag coefficient. Thus, the drag function employed in the next iteration is a correctly modified value, and during the entire CFD computation, the energy is minimized for the drag function. This process thus combines the modified CSD drag model A with a CFD code.

W. Shuai et al. / Powder Technology 225 (2012) 176–189

183

1.0

The distribution of solid concentration and velocity was measured by Yan and Zhu [37,38] in a circulating fluidized bed. The internal diameter and height of the riser were 0.076 m and 10.0 m. FCC catalyst with a mean diameter of 67 μm and a particle density of 1500 kg/m 3 was used. The local solid concentration and axial velocity were measured with a reflective-type fiber-optic probe. The gas–solid suspension traveled up in the riser and passed through a smooth exit into the primary cyclone for gas–solid separation. Solids from the storage tank entered the riser bottom and were accelerated by air under near ambient conditions. Parameters used in the simulations are listed in Table 5. More details on the experimental apparatus can be found in Yan and Zhu [37,38]. For simplicity, solid particles are fed from the bottom of the riser at minimum fluidization conditions. At these conditions, an expected low value of granular temperature is assigned to the particulate phase. The simulated system is isothermal at 300 K. A preliminary study was initially conducted to aid the choice of the grid resolution for 2-D simulations. Three different grid resolutions were employed and their influence on concentration of particles was used as the criteria for selecting the grid resolution. Fig. 2 shows the concentration of particles along height at three different grids. A similar effect of the grid resolution on concentration of particles was found. The use of very coarse grids (i.e., 12 cells along the radial direction for the half diameter, 140 cells along the axial direction) leads to underestimation of concentration of particles. The numerical simulations for the mid mesh (18 × 188) and finer mesh (32 × 248) resolutions are very similar. Both simulations give the concentrations of particles decrease from the bottom. The concentration of particles is large near the wall. The grid size of the order of a few particle diameters used in this study is usually adequate for resolving the concentration of particles. Therefore, we choose the mid grid resolution of 18 × 188 in present simulations. Fig. 3 shows the simulated flow pattern of instantaneous concentration of particles at the gas velocity and mass flux of 3.5 m/s and 100 kg/m 2 s, respectively. Particles carried by gas travel up in the riser. A characteristic feature of the flow is the oscillating motion of solid particles with a high concentration near the walls and low at the center of the riser. It should be noted the concentration of particles is high near the bottom due to the continual collection of descending particles. While the concentration of particles near the top of the riser is low. The low instantaneous concentration of particles is found near the walls and high in the center of the riser. However, more complex combinations are possible, too. In order to compare simulation results with Yan and Zhu [37,38] experimental data, the time-averaged distributions of flow variables have been computed. Fig. 4 shows the computed time-averaged

0.03 Experiments (Yan and Zhu, 2005) Coarse grids (12x140) Mid grids (18x188) Finer grids (32x248)

0.8

0.6

Coarse grids Mid grids Finer grids z/H=0.6

0.02 0.01

u g=8.0 m/s

0.00

d s=67 μm

0.4

Coarse grids Mid grids Finer grids z/H=0.2

2

Gs=100 kg/m s

ρs=1500 kg/m

0.2

0.0 0.00

0.04

3

0.08 0.0

0.2

0.4

0.6

0.08

0.04

0.8

Yan and Zhu [37,38]

Present simulations

Radii of the riser R Height of the riser H Particle average diameter ds Particle density ρs Operating pressure pg Gas shear viscosity μg Gas density ρg Particle–particle coefficient of restitution e Wall–particle coefficient of restitution ew Specularity coefficient ϕ Maximum solid volume fraction εs,max

38.0 mm 10 m 67 μm 1500 kg/m3 1 atm 1.85 × 10− 5 kg/m s 1.2 kg/m3 /

38.0 mm 10 m 67 μm 1500 kg/m3 1 atm 1.85 × 10− 5 kg/m s 1.2 kg/m3 0.97

/

0.9

/ /

0.5 0.6

0.00 1.0

Concentration of particles Dimensionless distance x/R Fig. 2. Effect of grid resolution on concentration of particles.

concentration of particles at the gas velocity and solid mass flux of 3.5 m/s and 100 kg/m 2 s. It can be seen that the distribution is predicted with a high concentration at the wall and a low in the center. At the center of riser, the simulation results reach a local minimum point, similar to experimental data. This distribution reflects the establishment of a core-annular regime shown by the experiments and the computational results. Predicted concentrations of particles are close to measurements. Both simulations and experiments gave a maximum at the wall. Although the values have quantitative discrepancies between them near the walls, the trends are the same. Simulations are also performed with Ergun/Wen and Yu drag correlations. For porosities less than 0.8, the drag force between gas and particles is calculated by the Ergun equation [8]. For porosities greater than 0.8, Wen and Yu [3] equation is used. The predicted concentration of particles increases toward to the wall. However, at the wall, the simulation results obtained using the equations of Ergun/Wen– Yu are smaller than measurements. We see that the simulation results predicted using the equation of Ergun/Wen–Yu are different from that using equation of modified CSD drag model A. The values predicted by the equations of Ergun/Wen–Yu are smaller than that using modified CSD drag model A and experimental data. The predicted concentration of particles by means of CSD drag model [17]

Table 5 Parameters used in numerical simulations. Description

Concentration of particles

4.1. Comparison with Yan and Zhu experimental data

Dimensionless height z/H

4. Simulated results and discussion

t=35s 37

39

41

42

44

Fig. 3. Instantaneous concentration of particles.

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W. Shuai et al. / Powder Technology 225 (2012) 176–189

10.0

0.15

ds=67 μm,ρs=1500 kg/m

Experiments (Yan and Zhu, 2005) Modified CSD drag model A CSD drag model (Shuai et al., 2011) KTGF with Ergun/Wen-Yu correlations

Axial velocity of particles (m/s)

Concentration of particles

0.20

ds=67 μm 0.10

3

ρ =1500 kg/m

s

ug=3.5 m/s 0.05

0.00 0.0

2 Gs=100 kg/m s

0.2

0.4

0.6

0.8

8.0

2

ug=3.5 m/s, Gs=75 kg/m s

6.0 4.0 2.0 0.0 -2.0 -1.0

1.0

3

Modified CSD drag model A CSD drag model (Shuai et al., 2011) Modified CSD drag model B KTGF with Ergun/Wen and Yu correlations -0.5

0.0

0.5

1.0

Dimensionless distance x/R

Dimensionless distance x/R Fig. 4. Distribution of concentration of particles.

Fig. 6. Distribution of solid axial velocity.

without the wall friction is reduced in comparison to the simulations with the wall friction. We see that the computed results using modified CSD drag model A agree well with experimental data. Fig. 5 shows the comparison between the calculated axial velocity distribution with experimental data [37,38]. This figure shows a high axial velocity region in the center and a low axial velocity of particles at the walls. However, contrary to the experiments, the computational results showed a high velocity in the center regime and negative near the wall. The axial velocity of particles at the walls is predicted to be larger than the experiments. We see that the predicted axial velocity of particles using the equations of Ergun/Wen–Yu is different from that using modified CSD drag model A. The predicted velocity of particles using the equations of Ergun/Wen–Yu is larger than that using modified CSD drag model A and experimental data near the wall. While it is reverse in the center regime of the riser. The difference of the predicted axial velocity of particles between the modified CSD drag model A and the CSD drag model proposed by Shuai et al. [17] is obvious. The simulated results predicted by the modified CSD drag model A with wall friction is lower than that by means of the CSD drag model without wall friction. We see that the computed results using the modified CSD drag model A are in agreement with experimental data.

that particles flow upwards in the center, and downwards near the wall. However, the predictions using Ergun/Wen and Yu correlations give an up-flow of particles with low axial velocity in the riser. The predictions also show that the axial velocity of particles predicted by Ergun/Wen and Yu correlations is lower than that by means of the modified CSD drag model A. It has been well known that there are two different drag models in two-fluid model proposed by Gidaspow [1]: model A and model B, resulting from different treatments of the gas pressure term. Therefore, there are correspondingly two sets of governing equations, and their correct match is important to generate accurate quantitative results. In model A, the gas pressure acts on both the gas phase and the solid phase. While in model B the gas pressure is in the gas phase only. These two models do not result in much difference in fluidization simulation in two-fluid model, although Model B is considered to be well-posed while model A ill-posed [1]. The modified CSD drag model B is given in Appendix A. Simulated results using modified CSD drag model B is also shown in Fig. 6. The predicted axial velocity of particles using modified CSD drag model B is lower in the center regime and higher near the wall than that predicted by modified CSD drag model A. However, the trends are the same. Simulations by modified CSD drag model A and model B are smaller near the walls than that using Ergun/Wen and Yu correlations. However, it is reverse in the center regime of the riser. The comparisons of local concentration of particles with three different drag models are shown in Fig. 7. The distributions of particle concentration are low in the central region and rapidly increase near the

4.2. Radial distributions of particles velocity Fig. 6 shows the distribution of solid velocity with three different drag coefficient models. Both the modified CSD drag model A with wall friction and the CSD drag model without wall friction [17] show

0.3

Experiments (Yan and Zhu, 2005) Modified CSD drag model A CSD drag model (Shuai et al., 2011) KTGF with Ergun/Wen-Yu correlations

6.0

Concentration of particles

Axial velocity of particles (m/s)

8.0

4.0

ds=67 μ m

2.0

3 ρ =1500 kg/m s

Modified CSD drag model A CSD drag model (Shuai et al., 2011) Modified CSD drag model B KTGF with Ergun/Wen and Yu correlations 0.2

ds=67 μ m ρ =1500 kg/m

3

s

ug=3.5 m/s 0.1

2

Gs=100 kg/m s

ug=3.5 m/s

0.0

2 Gs=100 kg/m s -2.0 0.0

0.2

0.4

0.6

0.8

Dimensionless distance x/R Fig. 5. Distribution of axial velocity of particles.

1.0

0.0 -1.0

-0.5

0.0

0.5

Dimensionless distance x/R Fig. 7. Distribution of concentration of particles.

1.0

W. Shuai et al. / Powder Technology 225 (2012) 176–189

2

u g=3.5 m/s, G s=75 kg/m s d s=67 μ m, ρs =1500 kg/m

0.10

3

0.05

0.00 0.0

0.1

0.2

Modified CSD drag model B Modified CSD drag model A Gu and Chen (1996) Harris et al. (2003) Subbarao (2010) u g=3.5 m/s, d s=67 μ m

0.015

Cluster diameter (m)

Granular temperature (m/s)2

0.15

Modified CSD drag model B Modified CSD drag model A KTGF with Ergun/Wen and Yu correlations

0.010

2

G s=75 kg/m s ρ s=1500 kg/m

0.005

0.000 0.00

0.3

0.05

3

0.10

0.15

0.20

0.25

0.30

Concentration of particles

Concentration of particles

Fig. 10. Distribution of cluster diameter as a function of concentrations.

Fig. 8. Distribution of granular temperature as a function of concentrations.

wall. It can be seen that the simulation can reasonably predict a coreannulus flow of particles in the riser. The predicted concentration of particles using modified CSD drag model A and CSD drag model [17] are similar in the center regime. The difference is found near the wall. The concentration of particles is higher using modified CSD drag model A with wall friction than that by means of CSD drag model [17] without wall friction. For the CSD drag model A, the predicted concentration of particles is smaller than that by means of the CSD drag model B near the walls, while they are the same in the center regime of the riser. Both modified CSD drag models A and B give a high concentration of particles in the riser. Combining the results shown in Fig. 6, the concentration is low in the core regime with up-flow of particles and high in the annular region with down-flow of particles. The circulation of particles is formed in the riser. The predictions verify the annular-core flow structure in the riser. Fig. 8 shows the variation of granular temperature as a function of concentration of particles at the superficial gas velocity and solid mass flux of 3.5 m/s and 75 kg/m 2 s, respectively. Both the modified CSD drag models A and B show that the granular temperature increases, reaches a maximum, and then decreases with the increase of concentration of particles. In the dilute region, the granular temperature is proportional to the solid concentration raised to the power of 2/3 [39]. Thus, the granular temperature is increased as the concentration of particles increases. For the modified CSD drag model B and model A, the mean granular temperature are 0.0242 (m/s) 2 and 0.0235 (m/s) 2, respectively. Simulated granular temperature by Ergun/Wen and Yu correlations is also shown in Fig. 8. Compared to the modified CSD drag models, the drag model using Ergun/

185

Wen and Yu correlations gives the granular temperature increases with increase of concentration of particles. The mean granular temperature is 0.0202 (m/s)2 that is lower than that by means of the modified CSD drag models. The trends, however, are the same. Fig. 9 shows the distribution of solid shear viscosity as a function of concentration of particles at the superficial gas velocity and solid mass flux of 3.5 m/s and 75 kg/m 2s, respectively. For both the modified CSD drag model A and model B, the solid phase shear viscosity increases with the increase of concentration of particles. The solid phase shear viscosity consists of two parts, the kinetic term and the collisional component. In the kinetic regime, the solid viscosity is proportional to the concentrations of particles raised to the power of 1/3 [1]. In the collisional regime, the solid viscosity increases with the increase of the collision number of particles. For the modified drag model B and model A, the mean solid shear viscosity is 0.0255(Pa s) and 0.00215(Pa s), respectively. Simulated solid shear viscosity using Ergun/Wen and Yu drag model is also shown in Fig. 9. The predicted solid shear viscosity is increased with the increase of concentration of particles. The mean solid shear viscosity is 0.0018 (Pa s) that is smaller than that using the modified CSD drag models. An empirical correlation was given by Huilin and Gidaspow [40] for 75 μm FCC particles in a riser. Their proposed correlation was as follows: 1=3

μ s ¼ 0:0165εs g o :

ð22Þ

In the figure, the computed trend line from FCC experimental data correlation is also shown. The computed solid shear viscosity increases with increasing solid concentration. There is an agreement between the simulated shear viscosity and the correlation at low concentration of particles. 0.15

Particle-wall friction factor

Solids viscosity (Pa s)

0.03

Modified CSD drag model B Modified CSD drag model A KTGF with Ergun/Wen and Yu correlations Huilin and Gidaspow (2003) 2

u g=3.5 m/s, G s=75 kg/m s 0.02

d s=67 μ m, ρ s=1500 kg/m

3

0.01

0.00 0.0

0.1

0.2

Concentration of particles Fig. 9. Distribution of solid viscosity as a function of concentrations.

0.3

Capes et al. (1973) Kmiec et al. (1978) Breault et al. (1989) (ρs=0.01)

0.10

Stemerding (1962)

0.05

0.00 1.0

2.0

3.0

4.0

5.0

6.0

7.0

Solid velocity (m/s) Fig. 11. Particle–wall friction factor versus particle velocity in the riser.

W. Shuai et al. / Powder Technology 225 (2012) 176–189

Axial velocity of particles (m/s)

10.0

Experiments (Yan and Zhu, 2005) Modified CSD drag model B (Capes et al., 1973) Modified CSD drag model B (Kmiec et al., 1978) Modified CSD drag model B (Breault et al., 1989)

8.0 6.0 4.0

ds=67 μ m

2.0

ρ =1500 kg/m

3

s

ug=3.5 m/s

0.0

2 Gs=100 kg/m s

-2.0 0.0

0.2

0.4

0.6

0.8

1.0

Dimensionless distance x/R Fig. 12. Profile of radial profiles of axial velocity of particles.

4.3. Effect of wall friction

On the other hand, at high solid concentration, the simulated results deviate because this correlation is obtained for much lower solid fluxes. Experiments show that the particle clusters flow downward, stagnant and even upward in the risers. Clusters in the annulus tend to travel downward whilst clusters located in the core usually travel upward. Clusters form at the wall, descend, break-up, travel laterally from the annulus to the core and then be re-entrained in the upward flowing core. Gu and Chen [26] summarized the curve fitting correlations of diameter of clusters from other publications. Their results were shown to be in agreement with the experimental measurements. The particle cluster diameter is 6

dc ¼ ds þ ð0:027−10dÞεs þ 32εs :

ð23Þ

Harris et al. [27] presented a correlation for predicting the properties of cluster of particles traveling near the riser wall. The correlation is correlated as a function of concentration of particles from experimental data published in the literature for vertical risers ranging from laboratory to industrial scale. The cluster diameter is −1

dc ¼ εs ð40:8−94:5εs Þ

:

ð24Þ

Subbarao [41] proposed a cluster diameter correlation considering the effect of the column diameter. The correlation was derived from a conceptual model for the fully developed gas–solid flow, which assumes that ratio of the volume of cluster to that of void is equal to the ratio of the cluster to gas volume fraction. In this model, the void size is restricted to the column diameter and the rise velocity is restricted to slug rise velocity, which again depends on the column diameter. The final equation of the cluster diameter is h i −1 1=3 dc ¼ ds þ dv δc ð1−δc Þ

shows a similar behavior to each other, but the predicted particle cluster diameter is higher than that calculated by Subbarao's correlation, and lower than that predicted by Gu's and Harris's correlations. This figure also illustrates at the low concentration of particles the simulated cluster diameters from both the modified CSD drag models A and B are close to the values calculated by Gu's, Harris's and Subbarao's correlations. The difference between the simulations and calculations from Gu's and Harris's correlations exists in the range of high concentration of particles. Eq. (23) is of a form similar to Eq. (24), that is, they relate cluster diameter to averaged solid concentration. Note that these two correlations are fitted by experimental data collected near the walls. Clusters have been observed in both the core and annulus regions of risers. Clusters observed in the annulus tend to travel downward whilst clusters located in the core usually travel upward. Predicted cluster diameters using the modified CSD drag model B are in agreement with Gu's and Harris's correlations.

ð25Þ

where δc is the cluster fraction, and dv is the diameter of the void. Fig. 10 shows the distribution of predicted cluster diameter by modified CSD drag model and particle cluster diameter calculated from Gu's, Harris's and Subbarao's correlations as a function of concentration of particles. The cluster diameter increases with the increase of concentration of particles. These trends are consistent with the experimental data observed by many researchers. From Fig. 10, we see that the quantitative results predicted by both the modified CSD drag models A and B are in agreement with calculations from Gu's and Harris's correlations. While Subbarao's correlation give values smaller than that calculated by the Gu's and Harris's correlations, which can be attributed to lower value of the cluster fraction. The profile

Expressions for the friction factor reported in Table 1 are commonly used to evaluate the contribution of particle–wall friction to total pressure drop. Fig. 11 shows the profile of particle–wall friction factor as a function of solid velocity using four different friction factor correlations. The difference of predicted results using the different correlations of particle–wall friction factor is obvious. The predicted particle–wall friction factor using Capes et al. [22], Breault et al. [42] and Kmiec et al. [43] correlations decreases with the increase of particle velocity, except for Stemerding [44] who proposed a constant. At high solid velocity, one may observe that the particle–wall friction factor tends to a constant. Fig. 12 shows the radial profiles of particle velocity at the superficial gas velocity and solid mass flux of 0.35 m/s and 100 kg/m 2 s, respectively. Simulated axial velocity of particles using three different particle–wall friction factors show that the particle velocity in the center region of the riser remains nearly constant throughout the riser, and then decreases toward the wall. However, the difference for three different particle–wall friction factors is obvious. Simulated results using Breault et al. correlation [42] are largest in the center regime and smallest near the walls. Three different correlations by Capes et al. [22], Kmiec et al. [43] and Breault et al. [42] give the negative axial velocity of particles near the wall, which means particles flow-down near the wall. This indicates that a high wall friction leads to a high downward velocity of particles near the wall and a high upward velocity of particles in the center of the riser. Fig. 13 shows the distribution of concentration of particles at the superficial gas velocity and solid mass flux of 3.5 m/s and 100 kg/m 2 s. Simulations show that the radial profile has a flat center region, turning 0.3

Concentration of particles

186

Experiments (Yan and Zhu, 2005) CSD drag model B (Capes et al., 1973) CSD drag model B (Kmiec et al., 1978) CSD drag model B (Breault et al., 1989)

0.2

ds=67 μ m ρ =1500 kg/m

3

s

ug=3.5 m/s 0.1

2

Gs=100 kg/m s

0.0 0.0

0.5

Dimensionless distance x/R Fig. 13. Distribution of concentration of particles.

1.0

W. Shuai et al. / Powder Technology 225 (2012) 176–189

then smoothly increase toward the wall. The difference for these three models by means of Capes et al. [22], Breault et al. [42] and Kmiec etal. [43] is obvious. Compared to the middle region of the riser, more changes occur near the wall region. The particle–wall friction increases the dissipation of the particle fluctuation at the wall, and increases the dissipation of the particle fluctuation energy at the wall, leading to decrease granular temperature and increase solid concentration due to inelastic clumping. Thus, a high wall friction will gives a high concentration of particles near the wall. The predictions using Capes et al. correlation [22] are in agreement with experimental results. Simulated concentrations of particles using Breault et al. correlation [42] are highest in the riser. We believe that one of the main reasons for differences in predictions is due to the condition of the riser wall. In almost all the works published few studies reported the state of the wall. The particle–wall friction factor increases with an increase in the roughness of the wall surface. The pressure drop due to particle–wall friction has substantial impact on the hydrodynamics. Since the roughness of the wall surface is a dynamic mechanism, one cannot speculate about an exact value for the particle–wall friction factor. A boundary for the particle–wall friction factor, however, needs to be further determined. 5. Conclusions On the basis of conservation of momentum in the dense phase and the dilute phase, a modified cluster structure-dependent drag coefficient (modified CSD drag) model with consideration of wall friction is developed in combination with the stability criterion which means the tendency of gas phase to consume a minimum energy by heterogeneous drag (MEDHD). Both the modified CSD drag model A and model B are predicted with specified gas pressure gradient ∂pg =∂z, gas velocity ug, solid velocity us and porosity εg in the computational cells. The distribution of concentration and axial velocity of particles is predicted in combination the kinetic theory of granular flow with the modified CSD drag model and Ergun/Wen–Yu correlations. The flow behavior of particles is affected by the solid down-flow in the form of clusters near the walls of the riser. The particle–wall friction results in the concentration of particles increases near the walls. The predicted concentration and axial velocity of particles compares reasonably with experimental data. Present modified CSD drag model depends on the wall friction. In dense flow or in dilute flow with a layer of solids at the walls, this pressure drop may have to be corrected for the effect of the normal solid stress transmitted by that contact of particles. This is important to incorporate the effects of wall friction on the drag coefficient closure before attempting a comparison of the model predictions with experimental data. Many factors, including gas turbulence, accelerations of gas in the dilute and dense phases and restitution coefficient of particles, affect the predicted velocity and concentration of particles. As a future work, it would be interesting to see the effect of gas turbulence on flow behavior of particles in risers. The present model is further refined by more accurate and experimentally verified correlation of gas–wall and particle–wall interactions. Nomenclature a acceleration, m/s 2 CD drag coefficient ds particle diameter, m dc cluster diameter, m ðdp=dzÞf friction pressure drop, N/m 3 ðdp=dzÞgw pressure drop due to gas–wall friction, N/m 3 ðdp=dzÞpw pressure drop due to particle–wall friction, N/m 3 D riser internal diameter, m e restitution coefficient f volume fraction of dense phase fg gas–wall friction factor

187

fp particle–wall friction factor F force acting on each particle or cluster, N g gravity, m/s 2 go radial distribution function at contact H riser height, m ks conductivity of fluctuating energy, kg/m s nden number of particles in the dense phase per unit volume ndil number of particles in the dilute phase per unit volume Ndf energy dissipation by drag force, W/kg Nst energy consumed for transportation and suspension, W/kg Pg gas pressure, Pa Ps particle pressure, Pa Re Reynolds number ug gas velocity, m/s umf minimum fluidization velocity of particles, m/s us particle velocity, m/s ug,den, us,den velocities of gas and particles through the dense phase along vertical direction, m/s ug,dil, us,dil velocities of gas and particles through the dilute phase along vertical direction, m/s U superficial gas velocity, m/s Uden superficial slip velocity in dense phase, m/s Udil superficial slip velocity in dilute phase, m/s Uint superficial slip velocity of interface, m/s Vden volume of dense phase in the control volume, m 3 Vdil volume of dilute phase in the control volume, m 3 vg,den, vs,den velocities of gas and particles through the dense phase along lateral direction, m/s vg,dil, vs,dil velocities of gas and particles through the dilute phase along lateral direction, m/s x transverse distance from axis, m z height along vertical direction, m Greek letters β drag coefficient, kg/m 3 s γ collisional energy dissipation, kg/m s 3 εg,dil porosity in the dilute phase εg,den porosity in the dense phase εs,dil particle concentration in the dilute phase εs,den particle concentration in the dense phase εg porosity εmax maximum porosity for particle aggregating εs particle concentration εs,max particle concentration at packing θ granular temperature, m 2/s 2 μg gas shear viscosity, kg/ms μs solid shear viscosity, kg/ms ρg gas density, kg/m 3 ρs density of solid phase, kg/m 3 τg gas stress tensor, Pa τs particle stress tensor, Pa Subscripts c cluster den dense phase dil dilute phase g gas phase s particles phase Acknowledgments This work was supported by Natural Science Foundation of China through grant nos. 51076040 and 51176042. The authors are grateful to the referees for their useful comments, which helped improve the clarity of this manuscript.

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W. Shuai et al. / Powder Technology 225 (2012) 176–189

Appendix A In the two-fluid model, the gas phase is treated as a continuous phase, and its flow complies with the law of conservation of mass and momentum. Two sets of governing equations or two model formulations, referred to as models A and B, have been proposed [1]. Model A assumes that the pressure drop is shared between the gas and solid phases, and model B assumes that it applies to the gas phase only. Correspondingly, there are also two model formulations of drag coefficient to couple the gas and solid phases. The modified cluster structure-dependent drag coefficient model B (modified CSD drag model B) can be expressed as follows. A.1. Momentum equations of gas phase in the dilute and dense phases For the flow of gas in the dense phase, the momentum equation of gas at the steady state along vertical (z) direction in the absence of gas shear stresses is expressed by  ∂   ∂  f ε den ρg ug;den ug;den þ f εden ρg ug;den vg;den ¼ ∂z ∂x
 ∂pg dp −f δ : −nden Fden −f εden ρg g−f dz gw iw ∂z

ðA  1Þ

For gas phase in the dilute phase, the momentum equation at the steady state along vertical (z) direction in the absence of gas shear stresses is i ∂ h i ∂ h ð1−f Þεdil ρg ug;dil ug;dil þ ð1−f Þεdil ρg ug;dil vg;dil ¼ ∂z ∂x
 ∂pg dp −ð1−f Þ δ : −ndil Fdil −nint Fint −ð1−f Þεdil ρg g−ð1−f Þ dz gw iw ∂z ðA  2Þ For simplicity, the second term on the left hand side of Eqs. (A-1) and (A-2) is neglected. Substituting Eq. (A-1) in Eq. (A-2), we obtain nden F den ndil F dil nint F int ¼ þ þ ρg g ðεdil −εden Þ f ð1−fÞ ð1−f Þ  þ ρg εdil ag;dil −εden ag;den

" #   
dp ndil Fdil ¼ ð1−f Þ ð1−εdil Þ ρs −ρg g þ as;dil þ δiw : dz pw

  and ag;dil ¼ ∂ ð1−f Þεdil ug;dil ug;dil =ðð1−f Þ∂zÞ   ¼ ∂ f εden ug;den ug;den =ðf ∂zÞ, and are known as the accelerations

of gas in the dilute and dense phases. The third term on the right hand side is contributed by the gravitational force between the dilute phase and dense phase, and the last term is the rate of momentum. The pressure drop balance between the clusters and the dilute phase used by Nikolopoulos et al. [15] and Shah et al. [16] is without these two terms. A.2. Conservation equations of particles in the dense phase and dilute phase From Eq. (A-1), the momentum equation of particles in the dilute phase at the steady state along vertical (z) direction in the absence of solid stresses is i ∂ h i ∂ h ð1−f Þð1−εdil Þρs us;dil us;dil þ ð1−f Þð1−εdil Þρs us;dil vs;dil ¼ ∂z ∂x
 dp ndil Fdil;i −ð1−f Þð1−εdil Þρs g−ð1−f Þ δ : dz pw iw ðA  4Þ

ðA  5Þ

The expression for the superficial slip velocity in the dilute phase Udil is ρg ds 0:15 μg ¼

4:7 2 εdil ds

!0:687 1:687

U dil "

18μ g

þ U dil

#   
dp δiw : ρs −ρg g þ as;dil þ dz pw 1−εdil

ðA  6Þ

The momentum equation of particles in the dense phase at the steady state along vertical (z) direction in the absence of solid stresses is    nden Fden þ nint Fint ¼ f ð1−εden Þ ρs −ρg g þ as;den
 dp δ : þf dz pw iw

ðA  7Þ

Substituting Eq. (A-3) in Eq. (A-7), the equation for the superficial slip velocity of the dense phase Uden is ρg ds 0:15 μg

!0:687 1:687

U den þ U den

( h    i d2s ε4:7 den ¼ ð1−f Þρg εdil g þ ag;dil −ε den g þ ag;den ð1−εden Þ18μ g  h    i þ ρs −ρg ð1−f Þð1−εdil Þ g þ as;dil þ f ð1−εden Þ g þ as;den )
 dp δ þ : dz pw iw ðA  8Þ

ðA  3Þ

where ag;den

For a flow of gas and particles, (ρs − ρg) ≈ ρs. For simplicity, the second term on the left hand side of Eq. (A-4) is neglected. Thus, Eq. (A-4) is simplified

Substituting Eqs. (A-3) and (A-5) in Eq. (A-7), the expression for the superficial slip velocity of clusters Uint is expressed by ρg dc 0:15 μg ¼

!0:687

d2c ð1−f Þ5:7 18μ g

1:687

U int (

þ U int

h    i ρg εden g þ ag;den −ε dil g þ ag;dil

ðA  9Þ

)  h    i þ ρs −ρg ð1−εden Þ g þ as;den −ð1−εdil Þ g þ as;dil :

A.3. Modified cluster structure-dependent (modified CSD) drag coefficient From model B proposed by Gidaspow [1], the relationship between the structure parameters and the modified cluster structuredependent drag coefficient (modified CSD drag model B) can be written as follows εg  ½nden Fden þ ndil Fdil þ nint Fint : βCSD ¼   ug −us 

ðA  10Þ

W. Shuai et al. / Powder Technology 225 (2012) 176–189

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