A New Approach To Specify the Inlet Boundary Conditions for

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A New Approach To Specify the Inlet Boundary Conditions for Computational Fluid Dynamics (CFD) Modeling of Hydrodynamic Behavior of a Circulating Fluidized Bed (CFB) Riser B. Peng,† J. Zhu,‡ and C. Zhang† †

Department of Mechanical and Materials Engineering, and ‡Department of Chemical and Biochemical Engineering The University of Western Ontario, London, ON N6A 5B9, Canada ABSTRACT: A new approach to specify the inlet boundary conditions that considering the inlet air jet effect was proposed in this study to simulate gassolid two-phase flows in circulating fluidized bed (CFB) risers more accurately. A computational fluid dynamics (CFD) model based on EulerianEulerian approach coupled with kinetic theory of granular flow was adopted to simulate the flow using the proposed inlet boundary conditions. Simulation results were compared with experimental data. Good agreement between the numerical results and experimental data was observed under different operating conditions, which indicates the effectiveness and accuracy of the CFD model with the proposed inlet boundary conditions. The results show that the proposed inlet boundary conditions are not only physically more realistic but also result in more-accurate simulation results than commonly used inlet boundary conditions. Both experimental and numerical results illustrate a clear core annulus structure in the CFB riser under all operating conditions.

1. INTRODUCTION Given the advantages of good particle mixing and heat-transfer characteristics and continuous powder handling ability, gas solid circulating fluidized beds (CFBs) have been widely used in the chemical, petrochemical, energy, and metallurgical industries, including fluid catalytic cracking (FCC), combustion of lowgrade coal in power generation, and biomass gasification.14 A more efficient design for CFB reactors requires a quantitative understanding of the complex hydrodynamics in CFB reactors. Because of its importance, the hydrodynamics in CFB risers has been the subject of many studies, as reported in the literature.510 In principle, experimental data are considered more accurate; however, in many cases, conducting detailed measurements is expensive and difficult. Experimentally lacking of a thorough quantitative understanding of the hydrodynamics in the fluidization process, computational fluid dynamics (CFD) is thus considered a relatively more powerful, less expensive, and more convenient tool for predicting the hydrodynamic behavior of CFBs. With the fast development of computer technology, a CFD approach has been widely used to solve gassolid two-phase flows in CFBs and to investigate the relationship between solids concentration, operating conditions, and flow field. Generally, two approaches can be used to solve such flows, according to the method used to address the solid phase: the Eulerian Lagrangian (E-L) approach and the EulerianEulerian (E-E) approach. In the E-L approach, NavierStokes equations are solved for the gas phase (continuous phase). The solid phase is solved as a discrete phase, and each solid particle is tracked based on a Lagrangian force balance equation. By averaging the movement parameters of many of the tracked particles, the solid-phase flow and concentration distribution can be acquired. This method has many advantages, such as clear and simple physical mechanism r 2011 American Chemical Society

and escape from false numerical diffusion. However, its biggest drawbacks are the high computational cost and the neglect of solids pressure and solids viscosity, because of particle random motion and particleparticle interaction. With the increased number of particles, the calculation time will increase exponentially, and larger computer memory is required. Therefore, currently, this method can only be well-applied to some engineering cases for which the solids are so dilute that the solids viscosity and solids pressure might be neglected. In the E-E approach, the two phases are treated as interpenetrating continua, and the concept of phasic volume fraction is introduced since the volume of one phase cannot be occupied by the other phase. Mass and momentum conservation equations for both phases are solved. These equations are closed by constitutive relations that are obtained from empirical information, or, in the case of granular flows, by application of kinetic theory of granular flow. The E-E approach has been widely used for simulations of gassolid two-phase flows in fluidized beds.1123 When the E-E approach is used, the solid-phase volume fraction is an unknown to be solved. The governing equation for solids volume fraction is implicit in the mass conservation equation for the solid phase and the inlet boundary condition is of crucial importance to the numerical simulation results, especially to the solids volume fraction distributions. Moreover, in the recent work,24 it was theoretically and numerically proved that, unlike single-phase flows, the gassolid two-phase flow profiles in the fully developed region depend on the flow profiles at the Special Issue: Nigam Issue Received: April 28, 2011 Accepted: August 26, 2011 Revised: August 25, 2011 Published: August 26, 2011 2152

dx.doi.org/10.1021/ie200916c | Ind. Eng. Chem. Res. 2012, 51, 2152–2165

Industrial & Engineering Chemistry Research

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Figure 1. Schematic diagram of the experimental CFB setups.

inlet, i.e., different inlet flow profiles will result in different flow profiles in the fully developed region in a gassolid two-phase flow system. Therefore, correctly specifying the inlet boundary conditions is peculiarly important for the accuracy of numerical simulations of gassolid two-phase flows in CFB risers. Surprisingly, very few researchers have reported, in the open literature, the details regarding how to specify the inlet boundary conditions, especially the reasons why certain boundary conditions were used (if ever reported). The most commonly used expression for inlet boundary conditions for gassolid two-phase flows is “velocities and concentrations of both phases were specified” or “the solids volume fraction at inlet is determined by experimental measurements”. Given the importance of the inlet boundary condition on the simulation results for gassolid twophase flows, comprehensive and thorough analysis and discussion on how to specify the inlet boundary conditions used in the simulations of gassolids two-phase flows in CFBs are necessary. The main objectives of the present work are (1) to develop a new approach to specify the inlet boundary conditions, which are physically more realistic and reasonable, for gassolid two-phase flow simulations in CFBs, (2) to conduct comparative analysis of the effects of inlet boundary conditions on the simulation results, (3) to numerically investigate the detailed solids acceleration process and flow before fully developed profiles are formed, and (4) to investigate the effects of particle size on the flow in CFB risers. The simulation results are compared with the experimental data to validate the proposed inlet boundary conditions.

2. CONFIGURATION OF THE EXPERIMENTAL CFB SYSTEM Figure 1 illustrates the experimental CFB system. Compared with the experimental setups used by many other researchers,

this riser is much higher, and therefore, the gassolid flow can easily reach the fully developed stage. It was designed to incorporate both a riser and a downer, and allow the experimental studies in the riser and the downer to be carried out separately or simultaneously. The riser is 15.1 m in height with an inner diameter (ID) of 0.1 m, and the downer is 9.3 m in height with the same ID as the riser. To provide quick acceleration for the solids at the bottom of the riser, a nozzle-type gassolid distributor is employed for the riser. The gassolid distributor includes a perforated plate and a bundle of nozzles uniformly installed on the perforated plate. The main air enters the riser through a bundle of 13-mm-outer-diameter nozzles, extending 0.30 m into the riser bottom to carry the gassolid suspension up the riser. The perforated plate is present to allow auxiliary gas to fluidize the solids from the storage tank, while the nozzles are present for the main gas to carry the solids upward. During the operation, the main air enters the riser through nozzle tubes and the solids coming from the storage tank are fluidized by the auxiliary air at the riser bottom and then carried upward by the combination of the auxiliary and main gas stream along the riser column. Compared with the main air flow rate, the auxiliary air flow rate is negligible. At the top of the riser, the solids pass through a smooth elbow into the primary cyclone at the top of the downer for gassolids separation, and some escaped solids enter the secondary and tertiary cyclones and a bag filter for separation and eventually enter the storage tank. The solids are then recycled to the riser bottom from the storage tank, through a butterfly valve located in the inclined feeding pipe. Fluid catalytic cracking (FCC) particles are used in the CFB. The particulate material properties are given in Table1. The solids circulation rate is regulated by the butterfly valve and is 2153

dx.doi.org/10.1021/ie200916c |Ind. Eng. Chem. Res. 2012, 51, 2152–2165

Industrial & Engineering Chemistry Research Table 1. Material Properties of Solid Particles property

value

particle type

FCC

density, F

1500 kg/m3

particle mean diameter, dp

67 μm

measured by the measuring pipe shown in Figure 1. The fluidization gas used in the study is air at ambient temperature and pressure. An orifice plate is employed to measure the air flow rate. The local solids concentrations and velocities under various operating conditions in the riser are measured at different radial positions (r/R = 0.0, 0.158, 0.382, 0.498, 0.590, 0.670, 0.741, 0.806, 0.866, 0.922, 0.975) and at different height levels. The superficial gas velocity (Ug) ranges from 3.5 m/s to 5.5 m/s and the solids circulation rate (Gs) ranges from 100 kg/(m2 s) to 200 kg/(m2 s). All experiments are conducted under ambient temperature (25 °C) and pressure (101.3 kPa) by Huang et al.10 in our research group.

3. GASSOLID TWO-PHASE FLOW MODEL DESCRIPTION Sinclair and Jackson11 introduced the kinetic theory of granular flow to address the solids stress for the first time in 1989. Similar to the thermal motion of molecules in a gas, the random motion of the particles arising from particleparticle collisions is considered by the kinetic theory of granular flow. At present, the gassolid CFD models using the kinetic theory of granular flow can be classified as three main types: laminar model for each phase, turbulence mode for the gas phase, and turbulence model for each phase. The first method “laminar model for each phase” is the socalled standard model. In this method, the single particle motion is analogous to single molecule motion. This means that the molecular viscosity is analogous to solids viscosity, because of the single particle motion. However, Pita and Sundaresan25 found that the numerical results are extremely sensitive to the restitution coefficient, using the model of Sinclair and Jackson.11 An important phenomenon neglected in the laminar model for each phase is the turbulent effect from the gas phase. However, the gas-phase flow in a CFB is often highly turbulent. The neglect of gas-phase turbulence is believed to cause the issue of sensitivity to the restitution coefficient when employing the laminar model for each phase. Louge et al.26 included the gas-phase turbulence effect in their simulations. Similar to the work of Sinclair and Jackson,11 they adopted kinetic theory of granular flow to obtain a solid-phase viscosity. The turbulent viscosity of the gas phase was obtained by a turbulence model. The kinetic theory of granular flow takes into account the random motion (or the velocity fluctuation) of single particles. The solids mixing at the particle level is well-considered, but the solids mixing at the “clusters” level is not expressed well. However, the particle-phase turbulence is a very common phenomenon in CFBs in both the laboratory scale and the industrial scale. The existence of “clusters” (also called “collections of particles” or “particle packets”) has been observed by many experiments. The gas-phase motion can be classified using the local mean velocity, molecular random motion, and turbulence velocity fluctuation of the gas phase. Similarly, the particle motion includes three parts: local mean velocity, single particle velocity fluctuation and collective particle fluctuation. Therefore,

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the turbulence model for each phase, which is more complex, but more accurate, should be used for gassolid two-phase flow simulations. The effects associated with the collective motion of particle packets in dense gassolids flows were considered by Dasgupta et al.27 In their model, gas-phase turbulence is modeled by a kε model, and the cluster motion is modeled using the method similar to single-phase turbulent flow. Hrenya and Sinclair28 developed a hydrodynamic CFD model that incorporated two mechanisms: interactions associated with individual particles based on a kinetic theory and interactions associated with collections of particles (clusters) based on an analogy with singlephase turbulent flows. In this study, the turbulence model for each phase is used for gassolids two-phase flow simulations in the riser. The kε turbulence model is adopted to account for the gas-phase turbulence effect and the velocity fluctuation associated with particle packets is taken into account using the solids-phase kε model. 3.1. Governing Equations. In order to simulate the gassolid two-phase flow in a CFB riser, a CFD model based on the transient E-E approach with the kε turbulence model for each phase, coupled with the kinetic theory of granular flow, is used.29 The governing equations consist of mass and momentum conservation equations for both phases, as follows: Mass Conservation Equation of the Gas Phase: ∂ ðαg Fg Þ þ ∇ 3 ðαg Fg vFg Þ ¼ 0 ∂t Mass Conservation Equation of the Solid Phase: ∂ ðαs Fs Þ þ ∇ 3 ðαs Fs vFs Þ ¼ 0 ∂t

ð1Þ

ð2Þ

with the constraint αg þ αs ¼ 1

ð3Þ

where t is the time, α the volume fraction, F the density, andBv the velocity. The subscripts “g” and “s” represent the gas phase and solid phase, respectively. Momentum Conservation Equation of the Gas Phase: ∂ ðαg Fg vFg Þ þ ∇ 3 ðαg Fg vFg vFg Þ ¼ αg ∇p ∂t h i

F F þ ∇ 3 αg ðτmg þ τRe g Þ þ α g Fg g B þ Ksg ðvs vg Þ

ð4Þ

where τmg is the stress tensor, defined as d 2 τmg ¼ μg, m ∇ 3 ! v g I þ μg, m ∇! v g þ ∇! v Tg 3 τRe g the Reynolds stress tensor, defined as τRe g ¼

d 2 T v g I þ μg, t ∇! v g þ ∇! vg Fkg þ μg, t ∇ 3 ! 3

p the pressure, B g the gravitational acceleration, Ksg the interphase momentum exchange coefficient, μg,m the gas-phase molecular d viscosity, and I the identity tensor. Momentum Conservation Equation of the Solid Phase: ∂ ðαs Fs vFs Þ þ ∇ 3 ðαs Fs vFs vFs Þ ¼ αs ∇p ∇ps ∂t F F F þ ∇ 3 ½αs ðτmg þ τRe g Þ þ αs Fs g þ Kgs ðvg vs Þ

2154

ð5Þ


Industrial & Engineering Chemistry Research where τmg

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k is the turbulent kinetic energy, ε is the dissipation rate of the turbulent kinetic energy, and Gg,k is the production of turbulent kinetic energy for the gas phase. kε Turbulence Model for the Solid Phase:

d 2 ¼ λs μ s ∇ 3 ! v s þ ∇! v Ts v s I þ μs ∇! 3

and τRe g ¼

2 3

∂ ðαs Fs ks Þ þ ∇ 3 ðαs Fs vFs ks Þ ∂t μs, t ¼ ∇ 3 αs ∇ks þ ðαs Gs, k αs Fs εs Þ σk

d v s I þ μs, t ∇! v s þ ∇! v Ts Fks þ μs, t ∇ 3 !

The solids pressure (ps), solids shear viscosity (μs), and solids bulk viscosity (λs) in eq 5 are related to the granular temperature (Θs), based on the kinetics theory. The turbulent viscosities μg,t and μs,t are obtained by a modified standard kε turbulence model in which the interaction between the two phases is included. The granular temperature (Θs) is obtained by solving its transport equation. Granular Temperature Equation: 3 ∂ ðFs αs Θs Þ þ ∇ 3 ðFs αs vFs Θs Þ 2 ∂t d

¼ ð ps I þ τs Þ : ∇vFs þ ∇ 3 ðkΘs ∇Θs Þ γΘs þ ϕgs d

where g0,ss is the radial distribution function, ess the restitution coefficient for particleparticle collisions (the value is taken as 0.95 for the particles used in this research), and ew the restitution coefficient for particlewall collision (which is assigned a value of 0.9, from the work of Benyahia et al.20). kε Turbulence Model for the Gas Phase: ∂ ðαg Fg kg Þ þ ∇ 3 ðαg Fg vFg kg Þ ∂t μg, t ¼ ∇ 3 αg ∇kg þ ðαg Gg, k αg Fg εg Þ σk

þ

Ksg ðvFs

vFg Þ 3

μg, t αg σ g

αg σ g

∇αg

μs, t ∇αs αs σ s

μs, t ∂ ðαs Fs εs Þ þ ∇ 3 ðαs Fs vFs εs Þ ¼ ∇ 3 αs ∇εs ∂t σε ( εs þ C1ε αs Gs, k C2ε αs Fs εs ks " μg, t þ C3ε Kgs ðCgs kg Csg ks Þ Kgs ðvFg vFs Þ 3 ∇αg αg σg #) μs, t F F þ Kgs ðvg vs Þ 3 ∇αs αs σ s

ð10Þ

ð11Þ

where the turbulent viscosity is defined as μs,t = FsCμ(ks2/εs). The interaction between gas and solid has been mainly expressed in the form of a drag force, which is used to model the momentum exchange between gas and solid phases. The drag force has been studied extensively, and the drag coefficient is related to the flow regime and the properties of the two phases. The Syamlal and O’Brien correlation,31 which is based on the measurements of the particle terminal velocity in fluidized bed or settling beds, is commonly used to estimate the drag coefficient for flows in fluidized beds. Therefore, it is adopted in the present work.

ð12Þ

where A ¼ α4:14 g

ð8Þ

∂ ðαg Fg εg Þ þ ∇ 3 ðαg Fg vFg εg Þ ∂t μg, t εg n ¼ ∇ 3 αg ∇εg þ C1ε αg Gg, k C2ε αg Fg εg σε kg μs, t þ C3ε Ksg ðCsg ks Cgs kg Þ Ksg ðvFs vFg Þ 3 ∇αs αs σ s μg, t þ Ksg ðvFs vFg Þ 3 ∇αg ð9Þ αg σ g where μg,t is the turbulent viscosity, which is defined as ! kg 2 μg, t ¼ Fg Cμ εg

μg, t

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vr, s ¼ 0:5 A 0:06Res þ ð0:06Res Þ2 þ 0:12Res ð2B AÞ þ A2

μs, t ∇αs αs σs

∇αg

þ Kgs ðvFg vFs Þ 3

ð6Þ

where ϕgs is the energy exchange (ϕgs = 3KgsΘs), kΘs the d diffusion coefficient for granular energy, and τs the solids stress tensor. The collisional energy dissipation (γθs ) can be obtained as follows:30 12ð1 e2ss Þg0, ss pffiffiffi γΘs ¼ ð7Þ Fs α2s Θ3=2 s ds π

þ Ksg ðCsg ks Cgs kg Þ Ksg ðvFs vFg Þ 3

þ Kgs ðCgs kg Csg ks Þ Kgs ðvFg vFs Þ 3

B¼

8 < 0:8α1:28

for αg e 0:85

: α2:65 g

for αg > 0:85

g

and Res ¼

Fg ds jvFs vFg j μg

The interphase momentum exchange coefficient then is given as ! !2 3αs αg Fg 4:8 Res F F Ksg ¼ 0:63 þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi jv vg j 4v2r, s ds vr, s s Res =vr, s ð13Þ where ds is the solid particle diameter. 2155



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The solids pressure can be calculated using the following expression:32 ps ¼ 2Fs ð1 þ ess Þα2s g0, ss Θs

ð14Þ

The solids shear viscosity consists of collisional and kinetic terms given by the formula below:28 1=2 4 Θs μs ¼ αs Fs ds g0, ss ð1 þ ess Þ 5 π pffiffiffiffiffiffiffiffiffi αs Fs ds Θs π 2 1 þ ð1 þ ess Þð3ess 1Þαs g0, ss þ 6ð3 ess Þ 5 ð15Þ The solids bulk viscosity accounts for the resistance of the granular particles to compression and expansion. It has the following form:30 1=2 4 Θs λs ¼ αs Fs ds g0, ss ð1 þ ess Þ ð16Þ 3 π 3.2. Boundary Conditions. To carry out the numerical simulations, the boundary conditions at the inlet, wall, and outlet of a CFB riser must be specified. Boundary Conditions at the Inlet: The profiles of the solids volume fraction and velocities for both phases must be specified, which will be thoroughly discussed later in a separate section in this paper. Boundary Conditions at the Wall: The no-slip boundary condition is used for the gas phase, i.e., the tangential and normal velocities are zero at the wall. For the solid phase, the velocity normal to the wall is also set as zero, i.e., no solids mass flow across the wall, while the solid phase is allowed to slip at the wall. The tangential velocity for the solid phase, i.e., the solids slip velocity parallel to the wall, νt,w, is calculated as follows:20

vt, w ¼

6μs αs, max ∂vs, w pffiffiffiffiffiffiffiffi πϕFs g0, ss αs 3Θs ∂n

ð17Þ

where αs,max is the volume fraction for the particles at maximum packing. The general boundary condition for granular temperature at the wall takes the following form:33 qs ¼

! pffiffiffiffiffiffi πpffiffiffi αs πpffiffiffi αs 3ϕ 3 ð1 e2sw ÞFs g0 Θ3=2 Fs g0 Θs v2t, w s 6 4 αs, max αs, max

ð18Þ Here, ϕ is the specularity coefficient between the solids and the wall and g0 is the radial distribution function. Boundary Conditions at the Outlet: Fully developed exit conditions (outflow) are employed for both phases, since the riser is long enough.

4. COMPUTATIONAL MESH AND SOLVER Since the computational time is prohibitive for three-dimensional (3-D) gassolid two-phase simulations in a CFB riser, the axisymmetric model is employed to simulate the flow in the CFB riser. The mesh for the simulation is generated using Gambit.34 A quad grid system is used with finer mesh near the wall, as shown

Figure 2. Mesh for the computational domain of the CFB riser.

in Figure 2. The mesh in the inlet region is also refined, since the change in flow parameters is great at the inlet region. The axisymmetric, double-precision, segregated, implicit formulation and unsteady solver is used in this study. In addition, the results for a two-dimensional (2-D) model, instead of an axisymmetric model, are also presented, and will be discussed in detail later. The governing equations are solved using a finite-volume approach. Phase-coupled SIMPLE algorithm35 is employed in pressurevelocity coupling. Power-law schemes32 are used to discretize the governing equations for all unknowns, except the volume fraction, for which QUICK36 is used. The results of the grid-size-independent test and time-step-size-independent test are shown in Tables 2 and 3, respectively. Since the difference between the results from two different grid sizes is small enough, as shown in Table 2, the mesh with 6200 cells is chosen to conduct further simulations. Table 3 also shows that the difference between the results from two different time step sizes is very small. Therefore, the time step size of 0.0001 s with 100 iterations per time step is adopted. The commercial CFD software Fluent 6.329 is used to carry out the simulations.

5. PROPOSED APPROACH TO SPECIFY THE INLET BOUNDARY CONDITIONS The commonly used inlet boundary conditions for gassolid two-phase flow simulations in fluidized beds are uniform profiles for solids volume fraction and velocities of both phases. However, it is found in this study that the numerical results using those boundary conditions do not agree well with the experimental data. The main reason is, for example, in the CFB riser used in this study, the main air enters the riser through many nozzles, as shown in Figure 1b, generating several high-velocity air jets. 2156



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Table 2. Comparison of Solids Velocity under Different Grid Sizes

Table 3. Comparison of Solids Velocity under Different Time Steps Value

Value 6200

11500

parameter

grid cells

grid cells

solids axial velocity at the riser center (7.55, 0)

6.317 m/s

6.510 m/s

solids axial velocity

3%

at the riser center (7.55, 0) difference between

difference between different grid sizes

Those high-velocity air jets affect the solids fluidization significantly. Therefore, the use of a uniform air velocity profile at the riser inlet will cause a discrepancy between the numerical results and experimental data. Therefore, it is highly desirable to develop more physically realistic inlet boundary conditions for accurate numerical simulations of CFBs. Thus, a new approach to specify the inlet boundary conditions is proposed in this study in order to improve the accuracy of the numerical results. Based on the actual CFB riser operating conditions, it is more reasonable to specify the inlet air flow through many air jets as the velocity inlet boundary condition for the gas phase. The air flow rate in all jets is assumed the same and is determined based on the total air flow rate. The air velocity outside of the air jets is specified as zero, as shown in Figure 3. For each air jet, it is assumed that the air velocity is in the axial direction and is either uniform (flat profile), as shown in Figure 3a, or fully developed (law-of-logarithmic profile), as shown in Figure 3b. There are seven rows of air jets in the CFB riser considered in this study. For the solid phase, since a nozzle-type gassolids distributor is used at the riser inlet and the solids from the storage tank are initially fluidized by the auxiliary gas before the main gas carrying the solids upward, it is reasonable to assume the solids volume fraction equals 0.3, which is a typical concentration for the bubbling fluidized bed in the bottom region. Then, with the given solids circulation rate (Gs), the solids axial velocity at the riser inlet, which is assumed uniform, can be calculated by us ¼

Gs Fs α s

ð19Þ

The solids radial velocity is set as zero at the riser inlet.

6. RESULTS AND DISCUSSION The simulations are conducted for the CFB riser shown in Figure 1, using different approaches to specify the inlet boundary conditions to study the effect of inlet boundary conditions on the numerical results. The simulations are also carried out under different operating conditions to investigate the gassolid twophase flow patterns in the CFB riser. 6.1. Volume Fraction and Velocity Profiles. Figure 4 illustrates the comparison of the numerical results using the proposed approach to specify the inlet boundary conditions, where the flat profile is used at the air jet, as shown in Figure 3a with the experimental data for the distributions of solids volume fraction and axial velocity along the radial direction at a height of x = 10 m, which is within the fully developed region of the riser. The operating conditions are Ug = 5.5 m/s and Gs = 108.1 kg/(m2 s), which are in the middle of the experimentally investigated range of Ug from 3.5 m/s to 8.2 m/s and Gs from 50 kg/(m2 s) to 200 kg/(m2 s). Figure 4a clearly shows the coreannulus structure, based on the solids volume fraction distribution in the CFB riser, and such structure has been widely reported in most of the

parameter

time step

time

size = 0.0001s

step size = 0.00005s

6.317 m/s

6.323 m/s

\

0.03%

different time steps

related open literature on experimental works.5,710 Although some deviation between the experimental data and predicted results using the jets approach still exists, especially in the nearwall region, the overall agreement between the predicted results and the experimental data is acceptable. The deviation between the experimental data and numerical results might result from the modeling assumption that particles are spherical with uniform size and the solids agglomeration is ignored. 6.2. Effects of Velocity Profiles at the Air Jet. Generally, if the nozzle tubes in the riser gassolids distributor are long enough for the main air to be fully developed before entering the inlet of the riser, assuming a logarithmic profile for each of air jet might be more realistic. However, it becomes greatly complicated to specify the velocity profile for the nozzle tubes. On the other hand, if the nozzle tubes are very short, applying a flat profile for each air jet might be more reasonable. Moreover, the nozzle tubes might have the intermediate length, and, thus, the actual velocity profile might reside between those two typical profiles. Comparison between the numerical results using the two different velocity profiles (i.e., flat profile and law-of-logarithmic profile) at each air jet is shown in Figure 5, where the experimental results are also presented for comparison purposes. Note that the results using those two different velocity profiles at the air jet are almost identical. Therefore, to simplify the simulation process, a uniform velocity profile for each air jet, as shown in Figure 3a, is recommended for further simulations of the gassolid two-phase flow in CFB risers and is used for the rest of the simulations in this work. 6.3. Comparison of the Proposed and Commonly Used Inlet Boundary Conditions. The commonly used inlet boundary conditions for gassolid two-phase flow simulations in CFB risers are the uniform boundary conditions (i.e., uniform profiles for gas velocity, solids velocity, and solids volume fraction at the CFB riser inlet). Comparison for the numerical results from the proposed inlet boundary conditions and the commonly used inlet boundary conditions is given in Figure 6. They are also compared with the experimental results. From Figure 6, it can be seen that the predicted solids volume fraction profile is quite flat and the coreannulus structure, which has been observed in experiments, cannot be distinctly seen when the commonly used uniform boundary conditions are used. The solids volume fraction in the center core region is higher than the experimental data, while, in the near-wall-annulus region, it is much lower than the measured value if the commonly used uniform inlet boundary conditions are applied. It should be noted that a work37 also predicted near-flat radial profiles for the solids concentration distribution in a CFB riser with FCC particles when employing the uniform inlet conditions and the E-E approach, and found that the so-called coreannulus structure for solids concentration is not very markedly seen. Figure 6 also shows that if the commonly 2157



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Figure 3. Gas velocity profiles of the air jets for inlet boundary conditions.

used uniform inlet boundary conditions are applied, the solids axial velocity in the center core region is much lower than the experimental data, while the value near the wall-annulus region is quite higher than the experimental data. On the other hand, if the proposed inlet boundary conditions are used, the predicted results are much better than those from the uniform boundary conditions for both the solids velocity profile and solids volume fraction profile compared with the experimental measurements, as shown in Figure 6. Compared with the commonly used uniform inlet boundary conditions, the results using the proposed inlet boundary conditions show a clearer coreannulus structure of the gassolid two-phase flow in the CFB riser, and the accuracy of the numerical results is also improved significantly, using the proposed inlet boundary conditions since the proposed approach to specify the inlet boundary condition for a CFB riser adopts a more physically reasonable assumption, which validates the effectiveness and accuracy of the proposed inlet boundary conditions. 6.4. Comparison between Two-Dimensional (2-D) and Axisymmetric Models. The 2-D Cartesian simulations have been carried out for the flow in a cylindrical CFB riser by many researchers. In the 2-D approach, one is essentially describing a “slice” of the 3-D flow using 2-D conservation equations in Cartesian coordinates. As shown in Figure 7, the difference between the results from the axisymmetric model and 2-D model is not very distinct. The solids concentration from the 2-D model is slightly higher than that from the axisymmetric model, and the solids velocity from the 2-D model is slightly lower, since the mass flow rate is the same for both models. Although these two models give very similar results, the axisymmetric model is more realistic and attractive, because the area element in axisymmetric geometry is the better representation of the actual CFB riser tube; thus, in the following simulations, we have adopted the axisymmetric model. 6.5. Solids-Diluting Acceleration Process. Figure 8 shows the details of the numerical results for the solids acceleration process from the inlet to the fully developed region. Figure 8a

Figure 4. Comparison between the numerical results using the proposed inlet boundary conditions with the flat jet profile and the experimental data.

clearly shows how the solids are accelerated by the main air flow. Initially, the solids enter the bottom of the riser at the velocity of ∼0.25 m/s; immediately after, at x = 0.01 m, the solids velocity reaches 1.3 m/s; and solids velocity keeps increasing to ∼2.8 m/s at x = 0.03 m; and then solids continue to be accelerated to an average velocity of ∼4.5 m/s at x = 0.1 m and an average velocity of ∼5.2 m/s at x = 1 m. The solids velocity profile at x = 1 m is still relatively flat. After that, the flow comes to a stage where the solids velocity profile changes gradually from flat profile to nonflat profile, and to the fully developed profile, at x ≈ 3 m. Correspondingly, Figure 8b shows how the solids volume fraction profile develops gradually along the riser. The solids volume fraction profile has a wavy shape at the inlet region, and the solids volume fraction at the wall reaches a very high value (∼0.59). The solids volume fraction at the wall then decreases gradually during the solids acceleration process and finally reaches the value of ∼0.1 at the riser outlet. The solids volume fraction in the core region also decreases during the solids acceleration process and finally reaches a relatively stable state. And this phenomenon, the so-called solids diluting-acceleration process, is governed by the solids mass conservation, which has been observed both numerically and experimentally. 2158



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Figure 5. Comparison of the numerical results using the proposed inlet boundary conditions with different velocity profiles at the air jets.

Figure 6. Comparison of the numerical results using different inlet approaches with the experimental data.

6.6. Comparison between Simulation and Experimental Data in the Entire Riser. Figure 9 shows the numerical results

along the riser, which illustrates the solids acceleration process. However, the numerical results predicted a faster acceleration process and, thus, a shorter entrance region than the experimental data. For example, the predicted solids velocity at x = 0.95 m is higher than the experimental data. In the numerical results, the radial profiles of solids axial velocity almost do not change after x = 2.59 m; however, the experimental results show that, even at x = 6.34 m, the solids velocity still changes slightly along the riser. Again, the difference between the numerical and experimental results can be seen in the region between r/R = 0.65 and r/R = 0.9. In the numerical model, the particle diameter is assumed uniformly at the size of dp = 67 μm. The neglect of particle size distribution might result in these deviations between experiment and simulation, which will be thoroughly discussed in the following section. 6.7. Effects of Particle Size on the Simulation Results. As mentioned previously, the major deviations between experimental data and numerical results are located in the region between r/R = 0.65 and r/R = 0.9, and these deviations might result from

with experimental data in the entire CFB riser. The experimental results show that the solids concentration is higher in the lower section than in the upper section of the riser and is lower in the center than in the wall region of the riser. At the bottom of the riser, the solids concentration increases significantly near the wall. Toward the riser top, the solids concentration in the wall region decreases slightly. The change of the solids concentration profile becomes smaller toward the riser top and eventually, the flow becomes fully developed. The same trend can be found in the numerical results. However, compared with the experiment results, the numerical results predicted a solids distribution with a flatter radial profile in the core region and a steeper radial profile in the annulus region. In the region between r/R = 0.65 and r/R = 0.9, the difference between the numerical and experimental results can be seen. With regard to the solids velocity profiles, both experimental and numerical results show that the solids velocity increases

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Figure 7. Comparison of numerical results from two-dimensional (2-D) and axisymmetric simulations.

the assumption of uniform particle size. In the practical operation of a CFB riser, the particles are not merely of uniform diameter. The size of the FCC particles used in the experiments, which have a mean Sauter diameter of 67 μm, varies from ∼35 μm to ∼120 μm measured by the Brinkman Particle Size Analyzer. In addition, during the actual operation, it is very common that the particle clusters occur inside the riser. The agglomeration and breakup of particles might occur for various reasons (e.g., electrostatic charge, bad initial mixing), so the real particle sizes can always be changing and fairly big “particle packets” may occur during the process. However, at the current stage, it is not practical to include the particles with a size distribution in the CFD model; therefore, the simulations using larger particles are also carried out to investigate the effect of the particle size on the flow patterns. The comparison of the numerical results for dp = 67 μm and dp = 120 μm with the experimental data for the mean value (dp = 67 μm) in the fully developed region is given in Figure 10. Figure 10b shows that the predicted solids axial velocity for larger particles is lower than that for smaller particles. The predicted

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Figure 8. Solids volume fraction and velocity profiles in the entrance region predicted using the flat jet inlet profile.

terminal velocity (e.g., represented by interphase velocity difference in the fully developed region) is increased from ∼0.3 m/s to ∼1 m/s with an increase in particle diameter from dp = 67 μm to dp = 120 μm. The reason is that larger particles are more difficult to be entrained upward by the gas flow, and, thus, the larger particle requires higher interphase velocity difference to produce the sufficient drag force to balance its gravity. In the near-wall region, the predicted solids concentration for dp = 120 μm is higher than that for dp = 67 μm, as shown in Figure 10a. In the region between r/R = 0.65 and r/R = 0.9, the deviations between the simulation results of dp = 120 μm and the experiment data become smaller for both solids concentration and velocity profiles, compared with those for dp = 67 μm. Therefore, a CFD model including different particle sizes can further improve the modeling accuracy. 6.8. Effects of Superficial Gas Velocity and Solids Circulation Rate. Figure 11 compares the numerical results with experimental data for apparent air velocities of 3.5 and 5.5 m/s while keeping the same solids circulation rate of 100 kg/(m2 s). Figure 11 shows that the higher the air velocity, the lower the solids volume fraction, under the same solids circulation rate, 2160



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Figure 9. Solids volume fraction and velocity profiles in the entire riser.

since higher air velocity will accelerate solids faster, resulting in a lower solids concentration. The comparison for the results under different solids circulation rates—100 and 200 kg/(m2 s)—but the same apparent air velocity of 5.5 m/s is also shown in Figure 11. It is clear that the higher the solids circulation rate, the higher the solids volume fraction under the same superficial gas velocity. Figure 11 also illustrates clearly the coreannulus structure for the solids volume fraction in the CFB riser under all operating conditions. The numerical results agree very well with the experimental data in the tested range, which again proves the effectiveness and accuracy of the numerical model with the proposed inlet boundary condition used in the study. 6.9. Discussions on the Effect of the Inlet Air Jets. 6.9.1. Mathematical Analysis of Inlet Air Jets Effect on the Flow Structure. Generally, a nozzle-type gassolid distributor is always employed in CFB risers, where the main air enters the riser through several nozzles, generating many high-velocity air jets. It is wellknown that fluid leaves a nozzle in a diverging flow pattern, because of the abrupt expansion, as illustrated in Figure 12, which tends to produce a radial (lateral) velocity toward the wall, and

Figure 10. Solids volume fraction and velocity profiles at x = 10 m for different particle sizes.

this lateral gas velocity (v) will drag the solids particles flowing toward the wall correspondingly. When the particles impinge at the wall, which will result in more energy dissipation and higher flow resistance therein, the solid particles will accumulate and become denser therein. One can mathematically understand this mechanism by considering the mass conservation equation for the solid phase. To simplify the derivation, we consider here the steady state of eq 2 in the xy plane, where x and y are the axial and lateral directions, respectively. αs

∂us ∂αs ∂vs ∂αs þ us þ αs þ vs ¼0 ∂x ∂x ∂y ∂y

ð20Þ

As mentioned in previous section, the solids flow process in the CFB riser is an acceleration-diluting process. Therefore, generally we have ∂αs/∂x < 0. Since us is the axial velocity of the solids particles in the riser and of a positive value, the second 2161



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Figure 12. Lateral flow due to air jets: (a) one jet and (b) three jets.

Figure 11. Comparison of solids volume fraction and velocity profiles at x = 10 m, under different operating conditions.

term in eq 20 is negative. Also, the acceleration process in the near-wall region is very slow, because of the no-slip condition at the wall for the gas phase, so ∂us/∂x ≈ 0 at this region. The lateral velocity vs caused by the air jets at the inlet of the riser is positive toward the wall and will decrease toward the wall due to flow resistances and the wall effect, which means ∂vs/∂y < 0 in the near-wall region. Therefore, based on eq 20, vs(∂αs/∂y) > 0. Here, vs is positive. Therefore, ∂αs/∂y > 0 in the near wall region (i.e., the solids volume fraction in the near-wall region increases in the lateral direction). Physically, this means that the lateral solids flow will increase the nonuniformity of the solids volume fraction distribution in lateral direction, especially in the nearwall region (i.e., the lateral solids flow will drag the solids particles to move toward the wall), damp the particle diluting process and steepen the “center-dilute side-dense” profile. That is one of the reasons why the solids concentration increases dramatically toward the wall at the annulus region. 6.9.2. Numerical Verification of the Effect of the Inlet Air Jets. In this subsection, another two different approaches to specify

the inlet boundary conditions for the gas phase are employed to investigate the influence of radial (lateral) solids flow toward the wall caused by the high velocity air jets at the inlet of the riser on the coreannulus structure formation. In the first approach (the one-jet approach), only one air jet is used at the center of the riser inlet, as shown in Figure 12a. In the second approach (the threejet approach), three air jets are employed at the riser inlet, as shown in Figure 12b. For both approaches, the solids concentration αs and the solids velocity us are assumed to be uniform at the riser inlet, and the gas velocity ug has a uniform flat profile in each air jet, while ug is equal to zero outside of the air jets at the riser inlet. Together with the commonly used uniform condition and the proposed approach (seven-jet approach), four cases will be presented here for comparison. The operating conditions are the same for all four of those cases, where Ug = 5.5 m/s and Gs = 108.1 kg/(m2 s). The results for the flow and concentration fields at the entrance region are shown in Figures 1316. Figure 13 shows the numerical results for the solids acceleration and flow development process near the entrance region, using the uniform inlet condition. From Figure 13, one can see that the velocity vectors for both phases are along almost straight lines. In this case, the gassolids flow can be regarded as the socalled plug flow. The flow is almost in the axial direction, and the lateral flow is almost negligible. The solids concentration contour shows that the solids distribution is also very uniform in the lateral direction, and the solids concentration is decreasing in the axial direction along the riser. In general, the solids diluting process along the riser is quite laterally uniform, although the solids diluting process is slightly slowed down near the wall, and this fairly flat profile at the entrance region leads to the very flat profile in the fully developed region as shown in Figure 13. It should be noted that the uniform gas inlet condition does not represent the practical operation condition of CFB risers. Figure 14 shows the numerical results for the solids acceleration and flow development process on different cross-sectional planes at the entrance region using the one-jet approach. It can be seen that the velocity of the gas phase changes its direction, by as much as 45°, at the corner between the inlet and the wall. A high lateral flow is produced resulting from the so-called “jet effect”. Correspondingly, the solid-phase velocity pattern is similar to the gas velocity pattern, since the particles are driven by the air flow. 2162



Figure 13. Flow and concentration fields in the entrance region using the uniform inlet condition.

In the solids concentration contour, it can be clearly observed that the axial diluting process near the wall is greatly dampened by the lateral solids flow. In actual CFB risers, air always enters the riser through several nozzles or holes on a perforated plate. Thus, it is very interesting and important to investigate the jet effect when multiple air jets are employed. The numerical results for the solids concentration field and flow field at the entrance region using the three-jet approach are given in Figure 15. From the gas velocity vector field shown in Figure 15, it can be seen that each air jet flows in a diverging manner, and there exists a “converging zone”, where two adjacent jets meet together. In the “converging zone”, the flow branches produced by the adjacent two jets are not exactly neutralized, and it seems that the center air jet pushes the side air jet to flow slightly further more toward the wall, which is

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Figure 14. Flow and concentration fields in the entrance region using the one-jet approach.

expected. If all jets are combined and regarded merely as one “generalized” jet, the entire “generalized” jet is supposed to flow toward the wall due to the “generalized” abrupt expansion. This perhaps explains how the air jets result in the lateral solids flow toward the wall. From the solid-phase velocity vector field shown in Figure 15, it can be seen that, in the near-wall region, the lateral flow of air jets drives the solids particles to move toward the wall. Therefore, a dense near-wall region is formed, as shown in the solids concentration contour in Figure 15. Another interesting and important phenomenon is that, in the “converging zone” where two adjacent jets meet together, the diluting process is also greatly dampened by the net incoming lateral solids flows caused by the two jets. However, in the “converging zone”, the axial acceleration is still very strong, compared with the flow near the wall, so the axial diluting process thereafter still exists, but it is 2163



Figure 15. Flow and concentration fields in the entrance region, using the three-jet approach.

already slowed. That is why there always exists a dense region corresponding to the “converging zone”, but the difference between the solids concentration in the dense region and the average solids concentration in the cross-sectional plane is becoming smaller and smaller when the flow is further developed. The results using the seven-jet approach are also illustrated in Figure 16. In general, the results show phenomena similar to the results using the three-jet approach (i.e., the “converging zone”, where two adjacent jets meet together can be seen, the lateral air flow pushes the particles to move toward the wall, to form a very dense near wall region; and finally a center-dilute side-dense solids distribution is produced). The comparison between the uniform inlet condition and different jet inlet approaches revealed that different inlet conditions result in different flow and concentration fields in the riser entrance region. These numerical

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Figure 16. Flow and concentration fields in the entrance region using the seven-jet approach.

examples evidently demonstrate that the air jets at the riser inlet produce a lateral flow toward the wall, which has a strong effect on the flow structure in the riser. This explains why the uniform gas inlet condition failed to capture the actual flow physics in CFB risers, and the need to propose the new approach to specify the inlet boundary conditions to solve gassolids two-phase flow in CFB risers considering the effect of the air jets at the riser inlet.

7. CONCLUSION A new approach is proposed on how to specify the inlet boundary conditions for the simulations of gassolids two-phase flows in a circulating fluidized bed (CFB) riser. The proposed inlet boundary conditions can represent the actual flow conditions at the inlet of the CFB riser more realistically than the 2164


Industrial & Engineering Chemistry Research commonly used uniform inlet boundary conditions. The comparative analysis using both the proposed and uniform inlet boundary conditions is also carried out. Simulation results are compared with the experimental data. In addition, the hydrodynamics and flow structures in the CFB riser under different operating conditions are analyzed. The following conclusions can be drawn from this study: (1) Inlet boundary conditions play an important role in accurately simulating the hydrodynamics and flow structures in the CFB riser. The numerical results using the proposed inlet boundary conditions agree better with the experimental data than those using the commonly used uniform inlet boundary condition. (2) Both experimental and numerical results illustrate a clear coreannulus structure in the CFB riser under all operating conditions. Good agreement is obtained between the numerical and experimental results, which indicates the effectiveness of the computational fluid dynamics (CFD) model with the proposed inlet boundary condition. Numerical results also clearly show the detailed solids acceleration process before the fully developed region. (3) Higher air velocity results in lower solids volume fraction under the same solids circulation rate, and a higher solids circulation rate results in higher solids volume fraction under the same air velocity. (4) Particle size has an important effect on the flow in the CFB riser. A CFD model considering different particle sizes has the potential to further improve the modeling accuracy.

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A New Approach To Specify the Inlet Boundary Conditions for

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