Simulation of Granular Transport of Geldart Type-A,-B, and-D Particles

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Simulation of Granular Transport of Geldart Type-A, -B, and -D Particles through a 90° Elbow Subhashini Vashisth† and John R. Grace*,† †

Department of Chemical and Biological Engineering, University of British Columbia, 2360 East Mall, Vancouver, Canada V6T 1Z3 ABSTRACT: The flow behavior of three different classes of granular materials - pulverized coal particles (type-A), glass beads (type-B), and polypropylene beads (type-D) - is numerically investigated based on computational fluid dynamics. Three elbow geometries - square and circular (R/D = 1.5 and 3.0) - are employed to investigate turbulent gas-particulate flow in dilute and diluteto-dense phase modes of pneumatic conveying. The unsteady Eulerian-Discrete phase approach with renormalized group (RNG) k-ε model is adopted, with the effects of particulate phase velocity on the gas flow, turbulent dispersion, lift forces, and particle-wall collisions incorporated in the model. The phenomenon of particle roping is well captured by the computations. Significant phase separation and particle segregation are observed in the vicinity of the lower wall of the pipe. Different classes of particles exhibit very distinct flow dynamics in terms of particle roping, particle segregation, turbulent dispersion, and particle velocity fluctuations. Numerical predictions for pressure drop, particle, and gas velocity profiles are in satisfactory agreement with experimental data from the literature.

1. INTRODUCTION Pneumatic conveying is extensively used to transport dry and free-flowing powdered and granular materials. Owing to their wide application in the chemical, cement, food, pharmaceutical, mining, and power industries, proper design of pneumatic conveying systems is of great importance. When space is a constraint, bends and elbows find extensive application in pneumatic conveying systems to change the direction of flow to transfer granular material to the required location. Despite their several advantages, various associated problems such as particle segregation, rope or sheet formation, product degradation, pipe wall erosion, and pressure losses are unavoidable. However, these problems can be minimized by changing the bend geometry1 or by installing mixers to reduce choking.2 The scope of these options are, nevertheless, restricted due to the limited understanding of internal flow dynamics. Over the past decade, findings of several researchers have augmented previous knowledge of the phenomena of particle rope formation, mixing and dispersion in 90° bends.25 It is of considerable importance to understand these complex phenomena in gassolid flows and to measure and accurately predict the hydrodynamics, so as to successfully design and determine the operating conditions for pneumatic conveying of different types of granular materials. The mode of transporting a granular material can be classified as dilute or dense phase, depending on such parameters as solids mass inflow, imposed gas flow rate, solid density, and diameter.6 In dilute phase systems, the particles move as a suspension in the conveying gas, whereas bulk motion (e.g., moving dunes or slug flow) is observed in dense phase conveying.7 In addition, the gassolid flow is locally unsteady and heterogeneous in nature. Combinations of flow nonuniformity and the centrifugal effect at the bend not only lead to the formation of particle clusters or ropes, but also contribute to back-flow and slippage near the pipe wall.8 As a result, dilute phase conveying may require excessive power and cause significant attrition and erosion due to high velocities (>15 m/s).9 Industrial applications focus more on r 2011 American Chemical Society

lower specific energy requirements and hence commonly employ low-velocity, dense phase systems to reduce power consumption, pipe erosion, and particle attrition. However, performance is difficult to predict in terms of conveying line pressure drop, and also below a certain velocity (called the saltation velocity) the particles start to accumulate at the lower pipe wall.4 The distribution of particles across the pipe cross-section is particularly important when process applications involve heat transfer, mass transfer, and reactions.10 Modes of operation are discussed in detail by Jones and William.7 Still there is a lack of understanding on the relationships and transitions between dilute and dense phase conveying. Different types of granular material differ in their flow behavior. Geldart type-D particles are larger in size and/or have higher density.11 When pneumatically conveyed, they tend to move as a sequence of plugs, forming a dense phase in the pipe. Particle plugs fill the pipe cross-section and are pushed forward by the incoming flow. Slugs then collapse, and such a sequence is repeated.12 This class of flow has received considerable attention, both experimental and computational,13 due to its low power consumption and particle attrition. Smaller and/or less dense particles of Geldart type-B are generally suspended in the carrier gas and transported due to aerodynamic drag. In this case, energy losses mainly occur due to interparticle and particle-wall collisions, which may cause pipe wall erosion. Knowledge of this class formed the basis for the earliest pneumatic conveying models. If the particle diameter or density is further decreased to yield a Geldart type-A material, the particles move as a series of dunes.14,15 In contrast to the type-D and -B particles, limited Special Issue: Nigam Issue Received: March 30, 2011 Accepted: July 12, 2011 Revised: July 8, 2011 Published: July 12, 2011 2030

dx.doi.org/10.1021/ie200647e | Ind. Eng. Chem. Res. 2012, 51, 2030–2047

Industrial & Engineering Chemistry Research

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Table 1. Numerical Grid Resolution

Figure 1. Schematic of the geometry: (a) square elbow; (b) circular elbow; R/D = 1.5; (c) circular elbow, R/D = 3.0; (d) location of angular planes at the elbow; (e) cross-sectional grid.

literature is available for type-A.3,8,1621 Some authors have focused on either uniform particle size distributions or distributions skewed toward the lower end.22 In addition to the particle size and density, the dynamic phenomena of ropes, plugs, slugs, and dunes also depend on wall roughness, conveying gas velocity, solid mass flux, and bend radius. Flows may also vary depending on other operating and geometric parameters. The focus of this paper is to investigate the dynamics of conveying these three types of particles in three bend geometries. In recent years, many commercial CFD codes, open source and in-house, have been increasingly used to study gassolid flows based on the two-fluid model or discrete phase approach .2,2332 The details and range of applicability of these numerical modeling approaches have been extensively summarized.3336 The present work addresses the flow behavior of different classes of granular material (Geldart type-A, -B, and -D) in dilute and dilute-to-dense modes of operation in three different 90° elbows. A three-dimensional Eulerian-Discrete phase model is developed to simulate the flow of gassolid two-phase suspension in pneumatic conveying, based on a commercial computational fluid dynamics code (Ansys-Fluent 12.1). The simulation predictions are compared with the experimental data of Yilmaz and Levy,3 Kuan et al.,21 and Lee et al.28 The effects of various operating conditions, and geometric parameters on the flow profiles of both the granular and gas phases, particle roping, and solid concentration distribution are investigated. Finally, different types of granular material are compared. The results aid in optimizing the performance of existing pneumatic conveying equipment and assessing methods of monitoring conveying systems.

2. MODEL DEVELOPMENT Euler-Discrete phase methodology (DPM), together with the renormalization group (RNG) k-ε model, is employed to perform numerical calculations of the particle-laden gas flow in

geometry

number of grid points

square elbow (R/D = 1.5)

24,117

circular elbow (R/D = 1.5)

31,815

circular elbow (R/D = 3)

41,022

90° elbows of different geometries. This approach computes the NavierStokes equation for the gas phase and the motion of individual particles by the Newton’s equation of motion. Computations are performed for the gas phase followed by solution for the solid phase. Various hydrodynamics forces acting on the particles are determined based on prior knowledge of the predicted gas flow. The renormalization group (RNG) k-ε model is used to simulate the gas phase turbulence.37 DPM treats the interactions between the gas and solid phases using the particlesource-in-cell method.33 The conservation equations include appropriate source terms resulting from the dispersed phase (i.e., two-way coupling). A stochastic method is used to account for the influence of gas turbulence on the solid phase.38 Interactions between particles and the wall are modeled using a coefficient of restitution. The effect of particulate phase on the gas flow, turbulent dispersion, lift forces, and particle-wall collisions are incorporated in the model. Ansys-Fluent 12.1 uses the finite volume approach to discretize the governing equations: 2.1. Governing Equations. Time-Dependent Continuity and Momentum Equations for the Gas Phase:39. ∂ðFg εg Þ ∂t

þ ∇:ðFg εg ug Þ ¼ 0

∂ðFg εg ug Þ ∂t

ð1Þ

þ ∇:ðFg εg ug ug Þ

¼ εg ∇P Fpg þ εg ð∇:τg Þ þ Fg εg g

ð2Þ

where Fpg is the volumetric particle-fluid interaction force given by Fpg = Kpg(u Bg u Bp) where Kpg is the coefficient for momentum exchange from particle to gas phase. The interphase exchange model follows that of Yang and Yu44 3 εs εg Fg j uBs uBg j 2:65 CD εg where 4 dp i 24 h CD ¼ 1 þ 0:15ðεg Res Þ0:687 εg Res Kpg ¼

The stress tensor is given by 2 τg ¼ μg ð∇ug þ ∇uTg Þ ð∇:ug ÞI 3 where ug, εg, Fg, and μg are the velocity, void fraction, density, and shear viscosity (sum of laminar shear viscosity and turbulent viscosity) of the gas phase, respectively. RNG Model Equations. ! ∂ðFg kÞ ∂ðFg kui Þ ∂ ∂k þ ¼ Rk μeff þ Gk Fg ε ∂xj ∂xj ∂t ∂xi 2031

dx.doi.org/10.1021/ie200647e |Ind. Eng. Chem. Res. 2012, 51, 2030–2047


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Figure 2. Comparison of experimental and predicted results: (a) single-phase gas velocity profiles in the bend and postbend section (Kuan et al.21); (b) particle velocity profile (Yilmaz and Levy3); (c) particle concentration profile (Yilmaz and Levy3); (d) pressure drop for air-glass beads (Lee et al.28); (e) pressure drop for air-polypropylene beads (Lee et al.28).

∂ðFg εÞ ∂t

þ

∂ðFg εui Þ ∂xi

∂ ∂ε ¼ Rε μeff ∂xj ∂xj

!

ε ε2 þ C1ε Gk C2ε Fg k k

η Cμ Fg η3 1 ε2 ηo 3 k 1 þ βη

!

where Gk (=μt(2SijS ij)1/2) is the generation of turbulent kinetic energy due to the mean velocity gradients. Rk and R ε are the inverse effective Prandtl numbers for k and ε, respectively, and η = (2SijSijk)1/2 /ε. The effective viscosity is computed using the high Reynolds number form given by μt = FCμk2/ε. The model constants are as follows: C1ε = 1.42; C2ε = 1.68; Cμ = 0.0845; Cζ = 100; η0 = 4.38; β = 0.012. 2032



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Table 2. Physical Properties, Relaxation Time, and Stokes Number Range for Different Particles at Standard Conditions material

mean diameter ( SD (μm)

density (kg/m3)

A

1680

0.03

0.20.32

B

1755

1.4

914

500 ( 50

B

2700

2

14 22

2800 ( 500

D

1123

27

180288

Particle Motion Equation. The particulate phase is represented by computational particles whose trajectories are computed by simultaneous integration of d xBp ¼ uBp dt

Table 3. Operating parameters phase

ð3Þ

parameter

d uBp ¼ B FD þ B Fg þ B F pg þ B FA þ B F sl dt

solid

mass flow rate (kg/m2 s)

15.8, 29.0, 31.1, 46.3

velocity (m/s) density (kg/m3)

1016.0 1.225

viscosity (kg/m.s)

1.8 105

ð4Þ

This equation describes the balance of forces acting on the particles as it moves along its trajectory. The term on the left side is the inertia force acting on the particle due to its acceleration, and the right-hand side terms are external forces acting on the particle, with subscripts D, g, pg, A, and sl, respectively, denoting force components arising from drag, gravity, flow pressure gradient, added mass, and slip-shear lift. The most influential force acting on the particle is the viscous drag force exerted by the continuous phase. This force is predicted with the aid of the standard drag coefficient CD and relative velocity between the particle and the carrier fluid ug uP ð5Þ FD ¼ mp τr with ug = u + u0 and particle relaxation time, τr = Fpd2p/(18μfD), and the Schiller-Naumann drag correlation fD for a sphere ( 1 þ 0:15Re0:687 for Rep e 1000 p ð6Þ fD ¼ 0:01833Rep for Rep > 1000 24 fD Rep

As Rep increases beyond 1000, the corresponding fD leads to a constant drag coefficient CD of 0.44. Fp-g is the interaction term representing the effect of particle on the fluid field, whereas FD is the interaction drag force on the particle from the fluid. The volumetric fluid-particle interaction force in a computational cell can be determined by Fp-g= FD/ΔV, where ΔV is the volume of the computational cell. The fluid drag force acting on individual particles will react on the fluid phase from the particles in a computational cell as per Newton’s third law. The force components due to gravity Fg, added mass FA, and pressure gradient Fpg are, respectively, given by ! Fg Fg ¼ m p 1 g ð7Þ Fp 1 Fg dup FA ¼ m p 2 Fp dt

value

gas

and the equation of particle motion

CD ¼

Stokes number range (St)

70 ( 55

glass beads (GB)

mp

τr (s) (based on dp,mean)

500 ( 50

pulverized coal (PC) glass beads (GB) polypropylene beads (PP)

Geldart class

1 Fpg ¼ πdp2 ∇P 4

ð9Þ

The present model adopts a slip-shear model by Mei40 Fsl ¼

π 3 d F Csl ððug up Þ ωg Þ 8 p g

ð10Þ

where 4:1126 Csl ¼ pffiffiffiffiffiffi f ðRep , Res Þ Res f ðRep , Res Þ ¼

8 < ð1 0:3314γ1=2 Þ eð 0:1Rep Þ þ 0:3314γ1=2 for Rep e 40 : 0:0524ðγRep Þ1=2 for Rep > 40

ð11Þ with Res =

(Fgd2p|ωg|/(μ)

γ ¼ 0:5

Res ; ω g ¼ ∇ ug Rep

Assuming that all force components except drag are constant during time step Δt, eq 4 can be integrated analytically to yield up ¼ ug þ ðu0p ug ÞeΔt=τr þ

τr ð1 eΔt=τr Þ mp

ðFg þ FA þ Fsl þ Fpg Þ

ð12Þ

with superscript 0 indicating the start of a time step. Similarly eq 3 can be integrated analytically to yield xp ¼ x0p þ up Δt

ð13Þ

Equations 12 and 13 were solved within a given cell in the particle tracking calculations. The effect of gas turbulence is included within the particle transport model by using the instantaneous velocity of the carrier fluid as U BR ¼ uBg uBp ¼ ð uB uB0 Þ uBp

ð8Þ

The particle is described as a single point that moves at its velocity, with each particle being treated individually. Particle 2033



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Figure 3. Effect of the flow of Geldart type-A, -B, and -D particles on (a) gas velocity at t = 2 s; (b) time-averaged solid volume fraction in a circular elbow R/D = 1.5, Gs = 31.1 kg/m2 s, ug = 10.6 m/s.

Figure 4. Effect of the flow of Geldart type-A, -B, and -D particles on rope formation and particle velocity at t = 2 s in a circular elbow. R/D = 1.5, Gs = 31.1 kg/m2 s, ug = 10.6 m/s.

trajectories are computed individually at specified intervals during the fluid phase calculation. However, for a large number of particles, computational 'parcels’ are used where each parcel represents a cloud of many particles with the same characteristics

(diameter, velocity etc.) The parcel of particles is tracked through the domain, updating their positions at every time step, size and local flow conditions. The mass-flow of the parcel is equal to the mass-flow of the group of particles, to account for their number 2034



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Figure 5. (a) Pressure contours of gas-polypropylene flow at the bend (values in Pa) and (b) distribution of pressure coefficient of gas-polypropylene flow in the bend. t = 2 s, R/D = 1.5, Gs = 29 kg/m2 s, ug = 14 m/s.

on the momentum exchange with the gas phase. The parcel size only affects the collisions, but the hydrodynamic forces are based on the particle diameter. The strength of each particle stream (number of particles flowing along it per second) is calculated according to the specified mass flow rate, particle density and diameter, and number of streams. The number of particles in each parcel is then calculated based on the specified mass flow rate and particle diameter. In the present case, each parcel contained approximately 220 particles. Following the method of Gosman and Ioannides,38 the motion of particles is tracked as they interact with a succession of discrete turbulent eddies. Eddies are assumed to have constant velocity, length, and time scales during their interaction with particles. A particle is assumed to interact with an eddy for a time which is the smaller of either the eddy lifetime or the transit time required for the particle to cross the eddy. These times are estimated by assuming that the characteristic size of an eddy is the dissipation length scale 3=2 =ε Le ¼ C3=4 μ k

ð14Þ

2.2. Boundary Conditions. The flow geometry consisted of a horizontal pipe of length 5 pipe diameters, an elbow section, and a vertical pipe of length 20 pipe diameters. Uniform velocity conditions were imposed at the inlet. The outflow boundary conditions at the flow exits correspond to zero diffusion flux for all flow variables. The conditions of the outflow plane are extrapolated from within the domain and have no impact on the upstream flow. The extrapolation procedure updates the outflow velocity and pressure in a manner consistent with a fully developed flow assumption. Turbulence quantities were calculated from the relations

kinlet

3=2 k 3 uinlet 2 ¼ ; εinlet ¼ inlet 2 10 0:3D

The characteristic turbulent eddy velocity at the inlet is 10% of the mean flow velocity. Zero-slip was prescribed at the wall surfaces for the gas phase. The particles were randomly injected at the inlet at 220 locations whose coordinates were randomly sampled. The diameters of the computational particles at each starting location were stochastically sampled from a Rosin-Rammler distribution function " # dp n ð15Þ ProbðDiameter > dmean Þ ¼ exp dmean where dmean is the mean diameter of the distribution, and the exponent, n, represents the spread of the data from the mean; dp and dmean are given in Table 2, and n was taken to be 2.8. Particle-wall collisions were modeled through a coefficient of restitution, e, which is the ratio of normal velocities of the particle before and after the collision. The coefficient of restitution was set equal to 0.9 for all simulations performed in this study.3,31,43 The motion of relatively small particles are controlled by fluid motion and turbulent dispersion; thus, the influence of particlewall interaction is less important away from the wall since small particles promptly follow the carrier fluid. 2.3. Numerical Procedure. The governing equations described above are solved using the finite volume commercial CFD code Ansys-Fluent 12.1. The various geometries considered for the present study are shown in Figure 1. The square elbow geometry consisted of a 0.5 m long horizontal straight duct, a squared 90° bend with R/D = 1.5, and a 1 m long vertical straight duct (Figure 1a). The major feature of the squaresectioned duct flow is the strong curvature in the streamwise direction along the duct length. Similarly, the circular elbows in Figure 1(b) and (c) consists of a curved 90° bend with R/D = 1.5 and 3, respectively, keeping all other parameters same. The pipe diameter was 0.05 m. Figure 1(d) shows the measurement 2035



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Figure 6. Effect of particle density on particle roping and particle velocity for gas-glass beads flow at t = 2 s. R/D = 1.5, Gs = 31.1 kg/m2 s, ug = 10 m/s.

locations of different angular planes at the elbow section. All calculations are performed on structured hexagonal grids (see Table 1, for the number of grid points for different geometries). Figure 1(e) shows the grid at the cross-section of the pipe. The grid is sufficiently refined at the wall to produce a grid-independent solution (see Figure 2(b),(c)). Solid particles are released into the duct at the same velocity as the gas phase, i.e., zero slip, and from random locations on the inlet plane. A pressure-based segregated solver is used, with the SIMPLE scheme for pressure-velocity coupling. All diffusion terms are treated with second-order discretization. Second-order upwind interpolation is used for the convection terms. Under-relaxation factors are mostly left at the Fluent default settings. To handle the nonlinearity of the equations, the treatment of all transported variables involves two nested levels of iterations referred to as inner and outer iterations. Outer iterations are repeated until the problem satisfies a convergence criterion. A converged solution of the coupled two-phase flow system is obtained by successive solution of the Eulerian and discrete phase, respectively. For the simulations performed in this study, the convergence of outer iteration was judged by how accurately the continuity equation was satisfied by the current values of the dependent variables. The solution procedure is considered converged when the ratio of the summation of absolute mass source residuals to the total rate of mass inflow falls below a prescribed tolerance Ncell

∑ RM, i

i¼1

mair

eδ

ð16Þ

with δ was 1 106 for the simulations performed in this study. The convergence of the gas-particle flow solution is judged by monitoring the history of the residuals of the U, V, and W momentum equations and of the continuity equation during the

two-way coupling process. Initially, the flow field is calculated without the particle phase until a converged solution is achieved. Thereafter, a large number of particles are tracked through the flow field (typically in the range of 100,000 to 800,000 depending upon the solid mass inflow, as described in section 2.1). Each case was simulated for 2 to 5 s. Predicted particle statistics were collected at each measurement axial and angular locations and on the center-plane of the geometry in 10 equally spaced bins at the exit. The particle statistics were then averaged to produce a representative value41 based on

Up ¼

∑ upni ϕi ∑ ni ϕi i ¼ 1, 12

i ¼ 1, 12

ð17Þ

where ni is the number of particle tracks for size fraction i, and ϕi is the corresponding particle number flow rate per particle track.

3. RESULTS AND DISCUSSION The calculations of gas-solid two phase flow through a 90° bend are presented in this section for three types of granular material (Geldart A, B, and D) for dilute and dilute-to-dense phase conveying. Numerical results are then compared with experimental data of Yilmaz and Levy,3 Kuan et al.,21 and Lee et al.28 Simulation conditions were chosen to correspond to those experiments. Detailed information about the physical properties and operating conditions are presented in Tables 2 and 3, respectively. The predicted data correspond to the vertical central plane (z = 0) of the geometry and various axial and angular locations (as shown in Figure 1d) in the streamwise direction. For smaller particles like the pulverized coal particles (PC), the particle velocity essentially equals the local gas velocity. For 2036



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Figure 7. Effect of bend ratio on gas velocity contours in (a) square elbow; (b) circular elbow, R/D = 1.5; (c) circular elbow, R/D = 3.0, for gas-glass beads flow at t = 2 s. Gs = 15.8 kg/m2 s, ug = 11.9 m/s.

larger and heavier particles like polypropylene beads and glass beads, there is a significant slip velocity which is a function of the particle size, relative density, geometry, and velocity. Time-averaged velocities were obtained for the gas phase as well as the dispersed phase. All data are plotted against the nondimensional distance x ¼ x=r ðfor horizontal sectionÞ or y ¼ y=r ðfor vertical sectionÞ such that x* or y* = 1 at the inner wall and +1 at the outer wall. The predictions were taken at x/D = 5 and 8 and y/D = 1, 5, 10, 15, and 20 downstream of the bend exit in the vertical duct. As shown in Figure 1(d), gas and particle data were collected at six different angular locations: θ = 0°, 30°, 45°, 60°, 75°, and 90°. 3.1. Comparison with Experimental Results. The numerical calculations are compared with experimental data in three steps: (1) Comparison with the experimental data of Kuan et al.21 for nonisothermal gas phase flow in an elbow. (2) Comparison of predicted particle velocity and particle concentration profiles in 90° circular elbow with the experimental data of Yilmaz and Levy.3 (3) Comparison of pressure drop in 90° square elbow with Lee et al.28 experimental data. Predicted and experimental gas phase profiles are compared in Figure 2(a). The measured profiles21 indicate a growing layer of slow gas stream next to the outer wall of the pipe due to adverse pressure gradient on the outer wall, within the bend. However, the gas flow accelerates steadily near the inner wall. The numerical

prediction provides a good representation of the experimentally measured flow. Yilmaz and Levy3 experimentally investigated solid flow nonuniformities in a dilute system. They studied the flow of pulverized coal (PC) particles conveyed by air through a 90° circular elbow using a fiber optic probe. The dimensions and geometry created for our simulation match the experimental setup, with D = 0.154 m, horizontal section length = 10D, vertical section length = 20D, and R/D = 1.5. Parts (b) and (c) of Figure 2 compare the predicted and experimental particle velocity and particle concentration profiles, respectively. As can be seen from Figure 2(b), the inlet particle velocity profiles did not change as the solids loading ratio increased from 0.33 to 1.0. However, the particle concentration close to the outer wall increased with the solids loading (Figure 2(c)). As the solids loading is increased, particle ropes become denser than at lower mass loadings. The particle velocities in denser ropes are lower at the outer wall of the pipe due to higher frictional losses. Hence, the particle mass concentration increases toward the outer wall with increasing solids loading. This is responsible for the radial difference in concentration profiles with different solids loading. Reasonable agreement was obtained between the experimental data and CFD predictions. Lee et al.28 studied the characteristics of air-glass beads (GB) and air-polypropylene (PP) beads flowing through a 90° square elbow based on electrical capacitance tomography (ECT), particle image velocimetry (PIV), and phase Doppler particle analysis. The dimensions were D = 0.05 m, horizontal section length = 10D, vertical section length = 20D, and R/D = 1.5. Figure 2(d) shows the pressure drop in the postbend vertical 2037



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Figure 8. Effect of bend ratio on particle roping and particle velocity in (a) square elbow; (b) circular elbow, R/D = 1.5; (c) circular elbow, R/D = 3.0 gas-glass bead flow at t = 2 s. Gs = 15.8 kg/m2 s, ug = 11.9 m/s.

Figure 9. Secondary flow fields at bend exit and downstream of bend for polypropylene beads: (a) at bend exit, θ = 90°; (b) at y = 2D; (c) at y = 5D; R/D = 1.5, Gs = 15.8 kg/m2 s; ug = 14 m/s. Arrows represent direction of gas flow, and colors indicate corresponding magnitudes of velocity vectors in these planes.

section of the square elbow for air-glass beads (Geldart class-B) at solid mass fluxes, Gs, of 15.8 and 29.0 kg/m2 s. The pressure drop increased with increasing superficial gas velocity, as expected for dilute phase conveying. However, there was an abrupt decrease in pressure drop at ug = 12 and 15.6 m/s for Gs = 29.0 kg/m2 s. More simulations for these particular cases are required to draw conclusions. Reasonable agreement is obtained for Gs = 15.8 kg/m2 s. On the other hand, Figure 2(e) illustrates the pressure gradient for the polypropylene beads (Geldart class-D) at Gs = 15.8 and 31.1 kg/m2 s. As indicated in Figure 2(e), the opposite pressure gradient trend was obtained,

with pressure drop decreasing with increasing superficial gas velocity, indicating transition from dilute to dense phase pneumatic conveying. In general, calculation results agree well with the experimental data. A qualitative comparison of the predicted solid volume fraction distribution was also made with the ECT measurements for glass beads, but the results cannot be presented here due to the poor quality of the available ECT pictures. 3.2. Parametric Study. The influence of various factors affecting the flow behavior of gas-solid system such as particle size, particle density, bend radius ratio, square vs circular elbow geometry, gas velocity, and solid mass flux are discussed in this section. 2038



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Figure 10. Mean streamwise gas and glass bead velocity profiles in 90° circular elbow at (a) θ = 0°, (b) θ = 30°, (c) θ = 45°, (d) θ = 60°, (e) θ = 75°, (f) θ = 90°. t = 2 s, R/D = 1.5, Gs = 15.8 kg/m2 s, ug = 11.9 m/s.

3.2.1. Flow Behavior of Different Classes of Granular Material. Figure 3(a),(b) illustrates the effect of the type of granular material on the gas velocity at t = 2 s and the time-averaged solid volume fraction, respectively. As the gas-solid suspension flows through the horizontal section of the pipe, the two phases

tend to segregate, as depicted in Figure 3(a). A region of lower gas velocity near the outside of the pipe can be seen for all three types of particles, more pronounced for the glass beads, owing to their higher density. The maximum gas velocity is displaced toward the inner wall at the bend, as a result of the favorable 2039



Figure 11. Profiles of mean streamwise air and glass bead velocities in postbend vertical section at (a) y = 5D; (b) y = 10D; (c) y = 15D. t = 2 s, R/D = 1.5, Gs = 15.8 kg/m2 s, ug = 11.9 m/s.

streamwise pressure gradient there (see Figure 5). The unbalanced centrifugal force on the main flow induces secondary flow in the pipe bend. The secondary flow exhibits a current away from the outer wall toward the pipe center, providing the mechanism for removal of particles to particle-free regions in the pipe cross-section. This not only leads to significant phase separation, due to the density difference, but also favors particle

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segregation. The particles band together in a narrow localized stream, referred to as a particle rope. This leads to a large negative slip velocity between the gas and particulate flows near the outer wall. As seen from Figure 3(b), a region of high solid volume fraction appears near the lower wall, downstream of the bend, and occupies the outer wall at the bend. Rope formation is shown in Figure 4. From Figure 4(a) it can be seen that PC particles, being lighter and smaller, are accelerated more and hence gain higher axial velocity upstream of the bend compared with glass and polypropylene beads. On the other hand, the coarser polypropylene beads probably experience more particle-particle and particle-wall interactions, thereby significantly reducing the acceleration of particles in the rope (Figure 4c). The centrifugal forces induced by the bend direct the particles toward the outer wall, as is evident from the high particle volume fraction there (Figure 3b). In curved pipes, the more rapidly flowing central parts of the flow are forced outward by centrifugal action, while the slower parts along the wall migrate inward where the pressure is less, and a secondary flow develops at right angles to the main flow.42 The PC particles, being smaller and lighter, follow the gas phase dynamics and hence become completely dispersed in the horizontal section of the pipe before reaching the bend. On the other hand, the onset of phase separation takes place as early as x = 5D for the glass and polypropylene beads (Figure 4b,c). This is more pronounced for the GB, with a strong rope toward the lower wall at x = 5D. The glass beads, having the highest density among the three types of particles, tend to settle faster under the effect of gravity. Interestingly, the roping was found to abruptly disperse at the bend exit for the glass and polypropylene beads compared to the PC particles. As shown in Figure 4(a), particles within the rope occupied a smaller cross-section of the pipe toward the outer wall, upstream of the bend exit at y = 5D, indicating that the particle transport in the radial direction is less for lighter and finer particles. Particle dispersion takes place gradually due to turbulence, and particles are completely dispersed at y = 10D. Similar observations were made experimentally.2,3 The Stokes number (St) is the characteristic response time of the particle divided by the characteristic response time of the fluid. The relaxation time for smaller and lighter particles (like PC) is much less than for the larger and heavier particles (like GB and PP) for the same gas velocity. The ranges of St for PC, GB, and PP are presented in Table 2 for gas velocities of 10 to 16 m/s. St , 1 for PC, meaning that the particle motion is tightly coupled to the fluid motion. The particle dispersal is then virtually the same as fluid dispersal. This can also be seen from the formation of particle ropes and gas velocity contours in Figures 3 and 4. There is little relative motion between the two phases locally upstream of the bend entrance. St . 1 for PP, so that particles are not significantly influenced by the fluid. Their response time is longer than the time the fluid has to act on it. (The fluid time scale may be the rotation time of a characteristic eddy.) Hence particles pass through the flow with little deflection from their initial trajectories. As seen from Figure 4, the particles tend to settle toward the lower wall of the pipe before entering the bend. Higher St thus implies more collisions of the particles with the walls of the bend. The most interesting case is GB, where St is intermediate between the values for PC and PP, and so is its diameter. GB is denser than both PC and PP. GB particles migrate to the margins of the eddy and hence are 'unmixed’ and settle faster. Dynamic pressure contours for the air-polypropylene bead flow are displayed in Figure 5(a) at the central plane (z = 0). The 2040



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Figure 12. Particle segregation for (a) glass beads; (b) polypropylene beads. R/D = 1.5; Gs = 15.8 kg/m2 s, ug = 14 m/s.

Figure 13. Normalized fluctuating gas and particle turbulence in the vertical duct. t = 2 s, R/D = 1.5, Gs = 15.8 kg/m2 s, ug = 11.9 m/s.

dynamic pressure at the inner edge of the bend is higher than at the outer wall. As can also be seen from Figure 3(a), there is deceleration near the outer wall due to the unfavorable pressure gradient on the upper half of the bend. Figure 5(b) shows that the pressure coefficient, CP = ΔP/(0.5Fgu2g ), decreases toward the inner wall. Upon leaving the bend, axial flow stabilizes. However, the flow still recirculates due to Dean vortices formed at the bend. Hence, the pressure falls sharply over the bend but decreases slowly and nearly linearly with height in the downstream vertical section. The most influential force acting on the particles is the viscous drag force, FD, exerted by the continuous phase. The net gravity (weight minus buoyancy) force, Fg, has a similar magnitude of the

order of 106 N, in the present case study. There is a strong correlation of these forces with the particle flux toward the wall, especially at the outer wall of the elbow. This causes the particles to segregate, settle toward the lower wall of the pipe in the horizontal section, while deflecting toward the outer wall at the bend. The effect of gravity on more massive particles becomes significant in the postbend vertical section. A combination of secondary flow at the bend and the effect of gravity in the postbend section causes the particle rope to disperse as it moves downstream. The particles entering the wall layer with curvature at the bend may be re-entrained into the recirculation zone formed immediately downstream of the bend exit. The formation of particle ropes of smaller lighter particles is dominated by 2041



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Figure 14. Solid concentration contours at different axial and angular locations for pulverized coal particles. t = 2 s, Gs = 31.1 kg/m2 s; ug = 10.6 m/s.

Figure 15. Solid concentration contours at different axial and angular planes for glass beads. t = 2 s, Gs = 31.1 kg/m2 s; ug = 10.6 m/s.

diffusional deposition (due to the small fluctuations in the viscous wall region). The free-flight deposition mechanism then becomes important for increasing particle inertia. The other external forces that could influence particle trajectories are added mass, due to acceleration of carrier fluid in the vicinity of the particle, slip-shear, Basset force, and the pressure gradient force. However, for gas-particle flows with density ratio, Fp/F, of order 103, these forces are orders of magnitude smaller than the drag and gravity forces.

3.2.2. Effect of Particle Density. Figure 6 shows the effect of varying the density of the glass beads from 1755 to 2700 kg/m3, keeping the particle diameter constant (dp = 500 μm). Despite the uniform velocity distribution at the inlet, the heavier particles tend to move more slowly than the lighter ones. The glass beads with higher density tend to settle faster under the effect of gravity and a particle rope appears as early as x = 5D. Owing to their lower velocity, the dispersion coefficient of the heavier particles is higher than for the lower-density particles. As a result, the particle 2042



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Figure 16. Effect of gas velocity on gas-phase turbulence intensity at t = 2 s: (a) Gs = 31.1 kg/m2 s; ug = 11.9 m/s; (b) Gs = 31.1 kg/m2 s; ug = 16 m/s.

rope rapidly disperses as soon it leaves the bend (Figure 6b), whereas the particle rope persists up to y = 10D for the lighter particles. 3.2.3. Effects of Bend Radius Ratio and Elbow Shape. The predicted gas velocity and particle roping along the central plane (z = 0) for all three elbows are plotted in Figures 7 and 8, respectively. The effect of sharp 90° edges on the gas velocity contour in a square elbow appears in Figure 7(a). Two regions of very low air velocity can be seen near the outer and inner sharp edges due to high pressure gradients. On the other hand, for the circular elbows, a region of high air velocity can be seen at the inner wall due to higher pressure gradient (see Figure 5a). However, for a longer elbow (Figure 7c), the region of higher air velocity spreads over a larger region, occupying the complete inner surface of the elbow. The average residence time within the elbow plays a strong role as can be seen from Figure 8. The longer the particles experience the inertial effect within the elbow, the stronger the rope, i.e. the larger the peak rope concentration at the bend exit. Unlike the circular elbows, no particle roping is predicted for the square elbow. A particle rope created by an elbow starts to disperse once it leaves the elbow due to secondary flows and turbulence, possibly promoted by gas-particle flow instabilities. CFD predictions show rapid rope dispersion within three to four pipe diameters downstream of the elbow exit plane, accompanied by a rapid decrease in the intensity of the secondary flows and turbulence levels between y = 1D and 5D. Rope dispersion characteristics are predicted to be very different for the R/D = 1.5 and 3.0 elbows. Ropes within the latter move toward the center of pipe and disperse more quickly than for the R/D = 1.5 elbow, where the rope flow stayed close to the outer wall.

Figure 9 shows the secondary flow fields at the bend exit and upstream of the bend for gas-polypropylene beads at t = 2 s. The arrows represent radial gas velocity vectors. The formation of Dean vortices at the inner wall can be seen at the bend exit at θ = 90° and postbend at y = 2D. The secondary flows carry particles around the pipe circumference through the particle-lean regions, resulting in a spreading of the particles from within the rope. The secondary flows created in the elbows contribute to the dispersion of the particle ropes. 3.2.4. Velocity Profiles, Particle Segregation, and Turbulence. Figures 10 and 11 compare the normalized gas and particle velocity profiles in the 90° bend and in the postbend vertical section for the pulverized coal particles at ug = 10.6 m/s and Gs = 31.1 kg/m2 s, respectively. Figure 10(a) to (f) shows the development of the axial velocity profiles at different angular planes (θ = 0°, 30°, 45°, 60°, 90°). Upon reaching the bend, as seen from Figure 10(a), the gas flow is already shifted toward the inner wall and accelerates in the first half of the bend, reaching a maximum of 1.34ug,inlet at θ = 45°. It then begins to decrease over the second half of the bend. At the bend exit, θ = 90°, the gas velocity reaches its lowest value of 0.85ug,inlet near the inner wall region and the higher velocity region shifts toward the outer wall. Figure 11(a) to (c) shows the gas velocity profiles in postbend vertical section. At y = 5D, the maximum velocity has shifted toward the outer wall under the influence of unbalanced centrifugal forces on the main flow at the bend. As one moves further downstream (see y = 10D), the velocity profile gradually equalizes, and, upon reaching y = 15D, it is almost symmetrical. The particles appear to follow closely the trend of the gas velocity profile up to the bend entrance. At θ = 90° the particles start to congregate toward the outer wall, as noted in Figure 3. This particle maldistribution initiates particle roping. Particle 2043



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Figure 17. Effect of solid mass flux on gas-phase turbulence intensity at t = 2 s: (a) Gs = 31.1 kg/m2 s; ug = 11.9 m/s; (b) Gs = 46.3 kg/m2 s; ug = 11.9 m/s.

ropes become more concentrated toward the outer wall as gas and particles pass through the bend. As shown in Figure 10(c) to (f), a particle rope nearly occupies the lower half-section of the pipe and begins to disperse at the exit of the bend. The effect of recirculation and backflow of particles due to the secondary flow at the bend is seen up to y = 10D. Figure 11(a) and (b) shows that a group of particles descends near the inner wall. Beyond y = 10D, no downward-moving particles are seen, and the particle velocity profile gradually approaches symmetry by y = 15D. The extremely rapid development of the gas velocity profile, even at the bend entrance, is due to rope formation which already starts before the bend entrance, especially for heavier and larger particles. The particles exert higher resistance force on the gas flow in the region where the rope is formed, causing gas to escape and accelerate through the particle-free zone in the upper region of the pipe. A significant difference is observed in the gas and particle velocity profiles at the outer wall due to particle-wall collisions. The effect of particle-wall collisions is predicted to be most severe at θ = 60° where particles lag behind the gas phase and the velocity difference reaches a maximum. At the bend exit, particles were moving up to 30% more slowly than the gas phase. In the postbend vertical section, the particle velocity remains symmetrical, while the gas velocity gradually becomes fully developed by y = 15D. Figure 12 shows particle segregation, with coarser particles falling toward the lower wall, while finer particles remain in a dispersed state. The gas and solid turbulence profiles are presented in Figure 13 for the postbend vertical section. Figure 13 indicates that for the gas phase, two regions exist with high levels of turbulence at y = 5D. This confirms the formation of Dean vortices

due to secondary flow at the bend. At the same time, the particles experience high turbulence toward the inner wall of the pipe. In addition, the upward- and downward-pointing symbols denote particles traveling upward with the gas flow and downwardmoving particles, respectively. A uniform kinetic energy field was observed for both the gas and solid phase 15D downstream of the bend. The particles seem to follow the gas motion at the inner wall region, supporting the earlier result (Figure 12) that the coarser particles have already segregated from the gas flow. However, the fine particles are carried away in the gas stream near the inner wall. 3.2.5. Solid Concentration Profiles. Figures 14 and 15 show contour plots of the predicted particle concentrations in the pipe cross-section at different axial positions along the horizontal section (x/D = 5 and 8), angular planes (θ = 0°, 30°, 45°, 60°, 90°), and vertical section (y/D = 1, 5, 15, and 20) for pulverized coal particles and glass beads, respectively. Figure 14 shows that the PC particles are dispersed over the complete cross-section of the pipe in the horizontal section. Particle segregation and settling starts as early as x = 5D and gradually increases with distance until the bend entrance at θ = 0°. An abrupt, highly particle-concentrated region forms at θ = 30° marking the onset of particle roping. The rope region becomes more condensed as particles pass through the bend. However, the particle concentration decreases at the outer wall due to localized mixing in that region (see θ = 60° and 75°). The localized particle mixing probably occurs due to a combination of turbulence and particlewall and particle-particle interactions. Similar observations were made by Bilirgen and Levy2 in experimental work using a fiber optic probe. At the exit of the bend, the concentration of the 2044



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Figure 18. Axial velocity contours at the central plane (z = 0) of a circular elbow for (a) air alone; (b) air-glass beads; (c) air-polypropylene beads. t = 2 s, R/D = 1.5, Gs = 15.8 kg/m2 s; ug = 14 m/s.

particle rope becomes dilute under the action of secondary flow and gas turbulence. Rope dispersion is rapid as soon as it reaches the vertical section at y = 1D and is almost complete at y = 5D. However, a nonuniform particle distribution persists until y = 15D. The heavier and larger glass beads (dp,mean = 500 μm; Fp = 2700 μm) settle more quickly than the finer and lighter PC particles (dp,mean = 70 μm; Fp = 1680 μm) as can be seen from Figure 15. A stream of highly concentrated GB particle region is observed as early as x = 5D, dispersing a little as it travels downstream. The higher concentration region moves toward the lower outer wall as the rope becomes denser at θ = 45° and 60°. The rope remains intact until θ = 90° and then abruptly disperses beyond the bend. Thereafter, a nearly uniform distribution is predicted in the vertical section of the pipe. 3.2.6. Effects of Gas Velocity and Solids Mass Flux. The gas velocity and solids mass loading are major factors dictating flow characteristics and the flow regime of the system. In this study, the gas velocity was varied from 11.9 to 16.0 m/s, a typical operating velocity range for pneumatic conveying. The effect of varying gas velocity on the gas turbulent intensity ((u0 g)2/ug)) can be seen in Figure 16 for a constant solids mass flux of Gs = 31.1 kg/m2 s. Increasing the gas velocity does not affect much the gas turbulence intensity in a circular elbow. However, a region of high turbulence intensity was found near the inner sharp 90° edge of the square elbow. Intense turbulence occurs at the bend exit, due to secondary flow and formation of Dean vortices, and this increases with increasing gas velocity. Some particles become trapped in this recirculation zone before dispersing in the vertical pipe. When the solids mass flux increased from 31.1 to 46.3 kg/ m2 s at a constant gas velocity, ug = 11.9 m/s for the square elbow,

it can be seen from Figure 17 that the region of high turbulence intensity is reduced at the inner wall of the bend. When the solids mass loading is increased, the particles tend to form a dense phase while traveling along the pipe, thereby lowering the magnitude of particle velocity. Thus, the particle velocity is predicted to decrease from 11.1 to 4.7 m/s when the mass flux increases from 31.1 to 46.3 kg/m2 s. This leads to the formation of a dense particle rope region toward the outer wall of the pipe at the bend, with this rope occupying a major portion of the bend cross-section, thereby affecting the gas turbulence. 3.2.7. Effect of GasSolid Interaction. Figure 18 shows the effect of the presence of particles on the axial velocity contours of the gas phase in a circular elbow. The gas-phase predictions were performed in the absence of solid particles (Figure 18a). Upon reaching the bend, the flow near the inner wall accelerated under a favorable pressure gradient until the midbend. The gas has a maximum axial velocity of 17.6 m/s toward the inner zone of the bend close to θ = 45° (Figure 18). Beyond 45°, the flow gradually shifts toward the outer wall. The streamwise velocity declines next to the inner wall at θ = 60°, 75°, and 90°, whereas the streamwise velocity at the outer wall increases slightly due to the resistance from particles to the gas flow, reaching its maximum in the region where the peak of the axial velocity shifts. As seen in Figure 3, particles tend to aggregate toward the outer wall at the bend, offering maximum resistance to the gas flow there. Hence, gas tends to flow through the region where it experiences less resistance. The gassolid interaction force is the sum of drag forces.31 The drag force on the particles is large at the outer wall. The relative velocity between the two phases is large at the outer wall: 14.8 m/s (gas phase) 3.5 m/s (solid phase) = 11.3 m/s. Large gassolid interaction force within the rope causes less gas flow through the rope so that more gas flows along the sides from 2045


Industrial & Engineering Chemistry Research the outer to the inner wall. Beyond y = 5D, downstream of the bend exit, the low-velocity flow in the core region gradually recovers.

4. CONCLUSIONS The pneumatic conveying of three different classes of granular material (type-A: pulverized coal; type-B: glass beads; type-D: polypropylene) was numerically investigated in a 90° square and two circular elbows (R/D = 1.5 and 3.0) for dilute and dense modes of operation. The Eulerian-Discrete phase approach, including generalized wall-boundary conditions for the solid phase and a particle-wall collision model, was employed. The turbulent dispersion of particles was taken into account by stochastic tracking. Numerical computations generally compare favorably with the experimental fiber optic probe results of Yilmaz and Levy3 and the electrical capacitance tomography (ECT), particle image velocimetry (PIV), and phase Doppler particle analysis data of Lee et al.28 Fundamental features like particle roping, particle segregation, secondary flow, recirculation, and gas-particle interactions were reasonably captured and compared for different classes of particles for the square and circular bends. The results show that secondary flows disperse particle ropes by carrying particles around the pipe circumference, while turbulence disperses the ropes by localized mixing. Significant differences were found in the flow behavior of the different granular materials. Various parameters such as particle density, particle diameter, gas velocity, solid mass flux, and bend geometry affect the flow. ’ AUTHOR INFORMATION Corresponding Author

*Phone: 604-822-3121. E-mail: [email protected].

’ NOMENCLATURE CD drag coefficient () Cp particle mass concentration (kg/m3) Cp,avg average particle concentration within pipe cross-section (kg/m3) Cμ turbulence model constant (as in eq 14) (-) D pipe diameter (m) dp particle diameter (m) fD Schiller-Naumann drag coefficient (-) F force vector (N) e restitution coefficient () g acceleration due to gravity (m/s2) Gs solid mass flow (kg solid particles/s) k turbulence kinetic energy (m2/s2) Le characteristic size of an eddy (m) m mass flow rate (kg/s) n number of particle tracks (-) P pressure (Pa) R elbow turning radius (m) Re Reynolds number (-) Rep particle Reynolds number (-) Res particle Reynolds number based on fluid rotation (-) RM, i absolute mass source residual for computational cell i (kg/s) t time (s) τR particle residence time in elbow (s) u mean gas velocity (m/s)

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u0 fluctuating velocity vector (m/s) x, z transverse distances in pipe cross-section (m) y axial distance (m) Greek symbols

εp μ μg τr F θ ωg ε

particle volume fraction viscosity (kg/m.s) gas shear viscosity (kg/m.s) particle relaxation time (s) density (kg/m3) elbow turning angle (deg) fluid rotation vector (1/s) energy dissipation rate (m2/m3)

Subscripts

A g p D pg p-g sl

added mass gas phase solid phase drag pressure gradient fluid-particle interaction slip-shear lift

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Simulation of Granular Transport of Geldart Type-A,-B, and-D Particles

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