3604
Ind. Eng. Chem. Res. 2004, 43, 3604-3615
Effect of Interstitial Fluid on Particle-Particle Interactions in Kinetic Theory Approach of Dilute Turbulent Fluid-Particle Flow Kunn Hadinoto and Jennifer S. Curtis* School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907
A new two-fluid model for fluid-particle turbulent flow is developed in the present study. The model, which employs kinetic theory of dense gas concepts in describing momentum and kinetic energy transfer between colliding particles, incorporates the influence of the interstitial fluid on the random motions of the particles by introducing two distinct particle coefficients of restitution, ef and es, to characterize the inelasticity of particles colliding in a fluid and in a vacuum, respectively. When two particles collide in a fluid, a fraction of the particles’ fluctuating kinetic energy is dissipated as heat as a result of inelastic collisions and another fraction is dissipated into the fluid fluctuations, as the particles must exert work on the fluid to displace the interstitial fluid between the two particle surfaces. The values for ef have been shown experimentally to depend on the impact Stokes number, St, which characterizes the ratio of the particle inertia to the viscous force. The predictions of the model are compared with data from several experiments on dilute turbulent fluid-particle flow in a vertical pipe for a wide range of impact Stokes numbers (40-1600), including both gas-particle and liquid-particle flows. In general, good agreement is found between the model predictions and the experimental data for both the fluid and particle phases at the level of the mean and fluctuating velocity. Introduction Turbulent fluid-particle flow occurs in many industrial applications, such as pneumatic and hydraulic transport of particulates, circulating fluidized beds, cyclone separators, and chemical reactors. The design and optimization of these processes require the development and implementation of accurate two-phase flow models. To simulate a two-phase flow system, interactions both within each phase and between phases must be accurately modeled. There are two paths for modeling the particle phase in a fluid-particle flow system: the Lagrangian (particle trajectory) approach and the Eulerian (two-fluid) approach. In the Lagrangian approach, individual particle trajectories and velocities are calculated by integrating the equation of motion for particles dispersed in a fluid. The two-fluid model, which is the model that we employ here, treats the particle phase as a continuous medium in which appropriate descriptions for the averaged particle-phase properties are required. Fluid-particle flow behavior can be classified into three physical regimes, namely, the macro-viscous regime, transitional, and grain-inertia regimes. In the macro-viscous regime, the momentum and energy transfer between particles occurs indirectly through the fluid. In other words, momentum and energy of neighboring particles are imparted to a particle through the fluid medium. This kind of momentum and energy transfer is usually significant when the densities of the fluid and the particle are similar in their orders of magnitude. A slow, viscous liquid-particle flow is one example of a macro-viscous regime flow. On the other hand, in the grain-inertia regime, the random motions of the particles are governed by direct collisions between the particles and are unaffected by the interstitial fluid. In the case of a low ratio of fluid density to particle density, * To whom correspondence should be addressed. Tel.: (765) 494-2257. Fax (765) 494-0805. E-mail:
[email protected].
direct collision is the dominant mechanism for momentum and energy transfers. In the transitional regime, the random motions of the particles are affected by both mechanisms, i.e., through direct collisions and through the fluid viscosity. The kinetic theory of granular flow (Lun et al.1) has had some success in describing the momentum and kinetic energy transferred by velocity fluctuations of the particles in the grain inertia regime. In the work of Lun et al.,1 the effect of the interstitial fluid was neglected in deriving the constitutive theory for the particle-phase stress and the particle turbulent energy flux from the Maxwell transport equation. Hence, the theory is more appropriately referred to as the kinetic theory of dry granular flow. In their later work, Lun and Savage2,3 included the effect of the interstitial fluid into the existing kinetic theory of rapid granular flow. Their model introduced a “lubrication” approximation for modeling interparticle collisions in the presence of an interstitial fluid. The model takes into account additional stresses exerted by the fluid on the particle through “long-range” potential interactions. Sinclair and Jackson4 incorporated the kinetic theory of dry granular flow in a two-fluid model of a laminar gas-particle flow in which they assumed that the random motions of the particles in a fluid-particle system resembled those predicted for dry granular flow. Louge et al.5 were the first to incorporate both the kinetic theory of dry granular flow and gas turbulence into the two-fluid model. The effect of the gas turbulence on the particle fluctuating velocity and the effect of the particle on the turbulence modulation of the gas phase were considered in their work. Both are crucial in accurately predicting dilute fluid-particle flow, but are less important in dense fluid-particle flow (Hrenya and Sinclair6). A one-equation turbulence model is used to describe the gas turbulence. Bolio et al.7 extended the work of Louge et al.5 and described the gas turbulence as a two-equation low-Reynolds-number k- turbulence
10.1021/ie030478m CCC: $27.50 © 2004 American Chemical Society Published on Web 12/05/2003
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3605
model. The Bolio et al.7 model was originally validated with the data of Tsuji et al.8 and Tsuji9 for 200-µm particles and was later validated with the data of Jones and Sinclair10 for 70- and 200-µm particles. Very good agreement occurred between measurements and model predictions for both gas and particle mean velocities. However, Jones and Sinclair10 showed that the model predictions of Bolio et al.7 for 70-µm particles resulted in particle fluctuating velocities that were larger than the experimental measurements. Jones and Sinclair10 reasoned that an additional loss of particle fluctuating kinetic energy due to the lubrication force exerted by the fluid on the particle was significant for smaller particles. The objective of the present work is to build on the previous model of Bolio et al.7 by integrating the kinetic theory of granular flow, which incorporates the effects of the interstitial fluid on the random motions of the particles, into the existing two-fluid model. The framework of the previous modeling effort is retained. In particular, the present model seeks to correctly capture the effects of the interstitial fluid on the particle fluctuating velocity for fluid-particle flows that can be classified in the grain-inertia and transitional regimes. The model predictions are compared with turbulent gas-particle flow data of Tsuji et al.8 and Jones and Sinclair10 and turbulent liquid-particle flow data of Alajbegovic et al.11 Lun and Savage2,3’s Kinetic Theory of Granular Flow In this section, we provide a brief overview of the work of Lun and Savage2,3 on the modified kinetic theory of granular flow. In the work of Lun et al.,1 the constitutive equations for the particle-phase stresses were derived for interparticle collisions in a vacuum. Lun and Savage2,3 expanded this work by considering inelastic collisions between two smooth, spherical particles in a viscous fluid. When two particles collide in the presence of an interstitial fluid, two particle-particle interaction mechanisms occur. First, the particles can interact through direct collisions where momentum and energy are transferred through elastic-plastic deformations, which is similar to how particle interactions occur in a vacuum (Gondret et al.12). This type of mechanism is referred to as the hard-sphere collision (direct contact) or the short-range potential interaction in Lun and Savage.2 Second, the particles can interact with each other through the interstitial fluid. The presence of a fluid velocity gradient and the pressure disturbance generated by the fluctuating motions of neighboring particles will cause a particle to experience a sequence of softsphere collisions.2 In other words, the Brownian-like motions of the neighboring particles result in momentum and energy transfers through the interstitial fluid to the particle. Lun and Savage2 consider this type of mechanism as the soft-sphere collision (noncontact) or the long-range potential interaction. Lun and Savage2,3 proceeded by including the softsphere collision term in their formulation for the evolution of the single-particle velocity distribution function (f (1)). The Boltzmann-type of equation for the evolution of f (1) is given as
( )
∂f (1) ∂f (1) ∂f (1) + u‚∇f (1) + F‚ ) ∂t ∂u ∂t
C
(1)
where u is the instantaneous particle velocity and F is the body force per unit mass, which consists of the gravitational force and the mean interfacial drag force. (∂f(1)/∂t)C is the rate of change of f (1) due to particleparticle and fluid-particle interactions. Following the work of Rice and Allnatt,13 (∂f (1)/∂t)C is expressed as the sum of the soft-sphere collision term and the hard-sphere collision term in Lun and Savage2
( ) ∂f (1) ∂t
C
)
[
]
∂ ∂ (- F ˆ a)f (1) + (B f (1)) + φH (2) ∂ua ∂ub ab
where ua and ub are the instantaneous velocities of particles a and b, respectively. The first term on the right-hand side of eq 2 is the soft-sphere collision term, which is made up of two parts. The first part accounts for the rate of change of a particle’s momentum due to the fluctuating interfacial force (F ˆ a) acting on particle a. The second part accounts for the rate of change of the particle’s momentum due to fluctuating energy transfers from the fluid phase to the particle a, which is represented by the specific kinetic energy transfer tensor Bab. This fluid-induced fluctuating energy transfer can take place either via the inelastic collisions of the neighboring particles or via the Brownian-like motions of the neighboring particles, which create disturbances in the fluid phase that are transferred to particle a. Lun and Savage2 proposed that this Bab tensor is related to the fluctuating interfacial force term F ˆ a by a constant R, defined as the coefficient of fluid turbulent kinetic energy absorption, which is based on the kinetic theory of gas, where the energy term Bab can be related to the force term F ˆ a by considering the case of molecular equilibrium with no mean flow (Rice and Gray14). The hard-sphere binary collision term (φH) has been described in Lun et al.1 for slightly inelastic particles. Following the work of Chapman and Cowling15 in the kinetic theory of gases, the general transport equations for the particle phase are obtained. By multiplying the evolution equation for f (1) by ψ and integrating over the entire velocity space and then letting ψ be mp, mpu′s , and then mpu′s 2/2, where mp is the mass of the particle, Lun and Savage2,3 obtained the conservation equations of the particle’s mass, momentum, and fluctuating kinetic energy, respectively, as given in the following section. The fluid-phase and particle-phase constitutive relations of the modified kinetic theory of granular flow presented in this work were derived in Lun and Savage.3 Their derivations, however, were specifically intended for dilute gas-particle flow in which ef ≈ es; therefore, the concept of ef e es introduced by Lun and Savage2 was not explored. In other words, Lun and Savage3 considered only the fluctuating interfacial force as the main fluctuating energy transfer mechanism between the fluid and particle phases. The loss of the particle fluctuating energy to the fluid fluctuations due to inelastic collisions was not considered because the fluctuations between the gas phase and the particle phase were considered to be uncorrelated (Lun and Savage3). As a result, the fluid-phase transport equations presented later in this work have extra terms that are not present in the corresponding equations of Lun and Savage3 to account for the source of fluid fluctuations due to inelastic collisions, which are reduced to zero for ef = es.
3606 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Modeling Effort We consider the case of steady, fully developed turbulent flow of a fluid-particle mixture in a vertical pipe. The particle phase is treated as a continuum, so that the point mechanical variables, such as velocity and pressure, are replaced with their corresponding local mean variables (Anderson and Jackson16). The particles are assumed to be spherical, frictionless, cohesionless, and uniform in size. Governing Equations. The continuity equation for the fluid phase is given by
1 ∂ [F (1 - υ)Ufz] ) 0 r ∂r f
(3)
and correspondingly the continuity equation for the particle phase becomes
1 ∂ (F υU ) ) 0 r ∂r s sz
(4)
In these equations, Ufz and Usz represent the timeaveraged fluid and particle velocities, respectively; υ is the particle volume fraction; and Ff and Fs are the fluid and particle densities, respectively. Following the work of Lun and Savage2,3 and Bolio et al.,7 the vertical z component of the momentum equation for the fluid phase becomes
0)
[
]
∂Ufz ∂p 1 ∂ - β(Ufz - Usz) + r(µef + µT) r ∂r ∂r ∂z ηs - η f 1 ∂ ∂Usz rµ/SG1C + Ffgz (5) ηf r ∂r ∂r
(
)
and for the axial and radial momentum equations of the particle phase, one obtains
0)-
1 ∂ (rσ ) + β(Ufz - Usz) + Fsυgz r ∂r rz σθθ 1 ∂ 0)(rσrr) r ∂r r
(6) (7)
(8)
where µf is the intrinsic fluid viscosity A form for the drag coefficient is based on an expression suggested by Bolio et al.7
3 Ff υ C |U - Usz| 4 dp D(1 - υ)2.65 fz
(9)
where dp is the particle diameter and CD is expressed in Lun and Savage2 as
CD )
(
)
24 4 + + 0.4 (1 - υ - 0.33υ2)-2.5 Rep Re 1/2 p
Rep )
Ffdp|Ufz - Usz| µf
(10)
(11)
The fourth term on the right-hand side of eq 5 is the new collisional fluid stress term introduced by Lun and Savage2 to account for the source of the fluid-phase momentum due to numerous particle collisions. This term was neglected in the work of Bolio et al.,7 who followed the kinetic theory of dry granular flow of Lun et al.1 In dry granular flows, where the effect of interstitial fluid is neglected in interparticle collisions, the amount of energy dissipated because of the inelasticity of the contacts is characterized by es, which is defined as the ratio of the rebound velocity to the impact velocity. However, in real systems, where the effect of the interstitial fluid is not negligible, an effective coefficient of restitution, ef, that takes into account the viscous dissipation of the particle fluctuating kinetic energy to displace the fluid between surfaces in addition to the inelasticity of the contact (Joseph et al.17), is required. For the case of no interstitial fluid, ef is equal to es. Because the particle-phase viscosity and particlephase stress depend strongly on the particle velocity fluctuations, a balance for the particle fluctuating kinetic energy associated with the random particle motions, commonly referred to as the granular energy or pseudothermal energy balance, is developed following Lun and Savage2,3
0)-
µef is the effective fluid viscosity, µT is the eddy viscosity, p is the fluid pressure, β is the drag coefficient, and gz is the gravitational acceleration. σrz, σθθ, and σrr are the particle-phase stress tensors. ηs ) (1 + es)/2 and ηf ) (1 + ef)/2 are constants related to the particle coefficients of restitutions in a vacuum, es, and in fluid, ef, respectively. µ/S is the particle-phase shear viscosity in a fluid, and G1C is a constant related to the collisional contribution of the particle-phase stress σrz. The effective fluid viscosity, µef, is used because, as particles move through a dispersed mixture, they experience an additional resistance to their motions due to the presence of neighboring particles. The particles experience this as an increase in resistance that increases with increasing particle concentration as if the fluid viscosity had been increased. The form of the effective fluid viscosity is given in Lun and Savage2 as
µef ) µf (1 - υ - 0.33υ2)-2.5
β)
∂Usz 1 ∂ (rq ) - σrz - γ - 3βT + 3RβT (12) r ∂r PTr ∂r
where T is the pseudothermal or granular temperature (T ) 1/3u′siu′si) and u′si is the particle fluctuating velocity. The first four terms on the right-hand side of eq 12 represent pseudothermal energy conduction (qPTr); pseudothermal energy generation due to shear; dissipation (γ), which includes the loss of particle fluctuating energy to the thermal energy and to the fluid fluctuations due to inelastic collisions; and interfacial energy transfer due to the fluctuating component of the drag force (-3βT), respectively. The expression for the dissipative term is
γ)
Fs 48 ηf(1 - ηf)g0υ2 T 3/2 dp xπ
(13)
Note that the expression for the interfacial energy transfer due to the fluctuating drag force in this model is different from that used in the model of Bolio et al.,7 who followed the work of Louge et al.5 The model of Bolio et al.7 has an additional source term in the form of βu′fiu′si, which is due to the description of the fluctuating drag force that is derived differently in Lun and Savage.2,3
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3607 Table 1. Particle-Phase Stress and Pseudothermal Energy Flux Descriptions (Lun and Savage2,3) σrr ) σθθ ) Fs(ωG2K + G2C)T
∂Usz ∂r 8ηfυg0(3ηf - 2) 1 G1K ) 1+ 5 ηf(2 - ηf)g0
σrz ) - µ/S(ωG1K + G1C)
[
[
]
]
8ηfυg0(3ηf - 2) 768υ2g0ηf 8υ 1+ + 5 25π 5(2 - ηf) G2K ) υ G2C ) 4ηfυ2g0
G1C )
qPTr ) -λ*(ωG3K + G3C)
∂T ∂r
G3K )
8 12 1 + ηf2υg0(4ηf - 3) 5 ηf(41 - 33ηf)g0
G3C )
96υ 12 16 η υg (41 - 33ηf) 1 + ηf2υg0(4ηf - 3) + 5 15π f 0 5(41 - 33ηf)
g0 )
[
[
υ01/3 1/3
υ0
-υ
]
1 λmfp 1+ R 75 xπFsdpxT λ) 384 µS µi ) ηf(2 - ηf) ω)
1/3
5 xπFsdpxT 96 µi µ/i ) 2ζdµi 1+ υFsg0T µS )
λ/i )
]
λi 6ζdλi 1+ 5υFsg0T
λi )
λmfp )
8λ ηf(41 - 33ηf)
The last term on the right-hand side of eq 12 is a source of particle fluctuating energy due to long-range particle interactions through fluid, which was not considered in Bolio et al.7 model. This source term originates from the soft-collision Bab tensor, where R is the coefficient of fluid turbulent kinetic energy absorption employed to account for this transfer. For gasparticle flow, we assume that this contribution to the particle fluctuating energy is negligible; for liquidparticle flow, we employ an expression proposed by Lun and Savage2
R ) 1 + 0.88υ3/2
(14)
In general, R should depend on a number of factors such as fluid turbulence; particle concentration; both fluid and particle physical properties; and, more importantly, es and ef. However, because of the lack of theoretical or experimental investigations performed, its functional form in the present work for liquid-particle flow is assumed to be a function of the particle volume fraction only, following Lun and Savage.2 The dependence of R ∝ υ3/2 was obtained by Lun and Savage2 by taking the limit as υ f 0 of eq 12 such that the granular temperature T would remain finite for a single particle suspended in a fluid. The constant 0.88 was selected to yield granular temperature predictions closest to the experimental results. The granular kinetic theory constitutive relations for the particle-phase stresses and the pseudothermal energy flux of Lun and Savage2,3 are summarized in Table 1.The modified descriptions for the particle-phase shear viscosity and the pseudothermal energy conductivity given in Table 1 indicate that the values obtained by Lun et al.1 are reduced in Lun and Savage2,3 because the term in the denominator will always be larger than 1. In the kinetic theory of granular flow, the particle-
dp 6x2υ
µ/i )
µi 2ζdµi 1+ υFsg0T
λ/i )
λi 6ζdλi 1+ 5υFsg0T
ζd )
3 Ff CD |Ufz - Usz| 4 Fs d p
µ/S )
µ/i µS µi
λ* )
λ/i λ λi
phase viscosity is proportional to the square root of the granular temperature, which is similar to the squareroot dependency of the gas viscosity on the thermodynamic temperature in the kinetic theory of gas. Therefore, the decrease in the particle-phase viscosity when an interstitial fluid is present can be explained by the fact that an additional resistance to the particles’ motion due to the interstitial fluid results in a decrease in the granular temperature or the intensity of the fluctuating motion of the particles. As a result, the particle-phase viscosity is decreased. The ζd term in the denominator is the specific fluid friction coefficient analogous to the friction coefficient in the kinetic theory of gases (Rice and Gray14). It should be noted that we do not use the first-order approximation for ζd provided by Lun and Savage,2 who took the ensemble average of the difference between the instantaneous velocity of the fluid and the particle phase using a Maxwellian velocity distribution for f (1). We instead use the time-averaged velocities of the two phases following Lun and Savage3 to describe ζd, as shown in Table 1. To account for the effect of the fluid-phase turbulence on the particles’ random motions, a fluid-phase turbulent kinetic energy (k ) 1/2u′fiu′fi) balance, where u′fi is the fluid fluctuating velocity, is derived. A low-Reynoldsnumber k- turbulence model is used
[
(
) ]
( )
µT ∂k ∂Ufz 2 1 ∂ r(1 - υ) µef + + (1 - υ)µT r ∂r σk ∂r ∂r ηs - ηf / ∂Usz ∂Ufz µS G1C Ff(1 - υ) - F′KG + EW + + ηf ∂r ∂r γFKET - 3RβT (15)
0)
where σk is the turbulent Prandtl number for k and µT
3608 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Figure 1. Particle fluctuating energy transfer according to Lun and Savage.2
is the turbulent eddy viscosity, given by
µT )
cµfµFfk2
(16)
where the descriptions for cµ and fµ are given in Bolio et al.7 The first three terms on the right-hand side of eq 15 represent the diffusion, generation, and dissipation, respectively, of fluid turbulent kinetic energy. The fourth term represents fluid-particle interactions at the level of fluctuating energy described as an interfacial energy transfer due to a fluctuating drag force (Bolio et al.7)
F′KG ) -β(u′fiu′si) + 2βk
(17)
The correlation u′fi u′si is difficult to determine experimentally because it requires simultaneous measurements of the fluctuating velocities of both phases at a given point in the flow. In this work, we use a closure proposed by Sinclair and Mallo18 to represent a source of fluid turbulent kinetic energy due to the presence of the particles
u′fiu′si ) x2k3T
(18)
The fifth term represents an additional source term due to the presence of wakes behind the particles, which becomes important for flow with particle Reynolds numbers, Rep, larger than 300 as suggested in Lun.19 This is consistent with the experimental results of Hetsroni,20 who showed that fluid turbulence was always intensified by the vortex shedding of the particle phase for fluid-particle flow in this regime. In this work, the particle Reynolds number for the liquidparticle flow data used for comparisons is approximately
300; therefore, an additional source term should be included. The gas-particle flow data, on the other hand, have particle Reynolds number less than 50, so that the source due to wakes is negligible. EW is expressed in Lun19 for Rep < 310 as
EW ) 2πCWndpFfνtk
(19)
with
CW ) 16/3
and
νt ) νf0.017Rep
(20)
where νf and n are the kinematic fluid viscosity and particle number density, respectively. Other descriptions for the fluid turbulence modulation due to the wakes that are available in the literature (for example, Bolio and Sinclair21) would greatly overpredict the fluid turbulent kinetic energy. The sixth term in eq 15 is a generation term due to the collisional fluid stress. The last two terms are a collisional energy source term from the particle phase to the fluid phase and a collisional energy sink term from the fluid phase to the particle phase, respectively. A pictorial description on how the collisional energy is transferred between the particle phase and the fluid phase is provided in Figure 1. The collisional energy source term is given in Lun and Savage2 as
γFKET ) (1 - β)γ
(21)
where β is defined as the ratio of particle fluctuating energy dissipated as heat to the total energy dissipation, including losses both to the fluid fluctuations and to the thermal heat due to inelastic particle collisions. An expression for β related to ef and es is derived using the definitions for ef and es, which represent the changes in the particle kinetic energy due to inelastic collisions in
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3609
a fluid and vacuum, respectively
β)
KEthermal loss KEthermal loss + fluid loss
)
1 - es2 1 - ef2
(22)
The turbulent kinetic energy dissipation rate , where
) νf
∂u′fi∂u′fi ∂xj∂xj
pipe. They also measured the axial and radial mean and fluctuating velocities of both the liquid and particle phases. The values for ef used in this study are based on the results obtained in the experimental work of Gondret et al.,24 who studied particle-wall collisions in fluid with different viscosities. They investigated the bouncing motions of particle spheres onto a particle plate in an ambient fluid. In particular, they measured the coefficient of restitution, ef, as a function of impact Stokes number, St, defined in Joseph et al.17 as
is given correspondingly as
[
(
) ]
µT ∂ 1 ∂ 0) r(1 - υ) µef + + r ∂r σ ∂r ∂Ufz 2 2 - c2f2Ff(1 - υ) - c3f2F′KG + c1f1 (1 - υ)µT k ∂r k k ∂Usz ∂Ufz ηs - ηf / + µSG1C c3f2EW + c1f1 k k ηf ∂r ∂r c1f1 γFKET - c2f23RβT (23) k k
( )
where the model parameters (c1, c2, c3, f1, and f2) follow those used in Bolio et al.7 and σ is the turbulent Prandtl number for . Boundary Conditions. In summary, there are six coupled differential equations (eqs 5-7, 12, 15, and 23) describing the relationship between Ufz, Usz, υ, T, k, and . A complete formulation of the model requires boundary conditions at the pipe wall and at the centerline. In the present study, we use the same boundary condition formulations as described in Bolio et al.7 At the centerline, symmetry requires that the gradients of all variables be zero. At the pipe wall, we use no-slip boundary conditions for the fluid mean velocity and the fluid turbulent kinetic energy. For the particle phase, the boundary condition for the mean velocity is found by equating the particle-phase shear stress adjacent to the wall to the momentum flux transferred to the wall due to particle-wall collisions (Johnson and Jackson22). Similarly, the boundary condition for the pseudothermal temperature is obtained from an energy balance on a thin region adjacent to the wall, as described in Johnson and Jackson.22 Complete details of the boundary conditions used are described in Bolio.23 Experimental Data In the present study, we evaluate our model predictions with turbulent fluid-particle flow experimental data obtained by Tsuji et al.,8 Tsuji,9 Jones and Sinclair,10 and Alajbegovic et al.11 Tsuji et al.8 and Tsuji9 studied upward turbulent gas-particle flow in a vertical pipe and measured axial mean and fluctuating velocities of both the gas and particle phases. They considered different particle sizes (200-500-µm polystyrene particles) and varied the particle loading, m, which is defined as the ratio of the particle mass flux to the fluid mass flux. Jones and Sinclair10 studied downward turbulent gas-particle flow using glass-bead particles (70-500 µm) in a vertical pipe and measured both axial and radial mean and fluctuating velocities of both phases. Alajbegovic et al.11 studied large ceramic particles (2320 µm) suspended in water flowing upward in a vertical
St )
mpvimp 6πµfa2
(24)
where mp is the particle mass, vimp is the impact velocity of the particle, µf is the ambient fluid viscosity, and a is the particle radius. They carried out experiments for a wide range of particle materials, particle sizes, and fluid viscosities. The particle plate was made of an opticalquality glass to minimize its surface roughness. They found that all of their data collapsed well onto a single “master” curve of ef/es as a function of impact Stokes number, St. In their master curve, ef/es is 0 at small St and increases monotonically with St, reaching an asymptotic value of 1 at high St (>1500) corresponding to ef ) es. More importantly, their ef/es ) f (St) master curve shows that the impact Stokes number is the pertinent scaling parameter for particle collisions in a fluid. In this study, we used the results of Gondret et al.24 to estimate the values of ef in a turbulent fluid-particle flow by using the particle fluctuating velocity, u′sz , obtained experimentally as the particle impact velocity in eq 24. We used the lower limit of the particle fluctuating velocity data to calculate the impact Stokes number based on the observation made by Zenit and Hunt25 that the particles decelerate prior to contact. The experimental data mentioned earlier gave rise to three interesting case studies that we investigate with the present model. These case studies differ in how ef is related to es, and they are summarized in Table 2. Rec and Resf are the fluid Reynolds numbers calculated on the basis of the fluid centerline velocity and fluid superficial velocity, respectively. The goal of these comparisons is to evaluate the performance of the present model, which employs the effects of the interstitial fluid on the kinetic theory of granular flow, for a wide range of values of ef as compared to es and to compare the predictions of the new model relative to the predictions of the model of Bolio et al.,7 which neglected the role of the interstitial fluid. Model Solution The governing equations with their boundary conditions are solved using an adaptation of the implicit finite volume marching technique developed by Patankar.26 In this technique, the problem is formulated as pseudotransient and is integrated in time until the steady-state solution is obtained. We input a pressure gradient and a particle volume fraction at the centerline or at the pipe wall and iterate on these two variables until the specified values of Reynolds number and particle loading are satisfied. The details of the numerical technique are given in Bolio.23
3610 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 Table 2. Summary of Experimental Data ref
figure(s)
dp (µm)
Fs/Ff
m
φ
ew
St
Rep
Rec, Resf
0.15 0.94
1460 1370-1620
14 40
Resf ) 13 400 Rec ) 36 600
0.14 0.85
730-1150 530-800
8 11
Resf ) 13 400 Rec ) 21 300
0.50
40-90
300
Resf ) 67 200
case 1: ef ) es ) 0.94 Jones and Sinclair10 Tsuji et al.8
2-4 5
70 200
2100 860
Jones and Sinclair10 Tsuji et al.8
6-8 9
70 200
2100 860
1.0 1.0
0.008 0.002
case 2: ef ) 0.85, es ) 0.94 4.0 3.2
0.008 0.002
case 3: ef ) 0.50, es ) 0.94 Alajbegovic et
al.11
10-12
2320
2.45
The coefficient of restitution for particle-wall collisions in a fluid, ew, which is used to determine the particle-phase velocity boundary condition, is assigned to be 0.94 and 0.85 for cases 1 and 2, respectively, for the experimental data of Tsuji et al.,8 who performed gas-particle flow experiments in a glass pipe. Jones and Sinclair10 carried out their experiments in a copper pipe; therefore, lower values of ew equal to 0.15 and 0.14 for cases 1 and 2, respectively, are used. An ew value of 0.50 is assigned for the data of Alajbegovic et al.,11 as they carried out their liquid-particle flow experiments in a fluorinated ethylene propylene (FEP) pipe, which is an optical-quality glass pipe. These values are consistent with the coefficients of restitution for these particlewall materials given in Goldsmith.27 Furthermore, these values also account for the effect of the interstitial fluid on the values of the coefficient of restitution for particlewall collisions as given in Gondret et al.24 In the particle-phase mean velocity and pseudothermal temperature boundary conditions at the pipe wall, a parameter arises that describes the fraction of tangential momentum lost in particle-wall collisions depending on the wall roughness. This parameter is the specularity coefficient, φ, which varies from 0 to 1 for smooth to rough surfaces, respectively. The specularity coefficient is important because it influences the value of the particle slip at the wall. For the experimental data of Tsuji et al.,8 Bolio et al.7 assigned a value of φ equal to 0.002 for the glass pipe used in their work. This value did not vary over all operating conditions considered in the study with a glass pipe. Jones and Sinclair10 assigned a specularity coefficient value of 0.008 for a gas-particle flow in a copper pipe, a rougher surface. Again, this value did not vary for all data obtained in a copper pipe over a range of operating conditions. Following their methodology, a specularity coefficient of 0.002 is assigned for the Alajbegovic et al.11 data, who employed a glass pipe. As mentioned earlier, we compare the present model predictions with the predictions of Bolio et al.7 for the three case studies described in Table 2. From the particle axial fluctuating velocity data obtained by Tsuji et al.,8 Tsuji,9 Jones and Sinclair,10 and Alajbegovic et al.,11 we calculate their corresponding impact Stokes numbers, St, and determine the value of ef from the master curve provided in Gondret et al.24 We report the axial mean velocities and the root-mean-square fluctuating velocities of both the fluid and the particle phases in this paper. The solutions have been verified to be grid-independent. Results and Discussion The flow characteristics of the case studies considered in this work are summarized in Table 2. In an attempt to single out the effects on the model predictions of the
0.063
0.002
interstitial fluid introduced in the present model, the Bolio et al.7 model predictions shown in this work were obtained by using eq 18 to describe the correlation u′fi u′si, which is different from the expression employed in their original work, which used the closure description by Louge et al.5 Case 1. We first investigate the model prediction for the case in which the effect of the interstitial fluid on the random motion of the particles is negligible. Furthermore, the particle loading (m ) 1.0) used suggests that the frequency of collisions is low at this particle loading. Therefore, the particles have high fluctuation and impact velocities, which result in high impact Stokes numbers (1370-1620). We compare our model predictions with the data of Jones and Sinclair10 for 70µm particles and the data of Tsuji et al.8 for 200-µm particles. In the case of ef ) es, all of the energy loss is dissipated as heat, and β is equal to 1. For ef ) es, the new source and sink collisional energy terms (the sixth and seventh terms in eqs 15 and 23) introduced in the present model are 0. Thus, only the particle-phase shear viscosity and the pseudothermal conductivity are influenced by the presence of the interstitial fluid. Data of Jones and Sinclair.10 Because Jones and Sinclair10 measured both the axial and radial fluctuating velocity components, comparisons are made here for the gas turbulent kinetic energy instead of the axial gas fluctuating velocity only. The gas turbulent kinetic energy k is determined from the measurements k ) 1 /2(u′fzu′fz + 2u′fru′fr), assuming that the radial and azimuthal components are equal. u′fz andu′fr are the axial and radial fluid fluctuating velocities, respectively. Figure 2 shows that both models are capable of predicting the axial mean velocities of both the fluid and particle phases. U0 is the single-phase gas mean velocity, which is equal to 20 m/s. Both models are able to predict the flattening of the mean gas velocity profile compared to the single-phase flow. A slight improvement in the predicted particle velocity is observed. Figure 3 indicates that the present model predicts similar results for the gas turbulent kinetic energy values compared to the predictions of Bolio et al.7 model, as the k- equation in the present model is reduced to the model of Bolio et al.7 for ef ) es. A damping of the gas turbulent kinetic energy due to the presence of the particles is observed. Figure 4 shows that the predicted axial particle fluctuating velocity is reduced in the present model. The decrease is due to the difference in the expression for the fluctuating drag force between the present model and the Bolio et al.7 model. Also, the present model is able to capture the shape of the particle fluctuating velocity profile observed experimentally.
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3611
Figure 2. Mean gas and particle velocities, ef ) es, es ) 0.94. Data of Jones and Sinclair.10
Figure 4. Particle fluctuating velocity, ef ) es, es ) 0.94. Data of Jones and Sinclair.10
Figure 3. Gas turbulent kinetic energy, ef ) es, es ) 0.94. Data of Jones and Sinclair.10
Figure 5. Particle fluctuating velocity, ef ) es, es ) 0.94. Data of Tsuji et al.8
In Figure 5, the magnitude of the prediction of the axial particle velocity fluctuations for 200-µm particles is again reduced significantly in the present model, consistent with the results for 70-µm particles in the data of Jones and Sinclair.10 Case 2: ef e es w ef ) 0.85. In the case of ef ) 0.85 and es ) 0.94, the role of the interstitial fluid in the random motions of the particles is still relatively small yet not negligible. These experiments for comparisons with model predictions were carried out at higher particle loadings (m ) 4.0 and 3.2) than those for case 1, resulting in an increase in the particle-particle collision frequency and a reduced particle fluctuating velocity. Hence, lower particle impact velocities and
lower impact Stokes numbers (530-1150) result. The calculated value of β is approximately equal to 0.40, which implies that 40% of the particle fluctuating energy loss due to inelastic collisions is dissipated as thermal energy while the other 60% is lost to the fluid fluctuations. Data of Jones and Sinclair.10 Figure 6 shows that the mean velocity predictions from the two models agree well with the measurements and are able to predict the flattening of the gas velocity profile due to the addition of particles. The effect of the collisional fluid stress introduced in the present model is shown to be insignificant to the mean fluid velocity predictions.
3612 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Figure 6. Mean gas and particle velocities, ef ) 0.85, es ) 0.94. Data of Jones and Sinclair.10
Figure 7. Gas turbulent kinetic energy, ef ) 0.85, es ) 0.94. Data of Jones and Sinclair.10
Figure 7 shows that the model of Bolio et al.7 predicts the gas turbulent kinetic energy fairly accurately and the prediction of the present model improves the agreement with the measurements and is lower than that obtained with the Bolio et al.7 model. The magnitude of the k profile in the present model is mainly governed by the interfacial energy transfer due to the fluctuating drag force and the collisional energy transfer. The generation term due to the collisional fluid stress is relatively small near the centerline where the fluid mean velocity profile is flat, but it gradually increases closer to the pipe wall. The results show that a net transfer of collisional energy takes place from the particle phase to the fluid phase and not vice versa. This source of collisional energy from the particle phase and
Figure 8. Particle fluctuating velocity, ef ) 0.85, es ) 0.94. Data of Jones and Sinclair.10
a source of energy due to the fluctuating drag force are the major contributions in the fluid turbulent kinetic energy balance. Figure 8 shows that both models overpredict the particle fluctuating velocity measurements. The present model prediction is lower than the prediction of Bolio et al.7 and is able to capture the particle fluctuating velocity profile closely. The magnitude of the particle fluctuating velocity profile is largely determined by the dissipation and interfacial energy transfer terms. As a result, the present model, which includes both viscous and thermal dissipations in its kinetic theory approach, yields lower values for the particle fluctuating velocity. In addition, the lower predicted values for the particle fluctuating velocity result in a decrease in the source of energy due to the fluctuating drag force in the k balance, x2k3T, which might explain the lower predictions for the fluid turbulent kinetic energy in the present model. In Figure 9, the magnitude of the prediction of the axial particle velocity fluctuations for 200-µm particles is again reduced in the present model, similarly to the case for the 70-µm particles. Excellent agreement with the experimental data is also observed. Case 3: ef e es w ef ) 0.50. Now, we investigate the case where ef is much smaller than es. In this liquidparticle flow experiment, the fluid density and the particle density differ by a factor of 2.5. Therefore, the particle velocity at impact and the impact Stokes number (40-90) are much smaller than in the gasparticle flow experiments considered previously because of the high liquid viscosity. The value of β in this case is approximately 0.16, indicating that a larger fraction of the particle fluctuating energy loss due to inelastic collisions is dissipated to the fluid fluctuations. The value for R is determined using eq 14, and it is found to be approximately equal to 1. Data of Alajbegovic et al.11 In Figures 10-12, we show the model predictions for the data of Alajbegovic et al.11 The experimental data indicate that the axial particle fluctuating velocity is lower than the axial
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Figure 9. Particle fluctuating velocity, ef ) 0.85, es ) 0.94. Data of Tsuji et al.8
Figure 11. Liquid fluctuating velocity, ef ) 0.5, es ) 0.94. Data of Alajbegovic et al.11
Figure 10. Mean liquid and particle velocities, ef ) 0.5, es ) 0.94. Data of Alajbegovic et al.11
liquid fluctuating velocity, as opposed to all of the previous gas-particle flow data, in which the particle fluctuating velocity exceeds the gas fluctuating velocity. Therefore, the random motion of the particles is clearly influenced by the interstitial fluid. Figure 10 shows that the present model yields slightly improved predictions for the axial mean velocities of both phases. Figure 11 shows that the present model predicts lower values for the liquid turbulent kinetic energy near the pipe center than the model of Bolio et al.7 and results in improved agreement with the measurements. As in case 2, the dominant contributions in the fluid turbulent kinetic energy balance are the collisional and interfacial energy transfer terms. Because R is approximately equal to 1, a significant amount of the collisional energy is transferred from the
Figure 12. Particle fluctuating velocity, ef ) 0.5, es ) 0.94. Data of Alajbegovic et al.11
fluid phase to the particle phase, which might explain the lower predictions for k in certain regions of the pipe. In Figure 12, it is clearly shown that the present model captures the shape of the particle fluctuating velocity profile obtained in the experiment more accurately than the model prediction of Bolio et al.7 In addition, there is an improvement in the quantitative agreement with the experimental measurements. As in case 2, a higher value of the dissipation term results in lower predictions for the particle fluctuating velocities.
3614 Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004
Sensitivity of Model Predictions to ef and r. In this work, the values of ef are obtained by using the experimental data of Gondret et al.24 and the particle impact velocity, which is estimated from the particle axial fluctuating velocity. We know from the work of Zhang et al.28 that the actual impact velocity is lower than this value, thus resulting in a lower ef. In Figures 8, 9, and 12 we show the sensitivity of the model predictions to a lower ef value, which results in improved predictions for the particle fluctuating velocity. The sensitivity of the model predictions to R for the liquidparticle flow is shown in Figure 12. When R is decreased from 1.0, which is the estimated value obtained from eq 14, to 0.9, the prediction for the particle fluctuating velocity near the centerline is significantly improved. Figure 12 indicates that lower values for both ef and R yield the best predictions for the particle fluctuating velocity. Conclusions In the present study, we have expanded the work of Bolio et al.7 to incorporate the kinetic theory of granular flow formulation of Lun and Savage2,3 that includes the effect of the interstitial fluid on the random motions of the particles in their derivations of the particle phase stresses. The resulting model is used to simulate data from several experiments on dilute turbulent fluidparticle flows in a vertical pipe. Three experimental data sets are selected for the comparison based on the significance of the effect of the interstitial fluid on the particle fluctuating velocity, which is characterized by their impact Stokes numbers, St. Comparisons between the present model predictions and experimental data for the mean and fluctuating velocity profiles for the case in which the role of the interstitial fluid is not very significant (ef ≈ es, ef ) 0.85) indicate that the present model, which considers the decrease in the particlephase shear viscosity due to the presence of the fluid, is able to capture the particle fluctuating velocity profiles more accurately than the previous model both qualitatively and quantitatively. For the case where ef is significantly lower than es (ef ) 0.50), the present model shows a significant improvement in the shape of the profile of the particle fluctuating velocity and also in the magnitude of the prediction. A sensitivity analysis shows that decreasing the value of ef improves the predictions for the particle fluctuating velocity in all cases. This indicates that the method of estimating ef using the data of Gondret et al.24 might be yielding values for ef that are too large. For the fluid fluctuating velocity predictions, the present model slightly improves the predictions for gas-particle flow (ef ≈ es). The proposed correlation for liquid-particle flow, which includes the fluid turbulence enhancement due to the wakes and long-range particle interactions through the fluid, results in reasonable predictions for the liquid turbulent kinetic energy. Despite the overall improvement of the model predictions achieved for both gasparticle and liquid-particle flow, improved descriptions for the correlation between the fluid and particle fluctuating velocities are needed to capture accurately the results obtained in the experiments. Acknowledgment The authors thank Christopher Calderon for his contribution to the preliminary part of the project and
the American Chemical Society Petroleum Research Fund, Grant ACS-PRF#35117-AC9, for funding this work. Nomenclature a ) particle radius (µm) Bab ) fluctuating kinetic energy transfer tensor between particles a and b via fluid (m2/s3) CD ) parameter in the fluid-particle drag coefficient CW ) parameter in EW cµ, fµ, c1, c2, c3, f1, f2 ) parameters for the low-Reynoldsnumber k- turbulence model dp ) particle diameter (µm) ef,es ) coefficients of restitutions in the fluid and in a vacuum, respectively ew ) wall coefficient of restitution EW ) fluid turbulent kinetic energy enhancement due to wake (kg/m‚s3) f (1) ) single-particle velocity distribution function F ) body force per unit mass (m/s2) F ˆ a ) fluctuating interfacial force tensor acting on particle a (m/s2) F′KG ) fluctuating interfacial energy transfer due to drag (kg/m‚s3) g0 ) radial distribution function gz ) gravitational acceleration (m/s2) G1K, G2K, G3K ) kinetic contributions to the particle-phase stress tensor G1C, G2C, G3C ) collisional contributions to the particlephase stress tensor k ) turbulent kinetic energy (kg‚m2/s2) m, n ) particle mass loading and particle number density, respectively (m-3) mp ) particle mass (kg) p ) fluid pressure (Pa) qPTr ) granular temperature conduction (kg/s3) Rep ) particle Reynolds number Rec ) fluid Reynolds number based on the fluid centerline velocity Resf ) fluid Reynolds number based on the fluid superficial velocity St ) impact Stokes number T ) granular temperature (m2/s2) u ) instantaneous particle velocity (m/s) u′fz, u′sz ) fluid and particle axial rms velocities, respectively (m/s) u′fr ) fluid radial rms velocity (m/s) u′fi u′si ) fluid-particle velocity variance (m2/s2) Ufz, Usz ) fluid and particle axial time-averaged velocities, respectively (m/s) U0, Ufc ) single-phase and centerline fluid velocities, respectively (m/s) vimp ) particle impact velocity (m/s) z, r, θ ) axial, radial, and azimuthal coordinates, respectively Greek Letters R ) coefficient of fluid turbulent kinetic energy absorption β ) fluid-particle drag coefficient (kg/m3‚s) β ) fraction of particle fluctuating kinetic energy lost to thermal heat γ ) granular energy dissipation (kg/m‚s3) γFKET ) loss of particle fluctuating energy to fluid fluctuations (kg/m‚s3) ) turbulent kinetic energy dissipation rate (kg‚m2/s3) ζd ) kinetic theory specific fluid friction coefficient (kg/m3‚ s) η ) (1 + e)/2 µef ) effective fluid viscosity (kg/m‚s)
Ind. Eng. Chem. Res., Vol. 43, No. 14, 2004 3615 λmfp ) mean free path of a particle (m) µf, νf ) intrinsic (kg/m‚s) and kinematic (m2/s) fluid viscosities, respectively µi, λi ) particle-phase shear viscosity and granular energy conductivity in a vacuum, respectively (kg/m‚s) µ/i , λ/i ) particle-phase shear viscosity and granular energy conductivity in the fluid, respectively (kg/m‚s) µS, λ ) elastic particle shear viscosity and granular energy conductivity in a vacuum, respectively (kg/m‚s) µ/S, λ* ) elastic particle shear viscosity and granular energy conductivity in the fluid, respectively (kg/m‚s) µT ) turbulent eddy viscosity (kg/m‚s) Ff, Fs ) fluid and particle densities, respectively (kg/m3) νt ) parameter in EW σij ) particle-phase stress tensor (kg/m‚s2) σk, σ ) turbulent Prandtl numbers for k and , respectively υ ) particle volume fraction υ0 ) particle volume fraction at the particle packing limit (υ0 ) 0.65) φ ) specularity coefficient φH ) hard-sphere binary collision tensor ψ ) any transport variable ω ) kinetic contribution’s damping factor
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(11) Alajbegovic, A.; Assad, A.; Bonetto, F.; Lahey, R. T., Jr. Phase Distribution and Turbulence Structure for Solid/Fluid Upflow in a Pipe. Int. J. Multiphase Flow 1994, 20, 453. (12) Gondret, P.; Hallouin, E.; Lance, M.; Petit, L. Experiment on the Motion of a Solid Sphere toward a Wall: From Viscous Dissipation to Elastohydrodynamic Bouncing. Phys. Fluids 1999, 11, 2803. (13) Rice, S. A.; Allnatt, A. R. On the Kinetic Theory of Dense Fluids. Singlet Distribution Function for Rigid Spheres with an Attractive Potential. J. Chem. Phys. 1961, 34, 2144. (14) Rice, S. A.; Gray, P. The Statistical Mechanics of Simple Liquids. Monogr. Stat. Phys. Thermodyn. 1965, 8. (15) Chapman, S.; Cowling, T. G. The Mathematical Theory of Non-uniform Gases; Cambridge University Press: Cambridge, U.K., 1970. (16) Anderson, T.; Jackson, R. A Fluid Mechanical Description of Fluidized Beds. Ind. Eng. Chem. Fundam. 1967, 6, 527. (17) Joseph, G. G.; Zenit, R.; Hunt, M. L.; Rosenwinkel, A. M. Particle-Wall Collisions in a Viscous Fluid. J. Fluid Mech. 2001, 433, 329. (18) Sinclair, J. L.; Mallo, T. Describing Particle-Turbulence Interaction in a Two-Fluid Modeling Framework. In Proceedings of FEDSM ‘98: 1998 ASME Fluids Engineering Division Summer Meeting, June 21-25, 1998, Washington, DC; ASME Press: New York, 1998; Vol. 4, pp 7-14. (19) Lun, C. K. K. Numerical Simulation of Dilute Turbulent Gas-Solid Flows. Intl. J. Multiphase Flow 2000, 26, 1707. (20) Hetsroni, G. Particles-Turbulence Interaction. Intl. J. Multiphase Flow 1989, 15, 735. (21) Bolio, E. J.; Sinclair, J. L. Gas Turbulence Modulation in the Pneumatic Conveying of Massive Particles in Vertical Tubes. Int. J. Multiphase Flow 1995, 21, 985. (22) Johnson, P. C.; Jackson, R. Frictional-Collisional Constitutive Relations for Granular Materials, with Application to Plane Shearing. J. Fluid Mech. 1987, 176, 67. (23) Bolio, E. J. Dilute Turbulent Gas-Solid Flow with Particle Interactions and Turbulence Modulation. Ph.D. Dissertation, Carnegie Melon University, Pittsburgh, PA, 1994. (24) Gondret, P.; Lance, M.; Petit, L. Bouncing Motion of Spherical Particles in Fluids. Phys. Fluids 2002, 14, 643. (25) Zenit, R.; Hunt, M. L. Mechanics of Immersed Particle Collisions. J. Fluid Eng. 1999, 121, 179. (26) Patankar, S. V. Numerical Heat Transfer and Fluid Flow; Hemisphere Publishing Co: New York, 1980. (27) Goldsmith, W. Impact: The Theory and Physical Behavior of Colliding Solids; Edward Arnold Ltd Publishers: London, 1960. (28) Zhang, J.; Fan, L. S.; Zhu, C.; Pfeffer, R.; Qi, D. Dynamic Behavior of Collision of Elastic Spheres in Viscous Fluids. Powder Technol. 1999, 106, 98.
Received for review June 4, 2003 Revised manuscript received August 29, 2003 Accepted September 2, 2003 IE030478M
