Particle–gas turbulence interactions in a kinetic theory approach to granular flows

doi:10.1016/S0301-9322(01)00039-8

International Journal of Multiphase Flow

Volume 27, Issue 11, November 2001, Pages 1945-1964

https://doi.org/10.1016/S0301-9322(01)00039-8 Get rights and content

Abstract

We describe a new two-fluid model for gas–solid flows that incorporates the gas turbulence influence on the random motion of the particles via a generalised kinetic theory, as well as a new gas turbulence modulation model. Simulation results for fully developed steady vertical pipe flows are in good quantitative agreement with experimental measurements. Our results show that the influence of gas turbulence on the particle microscopic flow behaviour cannot be ignored for relatively dilute flows, especially with smaller particles. Our new turbulent model captures the essential characteristics of gas turbulence modulation in the presence of particles, and can be solved for a wide range of particle–particle restitution coefficients.

Introduction

Gas–solid two-phase flows are important in many engineering applications and industrial processes, e.g., materials-handling engineering, circulating fluidised beds, pneumatic conveying and nuclear reactor cooling. Also, they are relevant in various natural phenomena, e.g., sandstorms, moving sand dunes, cosmic dusts, snow avalanches, dust explosions and settlement. Understanding the physical mechanisms governing these flows is essential for the optimal design of these industrial processes, and for modelling these natural phenomena.

Numerical models for gas–solid flows play a key role in both fundamental research and engineering applications. There are two parallel paths for modelling particles flowing with a carrier gas, according to the way the particle phase is treated: the Lagrangian (or particle trajectory) approach, and the Eulerian (or two-fluid) approach. The Lagrangian approach, which includes distinct element and direct simulation methods, describes the motion of each solid particle by a separate equation. These methods have distinct advantages when considering flows with e.g. particles of different sizes, and yield detailed information on the particle flow but at the cost of long computational time and high computer memory demands. In two-fluid models, the particle phase is treated as a continuous medium so that the governing equations are in a similar form to the hydrodynamic equations. Appropriate averaged properties of the particle phase are required. The advantages of a two-fluid model are: it requires less computational effort, it is more suitable for engineering applications where no details of the individual particle motion are necessary, and it is perhaps the only practical way of approaching modelling denser flows.

Elghobashi (1994) classified gas–solid flows into two regimes: dilute and dense flows. In the dilute regime, where the solid volume fraction is less than 0.1%, the collisions between particles have negligible effect on the carrier gas turbulence. In the dense regime, where the solid volume fraction is greater than 0.1%, the collisions begin to play an important role in the flows. This so-called “four-way coupling effect” should be considered.

On the other hand, when the solid volume fraction is very high, collisions between particles dominate the flows, and the carrier phase can be neglected. Many competing kinetic theories of granular flow (see, e.g., Jenkins and Savage, 1983; Lun et al., 1984; Jenkins and Richman, 1985; Abu-Zaid and Ahmadi, 1990; Gidaspow, 1994) have been proposed following the pioneering work of Bagnold (1954), Ogawa et al. (1980), and Savage and Jeffrey (1981). These theories have been reviewed by Savage (1984) and Campbell (1990).

The kinetic theory of granular flow has had some success in modelling the interaction mechanism between the mean and fluctuating solid velocities. Sinclair and Jackson (1989) incorporated this theory in a two-fluid model, where they assumed that the mechanism of the interaction between the mean and fluctuating particulate phase is the same as in the kinetic theory of granular flow alone. The interaction between the particulate phase and the gas phase was a mutual drag force, and the gas flow was assumed laminar. Although certain important physical mechanisms were not included, their results revealed many experimentally observed flow phenomena: co-current upflow, co-current downflow, counter-current flow, non-homogeneous radial distribution of solid concentration, etc. Since then, kinetic theories of granular flow have been widely used to model the solid phase motion in a multiple-phase flow system (e.g., Louge et al., 1991; Bolio and Sinclair, 1995; Bolio et al., 1995; Pita and Sundaresan, 1993; Nieuwland et al., 1996; Neri and Gidaspow, 2000; Mathiesen et al., 2000).

In the regime where the solid volume fraction is greater than 0.1% (so that the collisions between particles should be considered, but not dense enough that the gas turbulence can be neglected), we may need to investigate how the interstitial turbulent gas affects the constitutive equations of the solid phase – which is always neglected in a kinetic theory of “dry” granular flow. Also, the question of how the presence of particles modulates the gas turbulence should be addressed. This paper examines these issues arising in this regime of solid volume fraction.

Louge et al. (1991) began to incorporate both the kinetic theory of dry granular flow and gas turbulence into a two-fluid model for relatively dilute flows. The gas turbulence was described by a one-equation turbulence model, and the same characteristic length scale of the turbulence as in pure gas flow was adopted. Yuan and Michaelides (1992) proposed a turbulence modification model by arguing that the wake is responsible for the augmentation of turbulence and the work done on the particles is responsible for the attenuation of turbulence. Their results are in good agreement with experimental data. Later, Yarin and Hetsroni (1994) proposed a more detailed expression for the wake. Bolio et al. (1995) extended the work of Louge et al. (1991) to a two-equation turbulent model: the k–ε eddy viscosity model. They found that the gas turbulence intensity was somewhat underestimated. Bolio and Sinclair (1995) improved the model by adopting the turbulence modification model of Yuan and Michaelides (1992) and confirmed that the wake is responsible for the enhancement of gas turbulence. Kenning and Crowe (1997) suggested that the turbulence enhancement may be associated with the work done by the particle drag, and the inter-particle spacing is an additional physical restriction to the flow which is responsible for the attenuation of gas turbulence. Crowe and Gillandt (1998), Crowe and Wang (2000) and Crowe (2000) derived and improved a detailed turbulent modulation model. These authors argued that the common approach to the derivation of the turbulent kinetic energy balance equation (e.g. Louge et al., 1991) treats the averaged velocity as if it were a local velocity in the momentum equations of both phases.

In a gas–solid flow, whether the flow is governed by gas turbulence or particle collisions can be classified by three characteristic time-scales: the characteristic time-scale of eddies, t₁^t, the mean particle relaxation time, t₁₂^x, and the mean particle collision time, t₂^c. If t₁₂^x≪t₁^t, particle motion is controlled by the gas flow, and if t₂^c≪t₁₂^x, the flow is governed by particle collisions. But, when the particle relaxation time is not much larger than the particle collision time, the random motion of the particles will be affected by the gas-phase turbulence. Peirano and Leckner (1998) derived a competing kinetic theory of granular flow with an interstitial gas, based on the work of Jenkins and Richman (1985), which incorporated the influence of the gas turbulence on the particle random motion. As a result, the gas turbulence appears in their constitutive equations for the solid phase. If the interstitial gas were omitted, their results are identical to the work of Jenkins and Richman (1985) and Lun et al. (1984). Although their model is still restricted to slightly inelastic particle–particle collisions, it removed the assumption in previous kinetic theories that the interstitial gas can be neglected.

Crowe and his colleagues' new turbulent modulation model needs to be investigated numerically, which is the first aim of this paper. Second, we will examine the influence of the interstitial turbulent gas on the random motion of particles in a relatively dilute flow regime. All the simulation results will be compared with the comprehensive experimental measurements of Tsuji et al. (1984) and Tsuji (1993) for fully developed gas–solid flows. Throughout we will assume that the fully developed turbulent gas–solid flow is steady and axi-symmetric. This may be an appropriate assumption for fully developed flow in the small-diameter pipes Tsuji and his colleagues examined, where the spatial inhomogeneity of the particulate phase flow structure is likely to be small. Finally, we will assess the sensitivity of our new model to the particle–particle restitution coefficient.

Section snippets

The mathematical model

We assume that particle–particle collisions are nearly elastic and treat the coefficients of restitution of particle–particle and particle–wall as constants in the collisions. The governing equations for a dispersed solid phase and a carrier gas phase are locally averaged. The momentum equations are given by Anderson and Jackson (1967)
Solid phase: $ε_{2} ρ_{2} D D t U =−ε_{2} ∇P−∇·τ_{2} + F_{drag} +ε_{2} ρ_{2} g,$ Gas phase: $ε_{1} ρ_{1} D D t V =−ε_{1} ∇P−∇·τ_{1} − F_{drag},$ where subscripts 1 and 2 represent the gas phase and solid phase, respectively,

Boundary conditions

For a gas–solid flow, the particle diameter may not be negligible when compared to the width of the gas viscous boundary layer. Therefore, when we set the boundary conditions at the wall for the solid phase, we establish momentum and energy balance at a thin layer of particles, the thickness of which is the same order as the gas viscous layer. The gas phase boundary condition should also be set on the same thin layer, as suggested by Sinclair and Jackson (1989). As a result, it may not be

Results and discussion

There are few non-intrusive experimental measurements of particle–gas turbulence interaction, even for relatively simple flows such as up-flow in a vertical pipe. Tsuji et al. (1984) reported their laser Doppler velocimetry measurements of fully developed steady gas–solid flows in a vertical pipe. In 1993, Tsuji revisited the data and his profiles of the axial fluctuational velocity of the solid phase were cited in the work of Bolio et al. (1995). Currently, these experimental data produced by

Conclusions

The gas turbulence has an impact on the microscopic motion of the particles which cannot be neglected for smaller particles in relatively dilute flows. However, it may have a negligible effect on larger particles, even though the particle volume fraction is small. The turbulence modulation model of Crowe and his colleagues can predict the gas turbulence modification due to the presence of the particles reasonably well, and displays few problems of undue sensitivity to the particle–particle

Acknowledgements

The authors are grateful to Professor Crowe for offering copies of his colleagues' and his recent works and for the helpful discussions. We also thank the referees for their useful comments, which helped improve several aspects of this paper.

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Particle–gas turbulence interactions in a kinetic theory approach to granular flows

Abstract

Introduction

Section snippets

The mathematical model

Boundary conditions

Results and discussion

Conclusions

Acknowledgements

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Powder Technol.

Powder Technol.

Prog. Energy Combust. Sci.

Adv. Appl. Mech.

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Int. J. Multiphase Flow

Simple kinetic model for rapid granular flows including frictional losses

J. Eng. Mech. ASCE

Fluid mechanical description of fluidized beds: comparison with theory and experiment

I&EC Fund

Experiments on a gravity-free dispersion of large solid particles in a Newtonian fluid under shear

Proc. R. Soc. London A

Dilute turbulent gas–solid flow in risers with particle interactions

AIChE J.

Rapid granular flows

Annu. Rev. Fluid Mech.

A turbulence closure model for dilute gas–particle flows

Can. J. Chem. Eng.

On predicting particle-laden turbulent flows

Appl. Sci. Res.