Numerical study on flow dynamics of gas-liquid-solid three phase flow through a channel

Zhihong Li, Shi Liu School of Energy Power and Mechanical

Engineering North China Electric Power University Beijing,

102206, China lizhihong@ncepu.edu.cn

Yanwei Hu, Wentie Liu School of Energy Science and Engineering

Harbin Institute of Technology Harbin, 150001, China

hywhit@qq.com

Abstract—An Euler-Euler gas-liquid-solid three-phase flow model is developed to study the flow dynamics through a channel. Granular kinetic theory is applied in the work to deal with the particle phase. Effects of different inlet velocities and inclined angles of the channel are investigated. Distributions of solid volume fractions and velocities are gotten in this work. Numerical results are compared with limited experimental results and a same tendency is obtained.

Keywords: three-phase flow, granular kinetic theory, flow dynamics

1. INTRODUCTION The unique dynamics of the granular flow

have been learned by many natural sciences and the civil engineering. In the last decades, the granular flow mechanics is widely applied in the dynamics of debris flow. Debris flows are granular flows characterized by the presence of a liquid as interstitial fluid: water or mixtures of water, clay and mud [1]. Bagnold [2] takes the first attempt to formulate a mechanical approach to granular flow. Recent findings lead to a better understanding of the interaction between particles. With the physical similarity theory developing, a great deal of work in the application of kinetic theories to granular materials is based on the interpretation of the grain motion in the light of the dense gas-solid theory.

The granular kinetic models are based on the assumption that particles interact by instantaneous collisions, implying that only binary or two-particle collisions need to be considered. Furthermore, when the granular concentration is not low, the effect of the granular surface friction becomes evident.

In this paper, we will learn the mixtures flow of water, particles and air in a channel. The granular kinetic theory is applied to deal with the particle phase. We use different equations to deal with the particles interact, collisions or surface friction.

2. NUMERICAL METHODOLOGY

2.1. Governing equations Continuous equation

( ) ( ) 0i i i i it∂ ε ρ ε ρ∂

+ ∇ ⋅ =v (1)

Where i represents the gas, liquid or solid phase, ε is the concentration of gas, liquid or solid, v is the velocity vector and ρ is the density.

Momentum equation

Just as the N-S equation, and considering the interaction between liquid and particles, the equation is:

( ) ( )

( )

f f f f f f f

f f f f f f s

tp

∂ ε ρ ε ρ∂ε ε ρ ε β

+ ∇ ⋅ =

∇ ⋅ + − ∇ − −

v v v

τ g v v

(2)

,f l g= represents the liquid and gas phase, g is the gravity acceleration, β is the drag coefficient between liquid or gas and solid, fτ is the stress of the liquid or gas, the equation is:

2[ ( ) ] ( )3

Tf f f f f fμ μ= ∇ + ∇ − ∇ ⋅τ v v v I (3)

The momentum equation of solid is:

( ) ( )

( )

s s s s s s s

s s s s f s

tp

∂ ε ρ ε ρ∂ε ε ρ ε β

+ ∇ ⋅ =

∇ ⋅ + ∇ + −s

v v v

τ g - v v (4)

The solid phase stress sτ can be denoted like this [3]:

( )2{[ ( ) ] ( ) }3

s s s s s

s s sT

P ξ μ= − + ∇ ⋅ +

∇ + ∇ − ∇ ⋅

τ v I

v v v I (5)

978-1-4244-6255-1/11/$26.00 ©2011 IEEE

Energy equation

Granular kinetic equation is:

3[ ( ) ( )]2( ) : ( )

s s s s s

s s s s

s s fs

tp k

D

∂ ε ρ θ ε ρ θ∂

τ θγ ϕ

+ ∇ ⋅ =

−∇ + ∇ + ∇ ⋅ ∇− + +

v

I v (6)

sk is the coefficient of heat transfer in the solid phase, sγ is the dissipation of the particles velocity fluctuating energy, sφ is the velocity fluctuating energy transfer between particles and gas or liquid.

The dissipation of the kinetic energy is:

2 2 43(1 ) ( )

s

s s o se gd

γ

θε ρ θπ

=

− − ∇ ⋅ v (7)

The particles radial distribution function 0g is given out by the Bagnold function [2]:

1/3 10

,max

[1 ( ) ]s

s

g εε

−= − (8)

Based on the kinetic theory, Koch’s gave the dissipation function of the particles energy [4]:

222

18( )

4fs

fs f ss

dDd

μρρπθ

= −v v (9)

The velocity fluctuating energy transfer between particles and gas or liquid is:

3sφ βθ= − (10)

When the liquid concentration or effective porosity is less than 0.8, the particle bed pressure drop is gotten by the Ergun function [5], then the momentum transfer coefficient between particle and liquid or gas is:

2

2 2150 1.75s f f sf s

f fd dε μ ρ ε

βε ε

= + −v v (11)

When the liquid concentration or effective porosity is larger than 0.8, the particle bed pressure drop is gotten by the Wen & Yu function[6], then the momentum transfer coefficient between particle and liquid or gas is:

2.653

4d f s f f s

f

Cd

ε ε ρβ ε −

−=

v v (12)

0.68724(1 0.15Re ) / Re Re 10000.44 Re 1000dC

⎧ + ≤= ⎨

>⎩ (13)

Re is the particle Reynolds number:

/f f f s fRe dε ρ μ= −v v (14)

The interaction between liquid and gas is gotten by the schiller-naumann function [7], the momentum transfer coefficient between liquid and gas is:

0.75 Rel g l dCβ ε ε μ= (15)

And 0.68724(1 0.15Re ) / Re Re 1000

0.44 Re 1000dC⎧ + ≤

= ⎨>⎩

(16)

Re is the liquid Reynolds number:

/g g g l gRe dε ρ μ= −v v (17)

The interfacial force between liquid and gas is 0.07.

2.2. Problem geometrical configuration The geometrical configuration in this

simulation work is shown in Figure 1. It is about a channel with a inclined angle α . The angle α can be changed. The channel has a length of 6m and a height of 1m. The granular phase takes a proportion of 10% of the mixture. In the beginning, the air is full of the channel and the mixture accesses the channel through the inlet. In the end of the channel, there is a weir which has a height of 0.75m. The particles have a diameter of 6D mm= and a

relative specific gravity / 2.21p wρ ρ = .

a)

b) Figure 1 Schematic representation of the

research: a) geometrical configuration, b) mesh

3. RESULT Figure 2 gives the instant contours of the

solid concentration along the channel.

0.4s

0.8s

1.2s

2.0s

3.0s

4.0s

Figure 2 The solid concentration contours at different times

From the contours we know that the mixture flows along the channel from the inlet and the particles deposit because of the gravity.

When the mixture meets the weir, it creates an upstream backwater effect and when the mixture is enough, it overflows from the channel.

a)

b)

Figure 3 Distribution of the velocity vector: a) particles, b) liquid (u=5m/s)

In Figure 3, we see that the liquid velocity vector is larger than that of the particles. Under the dragging effect of the liquid, particles run along the channel and slop over the channel.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

/u gh

y/h

simulationexperiment

Figure 4 The distribution of the particles velocity (u=5m/s)

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.2

0.4

0.6

0.8

1.0

y/h

concentration

c-simulation c-experiment

Figure 5 The distribution of the particles concentration

Figure 4 and Figure 5 give the distribution comparison of the particles velocity and concentration between the simulation and the experiment. From Figure 4, we know that the velocity of the particles turns larger along the altitude-direction of the channel, while, the concentration turns smaller. The simulation and experiment data have the same tend, so we know that the model is applied to the fact, the difference between them is because of the simulation cannot be corresponding completely to the experimental solution.

a)

b)

c)

Figure 6 Particles concentration of different inlet velocities

a) u=1.0m/s, b) u=3.0m/s, c) u=5.0m/s

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

/u gh

y/h

1m/s 3m/s 5m/s

Figure 7 Distribution of the particles velocity at different inlet velocities

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

y/h

concentration

1m/s 3m/s 5m/s

Figure 8 Distribution of the particles concentration at different inlet velocities

Figure 7 and Figure 8 give the comparison of particles velocity and concentration at different inlet velocities. They have little distinction, so we know that the inlet velocity has little effect on the entirely depositing area.

a)

b)

Figure 9 Particles concentration of different inclined angles:

a) 15α = degree, b) 10α = degree

Figure 9 show that different inclined angles make a difference on the location of the upstream backwater effect. The larger inclined angle gives the bigger particles momentum because of the larger gravity component along the channel and makes the particles more disorganized. Figure 10 and Figure 11 give the comparison of the particles velocity and concentration in the entirely depositing area of different inclined angles. The different inclined angle also has little effect on that area.

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

/u gh

y/h

15 degree10 degree

Figure 10 Distribution of the particles velocity of different inclined angles

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

y/h

concentration

15 degree 10 degree

Figure 11 Distribution of the particles concentration of different inclined angles

4. CONCLUSIONS An Euler-Euler gas-liquid-solid three-

phase flow model is developed to study the flow dynamics through a channel. The effect of the inlet-velocity and inclined angle on the flow has been studied, and the influence of the weir in the end of the channel is also considered. Compared with the experiment data, it turns out that the model has a same tend with the experiment. The following conclusions are obtained for this model:

1) The Euler-Euler model is applied well with the gas-liquid-solid three-phase flow; it can be used to learn for the numerical calculation of the granular flow.

2) The inlet-velocity and the inclined angle have effects on the gas-liquid-solid three-phase flow, especially on the upstream

backwater effect, while in the entirely depositing area, the effects turn little.

Future work is needed to know more about the particles interaction, and calculations will be carried out to take the particles as discrete phase.

ACKNOWLEDGEMENT This work is financially supported by the

National Science Foundation in China through Grant No. 50736002, Program for Innovative Research Team in University (IRT0952).

REFERENCE [1] R.M. Iverson, The physics of debris flow, Rev.

Geophys. 35 (1997)245–296 [2] R.A. Bagnold, Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid

under shear, Proc. R. Soc. Lond., A 225 (1954) 49–63.

[3] Gidaspow D. Multiphase flow and fluidization: Continuum and kinetic theory descriptions. Academic Press Inc, 1994

[4] Koch D. L., Sangani A. S. Particle Pressure and Marginal Stability Limits for a Homogenous Monodisperse Gas-Fluidized Bed: Kinetic Theory and Numerical Simulations. J. Fluid Mech., 1999, 400: 229~263

[5] Ergun S. Fluid Flow through Packed Columns. Chemical Engineering Progress, 1952, 48(2): 89~94

[6] Wen C. Y., Yu Y. H. A Generalized Method for Predicting the Minimum Fluidized Velocity. J. AICHE. 1966, 12(3): 610~612

[7] L. Schiller, Z. Naumann, “A Drag Coefficient Correlation.” Z. Ver. Deutsch. Ing., 77:318, 1935. F.