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Pipe flow of suspensions in turbulent fluid

Electrostatic and gravity effects

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Abstract

The general case of a fully developed pipe flow of a suspension in a turbulent fluid with electrically charged particles or with significant gravity effect, or both, and for any inclination of the pipe with the direction of gravity, is formulated. Parameters defining the state of motion are: pipe flow Reynolds number, Froude number, electro diffusion number, diffusion response number, momentum transfer number and particle Knudsen number. Comparison with experimental results is made for both gas-solid and liquid-solid suspensions. It is shown that the gravity effect becomes significant in the case of large pipe diameters and large particle concentrations.

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Abbreviations

a :

radius of a particle

c 1, c 1(ϕ), c 1(r):

a constant of integration, as a constant or as a function of ϕ or r

C D :

drag coefficient for a sphere

D p :

particle diffusivity

E :

electric field

F :

inverse of relaxation time or relaxation time constant for momentum transfer

F*:

coefficient for deviation from Stokes law

ℱL :

radial lift force

ℱp :

vectorial field force acting on a particle

Fr :

Froude number

g :

gravitational acceleration

J F :

flux of particles due to field force

Kn p :

Knudsen number of the particle phase

L p :

particle-fluid interaction length

m :

mass of a particle

m :

total flow rate of particles

m*:

characteristic mass flow ratio of particle to fluid

m*:

characteristic mass ratio of particle to fluid

N ED :

electro diffusion number

N DF :

diffusion response number

N m :

momentum transfer number

P :

static pressure

q :

electric charge per particle

R :

pipe radius

R e :

Reynolds number

r :

radial coordinate

t :

time

u, v, w :

axial, radial, and tangential components of velocity of the fluid

u p, v p, w p :

axial, radial, and tangential components of velocity of the particle

u :

vectorial velocity of the particle cloud

u p :

vectorial velocity of the fluid

u 0 :

maximum or core velocity of the fluid phase

u pw :

velocity of particles at the wall

V :

electric potential

y :

coordinate, R−r

y s :

thickness of laminar sublayer

z :

axial coordinate

〈(Δu)2)〉1/2 :

intensity of motion of particles relative to the fluid

α, β, γ, η :

dimensionless groups as defined

ε 0 :

permittivity of free space

θ :

angle of inclination of the pipe with the direction of gravity

\(\bar \mu\) :

viscosity of the material constituting the fluid phase

ρ :

density of the fluid phase

\(\bar \rho\) :

density of the material constituting the fluid phase

ρ p :

density of the particle cloud

\(\bar \rho _p\) :

density of the material constituting the particles

τ, τ рzr , τ zϕ :

shear stress in the fluid and components in the z direction and the r, ϕ planes

τ р, τ рzr , τ рzϕ :

shear stress of the particle cloud and components in the z direction and the r, ϕ planes

ϕ :

polar angle

*:

dimensionless quantities as defined

p:

for particle phase

w:

for wall condition

References

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Authors and Affiliations

  1. Dept. of Mechanical Engineering, University of Illinois, 61801, Urbana, Ill., USA

    S. L. Soo & S. K. Tung

Authors
  1. S. L. Soo
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  2. S. K. Tung
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Soo, S.L., Tung, S.K. Pipe flow of suspensions in turbulent fluid. Appl. Sci. Res. 24, 83–97 (1971). https://doi.org/10.1007/BF00411707

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  • DOI: https://doi.org/10.1007/BF00411707

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