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Momentum and energy balances for dispersed two-phase flow

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Summary

The equations of motion and the mechanical energy balances for two-phase flow systems are derived by integration over a volume containing a large number of elements of the dispersed phase.

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Abbreviations

A, A′ :

boundary of volumes V, V′

dA, dA′ :

surface element of A, A′

A s :

boundary of particles in V

dA s :

surface element of A s

F :

force per unit volume of the system

g :

−g▽z=gravity vector

g :

acceleration by gravity

I :

unit tensor

p :

pressure

Q :

dissipation in the continuous phase

Q s :

dissipation in the dispersed phase

R :

compression work in the continuous phase

R s :

compression work in the dispersed phase

t :

time

u :

velocity of continuous phase

u s :

velocity of dispersed phase

u :

magnitude of u

u s :

magnitude of u s

V :

volume in the two-phase system

V′ :

part of V occupied by the continuous phase

W :

work done by F

z :

vertical coordinate

α :

local volume fraction of the dispersed phase

Π :

pI−Ψ=stress tensor of the continuous phase

Π s :

turbulent particle stress tensor

ρ :

density of the continuous phase

ρ s :

density of the dispersed phase

Ψ :

shearing-stress tensor of the continuous phase

Ψ s :

turbulent particle shearing-stress tensor

▽ :

nabla operator

▽ u, ▽u s :

velocity gradient tensor

\(\frac{D}{{Dt}} = \frac{\partial }{{\partial t}} + u \cdot \nabla \) :

substantial derivative

References

  1. Leva, M., Fluidization, New York, 1959.

  2. Santalo, M. A., Appl. Mech. Rev. 11 (1958) 523.

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Authors and Affiliations

  1. Koninklijke/Shell-Laboratorium, Amsterdam

    J. J. van Deemter

Authors
  1. J. J. van Deemter
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  2. E. T. van der Laan
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Additional information

(Shell Internationale Research Maatschappij N.V.)

(Bataafse Internationale Petroleum Maatschappij N.V.)

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van Deemter, J.J., van der Laan, E.T. Momentum and energy balances for dispersed two-phase flow. Appl. sci. Res. 10, 102 (1961). https://doi.org/10.1007/BF00411902

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  • DOI: https://doi.org/10.1007/BF00411902

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