Summary
As part of a study on the hydrodynamics of a cyclone separator, a theoretical investigation of the flow pattern in a flat box cyclone (vortex chamber) has been carried out. Expressions have been derived for the tangential velocity profile as influenced by internal friction (eddy viscosity) and wall friction. The most important parameter controlling the tangential velocity profile is λ = −u 0 R/(v+ ε), where u 0 is the radial velocity at the outer radius R of the cyclone, ν the kinematic liquid viscosity and ε is the kinematic eddy viscosity. For values of λ greater than about 10 the tangential velocity profile is nearly hyperbolic, for λ smaller than 1 the tangential velocity even decreases towards the centre. It is shown how λ and also the wall friction coefficient may be obtained from experimental velocity profiles with the aid of suitable graphs. Because of the close relation between eddy viscosity and eddy diffusion, measurements of velocity profiles in flat box cyclones will also provide information on the eddy motion of particles in a cyclone, a motion reducing its separation efficiency.
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Abbreviations
- A :
-
cross-sectional area of cyclone inlet
- h :
-
height of cyclone
- p :
-
static pressure in cyclone
- Δp :
-
static pressure difference in cyclone between two points on different radius
- r :
-
radius in cyclone
- r 1 :
-
radius of cyclone outlet
- R :
-
radius of cyclone circumference
- u :
-
radial velocity in cyclone
- u 0 :
-
radial velocity at circumference of flat box cyclone
- v :
-
tangential velocity
- v 0 :
-
tangential velocity at circumference of flat box cyclone
- w :
-
axial velocity
- z :
-
axial co-ordinate in cyclone
- α :
-
friction coefficient in flat box cyclone (for definition see § 5)
- α 1 :
-
value of friction coefficient for σ 1<σ<σ 2
- α 2 :
-
value of friction coefficient for σ 2<σ<1
- β :
-
=λα
- β 1 :
-
value of β for σ 1<σ<σ 2
- β 2 :
-
value of β for σ 2<σ<1
- δ :
-
thickness of laminar boundary layer
- δ′ :
-
=δ/h
- ε :
-
turbulent kinematic viscosity
- η :
-
ratio of z to h
- k :
-
ratio of height of cyclone to radius R of cyclone
- λ :
-
parameter describing velocity profile in cyclone =−u 0 R/(ν+ε)
- ν :
-
kinematic viscosity of fluid
- ρ :
-
density of fluid
- σ :
-
ratio of r to R
- σ 1 :
-
value of σ at outlet of cyclone
- σ 2 :
-
value of σ at inner radius of cyclone inlet
- τ w :
-
shear stress at cyclone wall
- ϕ :
-
angular momentum in cyclone/angular momentum in cyclone inlet
- ϕ 1 :
-
value of ϕ at σ=σ 1
- ϕ 2 :
-
value of ϕ at σ=σ 2
References
Broer, L. J. F., Symposium Cyclonen. I. Stromingsverschijnselen in cyclonen, Ingenieur 65 (1953) Ch. 77.
Einstein, H. A. and Huon Li, Steady Vortex Flow in a real Fluid, Reprints of Symposium on Heat Transfer and Fluid Mechanics, 1951.
Driessen, M. G., Trans. Amer. Inst. Min. Metall. Engrs, Coal Div. 1948 p. 177, 240.
Driessen, M. G., Rev. ind. minérale 31 (1951) 482.
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Rietema, K., Krajenbrink, H.J. Theoretical derivation of tangential velocity profiles in a flat vortex chamber-influence of turbulence and wall friction. Appl. sci. Res. 8, 177–197 (1959). https://doi.org/10.1007/BF00411748
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DOI: https://doi.org/10.1007/BF00411748