CHEE2945 – Lecture 14
Hindered particle settling:
- There are two important effects:
o Hydrodynamic forces,
o Colloidal forces.
- Particle volume fraction is defined as:
volume of particles
ϕ=
volume of particles + volume of liquid
- Liquid volume fraction is:
ε =1−ϕ
- Due to the way the particle volume fraction is defined, it may change as
particles settle under gravity.
- The effective particle density is given by:
ρe =ρ p ϕ+ ρ f ( 1−ϕ )
- The effective suspension viscosity for ϕ <0.01 is given by Einstein’s equation.
μe =μ( 1+ 2.5 ϕ)
- Quemada’s equation gives viscosity for ϕ <0.3 .
( )
−2
ϕ
μe =μ 1−
ϕm
where ϕ m =¿ particle volume fraction at maximum packing (0.63 for rigid
spheres).
- The general correlation for viscosity is:
μe =μf (ϕ)
- Under Stokes’ law, relative velocity of a particle relative to the fluid is given by:
2
D g( ρ p−ρe )
V rel =
18 μ e
2
D g( ρ p−ρ f ) 1−ϕ
V rel =
18 μ f f (ϕ)
- The leftmost fraction of the above equation is the effect of particle
concentration on particle settling velocity. This equation can be written more
simply as:
Hindered particle settling:
- There are two important effects:
o Hydrodynamic forces,
o Colloidal forces.
- Particle volume fraction is defined as:
volume of particles
ϕ=
volume of particles + volume of liquid
- Liquid volume fraction is:
ε =1−ϕ
- Due to the way the particle volume fraction is defined, it may change as
particles settle under gravity.
- The effective particle density is given by:
ρe =ρ p ϕ+ ρ f ( 1−ϕ )
- The effective suspension viscosity for ϕ <0.01 is given by Einstein’s equation.
μe =μ( 1+ 2.5 ϕ)
- Quemada’s equation gives viscosity for ϕ <0.3 .
( )
−2
ϕ
μe =μ 1−
ϕm
where ϕ m =¿ particle volume fraction at maximum packing (0.63 for rigid
spheres).
- The general correlation for viscosity is:
μe =μf (ϕ)
- Under Stokes’ law, relative velocity of a particle relative to the fluid is given by:
2
D g( ρ p−ρe )
V rel =
18 μ e
2
D g( ρ p−ρ f ) 1−ϕ
V rel =
18 μ f f (ϕ)
- The leftmost fraction of the above equation is the effect of particle
concentration on particle settling velocity. This equation can be written more
simply as:
