Question

For the sedimentation of a suspension of uniform particles in a liquid, the relation between observed sedimentation velocity Up and fractional volumetric concentration C is given by Up = UTɛ 4.8 where UT is the free-falling velocity of an individual particle. Calculate the concentration at which the rate of deposition of particles per unit area will be a maximum, and determine this maximum flux for 0.15 mm spheres of glass (density 2500 kg/m3 ) settling in water (density 1000 kg/m3 , viscosity 1 mNs/m2 ). It may be assumed that the resistance force F on an isolated sphere is given by Stokes’ law.

Answer 1

Given that;

The sedimentation velocity (U_p) is related to fractional volumetric concentration as :

U_p = U_T E^4.8

where;

E = fractional volumetric concentration

But we know that;

E = 1 - C

where C is the concentration of solid particles.

Therefore;

U_p = U_T (1 - C)^4.8

We know that;

Mass Flux = Concentration X Velocity

The rate of deposition of particles per unit area; ie; flux is given by :

J = U_p C

J = U_T C (1 - C)^4.8

To get the maximum value of J; we need to find its derivative dJ / dC.

dJ / dC = U_T (1 - C)^4.8 - 4.8 U_T C (1 - C)^3.8

For maxima;

dJ / dC = 0

U_T (1 - C)^4.8 - 4.8 U_T C (1 - C)^3.8 = 0

U_T (1 - C)^3.8 (1 - C - 4.8 C) = 0

But only the bracketted quantity can be equal to zero.

(1 - C - 4.8 C) = 0

1 - 5.8 C = 0

C = 1 / 5.8

C = 0.172

The concentration at which the rate of deposition of particles per unit area will be a maximum; C = 0.172

To calculate the maximum flux we need to find the free-falling velocity.

For a spherical particle whose resistance force is given by Stokes’ law ;

U_T = D² g (p_s - p_l) / 18u

where;

D = particle diameter = 0.15 mm = 0.00015 m

g = 9.81 m² / s

p_s = Density of glass = 2500 kg / m³

p_l = Density of water = 1000 kg / m³

u = viscosity of water = 1 mNs / m² = 0.001 Ns / m²

Therefore;

U_T = (0.00015)² X 9.81 X (2500 - 1000) / (18 X 0.001)

U_T = 0.0184 m/s

Therefore;

The maximum flux; J_max = 0.0184 X 0.172 = 0.003165 m³ / m².s