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1. Motion of particles in fluid
Processes for the separation of particles of various sizes and
shapes often depend on the variation in the behavior of the
particles when they are subjected to the action of a moving
fluid. Further, many of the methods for the determination
of the sizes of particles in the sub-sieve ranges involve
relative motion between the particles and a fluid.

2. Drag force
• The force in the direction of flow exerted by the fluid on the solid is called drag.
• For a non-viscous fluid flowing past a cylinder the velocity and direction of flow
varies round the circumference is shown in figure.
• At A and D the fluid is brought to rest and at B and C the velocity is at a maximum.
• Since the fluid is non-viscous, there is no drag, and an infinite velocity gradient
exists at the surface of the cylinder.
• If the fluid is incompressible and the cylinder is small, the sum of the kinetic energy
and the pressure energy is constant at all points on the surface.
• The kinetic energy is a maximum at B and C and zero at A and D, so that the
pressure falls from A to B and from A to C and rises again from B to D and from C to
D; the pressure at A and D being the same.
• No net force is therefore exerted by the fluid on the cylinder.

3. • For the case of viscous fluid flows over a surface, the fluid is retarded in the
boundary layer which is formed near the surface and that the boundary layer
increases in thickness with increase in distance from the leading edge.
• If the pressure is falling in the direction of flow, the retardation of the fluid is less
and the boundary layer is thinner in consequence.
• If the pressure is rising, however, there will be a greater retardation and the
thickness of the boundary layer increases more rapidly.
• The force acting on the fluid at some point in the boundary layer may then be
sufficient to bring it to rest or to cause flow in the reverse direction with the result
that an eddy current is set up.
• The velocity rises from zero at the surface to a maximum negative value and falls
again to zero. It then increases in the positive direction until it reaches the main
stream velocity at the edge of the boundary layer.

4. • At low rates of flow no separation of the boundary layer takes place.
• As the velocity is increased, separation of boundary layer occurs.
• If the velocity of the fluid is very high, the flow within the boundary layer will
change from streamline to turbulent before separation takes place.

5. Reynolds number and Stoke’s law
• For the case of motion of particle in a fluid or flow of fluid over a particle,
the flow is characterized by the Reynolds number:
Re’ = udρ/μ
in which ‘ρ’ is the density of the fluid,
‘μ’ is the viscosity of the fluid,
‘d’ is the diameter of the sphere
and ‘u’ is the velocity of the fluid relative to the particle.
• For the case of creeping flow, that is flow at very low velocities relative to
the sphere, the drag force ‘F’ on the particle was obtained in 1851 by
STOKES who solved the hydrodynamic equations of motion, the Navier–
Stokes equations, to give:
F = 3πμdu
• This known as Stokes’ law and is applicable only at very low values of the
particle Reynolds number and deviations become progressively greater as
‘Re’ increases.
• Surface friction constitutes two-thirds of the total drag on the particle thus,
the total force ‘F’ is made up of two components:
(i) surface friction: 2πμdu
(ii) form drag: πμdu

6. • As Re’ increases, surface friction becomes proportionately less.
• At values greater than about 20, flow separation occurs with the formation of
vortices in the wake of the sphere.
• At high Reynolds numbers, the size of the vortices progressively increases.
• At values of between 100 and 200, instabilities in the flow give rise to vortex
shedding.

7. Drag coefficient
• The dimensional less group R’/ρu2 is called drag coefficient often denoted
by the symbol C’D.
• R’ is the force per unit projected area of particle in a plane perpendicular to
the direction of motion.
• For a sphere, the projected area is that of a circle of the same diameter as
the sphere.
• Thus:
R = F/(πd2/4)
R/ρu2 = 4F/πd2ρu2 = C’D
• When the force ‘F’ is given by Stokes’ law , then:
R/ρu2 = 12 μ/udρ = 12Re’−1
• The above equation is applicable only at very low values of the Reynolds
number i.e. less than 1.

9. • Region (a) (10−4 < Re’ < 0.2)
In this region, the relationship between R’/ ρu2 and Re’ is a straight
line of slope −1 represented by equation:
R’/ρu2 = 12Re’−1
• Region (b) (0.2 < Re’ < 500–1000)
In this region, the slope of the curve changes progressively from −1
to 0 as Re’ increases.
R’/ρu2 = 12Re’-1(1 + 0.15Re’0.687)
• Region (c) (500–1000 < Re’ < 2 × 105)
In this region, Newton’s law is applicable and the value of R’/ρu2 is
approximately constant giving:
R’/ρu2 = 0.22
• Region (d) (Re’ > 2 × 105)
The flow in the boundary layer changes from streamline to
turbulent and the separation takes place nearer to the rear of the
sphere and:
R’/ρu2 = 0.05

10. Total force on a particle
• The force on a spherical particle may be expressed for
each of the regions a, b, c and d as follows:
• In region (a): R’ = 12ρu2(μ/udρ) = 12uμ/d
F = 12(uμ/d) ¼ πd2 = 3πμdu
This is the expression originally obtained by Stokes.
• In region (b): R’ = 12uμ/d(1 + 0.15Re’0.687)
F = 3πμdu(1 + 0.15Re’0.687)
• In region (c): R’ = 0.22ρu2
F = 0.22ρu2 ¼ πd2 = 0.055πd2ρu2
This relation is often known as Newton’s law.
• In region (d): R’ = 0.05ρu2
F = 0.0125πd2ρu2

11. Terminal falling velocity
• If a spherical particle is allowed to settle in a fluid under gravity, its
velocity will increase until the accelerating force is exactly balanced
by the resistance force.
• Although this state is approached exponentially, the effective
acceleration period is generally of short duration for very small
particles.
• The accelerating force due to gravity:
Fg = (πd3/6)(ρs − ρ)g
where ρs is the density of the solid.
• The terminal falling velocity u0 corresponding to region (a) is given
by:
(16πd3)(ρs − ρ)g = 3πμdu0
u0 = d2g/18μ(ρs − ρ)
• The terminal falling velocity corresponding to region (c) is given by:
(16πd3)(ρs − ρ)g = 0.055πd2ρu0
2
2= 3dg(ρs − ρ)/ρ
uo
• Similarly you can find terminal falling velocities for region (b) and
(d).

12. It is seen that terminal falling velocity of a particle in a given fluid becomes
greater as both particle size and density are increased.
For two material, material A of diameter dA and density ρA, and material B of
diameter dB and density ρB.
If Stoke’s law is applied the terminal
falling velocity u0A is given by:
• For particle of material A:
2g(ρA − ρ)/18μ
u0A = dA
• For particle of material B:
2g(ρB − ρ)/18μ
u0B = dB
• The condition for the two
terminal velocities to be equal is
then:
dB/dA = [(ρA − ρ)/(ρB − ρ)]1/2
If Newton law is applied the terminal
falling velocity u0B is given by:
• For particle of material A:
2= 3dAg(ρs − ρ)/ρ
uoA
• For particle of material B:
2= 3dBg(ρs − ρ)/ρ
uoB
• The condition for the two
terminal velocities to be equal is
then:
dB/dA = (ρA − ρ)/(ρB − ρ)
So we can write that in general relationship for equal settling velocity is:
dB/dA = [(ρA − ρ)/(ρB − ρ)]S
Where S = ½ for Stoke’s law region while S = 1 for Newton’s law region.

13. Exercise – From Coulson and Richardson’s

16. Exercise

17. Exercise

19. Exercise