Transcript Settling and Sedimentation

Lecturer: Dr Hairul Nazirah bt Abdul Halim

In most of technical processes, liquids or gases flow through
beds of solid particles.

Example:
i) A single fluid flow through a bed of granular solid such as
ion-exchange and catalytic reactors.
ii) two phase countercurrent
flow of liquid and gas through
packed columns.
Single fluid flow through a granular bed or
porous medium involves in;
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fixed bed reactor
filtration
adsorption
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The general structure of a bed of particles can often be
characterized by:
i) the fractional voidage of the bed
ii) the specific surface area of the bed

Voidage/porosity (ε) -The fraction of the volume of the
bed not occupied by solid material. It is dimensionless
and given by;

Specific surface area of the particles (av)
-
The surface area of a particle divided by its volume.
Sp : surface area of a particle in m2
υp : volume of particle in m3
For a spherical particle,
Dp : diameter in m

The volume fraction of particles in the bed: 1   
6
1   
a  av 1    
Dp
where:

a = the ratio of total surface area in the bed to
total volume of bed in m-1.
For packed bed, Reynolds number for a packed bed can
be defined as follows:
Dpu
GDp
N Re 

1    1   
Pressure drop
At a steady state, and negligible gravity effect,
The pressure drop is given by;
change to
Eq. 7.15
Eq. 7.16
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However, the experiments give an empirical constant of
150 for 72λ1, which gives the Kozeny-Carman equation for
laminar flow.
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For flow through beds at particle Reynolds number up to
about 1.0:
Eq. 7.17
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Flow is proportional to the pressure drop and inversely
proportional to the fluid velocity.
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At high Reynolds number (Rep > 1000), the BurkePlummer equation is applied:
Eq. 7.20

Ergun equation:
Eq. 7.22
Eq. 7.22 fitted data for spheres, cylinders, and crushed solids
over a wide range of flow rates (for low, intermediate and high
Reynolds numbers).

For a rigid particle moving through a fluid, there are 3 forces
acting on the body:
- The external force (gravitational or centrifugal force)
- The buoyant force (act parallel with the external force but in
opposite direction)
- The drag force (acts to oppose the motion and act parallel
with the direction of movement but in the opposite direction.)
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Consider a particle of mass m moving through a fluid under
the action of an external force Fe.
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Let the velocity of the particle relative to the fluid be u
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Let the buoyant force on the particle be Fb
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and let the drag be FD
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Then, the resultant force on the particle is:
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The acceleration of the particle is du/dt, and mass (m)
is constant:
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The external force (Fe ) is expressed as a product of
the mass (m) and the acceleration (ae) of the particle
from this force:
The buoyant force (Fb)
The buoyant force is given by
where
is the density of the fluid.
The drag force (FD)
where CD is the drag coefficient, Ap is the projected area of the
particle in the plane perpendicular to the flow direction.
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By substituting all the forces in the Eq. (1)
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Motion from gravitational force

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In gravitational settling, g is constant ( 9.81 m/s2)
The drag coefficient (CD) always increases with velocity (u).
The acceleration (a) decreases with time and approaches
zero.
The particle quickly reaches a constant velocity which is the
maximum attainable under the circumstances.
This maximum settling velocity is called terminal
velocity.

For spherical particle of diameter Dp moving through
the fluid, the terminal velocity is given by

Substitution of m and Ap into the equation for ut gives
the equation for gravity settling of spheres
ut 
4 gD p (  p   )
3CD 
Frequently
used
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Drag coefficient (CD) is a function of Reynolds number (NRE).
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The drag curve applies only under restricted conditions:
i). The particle must be a solid sphere;
ii). The particle must be far from other particles and the
vessel wall so that the flow pattern around the particle is not
distorted;
iii). It must be moving at its terminal velocity with respect to
the fluid.
Figure7.7 Drag coefficients (CD) for spheres and irregular particles
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Particle Reynolds Number
u : velocity of fluid stream
Dp : diameter of the particle
: density of fluid
: viscosity of fluid
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For Re < 1 (Stokes Law applied - laminar flow)
Thus,

For 1000 < Re < 200 000 (Newton’s Law applied –
turbulent flow)
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Newton’s law applies to fairly large particles falling in
gases or low viscosity fluids.
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To identify the range in which the motion of the particle
lies,
the velocity term is eliminated from the Reynolds number
(Re) by substituting ut from Stokes’ law and Newton’s
law.
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Using Stoke’s Law;
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Eq. 7.44
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To determine the settling regime, a convenient
criterion K is introduced.
Eq. 7.45
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Substituting Eq. 7.45 into Eq. 7.44, Re = K3/18.
Set Re = 1 and solving for K gives K=2.6.
If K < 2.6 then Stokes’ law applies.
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Using Newton’s Law;
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Substitution for ut by criterion K,
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Thus,
Set Re = 1000 and solving for K gives K = 68.9.
Set Re = 200,000 and solving for K gives K = 2,360.

Newton’s Law applies for 68.9 < K < 2360.
THUS;
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Stokes’ law range: K < 2.6
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Newton’s law range: 68.9 < K < 2,360

Intermediate range : when K > 2,360 or 2.6 < K < 68.9,
ut is found from;

ut is calculated using a value of CD found by trial from
the Fig. 7.7.
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In general case, the terminal velocity, can be found
by trial and error after guessing Re to get an initial
estimate of drag coefficient CD .
Normally for this case the particle diameter Dp is known
Fig. 7.7 Drag coefficients (CD) for spheres and irregular particles
Free settling
 When a particle is at sufficient distance from the wall of
the container and from other particle, so that its fall is
not affected by them, the process is called free settling.
 Terminal velocity is also known as free settling velocity.
Hindered settling
 When the particles are crowded, they settle at a lower
rate and the process is called hindered settling.
 The particles will interfere with the motion of individual
particles
 The velocity gradient of each particle are affected by the
close presence of other particles.

The velocity for hindered setting can be computed by
this equation:
Stokes Law
Correction
factor

where, ε is volume fraction of the slurry mixture and Ψp
is empirical correction factor.
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Bulk density of mixture –

Empirical correction factor -
The Reynolds number is then based on the velocity
relative to the fluid is
Where the viscosity of the mixture µm is given by;
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Fluidization is a process whereby a granular
material is converted from a static solid-like state to a
dynamic fluid-like state.
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This process occurs when a fluid (liquid or gas) is
passed up through the granular material.
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The most common reason for fluidizing a bed is to
obtain vigorous agitation of the solids in contact with the
fluid, leading to an enhanced transport mechanism
(diffusion, convection, and mass/energy transfer).
See this video:
https://www.youtube.com/watch?v=lFhrpSJZzck
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When a gas flow is introduced through the
bottom of a bed of solid particles, it will move
upwards through the bed via the empty spaces
between the particles.
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At low gas velocities, aerodynamic drag on
each particle is also low, and thus the bed
remains in a fixed state.
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Increasing the velocity, the aerodynamic drag forces will
begin to counteract the gravitational forces, causing the
bed to expand in volume as the particles move away from
each other
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Further increasing the velocity, it will reach a critical
value at which the upward drag forces will exactly equal
the downward gravitational forces, causing the particles to
become suspended within the fluid. At this critical value,
the bed is said to be fluidized and will exhibit fluidic
behavior.
See this video:
https://www.youtube.com/watch?v=1HXcq54NuNM
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By further increasing gas velocity, the bulk density of
the bed will continue to decrease, and its fluidization
becomes more violent, until the particles no longer form a
bed and are “conveyed” upwards by the gas flow.
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When fluidized, a bed of solid particles will behave as a
fluid, like a liquid or gas.
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Objects with a lower density than the bed density will
float on its surface, bobbing up and down if pushed
downwards, while objects with a higher density sink to
the bottom of the bed
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The fluidic behavior allows the particles to be
transported like a fluid, channeled through pipes, not
requiring mechanical transport
Based on the Figure:
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If the superficial velocity, VO is gradually increased,
the pressure drop will increases, but the particles do
not move and the height (L) remains the same.
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At a certain velocity, the pressure drop across the
bed counterbalances the forces of gravity on the
particles or the weight of the bed
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At point A = Any further increase in velocity, causes
the particles to move
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At point B = Further increase in velocity, the particles
become separate enough to move about in the bed
and true fluidization begins.
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From point B to point C = Once bed is fluidized, the
pressure drop across the bed stays constant, but the bed
heights continues to increase with increasing velocity.
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From point C to B = If the velocity is gradually reduced,
the pressure drop remains constant and the bed height
decreases.
*The pressure drop required for the liquid or the gas to flow
through the column at a specific flow rate
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Minimum velocity of fluidization took place at incipient
(beginning) fluidization.
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During this stage, the ratio of pressure drop to the
vessel height (L) is given by;
where
is the minimum porosity
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The minimum fluidization velocity
by this equation;
can be obtained
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For roughly spherical particles,
is generally between
0.4 and 0.45 (commonly taken as 0.45) which
increasing slightly with decreasing particle diameter.
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If the Reynolds number is used, then,