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The second lecture in the module Particle Technology, delivered to second year students who have already studied basic fluid mechanics. Dilute particle systems is mainly about sedimentation of single particles and dilute suspensions. The Particle Reynolds number determines the degree of turbulence in the fluid and techniques are provided for settling in laminar and turbulent systems. Industrial clarification is included.
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Dilute Particulate SystemsChapter 5 in Fundamentals
Professor Richard Holdich
R.G.Holdich@Lboro.ac.uk
Watch this lecture at http://www.vimeo.com/10200970
Visit http://www.midlandit.co.uk/particletechnology.htm
for further resources.
Dilute Particulate Systems
field force(s) and drag Stokes’s settling equation Particle Reynolds number Drag coefficient/Friction factor
plot what to do when Re'>0.2
• (Heywood tables) industrial clarification
Forces
Newton:
maF
Weight Force
Tedious to weigh small particles, hence we use the particle diameter and convert to mass, then weight.
sxm 3
6
Archimede’s Principle
When a body is wholly, or partially, immersed in a fluid it experiences an upthrust equal to the weight of fluid displaced.
Discovered in his bath? Buoyancy - hence buoyed weight is:
gxF sW )(6
3
Stokes’s Drag Expression
Solution to Navier-Stokes equation valid for no inertia
tD xUF 3
Inertia
Rate of change of momentum
t
Ux
t
Um
t
mUF sI d
d
6d
d
d
)(d 3
Centrifugal Force
Note the weight is:
gxF sW )(6
3
The centrifugal force is:
23 )(6
rxF sC
where r is radial position and omega is the angular speed (s-1).
Electro- and Thermo-phoretic
Due to electric field or temperature gradients
Mainly applicable to small particles (less than 10 microns) is gases
See gas cleaning notes
Gases
Small particles can ‘slip’ between the molecules of gases - hence there is a slip correction due to Cunningham applicable to particles less than 2 microns and based on the mean free path of the gas.
The settling velocity will be…?
Liquids
Small particles are subject to bombardment by liquid molecules giving rise to Brownian motion. Hence they might not settle at all!
Significant with particles 1 micron in diameter and less.
N.B. they will still settle in a centrifuge.
Stokes’s settling equation
Single particle settlinggxF sW )(
63
tD xUF 3
Ut is terminal settling velocity.
Stokes’s settling equation
18
)(2 gxU st
The Stokes settling equation:
Stokes law valid for no inertia present
AND a low concentration/single particle. Note that bigger particles settle faster - Galileo and that tower in Pisa?
Stokes’s settling equation
Free settling
Stokes’s settling equation
18
)(2 gxU st
Stokes’s settling equation:
Note that bigger particles settle faster. Industrially we often enhance settling rates by causing the particles to coagulate or flocculate together.
Stokes’s settling equation
Colloid stability important in filtration and sedimentation.
Often assessed by the Zeta potential
Surface forces can predominate at iso-electric point.
Particle Reynolds number
Particle Reynolds number:
still a ratio of inertial to viscous forces - note it is based on the FLUID properties of density and viscosity.
txU
Re
Must be less than 0.2 for Stokes’s law
AND a low concentration/single particle
Particle Reynolds number
Drag coefficient/Friction factor plot
What to do when Re'>0.2
Drag force = RAp
Drag force =
2UCR d2
4xAp
pd AUC 2 Weight = gx s )(6
3
cf Friction factor: shear stress over density and velocity2
What to do when Re'>0.2
Drag force = RAp
Drag force =
2UCR d2
4xAp
pd AUC 2 Weight = gx s )(6
3
d
s
C
xgU
)(
3
2 …check that this reduces
to Stokes law in laminar region.
What to do when Re'>0.2
Numerous correlations between friction factor and Reynolds number @ Re’>0.2
Above can be used to give settling velocity =f(diameter) or vice-versa.
Recommend a simple Tabular scheme developed by Heywood - now fully automated on the www (freely available): http://www.filtration-and-separation.com
Industrial Clarification
field force(s) and drag Stokes’s settling equation Particle Reynolds number Drag coefficient/Friction factor
plot what to do when Re'>0.2
• (Heywood tables) industrial clarification
Industrial Clarification
Simple Camp-Hazen clarification model
t
AHQ
t
HU t
tU
H
Q
AHt or tU
QA
Industrial Clarification - long tube test
Summary
field force(s) and drag Stokes’s settling equation Particle Reynolds number Drag coefficient/Friction factor plot what to do when Re'>0.2 industrial clarification
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