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Particle Technology- Fluid Flow in Porous Media
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Fluid Flow in Porous MediaProfessor Richard HoldichR.G.Holdich@Lboro.ac.ukChapter 2Darcy’s lawKozeny CarmanModified Reynolds numberFriction factor plot - Carman & ErgunDeep bed filtrationFluidisationWatch…
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- Fluid Flow in Porous Media
Professor Richard Holdich
R.G.Holdich@Lboro.ac.uk
Chapter 2
Darcy’s law
Kozeny Carman
Modified Reynolds number
Friction factor plot - Carman & Ergun
Deep bed filtration
Fluidisation
Watch this lecture at http://www.vimeo.com/10201454
Visit;http://www.midlandit.co.uk/particletechnology.htm for further resources. - Darcy’s law
- Porosity/voidage
- Solid Concentration
- Superficial velocity
- Interstitial velocity
- Darcy’s law
Darcy’s law:
At constant pressure drop:
Q is constant - permeation
Volumepassed
Pressure gradient is equal to the liquid viscosity multiplied by the superficial velocity and divided by the hydraulic permeability. Permeability in S.I. is m2.
Time - Darcy’s law
Darcy’s law:
At constant bed depth:
Pressure
Empirically derived by Darcy in 1856:
Driving potential = resistance x flow
Flow rate
Similar to Ohm’s law, heat conduction, Hagen-Poiseuille, etc. - Darcy’s law
- Darcy’s law
Darcy’s law:
In calculations - how do we know what to use for permeability in order to predict pressure drop for given flow rate? - Kozeny-Carman
- Darcy’s law:
- Kozeny-Carman equation:
- Kozeny-Carman
- Kozeny-Carman equation:
A: from a knowledge of the particle size and an estimate of the bed porosity, assuming K is 5. - Kozeny-Carman
- Derivation from Poiseuille’s equn:
- Kozeny-Carman
- Kozeny proposed (equn 3.2):
dm =
Wetted surface area
Bed volume cancels from top and bottom of above equation - Kozeny-Carman
Rest of derivation comes from putting Kozeny’s definition of equivalent diameter into Poiseuille’s law and using a dimensionless constant instead of 32, assuming that the channel length is proportional to the bed depth and converting between pore velocity (interstitial) and superficial by: - Modified Reynolds number
- Reynolds number:
in our expression.
Need an equivalent
Note velocity is interstitial. - Modified Reynolds number
- Kozeny proposed:
dm =
Wetted surface area
Bed volume cancels from top and bottom of above equation - Modified Reynolds number
- Reynolds number (N.B. velocities):
- Reynolds number < 2 - streamline flow
- Friction factor plot – p. 24
- Friction factor plot
- Friction factor plot
- Need to convert from shear stress into pressure drop
- Friction factor plot
Shear Stress and a force balance:
drag force =
R . particle surface area (N) - Friction factor plot
Shear Stress and a force balance:
drag force =
surface area of particles =
R . particle surface area (N)
(m2) - Friction factor plot
Shear Stress and a force balance:
drag force =
surface area of particles =
pressure drop on fluid =
R . particle surface area (N)
(m2)
(N m-2) - Friction factor plot
Shear Stress and a force balance:
drag force =
surface area of particles =
pressure drop on fluid =
force by the fluid =
R . particle surface area (N)
(m2)
(N m-2)
(N) - Friction factor plot
Shear Stress and a force balance:
drag force =
surface area of particles =
pressure drop on fluid =
force by the fluid =
R . particle surface area (N)
(m2)
(N m-2)
(N)
Therefore, - Friction factor plot
Therefore,
and Reynolds number, - Friction factor plot
- If streamline: use Kozeny-Carman
- If not, calculate velocity from flow rate
- Calculate Modified Reynolds number
- Calculate friction factor (Carman/Ergun)
- Calculate shear stress
- Calculate pressure drop
- If Re slightly > 2 then pressure drop will be?
- Deep Bed Filtration – p.29
- Deep Bed Filtration
- Beer
- wine
- effluent
- sea-water
- potable water
- etc.
- Influent <500 mg/L
- outflow <0.1 mg/L
- 0.5 to 3 m high
- 0.6 to 5 mm sand, etc
- 15 m3 m-2 h-1 feed
- virus removal:
- 0.1 m h-1
- Deep Bed Filtration
Normally batch (in duplicate) but some continuous ones:
Image supplied by DynaSand and Hydro International (Wastewater) Ltd. - Deep Bed Filtration
- Deep Bed Filtration
- Cleaning by backflushing
- often fluidised
- with air scour
- up to 36 m h-1
- up to 8 minutes, using 5% of filtrate
- timed cycle or on pressure drop monitor
- Simple design equation:
Deep Bed Filtration
Lamda is a filtration constant - only true at start of filtration. In reality: - Head loss by Kozeny-Carman:
Deep Bed Filtration - Fluidisation
Bed expansion during fluidisation:
Particles in bed moving apart as fluid flow rate is increased
Distributor plate - Fluidisation
When the bed weight is equal to the fluid drag the entire bed is supported by the fluid and fluidisation occurs. Little noticeable increase in pressure drop beyond this point. - Fluidisation
Bed weight (per unit area):
Fluid drag: - Fluidisation
Minimum fluidising velocity (Umf): - Fluidisation
During fluidisation superficial velocity for given porosity (Uo):
Richardson and Zaki equation - valid for particulate fluidisation only. - Fluidisation
Note bubbles of gas rising in the fluidised bed - these occur spontaneously and this type of fluidisation is called aggregative or bubbling. - Fluid Flow in Porous Media
Darcy’s law
Kozeny Carman
Modified Reynolds number
Friction factor plot - Carman & Ergun
Deep bed filtration
Fluidisation - This resource was created by Loughborough University and released as an open educational resource through the Open Engineering Resources project of the HE Academy Engineering Subject Centre. The Open Engineering Resources project was funded by HEFCE and part of the JISC/HE Academy UKOER programme.
Slide 27. Image of a DynaSand® is provided courtesy of Hydro International (wastewater) Limited. See http://www.hydro-international.biz/irl/wastewater/dynasand.php for more details.
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