particle technology- fluid flow in porous media
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The forth lecture in the module Particle Technology, delivered to second year students who have already studied basic fluid mechanics. Fluid flow in porous media covers the basic streamline and turbulent flow models for pressure drop as a function of flow rate within the media. The Modified Reynolds number determines the degree of turbulence in the fluid. The industrial processes of deep bed (sand) filtration and fluidisation are included.TRANSCRIPT
Fluid Flow in Porous MediaChapter 2
Darcy’s lawKozeny Carman
Modified Reynolds numberFriction factor plot - Carman & Ergun
Deep bed filtrationFluidisation
Professor Richard Holdich
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
dV
dt
1
A
Porous m edia
void + solid = unityfraction fraction
Volum e fractions:
U =o
U o
U
Superficial velocity:
+ C = 1
Darcy’s law
Darcy’s law:
At
V
kL
P 1
d
d
Q is constant - permeation
Time
Volumepassed
At constant pressure drop:
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.
Darcy’s law
Darcy’s law:
At
V
kL
P 1
d
d
Flow rate
Pressure
At constant bed depth:
Empirically derived by Darcy in 1856:
Driving potential = resistance x flow
Similar to Ohm’s law, heat conduction, Hagen-Poiseuille, etc.
Darcy’s law
20 kPa
P = dV 1 L k dt A
1.5 V
V = R I
0.75 V10 kPa
0 kPa 0 V
Darcy’s law
Darcy’s law:
At
V
kL
P 1
d
d
In calculations - how do we know what to use for permeability in order to predict pressure drop for given flow rate?
Kozeny-Carman
Kozeny-Carman equation:
At
V
kL
P 1
d
d
Darcy’s law:
At
VSK
L
P V 1
d
d)1(3
22
The term in the square bracket is inverse permeability, SV is specific surface and K is the Kozeny constant (often 5).
Kozeny-Carman
In calculations - how do we know what to use for permeability in order to predict pressure drop for given flow rate?
A: from a knowledge of the particle size and an estimate of the bed porosity, assuming K is 5.
Kozeny-Carman equation:
At
VSK
L
P V 1
d
d)1(3
22
Kozeny-Carman
Where d is channel diameter. Assume the porous medium is a bed of parallel channels of hydraulic mean diameter dm.
Derivation from Poiseuille’s equn:
32
2d
L
PU
Kozeny-Carman
Kozeny proposed (equn 3.2):
)1(
V
m Sd
dm = Volume of voids filled with fluid
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:
oUU
Modified Reynolds number
Reynolds number:
dU
Re
dm Need an equivalent in our expression.
Note velocity is interstitial.
Modified Reynolds number
Kozeny proposed:
)1(
V
m Sd
dm = Volume of voids filled with fluid
Wetted surface area
Bed volume cancels from top and bottom of above equation
Modified Reynolds number
Reynolds number (N.B. velocities):
o
V
U
S )1(Re
)1(Re
V
o
S
U
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)
LAS )1( (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)
LAS )1( P (N m-2)
(m2)
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)
LAS )1( PPA
(N m-2)
(m2)
(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)
LAS )1( PPA
(N m-2)
(m2)
(N)
LARS )1( PATherefore,
Friction factor plot
L
P
SR
)1(
Therefore,
)1(Re
V
o
S
U
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:
cz
c d
d
)exp( zcc oo
Deep Bed Filtration
Lamda is a filtration constant - only true at start of filtration. In reality:
.),,,( etcf os
Head loss by Kozeny-Carman:
gU
SK
z
ho
V
1)1(
d
d3
22
Deep Bed Filtration
Fluidisation
Bed expansion during fluidisation:
Distributor plate
Particles in bed moving apart as fluid flow rate is increased
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):
)1()( gLP s
Fluid drag:
oV USK
L
P
3
22)1(
Fluidisation
Minimum fluidising velocity (Umf):
2
3
)1(
)(
V
smf
SK
gU
Fluidisation
During fluidisation superficial velocity for given porosity (Uo):
nto UU
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 lawKozeny CarmanModified Reynolds numberFriction factor plot - Carman & ErgunDeep bed filtrationFluidisation
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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