particle technology- fluid flow in porous media

Fluid Flow in Porous MediaChapter 2

Darcy’s lawKozeny Carman

Modified Reynolds numberFriction factor plot - Carman & Ergun

Deep bed filtrationFluidisation

Professor Richard Holdich

R.G.Holdich@Lboro.ac.uk

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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