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# 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.

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### Particle Technology- Fluid Flow in Porous Media

1. 1. Fluid Flow in Porous Media<br />Professor Richard Holdich<br />R.G.Holdich@Lboro.ac.uk<br />Chapter 2<br />Darcy’s law<br />Kozeny Carman<br />Modified Reynolds number<br />Friction factor plot - Carman & Ergun<br />Deep bed filtration<br />Fluidisation<br />Watch this lecture at http://www.vimeo.com/10201454<br />Visit;http://www.midlandit.co.uk/particletechnology.htm for further resources.<br />
2. 2. Darcy’s law<br /><ul><li>Porosity/voidage
3. 3. Solid Concentration
4. 4. Superficial velocity
5. 5. Interstitial velocity</li></li></ul><li>Darcy’s law<br />Darcy’s law:<br />At constant pressure drop:<br />Q is constant - permeation<br />Volumepassed<br />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.<br />Time<br />
6. 6. Darcy’s law<br />Darcy’s law:<br />At constant bed depth:<br />Pressure<br />Empirically derived by Darcy in 1856:<br />Driving potential = resistance x flow<br />Flow rate<br />Similar to Ohm’s law, heat conduction, Hagen-Poiseuille, etc.<br />
7. 7. Darcy’s law<br />
8. 8. Darcy’s law<br />Darcy’s law:<br />In calculations - how do we know what to use for permeability in order to predict pressure drop for given flow rate?<br />
9. 9. Kozeny-Carman<br /><ul><li>Darcy’s law:
10. 10. Kozeny-Carman equation:</li></ul>The term in the square bracket is inverse permeability,SVis specific surface andKis the Kozeny constant (often 5).<br />
11. 11. Kozeny-Carman<br /><ul><li>Kozeny-Carman equation:</li></ul>In calculations - how do we know what to use for permeability in order to predict pressure drop for given flow rate?<br />A: from a knowledge of the particle size and an estimate of the bed porosity, assuming K is 5.<br />
12. 12. Kozeny-Carman<br /><ul><li>Derivation from Poiseuille’s equn:</li></ul>Where d is channel diameter. Assume the porous medium is a bed of parallel channels of hydraulic mean diameter dm.<br />
13. 13. Kozeny-Carman<br /><ul><li>Kozeny proposed (equn 3.2):</li></ul>Volume of voids filled with fluid<br />dm = <br />Wetted surface area<br />Bed volume cancels from top and bottom of above equation<br />
14. 14. Kozeny-Carman<br />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:<br />
15. 15. Modified Reynolds number<br /><ul><li>Reynolds number:</li></ul>dm<br />in our expression.<br />Need an equivalent<br />Note velocity is interstitial.<br />
16. 16. Modified Reynolds number<br /><ul><li>Kozeny proposed:</li></ul>Volume of voids filled with fluid<br />dm = <br />Wetted surface area<br />Bed volume cancels from top and bottom of above equation<br />
17. 17. Modified Reynolds number<br /><ul><li>Reynolds number (N.B. velocities):
18. 18. Reynolds number < 2 - streamline flow</li></li></ul><li>Friction factor plot – p. 24<br />
19. 19. Friction factor plot<br />
20. 20. Friction factor plot<br /><ul><li>Need to convert from shear stress into pressure drop</li></li></ul><li>Friction factor plot<br />Shear Stress and a force balance:<br />drag force =<br />R . particle surface area (N)<br />
21. 21. Friction factor plot<br />Shear Stress and a force balance:<br />drag force =<br />surface area of particles =<br />R . particle surface area (N)<br />(m2)<br />
22. 22. Friction factor plot<br />Shear Stress and a force balance:<br />drag force =<br />surface area of particles =<br />pressure drop on fluid =<br />R . particle surface area (N)<br />(m2)<br />(N m-2)<br />
23. 23. Friction factor plot<br />Shear Stress and a force balance:<br />drag force =<br />surface area of particles =<br />pressure drop on fluid =<br />force by the fluid =<br />R . particle surface area (N)<br />(m2)<br />(N m-2)<br />(N)<br />
24. 24. Friction factor plot<br />Shear Stress and a force balance:<br />drag force =<br />surface area of particles =<br />pressure drop on fluid =<br />force by the fluid =<br />R . particle surface area (N)<br />(m2)<br />(N m-2)<br />(N)<br />Therefore,<br />
25. 25. Friction factor plot<br />Therefore,<br />and Reynolds number,<br />
26. 26. Friction factor plot<br /><ul><li>If streamline: use Kozeny-Carman
27. 27. If not, calculate velocity from flow rate
28. 28. Calculate Modified Reynolds number
29. 29. Calculate friction factor (Carman/Ergun)
30. 30. Calculate shear stress
31. 31. Calculate pressure drop
32. 32. If Re slightly > 2 then pressure drop will be?</li></li></ul><li>Deep Bed Filtration – p.29<br />
33. 33. Deep Bed Filtration<br /><ul><li>Beer
34. 34. wine
35. 35. effluent
36. 36. sea-water
37. 37. potable water
38. 38. etc.
39. 39. Influent <500 mg/L
40. 40. outflow <0.1 mg/L
41. 41. 0.5 to 3 m high
42. 42. 0.6 to 5 mm sand, etc
43. 43. 15 m3 m-2 h-1 feed
44. 44. virus removal:
45. 45. 0.1 m h-1</li></li></ul><li>Deep Bed Filtration<br />Normally batch (in duplicate) but some continuous ones:<br />Image supplied by DynaSand and Hydro International (Wastewater) Ltd.<br />
46. 46. Deep Bed Filtration<br />
47. 47. Deep Bed Filtration<br /><ul><li>Cleaning by backflushing
48. 48. often fluidised
49. 49. with air scour
50. 50. up to 36 m h-1
51. 51. up to 8 minutes, using 5% of filtrate
52. 52. timed cycle or on pressure drop monitor</li></li></ul><li>Simple design equation:<br />Deep Bed Filtration<br />Lamda is a filtration constant - only true at start of filtration. In reality:<br />
53. 53. Head loss by Kozeny-Carman:<br />Deep Bed Filtration<br />
54. 54. Fluidisation<br />Bed expansion during fluidisation:<br />Particles in bed moving apart as fluid flow rate is increased<br />Distributor plate<br />
55. 55. Fluidisation<br />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.<br />
56. 56. Fluidisation<br />Bed weight (per unit area):<br />Fluid drag:<br />
57. 57. Fluidisation<br />Minimum fluidising velocity (Umf):<br />
58. 58. Fluidisation<br />During fluidisation superficial velocity for given porosity (Uo):<br />Richardson and Zaki equation - valid for particulate fluidisation only.<br />
59. 59. Fluidisation<br />Note bubbles of gas rising in the fluidised bed - these occur spontaneously and this type of fluidisation is called aggregative or bubbling.<br />
60. 60. Fluid Flow in Porous Media<br />Darcy’s law<br />Kozeny Carman<br />Modified Reynolds number<br />Friction factor plot - Carman & Ergun<br />Deep bed filtration<br />Fluidisation<br />