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.

1. 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.

3. Solid Concentration

4. Superficial velocity

6. 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.

7. Darcy’s law

8. 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?

10. Kozeny-Carman equation:The term in the square bracket is inverse permeability,SVis specific surface andKis the Kozeny constant (often 5).

14. 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:

19. Friction factor plot

21. Friction factor plot Shear Stress and a force balance: drag force = surface area of particles = R . particle surface area (N) (m2)

22. 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)

23. 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)

24. 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,

25. Friction factor plot Therefore, and Reynolds number,

27. If not, calculate velocity from flow rate

28. Calculate Modified Reynolds number

29. Calculate friction factor (Carman/Ergun)

30. Calculate shear stress

31. Calculate pressure drop

34. wine

35. effluent

36. sea-water

37. potable water

38. etc.

39. Influent <500 mg/L

40. outflow <0.1 mg/L

41. 0.5 to 3 m high

42. 0.6 to 5 mm sand, etc

43. 15 m3 m-2 h-1 feed

44. virus removal:

46. Deep Bed Filtration

48. often fluidised

49. with air scour

50. up to 36 m h-1

51. up to 8 minutes, using 5% of filtrate

53. Head loss by Kozeny-Carman: Deep Bed Filtration

54. Fluidisation Bed expansion during fluidisation: Particles in bed moving apart as fluid flow rate is increased Distributor plate

55. 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.

56. Fluidisation Bed weight (per unit area): Fluid drag:

57. Fluidisation Minimum fluidising velocity (Umf):

58. Fluidisation During fluidisation superficial velocity for given porosity (Uo): Richardson and Zaki equation - valid for particulate fluidisation only.

59. Fluidisation Note bubbles of gas rising in the fluidised bed - these occur spontaneously and this type of fluidisation is called aggregative or bubbling.

60. Fluid Flow in Porous Media Darcy’s law Kozeny Carman Modified Reynolds number Friction factor plot - Carman & Ergun Deep bed filtration Fluidisation

61. 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. © 2009 Loughborough University This work is licensed under a Creative Commons Attribution 2.0 License. The name of Loughborough University, and the Loughborough University logo are the name and registered marks of Loughborough University. To the fullest extent permitted by law Loughborough University reserves all its rights in its name and marks, which may not be used except with its written permission. The JISC logo is licensed under the terms of the Creative Commons Attribution-Non-Commercial-No Derivative Works 2.0 UK: England & Wales Licence. All reproductions must comply with the terms of that licence. The HEA logo is owned by the Higher Education Academy Limited may be freely distributed and copied for educational purposes only, provided that appropriate acknowledgement is given to the Higher Education Academy as the copyright holder and original publisher.