The eighth lecture in the module Particle Technology, delivered to second year students who have already studied basic fluid mechanics. Two phase flow, rheology and Powders covers flow of dispersions of powders in liquids and gases, as well as the storage of powders and why they sometimes do not flow. Equations to predict the pressure drop in pumped systems are provided, for both streamline and turbulent flows.

1. Two Phase Flow, Rheology and Powder Flow Chapters 6, 9 & 10 in Fundamentals Watch this lecture at www.vimeo.com Also visit; http://www.midlandit.co.uk/particletechnology.htm for further resources. Course details: Particle Technology, module code: CGB019 and CGB919, 2nd year of study. Professor Richard Holdich R.G.Holdich@Lboro.ac.uk

3. Homogeneous systems; Newtonian and non-Newtonian, laminar/turbulent

4. Homogeneous but with slip

5. pneumatic conveying - dilute phase

6. Heterogeneous systems

7. pneumatic conveying - dense phase

8. hydraulic conveying

10. Rheograms Non-time dependent Newtonian: Power law:

11. Rheograms Time dependent

12. Apparent viscosity Is the viscosity of a Newtonian fluid that flows under the same conditions of shear rate and stress as the non-Newtonian fluid. R (Pa) Apparentviscosity dv/dr (s-1)

13. Apparent viscosity In order to use Newtonian flow equations we really need “apparent viscosity for pipe flow” - from the “flow characteristic”, etc. In order to predict flow rate and pressure drop use simpler approach - appropriate to power law fluids. Force balance on a wall gives:

14. Wilkinson’s equation Combine the power law viscosity equation with the shear stress on the wall - much like the derivation of Hagen’s equation and integrate to give: Laminar flow of non-Newtonian power law fluids and suspensions.

16. Turbulent flow Need a Reynolds number that reduces to Newtonian equation when n=1, and the turbulent friction expression should reduce to Wilkinson’s equation given f=16/Re* - i.e. for laminar flow.

17. Turbulent flow The Generalised Reynolds number - threshold value of 2000 for laminar to turbulent flow.

18. Turbulent flow - Q from known pressure drop Solution to turbulent equation - note that f occurs on both sides of equation: estimate Q from laminar equation, calculate v and Re, calculate f from wall shear and friction factor equations, then square root of f, calculate RHS of D&M correlation, and check agreement, if doesn’t then …………. the flow rate - iterate until it agrees.

20. Summary for suspensions For Newtonian: Use Krieger for viscosity f(C) and use mean suspension density, then Treat as homogeneous fluid (i.e. CGA001) For non-Newtonian Wilkinson’s equation for LAMINAR Dodge & Metzner for TURBULENT

23. Saltation velocity Comes from Rizk correlation: Dimensional constants in SI units Ms is mass flow rate (kg/s) and D is pipe diameter (m).

24. Slip velocity (solid-gas) Solids will slip in the gas flow: Dimensional constants in SI units, empirical equation relating solid velocity to superficial gas velocity.

25. Dense phase design Difficult! Dense phase design: http://www.cheresources.com/pnuconvey.shtml

26. Hydraulic transport Firstly, identify occurrence of boundary between homogeneous and heterogeneous transport. Empirical correlation due to Kim et al, 1986, Int. Chem. Eng., p 731.

27. Hydraulic transport Secondly, use homogeneous non-Newtonian (or Newtonian) transport equations - if appropriate. If heterogeneous, correlation due to Durand (1953) but much better to empirically investigate own materials.

28. Powder Flow Powder flow issues Hopper failure Explosion Powder flood Hopper discharge Mass flow Core flow Wall and powder pressure - FRICTION Testing

29. Powder Flow & Storage Definitions: Hopper: Conical section, bottom Bin Cylindrical section, top Silo Used for both Interchangeable in use

30. Powder Flow Disasters Powder flood Silo failure Images removed from copyright reasons. For a suitable example please see http://www.jenike.com/Solutions/silofail.html Image created by R J Leask found at http://picasaweb.google.com/rjleask http://creativecommons.org/licenses/by/3.0/

31. Explosion Powder Flow Disasters Image removed from copyright reasons. For a suitable example please see http://www.teachersdomain.org/asset/lsps07_int_expldust/

32. Flow Patterns MASS FLOW: first in – first out CORE FLOW: first in – last out

33. Comparison of flow patterns Mass flow Core flow Flow is uniform and Erratic flow whichcan well controlled cause powderto aerate and flood (avalanche) No dead (static) regions Static zones at sides - no perishable spoilage - may empty at the end Channelling and bridging Piping may occurshould be absent Less segregation Particles roll in discharge Tall and thin May have higher capacity for capital cost High stress where Arrangement maydirection changes relieve wall stresses

34. Angle of Repose For a FREE FLOWING powder the hopper angle needs to be greater than the angle of repose for flow to occur. This is typically 30o BUT a different approach is required for COHESIVE powders. Angle of repose is difficult to measure - best to pour powder into an upside down glass funnel and carefully remove to leave heap in place.

35. Bulk Density Is the combined density of the powder and the void space. Remembering the definition of porosity: Porosity =  = void volume/total volume Hence the bulk density will be: the above densities are, in order: bulk, solid & fluid. If the fluid is air the furthest right term can be ignored.

36. Pressure transmission and powder discharge Unlike fluids there isn't a linear increase in pressure with height - for all heights. In fact, the pressure stabilises after a few metres and the rate of discharge from a hopper will, therefore, be remarkably constant. For free flowing powders the empirical equation: where D is the opening diameter. Note that this equation does not include powder height.

37. Pressure transmission Janssen’s analysis where Pvo is the pressure at z=0, called the 'surcharge' or uniform stress applied at the top of the powder. For Pvo=0 and at small values of z: as exp(-Az)  1 - Az for low z Thus, - a similar result to that of liquids BUT only for small values of z. At large values of z: as the exponential term disappears. i.e. pressure asymptotes to the above uniform value.

38. Importance of Janssen’s work Stress is not transmitted in a similar way to hydraulic head, and Wall friction has a very significant influence on the internal powder stresses.

39. Hopper design Mass flow discharge is based upon two factors: the hopper angle steep enough and the discharge opening wide enough to provide the flow. The Powder Flow Function (PFF or sometimes called the Material Flow Function), characterises the ease, or otherwise, of powder transport and storage.

40. Stable Arch Formation Thus the minimum hopper opening diameter needs to be The main stage is to identify the unconfined yield stress for a powder inside a hopper, and to know more about the functional relation H().

41. Mohr’s circle and principal planes The maximum principal plane stress for the circle formed by conditions of a and a is given by the Mohr's circle drawn through those points and is read off at the =0 axis. The unconfined yield stress is the stress (Pa) given by the Mohr's circle that goes through the origin AND is a tangent to the yield locus. It is the maximum principal plane stress for this circle.

42. Material or Powder Flow Function Obtained from a series of yield locii giving the maximum principal stress and unconfined yield stress; one data point from each yield locus. PFF Unconfined yield stress Maximum principal stress

43. Jenike shear cell Two rings are used. The powder in the rings has a consolidating (normal) load applied. This load is removed and a lower load used, together with a shear stress applied via the bracket on the side of the top ring. When the shear stress is sufficient the top ring will slide over the bottom, and the powder has sheared. This gives one value for shear and consolidating stress, that may be plotted on a Mohr circle.

44. Useful sites Description of Jenike and other techniques for yield locus determination – then how to use the data for hopper design. http://members.aol.com/SchulzeDie/grdle1.html Also, try the freeware program ‘spannung.exe’ A well known name and company with many useful resources: http://www.jenike.com On-line magazine for powder and bulk handling: http://www.powderandbulk.com/ Highly recommended article on different flow types: http://www.erpt.org/992Q/bate-00.htm and more generally on this subject: http://www.erpt.org/technoar/powdmech.htm http://www.erpt.org/technoar/powddyna.htm

45. 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. © 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.