1. Che5700 陶瓷粉末處理 Batching and Mixing Batch feed always involves mixing, objective: - high uniformity, high reliability (more judgment (experience) rather science; no proper equations to describe particle flow) Before mixing: action of feeding; next few graphs showing some common equipments, goal: good powder flowability. Most ideal state of mixing: random homogeneous mixture RHM

2. What is the mixer meant to do?  produce a good mix What do you mean by “good” then?  the mixer need to have the right quality (appearance, particle concentration)  well mixed, on a microscopic scale  the products should be free of long-range difference of concentrations  the mixture needs to be random  well mixed on certain scale of scrutiny  product must undergo a certain rate of reaction Purpose of Mixing

6. Bulk Solid Transport Che5700 陶瓷粉末處理 One way to avoid funnel flow: reduce friction from walls Mass flow – first in, first out, good results; Funnel flow – first in, last out (rat holing); not desirable

7. Properties of Mass of Particles Che5700 陶瓷粉末處理  Pressure is not the same in all directions: one applied pressure will create some pressures in other directions, but always smaller; related to particle shape and packing; define K’ = normal pressure/applied pressure  Shear applied at surface will be transmitted through a static mass of particles  Density of mass will vary  Before flow, mass of particles will increase its volume first (dilation)  When angular solids are piled up on a flat surface, there will be an angle of repose; (free flowing solids: this angle is between 15 and 30 o )

8. Angle of Friction Angle of storage tank and angle of friction of particle  whether particle can free flow, i.e. mass flow or funnel flow; Arching: state when particle can not flow at all Common experience: flow of powder decrease if size of flow unit > 15% of opening size

9. Cohesive & Noncohesive Solids Non-cohesive solids (free flowing; cohesionless): K’ 0.35 – 0.6 Cohesiveness: often sensitive to moisture Che5700 陶瓷粉末處理

10. Angle of friction: influenced by particle size, shape, or even water content, it often increase cohesive force between particles  increase angle of friction  more difficult to free flow Abrasion: another possible problem with ceramic particles during transportation

11. Powder Mixing Che5700 陶瓷粉末處理 More art than science; Can never achieve perfect mixing like that in fluid phase; Complete mixing: often refers to specific structure, not attainable from a random process;  Characterization of mixture: I.e. degree of homogeneity - (1) a statistical problem; (2) sample size (scale of scrutiny) consideration – need to be “proper”, too large or too small: little value; e.g. sintering after mixing – then consider diffusion distance during sintering, choose appropriate size for sampling ， can be considered as single sample within that size.  lack of scale-up principle: due to poor understanding of particle behaviour during motion;

12. Scale of Segregation The length, area or volume of the largest region of each component in the mixture is referred to “scale of segregation” of that component In a liquid solution: minimum scale of segregation – size of largest molecules In a particle system: largest particle size

13. Completely random vs completely dispersed Degree of segregation larger in the former case Sampling size: e.g. adding carotene into powder milk (1/1000), how should we sampling?

14. Degree of Mixedness p,q true fractions; x measured fraction Che5700 陶瓷粉末處理 Use statistical numbers as index of degree of mixing, to discuss uniformity of sample, to compare different mixing equipment and operation conditions. Statistically, bimodal distribution for mixtures, often use Gaussian or Poisson distribution as examples E.g. A, B equivalent, except color different, then 

15. Mixing Indices Che5700 陶瓷粉末處理 * Sampling analysis: Standard deviation s (s 2 = variance) ; where  o = standard deviation of original segregated mixture;  r = standard deviation of ultimate completely random mixture; s = standard deviation of current sample; N = number of analyzed samples; n = particle number in sample

16. More Indices Che5700 陶瓷粉末處理 For example: Rose – M = 1 – s/  o ; (unmixed 0  mixed 1- 1/n 1/2 ) Lacey – M = (  o 2 – s 2 )/(  o 2 -  r 2 ) (from 0  1) Kramer – M = (  o – s)/(  o -  r ) (from 0  1) Hixon-Tenney-Harvey index: if x > p  D1 = (1-x)/(1-p) x < p  D2 = x/p x = p  D3 = 1  average degree of mixing Da = (N1 D1 + N2 D2 + N3 D3)/N ( x = value of some component in sample, p= expected value)

17. For different material and different size, use this equnation to calculate RHM 的 variance; C 1, C 2 = fractional concentration of each component Ms = mass of sample  f w  sum of product of the weight fraction f of particles in each size class and the mean particle weight W in the class (could it be f * w; not f w )

18. Mixing Analysis Che5700 陶瓷粉末處理 Previous mixing index to evaluate mixing process or effect of parameters, try to minimize error in sampling and analysis. Macro-scale mixing: by chemical analysis, phase analysis, etc.; Micro-scale mixing: observation by microscopy technique E.g. use M index: to study time effect, determine optimal condition; indicating de-mixing behavior; mixing – by relative movement of particles: convection, shear, diffusion (three mechanisms). Different equipment provide different mechanisms. Inverse of mixing  segregation (percolation of fines, trajectory segregation, rise of coarse upon vibration)

19. Mixing yeast into dough: an example  source: Unit operations in food processing  5 kg yeast mix into 95Kg dough  after 5 min, 100 gm sample, yeast analysis: 0, 16.5, 3.2, 2.2, 12.6, 9.6, 0.2, 4.6, 0.5, 8.5;  After 10 min, 3.4, 8.3, 7.2, 6.0, 4.3, 5.2, 5.7, 2.6, 4.3, 2.0;  M = (so^2 – s^2)/(so^2 – sr^2)  so^2 = 0.048; sr^2 ~0; s5^2 = 0.003, s10^2 = 0.00038  M5 = 0.93; M10 = 0.99

20. Important Parameters Che5700 陶瓷粉末處理 Mixing effect affected by:  type of equipment  energy input  flowability and composition of sampe (size, shape, density, surface characteristics) For complete description of mixing, one need:  sample variance (intensity of segregation)  scale of segregation (to micro-scale)  long range structure

21. Microscale and Macroscale Mixedness Microscale analysis  provide information on microscale mixedness Macroscale mixedness  can be analyzed by many technqiues (chemical or physical) Sampling and analysis error should be kept to a minimum uncertainty in the standard deviation (s) become low when a large number of samples are taken

22. Taken from JS Reed, 2nd ed.

23. Taken from JS Reed, 1995 Commercial mixers: usually with two or more mixing elements to produce: high shear mixing in a local region & low shear bulk mixing

24. One example of two mixing elements To avoid vortex, we may add baffles Turbulence and cavitation – for diffusion (micro-scale mixing)

25. Mechanism in Horizontal Drum Mixer Che5700 陶瓷粉末處理 Path of circulation: particle move with rotating cylinder, mixing only if change in path of particles Radial mixing: due to mixing in the gravity direction (drop to a void); velocity gradient important; Radial de-mixing: core formation, small and heavy particles gradually go to bottom Axial mixing: diffusion mode Axial de-mixing: band formation, effect from mixer walls;

26. Change drum mixer into cone shape, beneficial to mixing; (taken from JS Reed, 1995) cement mixer!

27. Rate Process Che5700 陶瓷粉末處理 dM/dt = A (1-M) – B  (where  is segregation potential; M = 1 -  2 )  unmixing process In principle: B – effect from equipment;  - effect from particle characteristics

28. Equipment used in mixing of viscous paste (Taken from JS Reed, 1995); (a) helical mixer; (c) double planetary mixer Sigma blade mixer

29. 取自 JS Reed, 1995

30. Extent of reaction depend on degree of mixing and uniformity

31. Types of Mixers J. Bridgewater, Particuology, 10(4), 397-427, 2012 Mixing mechanism: convection, diffusion, shear

34. Aeration Blending Che5700 陶瓷粉末處理 Or named fluidized blending, for mixing of different particles. Some advantages:  can obtain uniform mixtures, even different in density  can be precisely controlled, energy cost/unit weight sample lower  easy to operate and maintain  large capacity  fast and easy loading; many different methods for feeding (e.g. pneumatic, mechanical, gravity)  Can use other gases, in addition to air

35. Dispersion of Powder into Liquid Che5700 陶瓷粉末處理  Steps involved: adhesion (a  b), immersion (b  c) and spreading (c  d);  Work involved with each step (equilibrium consideration):  W a =  SL – (  LV +  SV ) = -  LV (cos  +1)  negative for any   W i = 4  SL - 4  LV =-4  LV cos    < 90 o, negative value  W s = (  SL +  LV ) -  SV = -  LV (cos  -1)  positive value, at  =0 o, this number is zero  In summary: work must be done to obtain spreading

36. Contact Angle Che5700 陶瓷粉末處理  cos  = (  SV -  SL ) /  LV  Wetting implies contact angle  < 90 o

37. Macroscale Mixing and Microscale Mixing Che5700 陶瓷粉末處理  Mixing of viscous slurry with a single impeller: difficult to achieve both ** macrocsale mixing  high pumping capacity ** microscale mixing  turbulence  Look at two parameters, Re & P (power requirement) ** Re = N  s (dia.) 2 /  …. turbulence ** P/Re = C D  N 2 (dia.) 3 …. Pumping capacity  C D : drag coefficient;  For high viscosity, same power, but need large Re (microscale mixing)  need small size propeller, or increase velocity; yet for small propeller, non-uniformity occur (macroscale mixing)

38. Mixing and EMI Performance * ABS + Ni powder or fiber  mixing (Barbender mixer or dry mixing)  composite for EMI measurements

39. Dry mixing produced better shielding effect than Barbender mixer (in terms of low threshold value)

40. (a) Barbender mixer (powder; 20%); (b) dry mixing (powder 7%); (c) dry mixing (filament, 7%) Barbender mixer produced perfect mixing, not necessary good for EMI purposes; dry mixing produced macro-scale uniformity, not micro-scale uniformity

41. Correlation between percent measureable (electrical resistance) of composite of various samples Electrical resistance measurement by 4 point probe