References
29
6. Anderson TB, Jackson R (1967) Fluid mechanical description of fluidized beds. Equations of
motion. Ind Eng Chem Fundam 6(4):527–539
7. Aoki KM, Akiyama T (1996) Spontaneous wave pattern formation in vibrated granular
materials. Phys Rev Lett 77(20):4166
8. Bizhaem HK, Tabrizi HB (2013) Experimental study on hydrodynamic characteristics of
gas–solid pulsed fluidized bed. Powder Technol 237:14–23
9. Bizon C, Shattuck MD, de Bruyn JR, Swift JB, McCormick WD, Swinney HL (1998a)
Convection and diffusion in patterns in oscillated granular media. J Stat Phys 93(3):449–465
10. Bizon C, Shattuck MD, Swift JB, McCormick WD, Swinney HL (1998b) Patterns in 3D
vertically oscillated granular layers: simulation and experiment. Phys Rev Lett 80(1):57
11. Boivin M, Simonin O, Squires KD (2000) On the prediction of gas–solid flows with two-way
coupling using large eddy simulation. Phys Fluids 12(8):2080–2090
12. Bokkers G, van Sint Annaland M, Kuipers JAM (2004) Mixing and segregation in a bidisperse
gas–solid fluidised bed: a numerical and experimental study. Powder Technol 140(3):176–186
13. Bougie J, Duckert K (2011) Continuum simulations of shocks and patterns in vertically
oscillated granular layers. Phys Rev E 83(1):011303
14. Bougie J, Kreft J, Swift JB, Swinney HL (2005) Onset of patterns in an oscillated granular
layer: continuum and molecular dynamics simulations. Phys Rev E 71(2):021301
15. Bokun IA, Zabrodski SS (1966) Experimental study of heat transfer between a pulsating bed
and the heating surface. Nauk Tekh pp 27–32
16. Busciglio A, Vella G, Micale G, Rizzuti L (2009) Analysis of the bubbling behaviour of 2D
gas solid fluidized beds: part II. Comparison between experiments and numerical simulations
via digital image analysis technique. Chem Eng J 148(1):145–163
17. Cerda E, Melo F, Rica S (1997) Model for subharmonic waves in granular materials. Phys
Rev Lett 79(23):4570
18. Chialvo S, Sun J, Sundaresan S (2012) Bridging the rheology of granular flows in three
regimes. Phys Rev E 85(2):021305
19. Chialvo S, Sundaresan S (2013) A modified kinetic theory for frictional granular flows in
dense and dilute regimes. Phys Fluids 25(7):070603
20. Christensen D, Nijenhuis J, van Ommen JR, Coppens M-O (2008a) Influence of distributed
secondary gas injection on the performance of a bubbling fluidized-bed reactor. Ind Eng Chem
Res 47(10):3601–3618
21. Christensen D, Nijenhuis J, van Ommen JR, Coppens M-O (2008b) Residence times in
fluidized beds with secondary gas injection. Powder Technol 180(3):321–331
22. Clément E, Labous L (2001) Pattern formation in a vibrated granular layer. In: Pöschel T,
Luding S (eds) Granular Gases, Springer, Berlin, pp 233–243
23. Coppens M-O (2001) Method for operating a chemical and/or physical process by means of
a hierarchical fluid injection system. US6333019B1
24. Coppens M-O, Regelink MA, van den Bleek CM (2002) Pulsation induced transition from
chaos to periodically ordered patterns in fluidised beds. In: Proceedings of 4th World Congress
on Particle Technology (WCPT), Sydney, pp 1–8
25. Coppens M-O, van Ommen JR (2003) Structuring chaotic fluidized beds. Chem Eng J
96(1):117–124
26. Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys
65(3):851
27. Cruz MAA, Passos ML, Ferreira WR (2005) Final drying of whole milk powder in vibratedfluidized beds. Dry Technol 23(9–11):2021–2037
28. Cundall PA, Strack OD (1979) A discrete numerical model for granular assemblies.
Geotechnique 29(1):47–65
29. Daw CS, Finney CE, Vasudevan M, Van Goor NA, Nguyen K, Bruns DD, Kostelich EJ,
Grebogi C, Ott E, Yorke JA (1995) Self-organization and chaos in a fluidized bed. Phys Rev
Lett 75(12):2308
30. de Martín L, Ottevanger C, van Ommen JR, Coppens M-O (2018) Universal stability curve
for pattern formation in pulsed gas-solid fluidized beds of sandlike particles. Phys Rev Fluids
3(3):034303
29
6. Anderson TB, Jackson R (1967) Fluid mechanical description of fluidized beds. Equations of
motion. Ind Eng Chem Fundam 6(4):527–539
7. Aoki KM, Akiyama T (1996) Spontaneous wave pattern formation in vibrated granular
materials. Phys Rev Lett 77(20):4166
8. Bizhaem HK, Tabrizi HB (2013) Experimental study on hydrodynamic characteristics of
gas–solid pulsed fluidized bed. Powder Technol 237:14–23
9. Bizon C, Shattuck MD, de Bruyn JR, Swift JB, McCormick WD, Swinney HL (1998a)
Convection and diffusion in patterns in oscillated granular media. J Stat Phys 93(3):449–465
10. Bizon C, Shattuck MD, Swift JB, McCormick WD, Swinney HL (1998b) Patterns in 3D
vertically oscillated granular layers: simulation and experiment. Phys Rev Lett 80(1):57
11. Boivin M, Simonin O, Squires KD (2000) On the prediction of gas–solid flows with two-way
coupling using large eddy simulation. Phys Fluids 12(8):2080–2090
12. Bokkers G, van Sint Annaland M, Kuipers JAM (2004) Mixing and segregation in a bidisperse
gas–solid fluidised bed: a numerical and experimental study. Powder Technol 140(3):176–186
13. Bougie J, Duckert K (2011) Continuum simulations of shocks and patterns in vertically
oscillated granular layers. Phys Rev E 83(1):011303
14. Bougie J, Kreft J, Swift JB, Swinney HL (2005) Onset of patterns in an oscillated granular
layer: continuum and molecular dynamics simulations. Phys Rev E 71(2):021301
15. Bokun IA, Zabrodski SS (1966) Experimental study of heat transfer between a pulsating bed
and the heating surface. Nauk Tekh pp 27–32
16. Busciglio A, Vella G, Micale G, Rizzuti L (2009) Analysis of the bubbling behaviour of 2D
gas solid fluidized beds: part II. Comparison between experiments and numerical simulations
via digital image analysis technique. Chem Eng J 148(1):145–163
17. Cerda E, Melo F, Rica S (1997) Model for subharmonic waves in granular materials. Phys
Rev Lett 79(23):4570
18. Chialvo S, Sun J, Sundaresan S (2012) Bridging the rheology of granular flows in three
regimes. Phys Rev E 85(2):021305
19. Chialvo S, Sundaresan S (2013) A modified kinetic theory for frictional granular flows in
dense and dilute regimes. Phys Fluids 25(7):070603
20. Christensen D, Nijenhuis J, van Ommen JR, Coppens M-O (2008a) Influence of distributed
secondary gas injection on the performance of a bubbling fluidized-bed reactor. Ind Eng Chem
Res 47(10):3601–3618
21. Christensen D, Nijenhuis J, van Ommen JR, Coppens M-O (2008b) Residence times in
fluidized beds with secondary gas injection. Powder Technol 180(3):321–331
22. Clément E, Labous L (2001) Pattern formation in a vibrated granular layer. In: Pöschel T,
Luding S (eds) Granular Gases, Springer, Berlin, pp 233–243
23. Coppens M-O (2001) Method for operating a chemical and/or physical process by means of
a hierarchical fluid injection system. US6333019B1
24. Coppens M-O, Regelink MA, van den Bleek CM (2002) Pulsation induced transition from
chaos to periodically ordered patterns in fluidised beds. In: Proceedings of 4th World Congress
on Particle Technology (WCPT), Sydney, pp 1–8
25. Coppens M-O, van Ommen JR (2003) Structuring chaotic fluidized beds. Chem Eng J
96(1):117–124
26. Cross MC, Hohenberg PC (1993) Pattern formation outside of equilibrium. Rev Mod Phys
65(3):851
27. Cruz MAA, Passos ML, Ferreira WR (2005) Final drying of whole milk powder in vibratedfluidized beds. Dry Technol 23(9–11):2021–2037
28. Cundall PA, Strack OD (1979) A discrete numerical model for granular assemblies.
Geotechnique 29(1):47–65
29. Daw CS, Finney CE, Vasudevan M, Van Goor NA, Nguyen K, Bruns DD, Kostelich EJ,
Grebogi C, Ott E, Yorke JA (1995) Self-organization and chaos in a fluidized bed. Phys Rev
Lett 75(12):2308
30. de Martín L, Ottevanger C, van Ommen JR, Coppens M-O (2018) Universal stability curve
for pattern formation in pulsed gas-solid fluidized beds of sandlike particles. Phys Rev Fluids
3(3):034303