1. A new model for the drying of droplets containing suspended solids
C.S. Handscomb, M. Kraft and A.E. Bayly
Wednesday 19th September, 2007

2. outline Motivation Model Description
Industrial Application
The Drying Process
Model Description
Results for a Sodium Sulphate Droplet

3. motivation - spray drying
An important technology in industry
Used to produce, for example:
Pharmaceuticals
Food stuffs (e.g. milk powder and coffee)
Detergents
Unique drying technology combining moisture removal and particle formation

4. motivation – spray drying
Consider droplet drying in a spray dryer
Droplets contain suspended solids
Continuous phase may be either single- or multi-component
Droplets dry by atomisation and contact with hot drying air
Consider a single droplet

5. particle morphologies
Re-inflation
No particle formation
Collapse
Solid Particle
Low solids concentration <1%w/w
‘Puffed’ Particle
A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particles, unpublished, 2003
High temperature
‘Dry Shell’
A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particle, unpublished, 2003
‘Wet Shell’
Internal Bubble Nucleation
Crust Formation
Saturated Surface Drying
Initial Droplet
Inflated, Hollow Particle
Blistered (Burst) Particle
Shrivelled Particle

6. particle morphologies
Re-inflation
Collapse
Solid Particle
‘Puffed’ Particle
A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particles, unpublished, 2003
High temperature
‘Dry Shell’
A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particle, unpublished, 2003
‘Wet Shell’
Internal Bubble Nucleation
Crust Formation
Saturated Surface Drying
Initial Droplet
Inflated, Hollow Particle
Blistered (Burst) Particle
Shrivelled Particle

7. particle morphologies
Re-inflation
No particle formation
Collapse
Solid Particle
Low solids concentration <1%w/w
‘Puffed’ Particle
High temperature
‘Dry Shell’
A. Lee and C.Law. ‘Gasification and shell characteristics in slurry droplet burning’ Combust. Flame, 85(1): 77-93, 1991
‘Wet Shell’
Internal Bubble Nucleation
Crust Formation
Saturated Surface Drying
Initial Droplet
Inflated, Hollow Particle
Blistered (Burst) Particle
Shrivelled Particle
Tsapis et al. ‘Onset of buckling in Drying Droplets of Colloidal Suspensions’ Phys. Rev. Let. 94(1), 2005

8. particle morphologies
Re-inflation
Collapse
Focus on drying prior to shell formation in this paper
Solid Particle
‘Puffed’ Particle
High temperature
‘Dry Shell’
‘Wet Shell’
Internal Bubble Nucleation
Crust Formation
Saturated Surface Drying
Initial Droplet
Inflated, Hollow Particle
Blistered (Burst) Particle
Shrivelled Particle
Demonstrates the core features of the new model

9. particle drying with a shell
Ideal Shrinkage: r2a t S
R(t)
Shell formation
S(t)
Shrinkage stops upon shell formation
Shell ‘grows’ inwards

10. wet shell Hollow Shell Burst Shell Buckled Shell
Lee and Law. (1991) Combustion and Flame.
Particles produced by burning coal slurries
Tsapis et al. (2005) Physical Review Letters

11. new drying model Assumptions in the present model:
Three component system:
A – solvent;
B – solute;
D – solid
Spherical particles, 1D model
Small Biot number  uniform particle temperature
Allow for a single centrally located bubble
Assumed ideal binary solution

12. discrete phase Population balance for solids
Spherical symmetry  reduce to 1-D
One internal and one external coordinate
internal coordinate
external coordinate
advection term
diffusion term
Solve for the moments of this equation

13. discrete phase
Principle variable of interest is solids volume fraction
Related to the moments of the population balance equation by:
Integer moments of the internal coordinate

14. discrete phase
Stokes-Einstein equation for solids diffusion coefficient
Moment evolution equation
Equation system is unclosed with size dependent diffusion coefficient
Particle nucleation rate per unit volume

15. 4 PDEs required to describe the discrete phase
Moment hierarchy closed by linear extrapolation on a log-scale
4 PDEs required to describe the discrete phase

16. continuous phase Volume averaged equations for the continuous phase
Assume Fickian diffusion is primary transport mechanism
Volume Averages
Superficial
Intrinsic
Total
evolution
diffusion
advection
crystallization

17. continuous phase
Advection velocity arises due to density difference between the solute and solvent

18. continuous phase
Effective diffusion coefficient is a strong function of local solids fraction and solute mass fraction
Diffusion coefficient must be obtained from experiments

19. 1 PDE required to describe the continuous phase
 5 coupled PDEs in total
Continuous phase equation coupled to the population balance through the last term

20. continuous phase

21. boundary conditions Consider only low temperature drying
Initially ideal shrinkage
Droplet radius decreases as particles are free to move
At some point, shell formation occurs

22. boundary conditions Zero solute mass flux following receding interface
External solute boundary condition

23. boundary conditions Droplet shrinkage rate
Solvent mass flux to the bulk calculated using standard correlations based on a partial pressure driving force

24. boundary conditions
Solids remain wetted and are drawn inwards by capillary forces between particles
Population balance boundary condition…
…which gives BCs for the moments ;

25. numerical implementation
Apply coordinate transformation to all equations
Time derivatives are transformed according to
A virtual flux is introduced into all evolution equations

26. sodium sulphate droplet
Simulate the drying of a droplet of sodium sulphate solution
Initial conditions:
Solute content: 14 wt% (near saturated)
Droplet temperature: 20 C
Solids volume fraction: 1.1 x 10-12

27. sodium sulphate droplet
Crystallisation kinetics
D. Rosenblatt, S. Marks and R. Pigford ‘Kinetics of phase transitions in the system sodium sulfate-water’ Ind Eng Chem 23(2): , 1984
Nucleation kinetics (heterogeneous)
J. Dirksen and T. Ring. ‘Fundamentals of crystallization: Kinetic effect on particle size distributions and morphology. Chem Eng Sci, 46(10): , 1991

28. sodium sulphate droplet
Experimental data taken from: S. Nesic and J. Vodnik. ‘Kinetics of droplet evaporation’ Chem Eng Sci, 46(2): , 1991

29. sodium sulphate droplet
Radial solute profiles
Profiles plotted at 5 s intervals
Saturated solute mass fraction = 0.34

30. sodium sulphate droplet
Integrated moments

31. sodium sulphate droplet
Spatially resolved particle number density
Profiles plotted at 5 s intervals

32. sodium sulphate droplet
Spatially resolved solids volume fraction
Profiles plotted at 1 s intervals

33. conclusions
Spray dying to form particles is an important and complex industrial process
Outlined droplet drying model incorporating a population balance to describe the solid phase
New model capable of enhanced morphological prediction

34. acknowledgements