a new model for the drying of droplets containing suspended solids

vapour bubble formation

water removed by evaporation

‘blown shell’

A new model for the drying of droplets containing suspended solids

C.S. Handscomb, M. Kraft and A.E. BaylyWednesday 19th September, 2007

Christopher Handscomb([email protected])

outline

• Motivation– Industrial Application– The Drying Process

• Model Description

• Results for a Sodium Sulphate Droplet


• 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

motivation - spray drying


motivation – spray drying

• Consider droplet drying in a spray dryer

• Droplets dry by atomisation and contact with hot drying air

• Consider a single droplet

• Droplets contain suspended solids

• Continuous phase may be either single- or multi-component


particle morphologies

Initial Droplet Saturated Surface Drying

‘Puffed’ Particle

High temperature

Crust Formation Internal Bubble Nucleation

‘Dry Shell’

Solid Particle

‘Wet Shell’

Inflated, Hollow Particle

Blistered (Burst) Particle

Shrivelled Particle

Collapse

Re-inflation

A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particle, unpublished, 2003

A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particles, unpublished, 2003



Initial Droplet

No particle formation

Low solids concentration

<1%w/w

Saturated Surface Drying


High temperature


‘Dry Shell’

Solid Particle

‘Wet Shell’



Shrivelled Particle

Collapse

Re-inflation

A. Lee and C.Law. ‘Gasification and shell characteristics in slurry droplet burning’ Combust. Flame, 85(1): 77-93, 1991

Tsapis et al. ‘Onset of buckling in Drying Droplets of Colloidal Suspensions’ Phys. Rev. Let. 94(1), 2005



Initial Droplet Saturated Surface Drying


High temperature


‘Dry Shell’

Solid Particle

‘Wet Shell’



Shrivelled Particle

Collapse

Re-inflation

Focus on drying prior to shell formation in this paper

• Demonstrates the core features of the new 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

new drying model

Assumed ideal binary solution


discrete phase

• Spherical symmetry reduce to 1-D• One internal and one external coordinate

• Solve for the moments of this equation

• Population balance for solids

advection term diffusion term

external coordinateinternal coordinate


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


discrete phase

• Stokes-Einstein equation for solids diffusion coefficient

• Moment evolution equation

Particle nucleation rate per unit volume• Equation system is unclosed with size dependent diffusion coefficient


discrete phase

• Moment hierarchy closed by linear extrapolation on a log-scale

4 PDEs required to describe the discrete phase


continuous phase

• Volume averaged equations for the continuous phase

• Assume Fickian diffusion is primary transport mechanism

crystallization

diffusionevolution

advection

Volume AveragesSuperficial

Intrinsic

Total

R(c)

S

z


continuous phase

• Advection velocity arises due to density difference between the solute and solvent


continuous phase

• Effective diffusion coefficient is a strong function of local solids fraction and solute mass fraction

• Diffusion coefficient must be obtained from experiments


continuous phase

• Continuous phase equation coupled to the population balance through the last term

1 PDE required to describe the continuous phase

5 coupled PDEs in total


continuous phase


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


boundary conditions

• Zero solute mass flux following receding interface

• External solute boundary condition


boundary conditions

• Droplet shrinkage rate

Solvent mass flux to the bulk calculated using standard correlations based on a partial pressure driving force


boundary conditions

• Population balance boundary condition…

…which gives BCs for the moments

• Solids remain wetted and are drawn inwards by capillary forces between particles

;


numerical implementation

• Apply coordinate transformation to all equations

• Time derivatives are transformed according to

A virtual flux is introduced into all evolution equations


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



• Crystallisation kinetics

D. Rosenblatt, S. Marks and R. Pigford ‘Kinetics of phase transitions in the system sodium sulfate-water’ Ind Eng Chem 23(2): 143-147, 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): 2389-2427, 1991



Experimental data taken from: S. Nesic and J. Vodnik. ‘Kinetics of droplet evaporation’ Chem Eng Sci, 46(2): 527-537, 1991



• Radial solute profiles

Saturated solute mass fraction = 0.34

Profiles plotted at 5 s intervals



• Integrated moments



• Spatially resolved particle number density




• Spatially resolved solids volume fraction



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


acknowledgements

a new model for the drying of droplets containing suspended solids

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