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**A new model for the drying of droplets containing suspended solids**

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

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**outline Motivation Model Description**

Industrial Application The Drying Process Model Description Results for a Sodium Sulphate Droplet

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**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

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**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

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**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1%w/w ‘Puffed’ Particle A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particles, unpublished, 2003 High temperature ‘Dry Shell’ A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particle, unpublished, 2003 ‘Wet Shell’ Internal Bubble Nucleation Crust Formation Saturated Surface Drying Initial Droplet Inflated, Hollow Particle Blistered (Burst) Particle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

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Collapse Solid Particle ‘Puffed’ Particle A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particles, unpublished, 2003 High temperature ‘Dry Shell’ A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particle, unpublished, 2003 ‘Wet Shell’ Internal Bubble Nucleation Crust Formation Saturated Surface Drying Initial Droplet Inflated, Hollow Particle Blistered (Burst) Particle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

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**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

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**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

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**particle drying with a shell**

Ideal Shrinkage: r2a t S R(t) Shell formation S(t) Shrinkage stops upon shell formation Shell ‘grows’ inwards

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**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

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**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

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**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

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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

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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

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**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

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**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

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

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

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**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

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continuous phase

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**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

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**boundary conditions Zero solute mass flux following receding interface**

External solute boundary condition

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**boundary conditions Droplet shrinkage rate**

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

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

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**numerical implementation**

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

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**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

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**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

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**sodium sulphate droplet**

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

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**sodium sulphate droplet**

Radial solute profiles Profiles plotted at 5 s intervals Saturated solute mass fraction = 0.34

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**sodium sulphate droplet**

Integrated moments

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**sodium sulphate droplet**

Spatially resolved particle number density Profiles plotted at 5 s intervals

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**sodium sulphate droplet**

Spatially resolved solids volume fraction Profiles plotted at 1 s intervals

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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

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acknowledgements

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