A new model for the drying of droplets containing suspended solids C.S. Handscomb, M. Kraft and A.E. Bayly Wednesday 19th September, 2007
outline • Motivation • Industrial Application • The Drying Process • Model Description • Results for a Sodium Sulphate Droplet
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
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 Solid Particle Collapse ‘Puffed’ Particle Re-inflation ‘Dry Shell’ High temperature ‘Wet Shell’ A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particles, unpublished, 2003 A. Cheyne, D. Wilson and D. Bridgwater, Spray Dried Detergent Particle, unpublished, 2003 Internal Bubble Nucleation Crust Formation Saturated Surface Drying Initial Droplet Blistered (Burst) Particle Shrivelled Particle Inflated, Hollow Particle
particle morphologies No particle formation Solid Particle Collapse Low solids concentration <1%w/w ‘Puffed’ Particle Re-inflation ‘Dry Shell’ High temperature 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 Tsapis et al. ‘Onset of buckling in Drying Droplets of Colloidal Suspensions’ Phys. Rev. Let. 94(1), 2005 Blistered (Burst) Particle Shrivelled Particle Inflated, Hollow Particle
Demonstrates the core features of the new model particle morphologies • Focus on drying prior to shell formation in this paper Solid Particle Collapse ‘Puffed’ Particle Re-inflation ‘Dry Shell’ High temperature ‘Wet Shell’ Internal Bubble Nucleation Crust Formation Saturated Surface Drying Initial Droplet Blistered (Burst) Particle Shrivelled Particle Inflated, Hollow Particle
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
discrete phase • Population balance for solids • Spherical symmetry reduce to 1-D • One internal and one external coordinate external coordinate internal coordinate diffusion term advection term • Solve for the moments of this equation
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 • Equation system is unclosed with size dependent diffusion coefficient Particle nucleation rate per unit volume
discrete phase • Moment hierarchy closed by linear extrapolation on a log-scale • 4 PDEs required to describe the discrete phase
continuous phase Volume Averages Superficial Intrinsic Total • Volume averaged equations for the continuous phase • Assume Fickian diffusion is primary transport mechanism evolution diffusion crystallization advection
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
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 boundary conditions Solvent mass flux to the bulk calculated using standard correlations based on a partial pressure driving force
Population balance boundary condition… …which gives BCs for the moments Solids remain wetted and are drawn inwards by capillary forces between particles boundary conditions ;
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
sodium sulphate droplet • Crystallisation kinetics D. Rosenblatt, S. Marks and R. Pigford ‘Kinetics of phase transitions in the system sodium sulfate-water’ Ind Eng Chem23(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
sodium sulphate droplet Experimental data taken from: S. Nesic and J. Vodnik. ‘Kinetics of droplet evaporation’ Chem Eng Sci,46(2): 527-537, 1991
sodium sulphate droplet • Radial solute profiles Profiles plotted at 5s intervals Saturated solute mass fraction = 0.34
sodium sulphate droplet • Integrated moments
sodium sulphate droplet • Spatially resolved particle number density Profiles plotted at 5s intervals
sodium sulphate droplet • Spatially resolved solids volume fraction Profiles plotted at 1s intervals
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
