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Structure Formation within Spray-Dried Droplets; Mathematical Modelling of Spray Polymerisation

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Process-Spray

Abstract

Spray drying is a basic operation for the manufacturing of small, tailored particles. To obtain a specific porous structure of the particles, usually considerable experimental work is needed. Within this project, we developed a novel approach for modelling single droplet drying which accounts for the morphology evolution inside the particle from first principles. The underlying physical processes affecting structure formation, especially surface tension, wetting, and primary particle interactions, are taken into account within a detailed CFD approach. Simulation of suspension drying makes it necessary to represent the evolution of many interfaces. For that reason, we used the mesh-free simulation method “Smoothed particle hydrodynamics” (SPH), which represents the continuum by interpolation points moving according to a Lagrangian point of view. To our knowledge, SPH has been applied in this field for the first time. Several adaptions to the state of the art had to be made for the application in morphogenesis modelling. For the simulation of diffusion-driven drying, a hybrid simulation method was developed: liquid and solid phases are represented by SPH-particles, whereas a standard grid-based method accounts for the surrounding gas.

Simulations of the drying of suspended particles in a liquid show how the formation of a solid crust in the first drying period is affected by the drying rate and by interaction between liquid with solid particles. During the second drying period, vaporisation takes place: We propose a simple scheme to consider vapour diffusion and boiling and are able to model the formation of a hollow sphere. The receding liquid level in a porous structure has been simulated as well. The model is able to show the effects of surface tension and contact angle on diffusion-driven drying. All simulation results are 2D only, but agree qualitatively with experimental findings. Extension of our model to 3D is straight forward, but requires code parallelisation.

Spray polymerisation can be seen as an extension of spray drying in which the solid is formed by polymerisation reactions. Literature models of spray polymerisation treat droplets as fully mixed. Mathematical models accounting for spatial resolution are lacking. Based on established droplet drying approaches, we derived a 1D model for reactive spray drying processes with special emphasis on polymerisation reactions. Polymer properties, such as the chain length distribution, are represented by moments. Using the Maxwell-Stefan diffusion approach, we derived an advanced formulation for spatial transport of moments that accounts for the low mobility of the polymer chains. The molar mass distribution of the polymer can be simulated along the radius as a function of reaction rate, transport parameters, and operation conditions. Moreover, it can be shown that, against common assumptions respecting spray polymerisation, drying and chemical reactions are rather subsequent than concurrent processes. As a consequence, polymerisation reactions are typically performed as bulk polymerisation within the spray.

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Abbreviations

α :

Heat transfer coefficient [W m−2]

β :

Mass transfer coefficient [kg m−2]

c :

Colour function [variable]

Concentration [mol m−3]

c p :

Specific heat [J kg−1 K−1]

D :

Diffusion coefficient [m2 s−1]

δ :

Dirac delta function [m−3]

f :

Arbitrary quantity [variable]

f :

Acceleration due to external force [m s−2]

h :

Smoothing length [m]

Δh ev :

Specific heat of evaporation [J kg−1]

J N :

Molar diffusive flux [mol m−2 s−1]

k :

Reaction constant [variable]

k d :

Receeding rate in simplified drying model [kg m−2]

l :

Particle spacing [m]

λ :

Heat conductivity [W m−1 K−1]

λ k :

kth moment [mol m−3]

MW:

Molar weight [kg mol−1]

Μ :

Dynamic viscosity [Pa s]

Ω j N :

Molar flux of component j due to evaporation [mol m−2 s−1]

p :

Pressure [Pa]

p v :

Vapour pressure [Pa]

[P s ]:

Concentration of polymer with chainlength s [mol m−3]

φ :

Water content in vicinity SPH particle [−]

r :

Distance between particle [m]

Radial coordinate [m]

r j F :

Rate of formation for component j

r cutoff :

Cutoff radius for SPH neighbourhood

R :

General gas constant [8.315 J mol−1 K−1]

Droplet outer radius [m]

ρ :

Density [kg m−3]

s :

Interaction strength in pairwise forces (with index l for liquid, s for solid) [−]

t :

Time [s]

T :

Temperature [°C, K]

T boil :

Boiling temperature [°C]

θ :

Contact angle [°]

x :

Molar fraction [−]

x :

Position vector [m]

v :

Velocity vector [m s−1]

V :

Volume [m3]

w :

Mass fraction [−]

W :

Kernel function [m−3]

A:

Area-based

D:

Superscript for diffusive

i, j :

Particle indices

j :

Component index

N:

Superscript for molar

R:

Superscript for reaction related

V:

Volume-based

0:

Index for initial conditions/setup or for reference value

∞:

Index for surrounding gas, bulk

I:

Initiator

IC:

Consumed initiator

M:

Monomer

P:

Polymer (dead chain)

R:

Polymer (living chain, radical)

S:

Solvent

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Authors and Affiliations

  1. Institut für Chemische Verfahrenstechnik, Universität Stuttgart, Stuttgart, Germany

    Winfried Säckel & Ulrich Nieken

Authors
  1. Winfried Säckel
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  2. Ulrich Nieken
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Correspondence to Ulrich Nieken .

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Editors and Affiliations

  1. Institute of Materials Science, University of Bremen, Bremen, Germany

    Udo Fritsching

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Säckel, W., Nieken, U. (2016). Structure Formation within Spray-Dried Droplets; Mathematical Modelling of Spray Polymerisation. In: Fritsching, U. (eds) Process-Spray. Springer, Cham. https://doi.org/10.1007/978-3-319-32370-1_3

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