European Drying Conference - EuroDrying'2011
Palma. Balearic Island, Spain, 26-28 October 2011
MODELING OF PARTICLE FORMATION DURING SPRAY DRYING
D. Huang1
1
Novartis Pharmaceuticals Corporation
150 Industrial Road, San Carlos, CA 94070 USA
Tel.:+1 650 622 1535, E-mail: [email protected]
Abstract: A mathematical model for the evaporation of a single droplet, subjected to convective drying
in a spray dryer was developed to study the transient phenomena of particle formation. During the
constant-rate drying period, the governing equations and boundary conditions were transformed into a
form containing Peclet number, Pe, as the main controlling parameter. A numerical solution was
developed to solve the diffusion equation in spherical coordinates with a moving boundary (Stefan
Problem). The solution provides the concentration fields in a droplet as a function of time and space.
This information enables the prediction of the tendencies of droplet to form hollow or solid particles as
well as structure composition and morphology.
Keywords: Spray drying, Particle engineering, Mathematical modeling, Numerical simulation, Peclet
number.
INTRODUCTION
Spray drying is widely used in the food, detergent,
minerals, pharmaceutical, and many other
industries because of its ability to efficiently
transform a liquid feed into a dry powder. This
single unit operation comprises three processes:
atomization, drying, and particle collection. Of
these, drying kinetics is the key mechanism that
controls particle morphological properties (Walton
and Mumford, 1999; Vehring 2008).
In numerous studies, it has been demonstrated that
process conditions critically impact final particle
properties such as size, density, porosity, surface
roughness, and surface composition.
These
conditions include both the spray drying process
parameters and the liquid feedstock properties
(Vehring et al., 2007). The interplay between
process parameters such as temperature and flow
rate of drying gas, and feedstock properties such as
solute diffusion coefficient and solvent latent heat
govern the final particle morphology.
The objective of this study is to develop and test a
theoretical model that describes the transport
phenomena of a liquid droplet during spray drying.
This model will enable study of the time-dependent
character of a drying process which includes:
composition profiles within the droplet, the onset of
particle formation in a droplet, and dried particle size
and density (porosity). Such a model will enable
identification of critical process parameters and guide
excipient selection and formulation development.
DROPLET DRYING HISTORY
A schematic diagram of the drying history of a
single droplet inside a spray dryer is shown in
Figure 1. During the sensible heating period, the
droplet temperature increases to its wet-bulb
temperature and no appreciable solvent evaporation
takes place.
During the constant-rate drying
period, the droplet behaves like pure solvent;
evaporation rate is dictated by wet-bulb drying
kinetics. At the wet-bulb temperature, the droplet
shrinks as the solvent is rapidly lost through
evaporation. As evaporation progresses, solute
molecules arrange themselves within the droplet
according to diffusion rates. When solidification
occurs (also called skin formation), it is the
beginning of falling-rate drying period. At this
stage, little further shrinkage can occur, but the
skin may collapse. The skin temperature increases
as liquid boundary moves inward. At this point,
solidification slows the transport of solvent to the
surface for evaporation and drying becomes
diffusion rate-limited.
Figure 1. Schematic of droplet drying history and
morphological changes
THEORETICAL ANALYSIS
Modeling the droplet drying history requires
material and energy balances formulated in a
spherical shell element of the droplet (Figure 2).
Because the concentration and temperature fields
are assumed to be radially symmetric (no
variation in the θ and φ directions), this simplifies
the governing equations to the radial direction.
Because heat conduction inside the droplet is much
faster than heat convection away from its surface,
the temperature gradient within the droplet is
negligible.
The nominal spray-drying conditions include
• inlet and outlet temperatures (150°C and 70°C)
• initial and critical supersaturation concentrations
(5% and 90% w/v)
• Initial droplet radius and density of liquid (5 μm
and 1100 kg/m3)
RESULTS AND DISCUSSION
Figure 2. A liquid droplet expressed in spherical
coordinates
During the constant drying-rate period, the
evaporation of a liquid droplet of diameter d, is
proportional to its surface area. The evaporation
rate κ, is described to follow the “d2 Law” (Law
and Law, 1982). After dimensionless and similarity
transformation (Brenn, 2004; Shabde et al., 2005),
the diffusion equation and surface boundary
condition become:
Ci
C  2 Ci 2 Ci
1
(
)(  Pe  r  i 

) (1)
t
1  2  Pe  t
r
r r
r 2
C i
 Pe  C i (2)
r
where Pe is the Peclet number, which is defined
as:
Pe 

D

evaporation rate
(3)
diffusion rate
The above equations demonstrate that the droplet
drying process is mainly controlled by the Peclet
number.
Model calculations are able to predict the droplet
radius, mass, density, and water content during
drying. Most importantly, the solute concentration
profiles as a function of space and time can be
obtained to study the detailed mechanism of drying
kinetics.
Based on the pre-determined critical
supersaturation concentration of the system (He et al.,
2006), the induction time for shell formation on the
surface of a droplet can be predicted. From such
calculations, the particle size and density of spraydried powder can be estimated. Figure 3 illustrates
an example of the drying process of an aqueous
solution containing solute with an initial radius of 5
µm and an initial concentration of 5%. A Peclet
number equals to 1 means the evaporation rate of
solvent is of the same magnitude as the diffusion rate
of the solute. Each curve in Figure 3 represents the
concentration profile of the droplet at a given time.
The time required for the surface concentration to
reach the critical saturation concentration of 90% (in
this example, solid precipitation is known to occur at
this concentration) is approximately 21 milliseconds.
From the onset of skin formation and material
balance, the geometric particle size and density of the
spray-dried particle were predicted to be 2.0 µm and
0.7 g/cm3.
Numerical solution approach
A “Method of Lines” numerical technique was used
to transform the above partial differential equation
into a system of ordinary differential equations
(ODEs). The ODEs are solved through integration
using Matlab (Schiesser and Griffiths, 2009). At
each time point, conservation of mass is checked to
ensure that the numerical solution is correct. The
numerical solution developed is valid for a wide
range of Peclet number.
Model parameters
The physical and thermal parameters that are
necessary to solve the system of equations include
•
•
•
•
diffusion coefficient of solute (5x10-10 m2/s)
thermal conductivity of H2O (1.90x10-5
kW/m·K)
latent heat of evaporation (2200 kJ/kg)
external heat and mass transfer coefficients
(Ranz-Marshall correlation)
1
2
3
 2 (4)
1
2
 2 (5)
Nu  2  0.6  Re 3  Pr
Sh  2  0.6  Re 3  Sc
3
Where Nu is the Nusselt number, Sh is the Sherwood
number, Re is the Reynolds number, Pr is the Prandtl
number, and Sc is the Schmidt number.
Figure 3. Predictions of concentration profiles and
shell formation
The theoretical analysis has demonstrated that the
Peclet number controls particle formation during
drying. Figure 4 illustrates how particle size and
density are influenced by the Peclet number. When
Pe<1, both particle size and density change
gradually. This is because the evaporation rate is
slow compared to the diffusion rate. This allows
solute molecules to diffuse toward the center of the
droplet resulting in formation of a small solid
particle. Under this circumstance, the particle will
form a dense structure close to the theoretical density
of the material. However, when Pe>1, the solute
molecules do not have enough time to distribute
within the droplet. This results in solute enrichment
on the droplet surface. The faster the evaporation
rate, the sooner the surface reaches its critical
supersaturation, causing early skin formation. This
condition will lead to a larger particle size and a
lower-density wrinkled and/or hollow particle.
D
k
Nu
Pe
Pr
r
Re
Sc
Sh
t
Diffusion coefficient, m2s-1
Evaporation rate, m2s-1
Nusselt number
Peclet number
Prandtl number
radial coordinate, m
Reynolds number
Schmidt number
Sherwood number
time, s
ACKNOWLEDGEMENTS
Figure 4. Particle size and density versus Peclet
number
Figure 5 shows morphological difference of
maltodextrin particles produced from spray drying
conditions of two inlet/outlet temperatures: (A)
110/74°C and (B) 200/173°C (Alamilla-Beltran, L.,
2005). By estimation, the Pe is approximately 0.7 for
the low-temperature and 2.1 for the high-temperature.
Drying at low Pe resulted in small solid particles of
12 μm, whereas large hollow particles of 37 μm were
formed while drying at high Pe. The transformation
of droplet to dried particle could be more complex
due to other events such as breakage (Mezhericher,
M., et al., 2008) and buckling (Tsapis, N., et al.,
2005) of surface crust arising from thermal stress and
capillary forces. Description of these phenomena
would require more sophisticated mathematical
models, and will be pursued in the future.
The author would like to acknowledge many fruitful
discussions and comments from his Novartis
colleagues, D. Miller, J. Weers, and A. Clark and
Professor W. Schiesser at Lehigh University.
REFERENCES
Adhikari, B., Howes, T., Bhandari, B., 2007, Use of
solute fixed coordinate system and method of lines
for prediction of drying kinetics and surface
stickiness of single droplet during convective drying,
Chemical Engineering and Processing 46, 405-419.
Alamilla-Beltran, L., Canona-Perez, J., JimenezAparicio, A., Gutierrez-Lopez, G., 2005, Description
of morphological changes of particles along spray
drying, Journal of Food Engineering 67, 179-184.
Brenn, G., 2004. Concentration fields in drying
droplets, Chemical Engineering Technology 27 (12),
1252-1258.
He, G., Bhamidi, V., Tan, H., Kenis, P., Zukoski, C.,
2006, Determination of critical supersaturation from
microdroplet evaporation experiments, Crystal
Growth & Design 6 (5), 1175-1180.
Law, C., Law, H., 1982, A d2-law for
multicomponent
droplet
vaporization
and
combustion, AIAA Journal 20 (4), 522-527.
A. Pe ≈ 0.7
B. Pe ≈ 2.1
Figure 5. Morphological difference of particles due
to differences in Peclet number, as governed by
process conditions
CONCLUSIONS
A theoretical model of heat and mass transfer during
drying of a single droplet was developed to quantify
the solute concentration profiles inside the droplet
during evaporation. The Peclet number plays a
critical role in controlling particle formation. The
concentration profile during evaporation provides the
key information for predicting the onset of shell
formation. The spray-dried particle size and density
are related to the time of shell formation event. This
model provides valuable means to predict particle
formation mechanism over a broad range of process
conditions and material properties.
NOMEMCRATURE
Ci
d
mass concentration of solid fraction, kgkg-1
Diameter, m
Mezhericher, A., Levy, A., Borde, I., 2008,
Modelling of particle breakage during drying,
Chemical Engineering and Processing 47, 1404-1411.
Schiesser, W., Griffiths, G., 2009, A compendium of
partial differential equation models – method of lines
analysis with Matlab, Cambridge University Press,
New York, NY.
Shabde, V., Emets, S., Mann, U., Hoo, K., Carlson,
N., Gladysz, G., 2005, Modeling a hollow microparticle production process, Computers & Chemical
Engineering 29, 2420-2428.
Tsapis, N., Dufresne, E., Sinha, S., Riera, C.,
Hutchinson, J., Mahadevan, L., Weitz, D., 2005,
Onset of buckling in drying droplets of colloidal
suspensions, Physical Review Letters 94, 018302 1-5.
Vehring, R., 2008, Pharmaceutical particle
engineering via spray drying, Pharmaceutical
Research 25 (5), 999-1022.
Vehring, R., Foss, W., Lechuga-Ballesteros, D.,
2007, Particle formation in spray drying, Journal of
Aerosol Science 38, 728-746.