ECI International Conference on Heat Transfer and Fluid Flow in Microscale Castelvecchio Pascoli, 25-30 September 2005

INTRODUCTION

Spray freeze drying is a promising new method to obtain high-quality porous particles primarily for pharmaceutical use. Micro-sized droplets are sprayed to contact with a freezing medium, e.g., liquid nitrogen or cold gas stream, and allowed to develop into rapidly frozen micro-droplets. Subsequently, they are collected, packed in layer, and freeze-dried at the operation condition below the triple point of water (sub-zero temperature and vacuum pressure). Currently, many researchers are working on the applications of spray freeze-dried particles, e.g., pulmonary drug delivery [1], preparation of particles for micro-encapsulation [2,3], processing of low water soluble drugs [4], and epidermal powder immunization [5,6].

A collection of spherical particles is a continuous porous medium whose mass transport properties are dependent on the particle size and the packing porosity. At the same time, each spherical particle itself is also another porous medium whose mass transport properties are dependent on the ice crystal size and initial formulation of solution. The interaction between the two porous media with difference length scales should be carefully taken into account to obtain physically reliable results. The present study aims to develop a physically consistent model for the accurate prediction of the freeze drying of sprayed particles packing.

The prediction model for the spray freeze drying was developed considering the heat and mass transfer through the random packing of spherical particles, and the mass transfer and the evolution of ice front (sublimation front) in those micro-sized particles. We have previously developed a fixed grid calculation method [7] for general freeze drying problems by extending the sorption-sublimation models [8-10]. In this study, the fixed grid calculation procedure was modified to include the additional mass transfer resistances due to the fine

microscale structure inside the particles. The effects of the product height, particle diameter, and packing porosity on the freeze drying characteristics were investigated. The effect of volumetric heating was also studied.

EXPERIMENTAL

A preliminary experiment on the spray-freeze drying was conducted by spraying a solution of 10 % (w/v) bovine serum albumin (BSA) into liquid nitrogen. An ultrasonic atomizer with the specified mean droplet diameter of 12.5 µm (model SP120K50S, Zaxis Inc.) was used. The sprayed micro-droplets readily formed frozen slurries on contacting stirred liquid nitrogen contained in a stainless steel tray. Then, the tray was transferred into a programmable freeze-dryer (model PVTFD10R, Ilshin Lab Co.) where the frozen slurries were freeze-dried for 50 hours. The temperature was maintained at −20°C for 25 hours for the sublimation of ice (primary drying) and then at +20°C for another 25 hours for the desorption of residual moisture (secondary drying). Then, the micro-structure of the spray freeze-dried BSA was investigated using a scanning electron microscope.

Figure 1 shows the scanning electron micrographs of BSA before and after the spray freeze drying experiment. The initial BSA particles are relatively coarse as shown in Fig. 1(a). However, after the spray-freeze drying, the particle diameter is reduced as shown in Fig. 1(b). Small particles with large volumetric surface area are helpful for the enhancement of the bioavailability and the solubility of drugs. A broken BSA micro-particle, shown in Fig. 1(b), reveals that another fine micro-structure exists in the porous particles. The pore length scale is observed much smaller than the particle diameter.

A NUMERICAL STUDY ON

THE FREEZE DRYING OF SPRAYED PARTICLE PACKING

Chi Sung Song*, Jin Hyun Nam°

* Korea Institute of Machinery and Materials, 171 Jang-dong, Yusong-gu, Daejeon 305-343, Korea ° Ilshin Lab Co. 82 Ibam-ri, Nam-myeon, Yangju-si, Gyeonggi-do 482-872, Korea

ABSTRACT Spray freeze drying is an emerging new method to produce micro-sized porous particles for pharmaceutical use. In this study,

the primary drying of the freeze drying of micro-sized particles in a tray was studied based on the preliminary experimental observation. For accurate prediction of the freeze drying of sprayed particle packing, a freeze drying analysis model was developed considering the transport in multi-length scale porous media. The simulation results showed that the freeze drying of spherical particles required relatively long drying time, compared with the time required for the freeze drying of frozen solution, primarily due to the limitation in heat transfer. In addition, the dried layer that grew near the bottom surface was found to contribute to the long drying time. Parametric studies were conducted, which showed that smaller product height, larger particle diameter, and higher packing porosity were favorable for the enhancement of overall drying rate. The effect of volumetric heating was briefly studied to assess the possibility of its application in the freeze drying of spherical particles.

Figure 1. Scanning electron micrographs of BSA; (a) as received, and (b) after spray freeze drying.

MODEL DESCRIPTION

The freeze drying of sprayed particle packing should take into account the mass transfer in multi-length scale porous media. The idealized model for this freeze drying problem is presented in Fig. 2, where the domain is divided into a packing phase (1) and a particle phase (2). The role of the mass transfer in the packing phase is to transport the vapor from the particles out of the packing or the product. However, the role of the mass transfer in the particle phase is to exhaust the vapor, generated at the ice front by sublimation, out of the particles.

Figure 2. Physical model for the freeze drying of spherical particle packing; (1) packing phase, and (2) particle phase.

The structure of the packing phase is defined by the particle

diameter 1,pd and the packing porosity 1ε . The micro-

structure of the particle phase is defined by the pore diameter

2,pored and the porosity 2ε .

MODEL FORMULATION

Ice Saturation

The governing equations for the primary drying stage were derived, based on the sorption-sublimation model [7-10]. Ice saturation in a particle s is defined as the fraction of ice volume to the pore volume in the particle as

3

1,3

1,2

32

===

p

ice

p

ice

pore

ice

d

d

d

d

V

Vs

πεπε

, (1)

where iced denotes the diameter of an ice core in the particle,

shown in Fig. 2. From the ice saturation s , volume-averaged properties for

the packing phase can be defined as follows. The volume- averaged density is defined as

DF ss ρρρ )1( −+= , (2)

the volume-averaged heat capacity as

pDDpFFp cscsc ρρρ )1( −+= , (3)

and the thermal conductivity as

DF ksskk )1( −+= , (4)

where subscript F and D denote the effective properties of the spherical particle packing for s =1 and s =0, respectively. For example, the densities Dρ and Fρ are defined as mD ρεερ )1)(1( 21 −−= , (5a)

])1)[(1( 221 icemF ρερεερ +−−= , (5b) where mρ denotes the density of solid drying material (when

packed to 0 porosity) and iceρ is the density of ice. Similarly,

pDDcρ and pFFcρ are defined as

pmmpDD cc ρεερ )1)(1( 21 −−= , (6a)

])1)[(1( 221 piceicepmmpFF ccc ρερεερ +−−= , (6b)

and Dk and Fk as (negligible gas-phase conduction) mD kk )1)(1( 21 εε −−= , (7a)

])1)[(1( 221 icemF kkk εεε +−−= . (7b) Heat Transfer

Figure 2 shows the conventional heating method, the combined conduction at the bottom surface bq and radiation at

the top surface tq . Ignoring convective heat flux through the packing phase, the conservation of energy is written as

( ) ( ) ( )t

shTkTc

t DFsp ∂∂−∆=∇−⋅∇+

∂∂ ρρρ , (8)

where the last term denotes the latent heat source due to the sublimation of ice. The finite volume discretization of Eq. (6) results in

( ) ( )

( )t

ssh

TTdl

AkV

t

TTc

oPP

DFs

EWNSjjP

j

jo

PPop

∆−−∆=

−+∆∆−

∑=

ρρ

ρ , (9)

and the subscript P denotes the present finite volume and j denotes the east, west, north and south neighbour finite volumes. And jA and jdl are the face area and the distance between the

present and j finite volume. Using opc )(ρ instead of pcρ

enhances the stability of the fixed grid calculation as the heat

capacity is a function of oPs and remains constant during an iterative calculation. The boundary conditions for the heat transfer are also given in Fig. 2, those are )( TTkq hpfb −= , (10a)

)( 44 TTq hpt −= σ , (10b)

where fk is the film heat transfer coefficient at the bottom of

the tray, hpT is the temperature of heating plates, and σ is the

Stefan-Boltzmann constant (5.67×10−8). Mass Transfer in Packing Phase

As shown in Fig. 2, the packing phase serves as a transport path for the water vapor generated from particles. Then, the conservation equations for water vapor and inert gas are written as

genwpwww mn

T

p

tR

M,1 &=⋅∇+

∂∂

mε , (11a)

01 =⋅∇+

∂∂

iii

T

p

tR

Mmε . (11b)

Here, wm and im are the mass flux of water vapor and that of

inert gas through the packing phase, genwm ,& is the water vapor

generation from a particle, and pn is the number density of

particles defined as

3

1,

116

pp

dn

επ

−= . (12)

The mass flux equations for wm and im in the sorption- sublimation model are derived from the dusty-gas model [11] considering binary diffusion, Knudsen diffusion, and viscous flow as

( )twww

w ppkpkRT

M ∇+∇−= 21m , (13a)

( )tiii

in ppkpkRT

M ∇+∇−= 43m , (13b)

where the coefficients are ( εεεK12K21K DxDxDm += )

µεε

εε

εε

εε

εε

εε K

DD

DD

pkk

DD

DDk

DD

DDk

mwi

iw

tmwi

iwi

mwi

wwi ++

==+

=+

=K

KK42

K

K3

K

K1

1,,

.(14) The Carman-Kozeny correlation is used to determine flow permeability of the packing phaseK as [12]

( )2

2

3

1180pdK

εε

−= . (15)

The effective binary diffusivity εwiD is given as [13]

t

wi p

TD

334.261034.4 −×=τεε [m2/s], (16)

and here τε / is included to consider the reduction of diffusion area and the elongation of diffusion path length. For general

porous media, 5.1/ ετε = . The effective Knudsen diffusivity for species is defined as

2/1

K 5.48

=

iporei M

TdD

τεε , (17)

where the mean pore diameter pored is defined as

ppore ddε

ε−1

=3

2. (18)

Boundary conditions for mass transfer in the packing phase

are given as

∞= ,ww pp , ∞= ,ii pp for top surface, (19a)

0== iw mm for bottom surface. (19b)

Figure 3. Contribution of Poiseuille flow, slip flow and Knudsen flow to total mass flux

Mass Transfer in Particle Phase

The transport of inert gas in the particle phase can be neglected. Then, the mass flux of water vapor through the particle phase can be expressed as [14,15]

ww

w pRT

MD ∇Ω−= Knτ

εm , (20)

and Ω is a correction factor. Figure 3 shows the contribution of Poiseuille flow, slip flow, and Knudsen flow to total mass flux with respect to the Knudsen number. It is observed that the correction factor Ω approaches to 1 when the Knudsen number is larger than 1. The mean free path for water vapor at −20°C and 100 Pa is around 37 µm, which is considerably larger than the pore size in the particle phase, observed in Fig. 1(b). In this study, the pore diameter 2,pored was assumed to be 1 µm. Then, the Knudsen

number easily exceeds 10 in the particle phase, and the mass flux of water vapor is the same as the Knudsen flow with Ω=1. If the isothermal diffusion in the particle phase and the thermodynamic equilibrium in the ice front are assumed, the vapor generation from a particle genwm ,& can be derived as

(from the mass transfer resistance in a spherical coordinate)

( )

−

−=

1,

Kn,25.1

2

,11

)(2

pice

wsatw

w

genw

dd

pTpRT

MD

mπε

& , (21)

where the saturation vapor pressure satwp at the ice front is [16]

537.12/5.266310)( +−= Tsatw Tp [Pa]. (22)

RESULTS AND DISCUSSION

Freeze Drying Characteristics

The freeze drying characteristic of particle packing was studied first by choosing a skim milk solution [8] as a model drying material. The properties of the skim milk solution and the simulated operation conditions are summarized in Table 1, where the parameters with underlines are the base condition.

Table 1. Physical properties of a skim milk solution [8] and simulated operation conditions

1ε 0.4, 0.5, 0.6 l 5, 10, 15 mm

2ε 0.785 1,pd 10, 15, 20 µm

Dρ )1(328 1ε− kg/m3 2,pored 1 µm

Fρ )1(1048 1ε− kg/m3 sh∆ 2840000 J/kg

pDc 2590 J/kg-K ∞,wp 1 Pa

pFc 1930 J/kg-K ∞,ip 4 Pa

Dk )1(05.0 1ε− W/m-K hpT 253.15 K

Fk )1(12.2 1ε− W/m-K fk 10 W/m2-K

µ )]650/([1048.18 5.17 +× − TT kg-m/s

Figure 4. The simulated results for the primary freeze drying of a spherical particle packing; (a) drying curve, (b) temperature histories, and (c) pressure histories. Figure 4(a) shows the reduction of free water content during the primary drying stage. The primary drying is observed to be completed in about 24 hours when l = 10 mm. That primary drying time is relatively larger than expected, considering the packing porosity. The 1ε of 0.5 means that the amount of free water that should be removed during the primary drying stage is half of that in the freeze drying of frozen solution, e.g., freezing drying of a skim milk solution in a tray. The relatively long primary drying time is mainly due to the limitation of conductive heat transfer in the particle packing. The 1ε of 0.5 reduces the effective thermal conductivity to half from dense product, as indicated in Eq. (7). In Fig. 4, the drying is rather fast at the beginning but it becomes slower and slower later on. Considerable change in the drying rate is different from the relatively constant drying rate

observed during the freeze drying of frozen solution. This is related with the development of dried region near the bottom of the tray observed in Fig. 5. The particles near the bottom of the tray dry relatively fast and form a less conductive dried region. Note that the primary source of sublimation energy is the conduction from the bottom heating plate, and also the conductivity of frozen region is about 40 times larger that that of dried region. Thus, the dried region near the bottom surface acts as an insulating layer blocking the heat transfer from the bottom heating plate.

The average product temperature is plotted in Fig. 4(b) along with the temperature at the top and the bottom surfaces. All the temperatures are asymptotically approaching the heating plate temperature of 253.15 K. The temperature at the top surface is highest in spite of relatively small radiation heat transfer there, because the insulating dried region grows. However, the temperature difference inside the product seems to be small, less than 5°C throughout the process.

The partial pressures of water vapor and inert gas at the bottom surface are shown in Fig. 4(c). The inert gas pressure remains almost constant, similar to its pressure in the drying chamber ∞,ip = 4 Pa. However, the water vapor pressure

continuously increases with time to overcome the increased mass transfer resistance according to the thickness of the dried region near the top surface. The vapor pressure in turn increases the sublimation temperature or the product temperature.

The distribution of ice saturation at several instances is shown in Fig. 5(a), where a thin dried layer is observed to grow near the bottom surface (at 0.01 m). The water vapor generated in the small dried layer near the bottom surface seems to be redistributed as observed by the ice saturation higher than 1 there. Note that 1>s may be interpreted as the deposition of ice around particles.

The vapor pressure distribution in Fig. 5(b) is composed of two segmented lines, one for the dried region to the left and one for the frozen region to the right. Simulation shows that the water vapor pressure in the frozen region is almost similar to the saturation vapor pressure, which corresponds to the local temperature, and the difference is less than 0.1 Pa.

The temperature distribution in the packing phase is plotted in Fig. 5(c), where two or three lines comprise a temperature distribution. It is observed that a short but steep line exists near the bottom surface for 12>t hr, and this is correspondent to the development of the dried layer there. The very small thermal conductivity of the dried layer seems to induce very steep gradient in temperature distribution, or reduces the heat transfer rate significantly. The maximum temperature difference inside the product is assured to be less than 5°C.

Effect of Structural Parameters

The effects of structural parameters on the drying rate were studied, which were the product height, the particle diameter, and the packing porosity. In Fig. 6, larger product height requires longer drying time. At the same time, long drying time results in large thickness of the dried layer near the bottom surface of the product, as shown in Fig. 5(a). And this in turn decreases the drying rate. Due to these feed-back effects, the drying time is a non-linear function of the product height. For example, the primary drying time for l = 5 mm is about 8 hours but that for l = 10 mm is about 24 hours. That is, the overall primary drying rate decreases by about 33 % during the height change from 5 to 10 mm.

Figure 5. The simulated results for the primary freeze drying of a spherical particle packing; (a) the distribution of ice saturation, (b) vapor pressure, and (c) temperature.

Figure 7 shows the effect of particle diameter on the freeze

drying rate. In case of the freeze drying of frozen solution, the pore diameter is dependent on the ice crystal size which even reaches 50 µm due to relatively low freezing rate or long freezing stage. In case of freeze drying of spherical particles, however, the pore diameter is dependent on the particle diameter by Eq. (18) and thus the mass transfer resistance in the packing phase is a function of the particle diameter. In Fig. 7, the effect of the particle diameter is significant, e.g., increasing the particle diameter from 10 to 20 µm reduces the primary drying time by about 8 hours from 29 to 21 hours. However, too large particle diameter will not much enhance the mass transfer rate once the binary diffusion becomes dominant with small Knudsen number. In addition, at larger particle diameter, the mass transfer resistance inside particles becomes limiting.

Figure 8 shows the effect of packing porosity on the primary

drying rate. The observed shorter primary drying stage is partly due to the reduction of absolute amount of product, as shown by initial free water content at t = 0. However, the overall drying rate is also reduced when the packing porosity is decreased. Similar to the effect of the particle diameter, larger packing porosity generally results in smaller mass transfer resistance, and thus help increase the drying rate.

Figure 6. The effect of product height l on the freeze drying rate.

Figure 7. The effect of particle diameter 1,pd on the freeze

drying rate.

Figure 8. The effect of packing porosity 1ε on the freeze drying rate.

Effect of Volumetric Heating

As observed in previous simulations, the freeze drying of particle packing with conventional heating method generally experiences the limitation in the heat transfer rate. This is due to the high packing porosity and also due to the development the dried layer near the bottom surfaces. Volumetric heating such as microwave or infrared heating can enhance the heat transfer rate by skipping the conduction through the low conductivity regions.

In Fig. 9, the effect of volumetric heating was studied by varying the amount of heating, where lines without indices denote the base simulation results. Figure 9(a) shows that the drying time can be reduced significantly by using the volumetric heating method. However, higher volumetric heating results in higher product temperature and lower product quality. It is observed that the volumetric heating should not exceed more than 2500 W/cm2 to maintain the product temperature below −20°C for the base case.

Figure 9. The effect of volumetric heating on (a) the freeze drying rate, and (b) the maximum product temperature.

CONCLUSION

The freeze drying of spherical particle packing was studied using an analysis model, developed to taken into account the heat and mass transport though the porous media with different length scales, the packing phase and the particle phase. The freeze drying of spherical particles required relatively long drying time due to the limitation in heat transfer. In addition, the dried layer that grew near the bottom surface also reduced the heat transfer rate or increased the drying time. The parametric

studies on the effect of several structural parameters showed that smaller product height, larger particle diameter, and higher packing porosity were favorable for the enhancement of overall drying rate. The volumetric heating was found to increase the drying rate but it also increased the product temperature. The present numerical prediction is believed to be a useful tool for the process design of the freeze drying of spherical particles.

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