SIMULATION OF BOVINE PERICARDIUM FREEZE-DRYING USING MATHEMATICAL
MODELING
Camila Figueiredo Borgognoni1*, Joyce da Silva Bevilacqua2, Ronaldo N. M. Pitombo1
1
School of Pharmaceutical Sciences– University of Sao Paulo, Brazil; 2Institute of Mathematics and
Statistics – University of Sao Paulo
Abstract: Freeze-drying is a drying method used for heat-sensitive products where the
water is removed from the frozen material by sublimation. Mathematical modeling
applied to freeze-drying has been used to simulate the process and predict the behavior of
the product during its drying. Moreover, it is useful for predicting variables that cannot be
measured by experiment. Bovine pericardium was freeze-dried to obtain temperature
profiles. Microscopic observations of the bovine pericardium dried layer were employed
to determine sublimation rates. The effects of the chamber temperature on sublimation
rates were verified. A one-dimensional differential-equation model based on heat and
mass transfer was fitted to experimental data. It was shown that theoretical data are in
agreement with those obtained by experiment.
Key words: Freeze-drying, simulation, mathematical modeling, sublimation rate, bovine
pericardium.
1.
INTRODUCTION
Freeze-drying is a gentle method for the dehydration of sensitive products such as pharmaceuticals,
foodstuffs and biological materials (Liapis et al., 1996; Aimoli, 2007). It is the process where freezable
water is removed from frozen material first by sublimation then by desorption of unfrozen water under
reduced pressure. Hence, freeze-drying comprises three stages: freezing, primary drying and secondary
drying.
Freeze-drying is a lengthy process because of the slow drying rate and the slow pressures applied. As
such, a nonoptimized cycle can use up much time, energy and, consequently, money.
Mathematical modeling could help improve understanding of the freeze-drying process. It is used to
analyze drying rates and to predict the surface-temperature profile (Liapis and Litchfield, 1979; Millman,
1985). A series of works using heat- and mass-transfer modeling of the freeze-drying process are
presented in the literature (Liapis and Litchfield, 1979; Pikal, 1985; Millman, 1985; George, 2002).
Determinations of sublimation rates in primary drying can be achieved by various techniques. Some
authors determined the primary drying rates by stopping the freeze-drying process at a fixed point and
measuring the weight lost during this partial drying (Overcashier et al., 1999; Searles et al., 2001;
Chakraborty et al., 2006). Pikal (1983) described a microbalance technique to determine sublimation rates
by evaluating the resistance of different dried materials. Zhai (2003) demonstrated a method to determine
rates of sublimation by freeze-drying microscopy – a technique that permits visual observation of the
product during freeze-drying. They determined values of effective diffusion coefficients (Deff) by
observing drying rates and comparing to their theoretical data.
The aim of this work was to determine the drying rates during the primary drying of bovine pericardium
freeze drying from the speed of the moving interface between the dried and the frozen layer. In addition,
the effect of chamber temperature on the drying rate was taken into account and, finally, data predicted by
mathematical modeling and experimental data were compared.
2.
MATERIALS AND METHODS
2.2 Methods
2.2.1 Freeze-drying
Freeze-drying runs were performed in an FTS Systems model TDS-00209-A microprocessor-controlled
tray dryer (Dura-Stop, Dura-Dry MP). Samples 3 cm in diameter were placed on Petri dishes and frozen
to –50°C (a thermocouple was inserted into each sample to determine its final temperature). Thermal
annealing was applied by heating the frozen samples to –20°C and holding this temperature for one hour.
Finally, the samples were cooled to –50°C and the samples freeze dried. Primary drying was conducted at
a shelf temperature of –5°C (heating rate 1.5°C.min–1 /min) and pressures of 160, 320 and 480mTorr.
Temperature was measured by thermocouples inserted into the samples.
2.2.2 Freeze-drying microscopy
The freeze-drying microscope (Biopharma Technology Ltd, UK) consisted of a small freeze-drying
chamber containing a temperature-controlled stage, a vacuum pump and a microscope (Olympus BX-51)
equipped with a condenser extension lens (Linkam) and a monochrome video camera (Imasys, Suresnes,
France). A heating rate of 1.5°C.min–1 and pressures of 160, 320 and 480mTorr were used to evaluate the
effect of theses parameters on sublimation rates. A 3-mm-diameter sample of bovine pericardium was
placed on the temperature-controlled stage between two glass cover slides.
Thermal annealing was achieved by heating the frozen samples to –20°C and holding this temperature for
one hour. Finally, samples were cooled to –50°C and freeze dried. Sublimation occurred from the outer
margins to the centers of samples. The sublimation rate (Nt) was determined by measurement of the dried
layer every minute.
2.2.3 Mathematical model
The model, based on the work of Liapis and Litchfield, 1979, consists of an unsteady state energy balance
in the dried and frozen region.
TI
t
TII
x
2
(1)
TII , X≤x≤L
x2
(2)
N t c pv (TI )
x
I c pI
2
I
I
TI , 0≤x≤X
x2
The one-dimensional system is shown in Figure 1. The sample was placed on one glass slide and covered
by a second, exposing only the edge of sample to drying. In the freeze-drying process, a drying layer
grows creating a sublimation interface – a region that separates dried and frozen regions of the material –
that moves toward the center of the sample (x=L). The heating shelf provides energy for sublimation and
determines the rate of heat transmission (q) to the product; the heat flux passes through the frozen layer
and pressure is kept constant during drying. Water vapor diffused through the porous dried layer to be
collected in the condenser. The primary drying finishes when there is no drying layer remaining in the
matrix.
The mathematical model uses the following assumptions:
1) One-dimensional heat and mass flows, normal to the interface and the surface, are considered.
2) Sublimation occurs at an interface parallel to, and at a distance X from the surface of the sample.
3) The thickness of the interface is taken to be infinitesimal.
4) The frozen region is considered to remain at a temperature equal to the interface temperature, of
uniform thermal conductivity, density and specific heat.
5) In the porous region, the solid matrix and the enclosed water vapor are in thermal equilibrium.
6) Only the borders of the sample are not insulated to heat and mass transfer.
x=L
I
I
I
x=0
q
Nt
Fig. 1. Schematic of the system under consideration.
In the dried layer effective parameters have been considered to be independent of space.
The initial and boundary conditions are:
TI
TII
T0
at t=0, 0≤x≤L
(3)
q
kI
TI
x
at x=0, t>0
(4)
TI
TII
at x=X, t>0
(5)
Tx
The heat transfer from the shelf to the edge of the sample is due to thermal radiation only, since the
pressure applied is very low.
For radiation only,
q
F (TS4 TI4 ) at x=0, t>0
(6)
The sublimation rate is modeled by the expression:
Nt
n
II
X
t
(7)
The sublimation rate (Nt) was calculated from the thickness of the dried layer by freeze-drying
microscopy. The densities ( ) were measured using a pycnometer and a digital balance. The porosity (n)
was determined using a Hydrosorb 1000 (Quantachrome Instruments). The bovine pericardium thermal
properties were measured using a differential scanning calorimeter (DSC–50; Shimadzu).
3.
RESULTS AND DISCUSSION
The thickness of the dried layer was determined as a function of time for different chamber pressures by
visual observations of bovine pericardium freeze-drying (Figure 2). The interface velocity could be
calculated from derivation of thickness data. Figure 3 shows interface velocities as a function of time for
different chamber pressures. As the pressure was raised, the interface velocity decreased.
Sublimation rates were obtained by equation 6 for different chamber pressures (Figure 4). It was observed
that the sublimation rates are similar for the three chamber pressures. The sublimation rate is a variable
mainly influenced by the porosity and the density of the material. Small variations in interface velocity
had little effect on sublimation rate. These data were used in the partial differential equation that was
solved using MATLAB software.
Thickness of the dried layer x 103 (m)
1,60
160 mTorr
320 mTorr
480 mTorr
1,40
1,20
1,00
0,80
0,60
0,40
0,20
0,00
0
500
1000
1500
2000
2500
Time (s)
Fig. 2. Thickness of the dried layer versus time.
0,07
160 mTorr
320 mTorr
480 mTorr
Interface velocity x 105 (m/s)
0,06
0,05
0,04
0,03
0,02
0,01
0,00
0
100
200
300
400
500
Time (s)
Fig. 3. Interface velocity versus time.
0,000035
160 mTorr
320 mTorr
480 mTorr
Sublimation rate x 105 (kg/(m2 s))
0,00003
0,000025
0,00002
0,000015
0,00001
0,000005
0
0
500
1000
1500
Time (s)
Fig. 4. Sublimation rate versus time
The freeze-drying of bovine pericardium was simulated. Figure 5 shows the bovine pericardium
temperatures during freeze-drying at 160mTorr obtained by mathematical modeling. These data have
been verified by comparing with the experimental result. The comparison between the temperature
profiles predicted by the model and those obtained experimentally is shown in figure 6. The data
correlated well. Table 1 gives the parameters used in the simulations.
Fig. 5. Bovine pericardium temperature versus time during freeze-drying by mathematical modeling.
Fig. 6. Comparison between experimental and simulated temperature profiles for bovine pericardium.
Table 1. Parameters values.
Parameters
cpv
cpI
cpII
ρI
ρII
αI (=kI/ (ρI cpI))
n
kI
kII
1
Values
1.883 kJ/(kg K) 1
0.02186 kJ/(kg K)
0.01025 kJ/(kg K)
0.38 103 kg/m3
1.30 103 kg/m3
0.000065 m2/s
0.16
0.00054 kJ/(m s K)
0.00131 kJ/(m s K)
Results published by Bhattacharya et al., 2003 and Chakraborry et al., 2006.
4.
CONCLUSIONS
It has been observed that the sublimation rate is a little influenced by the chamber pressure of bovine
pericardium freeze-drying. The mathematical model used to predict the temperature profile during the
primary drying of the bovine pericardium freeze-drying fitted well with the experimental data.
5.
ACKNOWLEDGMENTS
The financial support of the Foundation for Research Support of the State of Sao Paulo (FAPESP) is
gratefully acknowledged.
6. NOMENCLATURE
Nt
T
t
x
X
cp
L
k
n
q
F
sublimation rate (kg/(m2 s))
temperature (K)
time (s)
space coordinate
interface position (m)
specific heat (kJ/(kg K))
sample radius
thermal conductivity (kJ/(m s K))
porosity
heat flux (kg/(m2 s))
emissivity factor
Subscripts
I
dried region
II
frozen region
v
water vapor
o
initial value
s
shelf
Greek Symbols
ρ
density (kg/m3)
α
thermal diffusivity (m2/s)
σ
Stefan-Boltzman constant
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