Determination of Mass and Heat Transfer Parameters During Freeze-Drying Cycles of Pharmaceutical Products

16 Pages·00.194 MB·English

by  Hottot

RESEARCH
Determination of Mass and Heat Transfer Parameters
During Freeze-Drying Cycles of Pharmaceutical Products
A. HOTTOT, S. VESSOT, AND J. ANDRIEU*
Laboratoire d’Automatique et de Ge´nie des Proce´de´s –LAGEP-UMR Q 5007 CNRS UCB Lyon1-CPE, Baˆt.
308G, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France.
ABSTRACT: The principal aim of this study was to evaluate the water vapour mass transfer resistance of the dried
layer and the vial heat transfer coefficient values of a pharmaceutical product during the primary drying period.
First, overall vial heat transfer coefficient values, Kv, were determined by a gravimetric method based on pure ice
sublimationexperiments.Thus,itwaspossibletosetupamapofthetotalheatfluxreceivedbyeachvialthroughout
the plate surface of our pilot scale freeze-dryer. Important heterogeneities were observed for the vials placed at the
plate edges and for the vials placed at the center of the plate.
As well, the same gravimetric method was also used to precisely determine the influence of main lyophilization
operating parameters (shelf temperature and gas total pressure) or the vial types and sizes on these overall heat
transfer coefficient values. A semi-empirical relationship as a function of total gas pressure was proposed.
The transient method by pressure rise analysis (PRA method) after interrupting the water vapour flow between the
sublimation chamber and the condenser, previously set up and validated in our laboratory, was then extensively used
withanamorphousBSA-basedformulationtoidentifythedriedlayermasstransferresistancevalues,R ,theicefront
p
temperature,andthetotalheattransfercoefficientvalues,Kv,withorwithoutannealingtreatment.Itwasprovedthat
this method gave accurate and coherent data only during the first half of the sublimation period when the totality of
the vials of the set was still sublimating.
Thus, this rapid method allowed estimation of, on line and in situ, the sublimation front temperature and the
characterizationofthemorphologyandstructureofthefreeze-driedlayer,allalongthefirstpartofthesublimationperiod.
TheestimatedsublimationtemperaturesshownbythePRAmodelwereabout2°Clowerthantheexperimentalvalues
obtained using thermocouples inserted inside the vial, in accordance with previous data given by this method for
similar freeze-drying conditions.
As well, by using this method we could confirm the homogenization of the dried layer porous structure by annealing
treatment after the freezing step.
Furthermore, frozen matrix structure analysis (mean pore diameter) using optical microscopy and mass transfer
modelling of water vapour by molecular diffusion (Knudsen regime) allowed, in some cases, to predict the
experimental values of this overall mass transfer resistance directly related to the freeze-dried cake permeability.
KEYWORDS: Freeze-drying, Pharmaceutical freeze-drying, Annealing, Heat and mass transfer, PRA model
Introduction timization of the whole process cycle is based mainly
on a deep knowledge and thorough analysis of subse-
Freeze-drying is a complex process based on coupled quent processes such as nucleation and state diagram
mass and heat transfer phenomena. Consequently, op- properties. During the primary drying period, the sub-
limation kinetics are controlled either by heat transfer
flux from the shelf and from the surroundings towards
the ice sublimation front inside the vial, or by the
Corresponding author: [email protected]
water vapour mass transfer through the dried layer.
138 PDA Journal of Pharmaceutical Science and Technology
The global parameter characterizing the overall heat Generally, heat transfer by convection was negligible
transfer to one vial is the overall heat transfer coeffi- due to small pore sizes so that only radiation and
cient, named Kv, whereas the total mass flux of water conduction mechanisms were considered in our stan-
vapour is expressed with the mass transfer resistance, dard freeze-drying conditions regarding thermosen-
noted as Rp. sible pharmaceuticals materials. During this study we
first used pure ice sublimation experiments to check
The aim of this study was to determine the values of the uniformity of the actual heat fluxes received by
the heat transfer parameters by means of the PRA each vial all through the plate surface. The mean
method (1) derived from MTM method first proposed values of Kv were then determined for the two main
by Milton et al. (2). As well, this transient method freeze-drying parameters, namely, the shelf tempera-
leads to the ice sublimation front temperature without ture and the total gas pressure for different vial types
anysensors(thermocouples),aninsertionthatstrongly and sizes. Next, during the second part of this work,
modifies the nucleation phenomena, as well as the ice we investigated the applicability of the pressure rise
and dried-layer morphology and the duration of the analysis method, called the PRA method, to noninva-
drying periods. sively identify two important freeze-drying variables,
namely the ice sublimation front temperature and the
The resistance to water vapour flow has also been dried layer mass transfer resistance directly propor-
characterized as a function of dried layer thickness by tional to the water vapour permeability through the
using a microbalance (3). In the case of the KCl dried layer.
system,theseauthorsreportedanonlinearevolutionof
Rpwithrespecttodriedlayerthickness.Kuuetal.(4) Materials and methods
estimated mass transfer coefficients values by using
Powell’s algorithm, proving that only thermal profiles Three types of glass vials were used for determining
and cumulative mass sublimation were sufficient. the overall heat transfer coefficient values: a set of
100, 4 ml tube glass vials (Verretubex); a set of 120,
In a comprehensive study, Overcashier et al. (5) pro- 7 ml vials (Dominique Dutscher) and another set of
vided relationships between water vapour mass trans- 120, 5 ml moulded glass vials (Fisher Bioblock) with
fer resistance and slight product collapse. These au- a more sharply bent and thicker bottom than the two
thors observed the cake structure by SEM and by previous ones.
fluorescence microscopy. The holes fraction (micro-
collapse), which increased with the process tempera- For one freeze-drying cycle with a given vial type, a
ture, decreased the mass transfer resistance during the set of 100 vials (weighed before the cycle) was filled
primary drying stage. with 1 ml of water each. They were then semi-stop-
pered and finally carefully placed in rows on the
Recently, Bru¨lls et al. (6) calculated Kv values by aluminium tray.
identifying product temperatures, based on a physical
2D-model.Theseauthorsunderlinedtheimportanceof For the gravimetric method, the overall heat transfer
the heat transfer mode from shelf to vials whether at coefficient for one vial was calculated based on the
lowpressurewhentheradiativemechanismdominates heat balance after measuring the mass loss after 5
(up to 60%) or at higher pressures from 16 Pa and hours of sublimation and the temperature difference
beyond when heat conduction is predominant. betweentheproductbottomandtheshelftemperature.
The Kv values were then determined by applying
The overall heat transfer coefficient is dependent on equation 2 which neglects the heat accumulation term
three elementary mechanisms (7–11): the radiation in frozen and in dried layers (stationary state hypoth-
phenomena, the heat contact between the vial bottom esis).TheutilisationofpureiceinexperimentsforKv
and the shelf or the plate, and the transfer by conduc- determinations eliminated the dried layer mass trans-
tionthroughthegaslayersurroundingthevialbottom. fer resistance and, consequently, these conditions cor-
Nail (12) first underlined the total pressure effects on responded to a constant sublimation front temperature
global heat transfer fluxes values. This author ex- all along the duration of the sublimation step. It
plained that the gas layer under the bottom vial fre- proved that the reproducibility of the calculated Kv
quently presented resistance to heat transfer limiting. values was around 2%. The repeatability was mea-
Vol. 59, No. 2, March-April 2005 139
Figure 1
Arrangement of vials and thermocouples for PRA measurements
suredbythreerepetitionsofthesamecycle(P(cid:1)6Pa The glass mass in contact with the frozen layer was
and T (cid:1) (cid:2)5 °C) with a set of 200 vials of 4 ml. considered equal to 1 g.
shelf
This coefficient of variation takes account the exper- Eachofthe525unstopperedvialswasfilledwith1ml
imental uncertainty regarding temperature and the of liquid solution which represented 0.084 m2 of total
weight measurements and also regarding all the other sublimation area for a chamber volume of 120 litres;
errors involved during the freeze-drying runs. thus, the ratio volume/sublimation surface was equiv-
alent to 1.43 m, i.e., of the same order of magnitude
The previously described PRA method (1) was imple- thanintheMiltonstudy(2).Thisratiovalueprovedto
mented with our laboratory pilot freeze-dryer, type be sufficient to lead to a significant and a reliable
SMH45(Usifroid,France).Thefreeze-dryerchamber pressure rise up to the end of the primary drying
had a volume equal to 120 L and the total pressure period.
values were measured using a capacitance manometer
MKS Baratron 622 (MKS Instruments, USA). This Twodifferentpressurelevels(P(cid:1)10PaandP(cid:1)26Pa)
sensor was connected to an acquisition system 2700 during the primary drying period were selected. Type
(Keithley Instruments, USA) in order to record all Kthermocouples(typeKthermocouplesweremadeof
necessarydataduringpressurerisetests.Arapidclos- Nickel-Chrome for the positive part and Nickel-Alu-
ing pneumatic butterfly type valve (response time in- minium for negative part) were introduced inside 6
ferior at 0.5 s) was used to isolate the condenser from vials located on the plate array as indicated on Figure
the sublimation chamber. The PRA tests were always 1below,andasevenththermocouplewasplacedinthe
carried out using 1 ml formulations placed in 4 ml gas space of the freeze-dryer chamber. Each thermo-
tubing glass vials having the following configuration couple was carefully fixed at the internal face of each
parameters: vialbottom.Totakeintoaccountthepossibleheatflux
heterogeneities, an average value was adopted as the
● Empty mass: 4.76 g mean experimental temperature value at the glass-
bottom/frozen-layer interface.
● Total height: 37mm
Freeze-drying protocol
● Exterior diameter: 16.25 mm
The two freezing protocols described below were ap-
● Inner diameter: 14.25 mm plied:
140 PDA Journal of Pharmaceutical Science and Technology
● Freezing down to (cid:2)45 °C at (cid:2)1 °C/min and hold- ond term, Kr, expresses the radiative equivalent con-
ing for 2 hours ductance,andthethirdterm,notedKg,correspondsto
the conductance of the equivalent gas layer located
● Freezing down to (cid:2)45 °C at (cid:2)1 °C/min, holding betweenthevialbottomandthefreeze-dryerplate.For
during2hoursandthenannealingtreatmentat-10 the experimental conditions of our work, Kr and
°C during 3 hours (only with PRA method). K values were quite constant and for this reason
contact
were gathered in a single constant, noted K .
Forgravimetricmethodexperiments,theshelftemper- v
ature increased from (cid:2)45 °C to (cid:2)5 °C at 0.5 °C/min
Furthermore, as pointed out by different authors for
and a gas total pressure P (cid:1) 6 Pa was maintained
standard conditions of pharmaceutical protein freeze-
throughout the drying period.
drying (8), Kv values were strongly dependent on the
PRArunswerecarriedoutonlyduringtheprimary total gas pressure and could be correlated by the
drying period with the following parameters val- following semi-empirical relationship:
ues: gas total pressure (cid:1) 26 PA; heating rate at
0.25 °C/min from (cid:2)45 °C to (cid:2)5 °C ; holding time Kp (cid:1) Pchamber
Kg(cid:1) , (4)
until all thermocouples reached the preset shelf 1(cid:3)Kd (cid:1) Pchamber
temperature.
withKp(cid:1)(cid:6)(cid:1)(cid:7)o, (5)
Results and Discussion
andKd(cid:1)L (cid:1)(cid:3)(cid:6)(cid:1)(cid:7)o/(cid:8)o(cid:4), (6)
The vial mean heat transfer coefficient, Kv, is defined sep
by equation 1 below from the total heat flux dQ/dt
where P represents the total gas pressure of the
received by one vial: chamber
freeze-drying chamber (Pa), (cid:6) an adimensionnal form
factor, characterizing the heat transfer efficiency,Kg
dQ
dt (cid:1)KvAs(cid:3)Tshelf(cid:2)Tbottom(cid:4), (1) theheattransfercoefficientbyconductionthroughthe
gas layer (W/m 2 /K), (cid:7)o the free molecular gas heat
where T and T represent, respectively, the conductivity at 0°C (W/m2/Pa/K) equal to 2.06 W/m2/
shelf bottom
temperature of the coolant fluid circulating in the Pa/K for pure water vapour, (cid:8)o the thermal conduc-
shelves and the mean temperature measured by the tivity of gas at atmospheric pressure equal to 0.018
thermocouples fixed at the bottom of the monitored W/m/K for water vapour,and L the equivalent gas
sep
vials as previously indicated. layer thickness between the bottom of the vial and the
plate (m).
Adopting the steady state hypothesis, the overall heat
transfer coefficient value, Kv, could be calculated by
1. Experimental Kv Values by Gravimetric Method.
equation 2 based on the overall sublimation rate for
one vial, noted m(cid:1), obtained by weighing.
1.1 Heat flux cartography inside the sublimation
chamber.
(cid:5)H (cid:1)(cid:3)m(cid:1)(cid:4)
Kv(cid:1) s , (2)
A (cid:1)(cid:3)T (cid:2)T (cid:4)
s shelf bottom After having checked the axial symmetry of the plates
ofthefreeze-dryingchamberandtheinvarianceofthe
where (cid:5)H , is the latent sublimation heat equal to 2
s Kv values according to the different shelves, experi-
323 KJ/kg.
ments were carried out to determine the Kv values at
different locations of the plate and, then, for different
As well, as a first approximation, the Kv coefficient
temperatures and different pressures for the three
could be decomposed as follows in three elementary
conductances (7) : types of vials studied.
Kv(cid:1)K (cid:3)Kr(cid:3)Kg(cid:1)Kc(cid:3)Kg. (3) Figure2belowdepicts ahistogramoftotalheat trans-
contact
fer coefficient values with respect to their location on
The first term, K , corresponds to the contact the plate for tubing glass vials of 7 ml at the total
contact
conductance of the vial with the plate, while the sec- pressureP(cid:1)6PaandatashelftemperatureT(cid:1)-5°C.
Vol. 59, No. 2, March-April 2005 141
Figure 2
Distribution of overall heat transfer coefficient values. Tubing vials of 7 ml Aluminium tray (Total pressure (cid:1)
6 Pa; Shelf temperature (cid:1) (cid:2)5°C)
First, we can observe the equality of Kv values be- Furthermore, two thermocouples were inserted and
tween the front and the back of the freeze-dryer plate, carefully fixed as already indicated. The sublimated
though these values are higher than the ones at the ice mass during these runs varied between 20 to 40%
plate center where they remain around 10 W/m2/K. for the different freeze drying cycles and for the
These differences could result mainly from the wall different vial types.
significant radiative flux existing in laboratory pilot
freeze-dryers. On average, for the standard conditions Table 1 gathers the Kv values for the different exper-
described above and for the 7 ml glass vial experi- imental conditions.
ments,thevialslocatedattheshelfcenterpresentheat
A maximum variation of 4% of Kv values was ob-
transfer coefficient values 20% lesser than those lo-
served for each type of vial. However, this deviation
cated on the edge of the plate. These data are in
tended to increase with the shelf temperature (in spite
accordance with those obtained by Rambhatla et al.
of the the measurement error). This observation could
(13).
be explained by the fact that the freeze-drying cham-
ber’s mean temperature increased slightly when the
1.2 Overall heat transfer coefficient for different
experimental conditions
TABLE I
The Kv values for one vial were calculated from
equation2andtheexperimentalvalueofm(cid:1).However, Heat Transfer Coefficient Values, Kv (W/m2/K) as
a Function of Shelf Temperature for Three Types
thelinearizationassumptionoftheradiativetermmust
of Vials. P (cid:1) 6 Pa
be checked because the applicability of this last rela-
tionship relies on it. This is why, in order to validate Tubing
this hypothesis, the Kv values were investigated by Types of Vials Moulded Vials Vials
carrying out experiments at three different shelves
Shelf Temperature (°C) 5 ml 7 ml 4 ml
temperatures, namely (cid:2)18 °C, (cid:2)5 °C,(cid:9) 5 °C for 3
vials types at the total gas pressure P (cid:1) 6 Pa. All the (cid:2)18 14.1 8.4 10.8
(cid:2)5 15.1 8.7 11
vials were carefully weighed after freeze-drying runs
5 15.2 9.1 11.1
to obtain the sublimation rates.
142 PDA Journal of Pharmaceutical Science and Technology
Figure 3
Overall heat transfer coefficient values as a function of total pressure. Three types of vials (T (cid:1) (cid:2)5°C).
shelf
shelf temperature increased. Thus, the collision inten- Figure3representstheexperimentalandtheestimated
sity between the gas molecules increased which re- Kv values as a function of total gas pressure. We
sulted in an increase of heat transfer by conduction observed that these Kv coefficient values for the tub-
through the gas layer. Besides, it is worth noting that, ing vials are in close agreement with those found by
regarding the Kc values from Table II and the Kv Pikal, namely Kv (cid:1) 8.5 W/m2/K, close to Kv (cid:1) 7
values from Figure 3, for the mean value of the total W/m2/Kforthesamevialsizeandthesametypeasin
gas pressure (P(cid:1) 40 Pa), the gas-conduction thermal our study. These curves show a clear non-linear rela-
conductance in the gas layer between the vial bottom tionshipwithtotalpressure.Besides,theKvvaluesare
and the shelf (Kg term) represents 50% of the vial’s higher for the moulded glass vials than for the tube
total thermal conductance. This percentage varies glass vials. Nevertheless, according to the Pikal data
largely with the total pressure from its highest value, (7), moulded vials presented Kv values lower than
70%atP(cid:1)80Pa,toitslowestvalue,30%atP(cid:1)6Pa. those of the tube vials not because of the high curva-
So, for freeze-drying conditions at low total gas pres- ture of their bottoms but their tube glass vials did not
suresandduetothelowthermalconductivityofgases
have the same geometrical dimension as those in our
inthisdomain,themainheattransferresistanceforthe
study. Finally, these data indicate the importance of
heat flux towards the sublimation front is located in
precise consideration of vial type and size during
the thin gas layer between the vial bottom and the
freeze-drying optimisation and modelling. Table II
shelf. In the frequent case of sublimation times con-
gathers the estimated parameters values used for cor-
trolled by the heat transfer phenomena, the reduction
relating equations 4–6.
of the operating costs related to this drying time relies
on the optimisation of this thermal resistance by de-
termining an optimum for the total gas pressure or, if
TABLE II
possible, by improving the shape of the vial bottom.
Heat Transfer Parameters for Three Types of
Vials (Equation 3-6)
Finally, we concluded that, in our experimental con-
ditions, using equation 1 we can express the first Vials Volume 5 ml 7 ml 4 ml
approximation of the total heat flux, including the
Kc (W/m2/K) 10.4 7.1 7.9
radiative component, as a function of the difference
Kp (W/m2/K/Pa) 0.48 0.25 0.3
between the shelf temperature and the product tem-
Kd (Pa(cid:2)1) 0.0121 0.0048 0.0031
perature.
Vol. 59, No. 2, March-April 2005 143
TABLE III TABLE IV
L sep and Heat Transfer Form Factor Values for Experimental L sep Values for Two Tubing Vials
Three Type of Vials Sizes
Vials Volume 5 ml 7 ml 4 ml Tubing Vials Volume 7 ml 4 ml
L sep (mm) 0.47 0.36 0.2 L (mm) 1.4 0.7
max
(cid:6) (adimensional) 0.23 0.12 0.145 Experimental L (mm) 0.4 0.24
sep
Concerning the parameter Kc (contact and radiation motion to allow enough time for the hardening reac-
mechanisms), its values were higher in our case of tion. To evaluate the maximum gap between the vial
moulded vials than for Pikal’s corresponding experi- bottom and the horizontal plane, we measured the
mental condition (14). So, this characteristic could rounded bottom centre of the vial as well as the vial’s
explain the higher Kv values for glass moulded vials diameter on its basis. We then calculated the volume
than for tube vials because the lower resistance of the of the spherical segment which allowed estimation of
gas layer was compensated by the higher thermal the height of a cylinder, L , which presented the
sep
resistance by contact and radiation. This is also the sameareaasthesphericalsegment.Inthismanner,we
reason why the moulded vials presented sublimation found a height of 0.6 mm for the moulded vials.
rates faster than the tubing vials (15). Finally, we Concerning our tubing glass vials, it was difficult to
concluded that equation 4 represents a precise and define a spherical segment, so that for this reason the
adequatesemi-empiricalcorrelationforexpressingthe L value was calculated based on the correlation
sep
influence of total gas pressure on Kv values, a rela- proposed by Pikal (7).
tionshipthatwillbeusedinfutureworkforaccurately
implementing the PRA model and for the advanced
We observed that the previous experimental results
modelling of freeze-drying processes.
regarding the gap were generally in close agreement
with equivalent gas layer thickness data estimated
1.3 Evaluation of gas layer thickness and thermal
from Kv values.
efficiency:
2. Measurement of Rp and Kv Values by PRA
Thesetwoparameters’valueswerenecessaryformod-
Method
elling the heat transfer and are defined as follows:
ThePRAmethodproposedbyChouvencetal.(1),has
For the overall heat transfer efficiency, noted (cid:6):
been applied to an aqueous formulation based on
B.S.A(SigmaAldrich,SaintLouis,USA),whichcon-
Kp
(cid:6)(cid:1) , (7) stituted an amorphous system. Each 4 ml tubing glass
(cid:7)o vial (Verretubex, France) was filled with 1 ml of
formulation up to a depth of 7 mm. The PRA runs
and for the gas layer equivalent thickness at the vial were realized by using a rapid closing butterfly valve.
bottom, L :
sep
The ice sublimation front temperature, Ti, and the
Kd product resistance, Rp, were identified, and in a sec-
Lsep(cid:1)(cid:8)o . (8)
Kp ond step the total heat transfer coefficient values, Kv,
were calculated using the following hypothesis:
The mean gap distance between the vial bottom and
theplatewasestimatedbyusingpasteAquasilTMsoft — variation of Kv values during the pressure rise test
putty (Dentsply DeTrey GmbH) used in stomatology
(makingteethimpressions).Forthismeasurement,the — variation of frozen layer thickness during the pri-
paste was mixed with a catalyst and was shaped as a mary drying period
rectangular parallelepiped, and the vial bottom was
then pressed onto it. Next, the vials were carefully — by neglecting desorption term during the primary
removed from the paste that was kept 5 min without drying period
144 PDA Journal of Pharmaceutical Science and Technology
Figure 4
Comparison of experimental and identified ice front temperatures profiles (T (cid:1) (cid:2)5°C and P (cid:1) 26 Pa)
shelf chamber
— taking into account the mass of glass in contact following relationship was used to calculate the thick-
with the frozen product (mass variable with time) ness of the frozen layer, L (t), at different subli-
frozen
mation times.
2.1 Freeze-drying parameters without annealing
(cid:1)
(cid:5)t Kv(cid:3)T (cid:2)T (cid:4)
L (cid:3)t(cid:4)(cid:1)L (cid:2) acq shelf bottom , (9)
Figure 4 depicts the variation of ice front temperature frozen 0 (cid:5)H(cid:10)
s frozen
estimated using the PRA method and also the mean
product temperatures given by the thermocouples in- with the following notations:
troduced inside the vial bottom, at the interface with
the frozen layer. (cid:5)Hs: sublimation enthalpy (J/Kg)
After2hofsublimation,theidentifiedsublimationice L : initial frozen layer thickness (m)
0
front temperature stabilized around -28 °C, that is to
say, 2 °C above the mean product temperature given Kv: total heat transfer coefficient (W/m 2 /K)
by thermocouples. This tendency has been also previ-
ouslyobservedbyChouvencetal.(1)usngamannitol (cid:5)t : mean time between 2 acquisitions (s)
acq
system. This difference was consistently observed un-
til the product temperature reached the shelf temper- (cid:4) : ice density (kg/m3)
frozen
ature.
Figure5describestheevolutionofthecalculateddried
Due to artefacts associated with nucleation phenom- layer thickness as a function of time.
ena and ice morphology resulting from thermocouples
insertion, the vials with thermocouples should have This figure shows that for our investigated BSA based
lower temperature values than those monitored with- formulation, the cake thickness increases linearly as a
outthermocouples(lesssurpercooling,largericecrys- function of sublimation time at a constant rate equal to
tals and greater sublimation rate); therefore, the ob- 10-7 m/s for L(cid:11)0.25 .10(cid:2)2 m, with this rate then pro-
served temperature difference seems logical and in gressively slowing down.
agreement with other experimental observations.
Furthermore, the water vapour mass transfer resistance
Furthermore, the PRA method allowed us to estimate increases linearly as well during the first half of the
the thickness of the dried layer as a function of time sublimationperiod(figure6)withrespecttothedrylayer
and the product mass transfer resistance values. The thickness (if L(cid:11)0.25 .10(cid:2)2 m) with an intercept value
Vol. 59, No. 2, March-April 2005 145
Figure 5
Variationofdriedlayerthicknesswithsublimationtime(T (cid:1)-5°CandP (cid:1)26Pa).Tubingvialsof4ml
shelf chamber
whichisclosetozero.Then,beyondathicknessequalto edge of the plate finished their sublimation earlier than
0.25cm,theidentifiedRpvaluesstronglyincrease(Fig- the vials located at the centre of the plate.
ure 6) from 600 to 1000 kPa.m2.s/kg.
Theoretically, the pressure rise ‘dP/dt’ is proportional
Figure7showsthatthetotalpressureriseduringaPRA to ‘NA /Rp’ as indicated by the following equation of
s
run decreased with the sublimation time, mainly due to the PRA model (1):
an increase of dry layer thickness.Moreover, in the sec-
ondhalfofthesublimationperiod,thisattenuationeffect dP(cid:3)t(cid:4) NART
(cid:1) s v (cid:3)Pi(cid:3)t(cid:4)(cid:2)P (cid:3)t(cid:4)(cid:4)(cid:3)F , (10)
couldalsoresultfromthereductionofthetotalsublima- dt M VR water leak
H2O p
tion front surface due to heat flux heterogeneities inside
the sublimation chamber. As a matter of fact, as previ- where N is the vial number, Tv the ambient tempera-
ouslyindicated,duetoradiativeeffectsofthewallandof ture; P the water vapour pressure (Pa) in the
water
the shelves of the freeze-dryer, the vials located at the chamber, Pi the water vapour pressure at the ice in-
Figure 6
Experimentaldriedlayermasstransferresistanceversusdriedlayerthickness(Tshelf(cid:1)(cid:2)5°CandP (cid:1)26Pa)
chamber
146 PDA Journal of Pharmaceutical Science and Technology
Figure 7
Pressure rise kinetics during PRA run for two sublimation times ‘t’ (T (cid:1) (cid:2)5°C and P (cid:1) 26 Pa)
shelf chamber
terface (Pa), and F the leak rate of the freeze dryer middlevalueoftheinitialfrozenlayerthickness.Asa
leak
(Pa/s). matter of fact, the total mass of sublimated water,
m , was calculated based on the estimated Kv or Rp
exp
Thus,foragivenpressurerisesignal,ifthevialnumbers values by a simple mass balance or heat balance from
is overestimated, the corresponding Rp values are also the following relationships:
overestimated with respect to the real ones. Otherwise,
(cid:2)
the Kv values are proportional to 1/Rp: t(cid:2)end(cid:2)lyo Kv(cid:3)t(cid:4)NA(cid:3)T (cid:2)T (cid:4)
m (cid:1) s shelf bottom (cid:1) dt
exp (cid:5)H
(cid:3)Pi(cid:2)Pc(cid:4)(cid:5)Hs 0 s
Kv(cid:1) , (11)
Rp(cid:3)Tshelf(cid:2)Tbottom(cid:4) (cid:2)
t(cid:2)end(cid:2)lyo NA(cid:3)P (cid:2)Pi(cid:4)
Consequently,thecorrespondingcalculatedKvvalues (cid:1) s Rchapm(cid:3)bte(cid:4)r (cid:1) dt, (12)
should decrease, as shown in Figure 8, which could 0
confirm that the vials located at the edge of the tray
had finished their sublimation. where, ‘t-end-lyo’ is the time at the last PRA run.
Moreover, at the end of the sublimation period, the For a standard freeze-drying cycle corresponding to
thickness of the frozen layer was estimated at the thepreviousoperatingconditions,only200gofwater
Figure 8
Evolution of Kv values during sublimation (T (cid:1) (cid:2)5°C and P (cid:1) 26 Pa)
shelf chamber
Vol. 59, No. 2, March-April 2005 147