Drying Technology: An International Journal ...

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Mathematical Modelling of the Primary and Secondary Drying Stages of Bulk Solution Freeze-Drying in Trays: Parameter Estimation and Model Discrimination by Comparison of Theoretical Results With Experimental Data a

H. Sadikoglu & A. I. Liapis

a

a

Department of Chemical Engineering and Biochemical Processing Institute University of Missouri-Rolla , Rolla, Missouri, 65409-1230, U.S.A. Published online: 07 May 2007.

To cite this article: H. Sadikoglu & A. I. Liapis (1997) Mathematical Modelling of the Primary and Secondary Drying Stages of Bulk Solution Freeze-Drying in Trays: Parameter Estimation and Model Discrimination by Comparison of Theoretical Results With Experimental Data, Drying Technology: An International Journal, 15:3-4, 791-810 To link to this article: http://dx.doi.org/10.1080/07373939708917262

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DRYING TECHNOLOGY.15(3&4). 791-810 (1997)

Downloaded by [Gebze Yuksek Teknoloji Enstitïsu ] at 04:04 30 January 2014

MATHEMATICAL MODELLING O F THE PRIMARY

AND SECONDARY DRYING STAGES O F BULK SOLUTION FREEZE-DRYING IN TRAYS: PARAMETER ESTIMATION AND MODEL DISCRIMINATION BY COMPARISON O F THEORETICAL RESULTS WITH EXPERIMENTAL DATA

H. Sadikoglu and A. I. Liapis' Department of Chemical Engineering and Biochemical Processing Institute University of Missouri-Rolla Rolla, Missouri 65409-1230 U.S.A.

Kev Words: Freeze-drying of pharmaceuticals; lyophilization of pharmaceuticals.

ABSTRACT A mathematical model was constlucted and solved in order to describe quantitatively the dynamic behavior of the primary and secondary drying stages of the freeze-drying of pharmaceuticals in trays. The theoretical results were compared with the experimental data of the freeze-drying of skim milk, and the agreement between the experimental data and the theoretical results is good. Detailed model calculations have indicated that the contribution of the removal of hound (unfrozen) water to the total mass flux of the water removed during primary drying, is not significant. For this reason, it was found that one could not

'Author to whom correspondence should be addressed. 791 CopyrighlO 1997 by Marcel Dekkcr. Inc.


792

SADIKOGLU AND LlAPlS

use effectively the experimental data of the primary drying stage to perform studies on parameter estimation and model discrimination for determining the functional form of the mechanism that could be used to describe the removal of bound water. The constitutive equation and the values of the parameters of the mechanism that could describe, for a given material of interest, the removal of bound water, should be determined by using the experimental data of the secondary drying stage. The results of this work indicate that one could neglect the mechanism of the removal of bound water in the mathematical model during primary drying, without introducing a significant error in the theoretical predictions. Two different mechanisms for the removal of hound water were examined, and it was found that a tirstdrder rate desorption mechanism could describe the dynamic behavior of the removal of bound water satisfactorily. The model presented in this work has a very important practical advantage when compared with other models, because its expressions do not require detailed information about the structure of the porous matrix of the dried layer of the material being freeze-dried.

Many pharmaceutical products when they are in solution deactivate over a period of time. Such pharmaceuticals can preserve their bioactivity by being lyophilized soon after their production, so that the molecules are stabilized [1-4]. In the primary drying stage. the frozen solvent is removed by sublimation. As the solvent (ice) sublimes, the sublimation interface, which started at the outside surface (Figure I), recedes, and a porous shell of dried material remains. The vaporized solvent (water) vapor is transported through the porous layer of dried material. During the primary drying stage, some of the sorbed water (unfrozen water) in the dried layer may be desorbed [I-51. The time at which there is no more frozen layer (i.e., there is no more sublimation interface) is taken to represent the end of the primary drying stage. The secondary drying stage involves the removal of solvent (water) that did not freeze (this is termed sorbed or bound water) [I-51. The secondary drying stage starts at the end of the primary drying stage, and the desorbed vapor is transported through the pores of the material being dried. In this work, a mathematical model is presented and solved that employs the mass transfer mechanisms of the dusty-gas model [5-8) to describe the transport of the solvent (water) vapor and inert gas in the pores of the dried layer. The very important practical advantage of the dusty-gas model is that it does not require detailed information about the structure of the porous matrix of the


BULK SOLUTION FREEZE-DRYING IN TRAYS

793

dried layer, and thus, the structural parameters C,,, C,, and C, that characterize the mechanisms of intraparticle convective flow (convective flow of a gas in the pores of the dried layer), Knudsen diffusion, and bulk diffusion, respectively, could be obtained from the analysis of the data of independent experiments [8] of moderate complexity. Furthermore, two different rate mechanisms for the removal of bound (unfrozen) water are examined in this work. The theoretical results of this work are compared with the experimental freeze drying data of skim milk.

2. MATHEMATICAL MODEL This work considers bulk solution freeze-drying in trays [1.2.5.9] and in Figure I a material being freeze-dried in a tray is shown. The thickness of the sides and bottom of the tray, as well as the material from which the tray is made, are most often in practice such that the resistance of the tray to heat transfer could be considered to be negligible [1,2,5,9]. Heat q, could be supplied to the surface of the dried layer by conduction, convection, or radiation from the gas phase; this heat is then transferred by conduction to the frozen layer. Heat q,, is supplied by a heating plate and is conducted through the bottom of the tray and through the frozen material to reach the sublimation interface or plane. The magnitude of the amount of heat q, in the vertical sides of the tray is much smaller [1,2.5.9] than that of q, or q,; q, represents the amount of heat transferred between the environment in the drying chamber and the vertical sides of the tray. Because the contribution of q, is rather negligible when compared to the contribution of q, and q,, the contribution of q, to the drying rate will not be further considered [1,2,5,9]. The terms N, and N, in Figure I represent the mass flux of water vapor and the total mass flux, respectively, in the dried layer. The total mass flux is equal to the sum of the mass fluxes of water vapor and inert gas. N, = N, + N,. where N,,denotes the mass flux of the inen gas. In the following sections, the mathematical models that are used in this work to describe the dynamic behavior of the primary and secondary drying stages of the freeze drying process, are presented.

In the primary drying stage sublimation occurs as a result of heat being conducted to the sublimation interface through the dried (I) and frozen (0layers


x=


FREEZE-DRIED

LAYER I

0

I

x=x

\ \ FROZEN MATERIAL U

x=L

FIGURE 1; Diagram of a material on a tray during freeze-drying (the variable X denotes the position of the sublimation interface (front) between the freezedried layer (layer I)and the frozen material (layer IT)).

[1.2.51. Following the assumptions reponed in the work of Liapis and Bruttini [I] and Millman et a]. [51, the energy balances in the dried (0 and frozen (U) layers and the material balances in the dried layer (I),are as follows:

The term JC&t in Equations (I) and (3) accounts for the change in the concentration of sorbed or bound water with time. Different rate mechanisms have been considered [1.2.5,9] lo describe the change in the concenuation of


795


bound water with time. Two different rate mechanisms are considered in this work and their expressions are as follows:

In Equation (Sa), the variablec,; represents the weight fraction of sorbed water in the solid which would be in local equilibrium with the partial pressure of water vapor, p,. The study presented in this work uses skim milk as the model material. and the functional form of C.; used in the calculations was constructed from the procedure reported in the work of Aguilera and Flink [17]. The parameter k, in Equation (5a) represents the solid film mass transfer coefficient thal characterizes the linear driving force (C,; -C,,) of the rate mechanism for the removal of bound water. In Equation (5b). the parameter k, represents the desorption rate constant of the linear rate mechanism that could be used to describe the desorption (removal) of bound water. The dusty-gas model equations [S-81 were used to develop the following expressions for the mass fluxes N, and Ni,:

N. I"

= - -

ap, + api" k+ k,pi" RTI[3a~ [ a ax))

The contribution of thermal mass diffusion to the mass fluxes, is insignificant by comparison of the contributions of the mass transfer mechanisms [I-5. 101


796

SADlKOGLU AND LlAPlS

included in Equations (6) and (7). This has led us to exclude the mechanism of thermal mass diffusion from the expressions of the model presented in this work. The expressions for N, and N, given by Equations (6) and (7), respectively, are inserted in Equations (3) and (4). while the expression for N, (N,=N,+N,) in equation (I) is obtained from the sum of the expressions given by the right-handside of Equations (6) and (7). In the dried layer, effective parameters are considered which include the physical properties of both the gas and solid, which have been considered to be independent of space. The initial and boundary conditions of Equations (1)-(4) and (5a) or (5b) are as follows: T,=Tn=T,=To

C,,

= C&

at

at

t=0, O s x s L

t=O

, OsxsL

and,

for radiation heat transfer to the upper dried surface,

(8)

(10)



For radiation only, q, =

OF(G -(T,(I,L)Y)

,

t>o

For a thin film between the frozen material and lower plate,

The value of the film thermal conductivity, k,, could be estimated from the expressions in [9.1 I], and suitably adjusted [9] to account for the lowered pressure in the freeze-drying process.

The variable p, is the chamber water vapor pressure, usually determined by the condenser design and assumed constant within the drying chamber. The variable Po denotes the total pressure (Po=p,. + pi,) at x = 0, and is usually considered to be approximately equal to the total pressure in the drying chamber. The mathematical model is completely specified by a material balance at the interface (x = X) which defines the velocity of the interface, V, as

The initial condition for Equation (21) is as follows: X = O at t = O


798

SADIKOGLU AND LIAPIS

Equations ( I ) - (22) represent the mathematical model that was used, in this work, to describe the dynamic behavior of the primary drying stage, and this model involves a moving boundary which is represented by the time varying position of the sublimation interface (X=X(t)). External transpon resistances can be easily incorporated into this model by including the expressions developed by Liapis and Litchfield [12]. However, in a well-designed freeze-dryer [1,2], the external mass and heat transfer resistances are not significant [I-5,9,10]. The equations of the model presented above were solved by using the numerical method developed by Liapis and Litchfield 1131, and Millman 1141. Two limits may possibly be reached during the primary drying stage. First, the surface temperature. T,(t,O), must not become loo high because of the risk of thermal damage; the value of T,(t.O) must be kept below the scorch temperalum [1,2] of the material being dried. Second, the temperature of the interface. T,, and the temperature T,(t,L) (the temperature at x=L) must be kept below the melting point [1.2] of the frozen material being dried. The end of the primary drying stage occurs when the position of the moving sublimation interface is at X=L; this condition implies that at the end of the primary drying stage there is no frozen (Q layer, and therefore, there is no moving sublimation interface.

2.2 Seeondarv Dwine Staee

ln the secondary drying stage, there is no frozen ([I)layer, and thus, there is no moving sublimation interface. The secondary drying stage involves the removal of bound (unfrozen) water. The thickness of the dried layer is L, and the energy balance in this layer is is follows:

The initial and boundary conditions of equation (23) are T , = y ( x ) at I=[,., , OsxhL


799


for radiation heat transfer to the upper dried surface,

The continuity equations for water vapor and inen gas are given by Equations (3) and (4). while the equation for the removal of bound water is given either by expression (5a) or (5b). The initial and boundary conditions for the material balance equations of the secondary drying stage, are given by the following expressions:

pin = O(x)

at

t=tX=, ,

= v(x)

at

i=tx=,

C,

OsxsL

(29)

, OsxsL

(30)

The functions y(x), 6(x), O(x) and v(x) provide the profiles of T,, p,, p, and C, at the end of the primary drying stage or at the beginning of the secondary drying stage; these profiles are obtained by the solution of the model equations for the primary drying stage. It should be noted at this point that during the secondary


800


drying stage. the temperature everywhere in the sample (OsxsL) should be kept below the scorch temperature [l.2] of the material being dried. The model equations for the secondary drying stage were solved by the method of orthogonal collocation [13.15.16]. External transport resistances can be easily incorporated into the model equations by including the expressions developed by Liapis and Litchfield [12]. But, as was discussed in the previous section (for the model equations of the primary drying stage), in a welldesigned freeze-dryer [ 1.21 the external resistances should not be controlling in determining the drying rate

3. RESULTS AND DISCUSSION Skim milk is considered in the freeze-drying study of this work and was selected because it could be considered as a complex pharmaceutical product in the sense that it contains enzymes and proteins. The pilot plant freeze-dryer and the experimental procedure used for the freeze-drying experiments, are presented in the work of Liapis and Bmttini [I]. The theoretical results for the freeze-drying of skim milk were obtained by solving simultaneously Equations (I)+) and (21) for the primary drying stage, and Equations (3)-(5) and (23) for the secondary drying stage. The values of the parameters as well as the expressions employed in the evaluation of certain parameters of the theoretical model, are presented in Table I. The values of the parameters were estimated from data obtained from independent experimens [1.2.4.8.9.17] and by matching numerous sets of the experimental freeze-drying data with the predictions of the theoretical model. The melting. T,, and scorch. T,,,, temperatures of skim milk were -10•‹Cand 60•‹C, respectively [I]. In Figures 2 and 3 the experimental data and the theoretical results for the amount of water removed at various drying times during the primary drying stage. are presented, for the cases when (i) the removal of bound water during the primary drying stage is considered, as shown in Figure 2, and (ii) the removal of bound water is not considered during the primary drying stage, as indicated in Figure 3 where k, = k, = 0. The agreement between the experimental and theoretical data in Figures 2 and 3 is, for all practical purposes, good, when one considers the complexity of the freeze-drying process during the primary drying stage. By comparing the results in Figure 2 when Equation (5a) is used to describe the mechanism of the removal of bound water (k, = 6 . 6 1 10.'~") ~ and when expression (5b) is employed for the desorption of bound water (k,=


801


TABLE 1: Parameter values and expressions

Parameters C,,, m' Cl, m c 2

kJkg-K Cp,, kJkg*K C,,, kJkg.K C kg waterkg solid C kg waterkg solid D,,?, kg.m/sl k,, kW1m2.K k,, kW1m.K k,, kW1m.K L, m p:, Nlm2 ph. N/m2 p:, Nlm2 p,. Nlm2 Po. N/m2 TO, K Tm K Tw. K f(Tx), N/m2 C,

A , kJkg AH, i d k g E

kg/m.s PI.. kg/m3 PI. kg/ml Po* kg/m3 o. kWlm2.K'

P

Values and Ex~ressions 7.219~10"' 3.85583x104 0.921 1.6166 2.59 1.93 0.6415 exp (2.3[1.36-0.036(T-'P)])1100

0.000143931(~~(1/~.+1/~,.))~' 1.5358xIO"P 1 . 4 1 2 ~ 1 0 ~ ( ~ , + ~ , ) + 2 .lo4 165~ (0.48819~,)+0.4685xlO" 0.02 4.00 4.00 1.07 1.07 5.07 233.15 313.15 313.15 133.3224[exp(-2445.5646mx+8.2312lloglO(Tx)0.01677006~~+1.20514xl0~'~~-6.757169)~ 2840.0 2687.4 0.785 18.4858[~~'~1(~~+650)] 215.0 212.21 1030.0 5.676~10'"


802 0.8

PRIMARY DRYING STAGE

m

x


@

o

0.8

-

EXPERIMENTAL DATA THEORETICAL RESULTS WITH k;68lxlO.' s' THEORETICAL RESULTS WITH 1 6 = 6 . 4 8 ~ 1 ~ ' 9 . '

0

2

4

6

8

10

12

TIME, hours

FIGURE 2: Amount of water removed versus time during the primary drying stage of the freeze-drying of skim milk. In this case, the theoretical model accounts for the removal of bound water during primary drying

0.8 0

x OI

EXPERIMENT

d 0.6 0

5LL

58 gb-

0.4

0.2

0

I

4

00 0

2

4

6

8

10

12

TIME. hours

FIGURE 3: Amount of water removed versus time during the primary drying stage of the freeze-drying of skim milk. In this case, the theoretical model does not account for the removal of bound water during primary drying.


803


6.48~10-'s-'),it becomes apparent that the difference between the theoretical results is quantitatively not significant, and furthermore. the value of k, is only about 1.97% smaller than the value of k,, although k, and k, characterize different functional forms (mechanisms) for the removal of bound water. Even when an expression given by Equation (34)

is considered for the desorption rate constant, k,, the theoretical results obtained using the expression given by Equation (5b) are, for all practical purposes, similar to those obtained when k, is constant and equal to 6 . 4 8 ~10-'s-'. The results in Figure 3 indicate that when the removal of bound water is not considered during the primary drying stage, the agreement between the experimental and theoretical results is, for all practical purposes, as good as when the removal of bound water during the primary drying stage is considered (Figure 2). A comparison of the results in Figures 2 and 3 indicates that the contribution of the removal of bound water to the total mass flux of water removed during the primary drying stage, is not significant, and for this reason one could not use effectively the experimental data of the primary drying stage to perform studies on parameter estimation and model discrimination for determining the functional form of the mechanism that could be used to describe the removal of bound water. Furthermore, the results in Figures 2 and 3 indicate that if the removal of bound water during the primary drying stage is not considered and the term aCJat in Equations (I). (3). and (5a) or (5b) is set equal to zero, the predictions of the mathematical model with dCJ& = 0 would, for all practical purposes, agree with the experimental data as satisfactorily as when the term dCJ& is considered to be non-zero in Equations (1). (3). and (5a) or (5b). In Figures 4 and 5 the experimental data and the theoretical results for the amount of water remaining in the sample at various drying times during the secondary drying stage, are shown, for the cases when (a) the mechanisms of mass transfer of water vapor and inert gas in the pores of the dried layer consider Knudsen diffusion, bulk diffusion, and convective flow (Q,= 7.219x10~"m2), as shown in Figure 4, and (b) the mechanisms of mass transfer of water vapor and inert gas in the pores of the dried layer account for Knudsen and bulk diffusion but not for convective flow (C,, = O), as indicated in Figure 5. It should be noted again at this point that during the secondary drying stage, the water vapor in the


804 0.14

SECONDARY DRYING STAGE 0


\ \

I

0.00 12

EXPERIMENTAL OATA THEORETICAL RESULTS WlTH k,=8.2110.' 9.' THEORETICAL RESULTSWTH ~6=78xl0'1.' IN THIS CASE. THE VALUE OF THE PARAMETER C,. IS EQUAL TO 7 2 1 ~ x 1 0 . "m'

1 14

18

18

20

22

TIME. hours

FIGURE 4: Amount of r e s i d u a l w a t e r venus time during the s e c o n d a r y drying stage of the freeze-drying of s k i m milk. convective

In t h i s case, the c o n t r i b u t i o n of

flow in t h e p o n s of the d r i e d l a y e r i s c o n s i d e r e d

SECONDARY DRYING STAGE EXPERIMENTALOATA

- THEORETICAL RESULTS WlTH ~ 8 . 2 ~ 1 i 0" THEORETICAL RESULTS WlTH k=7.8rl0'

a.

IN THIS CASE. THE VALUE OF THE PARAMETER Co, IS EQUAL TO ZERO

0.00 I 12

14

18

18

20

TIME, hours

FIGURE 5: Amount of r e s i d u a l w a t e r v e n u s lime during the s e c o n d a r y drying s t a g e of the freeze-drying of s k i m milk. In t h i s case, the contribution of convective flow in the p o r e s of t h e dried l a y e r i s not c o n s i d e r e d .



805

pores of the dried layer is formed from the removal of bound (unfrozen) water from the phase of the solute (skim milk). Therefore, the contribution of the removal of bound water to the total mass flux of water removed during the secondary drying stage is dominant, and thus, the term aCJat in Equations (I). (3). and (5a) or (5b) cannot be zero through the duration of the secondary drying stage, and this implies that neither k, nor k, can be equal to zero during the secondary drying stage. The agreement between the experimental and theoretical data in Figures 4 and 5 is. for all practical purposes, good. The values of k, and k, in Figures 4 and 5 that provide a good fit between experiment and theory, are more than two orders of magnitude larger than the values of k, and k, obtained during the primary drying stage (Figure 2). and this clearly indicates the importance of the mass transfer mechanism of the removal of bound water during the secondary drying stage. By comparing the results in Figure 4 when Equation (5a) is used to describe the mechanism of the removal of bound water and when expression (5b) is employed for the desorption of bound water, it becomes apparent that the difference between the theoretical results is quantitatively not significant, and furthermore, the value of k, is only about 4.88% smaller than the value of k,, although k, and k, characterize different mechanisms for the removal of bound water. The results in Figure 4 suggest that, for the freeze-drying system studied in this work, there is insignificant difference in the predictions of the theoretical model when Equation (5a) or (5b) is used for the mechanism of the removal of bound water. I•’ one, for a given material of interest, could describe satisfactorily the removal of bound water during the secondary drying stage by using the mechanism described by Equation (5b), then one would not have to construct an expression for the equilibrium variable C,,: it should be noted here that the construction of an expression for c,; requires tedious and time consuming adsorptionldesorption equilibrium experiments [1,2,4,5]. The results in Figure 4 suggest that one, for a given material of interest, could use, as a first approximation, the expression given by Equation (5b) to describe the removal of bound water; the parameter k, could be considered to have a constant value or could be given by the expression in Equation (34). A comparison of the results in Figures 4 and 5 indicates that the effect of neglecting the convective flow (C,, = 0 for the theoretical results in Figure 5) of water vapor and inter gas in the pores of the dried layer through the duration of the secondary drying stage, is not significant in determing the dynamic behavior of the drying process during secondary drying. This result suggests that, for the system studied in this work, the three major mass transfer mechanisms during secondary drying were as


806

SADIKOGLU AND LIAPlS

follows: (I) removal of bound water from the phase of the solute. (2) Knudsen diffusion, and (3) bulk diffusion. In general, it is recommended that the mechanism of convective flow should be included in the inathematical model used to describe the dynamic behavior of the secondary drying stage, since one, for a given system of interest, cannot accurately estimate a priori the effect of the contribution of the convective flow on the drying rate during secondary drying; if, of course, numerous comparisons of theoretical results with experimental data from the system of interest indicate that the contribution of the mechanism of convective flow in the pores of the dried material is not significant during secondary drying, then one could neglect the mechanism of convective flow in the mathematical model used to describe the dynamic behavior of the secondary drying stage. 4. CONCLUSIONS AND REMARKS A mathematical model was constructed and solved in order to describe quantitatively the dynamic behavior of the primary and secondary drying stages of the freeze-drying of pharmaceuticals in trays. The theoretical results were compared with the experimental data of the freeze-drying of skim milk, and it was found that the agreement between the experimental data and the results obtained from the theoretical model is, for all practical purposes, good. The model presented in this work employs the mass transfer mechanisms of the dusty-gas model [5-81 to describe the transport of the water vapor and inert gas in the pores of the dried layer, and thus, this model has a very important practical advantage when compared with other models reported in the literature since the expressions of the dusty-gas model do not require detailed [ I 4 1 information about the structure of the porous matrix of the dried layer. Two different mechanisms for the removal of bound water were examined, and it was found that the first-order rate desorption mechanism given in equation (5b) could describe the dynamic behavior of the removal of bound water, for the freeze-drying system studied in this work, sksfactorily. Funhermore, detailed model calculations performed in this work have indicated that the contribution of the removal of bound water to the total mass flux of the water removed during primary drying, is not significant; therefore, one could neglect the mechanism of the removal of bound water in the mathematical model during primary drying, without introducing a significant error in the theoretical results obtained from the solution of the model.


807

ACKNOWLEDGMENT


We are pleased to acknowledge helpful discussions with Dr. R. Bruttini of Criofarma-Freeze Drying Equipment. Turin, Italy.

NOTATION C, C, C,, C, C,

, . D

weight fraction of bound water in dried layer heat capacity constant dependent only upon stmctlire of porous medium and giving relative D'Arcy flow permeability constant dependent only upon structure of porous medium and giving relative Knudsen flow permeability constant dependent only upon structure of porous medium and giving the ratio of bulk diffusivity within the porous medium to the free gas bulk diffusivity, dimensionless free gas mutual diffusivity in a binary mixture of water vapor and inen gas

D,; D d Ed activation energy (Equation (34)) f a ) water vapor pressure-temperature functional form presented in Table I thermal conductivity bulk diffusivity constant =C~D&KJ(C,D;, +L P ) self diffusivity constant = (K,KJ(Cp& + KmP)) + (COilpm) bulk diffusivity constant = Cp:hK,J(C2D;'+&P) desorption rate constant of bound water (Equation (5b)) constant in Equation (34) film thermal conductivity (Equation (17)) solid film mass transfer coefficient (Equation (5a)) Knudsen diffusivity, I(. = C,(RT, /M,) Knudsen diffusivity, Y, = C,(RT, /M,)OJ mean Knudsen diffusivity for binary gas mixture,

''

(k" = Y.Y.

+

y i m

sample thickness molecular weight total flux (N, = N, inen gas flux

+ Nan)


808 P

Po


P. Pi. q R

To T t

v X X

.Y Yi.

total pressure in dried layer drying chamber pressure at surface of dried layer partial pressure of water vapor partial pressure of inen gas energy flux universal gas constant sample temperature at t=o temperature time velocity of interface position of frozen interface space coordinate mole fraction of water vapor mole fraction of inen gas

Greek Letters

a E

AH,

AH, p

p o

o

*

e f in L LP

m

thermal diffusivity voidage fraction enthalpy of sublimation of frozen water enthalpy of vaporization of sorbed water viscosity density Stefan-Boltzmann constant

initial value at time zero equilibrium value

effective value film inen value at x =L lo,wer plate melting



mx scor w

X UP o I

U

mixture scorch water vapor interfacial value upper plate surface value dried region frozen region REFERENCES Liapis, A. 1.. and Bmttini. R., 1994, "A theory for the primary and secondary drying stages of the freeze-drying of pharmaceutical crystalline and amorphous solutes: Comparison between experimental data and theory", Separations Technology. 4, pp. 144-155. Liapis. A. I., and Bmtlini, R.. 1995. "Freeze Drying", in A S . Mujumdar (ed) "Handbook of Industrial Drying" (Second Edition). Marcel Dekker, h c . New York and Basel, pp. 309-343. Liapis, A. I., and Bmtlini, R.. 1995, "Freeze-drying of pharmaceutical crystalline and amorphous solutes in vials: Dynamic multi-dimensional models of the primary and secondary drying stages and qualitative features of the moving interface", Drying Technology, 13,pp. 43-72. Liapis, A. I., Pikal, M. J.. and Bmttini. R.. 1996, "Research and development needs and opponunities in freeze drying". Drying Technology. 14,pp. 1265-1300. Millman, M. J.. Liapis, A. I.. and Marchello, J. M., 1985. "An analysis of the lyophilization process using a sorption-sublimation model and various operational policies". AIChE J.. 2, pp. 1594.1604. Jackson. R., 1977, 'Transpon in porous catalysts", Elsevier Scientific Publishing Company, Amsterdam. The Netherlands. Mason, E. A., and Malinauskas, A. P., 1983. "Gas transpon in porous media- the dusty-gas model". Elsevier. New York, N. Y.. U.S.A. Gloor. P. I.. Crosser, 0.K., and Liapis, A. I.. 1987, "Dusty-gas parameters of activated carbon adsorbent particles", Chemical Engineering Communications. 59. pp. 95-105. Bmltini. R.. Rovero, G., and Baldi. G.. 1991. "Experimentation and modelling of pharmaceutical lyophilization using a pilot plant". Chemical Engineering Journal. 45. pp. B67-877.


SADIKOGLU AND LlAPlS Mellor. J. D.. 1978, "Fundamentals of freeze drying", Academic Press, London. England. Carslaw. H. S.. and Jaeger, J. C.. 1976, "Conduction of heat in solids", Clarendon Press, Oxford, England. Liapis. A. I., and Litchfield, R. 1.. 1979. "Optimal control of a frecze dryer I: Theoretical development and quasisteady-state analysis", Chemical Engineering Science. 3,pp. 975-981. Liapis. A. I.. and Lifchfield, R. J., 1979, "Numerical solution of moving bounduy transport problems in finite media by orthogonal collocation". 2, pp. 615-621. Millman. M. J., 1984. 'The modeling and control of freeze dryers", Ph.D. Dissertation. University of Missouri-Rolla, Rolla Missouri. U.S.A. Villadsen. J.. and Michelsen. M. L.. 1978. "Solution of differential equation models by polynomial approximation". Prentice Hall, Englewood Cliffs. New Jersey, U.S.A. Holland. C. D., and Liapis, A. L, 1983. "Computer methods for solving dynamic separation problems". McGraw-Hill. New York. N.Y., U.S.A. Aguilera, I. M., and Flink, 1. M., 1974, "Technical note: Determination of moisture profiles from temperature measurements during freeze drying", J. Food Tech., 9, pp. 391-395.