fem freeze drying

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LSMI R Comput. Methods Appl. Mech. Engrg. 148 (1997) 105-124

Computer methodsin appliedmechanics andengineering

A computational model for finite element analysis of thefreeze-drying process

W.J. Mascarenhasa, H.U. Akayavby*, M.J. PikalTechnaly sis Inc., 7120 Waldemar Dr ., Indi anapol is, IN 46268, USA

Dept . of M echanical Engineeri ng, Purdue School of Engineeri ng and Technol ogy, IUPUI , Indi anapol is, IN , 46202, USAEli Li ll y and Co., Lil ly Corporate Center, Indi anapolis, IN 46285, U SA

Received 4 June 1996; accepted 7 April 1997

AbstractA brief overview of the freeze-drying process is given, followed by presentation of the governing equations and the finite element

formulation in two-dimensional axisymmetric space. The model calculates the time-wise variation of the partial pressure of water vapor, thetemperature, and the concentration of sorbed water. An Arbitrary Lagrangian-Eulerian method is used to accurately model the sublimationfront of the freeze-drying process. Both the primary and secondary drying stages of the process are modelled. Examples are presented thatvalidate the model and demonstrate representative applications of such calculations.

0 Nomenclaturehc;

CccCO,Clc2Dw in

D .infLf)KwKinKk,k

equilibrium concentration of sorbed water (kg water/kg solid)heat capacity &J/kg K)concentration of sorbed water (kg water/kg solid)initial concentration of sorbed water (kg water/kg solid)constant dependent only upon structure of porous medium and giving relative DArcy flowpermeabilityconstant dependent only upon structure of porous medium and giving relative Knudsen flowpermeabilityconstant dependent only upon structure of porous medium and giving the ratio of bulk diffusivitywithin the porous medium to the free gas bulk diffusivity (dimensionless)free gas mutual diffusivity in a binary mixture of water vapor and inert gasDw.inpwater vapor pressure-temperature functional formheat transfer coefficient (kW/m K)Knudsen diffusivity C, (RT/IV,).~Knudsen diffusivity C, (RT/Mi,)o.5mean Knudsen diffusivity for binary gas mixture y,K,, + yi nK,,,Kwinternal miass transfer coefficient for desorption (s- )thermal conductivity (kW/m K)

* Corresponding author.

00457825/97/ 17.00 @ 1997 Elsevier Science S.A. All rights reservedPI1 SOO45-7825(96:~00078-9


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106 W.J. Muscarenhas et al. I Cornput. Methods Appl. Mech. Engrg. 148 (1997) 105-124

k,k,, k,kLMzN,P inp:PVJp:4rRtTint fTIT ITIPTToT,VxYYinYW

bulk diffusivity constant C,D~,i,Kw /(C,D~,i , + K,,,,P)self diffusivity constant Kw Ki ,/(C,D~,i, + K,,,,P) + (CO, ,u,,,,)bulk diffusivity constant C2Di ,i,Kin /(C,Dt ,,, + K,,,,P)sample thicknessmolecular weight (kg/kg mole)normal vector to a surfacemass flux in the dried layer (kg/m* s)total mass flux in the dried layer (kg/m* s) N, + Ni,total pressure in dried layerpartial pressure of inert gas (Pa)initial partial pressure of inert gas (Pa)partial pressure of water vapor (Pa)initial partial pressure of water vapor (Pa)heat flux (kW/m2)radius (m)gas law constant &J/mole K)time (s)interface temperature (K)temperature in the dried layer (K)temperature in the frozen layer (K)lower plate temperature (K)upper plate temperature (K)initial temperature (K)ambient temperature (K)interface velocity (m/s)spatial coordinate in the thickness directionspatial coordinate in the radial directionmole fraction of inert gasmole fraction of water vapor

Greek symbols4 heat of sublimation of ice (kJ / kg)AH, heat of vaporization of bound water (W/kg)& void fraction in the dried regionlumx viscosity of the binary mixture of water vapor and inert gas in the porous dried layer (kg/m.s)P density (kg/m3)Subscriptse effective valueI dry regionII frozen regionn normal componentX x-componentY y-component

1. IntroductionFreeze-drying, or lyophilization, is a drying process where the solution, normally aqueous, is first frozen,

thereby converting most of the water to ice, and the ice is removed by sublimation at low temperature and lowpressure during the primary drying stage of the process. Sublimation occurs at the interface between the frozenand dry material and starts at the top of the material. The interface moves through the material until only a dried


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porous material remains at the end of primary drying. Water vapor flows out of the material through the pores ofthe material and is then collected on a condenser operating at very low temperatures (i.e., = -65 C). At theend of the primary drying stage, all ice has been removed. Normally, a significant quantity of water remainsassociated with the solute phase and does not freeze. This unfrozen water is removed by desorption in thesecondary drying stage of the process, usually employing temperatures above ambient. Secondary drying iscontinued until the residual water content decreases to the desired moisture content. In reality, some sorbedwater is removed in the primary drying stage. Thus, the two stages occur concurrently.

Since freeze drying is a low temperature process, the process is often used in pharmaceutical and foodindustry to dry materials which suffer degradation or other loss of quality during high temperature drying. Inthe pharmaceutical industry, the solution is normally filled into glass vials, the vials are placed on temperaturecontrolled shelves in a large vacuum chamber, and the shelf temperature is lowered to freeze the product. Aftercomplete solidification, the pressure in the chamber is lowered to initiate rapid sublimation. Process times areoften quite long, and since commercial freeze drying plants are expensive, process costs are relatively high.However, since the commercial value of a batch may approach 1 000 000, maintaining product quality isnormally the most important concern. Although a higher ice temperature produces a shorter, more economicalprocess, excessive ice temperatures may result in severe loss of product quality and rejection of the batch [1,2].Product temperature control is critical to preserve product quality and yet minimize process time. However,product temperature is not normally controlled directly. Rather, the shelf temperature and chamber pressure arecontrolled to control heat and mass transfer, such that the optimum product temperature profile with time isobtained. In practice, the appropriate shelf temperature and chamber pressure conditions are frequentlyestablished empirically in a trial-and-error experimental approach. Theoretical modelling studies, which arepredictive, have corrsiderable potential to guide the experimental studies, thereby decreasing development timeand insuring the design of a process which is optimal and robust.

A number of freeze-drying models have been published in the literature [3-l l] to describe the freeze-dryingprocess. The sublimation model of Liapis and Litchfield [ 1 l] was seen to be more accurate than the uniformlyretreating ice front model of King [5]. The sublimation model was then improved upon [ 12-141 by includingthe removal of bound water in the equations. This model is commonly known as the sorption-sublimationmodel.

Tang et al. [15] extended the dynamic, one-dimensional model described in [14] to model two-dimensionalfreeze-drying in a vial. Ferguson et al. [16] have presented a two-dimensional model based upon the Luikovsystem of partial differential equations. Various numerical methods can be used to solve the governing equations.Liapis and co-workers used a one-dimensional method which immobilizes the moving interface by rewriting theequations in terms of normalized co-ordinates [17]. The method of orthogonal collocation [l&20] was thenused to solve the equations. Ferguson et al. [161 used the finite element method to solve the governing equations.However only one-dimensional examples have been presented in the paper. In all one-dimensional models, thegeometry of the material must be rectangular and also the shape of the interface cannot be a curve.

The two-dimensional model presented in this paper overcomes these disadvantages. The material geometryand the interface can be of any arbitrary shape. The finite-element method with an arbitrary Lagrangian-Eulerian (ALE) scheme for tracking the sublimation front and a two-step rational Runge-Kutta (RRK)integration scheme for unsteady calculations is used throughout the simulation of the process. The two-dimensional model developed in this paper is general and can be used to study a variety of freeze-dryingprocesses.

2. Governing equa.tionsThe governing equations are the coupled mass balance (continuity equations) and the heat balance equations

described by Millman et al. [21]. Following this work [21], the following assumptions were made in thedevelopment of the finite element model.

-Two dimensional mass transfer and heat transfer are considered.- The interface thickness is infinitesimal.- A binary mixture of water vapor and inert gas flows through the dried layer.- At the interface, the concentration of water vapor is in equilibrium with the ice.


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- In the porous region, the movement of moisture is sufficiently slow so that the solid matrix and the gas arein thermal equilibrium.

-The frozen region is considered to have uniform mass and heat transfer properties and to contain anegligible proportion of dissolved gases.

- Dimensional changes in the material are negligible and are neglected.2.1. ass transfer equations

The continuity equations for the dried (I) region shown in Fig. 1 are [13]:

whereDCsw = k&AC,*, C,,)Dt

(1)(2)

NW = -k,VC,, - k,CJPNi, = -k,VC,,, - k,C,JPP = P, + Pi

4)36)7)

In the above equations, D/Dt denotes the material time derivative and d/at denotes the time derivative withrespect to an arbitrary frame of reference. CzWis the concentration of frozen water which is in equilibrium withthe local partial pressure of water vapor and local temperature.

From Eqs. (1) and (4),DC DC,,E*+hr- -V* [k,VC,, + k,C,,,VP]

. I Lr5 TI f'1 -w

i ITop SurfaceY

FrozenMaterialII)

q - heat fluxN I mpu tluxX - axial direction (axis of symmetry)Y I radial directionL - aamplohIcknero

Fig. 1. Schematic of a typical freeze drying problem.


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From Eqs. (2) and (5),DcpinE - = v- [k3VCpin+ k,C,,VP]Dt (9)

Eqs. (1) and (2) are coupled through the mass flux Eqs. (4) and (5). The contribution of the thermal massdiffusion to the mass fluxes is insignificant compared to the contribution by the bulk diffusion terms (masstransfer mechanisms) [8]. Therefore, the thermal mass diffusion terms were neglected. This simplificationconsiderably reduces the computational time since Eq. (1) can now be solved independent of Eq. (2). In thefollowing discussion therefore only the finite element formulation of Eq. (1) is presented. The most general formof the equation however is still maintained.

In two-dimensional axisymmetric space, Eq. (8) can be expressed as

(10)In terms of the partial pressure of water vapor, P,, and partial pressure of inert gas, Pin, the equations are

DP DC,,A, -$+p,-==A 2 -++$(+)] +A3{$[P,(%+%)]an ax+$rPw(~+g]}

whereQYVA2= RT, k,M,A3= RT,

Letting ap, i3PF= ax 9 F2 ~2 a v

F4=Pw 2+-(ap apim \ay ay

Eq. ( 11) then becomes

2.2. Heat transfer equationsEnergy balance for the dried (I) region in two-dimensional axisymmetric space gives

Energy balance for the frozen (II) region in two-dimensional axisymmetric space givesa ar,, 1 a aT,,

+s 7 >I0

(11)

(12)

(13)

(14)

(15)

(lo)


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110 W.J. Mascarenhas et al . I Com put. Methods Ap pl. Mech. Engrg . 148 1997) 105-124

2.3. Concentration equationFollowing Millman et al. [21], the time dependence of the water concentration in the solute phase (i.e. dried

layer) is assumed to be of the form,

17)In reality, it appears [22] that Eq. (17) is only a rough first approximation, at least for many materials. Eq. (17)implies that isothermal water desorption from a freeze dried powder in high vacuum (i.e. C,*, = 0), will be firstorder in that a semi-log plot of water content vs. time will be linear. Experimental plots are, in fact, highlycurved with the desorption rate slowing with increasing time much faster than Eq. (17) would predict. However,if CTW is interpreted as an empirical parameter, dependent upon both temperature and surrounding partialpressure of water, Eq. (17) becomes a much better approximation. The details of this refinement are discussedin another paper to be published by the authors.

3. Boundary conditions,Eqs. (14)-( 17) were solved numerically using the finite-element method and the following general initial and

boundary conditions (Fig. 1):At time t = 0,

P,=PZPi, = Pp,T, = T,, = To (18)Cw = CL

For time t > 0,P, = B, on 4 (19)NW,= fiw on r, (20)T=f on4 (21)9, = 9, on 4 (22)q,+h T-T-)=0 on & (23)

Eqs. ( 19) and (21) represent boundaries with specified pressure and temperature respectively. Eqs. (20) and(22) represent boundaries with specified water vapor flux and heat flux respectively. Eq. (23) represents aboundary with convective heat transfer. The concentration equation is an initial value problem and does not needany boundary conditions.

4. Sublimation interface tracking schemeDuring freeze-drying, the latent heat of sublimation is given off at the sublimation interface. This boundary

condition has to be imposed while solving the set of governing equations. Hence, it is necessary to track thelocation of the interface and then to impose the interface boundary condition at the interface nodes.

The motion of the interface is obtained by studying the heat balance at the interface. In the absence of a heat


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source, the total heat flux into the interface from the dried region is equal to the heat flux from the interface intothe frozen region plus the heat absorbed at the interface due to sublimation. In mathematical form the heatbalance can be expressed as

where(24)

This interface velocity normal to the interface, u,, can be calculated upon solving Eq. (25) and hence the newinterface nodes location can be obtained. A number of numerical techniques have been developed for solvingproblems with moving boundaries. In this paper a moving-grid finite-element method is used. The sublimationinterface is tracked continuously and the interface is treated as a moving-boundary condition. The finite-elementmesh is moved continuously such that the interface position always coincides with an element node. In otherwords, the interface will always lie on an element edge and/or node but never within the element. The interfaceboundary condition is then imposed on the nodes lying on the interface. Hence, the equations for the dried andfrozen region are solved simultaneously coupled with the interface boundary condition.

Various methods, such as deforming grids, variable or coordinate transformation, using special algorithmsnear the interface, etc. can be used to model moving-grid problems. In this paper the arbitrary Lagrangian-Eulerian (ALE) description was used to incorporate the effects of the moving mesh upon the governingequations. This formulation treats the finite-element mesh as a reference frame that may be moving with anarbitrary velocity. In the ALE formulation, the advantages of pure Lagrangian and Eulerian systems arecombined into one description.

The ALE formulation has been widely used to solve a variety of moving boundary problems. Donea [23] andBelytschko [24] introduced the technique to model fluid-structure interaction. Liu et al. [25] and Gosh et al.[26,27] used the formulation in nonlinear solid mechanics problems.

The ALE description introduces a reference configuration that consists of grid points in arbitrary spatialmotion. The material time derivative of any time-dependent physical quantity p is related to the referential timederivative by the expression,

(26)where D/Dtl, is the material time derivative and a/dtl, is the referential time derivative. m, is the materialvelocity and g; is the grid velocity. Eq. (26) is substituted into Eqs. (14)-( 17) to obtain the governing equationsin terms of the referential time derivative a/ at.

5. Finite element formulationThe weak (or variational) form of Eqs. (14)-(17) over a finite element domain Or, respectively are

' ap--gx$$-gl%) +A&~ at +A,gF 2aw aw DC,,+A,~,F,+A,~~F,+P,D~w rhdy (27)w[(A,F, + A$&, + (A$2 + A zh,l dS


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- k,,~n,+k,$&, wdS>(29)

(30)where w represents a finite element weight function and r represents the total boundary of the domain 0. II,and np are the directional cosines of the outward normal IE on boundaries. The boundary terms in Eqs. (27)-(29)can be identified asNormal water vapor Jlux:

NW = NW??, + NW&= (A,F, + AJ&, + (A?, + AF&, (31)

Normal heat jlux:4,, = qxn, + q,n,

= -k,, 2 n, - k,, % n,, (for dry region)= -k,, 2 n, - k,, 2 n, (for frozen region)

(32)

Total normal jhu:N,,, = N,.rn : + N,.vn, (33)

The same order of interpolation is used for all the variables,

Upon substituting Eqs. (31)-(34) into the weak forms, the following finite element equations are obtained,[MPl{~,l = 00 (35)[MT](T) = {If} (36)Wf%&) = {Rc) (37)

whereM;= I A, (qrdxdy (38)P


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MC= I pleCpleq9 r dx dy (for dry region)ne= I no p,,Cp,,qTr dx dy (for frozen region)

M;= I ne q?$r dx dy

R;= - I[air_:Xx,x W DC,,R +N,,~J + PI it?

-A, gxx+gy(apu %)?]rhdy + r. V&dS

RF= I[aq aRe cL; ) x+q -. v y +CPJ, (

aqWxx+N,Iz >

DCsw+AH,P,D~?+P,,CP,, ,Z+P,%>I

% t-tidy

- (q, + Cp,N,,T,)q dS (for dry region)aq a aT,, aT,,I[r ,,qx+,,qy+P,,CP,, ( zx+gyay % r& dY>I

- I q,I J< d.S (for frozen region)re

(39)

(40)

(41)

(42)

(43)

6. Computational detailsFour-noded quadrilateral isoparametric elements with bi-linear interpolation functions are used for the results

presented in this paper. Gauss-quadrature is used in evaluating the nonlinear finite element matrices and vectors.For planar problems, the same equations are solved by setting r = 1 in the axisymmetric formulation.

For time-integration of the coupled systems of nonlinear equations (35)-(37), a two-step rational Runge-Kutta (RRK) scheme is used, which is an unconditionally stable explicit algorithm for linear parabolic equationsproposed by Hairer 11281.For a general discretized parabolic system in the form

tiijUj = Riwhere Mij is the lumped-matrix approximation of the mass matrices in Eqs. (38)-(40) the steps of RRK can bedescribed as in the following. Lettingn+liz - cl;I At =~ , R, =F,where n is the time step and At is the time increment.Step 1.

g& At Fi(u; )Step 2.

g; = At F,(u; + c2gt )

(44)

(45)

(46)


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114 W.J. Mascarenhas er al. I Comput. Methods Appl. Mech. Engrg. 148 (1997) 105-124

where c2 is a solution parameter.

Solution.u, +=u: [2g, (glg)-gj(gfgf)I/(gg) (47)

where gf = b,gf + b,gf, b, + b, = 1, and b,c, G - 1I2 for stability. The scheme is second-order accurate intime if b,c, = - l/2, e.g. c2 = 0.5 and b, = - 1 O.

In order to correctly start the calculations at the initial time t = 0, the interface is assumed to be located at 2%of the total sample thickness. This is needed to have elements of finite thickness in the dry region for the initialtime step. Initially, smaller time increments need to be used to avoid instabilities due to nonlinearities. As thedry region grows, time increments are gradually increased using an empirical formula based on the thicknessratio of the dry region.

7. Numerical examples7 1 Freeze-drying of skim milk: Example I -One-dimensional model

The freeze-drying of skim milk was simulated and compared with results published by Millman et al. [20].The examples presented in this section are one-dimensional problems. Using appropriate boundary conditions,the one-dimensional problems were solved as two-dimensional problems.

Primary and secondary drying of a 3 mm thick (test case I ) and a 6 mm thick (test case 2) slab of skim milkwere studied. The finite element mesh for the 3 mm case along with the boundary and initial conditions is shownin Fig. 2. A 15 X 1 mesh was used for the 3 mm case whereas a 25 X 1 mesh was used for the 6 mm thicksample. The material properties used for the skim milk are given in Table 1. Due to space limitations only the

Pg 52668. T - 303.15 K

T - 263.15 K (prlmaw dryinp). 303.15 K (secondary drying)

side walls are adiabatic @l.mked)Total ThidDles=3mm 0.0 * I15Mesh:15 1 0 S 10

laitbl Ce: T=241.8K, P=S.668Pa Csw=02283kg~t~/kes0lid rii(min)Fig. 2. Finite element mesh with boundary conditions for Example 1.Equidistant points 1-7 are locations used to plot Figs. 4-l 1.

Fig, 3. Example I-Interface position versus time during primary drying of skim milk. The y axis has been normalized with respect to thethickness (3 mm) of the sample.


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Table 1Material properties for skim milk [20]Material SI unitsPrODUtY

Value

C*Cp*CP,CPUC?;Ri.,k 1ek,,kRKWPWI,,

k.I,kg KkJlkg KIdrkg Kkg water/kg solidkg m/skWl m KkWl m Kl/SmlsPaK 263.15K 303.15Id/kg 2791.2k.Ilkg 2791.2_ 0.706kg/m 145.13kg/m2 1058.0

0.44281.67472.5951.9678

0.01 exp(2.3( 1.36 O.O36(T - T)))8.729 X lo-(T,, + T,,, x340.68(12.98 x IO-P + 39.806 X Wh)0.002111.08 x IO-1.429 X 10m4(T0 + TI, , 133.32

3 mm case results are discussed in this paper. The results for the 6 mm case are similar to the 3 mm case and areomitted.

The comparison between the primary and secondary drying times predicted by our model and those publishedin literature [21] is shown in Table 2. Good correlation is seen between the two results. The interface position isplotted as a function of time in Fig. 3 for the 3 mm case. The y-axis has been normalized with respect to thethickness of the sample. An initial interface position of 0.06 mm (2% of total length) was used to properly startthe analysis. Fig. 4 shows the temperature variation with time during the primary drying stage for the 3 mmthick sample at locations l-7 indicated in Fig. 2. The variation of the sorbed water concentration with time atthe same 7 locations is shown in Fig. 5. Since the top of the material (x = 0.0) dries the earliest, the sorbed waterconcentration at any given time is the lowest at this location. Note that although the mesh is moving, the aboveresults have been plotted at fixed locations by interpolating the nodal values. The temperature and sorbed waterconcentration variation along the thickness of the sample at the end of primary drying is shown in Figs. 6 and 7,respectively. The primary drying analysis was terminated when the interface position reached 98% of the totalthickness. The results were then extrapolated in time and space to obtain the results for the start of the secondarydrying stage.

The finite element mesh changes continuously during the analysis. Initially there are 3 elements in the dryregion and 12 elements in the frozen region. As the interface moves down, the thickness of the dry regionincreases. Hence, the thickness of the elements in the dry region increases while the thickness of the elements inthe frozen region decreases. If the element density is maintained constant, very soon elements with bad aspect

Table 2Comparison of test case results with those published in [20]. Drying times in secondsSample thickness Primary drying Secondary dryingL (mm)

Authors Ref. [20] Authors3 13.77 13.47 228.926 55.26 54.07 229.20

Ref. [20]231.82232.82


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5 10 15Tii (mill)

- x-o.0- x.O.167L- xdJ33L- xdJ5OL- xd.667L- x-o.83 L- x-l.OL

Fig. 4. Example l-Temperature versus time during primary drying of 3 mm thick sample of skim milk.

B3Po.22-B;o3MYdP5 0.21 -

i .

- x-O.167L- UonL- x.o.saL-f- pR667L- mo.63L-c- Pl.OL

0.20 -I0 5 10 1

li i nxi 0)Fig. 5. Example I-Sorbed water concentration versus time during primary drying of 3 mm thick sample of skim milk.

ratios will result. In order to avoid such elements, the element density is continuously monitored during theanalysis. Based upon the element thickness, the element density in the individual regions is changed at certainintervals. The total number of elements however is maintained constant. The initial solution for the remeshedgrid is obtained through interpolation of the nodal values of the primary variables from the previous grid. Duringthe secondary drying there is no moving interface and hence there is no remeshing involved.

At the end of the primary drying stage, the temperature of the bottom surface was raised to the materialscorch temperature (303.15 K). The temperature results are shown in Fig. 8. The top and bottom materialtemperature is maintained constant at 303.15 K throughout the analysis. Therefore, the entire material reaches asteady state temperature equal to the top (or bottom) material temperature. The variation of the sorbed waterconcentration with time is shown in Fig. 9. When the final water weight fraction at any point in the materialdropped below the allowable limit (0.05 kg water/kg solid), the analysis was terminated. The temperature andsorbed water concentration variation along the thickness of the sample at the end of secondary drying is shownin Figs. 10 and 11, respectively.


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ao3.1ot 1.90 1 2 1 2i3bnccfromdletop(mm~ DisBncc~themp@nm)

Fig. IO. Example l-Temperature profile at the end of the secondary drying stage.Fig. I 1. Example I-Sorbed water concentration profile at the end of the secondary drying stage.

Table 3Material properties for BST. Material propetties were estimated from expe~men~ data on BST or when BST data were not available, wereassumed to be the same as for skim milkMaterialpropertyC,Cp,CP,CPUCr,L>w.rnk ,eknk.b

SI units

kJ/kg KkJ/kg KkJ/kg Kkg water/kg solidkg ml?kW/m KkW/m Kl/Smis

Value

0.3182.012.595I 9595GAB Theory (Eqs. (48)-(51))8.729 X IO-@,, + T,,, a

0.68( 12.98 X IO-P + 39.806 X IO-*)0.0027639.0 X IO-7.0021 X 10-4(T,, + T,,, ,)05

Pa 133.32T,,, K 268 KAH, kJ/kg 2834.6AH, Id/kg 2499.6e _ 0.95PI kg/m 55.0PI1 kg/m 927.1

where the weight ratio of water to anhydrous solid, W, is given by the GAB equation as

(49)

where P is the partial pressure of water vapor at the interface. P* is the vapor pressure of liquid (super cooled)water at temperature T K (or t C), which in Torr (mm of Hg), is


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In P* = -5750.38/T + 22.573 + ln( 1 - 0.004%) (50)The parameters, C, and K are functions of absolute temperature, T K, while W, is a constant for a givenmaterial, i.e.

W, = constant for given material(51)

For BST, an analysils of experimental water sorption isotherm data gives: W, = 0.0667, k, = 679.0, k, = 70.4.A 18 X 11 mesh was used for the analysis. The finite element mesh along with the boundary conditions is

shown in Fig. 12. The material was freeze-dried in a cylindrical vial of radius 1 cm. The material thickness is2 cm. A two-dimensional axisymmetric analysis was performed to simulate the freeze-drying process. An initialinterface position of 0.04 cm was used to properly start the analysis.

The interface position versus time along three different cross-sections (r = 0.0, r = 0.5 cm, r = 1.0 cm) isplotted in Fig. 13. The y-axis has been normalized with respect to the shortest thickness of the sample (1.92 cm).The primary drying analysis was stopped when any point on the interface is within 2% of the total thicknessfrom the bottom. This was done to avoid elements with zero thickness. The solution was then extrapolated intime and space to obtain the final solution. The interface is seen to move faster near the walls towards the end ofthe analysis. This is because the heat flux into the material from the bottom is higher close to the wall ascompared to the heat flux at the center (Fig. 12).

The variation of temperature at seven different locations along r = 0.5 cm is shown in Fig. 14. The locationsare shown in Fig. 10. The temperature versus thickness at the end of the analysis along three differentcross-sections (r = 0.0, r = 0.5 cm, r = 1.0 cm) is shown in Fig. 15. There is very little temperature variationacross the radius of the sample.

The variation of the sorbed water concentration at seven different locations along r = 0.5 cm is shown in Fig.16. The water concentration along the thickness of the sample at three different cross-sections (r = 0.0,r = 0.5 cm, r = 1.0 cm) is shown in Fig. 17. The solution obtained at the end of the primary drying analysis was

P ' 24.0 9a (prhuy drying)- r1ur-dqmdslt (s=eonduy&yiag)h - 0.0036 Irr/m.r

flrvflrv1)- 0.0= 0.0

h - 0.0623 kU/r.X

Fig. 12. Example 2-Finite element mesh with boundary conditions for freeze drying of BST.


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10

0

g -10Bc20c

o.o+ . I . I 1 . 1 0 2 4 6 8 10 12Time (hours)

Fig. 13. Example 2--Interface position versus time along three different cross-sections during primary drying of BST.

- x4.0- uO.167L- xd33L- -050L- x4.667L-o- x4.83 L- Xd.OL

0 2 4 6 8 10 12Tiie hotus~

Fig. 14. Example 2-Temperature versus time along r = 0.5R during primary drying of BST.

used as the initial condition for the secondary analysis. The boundary conditions for secondary drying are thesame as those for primary drying except for the pressure at the top surface. To accurately reflect physical realityduring a process where total pressure is controlled by an inert gas bleed, the top surface partial pressure ofwater during secondary drying is flux dependent and is no longer fixed. The flux dependence of the water vaporpressure is given by the following equations:

p,=p J,=I, + Ji ,where

1OOO.ONWJw= Mw

(52)

(53)


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1V.J. M ascarenhas et al. I Comput . M ethods App l. Mech. Engrg. 148 (1997) 105-124 121

Fig. 15. Example 2-Temperature profile along three different cross-sections (r = 0, r = 0.5R, r = R) at the end of the primary drying stage.

0.000

--o- x-o.0- rrd.1677-c uo33Lr-xd.soL- xd.667L-o- rd.83Lv x=l.OL

54)

0 2 4 6 8 10 12

Ti OFig. 16. Example 2-Sorbed water concentration versus time along r = 0.5R during primary drying of BST.

is the molar flux of water vapor,

J=;Pis the molar flux of inerts. Eq. (54) is an expression of vacuum pump performance, where for a typical freezedryer, R6 is a conslant equal to 3.84 X lo5 Pasm/mol.

The temperature variation along r = 0.5 cm is shown in Fig. 18. The entire material reaches a constanttemperature after an hour of drying. The temperature distribution at the end of the secondary drying stage isshown in Fig. 19. The variation of the sorbed water concentration with time is shown in Fig. 20. The analysis


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122

0.3

3bf_ 0.22*Y

83P 0.131

0.00.

WJ. M ascarenhas et al . I Comput . M etho ds Appl . M ech. Engrg. 148 (1997) 105-124

Distaoee~thofop(m)

- x4.167L- x433L- XdJOL- x4.667L- x4.63 L

1 2 3 4

Tii (hours)Fig. 17. Example 2-Sorbed water concentration profile along three cross-sections at the end of the primary drying stage.

Fig. 18.Example 2-Temperature versus time along r = 0.5R during secondary drying of BST.29.7

29.5

8

1t- 29.5

20.40.

\

- rd.0I ImUOR- mnl-001

-I1 0.005 0.010 0.015 0.020

Discmcc~t&top(m)Fig. 19. Example 2-Temperature profile along three cross-sections at the end of the secondary drying stage.

was stopped when the average water weight fraction dropped below 2% kg water/kg solid. The sorbed waterconcentration distribution at the end of the secondary drying stage is shown in Fig. 21.

8. ConclusionsA finite-element model to simulate freeze-drying in two-dimensional axisymmetric space has been presented

and used to study the freeze-drying of skim milk and BST. The model accounts for the removal of free andsorbed water and can predict the position and geometric shape of the moving interface. The results presented inthis paper for 3 mm and 6 mm thick samples of skim milk show good agreement with the published results. The


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CKJ. M ascarenhas et al. I Comput . M ethods Appl . M ech. Engrg. 148 (1997) 105-124 123

- ti.167L16 - xdl33L- xd5OL

gt - xd667L$ 2 - x-o.63L

.qSW 6Y4s

8

4

0 0 1 2 3 4

Tii (hours)Fig. 20. Example 2-Sorbed water concentration versus time along r = 0.5R during secondary drying of BST.

0.06

a .8 on?-

0.000 0.005 0.010 0.015 0.020

Fig. 21. Example P.-Sorbed water concentration profile along three cross-sections at the end of the secondary drying stage.

developed finite element model for freeze-drying provides useful information on the entire primary andsecondary stages of the process.

References[I] M.J. Pikal, Freeze-drying of proteins Part 1: Process design, BioPharm 3(S) (1990) 18-27.[2] M.J. Pikal, Freeze-drying of proteins Part 2: Formulation selection, BioPharm 3(9) (1990) 26-30.[3] G. Lusk, M. Karel and S.A. Goldblith, Thermal conductivity of some freeze dried fish Food Technol. 18 (1964) 1625.[4] J.P. Clark and C.J. King, Convective freeze drying in mixed or layered beds, Chem. Engrg. Progress Symp. Series 67( 108) (1971) 102.


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124 W.J. M ascarenhas et al . I Comput . M ethod s Appl . M ech. Engrg. 148 (1997) 105-124

[5] C.J. King, Freeze-Drying of Foods (CRC Press, Cleveland, OH, 1971) l-54.[6] C.C. Cox and D.F. Dyer, Freeze-drying of spheres and cylinders, Trans. ASME C (February 1972) 57.[7] W.M. Massey and J.E. Sunderland, Heat and mass transfer in semi-porous channels with application to freeze-drying, Int. J. Heat Mass

Trans. 15 (1972) 493.[8] D. Meo, Masters Thesis, University of Rochester, New York, 1972.[9] P.F. Greenfield, Cyclic-pressure freeze drying, Chem. Engrg. Sci. 29 (1974) 2115.

[lo] J.D. Mellor, Fundamentals of Freeze Drying (Academic Press, London, UK, 1978).[I I] AI. Liapis and R.J. Litchfield, Optimal control of a freeze-dryer - I: Theoretical development and quasi steady state analysis, Chem.Engrg. Sci. 34 (1979) 975-981.[12] T-Y.R. Sheng and R.E. Peck, Rates for freeze drying, AIChE Symp. Series 73(163) (1977) 124.[13] R.J. Litchfield and A.I. Liapis, An adsorption-sublimation model for a freeze dryer, Chem. Engrg. Sci. 34 (1979) 1085.[I41 A.I. Liapis and J.M. Marchello, Advances in the modelhng and control of freeze drying, in: AS. Majumdar, ed., Advances in Drying, 3

(Hemisphere Publishing Corporation, Washington, DC, 1984) 217-244.[ 151M.M. Tang, AI. Liapis and J.M. Marchello, A multi-dimensional model describing the lyophihzation of a pharmaceutical product in avial, in: A.S. Mujumdar, ed., Proc. Fifth Int. Drying Symposium, 1 (Hemisphere Publishing Corporation, NY, 1986) 57-64.

[16] W.J. Ferguson, R.W. Lewis and L. Tomosy, A finite element analysis of freeze-drying of a coffee sample, Comput. Methods Appl.Mech. Engrg. 108 (1993) 341-352.

[17] A.I. Liapis and R.J. Litchfield, Numerical solution of moving boundary transport problems in finite media by orthogonal collocation,Comput. Chem. Engrg. 3 (1979) 615-621.

[18] J. Villadsen and M.L. Michelsen, Solution of Differential Equation Models by Polynomial Approximation (Prentice-Hall, EnglewoodCliffs, NJ, 1977).

[19] C.D. Holland and AI. Liapis, Computer Methods for Solving Dynamic Separation Problems (McGraw Hill Book Company, NewYork, NY, 1983).[20] M.J. Millman, The modeling and control of freeze dryers, Doctors Thesis, University of Missouri-Rolla, Rolla, MI, 1984.[21] M.J. Millman, A.I. Liapis and J.M. Marchello, An analysis of the lyophilization process using a sorption-sublimation model and

various operational policies, AIChE J. 31 (1985) 1594-1604.[22] M.J. Pikal, S. Shah, M.L. Roy and R. Putman, The secondary drying stage of freeze-drying: Drying kinetics as a function of

temperature and chamber pressure, Int. J. Pharmaceut. 60 (1990) 203-217.[23] J. Donea, Arbitrary Lagrangian-Eulerian finite element methods, in: T. Belytschko and T.J.R. Hughes, eds., Computational Methods

for Transient Analysis (North Holland, Amsterdam, 1983) 473-516.[24] T. Belytschko and J.M. Kennedy, Computer models for subassembly simulation, Nucl. Engrg. Des. 49 (1978) 17-38.[25] W.K. Liu, H. Chang, J.S. Chen and T. Belytschko, Arbitrary Lagrangian-Eulerian Petrov-Galerkin finite elements for nonlinear

continua, Comput. Methods Appl. Mech. Engrg. 68 (1988) 259-310.[26] S. Ghosh and N. Kikuchi, An arbitrary Lagrangian-Eulerian finite element method for large deformation analysis of elastic-

viscoplastic solids, Comput. Methods Appl. Mech. Engrg. 86 (1991) 127-188.[27] S. Ghosh, Arbitrary Lagrangian-Eulerian finite element analysis of large deformation in contacting bodies, Int. J. Numer. Methods

Engrg. 33 (1992) 1891-1925.[28] E. Hairer, Unconditionally stable explicit methods for parabolic equations, Numer. Math. 35 (1980) 57-68.[29] G. Zografi and M.J. Kontny, The interactions of water with cellulose- and starch-derived pharmaceutical excipients, Pharmaceut. Res.

3(4) (1986) 187-194.

fem freeze drying

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