hr. J. Helrr Mu.%\ Trms/er. Vol. 20, pp. 517426. Pergamon Pms 1977.

Printed in Great Britain

MICROWAVE FREEZE-DRYING OF FOOD:

A THEORETICAL INVESTIGATION

T. K. ANG,* J. D. FORD and D. C. T. PEI

Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario, Canada

(Received 24 March 1976 and in revised form 13 September 1976)

Abstraet-An unsteady-state analysis of two dimensional freeze-drying with microwave internal energy
generation is carried out, taking into account the differences of the transport parameters with respect
to grain orientation, such as is found in food products. The anisotropic character of the material strongly
influences the temperature profiles during drying. This importance is further amplified by a coupling
effect between mass-transfer resistance, specimen temperature, and the absorption of microwave energy.

NOMENCLATURE K mass flux at the interface;

concentration of water vapor [g/cm3]; WL, mass flux at the external surface;
concentration of water vapor in the vacuum & x-coordinate;
chamber [g/cm’]; Y* Y-coordinate;
concentration of water vapor at triple point X(r), x-interface position at time t;
km”1 ; Y(t), Y-interface position at time t;
heat capacity of the water vapor [cal/g”CJ; L, dimensionless coordinate in the dried layer
heat capacity of the dried meat [cal/g”C]; along x-direction;
heat capacity of the frozen meat [cal/g”C]; z,, d~ensionI~s coordinate in the dried layer
effective diffusivity along x direction [cm2,/s]; along y-direction.
effective dilTusivity along y direction [cm”/s];
effective Knudsen diffusivity [cm’/s]; Greek letters
peak electric field strength [V/cm]; A density of the ice [g/cm3];
field strength in the dielectric material PD* density of the dried meat [g,4mS];
[V/ml ; PF, density of the frozen meat [g/cm3];
microwave frequency [c/s]; e, porosity of the dried meat;
heat-tr~sfer coeflicient at external surface se, relative dielectric constant;
[cal/cm2 “C s] ; E”, relative loss factor;
heat of sublimation of ice [Cal/g]; 80, dielectric constant of vacuum;
thermal conductivity of dried beef along W, microwave power absorbed per unit volume
x-direction [Cal/cm “C s] ; [cal/cm” s];
the~al ~nductivity of dried beef along 4 initial dried layer thickness [cm];
y-direction [cal/cm T s]; % thermal diffusivity [crr?,/s];
thermal conductivity of frozen beef along 7, dimensionless time;
x-direction [Cal/cm “C s] ; 4 dimensionless temperature;
thermal condu~ivity of frozen beef along d~ensio~ess concentration of water vapor;
y-direction [Cal/cm ‘32s]; ; ratio factor of thermal ~ndu~iviti~;
dimensionless interface position along t ratio of effective diffusivities.
x-direction;
dimensionless interface position along Subscripts
y-direction; Q in the dried layer;
process time [s]; F, in the frozen layer;
temperature [“Cl; at the interface;
temperature of the vacuum chamber c”C]; f;, at initial time;
tem~rature at the external surface E”C]; x, in the xdirection;
temperature at the interface PC]; Y? in the y-direction.
initial temperature c”C];
dimensionless coordinate in the ice-core INTRODUCTION
along x-direction; FREEZE-drying is a process in which frozen water con-
UP dimensionl~s coordinate in the ice-core tained within the pores of a body is removed by sub-
along y-direction; limation to the vapour state, usually under high
vacuum. Despite its high cost, freeze-drying of food has
* Present address: J. C. Food Product Inc, Manila, gained acceptance as the method of drying which will
Republic of Philippines. gerieralty produce the highest quality product.
517
518 T. K. ANG. J. D. FORD and D. C. T. PEI

There are four potential rate-limiting steps, which the grain is always less than that parallel to the grain.
occur in series with one another in conventional freeze- At a pressure of 0.5 torr, the thermal conductivity
drying processes: (1) External heat transfer from the across fibres amounts to only two-thirds of that along
heat source to the outer surface of each piece of the fibres.
material; (2) internal heat transfer from the outer sur- There are four mechanisms which contribute to gas
face of each piece to the sublimation front through mass transfer in porous media: (1) bulk diffusion, (2)
the dried layer; (3) internal mass transfer of water Knudsen diffusion, (3) slip flow, and (4) Poiseuille flow.
vapour from the sublimation front to the outer surface; A complicated equation which accounts for all these
and (4) external mass transfer of water vapour from effects has been proposed by Wakao et al. [16]. How-
the sample surface to the condenser or other moisture ever, for freeze-drying at low chamber pressure,
sink. It is obvious that as the thickness of the dried Knudsen diffusion accounted for most of the mass flux,
layer continuously increases during the drying process, and surface diffusion has not been found to be a sig-
rate limiting steps (2) and (3) assume increasing im- nificant contributor [ll]. Therefore, it is possible to use
portance. Hence, the conventional freeze-drying pro- an effective diffusivity. Sandal1 et al. [ 171, Harper [12]
cess is usually too slow for effective commercial and Dyer and Sunderland [18] have reported effective
processing of food products, particularly meat. diffusivities parallel and perpendicular to the grain for
Dielectric heating using microwave power appears turkey breast and raw beef respectively.
to offer one of the best solutions for overcoming the Microwave properties of food products such as
heat conduction problem encountered in conventional dielectric constant and relative loss factor are unfortun-
freeze-drying [l-4]. Microwaves are generated by ately incomplete or even unavailable over the tempera-
oscillator tubes (klystrons or magnetrons) with fre- ture ranges of interest. For raw beef, the most widely
quencies which lie in the band 300 MHz to 300 GHz. used dielectric data are those of Kan and Yeaton [ 191
At microwave frequencies, electromagnetic energy is over the microwave frequency range of 500 and
absorbed in dielectricmaterials such as food in response 3000 MHz. They found that the dielectric properties of
to applied fields. As the various particles are accelerated raw beef are strongly dependent on temperature.
by the field, they give off heat due to friction. Previous work on microwave freeze-drying concen-
When applied to the freeze-drying process, micro- trated essentially on experimental investigations [l-6].
wave energy penetrates very well into ice, by-passing The earliest mathematical analysis was carried out by
the problem of heat conduction across the dried layer. Copson [l]. The analysis was limited to simplified
This gives essentially volumetric heating of the receding approaches using a quasi-steady state assumption. The
ice-core, and hence reduces the drying time by as much first general unsteady analysis with internal heat gener-
as 75%. This has been successfully demonstrated on an ation was carried out by Ma and Peltre [20] with an
experimental scale by Copson [l], Jackson [3], Hoover infinite slab model. Therefore, both models have
[5], Decareau [6], and Ang [7]. neglected the anisotropic character of food, which has
To provide a better understanding of the heat and a strong effect on the temperature profiles of the
mass transfer processes occurring during microwave material during drying.
freeze-drying, an unsteady state, two dimensional
freeze-drying model using microwave energy is de-
veloped. This two dimensional model is shown to be THEORETICAL ANALYSIS

a distinct improvement over the previous flat slab Injkience of geometry and anisotropic structure on
models. freeze-drying
Consider a slab and a square of the same material,
LITERATURE REVIEW both to be freeze-dried under the same conditions using
Numerous freeze-drying models have been reported microwave energy as shown in Fig. 1. The displacement
in the literature and several comprehensive reviews on rate of the interface in the slab is given by:
freeze-drying have been given by Harper and Tappel dl(t) @
[8], Burke and Decareau [9], Ginnette and Kaufman (1)
dt pa
[lo]. The most recent review given by King [ll],
covering all aspects of freeze-drying, is a valuable guide and similarly for the square case:
to the literature. However, it should be pointed out
dX(t)
-=-__. (IV, dp=Y(r) -__(I%),.
that none of these theoretical analyses deal with internal (2)
dt per ’ dt PO
generation of energy and the anisotropic character of
food. In this section, only literature pertaining to the where X(t) and Y(t) are locations of x and y interfaces
present investigation will be discussed. at time t respectively.
Thermal conductivity data are needed for analysis of Assume that the resistance to mass transfer is
internal heat transfer during freeze-drying. Harper [ 121 negligible, so that all sublimed vapour immediately
and Woodams and Nowrey [13] have shown that ther- reaches the outer face. For the slab I%$ = lI$, but for
mal conductivities of freeze-dried foods are very similar the square IV’ # (IV’), # (IV’), due to geometry and
even for materials as diverse as fruit and meat. For anisotropy. Therefore, dX(t)/dt will not be equal to
raw beef, Bralsford [14] and Gunn and King [15] d Y(t)/dt. In other words, at a given rate of displacement
showed that the thermal conductivity perpendicular to of the interface, the sublimation rates (in terms of the
 Microwave freeze-drying of food 519

layer is somewhat lower), the effect will still be

negligible.
In any microwave cavity, when the electric field is
parallel to the surface of the material, the field strength
is the same in both the immediate su~oundings and
within the material, regardless of dielectric constant.
When the field is perpendicular to the surface of the
material, then the field strength inside the dielectric is
related to the field strength. It is well known that the
true situation must lie between these two extreme
cases. In order to simplify the situation, it is assumed
that only the dominant mode is able to propagate and
the domin~t placation of the field is normal to the
surface of the sample. Moreover, the strength of the
electric found is assumed constant throughout the
sample and equal to the value at the surface. This is
justified since the penetration depth at the microwave
frequency of 2450MHz is much greater than the size
of the sample.

FIG. 1. Effect of geometry and anisotropy on the mass flux

at the external surface of the samples-slab (above) and Di~ere~tial eq~ti~~ of the feel
square (below). The differential equations which describe the physi-
cal system are:
Dried layer.
total amount of vapour removed from the specimen) (i) Heat transfer:
are not the same due to geometric and anisotropy
factors.
It has been established in the literature that specimen
tem~rature is critical in the analysis of microwave
(3)
freeze-drying. Under a given experimental condition,
the temperature of the specimen is controlled by the
mass transfer rate which is in turn controlled by the (ii) Mass transfer:
resistance of the dried layer and the rate of shrinkage a2c ao ac ao ac
Dx$+D,-+--+L,$. (4)
at the interface. Therefore, it is obvious that anisotropy aY2 ax ax ax ax
has a strong influence on the quality of freeze-dried
product obtained. Frozen layer.
:
II
(i) Heat transfer
Model description and assumptions
aKFx ST, aKFy ZT,
Consider a piece of meat of infinite length with a
square cross section, As freeze-drying proceeds, the
ice-front (assumed to be of zero thickness) retreats,
ax XC+ ay ay I
= P&+0~. (5)
the sublimed water vapour being removed by diffusion
through the porous dried layer. Heat is transferred by
conduction and convection in the dried region, while There will be no mass transfer in the frozen layer. The
only induction will be considered in the frozen zone. bound~y conditions are
If the grain is oriented along the y-axis, both thermal At the center, for t >, 0
conductivity and effective ditfusivity in the y-direction
will be greater than those in the x-direction, The rate x=o:- QF / =O; y=O:$/ = =O. (6a)
of ice-front shrinkage in the y-direction will, then, be ax x=0 Y 0
faster than that in the x-direction.
A certain degree of shrinkage (amounting to less than At the interface, thermodynamic equilibrium between
15% of the original volume) usually occurs during ice and vapour would exist. Therefore, the concen-
fr~~~rying [Zl], however to simplify the system, it tration of the water vapour can be related to the ice
is neglected. Margaritis and King [22] have shown t~~rature by an ~~libri~ relationship. Energy
that there is a finite absorption of moisture in the dried balances at the x and y interfaces will give:
layer (less than 3x), due to the flow of water vapour.
x = X(t) : (W&AH, = KF, [z] - K$$] (69
However, this absorption contributes a negligible
amount to the overall mass-transfer resistance. With
microwaves, although a larger amount of adsorption y = Y(t):(W$,AH, = KFy
may be expected (since the temperature of the dried

HMT Vol. 20. No. S-F

520 T. K. ANG, J. D. FORD and D. C. T. PEI
and To = TF = 7;. At the outer surface, for t > 0, the literature. The moisture, and fat content, density,
and porosity of beef were assumed to be constant and
x=y=L:h(7,-Ts)=KDx~=KDr$. (64 values as quoted by Awberry and Griffiths [23] were
used in the simulation. Functional relationships ex-
By assuming the mass-transfer resistance at the outer pressed as polynomials of temperature were used for
surface to be negligible (because of low chamber thermal conductivity [13, 15, 241, effective diffusivity
pressure): [17], ice-water vapour equilibrium relationship [25],
and dissipation coefficient in a microwave field [19].
C(L,t) = CR for t 5 0. (6d) Since the dissipation coefficient for frozen beef is an
At the initial state of freeze-drying when material is order of magnitude higher than for dried beef, the
introduced to the chamber, before the desired level of frozen beef absorbs the major fraction of the micro-
vacuum is reached and microwave power turned on, wave energy.
some water vapour is sublimed, giving an initial thick- The temperature and total pressure of the vacuum
ness of dried layer 6. Therefore, an initial dried layer chamber are taken to be constant. The variation in
thickness of 0.004L is used in this analysis. Since this total pressure is assumed to have a very small effect
initial freeze-drying is essentially conventional freeze- on the water removal process, since the gas near the
drying, in which the heat of sublimation comes from outer surface of the dried layer is in the Knudsen-flow
the surroundings, both temperature and concentration regime. The initial temperatures in the dried and frozen
profiles at this stage are assumed to be linear, and the layers are assumed to be - 25 and - 15°C respectively.
frozen zone temperature assumed constant. Then, the The dried layer is assumed to be cooler than the frozen
initial condition is as follows: layer based on the assumption that during start-up
T,=Tr for O<x<L-6, O<y<L--6 ice is sublimed. The rate of heat loss by this means is
appreciably greater than the rate of heat absorption
T,-T, L-x by the material from its immediate surroundings. The
-=- for L-G<x<L (74
G-T, 6 result of these effects will be a temperature drop in the
G-T, L-y dried layer analogous to the wet bulb process. There-
___=__ for L-G<y<L. fore, the exact initial temperature in both frozen and
Ts-T, 6
dried layers varies and is difficult to control. Fortun-
The concentration profile is likewise assumed to be ately, it has only a small effect on the outcome of the
linear : drying process.

1
In freeze-drying, the maximum temperature of both
C=Cs- g i(L-x) for L-6<x<L (7b) dried and frozen layers during the process is often more
I important than the total drying time required. The
(L-y) for L-G<y<L. maximum temperature reached in most cases deter-
mines the quality of the resulting food products and
hence the feasibility of a given drying process. There-
By assuming normal polarization, the power dissipated
fore, two temperature constraints are put into the
in a dielectric substance which responds to an applied
model. In the frozen zone, the temperature is kept
electric field is given by:
below - 3”C, while an upper limit of 60°C in the dried
layer is set to prevent thermal degradation of the dried
products.

Transformation of d@erential equations into Numerical solutions

dimensionless form Equations (3)-(5) are a set of parabolic partial differ-
The following dimensionless variables are used to ential equations describing heat and mass transfer with
normalize the differential equations: phase change at a moving boundary. Moreover, the
physical properties of the beef are functions of depen-
L-x X
Z,=-- dent variable (temperature). Therefore, an analytical
L-X(t) ’ u”=xct, solution to such a complicated situation is unlikely,
hence a numerical technique, the Crank-Nicholson
Z,=-- L-Y u, = Y (9’3 method, is employed. A five-point implicit, central
L- Y(t) ’ y(t)
difference scheme is used. Moreover, Gaussian elimin-
0
uFx t C-CR ation is used to solve the linear equations desired
r=-; I-=
L2 from the difference equations and the boundary condi-
tions. As the three partial differential equations are
coupled through the boundary conditions at the inter-
face, an iterative procedure is necessary to solve the
PHYSICAL PROPERTIES AND EXPERIMENTAL problem.
CONDITIONS USED IN THE SIMULATION Furthermore, from the physical observation of the
Raw beef is chosen as the material for simulation, ice-core at different stages of drying, the corner effect
since the physical properties are readily available in present in the system is obvious. Carslaw and Jaeger
 Microwave freeze-drying of food 521

DRIED REG0N DRIED REGION

F‘ROZEN REGION

T-70
s
G-
f 0

--10 e
.
-90
Dmin Om‘nr
$5 I I ’ I
0.8 Q6Q4 0.2 6 0.2 04 96 0.8 ?ik&k&
Along Y AXIS

FIG, 2. Temperature pr0file.s(field strength = 130V/cm).

3.0 Discussion of results

The temperature profiles obtained in the x and y
directions are parabolic in shape, for both the frozen
ALong X Axis - and dried zones as shown in Fig. 2. On the other
hand, the concentration profiles are essentially linear
at all times as shown in Fig. 3. This indicates that the
accumulation terms in the mass balance equations are
negligible compared to the space derivative. This result
is in agreement with previous studies by Copson [26],
Dyer et al. [25], and Ma and Peltre [20].
At the start of the process, the temperature of the
ice-core dropped to a minimum, because the mass-
transfer resistance of the initial dried layer is small. The
microwave energy absorbed by the specimen at this
stage is insufficient to provide the heat of sublimation
at the interface. However, as the thickness of the dried
layer gradually increases, its temperature starts to rise
to a maximum due to the increasing mass-transfer
resistance encountered. Since the mass-transfer rate is
greater, the dried zone temperature in the y-direction
at x = 0 is always lower than its x-direction counter-
1.0
t a8 part. After nearly half of the ice is sublimed, the tem-
DRIED REGION COORINATE perature in the frozen zone starts to decrease whiie the
FIG. 3. Water vapour concentration profiles (field dried zone temperatures in both the x and y directions
strength = 13OV/cm). increase continuously. This is because at this stage the
energy absorbed by the ice-core, due to reduction in
[26] in their analytical solution of conduction of heat volume to less than one-half of its original value, is
in a solid of ii&e rectangular shape also showed that no longer sufficient to ptovide the necessary heat of
due to the corner effect, the isotherms in that region sublimation. However, the rate of decrease in frozen
are rounded. Therefore to obtain normal convergence temperature during this stage is considerably less
of the numerical solution using the mesh size limited because the mass-transfer resistance of the dried layer
by computer storage capacity at the University of has become higher.
Waterloo, the interface mesh grid at the corner is Since microwave heating is volumetric, and the
adjusted in the simulation. The detail of the numerical frozen beef is absorbing the major fraction of the
solution as well as the computer program for the model microwave energy, the volume of the ice-core remain-
are given in reference [7]. ing during the drying cycle is critical in the process
522 T. K. ANG, J. D. FORDand D. C. T. PEI
analysis. It has been observed that the freeze-drying
cycle can roughly be divided into four periods. During
the initial period, the appearance of the retreating ice-
core will be in the form of a rectangle. However,
because of the corners, it will shrink to the form of an - 100 v/cm
- 130 v/cm
ellipse, with major axis perpendicular to the grain
orientation. This period is referred to as the second
period. During the third period of the cycle, the ice-core
shrinks to the shape of a thin plate of negligible
thickness. Finally after the ice-core disappears, the
residual moisture in the dried product is removed.
During period one, the external surface temperature
Ts is less than the chamber temperature TR. The heat
flux at the external surface at this stage is from the
surroundings to the sample. The actual power con-
sumed by the sublimation at this stage is slightly
greater than the power supplied by microwaves. How-
ever, the heat flux from the surroundings will decrease
I I I I I 1 I
as Ts increases. When Ts approaches TR(corresponding 0 1 2 4 5 6
to the transition from period one to period two), this DRYING TIfiE, h
heat flux approaches zero. From period two up to the FIG. 5. Effect of field strength on specimen temperatures at
end of the process, heat flows from the external surface various drying times.
of the specimen to the surroundings. The actual power
consumed by the sublimation is thus slightly smaller where 4 and IJ are the ratio factors. The ratio factors
than that supplied by microwaves during this period. are the properties of the materials which may be con-
As shown in Fig. 4, the electric field strength has stant or variable with respect to a given condition. The
appearance ofthe retreating ice-core will be in any form
l.OI\ I
as shown in Fig. 6, depending on the magnitude of th?
ratio factors. It should be noted that

depends on the characteristic of the material. For

example in the case of raw beef, at a pressure of
OSmmHg, 4 is approximately equal to 1.5, and $
1 approximately equal to 3, assuming the grain is orientec.
0 1 2 3 4 5 6 7
DRYING TIME, h
along the y-direction.
EFFECT OF FIELD STREtWH ON DRYING TIME
If 4 = 1, and II/ = 1, that is, the transport parameters
are equal in both directions, the appearance of the
FIG. 4. Effect of field strength on drying time.
uniformly retreating ice-core will be a perfect square
during the initial stage, and will be in the form of a
circle in the second period.
a strong effect on the drying process. An increase in
If 4 = 0, and II/ = 0, the freeze-drying process be-
electric field strength inside the microwave applicator
comes one dimensional, and the infinite slab model is
increases the drying rate. However, the temperatures
obtained. In other words, the infinite slab model 1s a
attained in both the dried and frozen regions are like-
special case of the two dimensional model, in which
wise proportional to the field strength, as shown in
the ratio factors both equal zero. For 4 and $ other
Fig. 5. In order to prevent melt back at the ice-front,
than these two special cases, the uniformly retreating
and also thermal degradation of the dried region, only
ice-core will be in the form of a rectangle in period
a certain allowable field strength is applicable under a
given condition. It is important to mention here that one. It will shrink to an ellipse in period II. In Fig. 6,
this is designated as Model IIa and IIb. The ratio
if the quality of the freeze-dried product is not of
between the major and minor axes will depend on the
critical importance, it is possible to use a higher field
strength to achieve a shorter drying cycle. In this case, relative magnitudes of the ratio factors.
the maximum applicable field strength will be limited If identical initial experimental conditions are
by plasma formation. assumed, there will not be a great difference between
the models as far as the total drying time is concerned.
Parameter analysis
This is because microwave heating is volumetric and
In general, the transport parameters of the material volume has no direction. The drying curves of the
with respect to grain orientation may be expressed different models under identical experimental condi-
mathematically as tions are shown in Fig. 7. However, it is important to
note that the relative drying rates are not the same
K, = 4K, and D, = ll/Dx from period to period. At the initial period of the
 Microwave freeze-drying of food 523

Dy = po,

SPEUAL CASE

FIG. 6. Effect of ratio factors on the shapes of the retreating

frozen front.

08 PEAK FIELD STRENGTH -130 v/cm

ii
_z 0.6
r”
e MODEL 111 (slab model )
< 04
t
z

DRYING TIME , h
FIG. 7. Effect of ratio factors on total drying time.

process, the one dimensional infinite slab model has thus it has the slowest drying rate. Model II is inter-
the fastest drying rate. It is followed by Model II. mediate between these two extremes. During the later
Model I, which has both ratio factors equal to 1 has period, Model I has the highest ice-core temperature.
the slowest initial drying rate. However, it overtakes Also it has the fastest drying rate. The one dimensional
the two eventually. At the end of the drying cycle, the model has the lowest ice-core temperature, and the
slow starting Model I predicts the shortest total drying slowest drying rate. Model II is again intermediate
time, while the fast starting one dimensional model between these two.
predicts the longest total drying time. The correlation between ice-core temperature and
The maximum ice-core temperature is plotted drying-rate is due to a coupling effect of: (a) mass
against the drying time as shown in Fig. 8. All models transfer resistance; (b) specimen temperature, and (c)
are assumed to have an initial ice-core temperature dissipation coefficient. The coupling effect may be
of - 15°C. At the initial stages of the process, the one illustrated as follows: At the initial stages of the process,
dimensionalmodelhas the highest ice-core temperature. the thickness of the porous dried layer, which is re-
Correspondingly, it has the fastest drying rate. Model I, sponsible for the mass-transfer resistance, is relatively
on the other hand, has the coolest ice-core temperature, small. Model I has the largest vectorial sum of dif-
 T. K. ANG. J. D. FORD and D. C. 7’. PCI

PEAK FIELD STRENGTH: 130 v/cm

10
PEAK FIELD STRENGTH:130 v/cm

DRYING TIME I h

FIG. 9. Effect of ratio factors on the drying curve, in the

I 1 I I I I absence of coupling effect.
0 1 2 3 4 5
DRYING TIME, h
Oc
FIG. 8. Effect of ratio factors &I the ice-core temperature.
MODEL I
MODEL ii
fusivities among the three. It will have the fastest mass- MODEL Illklob modei)
transfer rate. At the other extreme, the one dimensional
model has the highest mass-transfer resistance. When 0
a
the mass-transfer resistance is low, the specimen will -6
be at the lowest temperature. Since the dissipation
coefficient of raw beef in both frozen and dried region
increases rapidly with increase in temperature, the
amount of microwave energy absorbed in Model I
becomes lowest among the models. Drying rate in a
microwave process is essentially controlled by the
amount of energy dissipated, especially in the ice-core.
Thus Model I will have the slowest initial drying rate.
As the ice-core shrinks, the mass transfer resistance
due to the dried layer increases. Since a circle has the
least perimeter for a given area, Model I will have the
least sublimation front. Consequently, it will have the DRYING TIME, h
largest mass-transfer resistance. Its tem~rature will
become the highest, and the energy dissipated will FIG. 10. Effect of ratio factors on ice-core temperature, in
become the highest among the modes giving the fastest the absence of coupling effect.
final drying rate.
In microwave freeze-drying, the maximum tempera-
ture obtained in both dried and frozen layers during If the dissipation coefficient of the material is not a
the process is far more important than the total drying strong function of the temperature, the coupling effect
time required. The maximum temperature obtained will not exist. The drying curve as well as the maximum
actually determines the quality of the resulting product. ice-core temperature will be as shown in Figs. 9 and 10.
It likewise determines the fe~ibility of a given process. Similar to Fig. 7 and Fig. 9, it shows that if identical
As shown in Fig. 8, the ice-core temperature of Model initial experimental conditions are used, there will not
III, which is the one dimensional infinite slab model be a great difference between the models as far as the
reaches a maximum of -4.2% If -3°C is the maxi- total drying time is concerned. This is because micro-
mum allowable ice-core temperature, the conclusion wave freeze-drying is controlled essentially by the
will be that the process is feasible, and the product ability of the dielectric to absorb microwave energy
obtained will be of good quality. However, the maxi- volumetrically in response to the applied field strength.
mum ice-core temperature obtained by Model I is However, if the dissipation coefficient of the material
-29°C. The conclusion baaed on this will be that the is not a strong function of the temperature, the over-
process is not feasible, since it has over-shot the maxi- lapping of drying curves as shown in Fig. 7, and
mum allowable ice-core temperature. Under these maximum ice-core temperatures as shown in Fig. 8,
conditions, melt-back would occur, yielding a product among the three models will not occur. In fact the
of inferior quality. maximum ice-core temperatures of the specimens of
 Microwave freeze-drying of food 525

Model I and II (Curve I3 and C) are always below that electric field is usually limited to between 100-135 V/cm
of the one dimensional model (Curve A). to ensure the fastest drying rate without overheating
The coupling effect can also be used to explain the or melt-back condition; (6) the chamber pressure
influence of ambient pressure on the process. If freeze- should be maintained between 0.4-0.2 mmHg; (7) due
drying is carried out at a si~i~cantly higher ambient to the coupling effect, the size of the sample limits the
pressure, the total drying time required will become maximum applicable field strength for uniform heating
slightly shorter due to the following sequence: (a) at of the entire specimen.
higher ambient pressure, the mass transfer resistance Acknowledgement-The authors would like to express their
is higher; (b) the ice-core temperature then increases, sincere appreciation to J. C. Food Product Inc. of the
due to this higher mass transfer resistance; (c) the Republic of the Philippines for providing financial support
energy dissipated in the ice-core then becomes higher, to Dr. T. K. Ang.
due to the higher temperature; and (d) the drying rate REFERENCE
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However, as mentioned earlier, the maximum tem- Technol. 12,270 (1958).
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3. S. Jackson, S. L. Richter and C. 0. Chichester, Freeze-
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drvina of fruit. Food Technol. 11.468 (1957).
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able field strength has to be reduced severely in order freeze-drying, A.I.Ch.E., Symp. Ser. 69(132), 47 (1973).
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freeze-drying of foods, Food ~ec~~o~. 20,807 (1966).
gas plasma formation at the usual operating field 6. R. V. Decareau, Freeze-drying of ~oods~u~, edited by
strength increases. S. Cotson and D. B. Smith. Columbine Press,
From the above analysis, low operating chamber Manchester (1962).
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(1964).
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CONCLUSIONS
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influences the process; (4) microwave power shortens experimental study of the freeze-drying of raw beef,
the total drying time significantly when compared with Tech. Rept.
__ ,*?._-.72-12-DR, U.S. Army Natick Lab., Natick,
conventional freeze-drying; (5) a maximum applied Mass. \lY IL).
526 T. K. ANG, J. D. FORD and D. C. T. PEI

22. A. Margaritis and C. J. King, Factors governing terminal 25. D. F. Dyer, D. K. Carpenter and J. E. Sunderland,
rates of freeze-drying of poultry meat, Chem. Engng Equilibrium vapour pressure of frozen bovine muscle,
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gels and ice, Food Technol. 15,243 (1961). (1962).

LYOPHILISATION DES ALIMENTS EN PRESENCE DE

MICRO-ONDES-UNE ETUDE THEORIQUE

R&sum&-On effectue une analyse non-stationnaire du sechage avec congelation dam deux dimensions
avec production de micro-ondes d’energie interne, compte tenu des differences entres paramktres de
transport dues a l’orientation des grains, comme cela existe dans les prod&s ahmentaires. Le
caractere anisotrope du materiau influe fortement sur les pro& de temperature iors du stchage. Cette
importance est encore accrue par l’effet de couplage entre la resistance au transfert massique, la temperature
de l’tchantillon et l’absorption de l’energie des micro-ondes.

MIKROWELLEN-GEFRIERTROCKNUNG VON LEBENSMTTTELN-EINE

THEORETISCHE UNTERSU~HUNG

Z~8mm~~~ng-Der instationiireFall der zweidimensionalen Gefriertrocknung mit Mikrowellen-

beheizung wird analytisch untersucht, wobei die Unterschiede in den Transportparametern infolge der
Kornorientierung, wie sie bei Lebensmitteln gefunden wird, beriicksichtigt werden. Der anisotrope
Charakter des Materials beeinflugt stark das Temperaturprofil wahrend des Trocknens. Dieser EinfluB
wird noch verstarkt durch einen Kupplungseffekt zwischen Stoffaustauschwiderstand, Probentemperatur
und Absorption der Mikrowellenenergie.

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