highfrequency drying of porous material
TRANSCRIPT
This article was downloaded by: [University of Kiel]On: 27 October 2014, At: 04:10Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
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Highfrequency Drying of Porous MaterialHarald Bäder a & Ernst-Ulrich Schlünder aa Institut für Thermische Verfahrenstechnik, Universität Karlsruhe , Karlsruhe, 76128,GERMANYPublished online: 07 May 2007.
To cite this article: Harald Bäder & Ernst-Ulrich Schlünder (1996) Highfrequency Drying of Porous Material, Drying Technology:An International Journal, 14:7-8, 1499-1523, DOI: 10.1080/07373939608917161
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DRYING TECHNOLOGY. 14(7&8). 1499-1523 (1996)
HIGHFREQUENCY DRYING O F POROUS MATERIAL
Harald Bader and Emst-Ulrich Schliinder Instit111 fiir Thermische Verfahrenstechnik,
Universitl Karlswhe, 761 28 Karlsruhe, GERMANY
Key words: electric conductivity. dielectric drying; drying rate; focusing effect; radiofrequency drying
ABSTRACT
Highfrequency drying of a porous ceramic has been studied experimentally. The influence of electrode voltage. electrode distance, electric conductivity of the moisture and geometry of the product on the drying rate was investigated. There exists an optimal conductivity for maximum power dissipation and the geometry of the product has a focusing effect on the electric field. A theoretical discussion concludes, that the eddy current free quasi-stationary field is the type of field commonly present in highfrequency drying. With this type of field and the fact, that the complex permittivity of the product changes during drying, the experimental observations can be understood. From theory and experiment, conclusions are drawn in respect to a typical drying curve at highfrequency drying.
INTRODUCTION
Highfrequency drying can be considered as a dielectric drying technology, be-
cause the product behaves like a dielectric in an electromagnetic field. Dielectric
drying technologies share the imponant feature, that the energy is dissipated in the
whole volume of the product. This distinguishes dielectric drying from convective
Copyright Q I996 by Marcel Dckkcr. Inc.
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drying, contact drying or infrared drying, where the energy must be transferred
across the surface of the product.
Highfrequency drying has been worked on for a long time. Siemens [22] per-
formed experiments with Leyden jars and described the underlying basic physical
effect of dielectric heating in a speech before the Akademie der W;ssetischafie,~ in
Berlin. Germany even in 1864. There are publications about highfrequency drying
as early as 1938 from Morozov [20]. In general, highfrequency drying is expensive
compared to other drying technologies, due to the installation costs for the high-
frequency generator and the costs for the electric energy. Still it turned out to be
economical for certain products ([4]. [25]. [ I I]. [IS]) where the possible specific
advantages of highfrequency drying, as equalized moisture distribution, reduced
migration, prevention of cracks due to shrinkage in drying of ceramics and higher
product quality in general outweigh the costs. Another advantage is the higher
drying rate, and therefore the reduced drying time and space requirement, espe-
cially in products with large volume and poor inner heat and mass transfer proper-
ties. Highfrequency dryers can be found in the paper, textile, ceramics and wood
industries. Jones [ 141 gives an overview of the usage of highfrequency heating and
drying in industry.
Besides costs, insufficient understanding of the technology seems to be an im-
portant constraint for the use of highfrequency drying. Highfrequency drying re-
quires both knowledge in electrodynamics and drying, which makes the subject
difficult and interesting. As a consequence, a lot of publications are focused on one
of the two aspects. Gibson [I21 presents a good and compact introduction into the
physical understanding of highfrequency drying.
Even the basic question: "What is the typical drying curve in the case of high-
frequency drying?" seems to provoke contradictory answers in the literature.
Avramidis [2], Baginski [3] and Besser [6] reported steadily decreasing drying
rates from the beginning to the end of drying. Alexander [I]. Krixher [I61 and
Mann [I91 divided the drying curve into a warming-up period, a constant-rate pe-
riod and a decreasing-rate period, whereas Dostie [ lo] reported a maximum in the
drying curve and no constant-rate period.
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HIGH FREQUENCY DRYING OF POROUS MATERIAL
THEORY
From the standpoint of electrodynamics the product during highfrequency
drying is a dielectric in an electromagnetic field. To obtain information about the
dissipated energy due to the field, it is necessary to perform a field calculation first.
Generally. this means solving the Maxwell's equations:
When approaching this task. it is of great advantage to check first which sim-
plifications in the Maxwell's equations are permitted or which type of field is pre-
sent. Following a classification proposed by Dirks [S]. the condition
is generally true in the case of highfrequency drying. and the type of field is called
the slow changing field or quasi-stationary field. The additional condition
is generally true as well in the case of highfrequency drying and the type of field is
called the eddy current free quasi-stationary field. This type of field overlaps with
category II. capacitive heating, in the classification proposed by Vermeulen [24],
but is not defined by the same condition. The second condition means, that the skin
depth is big compared to the size of the product and therefore that the skin effect
can be neglected. These two conditions have the pleasant implication, that the
Maxwell's equations can be solved uncoupled and the interesting electric field can
be calculated by taking the divergence of the I. Maxwell's equation (I).
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Using the material equations
and the notation with relative permittivity
yields
At this point the notation with complex permittivity is introduced.
.G,=<-j< (12)
showing in connection with equations (8) and (9). that the real pan of the complex
permittivity can be understood as describing the part of the total current density
which is out of phase with the electric field strength and therefore a measure of the
storage of energy, while the imaginary part of the complex permittivity. also called
loss factor, can be understood as describing the pan of the total cument density
which is in phase with the electric field strength and therefore a measure of the
dissipation of energy. Equation (1 I), also called the continuity equation, yields
This is the equation which has to be satisfied in the case of eddy current free
quasi-stationary fields in order to calculate the electric field strength. If there are
sectors with different material properties in the interesting region, as in the case of
drying the product and the surrounding air, then additional conditions at the inter-
face have to be satisfied:
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HIGH FREQUENCY DRYING OF POROUS MATERIAL 1503
Analytical solutions of equations (13) to (15) exists for plates, cylinders and
spheres with homogeneous permittivity in an initially homogeneous field. It turns
out that even though the field outside the product is distorted, the electric field
strength inside the product is constant. With the additional conditio~ that the
product is surrounded by air
the solution is
where m is a geometry parameter and takes the value 0 for plates. I for cylinders.
and 2 for spheres. Now that we have the internal field strength, we can calculate
the dissipated power density with the equation cited in every paper about highfre-
quency drying:
Inserting equation (17) yields
Two imponant conclusions can be drawn from equation (19):
I . There is a focusing effect just from the geometry of the product. Being
g,,, >> m and keeping the complex permittivity of the product and the field
strength outside the product constant, the power density of plate : cylinder :
sphere behaves like 1 : 4 : 9.
2. The real pan and the imaginary pan of the complex relative pennittivity of the
product will decrease during drying with decreasing moisture content. which
will be discussed later. As a consequence, the field strength inside the product
will vary during drying which cannot readily be seen from equation (18).
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1504 BADER AND SCHLUNDER
Therefore, holding the outside field strength constant, the power density as well won't be constant due to the varying loss factor and the varying field
strength inside the product.
On first sight, one would think the greater the loss factor, the greater the
power density. A closer look at equation (19) reveals, that the loss factor turns up
again in the denominator of the electric field strength inside the produn. Given a
real pan of the complex permittivity, it turns out that there exists an optimum value
of the loss factor for maximum power density which has the value
This result is the consequence of the trade-off between increasing the loss fac-
tor and decreasing the field strength inside the product.
This fact cannot be described when calculating the field strength by using the
quasi-static field, as it is generally done in the literature (Van Dommelen (91.
Dostie [lo]). The quasi-static field is defined by the conditions o <<o, and
E;:, << E;,, . the second condition being a stronger condition than d << 6 and there-
fore the quasi-static field is a subset of the eddy current free quasi-stationary field.
Results are identical for E:: << E ; , . which is generally fulfilled in heating of dry
products as in Van Dommelen [9], but results diverge when this condition is vio-
lated. which can happen in highfrequency drying.
Another conclusion can be drawn from equation (18) in connection with q u a -
tion (14). When drying band-shaped products like textile and paper, electrode ge-
ometries like fringe field or staggered through field are used with the aim of getling
a high internal field strength by conducting the field strength outside the product
tangential to the surface of the product. When the external field strength is tangen-
tial to the surface. then the internal field strength is independent of the permittivity
of the product and the power density varies linearly with the loss factor.
To proceed in calculating the power density with equation (19). the permittiv-
ity of the product during drying and its dependence on moisture content and tem-
perature must be known. The product is a heterogeneous mixture of dry product.
moisture and gas, and the permittivity can be either measured or calculated with a
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HIGH FREQUENCY DRYING OF POROUS MATERIAL 1505
mixing rule from the permittivities of the components. Beek [5] and Priou [21]
give an overview of the mixing rules and the underlying theory Here. a simple
mixing rule, based on earlier measurements [23], is used.
The permittivities of the gas and the solid phase are given in table I
At 31 MHz, the real part of the complex permittivity of water is equal to the
static permittivity and is taken from Hasted [13]:
The imaginary part depends on which of the different polarization mechanisms
is dominant. The effect of dipole polarization of free water in the highfrequency
range can be estimated with the Debye equation (71. [I31 at 27 MHz and 25 "C:
This value must be compared with the conductivity, which would yield the
same loss factor due to ion polarization.
The minimal conductivity of pure water due to the dissociation equilibrium is K
= 0.055 pS/cm at 25 'C. K = 1.7 pS/cm is in the range of distilled water or water
purified with an ion exchanger. Tap water for comparison has a conductivity of
about K - 600 pS/cm. a value that can vary by a factor two.
Since the moisture in the product rarely is as pure as distilled water, ion polari-
zation is the dominant mechanism and dipole polarization of free water plays only a
minor role in the frequency range of highfrequency drying. With increasing tem-
perature, the comparison gets even more favorable towards ion polarization.
One can find regularly publications, where dipole polarization is postulated to
be the dominant mechanism in highfrequency drying (Alexander [I] at 41 MHz.
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TABLE I
Pennittivities of the Gas Phase and the Solid Phase
Avramidis [Z] at 13.56 MHz, Gault [ I I] at 14 MHz). Dipole polarization of free
water is the mechanism in microwave drying. which operates at 2.45 GHz. a fre-
quency about I00 times greater than in highfrequency drying. but is not the domi-
nant mechanism at highfrequency. There can be some contribution from dipole
polarization of bound water in the highfrequency range. The different frequency
ranges of ion polarization and dipole polarization are illustrated in figure I. The
frequency of the dryer used in this paper, 3 1 MHz, and the'frequency used at mi-
crowave drying. 2.45 GHz, are specially marked in the figure.
material
air
The conductivity of the moisture in the drying experiments was varied by
adding NaCl to distilled water and moistening the product with the solution. The
conductivity of the solution depends on the concentralion of NaCI. "
The temperature dependence can be expressed as following:
E : ' I
Equations (26) to (28) are fitted to experimental values in the literature ([17].
[I811 in the range 25 OC < T < I00 OC and 0.0005 mol/l< c <saturation.
&?
0
I Aerolith 5 4.95 0.032
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HIGH FREQUENCY DRYING OF POROUS MATERIAL
FIGURE I: Real and imaginary pan of the complex relative permittivity of water as a Function of frequency
An enerby balance around the product. assuming isothermal conditions, equal
distributed moisture and the surface of the product to behave like a free liquid sur-
face yields
where the power density is calculated with equation (19). When the product
reaches boiling point temperature, the energy balance reduces to:
With this simple balance, one cannot expect to describe effects due to internal
heat and mass transfer resistance or due to bound water, but i t can help to under-
stand some basic effects in highfrequency drying.
EXPERIMENTS
The highfiequency dryer is operating at a constant electrode voltage and a
fixed frequency of 3 1 MHz and is shown in figure 2.
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air
FIGURE 2: Schematic design of the highfrequency dryer
The product is connected to an electronic balance and is located between two
parallel vertical plate electrodes with the size 400 mm x 270 mm. The distance be-
tween the plate electrodes can be adjusted. In addition to highfrequency, there is a
convective air flow with constant temperature, velocity and humidity. The set-up is
surrounded by shielding.
Moisture content and drying rate are determined with the balance. The product
temperature is measured with a fiberoptic temperature sensor and the electrode
voltage with a high voltage probe.
Aerolith 5, a porous ceramic, was used as a product. Some material properties
are given in table 2, the dimensions of the samples in table 3.
The electric conductivity of the moisture in the drying experiments was varied
by adding NaCl to distilled water and moistening the product under vacuum with
the solution.
In all experiments, the temperature of the air flow was T, = 40 OC, the velocity
v, = 0.5 m/s and the partial vapor pressure pH.,., = 60 Pa. The heat and mass
transfer coefficients are taken from the constant rate period of convective drying
experiments and given in table 3. The radiation coefficient was CI2 = 0.9.Cb. The
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HIGH FREQUENCY DRYING OF POROUS MATERIAL
TABLE 2
Material Properties of Aerolith 5
DroDeRY 1 svmbol 1 Aerolith 5
constituents SiO: (ca. 85%) AI,O, (ca. 4.3%)
porosity
, caoillarv radius r, luml
heat capacity
hygroscopizity 1 weak
geometry of the product, the electrode voltage U, the distance between the elec-
trodes D and the electric conductivity of the moisture K are given at the respective
experiments. The temperatures are measured in the center of the product. unless
stated otherwise.
density 1 0 lkdm'l 11280
TABLE 3
Dimensions and Heat and Mass Transfer Coefficients of the Aerolith 5 Samples
RESULTS
geometry
sphere
cylinder
plate
A typical drying curve for the experiments conducted in this paper. is shown in
figure 3. The product is a sphere, the electrode voltage U = 10.3 kV, the electrode
dimensions a [w/(m2K)1
8.2
7.9
6.3
d [mml
I [mml d [mml
s [mml a [mml b [mml
I3 [ ~ s I 1.1 . 10'
8 .9 . 10'
6.7. 10'
63
100 40
20 100 100
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0 T , r = Omm A T, r =20mm
0 -
kg dry sub.
FIGURE 3: Drying rate and temperature at different radii during highfrequency drying. Sphere. U = 10.3 kV. D = 20 cm, r = 1000 pS/cm
distance D = 20 cm and the conductivity K = 1000 pS/cm. The dlying rate in-
creases with decreasing moisture content and passes through a maximum. The
temperatures increase from the initial temperature and a temperature profile from
the center to the edge of the product evolves. The temperatures at the inner radii
reach boiling point temperature and exceed boiling point temperature at a moisture.
content X < 0.05. The appearance of the maximum is due to the changing permit-
tivity of the product during drying and will be discussed later.
The influence of the electrode voltage on drying rate and temperature is shown
in figure 4. The product is a sphere, the electrode distance D = 20 cm and the con-
ductivity K = 100 pS/cm. The higher electrode voltage increases the outside field
strength and therefore the power density, equation (19). and the drying rate. The
boiling point temperature is reached quicker with the higher outside field strength.
The influence of the electrode distance on drying rate and temperature is
shown in figure 5. The product is a sphere, the electrode voltage U = 13.7 kV and
the conductivity K = 1000 pS/cm. Reducing the electrode distance has the same
effect as increasing the electrode voltage ie . increasing the outside field strength.
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0 0.1 0.2 0.3 0.4
X ] kg dry sub.
FIGURE 4: Influence of electrode voltage on drying rate and temperature Sphere. D = 20 cm. K = 100 pS/cm
FIGURE 5 : Influence of electrode distance on drying rate and temperature. Sphere, U = 13.7 kV, K = 1000 pS/cm
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1512 . BADER AND SCHL~JNDER
The general shape of the drying curve, that is the existence of a maximum and the
moisture content at the maximum drying rate, is unchanged by varying the outside
field strength.
It is interesting to compare these results with the ones obtained by Dostie [lo].
Dostie observed a change in the shape of the drying curve with increasing outside
field strength, due to non-uniformity of the moisture distribution in the product.
This effect was not observed in this work, where the product had a homogeneous
porosity.
The first maximum at moisture content X = 0.38 in the drying curve with
electrode distance D = 20 cm is due to moisture leaving the product in liquid form
because of increased pressure inside the product afler reaching boiling point tem-
perature. This effect can reduce the energy consumption, because the enthalpy of
vaporization of the moisture leaving the product liquid has not to be dissipated by
the highfrequency generator. The usage of this effect is limited by the mechanical
stress, that the product can withstand.
The influence of the conductivity of the moisture on drying rate and
temperature is shown in figure 6. The product is a sphere. the electrode voltage U
= 13.3 kV and the electrode distance D = 20 cm. The averaged drying rate in-
creases from K = 100 pS/cm to K = 1000 pS/cm and decreases from K = 1000
pS/cm to K = 10000 )rS/cm. This general effect was foreseen with equation (19).
because at K = 100 pS/cm, the imaginary part of the permittivity of the product is
smaller than its real part, at K = 1000 pS/cm, the imaginary part has about the same
value as the real pan and at K = 10000 pS/cm, the imaginary pan is bigger than the
real pan.
The general shape of the drying curve changes with electric conductivity and
the moisture content at the maximum drying rate decreases with increasing con-
ductivity.
The moisture content at the maximum drying rate of the experiment with K =
10000 pS/cm is X = 0.03, which is a remarkable small value compared with X,, =
0.15, the critical moisture content during convective drying ofthis material.
The influence ofthe geometry of the product on drying rate and temperature is
shown in figure 7. The electrode voltage is U = 13.4 kV, the electrode distance D
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HIGH FREQUENCY DRYING OF POROUS MATERIAL
FIGURE 6: Influence of conductivity on drying rate and temperature. Sphere, U = 13.3 kV, D = 20 cm
= 20 cm and the conductivity K = I000 pS/cm. To examine the effect of geometry
exclusively, one has to choose products with constant surfaceJvolume ratio, be-
cause on one hand, the total convective heat flow and the total evaporation flow
are proportional to the product surface while the dissipation of highfrequency en-
ergy is proportional to the volume, and on the other hand, the drying rate in the
figures is the total mass rate divided by the surface area.
The drying rates increase from plate to cylinder to sphere. The experiments
demonstrate the increasing focusing effect from plate to cylinder to sphere, as it
was anticipated by equation (19).
A comparison between calculation and the experiment of figure 3 is shown in
figure 8. The calculation reproduces qualitatively the maximum in the drying curve.
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BADER AND SCHLUNDER
FIGURE 7: Influence of the geometry of the product on drying rate and tem- perature U = 13 4 kV, D = 20 cm. K = 1000 pS/cm
FIGURE 8: Comparison o f measured and calculated drying rate and tempera- ture. Sphere. U = 10.3 kV. D = 20 cm, K = 1000 pSicm
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HIGH FREQUENCY DRYING OF POROUS MATERlAL 1515
The temperature of the isothermal calculation is between the measured center and
edge temperature. The appearance of a maximum will be illustrated with figure 9.
Equation (19) is represented graphically in figure 9 in the form of lines of con-
stant power density With all possible combinations of the real and the imaginary
part of the complex permittivity yielding a certain power density, it is possible to
draw lines of constant power density in the complex permittivity plane. The power
densities increase by a factor 1.3 from line to line towards the origin of the dia- gram. These lines can be seen as contour lines of the power density surface in 3-d
space that gets steeper towards the origin.
The calculated complex permittivity of the product as it varies during drying is
also drawn in figure 9. The permittivity of the wet product at the beginning of
drying is marked with X = X, and the permittivity of the dry product is marked
with X = 0. Between these limits, the real part of the complex permittivity of the
product decreases with decreasing volume fraction of the moisture. The imaginary
part first increases because of the increasing conductivity of the moisture with
temperature, see equation (28). Later. the decreasing volume fraction of the mois-
ture also dominates and the imaginary pan decreases with further drying.
The permittivity of the product goes uphill on the power density surface with
decreasing moisture content until it is tangent to a line of constant power density.
This point corresponds to the maximum in the drying curve in figure 8. Mewards.
the power density decreases, as can be seen in figure 9.
Figure 10 compares calculation and experiment in respect to the inkluence of
the conductivity. The experiments are the ones of figure 6. The calculation repro-
duces qualitatively the change in shape of the drying curves with increasing con-
ductivity. The shift of the moisture content at the maximum drying rate towards
smaller values with higher conductivity can be illustrated in figure 9 in the same
way as the experiment of figure 8 was illustrated.
Figure I I compares calculation and experiment in respect to the influence of
the geometry of the product. The experiments are the ones of figure 7. The calcu-
lation reproduces qualitatively the increasing focusing effect from plate to cylinder
to sphere.
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BADER AND SCHLUNDER
- lines ofeonaant powa density
FIGURE 9: Lines of constant power density and evolution of the permittivity of the product during drying in the complex permittivity plane
FIGURE 10: Influence o f conductivity on drying rate. Comparison of experiment and calculation. Sphere, U = 13.3 kV. D = 20 cm
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HIGH FREQUENCY DRYING OF POROUS MATERIAL
.. m rn sphere iI, cylinder
.~ . . . o m. plate - rh, cal.. sphere . - .- ,i,, -1,. cylinder - -iI, cal., plate
0 0.1 0.2 0.3 0.4 0.5
kgHP I [kg dry rub.
FIGURE I I : Influence of the geometry of the product on the drying rate. Com- parison of experiment and calculation. U = 13.4 kV. D = 20 cm. x = 1000 uSIcm
The quantitative deviation between experiment and calculation is quite big.
There are physical effects for which the simple calculation does not account for, as the temperature and moisture profiles in the product. the inhomogeneous
permittivity. the effect of gravitation on the moisture distribution, the internal heat
and mass transfer resistance. and the deviation of the geometry of the product from
the ideal cylinder and the ideal plate in respect to the calculation of the electric
field.
In spite of all these simplifications, the effect of the conductivity of the mois-
ture and the geometry of the product on the drying rate can be understood with
this simple calculation.
CONCLUSIONS
Some theoretical electrodynamic aspects of highfrequency drying were dis-
cussed and results of highfrequency drying experiments were reported.
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1518 8kDER AND SCHLUNDER
The theoretical discussion pointed out. that it is useful to identify the type of
field in order to be able to calculate the electric field strength. The eddy current
free quasi-stationary field has the feature, that for a given real part of the complex
permittivity of the product. there exists an optimal imaginary part of the permittiv-
ity for maximum power dissipation.
The imaginary pan is essentially determined by the volume fraction of the
moisture and the electric conductivity of the moisture. When the importance of the
conductivity of the moisture is realized, the next step is to look at it as a degree of
freedom. If the conductivity of the moisture does not happen to be in the right or-
der of magnitude by chance, perhaps one is allowed, in respect to product specifi-
cation, and able to influence it.
The influence of electrode voltage, electrode distance, electric conductivity of
the moisture and geometry of the product were investigated in the highfiequency
drying experiments. The drying curves at constant plate voltage exhibits a maxi-
mum. which results from the changing permittivity of the product during drying.
Varying the electrode voltage and the electrode distance are both basically
varying the outside field strength. The drying curves keep the same general shape
and are scaled with the field strength. When increasing the conductivity, the aver-
aged drying rate first increases and decreases after passing the optimal conductiv-
ity. The general shape of the drying curve changes and the moisture content at the
maximum drying rate decreases with increasing conductivity. The geometry of the
product has an increasing focusing effect on the electric field from plate to cylinder
to sphere. These effects can be understood and reproduced qualitatively with the
calculation.
Finally. the authors try to give an answer to the question raised in the intro-
duction: "What is the typical drying curve in the case of highfrequency drying?".
First, it has to be identified how the highfrequency generator operates.
If the generator operates with constant electrode voltage, one has to check if
the outside field strength is tangent to the surface ofthe product. If this is the case.
the energy dissipated in the product is proportional to the loss factor of the prod-
uct. Does the loss factor decrease with decreasing moisture content, then the dry-
ing rate is steadily decreasing from the beginning to the end of drying. If the out-
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HIGH FREQUENCY DRYING OF POROUS MATERIAL 1519
side field strength is not tangent to the product surface, which is the case with
bulky products between electrode plates, then the power density is a more compli-
cated function o f both the real and the imaginary pan o f the permittiviry. This can
result in a maximum in the drying curve in the case o f quasi-static fields and in the
case of eddy current free quasi-stationary fields. Dostie [ lo ] and this work are ex-
amples for this type o f drying curve.
If the highfrequency generator operates at constant power input, then a con-
stant rate period i s observed as long as the matching device is able to compensate
for the changing dielectric properties. Alexander [I] i s an example for this type of
drying rate. where the power input to the oscillator i s kept constant by continual
retuning o f the Lecher wire system. The highfrequency generator can be controlled
in other ways. Besser 161 controlled the product temperature at a fixed point by
varying the input voltage to the power unit. which resulted in a continuously de-
creasing drying rate.
It is therefore not possible to identify a single typical drying curve of highfre-
quency drying. But regardless o f the generator being controlled in some way or
not, the changing complex permittivity during drying is crucial for understanding
the resulting drying curve.
ACKNOWLEDGMENT
The authors gratefully acknowledge the financial support o f the AIF
(Arheirs~emei~r.chf lmrhsrrieller For.vchroi~n~ereir~ip~~~geri e. I'.. Kijhi) together
with the FLT ( / ~ o r . s c h ~ r ~ r y v ~ ~ r r ~ i ~ ~ i ~ ~ ~ i ~ ,fiir lhfl- rrrrd liockrirrrrpsrrchrrik e.1'.
l~~urrkfitrr).
NOMENCLATURE
c molll concentration
Jl(kg K) heat capacity
d m size o f the product; diamete~
f Hz frequency
~ h , Jlkg enthalpy o f vaporization
j fi, imaginary unit
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BADER AND SCHLUNDER
ks/(m2h)
W/ml
Pa
s
mls
m'lm'
m :
Vdm'
Wl(m2K')
W/(mzK4)
Aslma
m Vlm
Aim
Alrnl
kg kglmol
W
Alm2
K, OC
v ,1
kg H,O I kg dry sub.
W/(m2K)
d s
m
As I(Vm)
As IlVm) A s h '
pS/cm
wS1cm
mixing exponent. equation (21)
drying rate
geometry parameter
power density
pressure
time
propagation velocity in the medium
volume fraction of phase p
surface of the product
magnetic induction
radiation coefficient
Stefan Boltzmann constant
displacement current density
distance between electrodes
electric field strength
magnetic field strength
conduction current density
mass molecular weight
power total current density
temperature
electrode voltage
volume
moisture content
heat transfer coeficient
mass transfer coefficient
penetration depth
permittivity
permittivity of free space
density of volume charge
electric conductivity
electric conductivity of water at To = 25 .'C
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HIGH FREQUENCY DRYING OF POROUS MATERIAL
P vd(Am) permeability > moVm' molar density I s relaxation time
w I Is anyular frequency We I Is limiting frequency A Scm2/mol molar conductivity Ao Scm2/mol molar conductivity at infinite dilution
Subscripts
n I
I
n
0
stat
gas inside
liquid
normal
outside
phase index
relative
solid
static
tangential
water
sector I
sector 2
bulk: at infinite frequency. equation (23)
complex number
Supcncripts - molar I saturation
real part of a complex number
imaginary pan of a complex numbel
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1522 BADER AND SCHLUNDER
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HIGH FREQUENCY DRYING OF POROUS MATERIAL 1523
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