Drying Model for Calcium Alginate Beads†
Margaret E. Lyn*,‡and DanYang Ying§
US Department of Agriculture, Agricultural Research SerVice, Application & Production Technology and
Biological Control of Pests Research Units, StoneVille, Mississippi 38776, and CSIRO Food and Nutritional
Science, 671 Sneydes Road, Werribee, Melbourne, VIC3030, Australia
The dehydration of calcium alginate beads is observed to have a simple power-law time dependence. The
ratio of the water content, M(t), to the initial water content, M0, can be expressed as
The parameter tcis insensitive to the guluronic content of the alginate or the degree of cross-linking. Using
this simple model, tccan be estimated from the properties of water vapor, the initial water content, and the
relative humidity, in addition to an empirically derived correction factor, K, which has been found to be 2.8
Obtained from certain types of brown seaweed, the natural
polymer called alginate has widespread applications in the
food,1-3pharmaceutical,4,5cosmetics, medical,6and biotech-
nology7,8industries. One of the outstanding features alginate
offers is the instant gelling property when the alginate solution
is in contact with Ca2+or other di- and trivalent cations, which
induce instant cross-linking.5,9-11This feature is why alginate
is frequently used for cell immobilization and microencapsu-
lation of a wide range of bioactives.
Our particular interest is the use of this material to encapsulate
bioactives and especially, living beneficial microbiological
agents for the control of agricultural pests.12An important step
in such a formulation is the drying of calcium alginate beads
containing the bioactives to a low water content appropriate
for long-term storage. A simple model that describes most of
the drying process could be useful in this and other applications
requiring dried alginate beads. Research areas involving the
development of agricultural biocontrol formulations, drug
delivery systems,13and probiotic foods14-18where alginate
beads containing immobilized cells and other bioactive com-
pounds are air-dried would benefit from a drying model that
could describe the drying process to nearly 100% water mass
Availability of a model to estimate either the time duration
necessary to reach an optimal moisture content or the conditions
to achieve a desired drying time would be beneficial. Such a
model could contribute to the development of an efficient
evaluation phase in the research and development of air-dried
alginate products or to the optimization of processing times in
industrial applications. However, a review of the literature on
the drying of gels19-24suggests that no simple model can be
used to describe the entire drying process. In this paper, we
point out that for the specific case of calcium alginate beads,
nearly the entire process can be described with the simple model
Model of Bead Dehydration
It will be assumed that the loss of mass of a drying bead is
governed by the diffusion of saturated water vapor at the surface
of the bead into the surrounding air at ambient relative humidity.
As such, the water loss is given by the Fickian laws of diffusion
assuming spherical symmetry. If Csatis the concentration of
water vapor at the surface of a bead with radius R and Cambis
the water vapor concentration at a distance l from the center of
the sphere, then the steady solution to the diffusion equation
gives the concentration of water vapor, C(r), at an arbitrary
distance, r, as25
The quantity of water vapor leaving the surface per unit time
where Dvis the diffusion coefficient of water vapor in air. For
l . R, the condition to be used for experiments,
or in terms of relative humidity, Rh,
The mass of a bead or droplet of water in terms of its density
and an additional factor, f, is
where F is the density of beads, which is very close to that of
water, 1.0 g·cm-3. The origin of the factor f, a truncation factor,
is described below.
For comparison with theory, the bead in an experiment should
maintain a well-defined shape during drying. Ideally, that shape
should be spherical; however, a spherical shape cannot be
obtained if the bead is to be supported on a flat surface. Under
such circumstances, the shape can be approximated as a
†Mention of trade names or commercial products in this article is
solely for the purpose of providing specific information and does not
imply recommendation or endorsement by the U.S. Department of
* To whomcorrespondence
firstname.lastname@example.org. Phone number: (662) 686-3641. Fax
number: (662) 686-5281.
‡US Department of Agriculture.
§CSIRO Food and Nutritional Science.
)(r - l)R
(R - l)r
dM/dt ) -4πDv(Csat- Camb)Rl/(l - R)
dM/dt ) -4πDv(Csat- Camb)R
dM/dt ) -4πDvCsat(1 - Rh/100)R
M ) 4/3πR3Ff
Ind. Eng. Chem. Res. XXXX, xxx, 000
XXXX American Chemical Society
truncated sphere with the degree of truncation being determined
by the hydrophobicity of the support surface. A suitable surface
for use in experiment is Teflon for which the contact angle is
According to Lee et al., the volume of a sphere truncated at
a surface with contact angle θ is27
For a bead or water droplet with a contact angle of 110° on a
That is, for a bead or water droplet supported on a Teflon
surface, the appropriate value of the truncation factor, f, is 0.75.
Solving eq 2 for R and substituting into eq 1 gives
which, on integration, gives
where M0corresponds to an initial water mass when t ) 0 and
Here an additional factor, K, will be included to emphasize the
influence of several effects not explicitly included in our simple
model. Therefore, eq 4 may be rewritten as
The value of K, which can be viewed as an empirical scaling
factor, is assumed to be constant for all values of relative
humidity. This empirical scaling factor K accounts for the fact
that the bead is on a surface which will result in nonspherical
water vapor isobars. In addition, K accounts for a bead
temperature that is lower than ambient temperature due to
For water at a temperature of 25 °C and normal atmospheric
pressure, Dv) 0.256 cm2·s-1and Csat) 23 g·m-3.28,29Using
these values in eq 5 and K ) 2.8, Figure 1 shows how tccan be
expected to depend on M0and Rh. Such a plot may be used to
predict experimental tcvalues using measurable experimental
parameters. For example, beads with an initial water mass of
15 mg in air at 50% relative humidity is expected to reach mass
equilibrium within 150 < tc< 200 min or in approximately 175
In this paper, the dehydration kinetics of calcium alginate
beads will be experimentally investigated and the above drying
model will be applied to beads air-dried under different
experimental conditions of M0and Rh. Further, the validity of
the model for applications of alginate with different guluronate
content will be investigated and the effect of bead size will also
Materials and Methods
Preparation of Calcium Alginate Beads. A 1% (w/v)
sodium alginate solution was added dropwise to a dilute calcium
chloride solution from a 10 mL syringe fitted with a 16 gauge
needle 3.8 cm in length. Sodium alginate droplets freefell a
distance of ∼3.8 cm, measured from the needle tip to liquid
surface, into the unstirred dilute calcium chloride solution or
so-called curing bath. Specific details are described.
The 1% (w/v) sodium alginate solution was prepared as
follows: Approximately 5.0 g of Kelgin HV (ISP Alginates),
sodium alginate with a 39% guluronic content, was dissolved
in 500 mL of deionized water. The mixture was stirred overnight
with a stir bar to allow dissolution of the alginate salt.
A saturated calcium chloride CaCl2solution at 24 °C was
prepared by adding an excess of 75 g of CaCl2to 100 mL of
deionized water. Two 10.00 mL aliquots of the saturated CaCl2
solution were quantitatively transferred to a 1.0 L volumetric
flask and brought to volume with deionized water. The curing
bath was prepared by transferring 100 mL of the dilute CaCl2
solution to a 175 mL, 7.6 cm diameter crystallization dish. In
less than 2 min, approximately 1.7-2.0 mL 1% (w/v) sodium
alginate was added to the curing bath in droplets totaling 60-70
beads. Collision between incoming droplets and existing beads
was avoided by laterally moving the needle above the surface
of the dilute CaCl2solution.
The curing time, or cross-linking time, of 1% (w/v) sodium
alginate droplets in calcium chloride was monitored with a stop
watch. Sodium alginate droplets, now calcium-cross-linked
alginate beads, remained in the curing bath for 80 min without
stirring. After ∼6 min in the curing bath, the bead density
became greater than the CaCl2solution and the beads sank.
After 80 min in the curing bath, the beads were poured into
a 56 mm diameter Buchner funnel and rinsed with 250 mL
deionized water. The beads were then transferred to a weighing
boat where excess water was removed by blotting with kim-
wipes. Ten beads, qualitatively similar in size, were selected
for the dehydration measurements.
Samples A-E were all prepared using the above procedure.
To test the effect of curing time on the drying behavior, sample
F was prepared using the same procedure but cured for 24 h
instead of 80 min as used for curing sample A-F. Sample G
was prepared similarly as F but was allowed to swell in
deionized water where the mass of each bead nearly doubled.
Sample H was prepared using Manugel GHB, with 63%
V ) 4/3πR3B2(3 - 2B)
B ) 1/2(1 - cos θ)
V ) 4/3πR3× 0.75
dM/dt ) -4πDvCsat(1 - Rh/100)(4/3πFf)-1/3M1/3
8πDvCsat(1 - Rh/100)
8πDvCsat(1 - Rh/100)
Figure 1. Calculated dependence of tcon relative humidity and initial water
Ind. Eng. Chem. Res., Vol. xxx, No. xx, XXXX
guluronic content, following the same procedure for samples
A-E. A summary of sample properties and curing times is
shown in Table 1.
Dehydration Kinetics. Different relative humidities, Rh’s,
were created inside a Mettler Toledo AG 204 analytical balance
using ambient conditions or by sealing the draft doors with
paraffin and using different saturated salt solutions or phosphorus
pentoxide, P2O5. P2O5was used to create a 21% and 31% Rh
environment. Potassium chloride and potassium sulfate were
used to create an Rh of 61% and 75%, respectively. An
imperfectly sealed balance prevented the saturated salt solutions
equilibrium Rh’s from being obtained. The sealed balance was
allowed to come to a steady state Rhovernight. A VWR digital
hygrometer/thermometer housed inside the balance was used
to measure the air temperature and steady state Rh.
Ten beads, similar in size, were selected and transferred to a
clean Teflon surface with a known average weight. After
isolating the ten beads by a distance of ∼1.7 cm from each
other, the beads were quickly transferred to the balance. Mass
loss with respect to time was recorded using a Mettler Toledo
LC-P45 printer at various time intervals until ∆M/∆t ) 0.
Results and Discussion
The alginate backbone is characterized by varying block
lengths of D-mannuronic (M), L-guluronic (G), and MG acid
units whereas the M/G ratio and arrangement of the blocks
varies with seaweed type. Dissolution of the sodium salt of
alginic acid in water produces polyanionic alginate, which cross-
links with various di- and trivalent cations such as Ca2+to
produce calcium alginate hydrogels or herein referred to as
beads. According to the egg-box model proposed by Rees,9-11
gel formation occurs via complexation of the cation with
consecutive guluronate residues. Therefore to evaluate the
validity of the model and applicability to beads with either
different guluronic content or M0, sodium alginate of two
different guluronic contents (39% and 63%) were evaluated in
addition to beads with different M0. In all, eight different samples
labeled A-H were analyzed (Table 1).
Applicability of the model to the drying of beads with similar
guluronate composition and M0(samples A-C and E) in air at
different relative humidites was first evaluated. The loss of water
mass from alginate beads A-C and E as a function of time at
four different drying humidities is shown in Figure 2. Experi-
mental data points have been fitted with the 3/2 power law given
in eq 3 using the indicated value of tc. The best fit of these
experimental data limited tcto three significant figures. For each
set of beads, the model shows a good fit to the data over most
of the drying period. This result indicates that the model appears
adequate in describing the drying curve for beads of similar
size and guluronic composition under different drying humidities.
The experimental tcvalues indicated in Figure 2 were obtained
by fitting the curves for up to 99% water mass loss. From here
on, this fitting parameter will be denoted as tc(exp) to distinguish
it from the theoretical value, tc(calc), obtainable from eq 4. The
ratio tc(exp)/tc(calc), limited to two significant figures by either
M0or Rhthrough eq 4, is given as K. This empirical scaling
factor, K, was included into the model, as shown in eq 5, to
emphasize the influence of several effects not explicitly included
in the simple model and is necessary to bring tc(calc) into
agreement with experimental data.
Since a spherical bead or water droplet on a supporting
surface adopts a truncated spherical geometry, nonspherical
water vapor isobars and higher water vapor pressures in the
restricted space near the droplet-surface interface are to be
expected in experiments. In these experiments, the presence of
a supporting surface reduces evaporation rates and leads to larger
experimental tcvalues in comparison to calculated tcvalues in
which the presence of a surface was not entirely taken into
account (eq 4). Unfortunately, inclusion of either a dynamic
temperature to account for evaporative cooling or a supporting
surface into the model prohibits a simple solution to the diffusion
equations. However, semiempirical values of tcmay be predicted
using eq 5 with an assumed value of K ) 2.8 ( 0.3 (discussed
Applicability of the model to beads with either different
guluronic content or different M0 but similar guluronate
composition was also evaluated. Figure 3 shows the experi-
mental drying curves for samples D and F-H fitted with the
3/2 power law given in eq 3 using the indicated value of tc. For
these samples, the model shows a good fit to the data over most
of the drying period. This observation indicates that the model
may also be adequate in describing the drying behavior of beads
Table 1. Summary of Sample Properties and Curing Times
sample IDguluronic contentcuring timediameter
Figure 2. Drying curves for calcium alginate beads with similar guluronate
composition and M0 in air (samples A-C and E) at different relative
humidities. The solid lines are a 3/2-power-law fit using the indicated values
Figure 3. Drying curves for calcium alginate beads with either different
guluronic content or different M0in air (samples D and F-H) at different
relative humidities. The solid lines are a 3/2-power-law fit using the indicated
values of tc.
Ind. Eng. Chem. Res., Vol. xxx, No. xx, XXXX
with different guluronic composition or initial water mass when
air-dried at a mid to low relative humidity.
A summary of tc(exp) and the empirical scaling factor, K,
for the eight different bead samples A-H is presented in Table
2. Dehydration kinetics measurements for each sample were
taken at the indicated steady state Rh’s. Samples A-C and E
are similar in both guluronate composition (39%) and M0(∼15
mg) whereas samples D, F, and G are also similar in guluronate
content (39%) but differ in initial water mass ranging from 11
to 30 mg. In contrast to samples A-G, sample H was prepared
from an alginate with 63% guluronic content.
Despite differences among the samples, no particular trend
in either guluronic content or M0or Rhcould be observed for K
(Table 2). In other words, there does not appear to be any
relation between the empirical scaling factor K and sample
differences arising from alginate type and initial bead size.
Consequently, the spread in K as shown in Table 2 may be
attributed to indeterminate errors in sample measurements
whereas the magnitude of K can be attributed to the presence
of a supporting surface in the experiments and by the process
of evaporative cooling. As previously discussed, inclusion of
either a dynamic temperature to account for evaporative cooling
or a supporting surface into the model prohibits a simple solution
to the diffusion equations.
The distribution of K values in Table 2 may be averaged to
give a value of K ) 2.8 ( 0.3 (99% confidence interval) for
use in predicting tcfor any arbitrary set of M0and Rhusing eq
5. Figure 4 presents this confidence interval range for tc(calc)
values for any arbitrary M0 and Rh. Superimposed on the
predicted tcplot are tc(exp) values from Table 2, which are
represented by open circles. Overall, the semiempirical model
appears to predict tc reasonably well for each of the eight
different samples. However, use of the model appears to be more
appropriate when drying under mid to low humidity conditions
rather than at higher humidities. At mid to high Rh, the model
appears to underestimate tc. This observation can be explained
by the relation between tcand Rh. It can be seen from eq 4 that
tcis a strictly increasing function that is asymptotic to Rh) 1.
That is, as Rhapproaches 100%, tcapproaches infinity. Thus at
high relative humidities, small differences between true Rhand
measured Rhwill lead to larger differences between fitted tc
(based on true Rh) and predicted tc (based on measured Rh)
values in comparison to drying at lower humidities.
While the simple model described herein has been shown to
be applicable to calcium alginate beads, the most significant
result from this investigation is that all the drying curves for
the different calcium alginate beads studied follow the same
power law, which was derived using a model that should be
most appropriate for a pure water droplet since the properties
of alginate do not enter into the equations. In fact, an attempt
to investigate the drying of a water droplet revealed the
following problem. As a truncated spherical water droplet
evaporates, it has a tendency to remain pinned to a fixed contact
surface. If the contact surface were to remain unchanged while
the droplet evaporates, the shape of the droplet would tend
toward that of a pancake. Although pinning is only a metastable
condition caused by surface irregularities, it did make it difficult
to obtain reproducible data for water droplets. For the calcium
alginate beads, pinning did not prove to be a problem. The
surprising result is that the model developed for a water droplet
works better for beads than for water droplets.
More complex models have been necessary for other
hydrogels.30-32An explanation to account for our observation
may be that strong interchain interactions prevent collapse of
the barrier layer in the case of alginate hydrogels. It is unclear
at this point whether this simple model could be extended to
nonalginate hydrogels. This is the first study in a series to
develop a simple model to describe the drying of alginate beads
for different applications. Additional studies are underway to
investigate applicability of the model to alginate beads loaded
with starch solids.
It has been shown that the time dependence of water loss
from calcium alginate beads, over almost the entire drying range,
can be described by a single simple expression, (1 - t/tc)3/2. At
25 ( 1 °C, the model has been found to be valid for alginates
with 39% and 63% guluronic content, for beads ranging from
11-30 mg and for beads air-dried at 20-75% relative humidity.
The parameter tcis easily estimated from the physical parameters
of water vapor, the initial water content, and the relative
humidity, in addition to an empirically derived correction factor,
K. An explanation of this surprising fact might be that the
surface of the bead is covered by a thin layer of free water. As
this water evaporates, it is continuously renewed by fresh water
expelled from the interior of the shrinking beads. The calcium
alginate beads dry essentially as a water droplet.
The authors would like to thank Mr. Frank Fusiak from
International Specialty Products (ISP) for providing the sodium
alginate samples for experimental use.
Table 2. Best Fit of tcto Various Dehydration Curves for Beads of
Similar Size and Guluronic Content (Samples A-E), Beads of the
Same Guluronic Content and Different Sizes (Samples F and G),
and Beads with Different Guluronic Content (sample H) Air Dried
at the Indicated Relative Humidities at 25 ( 1 °Ca
M (mg) % moisture Rh(%) tc(exp) (min) K
Kelgin HV (39)
Kelgin HV (39)
Kelgin HV (39)
Kelgin HV (39)
Kelgin HV (39)
Kelgin HV (39)
Kelgin HV (39)
Manugel GHB (63)
21 ( 4
31 ( 4
51 ( 2
66 ( 2
75 ( 4
61 ( 2
20 ( 4
31 ( 4
aThe empirical scaling factor, K, is derived from the ratio tc(exp)/
Figure 4. Predicted tcvalues for any arbitrary set of M0and Rhwithin the
99% confidence interval limits (shaded zone) for K ) 2.8 ( 0.3. Open
circles represent experimental data points.
Ind. Eng. Chem. Res., Vol. xxx, No. xx, XXXX
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ReceiVed for reView September 15, 2009
ReVised manuscript receiVed December 30, 2009
Accepted January 11, 2010
Ind. Eng. Chem. Res., Vol. xxx, No. xx, XXXX