DYNAMICS OF WATER SORPTION ON COMPOSITES “CaCl2 IN
SILICA”: SINGLE GRAIN, GRANULATED BED, CONSOLIDATED
LAYER
YU.I.ARISTOV, I.S.GLAZNEV, L.G.GORDEEVA
Boreskov Institute of Catalysis, Pr.Lavrentieva 5, Novosibirsk,
630090 Russia
I.V.KOPTYUG, L.YU.ILYINA
International Tomography Centre, Institutskaya 3a, Novosibirsk,
630090 Russia
J.KÄRGER, C.KRAUSE
University of Leipzig, Linnéstraße 5, D-04103 Leipzig, Germany
B.DAWOUD
Aachen University (RWTH–Aachen), Schinkelstrasse 8, D-52062 Aachen,
Germany
1. Introduction
Development and study of novel nanoporous adsorbents is a very challenging
goal of current material science research. Among these materials are new
alumosilicates, alumo- and iron phosphates, MCMs, SBAs, MOFs, carbon
sieves, etc., with the pore size of few nanometers. Less progress is made in
developing new adsorbents with relatively large pores of 7-20 nm.
The idea to modify a common mesoporous adsorbent (silica, alumina, etc.) by
introducing inside its pores a hygroscopic salt (CaCl2, LiBr, etc.) was known at
least since 1929 [1]. Such materials were used as adsorbents in gas masks and
then were almost forgotten. New impact has recently been done in a set of
papers [2-6] which presented the result of systematic study of adsorption and
thermophysical properties of a family of composite materials “salt inside
porous host matrix” which were called “Selective Water Sorbents” (SWSs).
Some of the SWSs synthesised and studied are displayed in Table 1.
Despite of the short renaissance period these materials have already met
practical applications as gas drying agents and materials for active heat
insulation in aerocraft “black boxes”. For these and other possible applications
that are under development now (freonless solar driven chillers [7] and fresh
water production from the atmosphere [8]) the dynamics of water sorption is of
high interest.
2
Here we present the first results on kinetics of water sorption on composite
“CaCl2 in mesoporous silica KSK” (SWS-1L) that at the moment is the most
studied material of SWS-type [3,5]. Here we focus mainly on the dynamics of
water adsorption under vacuum conditions when water vapour is the only
component in the gas phase although few results are reported on water
adsorption from air flux as it takes place in air drying units. Three adsorbent
configurations will be considered, namely, a single grain, a granulated layer and
a consolidated layer prepared with a binder and pore-forming additives.
Three methods have been used to study the water transport and sorption: a) 1H
NMR microimaging experiments on the spatial distribution of sorbed water in
the sorbent and its temporal evolution, b) the PFG NMR method and c) the
kinetics of water sorption under constant volume-variable pressure conditions.
The first method is a direct one because it directly measures water
concentration profiles and can give the water transport diffusivity. The second
method is used for measuring water self-diffusivity, and the last one, in
principle, allows to estimate the water diffusivity from the uptake curves [9].
TABLE I. A list of some SWS materials
synthesised and studied so far
Host
Water sorption,
matrix
g/g
Silica gel KSK
0.65
Silica gel KSM
0.25
aerogels
1.0-1.50
Carbon Sibunit
0.57
Al2O3
0.52
Silica gel KSK
0.67
Silica gel KSM
0.25
Carbon Sibunit
0.60
Al2O3
0.55
Silica gel KSK
0.60
Remarks
SWS-1L
SWS-1S
SWS-1Aero
SWS-1C
SWS-1A
SWS-2L
SWS-2S
SWS-2C
SWS-2A
SWS-3L
[3]
[4]
[5]
[5]
[5]
[6]
[6]
[6]
[6]
[6]
LiCl
Silica gel KSK
0.60
SWS-4L
[6]
MgSO4
Silica gel KSK
Al2O3
0.65
0.50
SWS-5L
SWS-5A
[5]
NaSO4
Silica gel KSK
0.62
SWS-6L
[5]
CuSO4
Silica gel KSK
0.58
SWS-7L
[5]
Salt
CaCl2
LiBr
MgCl2
Reference
3
2. Experimental
1
H NMR microimaging experiments were performed at 300 MHz on an Avance
NMR spectrometer equipped with a microimaging accessory. A sample was
placed in the rf coil of the NMR probe with its axis oriented along the magnetic
field of the superconducting magnet, pumped down and connected to an
evaporator to maintain a fixed vapour pressure over the sample during the water
sorption process. More details can be found in [10,11].
The sample was either a single cylindrical grain (diameter 6 mm, length 6 mm)
or a bed of spherical grains 2-3 and 5-6 mm in diameter. The bed diameter was
25 mm, the length was 30 mm. The pellets of the commercial silica KSK (the
average pore diameter 15 nm, the specific surface 350 m2/g, the pore volume 1
cm3/g) were impregnated with a 38 wt.% CaCl2 aqueous solution and then dried
at 1500C.
Consolidated SWS layers were prepared using silica (KSK, Davisil 60 and
Davisil 150) or alumina A1 powder together with a binder (pseudoboehmite).
In order to analyse the relative importance of the diffusional resistance in
macropores between the adsorbent particles and in mesopores inside the
particles, we varied the size of powder particles (between 0.04 and 0.5 mm), the
size of mesopores (between 6 and 15 nm) and the amount of the binder (0-30
wt.%). A carbon Sibunit and an ammonia bicarbonate were used as poreforming additives. The silica gels and alumina were moulded with the binder as
a cylindrical tablet of 16 mm diameter and 4-8 mm thickness and then
impregnated with aqueous solution of CaCl2. Porous structure of the tablets was
determined by SEM, BET and mercury porosimetry. The sample holder allows
the water adsorption only through the upper flat surface of the tablet.
The PFG NMR experiments were performed using the home-built PFG NMR
spectrometer FEGRIS 400 NT operating at a 1H resonance frequency of 400
MHz [12]. The probe head equipment allows measurements at temperatures
from ~120 to 470 K (1H NMR resonance) and gradients up to 35 Tm-1.
The procedure of the kinetic measurement under constant volume-variable
pressure conditions is described elsewhere [13.
3. Results and discussion
3.1. SINGLE GRAIN
First experiments with a single SWS-1L grain were performed under the flow
regime when the dry cylindrical grain (radius R = 3 mm) was placed in a flux of
humid air [10]. 1D profiles of adsorbed water are presented in Fig. 2a. Sharp
front of water adsorption is observed. The front propagates inside the grain
4
with a constant rate  (shrinkage cylinder behaviour) so that the calculated
uptake is mt/m = (2/R) t – (/R)2 t2 and the rate of adsorption decreases with
the sorption time as (2/R) – 2(/R)2 t. This decrease is in line with our
experimental data on water breakthrough curves in a plug flow adsorber with
SWS-1L.
b
tim
e
a
-0.4
-0.2
0.0
0.2
0.4
distance, cm
Figure 2. Visualization of the dynamics of the water sorption by a cylindrical SWS-1L
grain. Air flow FL=390 l/h, relative air humidity RH=55%. a) 2D SPI images;
accumulation time per image was 13 min 39 s; b) 1D profiles of water content along the
diameter of the pellet, accumulation of each profile lasted 34 s, every 8 th profile is
shown. The overall time span of the experiment was 290 min. [10].
The physical reason of the rate decreasing could be a mass transfer resistance in
the liquid layer near the external surface of the grain (Fig. 2b). Indeed, the
surface transfer coefficient ks depends on the layer thickness  and the mass
transfer surface area S = 2 (R – ) L (where L is the grain length) and can be
defined as ks = Dl S/, where Dl is the water diffusivity in the liquid layer. This
coefficient decreases during the water sorption due to gradual reduction of S
with simultaneous raise of  (Fig. 2b). This model could explain the observed
effect only if the water diffusivity in the liquid phase (Dl) is lower than in the
gas phase (Dg). Indeed, for molecular diffusion in large transport pores of silica
the water diffusivity in air at P=1 atm equals Dg = 0.23 cm2/s while in a 3M
CaCl2 aqueous solution Dl = 1.2710-5 cm2/s. Such low gas diffusivities are
observed in small silica pores of 60 nm size where the Knudsen regime takes
place.
In general case, the diffusion through both gas and liquid phase can
take place and for these parallel routes the effective diffusivity D eff can be
5
calculated as 1/Deff = xl /(l Dl) + xg /(g Dg), where xl and xg are the pore
volume fractions occupied by the liquid and gas phase, respectively (xl + xg =
1); l and g are the tortuosity factors for these phases. Thus, to analyse the
water transport in SWS-composites at least two new effects should be taken
into considered, namely, a) the surface resistance due to formation of the
solutions liquid layer near the grain external surface, and b) the complex water
diffusivity both through gas and liquid phase in this layer. This pattern is
formally similar to the case of the gas and surface diffusion but it is more
general as the surface diffusion occurs along the geometrical surface (that
means mainly within the first monolayer, xl << 1) while for SWS materials the
pores can be completely filled with liquid solution (xl = 1) even at relative
humidity 40-70% << 100% (conventional condensation).
Considering a non-condensable gas in such wet porous media, the gas phase
diffusion should strongly depend on the configuration of free pore space
available for gas diffusion that is varied due to water adsorption. The liquid
diffusion is a function of the gas solubility, its diffusivity, the tortuosity of the
layer, etc. This class of problems is quite different from those typical for
microporous solids (hindered diffusion, single-file diffusion, etc.).
In the case of single component (water) diffusion the water
concentration profiles measured by 1H NMR microimaging technique for a
single pellet of the CaCl2/ (silica KSK) are presented in Fig. 3 at time moments
t = n t0, where t0 = 9 min. 40 s, n = 0 - 28. At water uptake w > 0.1 g/g the NMR
signal is proportional to the average concentration of sorbed water. The
adsorption front was found to reach the centre of the pellet approximately after
35-45 min (Fig. 3).
The effective water diffusivity D can be estimated from the temporal
evolution of water concentration profiles. If we assume that the sorbent pellet is
isothermal and the pellet length is infinite, the water transport can be described
by
 1  


 rD
t r r 
r

,

where  is the water uptake in the sample, r is the radial co-ordinate. To obtain
the mass diffusivity from the experimental water concentration profiles this
equation can be integrated with respect to r
1
D
r
r


0
r r  r 
r

dr .
t
6
The thus calculated diffusivity is presented against the sorbed water uptake in
Fig. 4. Despite of the boundary effects, this procedure allows an estimation of
the effective water diffusivity in a single SWS-1L pellet equal to (2.0  1.0)106
cm2/s at water uptakes w = 0.05-0.3 g/g which is close to the values of the
self-diffusivity Dself obtained by the PFG NMR (Fig. 4). For the KSK silica
fully saturated with a 40 wt.% CaCl2 solution this method gives for solution in
pores Dpself = 7.310-6 cm2/s. This value for the bulk solution can be estimated
by the extrapolation of the literature data to high salt concentration Dbself =
1.210-5 cm2/s. For solution filled pores the equation Dpself =  Dbself / is valid
(where  is the porosity of silica KSK single grain ( = 0.726) and  is the
tortuosity of the silica pore). The empirical parameter  can be estimated from
the last equation  = 1.19. Thus, this approach gives very convenient tool to
measure the tortuosity factor.
distance, cm
Figure 3. 1D water concentration
profiles inside a single grain of the
SWS-1L measured every 9 min 40 s
[11]. The sorbent temperature was
21oC, the pressure of water vapour was
17.5 mbar.
Figure 4. Effective water diffusivity in
SWS-1L single grain as a function of
the moisture uptake calculated from the
concentration profiles () (see Fig. 3)
together with the data on the water selfdiffusivity measured by a PFG NMR
().
3.2. PELLETIZED BED
The study of the water transport in a pelletised bed is obviously restricted to
model beds of small size (approximately 25-30 mm) close to the size of the
NMR measuring cell. Typical water concentration profiles inside the SWS-1L
bed were found to be strongly dependent on the pellet size (Fig. 5, a,b).
7
For the small (2-3 mm) pellets the formation of the adsorption front is observed
(Fig.5, a) that indicates the regime of the bed diffusion control. The other
extreme mode is realised for the bed of the large (5-6 mm) pellets where the
sorption is almost homogeneous in space (Fig.5, b). In this case the vapour
easily penetrates the bed so that its pressure is constant along the bed, hence,
the rate of the sorption is defined by the diffusion inside single pellets (the
intraparticle diffusion control). Indeed, it was demonstrated that the kinetics of
water adsorption (obtained by integrating the water concentration profiles) for
the single SWS-1L grain and the bed of such grains almost coincide [11].
a)
b)
distance, cm
Figure 5. Typical water concentration profiles inside the SWS-1L bed consisted of the
pellets of 2-3 (a) and 5-6 (b) mm size [11]. Vapour pressure 17.5 mbar.
The crossover between the bed and the pellet diffusion control depends on the
ratio of the diffusional time constants A= (Dlayer/L2)/ (Dpellet/R2)], where Dlayer
and Dpellet are the effective water diffusivities in the layer and in the pellet, L is
the layer thickness, R is the pellet radius [9]. Assuming that the cross-over
takes place at A  1, the effective water diffusivity in the layer can be estimated
as Dlayer = (Dpellet) (L/R)2  (48) 10-4 cm2/s.
This diffusivity can be considered as an effective diffusivity D layer  
Dlm /(1 - )K, where  is the layer porosity ( 0.4), Dlm is the molecular water
diffusivity in the layer and K is the water sorption equilibrium constant (for
linear systems w = K C, where C is the water concentration in the gas phase).
The Dlm can be written as Dlm = Dm / , where Dm is the molecular diffusivity of
water at 17.5 mbar (Dm = 13.1 cm2/s). Assuming   1.0-1.5 and calculating K
= 2.5 104 cm-3/cm-3 from the water sorption isotherm, one can obtain Dlm = 8.7-
8
13.1 cm2/s and Dlayer  (2.33.5) 10-4 cm2/s, that is close to the value estimated
from the cross-over point.
3.3. CONSOLIDATED SORBENT LAYER PREPARED WITH THE BINDER
Although a pelletised bed can provide sufficient mass transport properties, the
heat transfer is strongly restricted by a poor heat conductivity of the layer that
is not acceptable, for instance, for adsorption heat pumps. An efficient way to
solve this problem is to organise a sorbent as a consolidated layer where
primary sorbent particles of the size R are glued with a binder. This can greatly
enhance the heat transport properties although the mass transfer is now less
efficient, and a reasonable compromise should be reached by varying the
primary particle size, the nature and amount of the binder as well as by using
special pore forming additives. Following this approach we synthesised various
layers (tablets) based on SWS-1L and SWS-1A (see Experimental) and studied
the water adsorption in and transport through these layers by means of the
NMR microimaging.
The typical water distribution profiles in SWS-1L layers are presented in Fig. 6.
The sole difference between the two samples was the diameter of mesopores
inside the primary silica particles that was either 6 or 15 nm. Fig 6 shows that
neither the profile shape nor the profile dynamics are influenced by this size.
Thus, the mass resistance in mesopores can be neglected and bed diffusion
controls the total sorption rate.
thickness - 7 mm. Vapour comes from
the left.
10
8
8
6
6
4
2
4
0
0.0
0.2
0.4
0.6
0.8
2
distance, cm
0
Figure 6. Water distribution profiles in
the SWS-1L layers with various
mesopores, measured at time 11 min
(,); 1 h 37 min (,); 3 h 4 min
(,); 13 h 11 min (,) and 17 h 30
min (,) [11]. Mesopore diameter is 6
nm (open symbols) and 15 nm (solid
symbols). Binder content is 30 wt.%.
Vapour pressure – 6.5 mbar, sample
0
2
4
6
8
distance, mm
Figure 7: Water concentration profiles
in the SWS-1L layers measured at time
11 min (); 1 h 37 min () and 5 h 57
min () [11]. Open symbols - binder
content 20 wt.%, fraction 0.15-0.25
mm; solid symbols - binder content 30
9
wt.%, fraction 0.15-0.25 mm; other
symbols - binder content 20 wt.%,
fraction 0.04-0.06 mm. Vapour
pressure - 6.5 mbar. The
mesopore diameter is 15 nm.
silica
Fig. 7 demonstrates the faster diffusion in the layer consisting of the larger
SWS particles (0.15 - 0.25 mm compared to 0.04 - 0.06 mm) because of the
larger interparticle voids available for vapour transport. When the amount of
the binder is increased, the voids are getting smaller, hence, it takes a longer
time for the water sorption front to propagate inside the layer (Fig. 7). Thus, it
is profitable to decrease the binder content, at least until the layer thermal
conductivity and mechanical strength still remain acceptable. The use of poreforming reagents can also lead to the increase in the size of transport pores that
facilitates the mass transport but with some sacrifice in other two
characteristics [11].
The mechanical strength of SWS-1A layers is much higher than that for SWS1L. This allows a decrease in the binder content down to 2-5 wt.% and a
regulation of the pore structure in a wider range. Again, the water adsorption
becomes faster if the amount of the binder B decreases (Fig. 8). Moreover, the
crossover between the intraparticle diffusion (at B = 20.0 wt.%) and the
interparticle (or bed) diffusion (at B = 2.5 wt.%) is clearly observed (Fig. 8).
For interparticle diffusion control the value of the effective water diffusivity
can be obtained from the shape of the water concentration profiles by means of
the Matano-Boltzmann approach [14]. If in the equation for isothermal
diffusion in a flat geometry
w   w  ,

D

t x  x 
(w is the water uptake in the sample, x is the co-ordinate perpendicular to the
layer surface) substitute t by  = x·t-0.5, all the profiles at t > 183 min. almost
coincide to give a “universal diffusion profile” (Fig.9).
To obtain the mass diffusivity from the measured water concentration profiles
this equation can be integrated with respect to x, yielding
w
Deff  0.5 
1
dw
dw



 0
 d  w

The thus calculated diffusivity is presented against the sorbed water uptake in
Fig. 10. The effective water diffusivity shows the tendency to increase from
0.510-6 cm2/s to 3.410-6 cm2/s during water adsorption, which is in agreement
with the increase of the water self-diffusivity measured by PFG NMR.
10
4. Kinetics of water sorption at constant volume-variable pressure
conditions
The kinetics of water sorption on SWS-1L loose grains on an isothermal plate
(at T = 50oC) has been measured over a water uptake range 0 – 0.4 g/g. Data
obtained for various grain sizes, namely, 0.34-0.5; 0.71-1.0; 1.3-1.6, 3.0-3.2
mm, and various salt contents (0 – 33.6 wt.%) will be published elsewhere.
Here we mainly focus our attention on the kinetics of water sorption on SWS1L(33.6 wt.%) at low water uptake where the formation of CaCl2 hydrates takes
place.
The adsorption kinetic curves are found to be non-exponential with an
exponential tail so that (1 – mt/m) ~ exp(-Kt) at long times t (Fig. 11). To
describe the rate of
water sorption we use the characteristic time of this exponential ( = 1/K) as
well as the times 0.5 and 0.9 that correspond to the dimensionless water
loadings of 0.5 and 0.9, respectively. All these times are presented in Fig 12 at
various water loadings which correspond to appropriate points 1-11 of the
water sorption isotherm. The slowest transformation is observed when the
formation of the salt dihydrate CaCl2 2H2O takes place
CaCl2 + 2H2O  CaCl2 2H2O.
6
Figure 8. Water concentration profiles
in the SWS-1A layers with various
binder content B from 2.5 to 20 wt.%.
The profiles are measured every 86
min. The first one was measured at
time 11 min. Primary particle size is
0.25-0.5 mm. Vapour pressure is 7.2
mbar.
4
uptake mol/mol
2
0
6
4
2
0
6
4
2
0
0
2
4
Distance , mm
6
11
-5.0
-5.5
2
Log D , sm / s
Water content , g/g
0.2
0.1
0.0
0.000
0.002
-6.0
-6.5
0.00
0.004
0.05
0.10
0.15
0.20
uptake g/g
 sm s-0.5
Figure 10. Effective water diffusivity
in SWS-1A layer (20 wt. of binder)
against the moisture uptake calculated
from the “universal diffusion profile”
(Fig. 9). Vapour pressure 7.2 mbar.
Figure 9. Water sorption profiles
presented as a function of . SWS-1A
layer with 20 wt. of binder. Profiles at
3 h 03 min. < t < 7 h 20 min. are
treated. () - “universal diffusion
profile”.Vapour pressure – 7.2 mbar.
A similar dependence (w) is observed for the desorption run which does not
coincide with the adsorption one (Fig. 12). Thus, one more inportant feature of
the materials “salt inside porous matrix” is a slow chemical transformation due
to hydrate formation inside the pores, so that the diffusion with slow reaction
and hysteresis has to be considered. The effective water diffusivity estimated
from these data is changed from 4.210-11 m2/s to 1.810-10 m2/s for w = 0 – 0.11
g/g and from 8.710-11 m2/s to 5.610-10 m2/s at w = 0.11 – 0.47 g/g.
0
2
0.5
10
12
0.14
6000
5000
0.3
0.2
9
5
3
5
10
0.06
2400
3600
4800
6000
7200
Time [s]
Figure 11. Kinetic curves of water sorption
on SWS-1L (33.6 wt.%). Curve numbers
correspond to appropriate points of the
water sorption isotherm (see Fig. 12).
0.04
3
1000
1200
6
4
2000
0
0.08
7
3000
0.1
10
8
4000
1
0.1
0.09
0.08
0.07
0.06
0.12
11
6
Time [s]
1 - mt / m
8
Adsorption Isotherm, 50 oC
Desorption Isotherm, 50 oC
=1/K
0.5
0.9
0.4
0.05
Pressure [mbar]
6
4
7000
1
0.02
2
0
0
Water Uptake [g/g]
1
0.9
0.8
0.7
0.6
0.02
0.04
0.06
0.08
Water Uptake [g/g]
0.1
0
0.12
Figure 12. Characteristic sorption times
for SWS-1L (33.6 wt.% ) as a function
of the water uptake as well as the water
sorption and desorption isotherms(bold
symbols).
12
Acknowledgments. The authors thank the Russian Foundation for Basic
Researches (projects N 02-03-32304 and 02-03-32770), the Integration Grant
Program of the SB RAS (project N 166) and the NATO (grant
PST.CLG.979051) for partial financial support of this work.
13
References
1. US Patent N 1,740,351. (1929) Dehydrating substance, H.Isobe, Dec. 17.
2. Levitskii E.A., Aristov Yu. I., Tokarev M.M., Parmon V.N. (1996) Sol.
Energy Mater. Sol. Cells 44, N 3, pp.219-235. c. References
3. Aristov Yu.I., Tokarev M.M., Cacciola G., Restuccia G. (1996) Selective
water sorbents for multiple applications: 1. CaCl2 confined in mesopores of
the silica gel: sorption properties, React.Kinet.Cat.Lett. 59, N 2, pp.325334.
4. Aristov Yu.I., Tokarev M.M., Cacciola G., Restuccia G. (1996) Selective
water sorbents for multiple applications: 2. CaCl2 confined in micropores
of the silica gel: sorption properties, React.Kinet.Cat.Lett. 59, N 2, pp.335342.
5. Aristov Yu.I. (2003) Thermochemical energy storage: new processes and
materials, Doctoral Thesis, Boreskov Institute of Catalysis, Novosibirsk,
375p.
6. Gordeeva L.G., (2000) New materials for thermochemical energy storage,
Ph.D. thesis, Boreskov Institute of Catalysis, 146p.
7. Aristov Yu.I., Restuccia G., Cacciola G., Parmon V.N. (2002) A family of
new working materials for solid sorption air conditioning systems,
Appl.Therm.Engn. 22, N 2, pp.191-204.
8. Aristov Yu.I., Tokarev M.M., Gordeeva L.G., Snitnikov V.N., Parmon V.N.
(1999) New composite sorbents for solar-driven technology of fresh water
production from the atmosphere, Solar Energy 66, N 2, pp 165-168.
9. Kaerger J., Ruthven D.M. (1992) Diffusion in Zeolites and Other
Microporous Solids, Wiley, N.Y.
10. Koptyug I.V., Khitrina L.Yu., Aristov Yu.I., Tokarev M.M., Iskakov K.T.,
Parmon V.N., Sagdeev R.Z. (2000) 1H NMR microimaging study of water
vapor sorption by individual porous pellets, J.Phys.Chem. 104, pp.16951700.
11. Aristov Yu.I., Koptyug I.V., Glaznev I.S., Gordeeva L.G., Tokarev M.M.,
Ilyina L.Yu. (2002) 1H NMR microimaging for studying the water
transport in an adsorption heat pump // Proc. Int.Conf.Sorption Heat
Pumps, Sept. 23-27 Shanghai, China, pp.619-624.
12. Galvosas P., Stallmach F., Seiffert G., Kärger J., Kaess U., Majer G.
(2001) Generation and Application of Ultra-High-Intensity Magnetic Field
Gradient Pulses for NMR Spectroscopy, J. Magn. Reson. 151, pp. 260-268.
13. Dawoud B., Aristov Yu.I. (2003) Experimental study on the kinetics of
water vapour sorption on selective water sorbents, silica gel and alumina
under typical operating conditions of adsorption heat pumps,
Int.J.Heat&Mass Transfer 46, pp. 273-281.
14. Crank J. (1975)The Mathematics of Diffusion, Oxford Univ. Press, pp.230-
14
238