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International Journal of Refrigeration 86 (2018) 40–47
Contents lists available at ScienceDirect
International Journal of Refrigeration
journal homepage: www.elsevier.com/locate/ijrefrig
Revisiting the adsorption equilibrium equations of silica-gel/water for
adsorption cooling applications
Ramy H. Mohammed
a , b , ∗, Osama Mesalhy
a , b
, Mohamed L. Elsayed
a , b
, Ming Su
c
,
Louis C. Chow
a
a
Department of Mechanical and Aerospace Engineering, University of Central Florida, Orlando, FL 32816-2450, United States.
b
Department of Mechanical Power Engineering, Zagazig University, Zagazig 44159, Egypt.
c
Department of Chemical Engineering, Northeastern University, Boston, MA 02115, United States
a r t i c l e i n f o
Article history:
Received 29 July 2017
Revised 2 October 2017
Accepted 30 October 2017
Available online 7 November 2017
Keywo rds:
Adsorption cooling
Equilibrium models
Silica-gel
Uptake
Capillary condensation
Experimental setup
a b s t r a c t
This paper addresses the discrepancies among the common adsorption isotherms of silica-gel/water pro-
vided in the literature. It is reported that the Freundlich model and Tòth equation cannot be used to
estimate the uptake at relative pressure less than 0.15. In addition, inconsistencies are found among the
various models and equations used to describe the uptake of water vapor onto silica-gel. New coefficients
for the Dubinin–Astakhov (D–A) model are proposed to eliminate these disagreements. Due to the lim-
ited experimental measurements, an experimental setup is designed and built to measure the sorption
kinetics and equilibrium uptake of any working pairs. Experimental measurements show that the maxi-
mum uptakes of silica gel RD-2060 and Type-RD are 0.38 kg kg
−1
and 0.48 kg kg
−1
, respectively. Apparent
capillary condensation is observed at a relative pressure of 0.4 and 0.35 for silica-gel RD and RD-2060,
respectively. Also, it is found that the D–A model can fit the adsorption isotherms of silica-gels appro-
priately for the entire range of relative pressure when the characteristic energy is set as a function of
relative pressure instead of assuming constant values.
© 2017 Elsevier Ltd and IIR. All rights reserved.
1. Introduction
Adsorption cooling technology represents an attractive solution
to meet the growing worldwide demand of dehumidification, air
conditioning, refrigeration, heat pump and desalination. It is pow-
ered by low temperature heat sources such as solar energy, in-
dustrial/automobile waste heat and geothermal heat ( Alsaman et
al., 2017 ; Askalany et al., 2017 ; Younes et al., 2017 ), thus reduc-
ing the dependence on fossils. Adsorption cooling systems use en-
vironmentally friendly working fluids and do not require moving
components such as compressors or pumps for circulation of work-
ing fluids and hence can operate without vibration and noise. De-
spite these advantages, adsorption cooling has not been compet-
itive as mechanical vapor compression due to the high thermal
and mass resistances within the adsorption beds. These large re-
sistances directly lead to low coefficient of performance (COP), low
specific cooling power (SCP), large volume and high capital cost
( Mohammed et al., 2017 ).
∗Corresponding author.
E-mail addresses: rhamdy@zu.edu.eg , rhamdy@knights.ucf.edu (R.H. Mo-
hammed).
The SCP parameter depends on the capacity of the adsorbent
to adsorb vapor as well as on the rate at which the bed can ad-
sorb/desorb the working fluid ( Ammar et al., 2017; Teng et al.,
2016 ). Therefore, an adsorbent with large specific surface area is
preferable for providing a large adsorption capacity. Since the ad-
sorption process is an exothermic process, the adsorbent should
have high thermal conductivity to be able to dissipate the heat
of adsorption without considerable temperature rise. The relation-
ship between the equilibrium capacity (i.e. uptake) of the adsor-
bent and the pressure at constant temperature is called the ad-
sorption isotherm. Accordingly, measuring the adsorption isotherm
is a key parameter to investigate the performance of the adsorp-
tion cooling systems.
The Freundlich, Tòth and Dubinin–Astakhov (D–A) equations
have been mainly used to fit the experimental measurements
of the adsorbent uptakes at various pressures and temperatures
( Chihara & Suzuki, 198 3 ; Pan & Zhang, 2012 ; Tóth, 1995 ; Tóth,
198 1 ). The present study goes over the adsorption equilibrium
models to identify their differences. After reviewing the adsorp-
tion equations for different types of silica-gel/water, conflicting in-
formation is found among the equations for the same type of
silica-gel. It is also found that only limited data on experimental
measurements of silica-gel/water are available in the literature.
https://doi.org/10.1016/j.ijrefrig.2017.10.038
0140-7007/© 2017 Elsevier Ltd and IIR. All rights reserved.
R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47 41
Nomenclature
b, b
o adsorption equilibrium constant
D affinity coefficient
E characteristic energy, J mol
−1
P pressure, Pa
R universal gas constant, 8314 J kmol
−1 K
−1
RP relative pressure
r
p pore radius, m
SCP specific cooling power, kW kg
−1 of adsorbent.
T temperature, K.
t
m statistical thickness of the adsorbed layer, m
V molar volume, m
3 mol
−1
W adsorbate concentration, m
3 kg
−1 of adsorbent.
W
o maximum adsorbate concentration, m
3 kg
−1 of ad-
sorbent.
X uptake, kg kg
−1 of adsorbent.
X
o adsorbent capacity, kg kg
−1 of adsorbent.
y fractional surface coverage.
H heat of adsorption, J kg
−1
Greek symbols
βcontact angle
θfilling ratio.
σsurface tension, N m
−1
Subscripts
a adsorbent
b boiling
L liquid
s solid, saturation
v vapor
Therefore, an experimental set-up is built to measure the equi-
librium update of water vapor on to different types of silica-gels.
With the new data and those in the literature, the discrepancies
among the equilibrium uptakes predicted from the equations are
addressed. At the end, the present work shows that a modified D–
A model is best suited to accurately predict the entire adsorption
isotherm of water vapor onto silica-gel.
2. Adsorption isotherm of silica-gel/water
Silica-gel is used as an adsorbent due to its high uptake ca-
pacity, reliability, repeatability, and inexpensiveness as compared
to other adsorbents. Wate r is considered to be an ideal refrigerant
because it is environment-friendly and non-toxic. Thus, silica-gel/
water is the most appropriate working pair for adsorption cooling
systems. In fact, water/silica-gel systems have been commercial-
ized in Japan since the late 80
s ( Yonezawa et al., 198 9 ). However,
silica-gel is ineffective in producing cooling effect below 0 °C and
the cycle cooling capacity is relatively low due to its low uptake
( Mahesh, 2017 ). There are numerous types of silica-gel produced
from many manufacturers. Accordingly, identification of the differ-
ences between these sorts of silica gels is critical to pick the most
suitable one for adsorption cooling applications. The adsorption
equilibrium equation describes the amount of adsorbate molecules
at certain pressure and temperature. Equilibrium state is achieved
after a long time when adsorbate molecules are settled onto adsor-
bent surfaces at a given pressure and temperature and this amount
adsorbed is called the equilibrium concentration. Many equations
have been proposed to fit the experimentally measured silica-gel
equilibrium uptake. These equations are:
2.1. Langmuir model
The Langmuir model is the simplest one for monolayer adsorp-
tion ( Choi et al., 2001 ). This model was originally developed to
represent chemisorption and its basic assumptions are given in
( Ruthven, 1984, 2006 ):
a. Adsorbed molecules are stationary (immobile).
b. Each site can hold one adsorbed molecule.
c. No interaction between the adsorbed molecules at neighboring
sites.
d. All sites are energetically equivalent.
Based on these assumptions, the following equation is pro-
posed:
y =
b P
v
1 + b P
v
(1)
where y is the fractional surface coverage, P
v
is the vapor pressure,
and b is the adsorption equilibrium constant that follows the van’t
Hoof equation ( Do, 1998 ).
b = b
o
exp
−H
R T
s
(2)
where H is the heat of adsorption (kJ mol
−1
), R is the universal
gas constant, and T
s is the adsorbent temperature ( K ).
As Langmuir model is valid for monolayer adsorption, it is good
for very low pressure close to vacuum, and it has limitations to fit
the uptake at pressures above 10 kPa for heterogeneous adsorbents
( Cevallos, 2012 ). As a result, this model is not suitable to fit the
experimental data for multilayer adsorption.
2.2. Brunauer–Emmett–Teller (BET) isotherm
Brunauer–Emmett–Teller (BET) theory was proposed to extend
the Langmuir model to multilayer adsorption as ( Ruthven, 19 84
Ruthven, 2006 ):
P
v
( P
s
−P
v
) W
=
1
W
m
e
E
1
−E
L
R T
s
+
e
E
1
−E
L
R T
s −1
P
v
W
m
e
E
1
−E
L
R T
s P
s
(3)
where W is the concentration of adsorbate at certain pressure and
temperature per unit mass of the solid adsorbent (m
3 kg
−1
), P
s is
the saturation vapor pressure at adsorbent temperature T
s
, E
1 is
the heat of adsorption for the first layer, E
L is the heat of adsorp-
tion for the second and higher layers and W
m is the monolayer
capacity.
The BET model is mainly used for the measurement of the spe-
cific surface area and pore volume of porous materials, and is not
used to fit the experimental data of physical adsorption.
2.3. Freundlich isotherm model
Freundlich equation was proposed in 190 6 to overcome the
shortcoming of Langmuir adsorption model. It is an empirical
equation that is valid only when the adsorption isotherm is linear,
and it is written as ( Chihara & Suzuki, 198 3 ):
X = X
o
P
v
P
s
1 /n
(4)
where X is the uptake (kg of adsorbate per kg of adsorbent), X
o is
the maximum uptake (adsorbent capacity), 1/ n describes the sur-
face heterogeneity of the adsorbent and it varies from 0 to 1, and
P
s is the saturation vapor pressure at the adsorbent temperature
( T
s
).
The adsorption equilibrium for Fuji Davison A-type silica-gel
was measured by Chihara and Suzuki and the data was fitted us-
42 R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47
Tabl e 1
Coefficients of the S-B-K equation for RD silica-gel.
RD silica-gel ( Saha et al., 1995, 2003; Youssef et al., 2015 )
A
o
−6.5314 31.198
A
1 0.072452 −0.26650
A
2
−0.23951 ×10
−3 0.769 ×10
−3
A
3 0.25493 ×10
−6 −0.73898 ×10
−6
B
o
−15.587 41.581
B
1 0.15915 −0.35435
B
2
−0.50612 ×10
−2 0.10199 ×10
−2
B
3 0.53290 ×10
−6 −0.97034 ×10
−6
ing the Freundlich equation ( Eq. (4) ) with X
o
= 0.346 kg kg
−1 and
n = 1. 6 ( Chihara & Suzuki, 1983; Wang et al., 2005 ). Also, the ad-
sorption equilibrium of the Fuji Type-RD silica-gel was represented
by Freundlich equation with X
o
and n of 0.552 and 1.6 , respectively
( Cho & Kim, 1992 ).
This form of Freundlich model is not valid at low and high rel-
ative pressure. Saha, Boelman and Kashiwagi (S-B-K) equation is
a modified version of the Freundlich equation that gives better
fitting of the experimental data. The S-B-K equation is given by
( Saha et al., 1995 ):
X = A
(
T
s
)
p
v
p
s
B
(
T
s
)
(5)
where A ( T
s
) =
i =3
i =0
A
i
T
i and B ( T
s
) =
i =3
i =0
B
i
T
i
. Both the coeffi-
cients A ( T
s
) and B ( T
s
) are calculated based on fitting of experimen-
tal data.
Saha et al. ( Saha et al., 1995 Saha et al., 2003 ) fitted the ex-
perimental and manufacturer data of Type-RD silica-gel/water. Two
different set of coefficients of S-B-K equation were proposed as il-
lustrated in Table 1 . Although these coefficients describe the equi-
librium uptake of the same type of silica-gel and the same exper-
imental measurements, there is about 10% deviation between the
predicted uptakes using the two different coefficients.
2.4. Dubinin models
The description of adsorption equilibria for microporous adsor-
bents is based on the theory of volume filling of micropores. This
theory is based on the concept of temperature invariance of the
characteristic curve expressing the distribution of the degree of fill-
ing ( θ). The Polanyi theory was originally developed to study the
adsorption of gas molecules onto porous materials ( Pan & Zhang,
2012 ). Based on Polanyi adsorption theory, the degree of filling of
adsorption space can be written as:
θ=
W
W
o
= exp
−R T
s
E
ln
P
s
P
v
n
(6)
where θis the degree of filling, W is the adsorbate concentration
at certain pressure and temperature per unit mass of the adsor-
bent solid (m
3 kg
−1
), W
o is the maximum adsorption concentra-
tion which is the micropore volume of the adsorbent per unit mass
of adsorbent (m
3
/kg), R T
s
ln (
P
s
P
v
) is the differential work of adsorp-
tion, T
s is the adsorbent temperature, R is the universal gas con-
stant, and E is the characteristic energy of adsorption (J mol
−1
) and
is obtained from adsorption potential at θ= e
−1
= 0 . 368 .
When n equals 2, Eq. (6 ) is known as the Dubinin–
Raduschkevich equation (D–R) ( Ruthven, 2006 ). The D–R equation
has been widely used to calculate the equilibrium concentration of
hydrocarbons and other organics onto activated carbon adsorbents.
It was found that Eq. (6 ) with n from 3 to 6 satisfactorily describes
the experimental data of zeolites over the range of fillings, θ, from
≈0.1 to 1, while n of 2 is applicable only to the region of high
filling from 0.8 to 1 ( Dubinin & Astakhov, 1971 ). In addition, it was
Tabl e 2
Coefficients of D–A model for different silica-gels.
Reference Silica-gel type E (J mol
−1
) n W
o
(cc g
−1
)
( Thu et al., 2013 ) Type-A5BW 3585 1.25 0.455
Type-RD 2560 4384 1.3 5 0.327
Type A + + 3804 1.2 5 0.489
( Cevallos, 2012 ) Type A + + 3176.69 1. 4 9 8 6 0.4707
( Ng et al., 2012 ;
Thu et al., 2016 )
Type A + + 3105 1.1 0.592
pointed out that this equation is not thermodynamically consistent
with Henry’s law ( Talu & Myers, 1988 ).
Converting the fractional concentration ( W / W
o
) to the upload-
ing ratio ( X / X
o
) requires the knowledge of W
o which is usually
not known because the molecular volume of the adsorbed fluid is
not known. As a result, this equation was modified with the as-
sumption that the density of the adsorbed phase is constant. The
D–A equation is rewritten in terms of mass capacity as follows
( Tamainot-Telto & Critoph, 2003; Zhao et al., 2012 ):
X = X
o
exp
−D
ln
P
s
P
v
n
(7)
where D is the coefficient of affinity that is a function of the
adsorbent microstructure. Both D and n depend not only on the
adsorbate–adsorbent pair but also on the brand and type of the
adsorbent.
Trouton’s rule assumes that the isosteres on the ln P
v vs. 1/ T
plot are indeed linear:
ln
P
v
P
b
= a
1 −T
b
T
v
(8)
where a is a constant which depends on the adsorbate type and T
b
is the saturation temperature at atmospheric pressure ( P
b
).
Based on the Trouton’s rule, a simple equation of adsorption
state has been proposed as follows ( Critoph, 1988 ):
X = X
o
exp
−k
T
s
T
v
−1
n
(9)
Three types of commercially available silica-gels were investi-
gated ( Cevallos, 2012; Ng et al., 2012 ). Eq. (6 ), Thu et al., 2013 was
used to fit the experimental data of silica-gel RD 2560, A5BW,
and A + + . The values of these coefficients are summarized in
Table 2 . It should be noted that the characteristic energy of Type
A + + measured by Thu et al. ( Thu et al., 2013 ) is 20% higher than
that measured by Cevallos ( Cevallos, 2012 ). Also, the values of the
distribution order ( n ) are different for the same type of silica-gel.
The predicted uptake using these coefficients deviates by 16% from
the experimental data of silica-gel A + + at relative pressure of
0.3. Their experimental measurements showed that silica-gel Type
A + + reaches the maximum equilibrium uptake at a relative pres-
sure of 0.6, which do not match with the predicted uptake using
the set of coefficients shown in Table 2 ( Thu et al., 2013 ). Accord-
ingly, the D–A equation with these coefficients cannot be used to
predict correctly the uptake of the water vapor onto silica-gel Type
A + + .
Xia et al. ( Xia et al., 2008 ) investigated the adsorption isotherms
of the silica-gel/water. The Freundlich equation and D–A equation
were first used to fit the experimental data, and a large deviation
was observed between the two equations and the data for low wa-
ter uptakes. Then, the modified Freundlich equation was used to
minimize this deviation by 50%.
2.5. Tòth equation
The D–A model sometimes does not represent the adsorption
isotherm correctly at low relative pressure. This is due to the zero
R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47 43
Tabl e 3
Coefficients of Tòth equation for two types of Fuji Davison silica-gels.
Reference ( Chua et al., 2002 ; Wang et al., 2004 )
Type of silica-gel Type-A Type-RD
b
o 4.65 ×10
−10 7. 3 ×10
−10
t 10 12
X
o
(kg
v
kg
a
−1
) 0.4 0.45
H (kJ kg
−1
) 2710 2693
slope at zero loading ( Cevallos, 2012 ). Tòth ( Tóth, 1995, 198 1 ) pro-
posed an equation that is based on the state equation of multi-
layers. It could satisfy the monolayer coverage and the multilayer
adsorption. It corrects the wrong behavior at both the low and
high pressure ends of adsorbate concentration that the Langmuir
and Freundlich models cannot describe accurately ( Cevallos, 2012 ).
It is also the first choice for fitting the experimental data to get
the isotherm equation of several heterogeneous solid adsorbents
( Cevallos, 2012 ). This equation is given as ( Mitra et al., 2017; Thu
et al., 2009; Wuet al., 2016 ):
X =
b
o P
v
exp
H
R T
s
1 +
b
o P
v
X
o
exp
H
R T
s
t
1
t
(10)
where t is the adsorbent structural heterogeneity parameter and b
o
is the equilibrium constant based on the working pair. Both t and
b
o can be calculated from the experimental data.
The Tòth equation was used to fit the experimental measure-
ments of two grades of Fuji Davison silica-gels as shown in Table 3 .
It cannot cover the entire adsorption isotherm of silica-gel/water.
Rezk et al. ( Rezk et al., 2013 ) measured the adsorption character-
istics of Fuji RD-2060 silica-gel for relative pressure less than 0.5
using an advanced gravimetric dynamic vapor sorption test facil-
ity. The Tòth equation for Type-RD silica-gel/water ( Wang et al.,
2004 ), with X
o
= 0.45 kg kg
−1
, was used to fit the measured data of
RD-2060 that has maximum uptake of 0.37 kg kg
−1
. Although the
deviation between the experimental results and those predicted by
the Tòth equation was within 10% , the maximum uptake ( X
o
) had
to be replaced by 0.37 kg kg
−1
, which is the maximum uptake of
silica-gel RD-2060.
3. Comments on adsorption models
It was previously discussed that the Langmuir and Freundlich
models do not fit the experimental measurements at low and high
relative pressures, while Tòth model does not fit the experimental
isotherm at relative pressure less than 0.15. In addition, the D–A
model sometimes fails to describe the experimental data.
Chihara and Suzuki ( Chihara & Suzuki, 1983 ) measured the ad-
sorption properties of Fuji silica-gel type-A with water vapor. The
Freundlich equation with X
o
= 0.346 kg kg
−1 and n = 1. 6 was used
to fit the data as shown in Fig. 1 . It is clear that Freundlich
equation does not fit the experimental data correctly when the
relative pressure is less than 0.15. As a result, the D–A model
( Eq. (7) ) was implemented to fit the same experimental measure-
ments with coefficients of X
o
= 0.348 kg kg
−1
, n = 1.6, and D = 0.449
( Xia et al., 2008 ). Fig. 1 shows there is a big discrepancy between
the results of the D–A model and the experimental measurements
(about 20% at relative pressure of 0.6). The modified D–A adsorp-
tion equation, Eq. (9 ), with coefficients of X
o
= 0.346 kg kg
−1
, n = 1.6
and k = 5.6 was suggested to describe the same experimental mea-
surements. This equation with the aforementioned coefficients is
widely used in the numerical simulations of the adsorption cool-
ing systems ( Di et al., 2007; Solmu ¸s et al., 2015; Solmu ¸s et al.,
2012; Solmu ¸s et al., 2012 ). By comparing the predicted uptake from
Fig. 1. Comparison among the Freundlich, D–A models, and experimental data for
the same type of silica-gel.
Eq. (9 ) (with X
o
= 0.346 kg kg
−1
, n = 1. 6 , and k = 5.6) with the ex-
perimental measurements, the equilibrium uptake at relative pres-
sure of 0.01 (i.e. T
v
= 10 °C and T
s
= 10 0 °C) is about 50% of the
maximum uptake as shown in Fig. 1 . This cannot be physically cor-
rect because the filling ratio is less than 10% at the low relative
pressure (monolayer coverage) ( Wanget al., 2014 ). Thus, this set of
coefficients should not be used to describe the silica-gel/water up-
take.
The disagreement between the experimental data and the dif-
ferent versions of D–A models can be minimized by adjusting the
model coefficients. The values of X
o
= 0.346 kg kg
−1
, D = 0.64, and
n = 1. 0 for Eq. (7 ) and X
o
= 0.346 kg kg
−1
, k = 11, and n = 1.0 for Eq.
(9 ) or Eq. (11 ) are suggested and plotted in Fig. 1 . It is clearly
shown that these new sets of coefficients can fit the experimen-
tal data very well.
It is generally preferable in numerical simulation to use Eq. (9 )
instead of Eq. (6 ), because the uptake is a direct function of the
adsorbent temperature. This allows one to simulate the adsorption
bed with simple conduction models without the need to solve the
velocity and pressure field. By using Trouton’s rule ( Critoph, 198 8 ),
Eq. (6 ) can be rewritten in a general form as:
X
X
o
= exp
−a T
b
R
E n
T
s
T
v
−1
n
= exp
−k
T
s
T
v
−1
n
(11)
Based on the saturation data of water, the value of the constant
a is 14.
It is worth mentioning that the Trouton’s rule states that
the entropies of vaporization are almost the same for various kinds
of liquids at their boiling points, but there are some exceptions. For
example, the entropies of vaporization of methanol and ethanol are
far from the predicted values. This makes changing Eqs. (6 ) to (9 )
not possible for all working pairs, and Eq. (11 ) is only valid for wa-
ter in the range of operating conditions (5–90 °C).
Although many models have been proposed in the literature to
predict the adsorption uptake of water vapor onto different types
of silica-gels, not all equilibrium models can satisfactorily describe
the experimental data over the entire range of the filling ratio. Ad-
sorption models and experimental data for various silica-gels are
reviewed and a lot of discrepancies are found for the same type of
silica-gel. The widely used Langmuir, Freundlich and Tòth isotherm
models inappropriately predict the equilibrium uptake when the
relative pressure is less than 0.15. Also, it is found that the D–A
model may lead to an error of 16–20% which is rather high for
design of adsorption cooling systems. Accordingly, further adjust-
ments are needed for these models to fit the experimental mea-
44 R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47
Tabl e 4
Thermo-physical properties and surface characteristics
of tested silica-gels.
Type RD RD-2060
BET surface area (m
2
g
−1
) 780 707
Pore volume (m
3
g
–1
) 0.44 0.34
Pore size (nm) 2.24 1.9 2
Particles size (mm) 1–2 0.18–1
Bulk density (g cc
−1
) 0.72 0.77
Fig. 2. Kinetic measurement setup.
surements in the range of the operating conditions of the adsorp-
tion cooling cycles which operate at relative pressure from 0.05 to
0.3. Also, it is required to determine the most appropriate model
for adsorption of water vapor onto different types of silica-gel. This
directly affects the sizing and compactness of the adsorption cool-
ing system design.
Due to the limited adsorption experimental measurements of
silica-gel/water, an experimental set-up is designed and developed
to measure the kinetics and equilibrium uptakes of any working
pair. The equilibrium uptakes of Fuji silica-gels Type-RD and RD-
2060 (manufactured by Fuji Silysia, Japan), which are commonly
used in adsorption cooling and desalination systems, need to be
experimentally measured. The particle size, BET surface area, pore
size, pore volume and bulk density of these silica-gels are provided
by the manufacturer and presented in Table 4 . The experimental
data are analyzed and fitted using the appropriate adsorption mod-
els. The predicted uptakes for the models are compared with the
experimental measurements to define the adequate one for the ad-
sorption cooling and desalination systems.
4. Experimental work
4.1. Experimental setup
In the development of any adsorption system design, it is very
essential to study the characteristics of adsorption isotherm and
kinetics of the adsorbate/adsorbent pair. There are two method-
ologies for measuring the adsorption isotherm; volumetric ( Aristov
et al., 2008 ) and gravimetric ( Sapienza et al., 2014 ). The volumet-
ric approach is suitable only when the amount of adsorbent is
very small so that the adsorption process can be considered as
quasi-isobaric. The gravimetric approach uses a larger amount of
adsorbent and allows the possibility to directly weigh the adsorp-
tion uptake. Based on the gravimetric approach, the present ex-
perimental test rig is designed to be simple and can monitor the
adsorption rate at a desired pressure and temperature. It also can
be used to investigate the performance of adsorption beds with
different structures and various working pairs. It consists of two
insulated stainless steel vacuum vessels as shown in Fig. 2 . The
first vessel works as an evaporator/condenser during the adsorp-
tion/desorption processes, respectively. This vessel will be filled
with water as the source or sink of adsorbate vapor. The water
temperature is controlled by the water that flows inside the copper
coil that is connected to the thermal bath 1 (TB1). The second ves-
sel represents the measuring unit which contains adsorbent placed
on a flat plate heat exchanger (120 ×40 mm) mounted on a single
point load cell/load gauge. The load cell has a range of 600 g and
it is calibrated using a very precise balance of ±0.001 g accuracy.
The accuracy of the load cell is less than 1% with a time response
faster than 0.1 s. The heat exchanger is connected to TB2 and TB3
by flexible tubing to control the sample temperature by providing
external cooling/heating fluid passing through the inner tubes of
the heat exchangers to remove or provide heat to the adsorbent
throughout the isobaric adsorption/desorption process. The posi-
tion of the flexible tubes and flow velocity inside are adjusted be-
fore the test to eliminate the vibrations that could affect the load
cell response. When the pressure inside the tank is smaller than
that on the outside, the acrylic cover will push onto the silicon
gasket. The deformation makes the gasket contact the cover per-
fectly and prevent ambient air from leaking inside. With this seal-
ing method, a high vacuum level inside the evaporator/condenser
tank is achieved. The results of the leakage test indicate that the
measuring unit chamber and evaporator/condenser chamber gain
only 20 Pa day
−1
. The measuring unit is equipped with 10 T-type
thermocouples, with accuracy of ±0.5 °C, to measure the sample
temperature at different locations during the test. Two MKS pres-
sure transducers with accuracy of ±0.25% are used to measure the
pressure of each chamber. All measurements are connected to a
data acquisition system.
4.2. Testing procedure
The following steps describe the testing procedure in detail, in-
cluding preparation, desorption, and adsorption.
1. The evaporator/condenser is filled with water and thermal bath
1 is set at specific temperature and then the vacuum pump is
run until equilibrium condition is reached.
2. The sample of the adsorbent is placed on the heat exchanger
and its temperature is controlled by the thermal bath 2 or 3.
3. Small known dead loads are placed on the load cell and the
load cell response is monitored to ensure that the flow through
the flexible tubes does not affect the load cell response and the
accuracy is still less than 1%.
4. The sample is heated and the measuring chamber is evacuated
by the vacuum pump to reach a relative pressure less than
10
−4
. The sample is assumed to be completely dry when the
load cell response becomes constant for 2 h, then the degassed
dry sample is weighed.
5. The sample is cooled down to the desired adsorption temper-
ature and then the two chambers are connected to start the
adsorption process.
6. During the adsorption, the sample temperatures and the load
cell response are monitored over time. The adsorption process
is assumed to be completed when the uniform sample temper-
ature is achieved and equals the inlet water temperature.
The adsorption isotherms of Fuji silica-gel RD and RD-2060 are
measured by following the above procedure. Silica-gel RD and RD-
2060 are in granular form and have particle size ranges of 1.0–
2.0 mm and 0.18–1.0 mm, respectively. Each test is run more than
once to check the repeatability of the experimental set-up and the
repeated measurements for the same operating conditions vary by
no more than ±4%.
R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47 45
Fig. 3. Equilibrium uptake of RD-2060 at different pressures and temperatures.
Fig. 4. Equilibrium uptake of Type-RD at different pressures and temperatures.
5. Results and discussion
5.1. Adsorption equilibrium
The equilibrium uptakes of water vapor onto silica-gel RD-2060
and Type-RD at different pressure and temperature range from
20 °C to 60 °C are presented in Fig. 3 and Fig. 4 , respectively.
The measured uptakes of RD-2060 show a good agreement with
the previous available experimental measurements ( Rezk et al.,
2013 ). The present measurements indicate that the maximum up-
takes of silica-gel RD-2060 and Type-RD are about 0.38 kg kg
−1
and 0.48 kg kg
−1
, respectively. The isosteric heat of adsorption of
Fuji silica-gel Type-RD is calculated by plotting Clausius–Clapeyron
equation ( Eq. (12 )) as illustrated in Fig. 5 . The slope of the lines
represents ( −H ). The plot is also repeated for Fuji silica-gel RD
2060. The results shows that the heats of adsorption for silica-gel
RD-2060 and Type-RD are nearly the same and have a value of
2415 kJ kg
−1 with a deviation ranging from 0.65% to 10.56% from
published data as shown in Table 5 .
∂ ln P
v
d
(
T
s
)
x
= −H
RT
2
s
⇒
ln
(
P
v
)
= constant −H
R T
s
(12)
Fig. 5. Isosteres of Type-RD silica-gel/water.
Fig. 6. Data fitting of the experimental water sorption isotherm of silica-gel Type-
RD.
5.2. Data analysis and regression
By plotting equilibrium uptake ( X ) versus relative pressure, the
water adsorption isotherms of these two types of Fuji silica-gel
could be classified as Type IV ( Wang et al., 2014 ) as shown in
Figs. 6 and 7 . The experimental measurements show good agree-
ment with the data provided by the manufacturer. Although the
maximum uptakes of Type-RD and RD-2060 silica gels are not the
same, both of them have similar equilibrium uptake from a rela-
tive pressure of 0.05 till 0.5. Because the relative pressure of the
adsorption cooling cycles normally varies from 0.05 to 0.3 ( Saliba
et al., 2016 ; Wan g et al., 2014 ), both adsorbents are expected to
have the nearly the same change in the uptake under the same
operating conditions, but the silica-gel RD-2060 is recommended
as it has a smaller particle diameter which enhances the diffusion
process during the adsorption process and hence increases perfor-
mance.
The measured isotherm curves show an inflection point at rel-
ative pressure (RP) ≈0.4 and 0.35 for silica-gel RD and RD-2060,
respectively. This is attributed to the fact that the region of the wa-
ter adsorption isotherm where the relative pressure is more than
0.4 (RP > 0.4) is typically dominated by capillary condensation. De-
spite the limited validity of the corrected Kelvin equation for mi-
croporous materials, it is commonly used to describe the apparent
46 R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47
Tabl e 5
Isosteric heat of adsorption of Fuji RD silica-gels.
Reference H (kJ kg
−1
) Deviation (%)
( Rezk et al., 2013 ) 2430 0.62
( Wang & Chua, 2007 ; Wang et al., 2007 ) 2510 0.96
( Chua et al., 2002 ; Mitra et al., 2017 ; Wang et al., 2004 ) 2693 10.32
( Oh, 2013 ) 2700 10. 56
Tabl e 6.
Coefficients of D–A and Tòth model for the tested silica-gels.
Type Tòth model D–A model
X
o
(kg kg
−1
) t H (kJ kg
−1
) b
o X
o
(kg kg
−1
) n E (J mol
−1
)
Type-RD 0.48 8 2415 5 ×10
−9 0.48 1.6 3030 + 192 R P
−1
1 . 3
RD-2060 0.38 4 2415 5 ×10
−9 0.38 1.6 3980 + 103 R P
−1
1 . 2
Fig. 7. Data fitting of the experimental water sorption isotherm of silica-gel RD-
2060.
capillary condensation regime, and is written as (
Ruthven, 19 8 4 ;
Saliba et al., 2016 ):
P
v
P
s
= exp
−2 σV
L
cos
(
β)
(
r
p
−t
m
)
R T
s
(13)
where r
p is the pore radius, R is the universal gas constant, and T
s
is the adsorbent temperature, σis the surface tension of water, V
L
the molar volume of water, βis the contact angle of water on the
silica-gel pore walls. The parameter t
m is the statistical thickness
of the adsorbed water layer and is calculated by dividing the vol-
ume of water adsorbed prior to the capillary condensation by the
surface area of the adsorbent ( Saliba et al., 2016 ).
Here, the properties of water at 30 °C are used for the calcu-
lation. Based on this equation and the adsorbent properties, the
contact angle is found to of 44 °and 36 °for silica-gel Type-RD and
RD-2060, respectively. Interestingly, the contact angle of silica-gel
Type-RD deviates by 1 °from the previous reported data ( Alcaniz-
Monge et al., 2010; Huang et al., 2010 )
As shown in Figs. 6 and 7 , it is clear that by setting the charac-
teristic energy E as a function of relative pressure, the D–A model
can be used to fit the experimental measurements accurately over
all the entire adsorption relative pressure range. The dependence
of E on the relative pressure can be attributed to the fact that E
represents the affinity of the adsorbent surface for the adsorbate.
During the adsorption process, the surface changes from partially
dry to multilayer condition. Partially dry surface has a higher affin-
ity for vapor molecules than a surface covered with an adsorbate
layer. In other words, E should decrease with the relative pres-
sure. With the correlation of E = f(RP ) as shown in Table 6 , the D–
A model can fit the experimental measurements appropriately for
the entire range of relative pressure with a standard deviation of
0.006.
Also, it is found that the D–A model with constant values of
n and E cannot fit the experimental uptakes accurately for the
entire range of relative pressure. Furthermore, the experimental
measurements are also fitted using the Tòth model and plotted
in Figs. 6 and 7 . The coefficients of this model are presented in
Table 6 with standard deviation of 0.017. It is clear that the Tòth
model does not predict the uptake correctly at relative pressure
less than 0.15. Although it predicts the equilibrium uptakes accu-
rately when the relative pressure is more than 0.15, it cannot be
used to simulate the performance of adsorption chillers because
the relative pressure at the beginning of adsorption is about 0.05.
Therefore, the specific cooling power estimated by Tòth equation
can be 13% higher than the accurate one.
By using a simple relationship between the saturation pressure
and temperature of water ( T = 191.054
∗P
0.0554
) ( Mohammed et al.,
2017 ), the characteristic energy can be set as a function of ( T
v
/ T
s
)
instead of ( P
v
/ P
s
) as shown in Eqs. (14 ) and ( 15 ) and then Eq. (11 )
can be used to calculate the equilibrium uptakes.
T ype −RD ⇒ E = 3030 + 192
T
v
T
s
−13 . 89
(J mo l
−1
) (14)
RD −2060 ⇒ E = 3980 + 10 3
T
v
T
s
−15 . 04
(J mo l
−1
) (15)
So, it is recommended for the designers of the silica-gel/water
adsorption cooling systems to use the D–A model with a variable
characteristic energy E to determine the equilibrium uptake over
the entire relative pressure range. This approach needs X
o
, n , and
the coefficients for the correlation of E . By fixing the value of n to
be 1.6 and choosing X
o
to be the maximum uptake, the coefficients
for E can be determined by fitting the limited manufacturer data.
6. Conclusions
After reviewing the adsorption isotherm equations of silica-
gel/water working pair, it is found that the adsorption equilibrium
models cannot be used to predict the equilibrium uptakes for the
entire relative pressure range. Moreover, inconsistencies between
the available experimental measurements and equilibrium models
for silica-gel/water are observed and addressed in order to prevent
their propagation in the scientific literature. These inconsistencies
are eliminated by proposing new sets of coefficients for the D–A
model to fit the available experimental data from the literature for
silica-gel/water isotherm.
R.H. Mohammed et al. / International Journal of Refrigeration 86 (2018) 40–47 47
Due to the discrepancies among these models and the limited
availability of experimental measurements, a simple but accurate
experimental setup is designed and constructed to measure the
equilibrium uptake of any working pairs at any operating con-
ditions. Apparent capillary condensation is observed at a relative
pressure of 0.4 and 0.35 for silica-gel RD and RD-2060, respec-
tively. The estimated contact angles based on the corrected Kelvin
equation are 44 °and 36 °for silica-gel Type-RD and RD-2060, re-
spectively. To be able to successfully fit the adsorption isotherms
for the entire relative pressure, the characteristic energy in the D–A
model is proposed to be a function of the relative pressure instead
of a constant value. The characteristic energy decreases as the rela-
tive pressure increases, because the attraction force (surface effect)
decreases as the thickness of the adsorbed phase increases.
Acknowledgments
This work is supported by National Science Foundation (NSF)
through Grant No. 1603215 and the Egyptian Ministry of Research
and Higher Education.
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