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Heat and Mass TransferWärme- und Stoffübertragung ISSN 0947-7411Volume 48Number 2 Heat Mass Transfer (2012) 48:217-226DOI 10.1007/s00231-011-0875-8
A comparison of three adsorptionequations and sensitivity study ofparameter uncertainty effects on adsorptionrefrigeration thermal performanceestimationYongling Zhao, Eric Hu & AntoniBlazewicz
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ORIGINAL
A comparison of three adsorption equations and sensitivity studyof parameter uncertainty effects on adsorption refrigerationthermal performance estimation
Yongling Zhao • Eric Hu • Antoni Blazewicz
Received: 24 October 2010 / Accepted: 20 July 2011 / Published online: 3 August 2011
� Springer-Verlag 2011
Abstract This paper presents isosteric-based adsorption
equilibrium tests of three activated carbon samples with
methanol as an adsorbate. Experimental data was fitted into
Langmuir equation, Freundlich equation and Dubinin-
Astakov (D–A) equation, respectively. The fitted adsorp-
tion equations were compared in terms of agreement with
experimental data. Moreover, equation format’s impacts on
calculation of the coefficient of performance (COP) and
refrigeration capacity of an adsorption refrigeration system
was analyzed. In addition, the sensitivity of each parameter
in each adsorption equation format to the estimation of
cycle’s COP and refrigeration capacity was investigated. It
was found that the D–A equation is the best form for
presenting the adsorptive property of a carbon-methanol
working pair. The D–A equation is recommended for
estimating thermal performance of an adsorption refriger-
ation system because simulation results obtained using the
D–A equation are less sensitive to errors of experimentally
determined D–A equation’s parameters.
List of symbols
x Adsorbate concentration, kg kg-1
T Temperature, K
Tc Condensing temperature, K
Te Evaporating temperature, K
P Adsorption pressure or partial pressure, Pa
Ps Saturation pressure at local temperature, Pa
Qst Isosteric heat of adsorption, kJ/kg
R Gas constant, kJ kg-1K-1
KI0
Adsorption constant, Pa-1
K Adsorption constant
COP Coefficient of performance
x0 Maximum adsorption capacity, kg kg-1
1/n Parameter of Freundlich equation
n Parameter of D–A equation
D Parameter of D–A equation
q� Equilibrium concentration, kg kg-1
q�0 Adsorption capacity, kg kg-1
h Surface coverage of the micropore
Qr Refrigeration capacity, kJ
Lo Latent heat of vaporation, kJ kg-1
cr Specific heat capacity, kJ kg-1 K-1
Mac Mass of activated carbon, kg
1 Introduction
In recent years, the solid adsorption refrigeration technique
has attracted extensive interest and research [1–3] as a
result of a series benefits it offers, such as low energy
consumption, environmental-friendliness, quiet operation
etc. For an adsorption refrigeration system, the selection of
the working pair (i.e., adsorbent and adsorbate) is crucial
because the thermal performance of the system largely
depends on the adsorptive properties of the working pair
employed. Therefore, obtaining the accurate adsorptive
property of the working pair and choosing a suitable
adsorption equation is essential for design calculation and
thermal performance prediction.
Traditionally, the adsorptive property of an adsorbent
and an adsorbate pair is normally determined experimen-
tally using the isotherm and isobar method [4, 5]. However,
neither method is efficient and precise because these
experiments are normally conducted at a series of constant
Y. Zhao � E. Hu (&) � A. Blazewicz
School of Mechanical Engineering, The University of Adelaide,
North Terrace, Adelaide, SA 5005, Australia
e-mail: [email protected]
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DOI 10.1007/s00231-011-0875-8
Author's personal copy
temperatures or pressures that do not cover the entire
temperature and pressure range of sorption to be utilized in
practice. The isosteric-based method [6] could directly test
typical isosteric lines within the range to be used in a real
adsorption refrigeration cycle. Therefore, the isosteric-
based method is a very convenient and efficient approach
that can be used to obtain the adsorptive property of a
working pair for design calculation purposes.
On the other hand, in recent years the manufacturing
method (activation method) of activated carbons has
changed extensively, which has resulted in changes in
adsorptive properties of activated carbons. An increasing
number of new categories of activated carbons are now
available on the market and several new types have been
used in research [7]. Critoph and Hu et al. studied the
adsorptive properties of several types of charcoals in 1988
and 1992 [4, 8]. However, research into the adsorptive
properties of new activated carbons is limited. Therefore,
study of the adsorptive properties of these new activated
carbons is necessary and meaningful.
However, once the experimental data of equilibrium tests
have been obtained, the adsorption equation presenting the
data best still needs to be determined. In the area of
adsorption equation formats, Langmuir equation, Freundlich
equation and D–A equation are the dominant ones employed
in current design calculation and various mathematical
modelling. Freundlich equation was proposed as an empiri-
cal equation and is currently widely adopted in adsorption
equilibrium modelling of the solid adsorption system [9–11].
Compared to Freundlich equation, Langmuir equation is a
semi-empirical formula in which monolayer surface
adsorption assumption was introduced and is now popularly
employed in thermal performance estimation of the silica
gel-water adsorption chiller and desalination system
[12–14]. The D–A equation was proposed on the basis of
carbonaceous substance and volume filling theory [15], which
is currently the dominant adsorption equation used for car-
bonaceous adsorbent adsorption system [16–18]. Compared
to Freundlich and Langmuir equations, D–A equation has
three parameters, which may contribute to a better agreement
with experimental data due to stronger adjustment.
In this paper, the isosteric adsorption experimental data
of three activated carbon samples (207C, 207EA and
WS-480, which are from Calgon carbon Corporation) with
methanol as an adsorbate were fitted into three formats
(Langmuir, Freundlich and D–A equations), respectively to
compare their agreement. The calculation of COP and
refrigeration capacity based on each format was conducted
to assess the difference when the same activated carbon
and methanol were used as working pair in a practical
adsorption refrigeration system. Furthermore, the sensitiv-
ity of each parameter in each format to thermal perfor-
mance estimation was studied.
2 Experimental setup and testing procedure
The test rig was set up at Shanghai Jiao Tong University
(SJTU), which was modified from the test rig employed in
the study in the literature [6]. Compared to the original test
rig, connections of the current test rig were modified and
vacuum valves were used to better maintain vacuum. A
pressure transducer ranging from zero kPa to 50 kPa with a
precision of 50 Pa replaced one with a precision of 100 Pa.
In addition, four-wired PT1000 temperature sensors and an
electronic balance with a precision of 0.01 g were used for
pressure and weight measurement. The schematic diagram
of the experimental rig is shown in Fig. 1.
The principle of the isosteric-based adsorption equilib-
rium testing method is that the amount of adsorbed
adsorbate remains nearly unchanged once the adsorbent
was separated from the bulk of adsorbate after reaching
adsorption equilibrium. Procedures therefore include the
following major steps:
1. Degassed the system and then isolated the system by
closing V3. Opened V1 and V2 to initialize adsorption,
which was kept for twelve hours in order to reach
adsorption equilibrium.
2. Closed V1 and conduced measurement of the P–T-x
correlation of the isosteric line.
3. Measured the pressure in the generator using a
pressure transducer when pressure reached equilib-
rium. By setting a series of adsorbent temperatures
(through controlling temperature of water bath), cor-
responding equilibrium pressures was obtained. In
order to avoid condensation on the inner surface of the
Fig. 1 Schematic diagram of the testing system
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tube, tube connecting to the generator was wrapped
with heating wire, which was used to keep the tube
temperature equal to that of water bath.
4. Measured the mass of the generator and compared with
initial mass of generator to get mass variation and then
the amount of methanol adsorbed, i.e. concentration x,
was obtained.
5. Repeated step1 to step 4 to get various isosters and
corresponding temperature and pressure values by
setting appropriate temperatures of two water baths for
adsorption equilibrium.
6. Experimental data of tested three samples is listed in
Table 1.
3 Adsorption equations
(a) Langmuir equation
Langmuir equation was proposed based on a uniform
surface adsorption assumption and deemed one monolayer
adsorption generated on a uniform surface. The basic form
of the Langmuir equations is expressed in Eq. 1.
h ¼ KP=ð1þ KPÞ ð1Þ
where h ¼ q�=q�0
� �represents the surface coverage of the
micropore, K is an adsorption constant while P is the
partial pressure [5]. When the concentration is low, one
practical working form of the Langmuir equation can be
derived with Henry’s law and vant Hott equation, which is
expressed as follows:
x ¼ P � KI0 exp
Qst
RT
� �ð2Þ
where x is the concentration of adsorbed adsorbate; P is
adsorption pressure; KI0is the adsorption constant; Qst is
adsorption heat; R is gas constant and T is the temperature
of adsorbent.KI0 and Qst are two parameters normally
determined by experiments.
(b) Freundlich equation
Freundlich equation is an empirical equation but still
enjoys considerable accuracy in describing physical
adsorption within the range where the adsorbate concen-
tration is relatively high. For solid–vapor adsorption, Fre-
undlich equation can be given as [19]:
Table 1 Experimental data of
adsorption equilibrium tests
on samples 207C, 207EA
and WS-480
207C 207EA WS-480
x (kg/kg) T �C P (Pa) x (kg/kg) T �C P (Pa) x (kg/kg) T �C P (Pa)
0.13 19.8 4,500 0.24 19.9 4,050 0.22 29.7 8,520
0.13 29.7 7,500 0.24 29.8 6,650 0.22 39.9 14,050
0.13 39.9 11,900 0.24 39.9 10,700 0.22 44.8 18,150
0.13 49.8 19,250 0.24 49.6 17,600 0.22 49.7 22,950
0.13 59.9 30,350 0.24 59.8 26,400 0.22 54.8 28,800
0.11 19.6 3,250 0.20 29.7 4,450 0.19 29.8 5,850
0.11 29.8 4,950 0.20 39.8 7,300 0.19 39.9 10,050
0.11 39.9 8,050 0.20 49.7 12,450 0.19 49.7 16,900
0.11 49.7 13,350 0.20 59.8 18,750 0.19 54.9 21,300
0.11 59.8 20,300 0.20 69.6 26,500 0.19 59.7 26,600
0.08 39.6 4,150 0.17 39.6 5,750 0.15 34.9 4,900
0.08 49.5 7,900 0.17 49.8 8,950 0.15 44.7 8,500
0.08 59.7 12,250 0.17 59.9 13,600 0.15 54.8 14,050
0.08 69.8 17,500 0.17 69.5 20,750 0.15 59.9 18,200
0.08 79.7 27,050 0.17 79.9 29,350 0.15 64.9 22,700
0.06 49.5 4,450 0.11 49.6 4,150 0.09 39.6 3,450
0.06 59.8 6,900 0.11 59.7 7,950 0.09 49.8 5,700
0.06 69.8 10,650 0.11 69.6 10,200 0.09 59.7 9,650
0.06 79.9 18,550 0.11 79.8 17,250 0.09 64.5 12,950
0.06 89.8 28,500 0.11 89.9 24,250 0.09 69.7 15,500
0.04 49.8 3,250 0.06 54.7 3,550 0.05 49.8 3,000
0.04 59.7 5,050 0.06 59.8 4,850 0.05 59.6 5,050
0.04 69.9 7,250 0.06 69.7 6,800 0.05 69.7 8,600
0.04 79.7 10,300 0.06 79.9 9,250 0.05 79.8 13,800
0.04 89.7 16,350 0.06 89.9 14,500 0.05 89.9 20,450
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x ¼ x0
P
Ps
� �1=n
ð3Þ
where x0 is theoretical maximum adsorption capacity; Ps
and P are saturation pressure of adsorbate and system
adsorption pressure respectively. 1/n is model parameter. x0
and n are two parameters determined by experiments.
(c) D–A equation
D–A equation was proposed based on carbonaceous
substances and modified from Dubinin-Radushkevich
(D–R) equation which established volume filling theory.
Compared to Langmuir and Freundlich equations, D–A
equation is more popular on physical micropores adsorp-
tion and its format can be given as below [4].
x ¼ x0 exp �D T lnPs
P
� �n� �ð4Þ
where D is a constant that is determined by the adsorbent
microstructure and n is a parameter introduced to achieve a
better fit with experimental data. x0, D and n are three
parameters normally determined by experiments.
4 Equation fitting
Experimental data were processed in a Matlab environment
to fix the parameters of three adsorption equation formats,
respectively. The specific working form of the Langmuir
equation was used for curving fitting and sensitivity study.
This specific working form was chosen because this work
attempts to experimentally confirm the failure of the
working forming at high concentration condition. More-
over, this work attempts to point out the dramatic sensi-
tivity of parameters of the working form.
With respect to curve fitting method, multiple linear
regression technique was used to determine the parameters
of the Langmuir format. Simple linear regression was
employed to determine parameters of Freundlich and D–A
Formats. Parameters of each sample in three different
formats were determined and shown in Table 2.
5 Results discussions
5.1 Equations’ agreement
The parameters in three equations that were found by fit-
ting experimental data into Eqs. 2, 3 and 4, are listed in
Table 2. The corresponding P–T–x diagrams of three
samples plotted with experimental data and isosteric lines
plotted from determined equations are presented from
Figs. 2, 3 and 4.
The values of R2 in Table 2 indicate that D–A format
enjoys the best agreement with experimental data while
Langmuir format has a relative poor agreement. From three
groups of figures, it’s not difficult to see two features of
those three adsorption equations. One is that Langmuir
equation only agrees well with experimental data in the
range of mediate and low concentration. The other is that
Freundlich equation drops its precision when the concen-
tration (x) decreases, which agrees with the tendency dis-
covered on silica gel-water working pair [19]. However, it
is still accurate for design calculation purposes since in
most cases the adsorption refrigeration system doesn’t
reach the extremely low concentration zone.
5.2 Format impact on COP and refrigeration capacity
When designing a carbon methanol refrigeration system,
estimating the refrigeration capacity and the system COP is
essential. However, for a given working pair and the
adsorption equilibrium test data, using a different adsorp-
tion equation format may result in a huge difference.
Figures 5 and 6 show the refrigeration capacity and COP,
respectively calculated using the three adsorption equation
formats based on tested three activated carbon samples.
Detailed thermodynamic analysis and calculation method
of ideal refrigeration capacity Qr and COP of a basic
adsorption refrigeration system can be found from litera-
ture [8]. In the studied base-case, the system is assumed to
work under the conditions that the condensing temperature
and evaporating temperature are 30�C and -5�C, respec-
tively. The adsorbent bed finally reaches the temperature of
the heat source and the highest heat source temperature is
assumed below 120�C for the consideration of avoiding
methanol decomposition [20].
Figure 5 shows the refrigeration capacity calculated
with different adsorption equations on each sample.
Parameters used in the numerical calculation are listed in
Table 3. Figure 5a, b, c clearly show that refrigeration
capacity Qr calculated from D–A format always achieve
the highest values for each tested samples while those
calculated from Langmuir format always have the lowest
values in the temperature range 85–120�C. Moreover, in
the typical working temperature range, the difference of Qr
calculated from D–A and Freundlich format is significant,
almost reaching 200% for each sample at the temperature
85�C. Therefore, from the view of design calculation, these
adsorption formats need to be treated carefully.
From Fig. 6, it is not difficult to see the variation of
system COP calculated using different formats. Generally,
D–A format gives the highest COP values. The difference
among the COP values calculated at a given heat source
temperature can be quite significant. For instance, there is
an almost 170% absolute difference between results from
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D–A and Freundlich equation when the values are nearly
stable. Besides the difference in COP values, the minimum
heating source temperature required to run the refrigeration
cycle calculated from those three adsorption equations
varies considerably. The minimum temperatures calculated
from D–A equation and Freundlich equation don’t vary
much for each carbon sample while that calculated from
Langmuir equation are quite different for different carbon
samples.
5.3 Parameters’ sensitivity
It is inevitable that some errors may be generated during
the experiments and when fitting experimental data to
isotherm equations, which will produce errors in the
parameters of each isotherm equations. Therefore, a sen-
sitivity analysis of each parameter in the three isotherm
equations to the system COP and refrigeration capacity has
been conducted to reveal how tolerant the calculated sys-
tem performance is to errors in parameters when each
isotherm is used. The scenario used is that of activated
carbon 207C and methanol as a working pair in a refrig-
eration system that is operated with heating source at
100�C and condensing temperature at 30�C to generate
evaporating temperature of -5�C. In order to study
parameters’ sensitivity to system’s refrigeration capacity
Qr and COP, up to ± 10% errors are assumed for each
parameter to be caused by the experiment and data pro-
cessing. Figures 7 and 9 show the results of the sensitivity
study.
Figure 7 shows that for Langmuir equation, error in the
parameter Qst has a significant impact on refrigeration
output and system COP while parameter KI0 only has a
relative slight effect. When KI0 has 10% error, the impact
on the values of refrigeration power output or system COP
is in a similar range, i.e., approximately 10 and 5%,
respectively. However, the impact of the error in Qst is
huge. When the error in Qst is ?1%, the variation of cal-
culated COP and refrigeration output is about 5–10%. If the
error in Qst was up to 10%, variation in calculated refrig-
eration output could be as high as 340% while the variation
in calculated COP could be about 55%. Moreover, positive
error in Qst could affect Qr more significantly compared to
that caused by negative error.
For Freundlich equation, errors in both the parameters
(i.e., x0 and n) apparently have little influence in the cal-
culations of refrigeration output Qr and system COP as
shown in Fig. 8. Figure 8 shows that the calculated values
of Qr and COP are less sensitive to parameter n than to
parameter x0.
In D–A equation, parameter n is more sensitive to the
calculations of Qr and COP as shown in Fig. 9. A -10%
error in the parameter n could lead to about -48 and -34%
deviations in refrigeration output Qr and system COP,
respectively. Figure 9 also shows that the calculated values
of Qr and COP is less sensitive to errors of parameters x0
and D.
6 Conclusions
The experimental data of isosteric-based adsorption equi-
librium testing for three activated carbon and methanol
were fitted into three classical adsorption equations. The
parameters in the three equations were found and their
sensitivities to the calculations of system performance i.e.,
system COP and refrigeration capacity were studied.
The following conclusions can be drawn from the study:
1. D–A equation is the most suitable format to present the
adsorptive properties of the tested activated carbon/
methanol working pairs, in terms of its agreement with
experimental data. Two factors may contribute to the
superiority of D–A equation for presenting adsorptive
property of activated carbon/methanol working pairs.
One is that D–A equation was proposed based on the
theory of volume filling of micropores (TVFM), which
is more realistic compared to monolayer adsorption
Table 2 Determined parameters of each adsorption equation formations
Carbon sample Format Determined parameters R2
207C Langmuir – KI0 ¼ 5:68� 10�6 Qst ¼ 1007:30 0.9272
207EA Langmuir – KI0 ¼ 7:12� 10�6 Qst ¼ 1040:90 0.9558
WS-480 Langmuir – KI0 ¼ 7:87� 10�7 Qst ¼ 1196:30 0.9323
207C Freundlich x0 ¼ 0:25 – n ¼ 1:65 0.9777
207EA Freundlich x0 ¼ 0:54 – n ¼ 1:47 0.9723
WS-480 Freundlich x0 ¼ 0:43 – n ¼ 1:39 0.9641
207C D–A x0 ¼ 0:15 D ¼ 9:67� 10�6 n ¼ 1:72 0.9901
207EA D–A x0 ¼ 0:28 D ¼ 8:45� 10�7 n ¼ 2:08 0.9865
WS-480 D–A x0 ¼ 0:27 D ¼ 9:08� 10�6 n ¼ 1:78 0.9942
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Fig. 2 Comparison between experimental data (markers) of sample
207C and corresponding isosteric lines (solid lines) plotted with
determined adsorption equations: (a) Langmuir equation, (b) Freund-
lich equation and (c) D–A equation
Fig. 3 Comparison between experimental data (markers) of sample
207EA and corresponding isosteric lines (solid lines) plotted with
determined adsorption equations: (a) Langmuir equation, (b) Freund-
lich equation and (c) D–A equation
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Fig. 4 Comparison between experimental data (markers) of sample
WS-480 and corresponding isosteric lines (solid lines) plotted with
determined adsorption equations: (a) Langmuir equation, (b) Freund-
lich equation and (c) D–A equation
Fig. 5 Qr versus heat source temperature calculated with Langmuir,
Freundlich and D–A equations, respectively: (a) Sample 207C,
(b) Sample 207EA and (c) Sample WS-480
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assumed by Langmuir equation. The other factor is
that D–A equation was just proposed based on
carbonaceous substances.
2. Freundlich equation is still in a good agreement with
experimental data, except in the area with a relatively
low concentration. This feature is similar to the finding
reported by Xia et al. [6]. One factor contributing to
the failure in low concentration is experimental errors,
because in this circumstance the errors of measuring
charged methanol become considerable.
3. Langmuir equation is not suitable to present the
adsorptive properties of the tested activated carbon/
Fig. 6 COP versus heat source temperature calculated with Lang-
muir, Freundlich and D–A equations respectively: (a) Sample 207C,
(b) Sample 207EA and (c) Sample WS-480
Table 3 Parameters used in thermal performance estimation
Symbol Value Unit Symbol Value Unit
Tc 30 �C Cr 2.53 kJ/kg K
Te -5 �C Mac 1 kg
L0 1,120 kJ/kg
Fig. 7 Impact on Qr and COP of a base-case cycle caused by error of
parameters KI0 and Qst of Langmuir equation: (a) refrigeration
capacity Qr and (b) refrigeration COP
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methanol working pair because it has the least
agreement with experimental data. However, when
the concentration is low, this working form of the
Langmuir equation enjoys a relatively good agreement
with experimental data. This is because the working
form of the Langmuir equation was derived with
Henry’s law, which should work better when the
adsorbate concentration is low. The present work
experimentally confirmed this theoretical point.
4. In the Langmuir equation, any error in the process of
experiment and data processing would cause huge vari-
ation in the system performance calculations. No matter
what working pairs are used, great effort is necessary to
determine the parameters accurately in Langmuir
equation, especially the Qst, otherwise the system
performance calculations would suffer considerably.
5. Freundlich isotherm format and D–A isotherm format
are recommended for carbon methanol adsorption
refrigeration system thermal performance calculations
since errors of their parameters have less impact on
calculation. However, the parameter n still needs to be
determined carefully in Freundlich and D–A equation
formats, especially in the D–A format.
Acknowledgments The tests were carried out in the Institute of
Refrigeration and Cryogenics at Shanghai Jiao Tong University
(SJTU). Special appreciation goes to Prof. R.Z. Wang for access to
their lab, and we would also like to extend our gratitude to Dr. T.X. Li
and L.X. Gong for their help in setting up the test rig.
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