Prediction of Gas-Phase Adsorption Isotherms
Using Neural Nets
Sukanta Basu1*†, Paul F. Henshaw1, Nihar Biswas1 and Hon K. Kwan2
1
2
Civil and Environmental Engineering, University of Windsor, Windsor, ON N9B 3P4, Canada
Electrical and Computer Engineering, University of Windsor, Windsor ON, N9B 3P4, Canada
F
or virtually any physical adsorption process, the capacity of an
adsorbent decreases as the temperature of the system increases. As
the temperature increases the adsorbed molecules acquire
sufficient energy to overcome the van der Waals’ attraction, holding
them in the condensed-phase and migrating back to the gas-phase.
Adsorption is an exothermic process. At low concentrations the heat
release is minimal and is quickly dissipated by the airflow through the
bed. At high concentrations considerable heating of the bed can occur,
which if not removed can cause the adsorbent efficiency to decrease
rapidly. In addition, in the case of granular activated carbon (GAC) beds,
heat accumulation in the interior of the bed can even cause auto-ignition of
the carbon bed. Thus, it becomes essential to monitor the bed temperature
and to understand its relationship to the adsorption capacity.
A large number of adsorption isotherm models presented in the literature
are generally able to describe the relationship between the adsorbate
concentrations in the fluid and adsorbed phases at a given temperature.
This necessitates an experimental study at each anticipated temperature
of the adsorption application. A few adsorption models have temperature
as a variable. This paper describes some of these models and a newly
proposed methodology for the prediction of gas-phase adsorption of
isotherms at different temperatures using artificial neural networks (ANNs).
Conventional Models
A brief overview is provided of some conventional adsorption isotherm
models capable of predicting adsorption equilibria at different temperatures,
so as to present a general view of the range of the features available
among the different models.
Modified Sips’ Model
Sips (1948) proposed the following semi-empirical isotherm (Rudzinski
and Everett, 1992) which is a combination of the Langmuir and
Freundlich equations:
(1)
*Author to whom correspondence may be addressed. E-mail address:
[email protected]
† Present Affiliation: St. Anthony Falls Laboratory, Department of Civil Engineering,
University of Minnesota, Minneapolis, MN 55414-2196, USA
The Canadian Journal of Chemical Engineering, Volume 80, August 2002
On examine dans cette étude plusieurs modèles (le
modèle Sips modifié, le modèle de Dubinin-Astakhov,
la théorie VSM, le modèle de Khan et al. généralisé et
un réseau neuronal artificiel simple (ANN)), afin de
prédire l’effet de la température sur l’adsorption à
l’équilibre des gaz et des vapeurs d’hydrocarbures sur
le carbone activé. On a utilisé des données publiées
sur l’adsorption du méthane, de l’éthane et du
propane sur du carbone activé à des températures
entre 311 K et 505 K pour estimer les paramètres des
modèles conventionnels et entraîner le réseau. Les
modèles conventionnels et l’ANN ont ensuite été
utilisés pour prédire l’isotherme à une seule température
pour chaque adsorbat, puis les résultats ont été
comparés aux données expérimentales. On a trouvé
que le modèle ANN avait une erreur relative moyenne
plus faible que les modèles conventionnels.
Keywords: activated carbon, adsorption, artificial
neural networks, isotherm, multilayer perceptron.
1
n
(ap)
v
=
1˘
v0
È
Í1+ (ap ) n ˙
ÍÎ
˙˚
This study investigated a number of models (the
modified Sips’, Dubinin-Astakhov’s, VSM theory, the
generalized Khan et al.’s model and a simple artificial
neural network (ANN)) to predict the effect of
temperature on equilibrium adsorption of hydrocarbon
gases and vapors on activated carbon. Published data
on the adsorption of methane, ethane and propane
on activated carbon at 311 K to 505 K were used to
estimate the parameters of the conventional models
and train the network. Then, the conventional models and
the ANN were used to predict the isotherm at a single
temperature for each adsorbate, and these results
were compared with experimental data. It was found
that the ANN model had a lower mean relative error
than the conventional models.
For adsorption of hydrocarbons on activated carbon,
Equation (1) was found to be superior to either the
Langmuir or Freundlich equations in data correlation
(Koble and Corrigan, 1952).
Koble and Corrigan (1952) used Sips’ equation to
analyze the adsorption isotherms of various hydrocarbons
on active charcoal as reported by Ray and Box (1950).
1
The constants v0 and n were found to be independent of
temperature. However, the semi-logarithmic plots of (a)1/n
versus (1/T) were straight parallel lines. Therefore, to incorporate
the temperature dependence, Sips’ equation is modified in the
present work as:
v
=
v0
a2 1
a1e T p n
a2 1 ˘
È
Í1+ a1e T p n ˙
Generalized Khan et al.’s Model
(2)
˙
˚
Í
Î
Dubinin-Astakhov (DA) Model
The potential theory of adsorption, originally developed by
Polanyi (1932), is based on rational physical and chemical
concepts rather than on the empirical fit of experimental data.
Using this, Dubinin and others (Dubinin and Astakhov, 1971;
Dubinin 1975; Golovoy and Braslaw, 1981; Werner and Winters,
1988) have shown that the adsorption isotherms of various
vapors on microporous adsorbents (e.g., activated carbon) can
be represented by the Dubinin-Radushkhevich (DR) equation.
The DR equation, however, fails for many gas–solid systems
(Doong and Yang, 1988). A more general equation, proposed
by Dubinin and Astakhov (DA), has been found to be useful
(Dubinin and Astakhov, 1971):
È Ï A n
¸Ô˘
ÔÊ ˆ
q = q00exp Í- ÌÁ ˜ + a(T - T0 )˝˙
Í ÔË E ¯
Ô˛˙˚
Î Ó
(3)
The limiting adsorption value (q00) corresponds to the initial
adsorption isotherm temperature T0.
The differential molar work of adsorption (A) is defined as:
A = RT ln
p*
p
(4)
The heterogeneity parameter (n) is equal to 2 for activated
carbon (Rudzinski and Everett, 1992).
The thermal coefficient of limiting adsorption (a) can be
calculated as (Dubinin, 1975):
Êr ˆ
logÁ *b ˜
Ë rcr ¯
a =
0.434(Tcr - Tb )
explicitly include the effect of temperature. Cochran et al. (1985)
proposed a new model based on the Vacancy Solution Model
of adsorption in conjunction with the Flory-Huggins activity
coefficient equations to incorporate the temperature dependency.
Khan et al. (1999) suggested a generalized isotherm with
temperature dependence for the equilibrium adsorption of pure
gases.
All the conventional models cited above have limited success
in adsorption equilibrium predictions due to several assumptions in
the model development. One way of avoiding this difficulty is
through the use of an assumption-free model. Artificial neural
networks operate directly on the data, thus avoiding the
necessity of a priori assumptions of conventional models with
all their consequent constraints.
Artificial Neural Network (ANN)
Artificial neural networks are highly parallel, flexible mathematical constructs that have been inspired by the workings of the
biological nervous system. ANNs have a natural propensity for
storing experiential knowledge and making it available for use
(Haykin, 1999). In approximating an input–output response, the
ANN mathematics involved is simple to implement yet difficult
to comprehend, and ANN is sometimes viewed as a ‘black box’.
In recent years, there has been a growing interest in the
application of ANNs in the chemical engineering field due to
various factors (Flood and Kartam, 1994a, b; Morshed and
Kaluarachchi, 1998).
The use of an artificial neural network requires following a
discrete set of steps outlined as follows (Gately, 1996).
1. Determination of ‘what is to be predicted’.
2. Data collection: Since artificial neural networks are not usually
able to extrapolate, the data patterns should go at least to
the edges of the problem domain in all dimensions (Flood
and Kartam, 1994a, b), and should be evenly distributed.
3. Preprocessing of data: To extract information from data sets
it is sometimes required to modify or preprocess the inputs
(Gately, 1996). For example two (or more) values can be
reduced into one input, which has more value to the
network than the original variables.
(5)
The adsorbate density at the critical temperature (rcr*) will be
(Dubinin, 1975):
*
rcr
=
M
1000b
(6)
where,
b =
1 Ê RTcr ˆ
8 ÁË pcr ˜¯
(7)
VSM Theory
Suwanayuen and Danner (1980) proposed the Vacancy
Solution Model (VSM) using the Wilson model for the activity
coefficients (Cochran et al., 1985). But this approach fails to
2
Figure 1. Two-layer perceptron network architecture.
The Canadian Journal of Chemical Engineering, Volume 80, August 2002
Table 1. The properties of adsorbates (Reid et al. 1977).
Adsorbate
Mol. Wt.
(g/mol)
Tb (K)
Tcr (K)
pcr (kPa)
rT (g/mL)
rcr* (g/mL)
a (1/K)
Methane (CH4)
16.043
111.7
190.6
4599.02
0.425111.7
0.372
1.67 ¥ 10–3
Ethane (C2H6)
30.070
184.5
305.4
4882.66
0.548183.0
0.462
1.40 ¥ 10–3
Propane (C3H8)
44.097
231.1
369.8
4244.47
0.582231.0
0.487
1.29 ¥ 10–3
Note: rT is bulk density at reference temperature T oK.
Figure 2. Processing at the neuron level.
4. Selection of network architecture: The most predominant
network architecture is Multilayer Perceptron (MLP). Figure 1
shows a typical two-layer perceptron for simulating an
input–output response. The network is made of nodes,
called neurons. The nodes are organized into input, hidden
and output nodal layers. The input layer is not considered to
be a neuron layer, as it does not process any signal. Each
node is connected to all the nodes in the adjacent layers.
Signal (preprocessed and scaled data) enters the network at
nodes of the input layer from where it is propagated to the
output layer through an intermediate hidden layer. Each
connection has associated with it a weight, which acts to
modify the signal strength. The nodes in the input and
output neurons perform two functions: the weighted inputs
and a bias term are summed and the resulting summation is
passed through an activation function (Figure 2). The
logistic function, a form of sigmoid nonlinearity (Haykin,
1999) is the most common choice of activation function.
Generally, there is no direct and precise way of determining
the most appropriate number of neurons to include in the
hidden layer. Increasing the number of hidden neurons
provides a greater potential for developing a solution
surface that fits closely to that implied by the data patterns.
However, a large number of hidden neurons can lead to a
solution surface that deviates dramatically from the trend or
that ‘overfits’ the data points. In addition, a large number of
hidden neurons slows down the operation of the network
(Flood and Kartam, 1994a, b).
5. Selection of a learning algorithm: The back-propagation
algorithm (BPA) is the most commonly used training
algorithm for neural nets.
6. Training: ANN is developed in two phases: the training
(accuracy or calibration) phase, and the testing (generalization
or validation) phase (Morshed and Kaluarachchi, 1998).
Following the specification of the network, inputs, outputs
and topology, the network is trained by successive presentations
of the input–output data pairs. One complete presentation
of the entire training set during the learning process is called
The Canadian Journal of Chemical Engineering, Volume 80, August 2002
an epoch. The learning process is maintained on an epoch-byepoch basis until the synaptic weights and bias levels of the
network stabilize and the average squared error over the entire
training set converges to some minimum value (Haykin, 1999).
In general, there are no well-defined criteria for stopping the
training. Some reasonable criteria may be number of
epochs, average rms error, maximum rms error or maximum
absolute error.
7. Testing: Normally, about 90 percent of the data are used in
the training set, while the rest is put aside for the testing set.
In the training process the network uses the training set of
data to recognize inherent patterns. The testing data set is
reserved to validate the trained network using information
the network has not seen during training.
Research Approach
In this study, the objective was to predict the gas-phase adsorption
isotherms at variable temperatures using conventional and
artificial neural network models. The experimental data for the
Table 2. Calculated saturation pressures (p*) and densities (r) of
adsorbates at different temperatures.
Adsorbate
T (K)
p* (kPa)
r (g/mL)
Methane (CH4)
310.92
338.70
366.48
394.20
422.00
37 196
48 803
61 449
74 871
88 927
0.305
0.291
0.278
0.265
0.253
Ethane (C2H6)
310.92
338.70
366.48
394.20
422.00
449.80
477.59
5423
8733
13 084
18 503
25 022
32 599
41 179
0.459
0.442
0.425
0.408
0.393
0.378
0.363
Propane (C3H8)
310.92
338.70
366.48
394.20
422.00
449.80
477.59
505.37
1295
2364
4003
6238
9163
12 387
17 288
22 532
0.525
0.507
0.489
0.472
0.455
0.439
0.424
0.409
3
Table 3. Range of data in training sets for the ANN.
Adsorbate
Methane (CH4)
Ethane (C2H6)
Propane (C3H8)
T (K)
p*/p
r (g/mL)
q (mol/g)
310.92–422.00
310.92–477.59
310.92–505.37
25–1812
3.7–1955
1.9–1416
0.253–0.305
0.363–0.459
0.40 –0.525
0.09–3.77
0.25–5.81
0.33–5.30
No. of Patterns
38
49
51
Table 4. Sets used in testing the ANN.
Adsorbate
Methane (CH4)
Ethane (C2H6)
Propane (C3H8)
No. of Patterns
7
9
8
T (K)
p*/p
r (g/mL)
q (mol/g)
394.20
422.00
366.48
51–1451
17–1251
6.3–1001
0.265
0.393
0.489
0.13–1.82
0.27–3.59
0.69–4.24
convergence problem is due to the double exponentiation. It
was our experience that it is also very difficult to have a
converging solution in the case of the Khan et al. generalized
model without an extremely accurate estimate of the initial
parameters. Although in some trials the fit converged with an
attractive plot, the confidence intervals of some parameters
were very wide (i.e., the standard errors were large), and this
means the fit was not unique (Basu et al., 2000). So, in this work,
the modified Sips’ model and Dubinin-Astakhov model were
selected for further study and comparison.
gas-phase adsorption of hydrocarbons on activated charcoal at
different temperatures were collected from the literature (Ray
and Box, 1950). The same data set had been reported by Koble
and Corrigan (1952) and Valenzuela and Myers (1989) in
different formats.
The properties of different adsorbates are given in Table 1
(Reid et al., 1977). The thermal coefficient of limiting adsorption
(a) and the adsorbate density at the critical temperature (rcr*)
have been calculated by means of Equations (5) and (6),
outlined by Dubinin and Astakhov (1971). The saturation vapor
pressures (p*) and the adsorption densities at different temperatures
are given in Table 2. Saturation vapor pressures below the
critical pressures were taken from Valenzuela and Myers (1989).
Effective values of p* for T > Tc were calculated using the
proposed method of Dubinin (1975). Adsorbate densities (r) for
T > Tb were evaluated by substituting T for Tb and solving for r
in place of rb in Equation (6), using the value of a given in Table 1.
Modified Sips’, DA, VSM and Khan et al.’s generalized
models were primarily selected for the present work based on
the principle of system identification. System identification
generally refers to the process of determining a suitable
mathematical model for an unknown system by correlating the
input–output data (Jang et al., 1996). After identifying the four
models, the Levenberg-Marquardt algorithm was applied for
estimating the parameters of the models. This nonlinear
regression-based algorithm was implemented using DataFit (version
6.1.10, Oakdale Engineering, Oakdale, PA). When determining the
goodness of fit for VSM, it was found that the solution did not
converge in many cases. According to Khan et al. (1999), this
ANN Development
ANN development for simulating the gas-phase adsorption
equilibrium requires identification of the input and output
Table 5. The properties of the ANN.
Training algorithm
Back propagation algorithm (BPA)
Activation function
Logistic function
Training properties
Learning rate
Momentum
Stopping criteria
Epoch
Avg. rms error
Max. rms error
Percent correct
0.1
0.0
200 000
0.01
0.01
100
Table 6. Comparison between experimental and model predictions based on the training data set.
Adsorbate
Temp. range (K)
Modified Sips’ model, Eq. (2)
vo
Dubinin-Astakhov model, Eq. (5)
a1
a2
1/n
MARE (%)
q00
E
MARE (%)
Methane (CH4)
310.92 – 422.00
141.9
7.57¥10-5
1849.1
0.884
4.70
6.332
2800.6
22.28
Ethane (C2H6)
310.92 – 477.59
168.7 6
44¥10-5
2086.9
0.626
4.83
5.974
34153
9.09
Propane (C3H8)
310.92 – 505.37
140.7
9.95¥10–5
2283.6
0.654
5.68
5.230
3945.6
9.81
4
The Canadian Journal of Chemical Engineering, Volume 80, August 2002
Table 7. Comparison between ANN and conventional model predictions based on the testing data set.
Adsorbate
model,
Experimental
q (mol/g)
v (mL/g)
ANN
Modified Sips’ model,
q (mol/g)
Equation (2)
v (ml3/g)
Dubinin-Astakhov
Equation (5)
q (mol/g)
Methane (CH4)
0.1257
0.2352
0.2355
0.7680
1.1780
1.5160
1.8230
3.0746 0
5.7542
5.7622
18.7899
28.8112
37.0788
44.5948
MARE (%)
1562
0.2332
0.2357
0.8414
1.3109
1.6873
1.9715
8.19
3.7167
6.4673
6.5318
20.5432
31.3342
40.5028
48.5228
11.82
0.0872
0.1768
0.1790
0.6653
1.0409
1.3617
1.6465
17.76
Ethane (C2H6)
0.2745
0.5149
0.7147
0.8845
0.8909
2.2470
2.9290
3.2260
3.5940
6.7151
12.5974
17.4869
21.6398
21.7963
54.9919
71.6523
78.9177
87.9369
MARE (%)
0.2781
0.4539
0.7209
0.9050
0.9176
1.9996
2.7437
3.0529
3.2659
5.68
9.4118
14.7887
19.6059
23.2362
23.5039
46.7735
62.7482
71.4131
78.9828
14.66
0.2383
0.4479
0.6476
0.8017
0.8131
1.8146
2.4926
2.8544
3.1662
12.36
Propane (C3H8)
0.6942
1.6770
2.3860
2.9540
3.0410
3.5840
3.9220
4.2440
16.9852
41.0270
58.3645
72.2643
74.4081
87.6863
95.9539
103.8457
MARE (%)
0.6682
1.6325
2.1515
2.9505
3.0301
3.9183
4.1360
4.2252
3.99
15.6545
38.0603
53.4046
69.6669
71.2441
93.9041
103.5570
108.9380
6.42
0.9582
1.9065
2.4574
3.0096
3.0625
3.8246
4.1543
4.3379
9.02
variables. From the knowledge of conventional models, the
amount of adsorbate adsorbed at equilibrium (q) can be
expressed as a function of temperature, density, equilibrium
pressure of the adsorbate (p) and saturated vapor pressure of
the adsorbate (p*).
Based on the principle of preprocessing, the input variables p*
and p were combined to extract more information from the experimental data. The prediction capability of the network was found to
be better with the (p*/p) variable than with the original variables.
The training and testing data sets are summarized in Tables
3 and 4, respectively. The architecture of the artificial neural
network depends on the number of input and output variables.
For the present problem, the two-layer perceptron network
consisted of three input nodes and one output node. However,
the number of hidden nodes in the hidden layer was not fixed.
The best number was determined by testing the network,
following the procedure adopted by Basheer and co-workers
(Basheer and Najjar, 1994, 1995, 1996; Basheer et al., 1996).
ANNs were developed using Qwiknet (version 2.23, Jensen
Software, Redmond, WA). The training was performed using
the default values in the package for the parameters of the
network, e.g., learning rate, momentum, input noise, weight
decay, error margin and pattern clipping. The back-propagation
algorithm (BPA) was used for learning. The properties of the
network are tabulated in Table 5.
The Canadian Journal of Chemical Engineering, Volume 80, August 2002
Results and Discussion
The parameters of the modified Sips’ and Dubinin-Astakhov models,
together with mean absolute relative errors (MAREs) between the
predicted and experimental values based on the training data set are
listed in Table 6. The MARE is defined as follows:
Ê 100 ˆ N qi ,experimental - qi ,predicted
MARE(%) = Á
˜Â
Ë N ¯ i =1
qi ,experimental
(8)
The various values of MARE for different neural network
architectures are shown graphically in Figure. 3. Since there is
no clear trend in Figure. 3, the minimum MARE criterion is
considered for the selection of the final network.
The comparisons based on the testing data set between the
conventional and artificial neural net models are listed in Table 7.
Clearly, the ANN prediction results in a more accurate representation
of the true isotherm (lower MARE values) than for the modified
Sips’ model or DA predictions. In general, the prediction errors
are highest for methane, lower for ethane and lowest for
propane. The exception to this tend is the modified Sips’ model,
which has the highest MARE for ethane, followed by methane
and propane.
The predicted isotherms for the hydrocarbons are shown in
Figure. 4. The steepness of the curves increases as carbon
5
Acknowledgements
The authors wish to acknowledge the financial support of the Natural
Sciences and Engineering Research Council of Canada.
Nomenclature
a
a1
a2
A
b
E
M
n
N
p
p*
Figure 3. Variation of mean relative absolute error with the number of
hidden nodes.
pcr
q
qi,experimental
qi,predicted
q00
R
T
T0
Tcr
Tb
v
v0
empirical parameter in Equation (1)
empirical parameter in Equation (2)
empirical parameter in Equation (2)
differential molar work of adsorption, (J/mol)
molar volume, (mL/mol)
characteristic free energy of adsorption, (J/mol)
molecular weight of adsorbate (g/mol)
empirical parameter in Equations (1), (2) and (3)
number of experimental data
equilibrium pressure of the adsorbate, (kPa)
saturated vapor pressure of adsorbate at test
temperature, (kPa)
critical pressure, (kPa)
moles of adsorbate adsorbed per unit mass of
adsorbent, (mol/g)
experimental adsorption capacity, (mol/g)
predicted adsorption capacity, (mol/g)
limiting adsorption value for temperature T0 (mol/g)
gas constant, (8.3144 J/mol·K)
temperature, (K)
reference adsorption isotherm temperature, (K)
critical temperature of adsorbate, (K)
boiling point of adsorbate, (K)
volume of gas adsorbed (usually measured at STP), (mL/g)
limiting adsorption capacity, (mL/g)
Greek Symbols
a
r
rb
rcr*
thermal coefficient of limiting adsorption, (1/K)
adsorbate density, (g/mL)
density of bulk liquid at Tb, (g/mL)
adsorbate density at critical temperature, (g/mL)
Abbreviations
Figure 4. Comparison between the experimental and ANN predicted
adsorption isotherms.
number increase. In addition, the ANN over-predicts the
adsorption capacities for propane and methane, but
under-predicts for ethane.
From Table 7, it is obvious that the neural network performance
was better than the conventional models in all respects.
Moreover, the processing time is significantly reduced once the
network is trained.
Conclusions
A two-layer ANN with three hidden nodes was used to model
the adsorption isotherms for methane, ethane and propane on
activated carbon at temperatures from 311 to 505K. The values
of the MARE were lower for the ANN predictions as compared
to the modified Sips’ model and the Dubinin-Astakhov model in
all cases. Values of MARE for ANN ranged from 3.99 to 8.19. In
general, the MARE values are lower as the number of carbon
atoms in the adsobate increases.
The use of ANNs to accurately model physical phenomena
can be exploited in used process control applications, since the
ANNs can adapt to changing conditions, faulty inputs and
unaccounted-for variables.
6
ANN
MARE
NH
Artificial neural network
mean absolute relative error
number of hidden nodes
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Manuscript received January 3, 2001; revised manuscript received
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