Prediction of Methane Uptake on Different Adsorbents in Adsorbed

Article pubs.acs.org/EF

Prediction of Methane Uptake on Different Adsorbents in Adsorbed Natural Gas Technology Using a Rigorous Model Ebrahim Soroush,† Mohammad Mesbah,‡ Amin Shokrollahi,‡ Alireza Bahadori,*,§ and Mohammad Hossein Ghazanfari‡ †

Department of Chemical Engineering, Sahand University of Technology, Tabriz, Iran Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran § School of Environment, Science and Engineering, Southern Cross University, Lismore, New South Wales 2480, Australia ‡

ABSTRACT: One of the most promising methods for transporting natural gas and overcoming its low energy density is adsorbed natural gas (ANG) technology. ANG technology is highly dependent on the perfect conception of adsorption isotherms in different operational conditions and on different adsorbents. In this study, the utilization of a novel mathematical model of least squares support vector machine (LSSVM) for accurate prediction of adsorption isotherm has been examined. The considered variables were temperature, pressure and type of adsorbents. A data set containing 670 experimental data points of methane adsorption on 10 different adsorbents in a broad range of temperature and pressure were used for training and testing of the LSSVM model. Results showed that the LSSVM model is capable to predict adsorption isotherm with an acceptable statistical parameters of 2.3058% and 0.9995 for AARD% and R2, respectively. In addition, the leverage statistical algorithm indicated that the suggested model is statistically authoritative for prediction of methane isotherm adsorption and no outliers have been detected in the data set.

1. INTRODUCTION Natural gas is a significant and abundant energy source, which mainly consists of methane. Decrease of carbon dioxide emissions and clean combustion (reduction in CO, NOx, and SOx emission), in comparison to petroleum-based fuels, makes natural gas an interesting alternative from the environmental point of view.1,2 One of the most important disadvantages of natural gas is its low energy density (defined as the combustion heat per volume unit) of 0.038 MJ/L (0.11% of gasoline values).1 This is the main barrier for commercialization of this fuel.3,4 Transporting natural gas from production sites to consumption markets, when pipelines are not economical, usually is done through compressed natural gas (CNG) or liquefied natural gas (LNG) processes, one in high pressures (20 to 30 MPa) and the other in extremely low temperatures (−162 C).5,6 Also, these are the conventional methods for overcoming of natural gas low energy density.7 LNG with volumetric energy density (VED) of 22.2 MJ/L (64% of gasoline, 34.2 MJ/L) is in need of expensive cryogenic vessels and suffers from boil-off losses.8 The VED for the other method, CNG, is only 9.2 MJ/L (27% of gasoline) and requires thick-walled cylindrical storage tanks and multistage compressors.8 One promising alternative to mentioned methods is adsorbed natural gas (ANG) . In this process, natural gas molecules adsorbed on a micro- and meso-pores of the solid adsorbent through van der Waals forces.1,8 This is done in a relatively low pressure of about 3.5−4 MPa,1,6,9which counts as an advantage because the lower the pressure the lower the cost of fuel and higher the safety.4,10 Also, it has much higher temperature (room temperature) than LNG but is lower in capacity.9 This technology has no need for compression to high pressures or liquefaction to low temperatures though metal consuming vessels © 2014 American Chemical Society

and compressor equipment or cryogenic facilities are not necessary, giving it the valuable advantage of low capital cost and also lower energy consumption.7,11 Despite its low specific mass (ratio of useful gas stored to total parasitic mass of activated carbon and the container), it can expect to provide 160 (v/v) in volumetric storage capacity.9 Understanding of the adsorption isotherms over a broad range of pressures and temperatures for different adsorbents is an indispensable need for development of ANG technology.12 In fact, adsorption isotherms are equilibrium correlations, describing the manner of adsorbate and adsorbent interaction.13 Numerous models could find in literature for determination of adsorption isotherms of different adsorption systems. Although there are many isotherm models available in the literature, complexity of models for a wide range of users and lack of a flexible general model capable of representing data for all recognized types of adsorption isotherms due to their variable nature make one desire a more appropriate, accurate and unifying model. Lately, because of progress in artificial intelligence systems, modeling in different problem areas has found new approaches.14−19 In this manner, alternative methods have been introduced for determining adsorption isotherms. One of the most popular methods, which has been subject of many adsorption behavior prediction papers, is artificial neural networks (ANNs).20−31 The advantages of this artificial intelligent method were so attractive for investigators. It is easy to use, nonlinear, and highly accurate.32 It has the capability for use of additional parameters20 in comparison to traditional isotherm models and it may identify which variable is more significant for the process.22 Received: July 9, 2014 Revised: September 9, 2014 Published: September 10, 2014 6299

dx.doi.org/10.1021/ef501550p | Energy Fuels 2014, 28, 6299−6314

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Table 1. Characteristics of Experimental Data adsorbents

temperature range (K)

pressure range (MPa)

Maxsorb II specimen9 Maxsorb III52,53 ACF (A-20)52 HSZ-3206 DAY6 RP-157 RP-207 activated carbon10 KT activated carbon from the east of Kalimantan12 CarboTech commercial activated carbon12

281.40−343 160−348.15 278.15−348.15 303.15−348.15 303.15−323.50 293.15−313.15 293.15−313.15 293.15−313.15 300−318 300−318

0.060−1.254 0.013−2.203 0.051−2.433 0.023−2.696 0.029−2.607 0.011−3.606 0.010−3.628 0.046−1.569 0.058−3.496 0.063−3.52

identification [0 [1 [1 [1 [0 [1 [0 [1 [0 [1

1 0 1 1 0 0 0 0 0 1

1 0 1 0 1 0 0 1 1 1

0] 1] 0] 1] 0] 0] 0] 0] 1] 1]

adsorption range (g/g) 0.0040−0.1310 0.0060−0.6460 0.0030−0.1370 0.0005−0.0197 0.0002−0.0284 0.0026−0.1080 0.0026−0.1289 0.0030−0.0681 0.0070−0.0510 0.0060−0.0590

Figure 1. Schematic of the CSA-LSSVM algorithm.

There is no need for fully detailed knowledge of the physical phenomena22,23 and also no restriction by any process assumptions. Also, it has very good accuracy even in the case of limited data.22 Still with all the advantages of ANNs, there are some deficiencies for their external predictions because they may result in random initialization of the networks and variation of the stopping criteria during the optimization of the model parameters.33 For determining model parameters, precise and dependable data are required. As a result, the experimental data should be investigated with a comprehensive model, which is capable of handling various experimental techniques. In this study, a novel mathematical technique called least square support vector machine (LSSVM) is presented for predicting methane adsorption isotherm behavior on different adsorbents. With use of statistical parameters, the validity of the model will be examined. Then we will compare the network with prior isotherm models for its accuracy. For avoiding doubtful data the statistical based Leverage approach is used.

Figure 2. Comparison between the results of the developed model for train data set and the database values. (a) Scatter plot and (b) relative deviation plot.

(methane uptake) depends on finding a suitable mathematical tool. Also, ANN-based models have presented excellent accuracy in different problems,34−40 they have a great disadvantage due to nonreproducibility of results. This problem is because of the random initialization of the network and stopping criteria during the model parameter optimization process.41,42 Besides, there are other problems like time-consuming network training procedure (high computational load), tendency to over fit and high dependency on ANN design parameters (number of layers, nodes, etc.), and also extrapolation is not recommended for ANN approach.43

2. LSSVM MODEL Creating a mathematical relation between experimental data (temperature, pressure, type of adsorbent) and eligible output 6300


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Table 2. Details of Statistical Parameters for Proposed LSSVM, Toth, and Langmuir Models MSE LSSVM model

Toth model Langmuir model

all train test

3.57 3.51 3.81 3.49 1.33

× × × × ×

−06

10 10−06 10−06 10−05 10−04

AARD%

STD

R2

N

2.3058 1.8422 4.1601 6.6905 6.5168

0.0871 0.0891 0.0788 0.0871 0.0871

0.9995 0.9996 0.9994 0.9954 0.9828

670 536 134 670 670

Table 3. Details of LSSVM Model Statistical Parameters on Each Adsorbent adsorbents

AARD%

STD

R2

N

1.02 × 10 1.28 × 10−05 1.29 × 10−07 1.33 × 10−07 9.1 × 10−08 5.24 × 10−07 1.02 × 10−06 1.43 × 10−07 4.06 × 10−06

1.3674 1.2708 0.7790 3.9300 7.0901 4.5573 7.8458 1.9396 2.6241

0.0329 0.1413 0.0341 0.0058 0.0087 0.0360 0.0416 0.0194 0.0139

0.9991 0.9994 0.9999 0.9963 0.9988 0.9996 0.9995 0.9996 0.9794

127 160 128 36 37 38 38 43 31

9.57 × 10−08

0.8846

0.0174

0.9997

32

MSE 9

Maxsorb II specimen Maxsorb III52,53 ACF (A-20)52 HSZ-3206 DAY6 RP-157 RP-207 activated carbon10 KT activated carbon from the east of Kalimantan12 CarboTech commercial activated carbon12

−06

regression problems has a high computational burden for applying the constraint of the mentioned convex optimization.47 For this reason, application of an SVM algorithm in a large scale problems data is limited due to memory and time consumed during optimization.47 An improved version of SVM, which called least squares support vector machine (LSSVM), has been developed by Suykens and Vandewalle.45 A modified version attempted to reduce the complexity and increase the speed of convergency. One of the main differences between LSSVM and SVM is that the inequality constraints of SVM change to equality constraints in LSSVM paradigm.45,47By this reformulation, the learning process has been done through solving a set of linear equations that can be solved iteratively.45,48 Thus, in large scale problems with large data set in which accuracy and time are important, application of LSSVM algorithm is better than SVM. Generally, in LSSVM algorithm, the optimization problem is expressed as follows:45 Figure 3. Comparison between the results of the developed model for test data set and the database values. (a) Scatter plot and (b) relative deviation plot.

n

1 1 || w ||2 + μ ∑ ei2 2 2 i=1

min J(w , e) =

(1)

subjected to the following linear constraints: Support vector machines have been introduced as powerful and robust tools for solving a variety of complex problems from pattern classification to nonlinear function approximation. Considering that the available measured data set,{(x1,y1), ..., (xn,yn)} where xi represent the input parameter and yi represent the desired output, SVM makes a nonlinear mapping in a manner that input space maps into a higher/infinite dimensional feasible space. In fact, the first goal of SVM is to find the best hyperplane that have minimum distance from all experimental data points.44,45 The details and equations of SVM theory can be found in previous studies.46,47 Application of SVM algorithm for regression problems or function approximation results in facing a convex optimization problem that is subjected to inequality constraints to find support vectors. Consequently, employing SVM method for

yk = ⟨w t , g (xk)⟩ + b + ek

k = 1, 2, ..., n

(2)

In eq 1, ek signifies error variables; μ ≥ 0 denotes regularization constant; g(x) represents the mapping function, b and w are bias terms weight vectors and weight vectors, respectively, and index t represents the transpose of the weight matrix. Incorporating the linear constraint into the objective function previously introduced in eq 1 provides45 n

L LSSVM

1 1 = || w ||2 + μ ∑ ei2 2 2 k=1 n

−

∑ βk{⟨wt , g(xk)⟩ + b + ek} k=1

6301

(3)


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Table 4. Maximum and Minimum Absolute Relative Deviation for Each Adsorbent in Different Models absolute relative deviation % LSSVM model adsorbents

N

Maxsorb II specimen9 Maxsorb III52,53 ACF (A-20)52 HSZ-3206 DAY6 RP-157 RP-207 activated carbon10 KT activated carbon from the east of Kalimantan12 CarboTech commercial activated carbon12

127 160 128 36 37 38 38 43 31 32

min 1.37 1.97 7.51 1.42 7.44 1.54 1.95 7.27 2.37 3.39

× × × × × × × × × ×

Toth model max

10−02 10−03 10−03 10−02 10−03 10−03 10−03 10−03 10−02 10−02

Langmuir model

min

10.85 20.40 6.46 44.19 81.51 81.68 198.97 15.50 21.31 7.76

4.92 2.18 1.83 2.45 2.45 2.22 1.53 3.97 3.31 1.75

× × × × × × × × × ×

10−03 10−01 10−02 10−01 10−01 10−02 10−02 10−02 10−02 10−01

max 18.57 77.82 19.55 72.74 72.74 188.72 37.72 39.33 95.97 5.38

min 2.11 1.16 2.77 3.13 7.81 1.53 9.81 1.48 5.19 2.08

× × × × × × × × × ×

10−02 10−01 10−03 10−02 10−02 10−01 10−02 10−01 10−02 10−01

max 8.19 66.35 22.53 16.36 148.06 24.95 48.54 93.39 18.62 24.33

Figure 4. Adsorption isotherms of CH4 on different adsorbents.

with Lagrangian multipliers βk ∈ R. Based on the method of Lagrangian multipliers, the below conditions are necessary for optimization: ⎧ ∂L ⎪ LSSVM ⎪ ∂b ⎪ ⎪ ∂L LSSVM ⎪ ∂w ⎪ ⎨ ⎪ ∂L LSSVM ⎪ ∂βk ⎪ ⎪ ⎪ ∂L LSSVM ⎪ ⎩ ∂ek

regression conditions, Kernel function has been contributed in eq 5 as follows: n

y=

n

=0→

k=1

∑ βk = 0 n

∑ βkg(xk)

K (x , xk) = g (x) ·g (xk)t

k=1

=0 (4)

By considering a linear regression between independent and dependent parameters of the LSSVM algorithm, the following equation can be written:45

∑ βkxk tx + b k=1

K (x , xk) = (1 + xT xk /c)d

(8)

K (x , xk) = exp(|| xk − x ||2 σ 2)

(9)

where d is the degree of polynomial and σ2 is squared bandwidth, which must be optimized through an optimization algorithm during the learning process.

n

y=

(7)

There are many Kernel functions (e.g., radial basis function (RBF), polynomial, linear, sigmoid, spline, etc.) that can be used in LSSVM algorithm. Though, the most common Kernel functions are polynomial (eq 8) and RBF (eq 9)47

= 0 → {⟨w t , g (xk)⟩ + b + ek − yk

= 0 → βk = μek , (k = 1, ..., n)

(6)

Kernel function K(xi, x) is defined as inner product of vectors g(x) and g(xi) as is shown below:

k=1

=0→w=

∑ βkK (xk , x) + b

3. LEVERAGE APPROACH Outlier diagnostic is necessary for constructing a mathematical model.49,50 This algorithm deals with identifying secluded data (or sets of data) that are possible to extravagant from the bulk of data.49,50 Experimental errors are the major reason for

(5)

It should be mentioned that, eq 5 can only use for linear regression problems. To extend its capability for nonlinear 6302


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Table 5. Optimized Parameters of Toth and Langmuir Models

( ΔRTH )P n ⎤1/ n ΔH P ( RT ) ) ⎦⎥

K 0exp q Toth model → = q0 ⎡ 1 + K 0exp ⎣⎢

(

P(MPa) and T (K)

( ΔH ) ( ( ))

K 0exp RT P q Langmuir model → = ΔH q0 1 + K 0exp RT P P(MPa) and T (K)

Toth model parameters adsorbents 9


q0

K0

ΔH/R

n

0.469 1.033 0.2709 0.0811 0.1772 0.1442 0.188 0.09262 0.06702

0.000992 0.001888 0.001999 0.003342 0.001275 0.001786 0.000469 0.001377 0.01926

1788 1733 1666 1714 1451 2105 2402 2103 1631

0.6543 0.3521 0.7161 0.3781 0.541 0.7195 0.6848 0.9756 0.6397

0.09318

0.000993

2271

0.6301

Langmuir model parameters 9


q0

K0

ΔH/R

0.257 0.5864 0.1943 0.02588 0.06523 0.1219 0.1495 0.09106 0.05538

0.001482 0.001812 0.002148 0.002776 0.01242 0.001108 0.000289 0.001441 0.01665

1791 1433 1684 1796 961.2 2196 2505 2088 1551

0.07009

0.002794

1888

Figure 5. Comparison between the results of the Toth model and the database values. (a) Scatter plot and (b) relative deviation plot.

“cut off” Leverage is generally 3, so the accepted points should be in the standard deviation range of ∓3.49,50 The accumulation of data points in the range of 0 < H < H* and −3 < R < 3, proving that model is statistically valid. The points which model could not predict at all, are called good high leverage. These points are in the range of H ≤ H* and −3 < R < 3. Bad high leverage or outliers are in fact the suspicious data. These points are in the range of R < −3 or R > 3.

outliers.37 These outliers may damage the model through the introduction of some uncertainties and lower the prediction accuracy.36,37 The leverage approach is a strong statistical algorithm for outlier detection.50,51 This method is based on a matrix, which its elements are the deviation between model predictions and experimental values.49,50it is called Hat matrix. Here is the definition of Hat indices49,50 H = X(X t X )−1X t

4. RESULT AND DISCUSSION 4.1. Designing the LSSVM Model. In order to implement the LSSVM algorithm, 670 experimental data points on the methane uptake by different adsorbents that cover wide range of pressure and temperature collected from literature.6,7,9,10,12,52,53 The type of adsorbent, temperature, and pressure range for these experimental data could be found in Table 1. In this communication, LSSVM modeling algorithm that presented by Packman et al.54 and Suykens and Vandewalle41 is used to make a precise relation between methane uptake and the adsorbent temperature, pressure, and type of adsorbent as follows:

(10)

Where X is a matrix with n rows (each row represents a single data) and k columns (each column as an indicator of a model parameter). In this definition, t denotes transpose matrix. The diagonal elements of H, presented as the Hat values. Williams plot with the use of standardized cross-validation residuals (R) and correlation of hat indices, graphically determines doubtful data.49,50 The H values are found through eq 10 and are used for sketching William plot. Through the definition of 3p/n the value of warning leverage is determined. Here, n determines the number of training points and p is representing the number of correlation input parameters.49,50 As an adequate

methane uptake by adsorbent = f (T , P , type of adsorbent) (11)

It should be mentioned that interpreting the adsorbent type for the mathematical algorithm is done by specifying each type as a 6303


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unique combination of 0 and 1 digits in the model. The mathematical interpretation of each adsorbent type could be found in Table 1. The most popular Gaussian radial basis function (RBF) kernel is used in LSSVM learning strategy.55−57 The mean square error (MSE) of the results of the LSSVM algorithm is defined as follows: n

MSE =

∑i = 1 (Ppred. − Pexp.) N

(12)

Where P is the total methane uptake, indices pred. and exp. represent the predicted and experimental methane uptake values, respectively, and N is number of sample data from the whole data set. For developing the model, whole database is divided into two main groups, namely the “training” and “test” sets. The data in training group is used for constructing the model; also, the data in test group is used for evaluating the validity and comprehensiveness of the presented model. Consequently, the ratios of 80% and 20% are considered for the training (536 data points) and the test (134 data points) groups. Combination of coupled simulated annealing (CSA) and standard simplex method are used to find the tuning parameters of developed LSSVM model.58 A typical flowchart of CSA−LSSVM strategy is provided in Figure 1. 4.2. LSSVM Accuracy. Model accuracy is examined with the use of graphical and statistical measures. Cross plot and error distribution plots are used as graphical error analysis methods which represent the accuracy of developed model. The reliability of the proposed model will checked by the cross plot while the error distribution shows error trend. Table 2 indicates statistical parameters of LSSVM model and corresponding experimental data of training and testing sets. These parameters include correlation coefficients (R2), mean

Figure 6. Comparison between the results of the Langmuir model and the database values. (a) Scatter plot and (b) relative deviation plot.

Figure 7. Temperature versus pressure plot for the methane uptake on Maxsorb II specimen, including the experimental data9 and the LSSVM model, Toth model, and Langmuir model predictions. 6304


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Figure 8. Temperature versus pressure plot for the methane uptake on the Maxsorb III, including the experimental data52,53 and the LSSVM model, Toth model, and Langmuir model predictions.

Robustness of the process model is illustrated by the accumulation of data about the first quadrant bisector (Figures 2a and 3a). The result shows great agreement between experimental values and LSSVM predictions. Relative deviation plots

square errors (MSE), and average absolute relative deviations (AARDs). Comparison of LSSVM predictions and corresponding experimental data for training and testing sets are presented in Figures 2 and 3, respectively. 6305


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Figure 9. Temperature versus pressure plot for the methane uptake on the ACF(A-20), including the experimental data52 and the LSSVM model, Toth model, and Langmuir model predictions.

Figure 10. Temperature versus pressure plot for the methane uptake on the HSZ-320, including the experimental data6 and the LSSVM model, Toth model, and Langmuir model predictions.

temperature considered in this work are shown in Tables 2 and 3. In addition, maximum and minimum absolute relative deviation for each adsorbent in different models is presented in Table 4. These tables indicate that, the

(Figures 2b and 3b) show a reasonable error for the whole the data. Details on the model predictions of methane uptake on different adsorbers in a broad range of pressure and 6306


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Figure 11. Temperature versus pressure plot for the methane uptake on the DAY, including the experimental data6 and the LSSVM model, Toth model, and Langmuir model predictions.

Figure 12. Temperature versus pressure plot for the methane uptake on the RP-15, including the experimental data7 and the LSSVM model, Toth model, and Langmuir model predictions.

Figure 4. The adsorption isotherm for all adsorbents is classified as Type I according to IUPAC classification, which shows typical isotherm for microporous materials with high surface area of over 1500 m2/g. In these adsorbents, attractive forces between the adsorbate and adsorbent are much stronger than

LSSVM model is strongly capable of reproducing experimental data. 4.3. Comparison of LSSVM with Conventional Models. Methane uptake isotherm data for each adsorbent is graphically presented on a pressure−concentration−temperature plane in 6307


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Figure 13. Temperature versus pressure plot for the methane uptake on the RP-20, including the experimental data7 and the LSSVM model, Toth model, and Langmuir model predictions.

Figure 14. Temperature versus pressure plot for the methane uptake on the activated carbon, including the experimental data10 and the LSSVM model, Toth model, and Langmuir model predictions.

between adsorbate molecules in the bulk state.59 It means that the adsorbent pore size is not much larger than the adsorbate molecular diameter. In case of chemisorption, this type can be fairly modeled by the Langmuir equation. However, this model shows deficiencies in case of physisorption. Therefore, it seems

that the modified semiempirical models such as Sips and Toth should be used for modeling of these data. The Toth isotherm model has a high thermodynamic consistency, incorporates the surface heterogeneity of the adsorbent, and also gives acceptable results in a wide pressure range.60 6308


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Figure 15. Temperature versus pressure plot for the methane uptake on KT activated carbon from the east of Kalimantan, including the experimental data12 and the LSSVM model, Toth model, and Langmuir model predictions.

Figure 16. Temperature versus pressure plot for the methane uptake on CarboTech commercial activated carbon, including the experimental data12 and the LSSVM model, Toth model, and Langmuir model predictions.

type of adsorbents. In another word, type of adsorbent is not a variable. Therefore, for using these models, the data from each adsorbent modeled separately. The optimized parameters of Toth and Langmuir models could be found in Table 5.

In this manner, for examining the quality of the proposed LSSVM model, three parameter Langmuir61 and Toth62 equations used for modeling the whole data set. The problem with traditional isotherm models is that you cannot use them for modeling of a data set combined from different 6309


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Table 6. Sips Optimized Parameters Models and Statistical Measures for Maxsorb III Sips model →

q bP1/ n = q0 1 + bP1/ n

P(MPa) and T (K) temperature (K)

q0 b n

160

179

199.8

219.7

3.321 0.2449 3.073

32.75 0.01461 3.253

0.9644 0.617 2.206

0.6198 1.018 1.613

temperature (K) 219.7 R2 LSSVM model Toth model Langmuir model Sips model

0.9990 0.9985 0.9874 0.9994

199.8

MSE × × × ×

6.72 5.79 1.21 3.90

R2

AARD% −06

MSE

1.1428 0.9963 11.9672 0.9974 19.2203 0.9890 0.8759 0.9979 temperature (K)

10 10−04 10−03 10−06

179 2

R LSSVM model Toth model Langmuir model Sips model

0.9889 0.9877 0.9658 0.9948

× × × ×

10 10−04 10−03 10−05

2.511 6.137 19.162 1.859

160

MSE 1.28 6.10 2.01 6.00

× × × ×

3.20 1.86 1.86 1.75

AARD% −05

10−04 10−04 10−03 10−05

2

AARD%

R

3.8995 7.6690 15.4334 2.6656

0.998 0.980 0.878 0.997

MSE 2.07 3.87 2.84 3.31

× × × ×

AARD%

10−05 10−04 10−03 10−05

0.8037 3.1847 8.3763 1.0469

Table 7. Details of Toth and Langmuir Models Statistical Parameters on Each Adsorbent Toth model adsorbents 9


MSE

AARD% −06

1.091 × 10 0.0001434 7.039 × 10−07 7.279 × 10−07 3.604 × 10−07 1.599 × 10−06 3.629 × 10−07 8.605 × 10−07 5.379 × 10−07 3.355 × 10−07

1.7420 12.8245 2.0882 16.7415 16.8065 4.3758 3.1542 7.1229 2.2247 1.7592

Langmuir model R

2

0.9990 0.9930 0.9994 0.9794 0.9959 0.9987 0.9998 0.9980 0.9971 0.9989

MSE 1.422 5.532 1.644 4.598 4.150 3.687 3.268 8.585 8.121 1.339

× × × × × × × × × ×

−06

10 10−04 10−06 10−07 10−07 10−06 10−06 10−07 10−07 10−06

AARD%

R2

1.8990 12.9464 3.0907 5.9182 10.8542 5.3947 6.8540 6.9813 3.5784 5.2128

0.9988 0.9742 0.9988 0.9860 0.9944 0.9975 0.9986 0.9979 0.9970 0.9965

The optimized parameters of the Sips model and detailed statistical measures of Figure 8b are provided in Table 6. Although both Sips and Toth models maybe do not have very different results from LSSVM,in this case, they need so many parameter adjustments and are time-consuming. In addition, due to the independence of this mathematical model from the physical meaning of the problem, it has no deficiencies in either case of physisorption or chemisorption and high pressure or low pressure. It is obvious from P−C−T figures that better fit of experimental data will result from the Toth model than from the Langmuir isotherm. This is due to accounting the heterogeneity parameter (n). Small value of the heterogeneity parameter (n) is a sign of high microporous and heterogeneous surface. Albeit the Toth model has good predictions on experimental data at low pressures, it did not give acceptable results at high pressures. This is because at low pressure it obeys Henry’s law, but at high pressure, the adsorbed amount approaches saturating loading. The correlation coefficients (R2) in both conventional models are relatively good, but with an eye on the LSSVM

It is not a problem for LSSVM. As mentioned before, one of the variables, which has been considered for LSSVM model, was the adsorbent type. In fact, the model is open to accept as many variables as the user defines. This is an obvious advantage and means you can investigate the effect of parameters, which may not be considered in conventional methods. The experimental data for each adsorbent has been modeled through a nonlinear regression approach with Toth and Langmuir isotherms. Optimized parameters of the models for each adsorbent are shown in Table 5. The cross plots in Figures 5a and 6a illustrate the state of agreement between prediction of traditional isotherms and experimental values. In addition, relative deviation between the predicted values of isotherm models and experimental data are shown in Figures 5b and 6b. Pressure−concentration−temperature plots are drawn for a better visual comparison between the LSSVM and conventional isotherm models (Figures 7−16). As is mentioned before in case of physisorption the semiempirical models such as Sips and Toth have better results in comparison with Langmuir. This can be seen from Figure 8b where the low temperature favors the physisorption. 6310


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Figure 17. Detection of probable outliers and applicability domain of the presented model for each adsorbent (a) Maxsorb II specimen, (b) Maxsorb III, (c) ACF (A-20), (d) HSZ-320, (e) DAY, (f) RP-15, (g) RP-20, (h) activated carbon, (i) KT activated carbon from the east of Kalimantan and (j) CarboTech commercial activated carbon. 6311


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outstanding correlation coefficient (R2), the supremacy of this model is obvious. To make an accurate statistical comparison, detailed statistical measures have been shown in Tables 2, 3, and 7. These results indicate that predictions of LSSVM model have a better agreement with experimental data in comparison to those conventional isotherm models. On the basis of graphical and statistic error analysis that performed in this study, it can be concluded that LSSVM algorithm can be a powerful tool for prediction of adsorption isotherms. 4.4. Outlier Detection. Table 2 shows that the deviation of model predictions from experimental values is appropriate for using the leverage algorithm. The H values have been evaluated by eq 10, and with use of 3(p + 1)/n, warning leverages (H*) have been found. Figure 17 shows Williams plots for each adsorbents. As the figures show, whole methane uptake data set is in the LSSVM applicability domain. The figures indicate that no outliers are in the data. The quality of the data used for modeling is different. The data with lower H values and the lower absolute values R may be identified as more authentic experimental data. 4.5. Sensitivity Analysis. In order to investigate the sensitivity of the LSSVM predictions to the independent variables or respective significance of each input/output relationship, a sensitivity analysis is conducted through the data. The outcome of this analysis provides a beneficial understanding of each parameter impact. In this study, the sensitivity analysis grounded on three different techniques: Pearson correlation, Spearman rank correlation, Kendall’s tau correlation. The difference in these techniques lies in their nonlinear or linear relation of the output assumptions and the input parameter.63 Figure 18 illustrates correlation coefficients

5. CONCLUSION In this study, the application of a novel method of least square support vector machine for prediction of methane adsorption isotherm behavior on different adsorbents in a broad range of pressure and temperature has been investigated. Using the advantage of LSSVM, which has no limitation in introducing any number of new variables (and assuming adsorbent type as an independent variable in the model), a unique model has been built on a data set combined from different type of adsorbents. The graphical measures and statistical parameters of the obtained model showed that the proposed model could estimate adsorption isotherm behavior with an excellent accuracy. In addition, it had a much better performance comparing with two conventional isotherm models. The leverage algorithm was used for identifying outliers and determining model applicability range. It showed statistical validity of LSSVM for predicting adsorption isotherm behavior. In addition, leverage approach determined that all of the data are in the model applicability domain.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +61 2 6626 9412. Notes

The authors declare no competing financial interest.

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Figure 18. Relative variable impacts on methane uptake.

between output variable, methane uptake, and input variables temperature and pressure. The correlation coefficients may receive a value from +1 to −1. A value of +1 demonstrate an increasing relation between variables, a value of −1 shows decreasing relation between variables, and if the value of correlation coefficient is 0, the two variables have no relationship. An increase in the absolute value of the correlation coefficient between any input and output variable shows that the influence of that input in determining the output’s value increases. Figure 18 indicates negative impact of temperature and positive impact of the pressure on methane uptake prediction. This is in complete agreement with known effect of these variables in methane uptake. With a quick look at Figures 7−16, it is obvious that the model mimics this behavior.

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NOMENCLATURE CNG = compressed natural gas LNG = liquefied natural gas VED = volumetric energy density ANG = adsorbed natural gas ANN = artificial neural networks LSSVM = least-squares support vector machine AARD = average absolute relative deviation,% R2 = correlation coefficient SVM = support vector machine T = temperature P = pressure ek = regression error γ = relative weight of the summation of the regression errors σ2 = squared bandwidth W = regression weight T = transpose matrix B = linear regression intercept of the model φ = feature map xk = input vector yk = output vector βi = Lagrange multipliers MSE = mean square error K(x,xk) = kernel function RBF = radial basis function H = hat matrix REFERENCES

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Prediction of Methane Uptake on Different Adsorbents in Adsorbed

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