Determination of Hydrate Stability Zones of Sour Gases Using a Two Parameter Model

In: Advances in Chemistry Research. Volume 24 ISBN: 978-1-63463-846-3

Editor: James C. Taylor © 2015 Nova Science Publishers, Inc.

Chapter 1

DETERMINATION OF HYDRATE STABILITY ZONES

OF SOUR GASES USING A TWO PARAMETER MODEL

Ali Eslamimanesh1, Farhad Gharagheizi

2

and Amir H. Mohammadi2,3

1Department of Chemical and Biomolecular Engineering,

Clarkson University, Potsdam, NY, US 2Thermodynamics Research Unit, School of Chemical Engineering,

University of KwaZulu-Natal, ‎Howard College Campus,

King George V Avenue, Durban, South Africa 3Institut de Recherche en Génie Chimique et Pétrolier (IRGCP),

Paris, France

ABSTRACT

In the present study, clathrate hydrate dissociation conditions of the methane +

carbon dioxide + hydrogen sulfide and methane + carbon dioxide/hydrogen sulfide

systems are correlated/predicted by applying the Least Squares Support Vector Machine

(LSSVM) algorithm. The Hybrid Genetic Algorithm (H-GA) is utilized to obtain the

optimal model parameters. The accuracy and reliability of the proposed model are proved

showing the average absolute relative deviations (AARD) of around 3% and squared

correlation coefficient of about 0.990. A computer program is finally developed for phase

equilibrium prediction of the investigated systems.

Keywords: Gas hydrate, support vector machine, model, methane, carbon dioxide, hydrogen

sulfide

Corresponding author: E-mail Address: [email protected] & [email protected]

Complimentary Contributor Copy

Ali Eslamimanesh, Farhad Gharagheizi and Amir H. Mohammadi 2

1. INTRODUCTION‎

Trapping of molecule(s) of a compound in a structure formed by the molecules of another

compound normally generates clathrate structures. [1, 2] The unique characteristics of the

water molecules result in formation of hydrogen-bonded three-dimensional networks able to

encage particular kinds of molecules. [3] The final compound is called the "hydrate", which

has been the subject of many studies from the 19th century. [1-3] At relatively high pressures

and low temperatures, water molecules form various crystalline structures generally

depending on the size and shape of the guest molecule(s). [1] Structures I (sI), II (sII), and H

(sH) are known to form as the three common structures of clathrate hydrates (or gas

hydrates).

The first industrial importance of the gas hydrate formation has been attributed to the

blockage of gas/oil transportation pipelines since this structure can form from association of

water with natural gas constituents in petroleum industry (or found abundantly in nature). [1-

4] More rigorously, gas hydrate formation may occur during the steps of natural gas

production and processing when traceable amounts of associated water exist (even in the form

of very low water content). On the other hand, there are indeed some positive applications of

gas hydrates, which have been generally proposed in the recent years and attracted much

attention. [1, 3] For instance, clathrate structures may be used as media for the storage and

transportation of natural gas and hydrogen, and CO2 capture and sequestration

process. [1, 3, 4]

In any case, reliable experimental phase equilibrium data of clathrate hydrates,

particularly those formed from natural gas components such as methane, carbon dioxide, and

hydrogen sulfide as well as accurate thermodynamic/numerical models are required to

prevent gas hydrate formation during processing and transportation of natural gas or use to

design separation/storage processes utilizing gas hydrate formation phenomenon.

Furthermore, since there is high probability of hydrogen sulfide clathrate hydrate formation in

the presence of liquid water, [5] accurate information on phase equilibria of gas hydrates

containing H2S is of importance during production, transportation, and processing of sour

gases.

A literature survey shows that comprehensive studies (both experimental and theoretical

ones) have been undertaken to date on phase equilibria of clathrate hydrates of hydrocarbons.

[1] However, phase equilibrium data regarding the mixed clathrate hydrates of CH4 + CO2 +

H2S are scarce and they are generally limited to the data reported by Robinson and Hutton [6]

and Sun et al. [7] Few experimental laboratories have the capability of experimental

measurements on hydrogen sulfide containing fluids mainly due to toxic and corrosive nature

of this compound.

Another point to mention is that the conventional thermodynamic gas hydrate models on

the basis of the van der Waals-Platteeuw [8] (vdW-P) theory for modeling the hydrate phase

and different equations of state for modeling the fluid phases may not normally lead to

acceptable phase equilibrium predictions for this system (it can be checked by commercial

softwares like HYSYS [9]). Furthermore, empirical correlations have not been demonstrated

to give reliable results for this purpose (mainly due to especial shape of the corresponding

phase boundary). In 1987, Baillie–Wichert [10] proposed some charts to predict the phase

behavior of the mixed clathrate hydrates of CH4 + CO2 + H2S. However, this method uses


Determination of Hydrate Stability Zones of Sour Gases … 3

graphical steps, which are not generally straight-forward and have some limitations in range

of applicability. A brief discussion about the aforementioned methods limitations have been

given elsewhere. [5] Additionally, the adaptive neuro fuzzy inference system (ANFIS)

mathematical method has been applied by ZareNezhad and Aminian [5] to calculate/estimate

the hydrate dissociation conditions of the aforementioned system. However, this

mathematical algorithm is generally complex with many parameters.

Therefore, it merits to check the capability of a new model for phase equilibrium

prediction of the system of interest. Since the Least Squares Support Vector Machine

(LSSVM) [11-13] methodology has been already employed in phase equilibrium predictions

of the systems containing gas hydrates, [14] herein, this method is applied to develop a

numerical network-based model for correlating/estimating the hydrate dissociation conditions

of the CH4 + CO2 + H2S and CH4 + CO2/H2S systems.

2. EXPERIMENTAL DATA

In the present study, we have treated the available experimental data in open literature for

the gas hydrate dissociation conditions of the CH4 + CO2 + H2S system. [6, 7] In addition, the

binary mixed clathrate hydrate dissociation data for the CH4 + H2S [15] and CH4 + CO2 [16-

21] and some simple hydrate dissociation data present in the data sets [16, 17] for the CH4

system have been used to extend the model for the cases where there are no CO2 and/or H2S

in the feed gas.

3. MODEL DEVELOPMENT

3.1. Features of the Support Vector Machine

The main purpose in this step is to establish nonlinear relationships between the

experimental hydrate dissociation temperature and molar composition of the gas mixture in

the feed [6, 7, 15-21] as inputs of the model and the desired output (hydrate dissociation

pressure). To achieve this goal, a network-based mathematical approach has been employed.

The Artificial Neural Network (ANN)-based models normally lead to high accuracy in

different engineering and scientific problems. [22] However, random initialization of the

networks, variation of the stopping criteria during optimization of the model parameters, and

large numbers of the parameters may discourage their use for external predictions (or

extrapolations). [14, 23-25]

The support vector machine is a capable mathematical tool developed from the machine-

learning community [11-14]. This approach analyzes a group of data and recognizes patterns,

applied for regression objectives. Furthermore, it can be stated that the SVM is a non-

probabilistic binary linear classifier [11-14]. The significant advantages of the SVM-based

methods over the conventional methods based on the ANNs have been already discussed. [14]

It should be pointed out that both linear and nonlinear regressions can be pursued for solving

the corresponding problems using the SVM algorithm. [11-14]



A considerable modification to the original SVM has been proposed by Suykens and

Vandewalle [11, 12] with the aim of facilitating the solution of set of nonlinear equations

(quadratic programming) in the original SVM algorithm. The new LLSVM strategy, brings

about an easier-to-implement calculation methods and faster solution-searches compared to

the traditional SVM method. [11-14]

3.2. Equations

The cost function of the applied model

has been calculated by the following

relation: [11-14]

N

kk

T

LSSVMewwQ

1

2

2

1 (1)

subject to the following constraint: [11-14]

kk

T

kebxwy )( k=1, 2,…, N (2)

where x is the input vector containing the input data (temperature and feed compositions of

the gas), y stands for the output vector (dependent parameter), b denotes the intercept of the

linear regression, w is the regression weight (slope of the linear regression), ek is the

regression error for N training objects, γ indicates the relative weight of the summation of the

regression errors compared to the regression weight (first right hand side of Eq. 1),

superscript T denotes the transpose matrix, and φ is the feature map, mapping the feasible

region (input space) to a high dimensional feature space.

The regression weight (w) is evaluated as: [11-14]

N

kkk

xw1

(3)

where,

kke 2 (4)

Therefore, Eq. (2) can be written as follows: [11-14]

N

k

T

kkbxxy

1

(5)

where, [11-14]

1)2(

)(

xx

byT

k

k

k (6)



Eq. 7 can be used as an alternative nonlinear regression applying the Kernel function as

follows: [11-14]

N

kkk

bxxKxf1

),()( (7)

where K(x,xk) is the Kernel function dependent on the inner product of the two vectors x and

xk in the feasible region defined by the inner product of the vectors Ф(x) and Ф(xk)

as follows: [11-14]

)(Φ.)(Φ),(k

T

kxxxxK (8)

where the radial basis function (RBF) Kernel has been applied as follows: [11-14]

)/exp(),( 22

xxxxKkk (9)

where σ is a decision variable, which is obtained by an external optimization method. The

mean square error (MSE) of the results of the final model can be determined by the following

expression:

nsMSE

i.

n

iirep./pred

2

exp1

)P(P

(10)

where P is hydrate dissociation pressure, subscripts rep./pred. and exp. indicate the

represented/predicted, and experimental hydrate dissociation pressure values, respectively,

and ns stands for the number of data from the initial population (generation). It is worth

knowing that the LSSVM algorithm developed by Pelckmans et al. [13] and Suykens and

Vandewalle [11] has been employed in the present work.

3.3. Calculation Steps

A same computational procedure described in our previous work [14] has been followed

here to obtain the optimal values of the parameters of the LSSVM [11-13] algorithm (i.e., γ

and σ2). The experimental datasets [6, 7, 15-20] have been divided into three sub-datasets

including the ―Training‖ set, the ―Validation (Optimization)‖ set, and the ―Test‖ set. In this

study, the‎‎ ―Training‖ set is used to generate the model structure, the ―Validation

(Optimization)‖‎ set is applied for optimization of the model, and the ―Test (prediction)‖ set is

used to investigate the prediction capability and validity of the proposed model. The division

of database into three sub-datasets has been conducted randomly. Thus, about 80%, 10%, and

10% of the main dataset are randomly selected for the ―Training‖ set (197 data points), the

―Optimization‖ set (24 data points), and the ―Test‖ set (24 data points), respectively. Many

distribution allocations have been taken into account to avoid the local accumulations of the



data in the feasible region of the problem. As a result, the acceptable distribution is the one

with homogeneous accumulations of the data on the domain of the three

sub-datasets. [14] The computational steps have been performed following the described

procedure. The two main parameters of this algorithm are σ2 and γ, which are supposed to be

optimized using an appropriate optimization strategy. Because of the high nonlinearity of the

SVM method, the robust Hybrid Genetic Algorithm (H-GA) method [26, 27] has been

implemented for this purpose. This optimization method leads to faster calculations compared

with the conventional linear optimization algorithms. The optimization toolbox of

MATLAB® [28] software has been employed for parallel computations. The number of

populations of the optimization problem has been herein set to 1000. To ensure that the value

of the final solution is very close to the probable global optimum of the problem, the

optimization procedure has been repeated several times. [14]

4. RESULTS AND DISCUSSION

The values of the probable global optima of the problems including σ2 and γ have been

calculated as 1.205 and 1473.852, respectively. The numbers of the reported digits of these

parameters have been obtained by performing sensitivity analysis of the overall errors of the

optimization procedure to the corresponding values.

Figures 1 and 2 show the represented/predicted results applying the developed LSSVM

model vs. experimental values [6, 7, 15-21] of hydrate dissociation conditions for the CH4 +

CO2 + H2S system. The experimental [6, 7, 15-21] and determined hydrate dissociation

conditions, the absolute relative deviations of results from experimental values [6, 7, 15-21]

and the status of the data points (whether they have been used as training, validation, or test

sets are reported in Table 1). Moreover, Table 2 reports the statistical parameters of the

proposed model. Anyone can easily apply the software to reproduce all our results and may

predict phase equilibria of the investigated systems at temperature conditions of interest (in

gas hydrate formation region).

It can be observed from the results that some of the represented/predicted values have

high deviations from experimental values. [6, 7, 15-21] A number of possible explanations for

these deviations can be expressed as follows:

1. We may not be able to define a domain of temperature and pressure conditions in

which the developed model leads to poorer results;

2. The high deviations may be due to some doubtful experimental phase equilibrium

data resulting from possible errors in the measurements. However, high deviations

may not be attributed to unreliable data for the system in the absence of hydrogen

sulfide (the CH4 + CO2 system) because we have already performed a statistical

evaluation method on such data and removed the probable doubtful data for

developing the proposed model in this work.

Significant points may not be omitted from our discussions.


Table 1. The detailed results of the developed model

Temperature / K Feed gas composition / Mole% Pressure / MPa ARD

a% Status Ref.

b

CH4 CO2 H2S exp.c rep./pred.

d

281.1 78.5 13.9 7.6 1.675 1.579 5.7 training [6]

282.5 80.3 13.0 6.7 2.275 2.018 11 training

286.6 81.0 13.0 6.0 3.868 3.917 1.3 validation

287.3 82.0 12.6 5.4 4.558 4.855 6.5 training

289.4 82.0 12.6 5.4 5.888 6.451 9.6 training

290.8 82.0 12.6 5.4 6.881 7.593 10 training

292.2 82.0 12.5 5.5 8.653 8.761 1.2 training

292.9 82.0 12.6 5.4 9.632 9.632 0 training

293.6 82.0 12.6 5.4 10.790 10.488 2.8 validation

294.7 82.5 12.1 5.4 12.341 11.933 3.3 training

295.6 82.0 12.6 5.4 14.079 13.665 2.9 training

296.4 82.0 12.6 5.4 15.707 15.254 2.9 validation

284.2 81.0 11.8 7.2 1.903 2.445 28 training

285.4 81.0 11.8 7.2 2.765 2.910 5.3 training

289.1 80.0 12.0 8.0 4.254 4.555 7.1 training

290.4 80.0 12.0 8.0 4.978 5.332 7.1 training

292.1 80.0 12.0 8.0 5.943 6.403 7.7 validation

293.1 81.6 11.1 7.3 6.984 7.776 11 training

294 83.9 9.4 6.7 7.529 8.933 19 training

279.2 68.6 24.9 6.5 1.475 1.466 0.6 training

282.1 69.9 24.1 6.0 2.034 2.179 7.1 training


Table 1. (Continued)


a% Status Ref.

b


d

284 70.5 23.5 6.0 2.771 2.662 3.9 training

286.4 71.5 22.8 5.7 3.744 3.626 3.2 training

288.4 72.5 22.0 5.5 4.930 4.899 0.6 training

290 72.5 22.0 5.5 6.185 6.331 2.4 training

290.9 72.5 22.0 5.5 7.550 7.359 2.5 training

293 72.0 22.3 5.7 11.225 10.282 8.4 test

293.7 72.3 22.2 5.5 12.011 12.030 0.2 training

287.4 69.9 12.7 17.4 2.020 2.003 0.9 training

289.9 70.0 12.3 16.7 2.648 2.640 0.3 training

291.5 72.0 12.0 16.0 3.330 3.333 0.1 training

293.3 72.0 12.0 16.0 4.392 4.378 0.3 training

294.7 72.0 12.0 16.0 5.123 5.479 6.9 training

295.4 71.1 11.9 17.0 6.495 6.149 5.3 training

296.5 70.8 12.1 17.1 7.384 7.278 1.4 training

297.6 72.5 11.9 15.6 8.405 8.572 2 training

297.6 68.8 13.6 17.6 8.005 8.411 5.1 validation

274.2 87.65 7.40 4.95 1.044 1.032 1.2 training [7]

277.2 87.65 7.40 4.95 1.580 1.576 0.2 training

280.2 87.65 7.40 4.95 2.352 2.473 5.1 validation

282.2 87.65 7.40 4.95 3.126 3.079 1.5 training

284.2 87.65 7.40 4.95 3.964 3.942 0.6 training



a% Status Ref.

b


d

286.2 87.65 7.40 4.95 5.121 5.214 1.8 validation

288.2 87.65 7.40 4.95 6.358 6.696 5.3 test

289.2 87.65 7.40 4.95 7.212 7.386 2.4 training

290.2 87.65 7.40 4.95 8.220 7.994 2.7 training

276.2 82.50 10.77 6.73 1.114 1.076 3.5 training

278.2 82.50 10.77 6.73 1.385 1.472 6.3 training

280.2 82.50 10.77 6.73 1.815 1.855 2.2 training

282.2 82.50 10.77 6.73 2.265 2.205 2.7 training

284.2 82.50 10.77 6.73 3.110 2.761 11 test

286.2 82.50 10.77 6.73 4.065 3.725 8.4 training

287.2 82.50 10.77 6.73 4.570 4.341 5 training

288.2 82.50 10.77 6.73 4.890 4.999 2.2 training

289.2 82.50 10.77 6.73 6.110 5.661 7.4 training

290.2 82.50 10.77 6.73 6.862 6.303 8.1 training

290.9 82.50 10.77 6.73 7.650 6.744 12 training

291.2 82.50 10.77 6.73 8.024 6.934 14 training

278.2 82.91 7.16 9.93 1.192 1.243 4.3 training

282.2 82.91 7.16 9.93 1.932 1.933 0 training

284.2 82.91 7.16 9.93 2.460 2.335 5.1 training

286.2 82.91 7.16 9.93 3.303 3.144 4.8 validation

288.2 82.91 7.16 9.93 4.212 4.328 2.7 training

289.7 82.91 7.16 9.93 4.930 5.254 6.6 training

291.2 82.91 7.16 9.93 5.868 6.061 3.3 training




a% Status Ref.

b


d

292.2 82.91 7.16 9.93 6.630 6.529 1.5 training

293.2 82.91 7.16 9.93 7.916 6.986 12 training

277.2 77.71 7.31 14.98 0.646 0.636 1.5 training

280.2 77.71 7.31 14.98 1.020 1.011 0.9 training

283.2 77.71 7.31 14.98 1.428 1.390 2.7 training

286.2 77.71 7.31 14.98 2.080 1.986 4.5 training

289.2 77.71 7.31 14.98 3.164 3.189 0.8 training

291.2 77.71 7.31 14.98 4.070 4.279 5.1 test

293.2 77.71 7.31 14.98 5.270 5.588 6 validation

294.7 77.71 7.31 14.98 6.698 6.781 1.2 training

295.7 77.71 7.31 14.98 7.910 7.701 2.6 training

282.2 75.48 6.81 17.71 0.950 0.987 3.9 training

284.2 75.48 6.81 17.71 1.244 1.378 11 validation

286.2 75.48 6.81 17.71 1.670 1.805 8.1 training

288.2 75.48 6.81 17.71 2.368 2.315 2.2 training

290.2 75.48 6.81 17.71 3.080 2.991 2.9 training

292.2 75.48 6.81 17.71 4.008 3.972 0.9 validation

294.2 75.48 6.81 17.71 5.314 5.429 2.2 training

295.2 75.48 6.81 17.71 6.310 6.363 0.8 training

295.8 75.48 6.81 17.71 6.880 6.985 1.5 training

296.6 75.48 6.81 17.71 7.825 7.872 0.6 training



a% Status Ref.

b


d

297.2 75.48 6.81 17.71 8.680 8.567 1.3 training

281.2 66.38 7.00 26.62 0.582 0.595 2.2 training

284.2 66.38 7.00 26.62 0.786 0.745 5.3 training

287.2 66.38 7.00 26.62 1.160 1.207 4.1 training

290.2 66.38 7.00 26.62 1.788 1.730 3.3 test

293.2 66.38 7.00 26.62 2.688 2.674 0.5 training

295.2 66.38 7.00 26.62 3.910 3.867 1.1 training

296.7 66.38 7.00 26.62 5.030 5.104 1.5 training

298.2 66.38 7.00 26.62 6.562 6.555 0.1 training

299.7 66.38 7.00 26.62 8.080 8.064 0.2 training

288.7 91.77 0.00 8.23 4.830 5.150 6.6 training [15]

284.3 90.49 0.00 9.51 2.590 2.584 0.2 training

282.3 93.70 0.00 6.30 3.030 2.886 4.8 validation

287.1 93.50 0.00 6.50 4.790 4.831 0.9 training

290.1 93.00 0.00 7.00 6.790 6.026 11 test

279.3 94.27 0.00 5.73 2.210 2.223 0.6 training

290.1 93.40 0.00 6.60 6.380 6.133 3.9 training

278.7 97.00 0.00 3.00 2.830 2.677 5.4 training

282.9 96.90 0.00 3.10 4.270 4.035 5.5 training

287.6 97.08 0.00 2.92 6.650 6.740 1.3 training

276.5 96.22 0.00 3.78 2.030 1.558 23 validation

278.4 99.00 0.00 1.00 3.240 3.641 12 training

282.3 98.96 0.00 1.04 4.620 5.003 8.3 training




a% Status Ref.

b


d

284.8 98.89 0.00 1.11 6.690 6.121 8.5 validation

287.6 78.00 0.00 22.00 2.100 2.079 1 training

295.4 80.20 0.00 19.80 5.070 5.064 0.1 training

279.8 78.60 0.00 21.40 1.030 1.028 0.2 training

281.5 90.50 0.00 9.50 2.070 2.110 1.9 training

287.3 89.00 0.00 11.00 3.590 3.463 3.5 training

292.1 88.50 0.00 11.50 6.000 6.149 2.5 training

273.56 38.31 61.69 0 1.500 1.495 0.3 test [16]

273.56 73.66 26.34 0 2.000 2.012 0.6 validation

275.86 43.52 56.48 0 2.000 2.010 0.5 training

273.16 100.00 0.00 0 2.600 2.604 0.2 validation

275.36 81.46 18.54 0 2.600 2.634 1.3 training

277.96 38.05 61.95 0 2.600 2.572 1.1 training

276.16 100.00 0.00 0 3.500 3.485 0.4 training

278.06 79.91 20.09 0 3.500 3.443 1.6 training

280.16 39.13 60.87 0 3.500 3.390 3.1 training

279.6 100.00 0.00 0 5.000 4.928 1.4 training

281.46 80.29 19.71 0 5.000 4.968 0.6 training

283.26 40.11 59.89 0 5.000 4.998 0 training

272.66 59.33 40.67 0 1.500 1.506 0.4 training [17]

273.56 38.31 61.69 0 1.500 1.495 0.3 training



a% Status Ref.

b


d

274.36 9.59 90.41 0 1.500 1.381 8 validation

273.56 73.66 26.34 0 2.000 2.012 0.6 test

274.36 66.25 33.75 0 2.000 2.032 1.6 training

275.86 43.52 56.48 0 2.000 2.010 0.5 training

276.56 20.46 79.54 0 2.000 2.046 2.3 test

273.16 100.00 0.00 0 2.600 2.604 0.2 training

275.36 81.46 18.54 0 2.600 2.634 1.3 training

276.76 60.28 39.72 0 2.600 2.598 0.1 training

277.96 38.05 61.95 0 2.600 2.572 1.1 training

278.26 21.57 78.43 0 2.600 2.514 3.3 training

276.16 100.00 0.00 0 3.500 3.485 0.4 training

278.06 79.91 20.09 0 3.500 3.443 1.6 training

279.26 57.35 42.65 0 3.500 3.313 5.3 training

280.16 39.13 60.87 0 3.500 3.390 3.1 training

280.76 23.83 76.17 0 3.500 3.419 2.3 test

279.6 100.00 0.00 0 5.000 4.928 1.4 training

281.46 80.29 19.71 0 5.000 4.968 0.6 test

282.56 59.11 40.89 0 5.000 4.969 0.6 validation

283.26 40.11 59.89 0 5.000 4.998 0 training

283.56 19.48 80.52 0 5.000 4.945 1.1 training

273.5 53.47 46.53 0 1.780 1.614 9.3 test [18]

274.2 53.40 46.60 0 1.830 1.771 3.2 test




a% Status Ref.

b


d

275.2 53.03 46.97 0 2.050 2.009 2 test

275.6 52.99 47.01 0 2.120 2.111 0.4 training

275.7 52.71 47.29 0 2.200 2.131 3.1 training

276.8 51.99 48.01 0 2.400 2.413 0.5 training

278.4 51.39 48.61 0 2.825 2.885 2.1 test

278.7 50.40 49.60 0 2.851 2.969 4.1 training

279.5 49.98 50.02 0 3.301 3.255 1.4 validation

280.1 49.60 50.40 0 3.370 3.493 3.6 training

281.8 49.58 50.42 0 4.410 4.313 2.2 training

283 49.42 50.58 0 5.001 5.022 0.4 training

283.1 48.38 51.62 0 5.070 5.065 0.1 training

280.3 0.0 100.0 0 3.040 3.027 0.4 training [19]

280.3 31.7 68.3 0 3.240 3.324 2.6 training

280.3 41.5 58.5 0 3.380 3.482 3 training

280.3 51.2 48.8 0 3.600 3.602 0.1 training

280.3 55.0 45.0 0 3.640 3.673 0.9 training

280.3 55.2 44.8 0 3.670 3.677 0.2 training

280.3 57.1 42.9 0 3.710 3.720 0.3 training

280.3 61.6 38.4 0 3.770 3.836 1.8 training

280.3 64.3 35.7 0 3.860 3.909 1.3 training



a% Status Ref.

b


d

280.3 69.8 30.2 0 3.980 4.052 1.8 training

280.3 69.0 31.0 0 4.000 4.031 0.8 training

280.3 68.9 31.1 0 4.010 4.029 0.5 training

280.3 71.2 28.8 0 4.060 4.088 0.7 training

280.3 70.7 29.3 0 4.070 4.075 0.1 training

280.3 73.2 26.8 0 4.150 4.140 0.2 training

280.3 75.5 24.5 0 4.200 4.206 0.1 test

280.3 75.9 24.1 0 4.220 4.218 0 training

280.3 78.5 21.5 0 4.310 4.305 0.1 training

280.3 78.3 21.7 0 4.320 4.298 0.5 validation

280.3 79.7 20.3 0 4.340 4.351 0.2 training

280.3 79.7 20.3 0 4.370 4.351 0.4 training

280.3 81.7 18.3 0 4.370 4.434 1.5 training

280.3 82.1 17.9 0 4.440 4.452 0.3 training

280.3 83.1 16.9 0 4.500 4.498 0.1 validation

280.3 85.6 14.4 0 4.570 4.621 1.1 training

280.3 85.9 14.1 0 4.580 4.636 1.2 training

280.3 85.7 14.3 0 4.630 4.626 0.1 training

280.3 89.6 10.4 0 4.750 4.829 1.7 training

280.3 91.0 9.0 0 4.850 4.899 1 training

280.3 93.5 6.5 0 4.990 5.012 0.4 training

280.3 100.0 0.0 0 5.460 5.177 5.2 training




a% Status Ref.

b


d

273.7 90 10 0 2.520 2.510 0.4 training [20]

275.8 91 9 0 3.100 2.975 4 test

277.8 92 8 0 3.830 3.777 1.4 training

280.2 92 8 0 4.910 4.896 0.3 training

283.2 92 8 0 6.800 6.723 1.1 training

285.1 92 8 0 8.400 8.355 0.5 training

287.2 91 9 0 10.760 10.637 1.1 training

274.6 86 14 0 2.590 2.567 0.9 test

276.9 87 13 0 3.240 3.230 0.3 training

279.1 87 13 0 4.180 4.118 1.5 training

281.6 87 13 0 5.380 5.423 0.8 training

284 87 13 0 7.170 7.210 0.6 test

286.1 88 12 0 9.240 9.395 1.7 training

287.4 87 13 0 10.950 10.914 0.3 training

273.8 75 25 0 2.120 2.106 0.6 training

279.4 78 22 0 3.960 3.901 1.5 training

283.4 78 22 0 6.230 6.105 2 training

285.2 79 21 0 7.750 7.748 0 test

287.6 75 25 0 10.440 10.413 0.3 training

273.7 56 44 0 1.810 1.698 6.2 validation



a% Status Ref.

b


d

276.9 58 42 0 2.630 2.583 1.8 training

280.7 60 40 0 4.030 3.972 1.4 training

283.1 61 39 0 5.430 5.380 0.9 training

285.1 61 39 0 6.940 7.022 1.2 training

275.6 50 50 0 1.990 2.047 2.8 training

278.5 53 47 0 2.980 2.950 1 validation

280.9 60 40 0 4.140 4.067 1.8 training

281.8 59 41 0 4.470 4.510 0.9 training

285.1 56 44 0 6.840 6.831 0.1 training

274.6 27 73 0 1.660 1.680 1.2 validation

276.4 30 70 0 2.080 2.078 0.1 test

278.2 32 68 0 2.580 2.587 0.3 training

280.2 32 68 0 3.280 3.289 0.3 training

282 33 67 0 4.120 4.138 0.4 training

273.7 21 79 0 1.450 1.482 2.2 test

275.9 22 78 0 1.880 1.907 1.4 training

277.8 24 76 0 2.370 2.401 1.3 training

279.6 25 75 0 2.970 2.971 0 training

281.6 26 74 0 3.790 3.826 0.9 training

282.7 15 85 0 4.370 4.406 0.8 training

284.2 73.6 26.4 0 5.290 6.287 19 training [21]

287.2 73.6 26.4 0 9.830 9.321 5.2 training




a% Status Ref.

b


d

289.2 73.6 26.4 0 11.620 12.218 5.1 training

279.1 72.8 27.2 0 3.600 3.638 1.1 training

284.8 51.0 49.0 0 5.820 6.158 5.8 training

289.9 51.0 49.0 0 12.410 12.375 0.3 training

284.9 50.0 50.0 0 5.880 6.195 5.4 training

279.1 49.6 50.4 0 2.960 3.107 5.0 training

280.6 27.0 73.0 0 3.160 3.369 6.6 training

281.9 27.0 73.0 0 4.020 4.012 0.2 training

289.1 27.0 73.0 0 13.060 13.040 0.2 training

a

b Reference of the experimental data. c Experimental value. d Represented/predicted value.

exp.(i)

exp.(i)ed.(i)rep.(i)/prARD

100%



A preliminary study shows that the ARD% values of the conventional gas hydrate

thermodynamic models utilizing the van der Waals-Platteeuw [8] (vdW-P) theory for

modeling the hydrate phase and different equations of state for modeling the fluid phases are

normally increased by increasing the concentration of the hydrogen sulfide in the feed gas.

Figure 1. Comparison between the represented (rep)/predicted (pred) results of the model and

experimental values of hydrate dissociation pressure for the CH4 + CO2 + H2S or CH4 + CO2/H2S

systems. [6, 7, 15-20].

The obtained hydrate dissociation pressure values indicate that the average absolute

relative deviations (AARD) of the developed LSSVM model results show generally the same

trend regarding the different concentrations of the hydrogen sulfide in the feed gas mixture.

Moreover, ZareNezhad and Aminian [5] have already stated that the AARDs of the results

through application of the famous Chen and Guo model [29] and CSMHYD software [1] vary

within 4 to 26% by increasing the concentration of the hydrogen sulfide in the feed gas while

the maximum ARD of the developed LLSVM model in this work is 5.5% for the H2S

concentration of 26.62%.

In the final analysis, it is worth mentioning that the developed LSSVM method is a

mathematical black-box. Although it leads to generally more accurate predictions than the

conventional thermodynamic models, we recommend the users to use it for interpolation of

the data. However, it may be applied for rough estimations for external predictions.



Table 2. Statistical parameters of the presented model for determination of hydrate

dissociation conditions of the CH4 + CO2 + H2S or CH4 + CO2/H2S systems

Statistical Parameter Value

training set

R2 a

0.991

Average absolute relative deviation b 2.9

Standard deviation error 0.24

Root mean square error 0.24

N c 197

validation set

R2 0.987

Average absolute relative deviation 4.8



N 24

test set

R2 0.981




N 24

total

R2 0.990




N 245 a: Squared correlation coefficient.

b , where n is the number of the model parameters.

c Number of experimental data.

N

i exp.(i)

exp.(i)ed.(i)rep.(i)/pr

nNAARD

100%



Figure 2. The relative deviations of the represented (rep)/predicted (red) results of the model and

experimental values of hydrate dissociation pressure for the CH4 + CO2 + H2S or CH4 + CO2/H2S

systems. [6, 7, 15-20].

CONCLUSION

A LSSVM network-based mathematical model was developed for correlating/estimating

the dissociation conditions of clathrate hydrates of methane + carbon dioxide + hydrogen

sulfide and methane + carbon dioxide/hydrogen sulfide.

The experimental data available in open literature [6, 7, 15-21] were applied for

developing and testing the model. The pattern search H-GA [26, 27] optimization algorithm

was used to optimize the LSSVM structure. The statistical parameters of the obtained model

demonstrate that it is a generally acceptable tool for representing and predicting the hydrate

dissociation conditions of the investigated systems compared to the conventional models or

softwares.

Finally, it should be noted that the proposed LSSVM model may not be able to predict

the hydrate dissociation conditions of the system of interest at higher concentrations of the

hydrogen sulfide than those applied for its development and testing.



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