# Development of Predictive Techniques for Estimating Liquid Water-Hydrate Equilibrium of Water-Hydrocarbon System

Authors: Amir H. Mohammadi; Dominique Richon
Journal: Journal of Thermodynamics
 (2009)

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## Abstract

In this communication, we review recent studies by these authors for modeling theLW-H
equilibrium. With the aim of estimating the solubility of pure hydrocarbon hydrate former in pure water in equilibrium with gas hydrates, a thermodynamic model is introduced based on equality of water fugacity in the liquid water and hydrate phases. The solid solution theory of Van der Waals-Platteeuw is employed for calculating the fugacity of water in the hydrate phase. The Henry's law approach and the activity coefficient method are used to
calculate the fugacities of the hydrocarbon hydrate former and water in the liquid water phase, respectively. The results of this model are successfully compared with some selected experimental data from the literature. A mathematical model based on feed-forward artificial neural network algorithm is then introduced to estimate the solubility of pure hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates. Independent experimental data (not employed in training and testing steps) are used to examine the reliability of this algorithm successfully.


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## Body

## 1. Introduction

Gas hydrates are ice-like structures in which water
molecules, under pressure, form structures composed of polyhedral cages
surrounding gas molecule “guests” such as methane and ethane [1–4]. The most common gas hydrate structures are those of
structure I (sI) and structure II (sII), where each structure is composed
of a certain number of large and small cavities formed by water molecules [1, 2]. For a molecule to
enter a cavity, its size should be smaller than a certain value. Large molecule
guests which can enter only a limited number of large cavities require smaller
“help gas” molecules to mainly fill some smaller cavities sufficiently to
stabilize hydrate crystals [1, 2]. It has been proved that gas hydrates occur in staggering
abundance in cold subsea, sea floor, and permafrost environments where
temperature and pressure conditions ensure their stability [1–5]. It is believed the amount of natural gas trapped in these deposits is much higher than the amount of natural gas existing in classical reserves [1, 2, 5]. Hydrate
technology has also been proposed as a means for CO2 separation from
industrial flue gases and sequestration in the deep ocean for reducing the
emission of greenhouse gases [1–5]. Gas hydrates can form as well in undersea piping and
above-ground pipelines, where they pose a major and expensive problem for the
petroleum industry [1, 2]. Hydrates are also being regarded as an alternate means of
gas transportation and storage [1, 2, 4]. It is believed that
the potential storage of gas in hydrate is comparable to gas storage in the
form of liquefied natural gas (LNG)
and compressed natural gas (CNG) [2, 6].The liquid water-hydrate (LW-H) equilibrium knowledge is necessary in the design
of gas transportation and storage process and should be destined for proposed
CO2 sequestration schemes [1–4, 6]. These factors, and
the potential widespread abundance of gas hydrates in the cold subsea, sea
floor, and permafrost environments [1–4, 6], warrant an understanding of LW-H equilibrium.Figure1 shows typical
solubility-temperature diagram for water-pure hydrate former (limiting
reactant) system [2]. As can be seen, the temperature
and pressure dependencies of the pure hydrate former solubility in pure water
being in the liquid water-vapor (LW-V) equilibrium region are
different from the corresponding dependency in the LW-H equilibrium region [1–4, 6–19]. The LW-V equilibrium is
strong function of temperature and pressure while the LW-H equilibrium is strong function of temperature but
very weak function of pressure [1–4, 6–19]. On the other hand, the pure hydrate former solubility in pure water being in the LW-V equilibrium region
increases with decreasing the temperature at a given pressure, while, the
corresponding solubility in pure water being in the LW-H equilibrium region decreases with decreasing the
temperature at the same pressure [1–4, 6–19]. Furthermore,
the metastable liquid water-vapor (LW-V)
equilibrium may extend well into the gas hydrate formation zone [1, 2, 6–19].Figure 1
Typical solubility
(x)-temperature (T) diagram for water-single (pure) hydrate former (limiting
reactant) system. LW: liquid water; H: hydrate; V: vapor; P: pressure [2]. Bold solid lines: LW-H equilibrium; solid lines: LW-V equilibrium; dashed lines: metastable LW-V equilibrium; bold dashed line: LW-V-H equilibrium.The experimental works done to describe theLW-H equilibrium are limited
mainly due to two factors: the possible extension of the metastable LW-V equilibrium into the gas
hydrate region and the experimental restraint that the existing analysis
methods require modifications [1, 2, 6–19]. Literature survey reveals the availability of only few sets
of experimental data for the LW-H equilibrium [1, 2, 7, 11, 16–19]. Consequently, few reliable models are available in the
literature for calculating the LW-H equilibrium [1–4, 6–8, 11, 14–16].The objective of this work is to review
recent studies by these authors for modeling theLW-H equilibrium [2–4]. A thermodynamic model based on
equality of water fugacity in the liquid water and hydrate phases for
determining solubility of pure hydrocarbon hydrate former in pure water being
in equilibrium with gas hydrates is first introduced [2]. The fugacity of water in the hydrate phase is calculated
using the solid solution theory of van der Waals-Platteeuw [20]. The hydrocarbon
hydrate former fugacity and water fugacity in the liquid water phase are
calculated using Henry’s law approach and the activity coefficient method,
respectively. The capability of this model is investigated by comparing its
predictions with some selected experimental data from the literature. It is demonstrated
that the results are in acceptable agreement demonstrating the capability of
the thermodynamic model developed in this work for estimating the solubility of
pure hydrocarbon hydrate former in pure water being in equilibrium with gas
hydrates. It is also shown that by assuming an average value equal to 1.2 for
the water activity coefficient, acceptable results can be obtained for
estimating the solubility of carbon dioxide in water at equilibrium with its
gas hydrate [3]. Finally, the capability of artificial neural network (ANN) algorithm, as an alternative tool,
for modeling the LW-H equilibrium is successfully demonstrated [4].


## 2. Liquid Water-Hydrate Equilibrium

The liquid water-hydrate
equilibrium of a system is calculated, by equating the fugacities of water in
the liquid water phase,fwL, and in the hydrate phase, fwH, as follows [1–3]:
(1)fwL=fwH.The fugacity of water in the
hydrate phase,fwH,
is related to the chemical potential difference of water in the filled and
empty hydrate cages
by the following expression [1, 2]:
(2)fwH=fwMTexpμwH−μwMTRT,
where fwMT is the fugacity of water in the hypothetical empty hydrate phase and μwH−μwMT represents the chemical potential
difference of water in the filled (μwH) and empty (μwMT) hydrates. R and T stand for universal gas constant and
temperature, respectively.The solid solution theory of van
der Waals-Platteeuw [20] can be employed for calculating (μwH−μwMT)/RT as follows [1, 2]:
(3)μwH−μwMTRT=−∑ivi′ln(1+∑jCijfj)=∑iln(1+∑jCijfj)−vi′,
where vi′ is the number of cavities of type i per
water molecule in a unit hydrate cell [1, 2], Cij stands for the Langmuir constant for hydrocarbon hydrate former’s interaction
with each type cavity, and fj is the fugacity of hydrate former [2].The
fugacity of water in the empty lattice can be expressed as follows [1, 2]:
(4)fwMT=PwMTφwMTexp∫PwMTPvwMTdPRT,
where PwMT, φwMT, vwMT,
and P are the vapor pressure of the
empty hydrate lattice, the correction for the deviation of the saturated vapor
of the pure (hypothetical) lattice from ideal behavior, the partial molar
volume of water in the empty hydrate [1, 2, 21], and pressure, respectively. The exponential term is a
Poynting type correction.Equation (4) may be simplified by two
assumptions: (1) that the hydrate partial molar volume equals to the molar
volume and is independent of pressure and (2) that PwMT is relatively small (on the order of 10−3 MPa), so that φwMT=1 [1, 2]. Therefore [1, 2],
(5)fwMT=PwMT×expvwMT(P−PwMT)RT.Using the previous expressions, the
following equation is obtained for the fugacity of water in the hydrate phase [2]:
(6)fwH=PwMT×exp[vwMT(P−PwMT)RT]×[(1+CsmallfHCL)−vsmall′×(1+ClargefHCL)−vlarge′],where fHCL is the fugacity of hydrocarbon hydrate former
in the liquid water phase.The Poynting correction term can be
ignored up to intermediate pressures and therefore, the following equation can
be obtained for calculating fugacity of water in hydrate phase [2, 3]:
(7)fwH=PwMT×[(1+CsmallfHCL)−vsmall′×(1+ClargefHCL)−vlarge′].
The fugacity of water in the liquid water phase can be expressed
by [2, 3, 22]:
(8)fwL=xwLγwLPwsat,where xwL and γwL are, respectively, the water mole fraction and the activity
coefficient of water in liquid water phase being in equilibrium with gas
hydrates. In the intermediate pressure range, the liquid water is
an incompressible fluid, hydrocarbon hydrate former solubility is very small
comparing to unity, and activity coefficient of water can be approximated to unity [22] (however it is necessary to be careful, as it is not the case at high pressures [22] where the nonideality of the liquid water phase and
solubility become important). Therefore, (8) can be satisfactorily written as follows [2]:
(9)fwL≅Pwsat.
Using (7)
and (9), the following expression is obtained [2]:
(10)Pwsat=PwMT×[(1+CsmallfHCL)−vsmall′×(1+ClargefHCL)−vlarge′]
or
(11)1−(PwMTPwsat)×[(1+CsmallfHCL)−vsmall′×(1+ClargefHCL)−vlarge′]=0,
where the fugacity of hydrocarbon
hydrate former in the liquid water phase up to intermediate pressures can be
calculated using the following equation [2, 22]:
(12)fHCL=xHCLH¯HC-w,
where H¯HC-w represents Henry’s constant for
hydrocarbon hydrate former-water system. Therefore, the following final
expression is obtained for estimating the solubility of pure hydrocarbon hydrate former in liquid water phase being in equilibrium with gas hydrates [2]:
(13a)1−(PwMTPwsat)×[(1+CsmallxHCLH¯HC-w)−vsmall′×(1+ClargexHCLH¯HC-w)−vlarge′]=0.Equation (13a) allows easy
calculation of the solubility of pure hydrocarbon hydrate former in the liquid
water being in equilibrium with gas hydrates.
Its main advantages are the availability of necessary input data and
the simplicity of the calculations, as the calculations can be done in
Excel spread sheets [2]. Furthermore, as can be
seen in (13a), almost all terms are temperature dependent while not
pressure dependent, indicating  the solubility of pure hydrocarbon hydrate former in liquid
water phase being in equilibrium with gas hydrates is strong function of
temperature and only very weak function of pressure. For the CO2 (and
other highly soluble gases like H2S)-water system, the solubility of
CO2 in the water phase cannot be ignored and consequently, the water
activity cannot be set to unity. Taking into account the CO2 solubility in the water phase and also the water activity, the following
equation can be derived [3]:
(13b)1−[PwMT(1−xCO2L)γwLPwsat]×[(1+CsmallxCO2LH¯CO2-w)−vsmall′×(1+ClargexCO2LH¯CO2-w)−vlarge′]=0.


### 2.1. Model Parameters

In (13a) and (13b), the following values of vi′ for structure-I hydrates can be used [1, 2]:
(14a)vsmall′=123,vlarge′=323,and for
structure II:
(14b)vsmall′=217,vlarge′=117.The Langmuir constants (they are generally functions
of temperature, pressure, composition and hydrate structure, while the effect
of pressure is normally ignored [23])
accounting for the interaction between the hydrate former and water molecules in the cavities were reported by Parrish and Prausnitz [24] for a range of temperatures and
hydrate formers [2]. However, the integration procedure
was already followed in obtaining the Langmuir constants for wider temperatures
using the Kihara [25] potential function with a
spherical core according to the study by McKoy and Sinanoğlu [26]. In this work, the Langmuir constants for hydrate former’s interaction with each type cavity have been determined using the equations of Parrish and Prausnitz [24]:For pentagonal dodecahedra (small cavity):(15)Csmall=aT×exp(bT),
for tetrakaidecahedra (large cavity):
(16)Clarge=cT×exp(dT),
where T is
in K and C has units of reciprocal MPa.
Constants a–d are reported in Table 1.Table 1
Constantsa–d in (15) and (16) [24].
Hydrate formera/(K⋅MPa−1)b/(K⋅MPa−1)c/(K⋅MPa−1)d/(K⋅MPa−1)Methane0.00372372708.80.0183732737.9Ethane000.0069063631.6Propane000.0123534406.1CO20.00119782860.50.0085073277.9The concept in (4) of universal
empty hydrate vapor pressure for each structure prompted Dharmawardhana et al. [27] to calculate the PwMT from a number of simple hydrate three-phase ice-vapor-hydrate equilibrium. By equating the
fugacity of water in the hydrate phase to that of pure ice at the three-phase
line, Dharmawardhana et al. [27] obtained the following equation for the vapor pressure of the empty
hydrate structure I [1, 2]:
(17a)PwMT=0.1×exp(17.440−6003.9T),
and for structure II:
(17b)PwMT=0.1×exp(17.332−6017.6T),
where PwMT is in MPa and T is in K. It should be
mentioned that the nonideality of water vapor pressure of the empty hydrate at
saturation seems to be negligible due to the small quantity (typically, 10−3 to 10−5 MPa).The following values for Henry’s constant of
hydrocarbon hydrate former-water can be used [2, 28]:
(18)H¯g,w(T)=(10A+B/T+C¯×log(T)+D×T)×0.1,
where T and H¯g,w(T) are in K and MPa, respectively. Constants A, B, C¯, and D are given in Table 2.Table 2
ConstantsA–D in (18) [28].
Hydrate formerAB/KC¯/K−1D/K−1TminTmaxMethane147.788−5768.3−52.29520.018616273.15373.15Ethane146.901−5768.3−51.85930.017410273.15343.15CO221.6215−1499.8−5.649470.0002062273.15373.15For the propane-water system the following equation was used:H¯propane,w(T)=10−3×exp{552.648+0.078453×T−(21334.4/T)−85.89736×ln(T)}.The water vapor pressure can be obtained using the
following expression [29]:
(19)Pwsat=10−6exp(73.649−7258.2T−7.3037ln(T)+4.1653×10−6T2),
where, T and Pwsat are, respectively, in K and MPa.


## 2.1. Model Parameters

In (13a) and (13b), the following values of vi′ for structure-I hydrates can be used [1, 2]:
(14a)vsmall′=123,vlarge′=323,and for
structure II:
(14b)vsmall′=217,vlarge′=117.The Langmuir constants (they are generally functions
of temperature, pressure, composition and hydrate structure, while the effect
of pressure is normally ignored [23])
accounting for the interaction between the hydrate former and water molecules in the cavities were reported by Parrish and Prausnitz [24] for a range of temperatures and
hydrate formers [2]. However, the integration procedure
was already followed in obtaining the Langmuir constants for wider temperatures
using the Kihara [25] potential function with a
spherical core according to the study by McKoy and Sinanoğlu [26]. In this work, the Langmuir constants for hydrate former’s interaction with each type cavity have been determined using the equations of Parrish and Prausnitz [24]:For pentagonal dodecahedra (small cavity):(15)Csmall=aT×exp(bT),
for tetrakaidecahedra (large cavity):
(16)Clarge=cT×exp(dT),
where T is
in K and C has units of reciprocal MPa.
Constants a–d are reported in Table 1.Table 1
Constantsa–d in (15) and (16) [24].
Hydrate formera/(K⋅MPa−1)b/(K⋅MPa−1)c/(K⋅MPa−1)d/(K⋅MPa−1)Methane0.00372372708.80.0183732737.9Ethane000.0069063631.6Propane000.0123534406.1CO20.00119782860.50.0085073277.9The concept in (4) of universal
empty hydrate vapor pressure for each structure prompted Dharmawardhana et al. [27] to calculate the PwMT from a number of simple hydrate three-phase ice-vapor-hydrate equilibrium. By equating the
fugacity of water in the hydrate phase to that of pure ice at the three-phase
line, Dharmawardhana et al. [27] obtained the following equation for the vapor pressure of the empty
hydrate structure I [1, 2]:
(17a)PwMT=0.1×exp(17.440−6003.9T),
and for structure II:
(17b)PwMT=0.1×exp(17.332−6017.6T),
where PwMT is in MPa and T is in K. It should be
mentioned that the nonideality of water vapor pressure of the empty hydrate at
saturation seems to be negligible due to the small quantity (typically, 10−3 to 10−5 MPa).The following values for Henry’s constant of
hydrocarbon hydrate former-water can be used [2, 28]:
(18)H¯g,w(T)=(10A+B/T+C¯×log(T)+D×T)×0.1,
where T and H¯g,w(T) are in K and MPa, respectively. Constants A, B, C¯, and D are given in Table 2.Table 2
ConstantsA–D in (18) [28].
Hydrate formerAB/KC¯/K−1D/K−1TminTmaxMethane147.788−5768.3−52.29520.018616273.15373.15Ethane146.901−5768.3−51.85930.017410273.15343.15CO221.6215−1499.8−5.649470.0002062273.15373.15For the propane-water system the following equation was used:H¯propane,w(T)=10−3×exp{552.648+0.078453×T−(21334.4/T)−85.89736×ln(T)}.The water vapor pressure can be obtained using the
following expression [29]:
(19)Pwsat=10−6exp(73.649−7258.2T−7.3037ln(T)+4.1653×10−6T2),
where, T and Pwsat are, respectively, in K and MPa.


## 3. Artificial Neural Network Algorithm

Artificial neural
networks have large numbers of computational units called neurons connected in
a massively parallel structure and do not need an explicit formulation of the
mathematical or physical relationships of the handled problem [4, 30–37]. The most
commonly used ANNs are the feed-forward
neural networks (FNN) [4, 36, 37], which are
designed with one input layer, one output layer and hidden layers [4, 33–35, 37]. The number of neurons in
the input and output layers equals to the number of inputs and outputs physical
quantities, respectively [4, 37]. The disadvantage
of FNNs is the determination of the
ideal number of neurons in the hidden layer(s); few neurons produce a network
with low precision and a higher number leads to overfitting and bad quality of
interpolation and extrapolation [4, 37]. The use of techniques such as Bayesian regularization, along with a Levenberg-Marquardt
algorithm [4, 38, 39], can help overcome this problem [4, 31, 32, 37].In theFNN method, the input layer of the
network receives all the input data and introduces scaled data to the network [4, 37]. The data from the input
neurons are propagated through the network via weighted interconnections [4, 37]. Every i neuron in
a k layer is connected to every neuron in adjacent layers [4, 37]. The i neuron within the hidden k layer performs the following tasks: summation of the arriving weighted inputs (input vector Ii=[Ii,1,…,Ii,Nk−1])
and propagations of the resulting summation through an  activation
function, f¯,
to the adjacent neurons of the next hidden layer or to the output neuron(s).
Three types of transfer functions are normally used: the exponential sigmoid,
tangent sigmoid and linear functions [4, 37]. In this work, the activation function is a linear
function [4]:
(20)f¯(x¯)=x¯,
where x¯ stands for parameter of linear activation
function. A bias term, b¯,
is associated with each interconnection in order to introduce a supplementary
degree of freedom. The expression of the weighted sum, S, to the ith
neuron in the kth layer (k≥2) is [4, 37]:(21)Sk,i=∑j=1Nk−1[(wk−1,j,iIk−1,j)+b¯k,i],
where w is the weight parameter between
each neuron-neuron interconnection. Using this feed-forward network with linear
activation function, the output, O, of the i neuron within the
hidden k layer is [4]:(22)Ok,i=Sk,i.To achieve a better
stability, the following scaling rule is applied to solubility of pure
hydrocarbon hydrate former being in equilibrium with gas hydrates (x) before normalization [4]:
(23)xNetwork=log(xexp),
where subscript exp represents experimental data.To develop theANN, the datasets are subdivided into
3 classes: training, testing and validation [4, 37]. After partitioning the datasets, the training
set is used to adjust the parameters. All synaptic weights and biases are first
initialized randomly. The network is then trained; its synaptic weights are
adjusted by optimization algorithm, until it correctly emulates the
input/output mapping, by minimizing the average root-mean-square error [4, 37]. The optimization method
chosen in this work was the Levenberg-Marquardt algorithm [38, 39], as mentioned earlier. The testing set is used during the
adjustment of the network’s synaptic weights to evaluate the algorithms
performance on the data not used for adjustment and stop the adjusting if the
error on the testing set increases. Finally, the validation set measures the
generalization ability of the model after the fitting process [4, 37].


## 4. Results and Discussion

### 4.1. Thermodynamic Model

Among theLW-H equilibrium
data reported in the literature for the solubility of methane in pure water
being in equilibrium with gas hydrates, those reported by Yang [16], Servio and Englezos [10] and Kim et al. [11] seem to be the most reliable. These data are reported in Tables 3, 4, and 5, respectively. As can be seen, the temperature range is from
274.15 to 281.7 K, and the pressures are up to 143.62 MPa. Other
experimental data reported in the literature for methane solubility in pure
water being in equilibrium with gas hydrates have not been considered in this
work, as they are not consistent with other literature data. Tables 3–5 also show the predictions of the models developed in this work and the absolute deviations (ADs). As can be seen, (13a) with no adjustable parameter shows
encouraging results. The predictions of this thermodynamic model for the
solubility of methane in pure water being in equilibrium with gas hydrates show
less than 18% absolute deviation and the average absolute deviation (AAD) among all the experimental and
predicted data is 7.3% [4].Table 3
Experimental [16] and predicted/calculated solubility of
methane in pure water being in equilibrium with gas hydrates (liquid water-hydrate
equilibrium) [4].
T/KP/MPaExperimental [16] methane solubility/mole fraction (×103)Predicted/calculated methane solubility
usingANN algorithm/mole fraction (×103)AD%Predicted methane solubility using
thermodynamic model/mole fraction (×103)AD%†276.1950.001.25**1.293.21.4012.0277.8550.811.501.444.01.553.3279.8550.981.65*1.650.01.745.5280.4751.261.751.721.71.813.4276.36101.001.251.304.01.4213.6278.71101.731.501.532.01.638.7280.11101.881.651.681.81.777.3280.65102.431.751.740.61.834.6276.67127.101.251.336.41.4415.2278.90127.221.501.553.31.6510.0280.39127.351.651.713.61.809.1281.08127.451.75**1.792.31.876.9276.91143.101.251.358.01.4717.6279.20143.621.501.585.31.6812.0280.59143.251.651.734.81.8210.3281.65143.241.751.866.31.9410.9Data* used for testing.Data** used for validation (the remaining data
were used for training).AD†=|(experimentalvalue−predicted/calculatedvalue)/experimentalvalue|.Table 4
Experimental [10] and predicted/calculated solubility of
methane in pure water being in equilibrium with gas hydrates (liquid water-hydrate
equilibrium) [4].
T/KP/MPaExperimental [10] methane solubility/mole fraction (×103)Predicted/calculated methane solubility
usingANN algorithm/mole fraction (×103)AD%Predicted methane solubility using
thermodynamic model/mole fraction (×103)AD%274.353.51.1701.142.61.267.7275.453.51.203*1.232.21.3411.4274.155.01.1901.125.91.255.0277.355.01.3601.392.21.5010.3275.256.51.2011.210.71.3310.7280.156.51.567**1.687.21.7713.0Data* used for testing.Data** used for validation (the remaining data
were used for training).Table 5
Experimental [11] and predicted/calculated solubility of
methane in pure water being in equilibrium with gas hydrates (liquid water-hydrate
equilibrium) [4].
T/KP/MPaExperimental [11] methane solubility/mole fraction (×103)Predicted/calculated methane solubility
usingANN algorithm/mole fraction (×103)AD%Predicted methane solubility using
thermodynamic model/mole fraction (×103)AD%276.25.01.331.293.01.405.3277.95.11.591.458.81.552.5279.95.11.751.665.11.750.0280.55.11.861.727.51.812.7276.410.11.33*1.311.51.416.0278.710.21.591.533.81.4210.7280.110.21.751.684.01.771.1280.710.21.861.755.91.841.1276.712.71.331.330.01.459.0278.912.71.59**1.552.51.653.8280.412.71.751.712.31.802.9281.112.71.861.793.81.881.1276.914.31.331.351.51.469.8279.214.41.591.580.61.685.7280.614.31.751.731.11.824.0281.714.31.88**1.870.51.943.2Data* used for testing.Data** used for validation (the remaining data
were used for training).Limited information is available
for theLW-H equilibrium
of ethane + water system. Yang [16] and
Kim et al. [11] measured ethane solubility in pure
water being in equilibrium with gas hydrates. These experimental data along
with the predictions of (13a) and ADs are shown in
Tables 6 and 7. As can be observed, the
temperature range is from 277.3 to 278.5 K, and the pressures are up to 151 MPa. This table also shows that the temperature change has no effect on experimental gas solubility
data [11, 16] indicating  the experimental data reported by Yang [16] and Kim et al. [11] for ethane solubility in pure
water being in equilibrium with gas hydrates are not reliable. The maximum AD between the experimental and predicted data for the ethane + water system is less than 32% and the AAD is less than 22% [4].Table 6
Experimental [16] and predicted solubility of ethane in pure water being in equilibrium with gas
hydrates (liquid water-hydrate equilibrium) [4].
T/KP/MPaExperimental [16] ethane solubility/mole fraction (×104)*Predicted ethane solubility usingANN algorithm/mole fraction (×104)AD%Predicted ethane solubility using thermodynamic model/mole fraction (×104)AD%277.31514.125.0021.44.9119.2277.821014.125.1725.55.1324.5278.461514.125.4031.15.4231.6Experimental* solubility data are not temperature dependent and cannot
be considered reliable. Therefore, the pseudo-experimental data generated from
the thermodynamic model were used for training.Table 7
Experimental [11] and predicted solubility of ethane in pure water being in equilibrium with gas hydrates (liquid water-hydrate equilibrium) [4].
T/KP/MPaExperimental [11] ethane solubility/mole fraction (×104)*Predicted ethane solubility usingANN algorithm/mole fraction (×104)AD%Predicted ethane solubility using thermodynamic model/mole fraction (×104)AD%277.310.14.375.0014.44.9012.1277.815.14.375.1718.35.1217.2278.520.14.375.4224.05.4324.3Experimental* solubility data are not temperature dependent and cannot
be considered reliable. Therefore, the pseudo-experimental data generated from
the thermodynamic model were used for training.Limited information is also available for the propane
solubility in pure water being in equilibrium with gas hydrates. Gaudette
and Servio [19] measured these data, which are reported in
Table 8. As can be seen, the temperature range is from 274.16 to 276.16 K, and
the pressures are up to 0.358 MPa. The AAD is equal to 8% and the maximum AD between
the experimental and predicted data is less than 17%. For
this system, the vapor pressure of the empty hydrate
lattice in the thermodynamic model was slightly
tuned, because it can be regarded as one of the deviations sources in the
thermodynamic model [1, 4].Table 8
Experimental [19] and predicted/calculated solubility of propane in pure water being in equilibrium with gas hydrates (liquid water-hydrate equilibrium) [4].
T/KP/MPaExperimental [19] propane solubility/mole
fraction (×104)Predicted/calculated
propane solubility usingANN algorithm/mole
fraction (×104)AD%Predicted
propane
solubility
using thermodynamic model/mole fraction(×104)†AD%274.160.3011.440*1.450.71.346.9274.230.2531.4391.461.51.328.3274.330.3581.5461.474.91.540.4275.200.3021.5721.560.81.542.0275.200.3521.5721.560.81.7913.9276.160.3551.642**1.661.11.3716.6Data* used for testing.Data** used for validation (the remaining data
were used for training).The† vapor pressure of the empty hydrate lattice in
the thermodynamic model for propane hydrate was slightly tuned.Among theLW-H equilibrium
data reported in the literature for the CO2 solubility in pure water at
equilibrium with its gas hydrate, those reported by Servio and Englezos [9] and Yang et al. [7] seem to be the most reliable. These data are reported in Tables 9 and 10, respectively. As can be seen, the temperature range is from 273.95 to 282.95 K, and the pressures are up to 14.2 MPa. Tables 9 and 10 also show the predictions of the thermodynamic
model used in this work and the absolute deviations. In this table, the
experimental data reported at 275.95 K and 2 MPa was used to adjust the
activity coefficient of water in (13b). An average value equal to 1.2 yielded very good results (AD = 0%). Using this value for the activity
coefficient of water, (13b) was applied to predict the CO2 solubility in pure
water at equilibrium with its gas hydrate. As demonstrated in Tables 9 and 10, (13b) shows encouraging results. The predictions show less than
14% absolute deviation and the average absolute deviation among all the experimental and predicted data is less than 7%. The average value equal to 1.2 eliminates
any need to use a model for the activity coefficient of water reconfirming
the simplicity of the model for predicting solubility of carbon dioxide in pure
water at equilibrium with its gas hydrate [3].Table 9
Experimental [9] and predicted/calculated solubility of CO2 in water at equilibrium with  gas hydrates (liquid water-hydrate equilibrium) [3].
T/KP/MPaExperimentalCO2 solubility [9]/mole fractionPredicted/calculatedCO2 solubility using thermodynamic model/mole fractionAD%275.952.00.0173*0.01730.0273.952.00.01640.014114.0277.053.70.01960.01836.6274.153.70.01580.01439.5278.154.20.01980.02001.0274.054.20.01560.01429.0274.055.00.01630.014212.9278.055.00.01980.01990.5282.855.00.02750.030510.9274.256.00.01630.01554.9276.456.00.01920.01872.6282.956.00.02810.03089.6This* experimental data was used to adjust the activity coefficient of water.Table 10
Experimental [7] and predicted solubility of CO2 in water at equilibrium with  gas hydrates (liquid water-hydrate equilibrium) [3].
T/KP/MPaExperimentalCO2 solubility [7]/mole fractionPredictedCO2 solubility using thermodynamic model/mole fractionAD%280.324.990.02560.02435.1280.596.080.02700.02497.8280.726.080.02680.02526.0280.746.080.02680.02526.0279.986.090.02620.02369.9278.716.100.02210.02114.5278.996.100.02310.02166.5279.016.100.02360.02178.1279.126.100.02320.02195.6279.326.100.02420.02237.9279.456.100.02390.02255.9279.496.100.02350.02263.8279.586.100.02490.02288.4279.946.100.02600.02359.6280.436.100.02620.02456.5280.986.100.02690.02584.1280.268.160.02680.02429.7280.3410.040.02630.02447.2278.3710.350.02270.02059.7278.3910.350.02240.02058.5278.3910.370.02230.02058.1279.1810.400.02260.02202.7278.6510.410.02350.021010.6279.4110.430.02480.02249.7279.1310.440.02410.02199.1279.6010.440.02500.02279.2280.3010.440.02580.02435.8280.3010.440.02570.02435.4280.4010.440.02600.02455.8280.0010.450.02560.02367.8277.8410.470.02010.01962.5280.3414.200.02600.02446.2In addition to the quality of the experimental data
reported in the literature, the deviations may be attributed to other factors,
as well: the Langmuir constants and the vapor pressures of the empty hydrate
lattice are consistent with initial data on the liquid water/ice-vapor-hydrate
equilibrium. Since the experimental conditions go far below the
initial hydrate formation conditions, the assumptions that are evidently valid
at the initial hydrate formation conditions may be invalid elsewhere [1]. Accurate determination of
fugacities has also important effect on the predictions. Henry’s law approach
for calculating fugacities of pure and heavy hydrocarbon hydrate formers may
not be very accurate, as Henry’s constants are normally developed based on
experimental data and measuring these data for heavy hydrocarbon hydrate formers
may not be easy [2–4].


### 4.2. Artificial Neural Network Algorithm

TheANN algorithm detailed in
Table 11 with one hidden layer was used to calculate/predict the logarithm of
solubility of pure hydrocarbon hydrate former in pure water being in
equilibrium with gas hydrates as a function of temperature and carbon number of hydrocarbon hydrate former
(which represents the type of hydrocarbon), because the logarithm of solubility is
approximately linear function of temperature for a given pure hydrocarbon
hydrate former. It should be
mentioned that plenty of data should generally be used for developing AAN algorithms, especially for highly
nonlinear systems. In our case, where the logarithm of solubility of pure
hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates
is approximately linear function of temperature, few sets of
data for training (data reported in Tables 3–8) can be used to develop the ANN algorithm. However, the more data
for training, the more reliable ANN  results.  Having this in mind, one neuron in the hidden layer yielded acceptable results
according to both the accuracy of the fit (minimum value of the objective
function) and the predictive power of the neural network [4].Table 11
Number of neurons, hidden layers, parameters, data and type of activation function used in theANN algorithm [4]. (Number of hidden layers = 1, number of parameters = 5, number of data used for training (and testing) = 44, type of activation function: linear, input neurons: temperature and carbon number of single hydrocarbon hydrate former representing the type of hydrocarbon, output neuron: logarithm of solubility of single hydrocarbon hydrate former being in equilibrium with gas hydrates.)
LayerNumber of neurons122131Tables3–8 show the predicted/calculated
solubility values using the ANN algorithm
developed in this work along with the absolute deviations. As can be seen in
Tables 3–5, the results obtained using the ANN algorithm for methane solubility in pure water being in
equilibrium with gas hydrates show less than 9% absolute deviation and the
average absolute deviation among all
the experimental and predicted data is 3.4%. The maximum AD and the AAD between the
experimental data and predicted data using the ANN algorithm for ethane solubility in pure water being in
equilibrium with gas hydrates are about 31% and 22%, respectively, as mentioned in
Tables 6 and 7. The deviations are attributed to experimental data [11, 16], which are not temperature dependent and cannot
be considered reliable.
Because of unreliability of the experimental data [11, 16] for the ethane + water system, the pseudo-experimental data generated from the  thermodynamic
model were used for training the ANN algorithm. Table 8
shows that using the ANN, the maximum AD between the experimental and predicted data for propane
solubility in pure water being in equilibrium with gas hydrates is below 5%
and the ADD equals to 1.6% [4].Using theANN algorithm, the solubility of i-butane (the heaviest pure hydrocarbon hydrate former) in pure
water being in equilibrium with gas hydrates, for which there is no
experimental data in the literature, was predicted. These data along with the
predictions of the ANN algorithm for
methane, ethane, and propane are shown in Figure 2. As can be observed, the ANN  predicts the true behavior of
solubilities of pure hydrocarbons hydrate formers in pure water being in
equilibrium with gas hydrates (the solubility decreases by increasing the
carbon number of pure hydrocarbon hydrate former and increases by increasing
the temperature), demonstrating the capability of the ANN algorithm for modeling the LW-H equilibrium of water + hydrocarbon systems [4].Figure 2
Estimated solubility of single hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates. Symbols: experimental data reported in Tables3–8. Lines: estimated values using ANN algorithm (from top to bottom: methane, ethane, propane, and i-butane (the heaviest simple hydrocarbon hydrate former)) [4].Finally,
it should be mentioned that theADs obtained using the thermodynamic
model are generally higher than the ADs
obtained using the ANN algorithm.
This was expected, as none of the parameters of this thermodynamic model were
adjusted [4].


## 4.1. Thermodynamic Model

Among theLW-H equilibrium
data reported in the literature for the solubility of methane in pure water
being in equilibrium with gas hydrates, those reported by Yang [16], Servio and Englezos [10] and Kim et al. [11] seem to be the most reliable. These data are reported in Tables 3, 4, and 5, respectively. As can be seen, the temperature range is from
274.15 to 281.7 K, and the pressures are up to 143.62 MPa. Other
experimental data reported in the literature for methane solubility in pure
water being in equilibrium with gas hydrates have not been considered in this
work, as they are not consistent with other literature data. Tables 3–5 also show the predictions of the models developed in this work and the absolute deviations (ADs). As can be seen, (13a) with no adjustable parameter shows
encouraging results. The predictions of this thermodynamic model for the
solubility of methane in pure water being in equilibrium with gas hydrates show
less than 18% absolute deviation and the average absolute deviation (AAD) among all the experimental and
predicted data is 7.3% [4].Table 3
Experimental [16] and predicted/calculated solubility of
methane in pure water being in equilibrium with gas hydrates (liquid water-hydrate
equilibrium) [4].
T/KP/MPaExperimental [16] methane solubility/mole fraction (×103)Predicted/calculated methane solubility
usingANN algorithm/mole fraction (×103)AD%Predicted methane solubility using
thermodynamic model/mole fraction (×103)AD%†276.1950.001.25**1.293.21.4012.0277.8550.811.501.444.01.553.3279.8550.981.65*1.650.01.745.5280.4751.261.751.721.71.813.4276.36101.001.251.304.01.4213.6278.71101.731.501.532.01.638.7280.11101.881.651.681.81.777.3280.65102.431.751.740.61.834.6276.67127.101.251.336.41.4415.2278.90127.221.501.553.31.6510.0280.39127.351.651.713.61.809.1281.08127.451.75**1.792.31.876.9276.91143.101.251.358.01.4717.6279.20143.621.501.585.31.6812.0280.59143.251.651.734.81.8210.3281.65143.241.751.866.31.9410.9Data* used for testing.Data** used for validation (the remaining data
were used for training).AD†=|(experimentalvalue−predicted/calculatedvalue)/experimentalvalue|.Table 4
Experimental [10] and predicted/calculated solubility of
methane in pure water being in equilibrium with gas hydrates (liquid water-hydrate
equilibrium) [4].
T/KP/MPaExperimental [10] methane solubility/mole fraction (×103)Predicted/calculated methane solubility
usingANN algorithm/mole fraction (×103)AD%Predicted methane solubility using
thermodynamic model/mole fraction (×103)AD%274.353.51.1701.142.61.267.7275.453.51.203*1.232.21.3411.4274.155.01.1901.125.91.255.0277.355.01.3601.392.21.5010.3275.256.51.2011.210.71.3310.7280.156.51.567**1.687.21.7713.0Data* used for testing.Data** used for validation (the remaining data
were used for training).Table 5
Experimental [11] and predicted/calculated solubility of
methane in pure water being in equilibrium with gas hydrates (liquid water-hydrate
equilibrium) [4].
T/KP/MPaExperimental [11] methane solubility/mole fraction (×103)Predicted/calculated methane solubility
usingANN algorithm/mole fraction (×103)AD%Predicted methane solubility using
thermodynamic model/mole fraction (×103)AD%276.25.01.331.293.01.405.3277.95.11.591.458.81.552.5279.95.11.751.665.11.750.0280.55.11.861.727.51.812.7276.410.11.33*1.311.51.416.0278.710.21.591.533.81.4210.7280.110.21.751.684.01.771.1280.710.21.861.755.91.841.1276.712.71.331.330.01.459.0278.912.71.59**1.552.51.653.8280.412.71.751.712.31.802.9281.112.71.861.793.81.881.1276.914.31.331.351.51.469.8279.214.41.591.580.61.685.7280.614.31.751.731.11.824.0281.714.31.88**1.870.51.943.2Data* used for testing.Data** used for validation (the remaining data
were used for training).Limited information is available
for theLW-H equilibrium
of ethane + water system. Yang [16] and
Kim et al. [11] measured ethane solubility in pure
water being in equilibrium with gas hydrates. These experimental data along
with the predictions of (13a) and ADs are shown in
Tables 6 and 7. As can be observed, the
temperature range is from 277.3 to 278.5 K, and the pressures are up to 151 MPa. This table also shows that the temperature change has no effect on experimental gas solubility
data [11, 16] indicating  the experimental data reported by Yang [16] and Kim et al. [11] for ethane solubility in pure
water being in equilibrium with gas hydrates are not reliable. The maximum AD between the experimental and predicted data for the ethane + water system is less than 32% and the AAD is less than 22% [4].Table 6
Experimental [16] and predicted solubility of ethane in pure water being in equilibrium with gas
hydrates (liquid water-hydrate equilibrium) [4].
T/KP/MPaExperimental [16] ethane solubility/mole fraction (×104)*Predicted ethane solubility usingANN algorithm/mole fraction (×104)AD%Predicted ethane solubility using thermodynamic model/mole fraction (×104)AD%277.31514.125.0021.44.9119.2277.821014.125.1725.55.1324.5278.461514.125.4031.15.4231.6Experimental* solubility data are not temperature dependent and cannot
be considered reliable. Therefore, the pseudo-experimental data generated from
the thermodynamic model were used for training.Table 7
Experimental [11] and predicted solubility of ethane in pure water being in equilibrium with gas hydrates (liquid water-hydrate equilibrium) [4].
T/KP/MPaExperimental [11] ethane solubility/mole fraction (×104)*Predicted ethane solubility usingANN algorithm/mole fraction (×104)AD%Predicted ethane solubility using thermodynamic model/mole fraction (×104)AD%277.310.14.375.0014.44.9012.1277.815.14.375.1718.35.1217.2278.520.14.375.4224.05.4324.3Experimental* solubility data are not temperature dependent and cannot
be considered reliable. Therefore, the pseudo-experimental data generated from
the thermodynamic model were used for training.Limited information is also available for the propane
solubility in pure water being in equilibrium with gas hydrates. Gaudette
and Servio [19] measured these data, which are reported in
Table 8. As can be seen, the temperature range is from 274.16 to 276.16 K, and
the pressures are up to 0.358 MPa. The AAD is equal to 8% and the maximum AD between
the experimental and predicted data is less than 17%. For
this system, the vapor pressure of the empty hydrate
lattice in the thermodynamic model was slightly
tuned, because it can be regarded as one of the deviations sources in the
thermodynamic model [1, 4].Table 8
Experimental [19] and predicted/calculated solubility of propane in pure water being in equilibrium with gas hydrates (liquid water-hydrate equilibrium) [4].
T/KP/MPaExperimental [19] propane solubility/mole
fraction (×104)Predicted/calculated
propane solubility usingANN algorithm/mole
fraction (×104)AD%Predicted
propane
solubility
using thermodynamic model/mole fraction(×104)†AD%274.160.3011.440*1.450.71.346.9274.230.2531.4391.461.51.328.3274.330.3581.5461.474.91.540.4275.200.3021.5721.560.81.542.0275.200.3521.5721.560.81.7913.9276.160.3551.642**1.661.11.3716.6Data* used for testing.Data** used for validation (the remaining data
were used for training).The† vapor pressure of the empty hydrate lattice in
the thermodynamic model for propane hydrate was slightly tuned.Among theLW-H equilibrium
data reported in the literature for the CO2 solubility in pure water at
equilibrium with its gas hydrate, those reported by Servio and Englezos [9] and Yang et al. [7] seem to be the most reliable. These data are reported in Tables 9 and 10, respectively. As can be seen, the temperature range is from 273.95 to 282.95 K, and the pressures are up to 14.2 MPa. Tables 9 and 10 also show the predictions of the thermodynamic
model used in this work and the absolute deviations. In this table, the
experimental data reported at 275.95 K and 2 MPa was used to adjust the
activity coefficient of water in (13b). An average value equal to 1.2 yielded very good results (AD = 0%). Using this value for the activity
coefficient of water, (13b) was applied to predict the CO2 solubility in pure
water at equilibrium with its gas hydrate. As demonstrated in Tables 9 and 10, (13b) shows encouraging results. The predictions show less than
14% absolute deviation and the average absolute deviation among all the experimental and predicted data is less than 7%. The average value equal to 1.2 eliminates
any need to use a model for the activity coefficient of water reconfirming
the simplicity of the model for predicting solubility of carbon dioxide in pure
water at equilibrium with its gas hydrate [3].Table 9
Experimental [9] and predicted/calculated solubility of CO2 in water at equilibrium with  gas hydrates (liquid water-hydrate equilibrium) [3].
T/KP/MPaExperimentalCO2 solubility [9]/mole fractionPredicted/calculatedCO2 solubility using thermodynamic model/mole fractionAD%275.952.00.0173*0.01730.0273.952.00.01640.014114.0277.053.70.01960.01836.6274.153.70.01580.01439.5278.154.20.01980.02001.0274.054.20.01560.01429.0274.055.00.01630.014212.9278.055.00.01980.01990.5282.855.00.02750.030510.9274.256.00.01630.01554.9276.456.00.01920.01872.6282.956.00.02810.03089.6This* experimental data was used to adjust the activity coefficient of water.Table 10
Experimental [7] and predicted solubility of CO2 in water at equilibrium with  gas hydrates (liquid water-hydrate equilibrium) [3].
T/KP/MPaExperimentalCO2 solubility [7]/mole fractionPredictedCO2 solubility using thermodynamic model/mole fractionAD%280.324.990.02560.02435.1280.596.080.02700.02497.8280.726.080.02680.02526.0280.746.080.02680.02526.0279.986.090.02620.02369.9278.716.100.02210.02114.5278.996.100.02310.02166.5279.016.100.02360.02178.1279.126.100.02320.02195.6279.326.100.02420.02237.9279.456.100.02390.02255.9279.496.100.02350.02263.8279.586.100.02490.02288.4279.946.100.02600.02359.6280.436.100.02620.02456.5280.986.100.02690.02584.1280.268.160.02680.02429.7280.3410.040.02630.02447.2278.3710.350.02270.02059.7278.3910.350.02240.02058.5278.3910.370.02230.02058.1279.1810.400.02260.02202.7278.6510.410.02350.021010.6279.4110.430.02480.02249.7279.1310.440.02410.02199.1279.6010.440.02500.02279.2280.3010.440.02580.02435.8280.3010.440.02570.02435.4280.4010.440.02600.02455.8280.0010.450.02560.02367.8277.8410.470.02010.01962.5280.3414.200.02600.02446.2In addition to the quality of the experimental data
reported in the literature, the deviations may be attributed to other factors,
as well: the Langmuir constants and the vapor pressures of the empty hydrate
lattice are consistent with initial data on the liquid water/ice-vapor-hydrate
equilibrium. Since the experimental conditions go far below the
initial hydrate formation conditions, the assumptions that are evidently valid
at the initial hydrate formation conditions may be invalid elsewhere [1]. Accurate determination of
fugacities has also important effect on the predictions. Henry’s law approach
for calculating fugacities of pure and heavy hydrocarbon hydrate formers may
not be very accurate, as Henry’s constants are normally developed based on
experimental data and measuring these data for heavy hydrocarbon hydrate formers
may not be easy [2–4].


## 4.2. Artificial Neural Network Algorithm

TheANN algorithm detailed in
Table 11 with one hidden layer was used to calculate/predict the logarithm of
solubility of pure hydrocarbon hydrate former in pure water being in
equilibrium with gas hydrates as a function of temperature and carbon number of hydrocarbon hydrate former
(which represents the type of hydrocarbon), because the logarithm of solubility is
approximately linear function of temperature for a given pure hydrocarbon
hydrate former. It should be
mentioned that plenty of data should generally be used for developing AAN algorithms, especially for highly
nonlinear systems. In our case, where the logarithm of solubility of pure
hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates
is approximately linear function of temperature, few sets of
data for training (data reported in Tables 3–8) can be used to develop the ANN algorithm. However, the more data
for training, the more reliable ANN  results.  Having this in mind, one neuron in the hidden layer yielded acceptable results
according to both the accuracy of the fit (minimum value of the objective
function) and the predictive power of the neural network [4].Table 11
Number of neurons, hidden layers, parameters, data and type of activation function used in theANN algorithm [4]. (Number of hidden layers = 1, number of parameters = 5, number of data used for training (and testing) = 44, type of activation function: linear, input neurons: temperature and carbon number of single hydrocarbon hydrate former representing the type of hydrocarbon, output neuron: logarithm of solubility of single hydrocarbon hydrate former being in equilibrium with gas hydrates.)
LayerNumber of neurons122131Tables3–8 show the predicted/calculated
solubility values using the ANN algorithm
developed in this work along with the absolute deviations. As can be seen in
Tables 3–5, the results obtained using the ANN algorithm for methane solubility in pure water being in
equilibrium with gas hydrates show less than 9% absolute deviation and the
average absolute deviation among all
the experimental and predicted data is 3.4%. The maximum AD and the AAD between the
experimental data and predicted data using the ANN algorithm for ethane solubility in pure water being in
equilibrium with gas hydrates are about 31% and 22%, respectively, as mentioned in
Tables 6 and 7. The deviations are attributed to experimental data [11, 16], which are not temperature dependent and cannot
be considered reliable.
Because of unreliability of the experimental data [11, 16] for the ethane + water system, the pseudo-experimental data generated from the  thermodynamic
model were used for training the ANN algorithm. Table 8
shows that using the ANN, the maximum AD between the experimental and predicted data for propane
solubility in pure water being in equilibrium with gas hydrates is below 5%
and the ADD equals to 1.6% [4].Using theANN algorithm, the solubility of i-butane (the heaviest pure hydrocarbon hydrate former) in pure
water being in equilibrium with gas hydrates, for which there is no
experimental data in the literature, was predicted. These data along with the
predictions of the ANN algorithm for
methane, ethane, and propane are shown in Figure 2. As can be observed, the ANN  predicts the true behavior of
solubilities of pure hydrocarbons hydrate formers in pure water being in
equilibrium with gas hydrates (the solubility decreases by increasing the
carbon number of pure hydrocarbon hydrate former and increases by increasing
the temperature), demonstrating the capability of the ANN algorithm for modeling the LW-H equilibrium of water + hydrocarbon systems [4].Figure 2
Estimated solubility of single hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates. Symbols: experimental data reported in Tables3–8. Lines: estimated values using ANN algorithm (from top to bottom: methane, ethane, propane, and i-butane (the heaviest simple hydrocarbon hydrate former)) [4].Finally,
it should be mentioned that theADs obtained using the thermodynamic
model are generally higher than the ADs
obtained using the ANN algorithm.
This was expected, as none of the parameters of this thermodynamic model were
adjusted [4].


## 5. Conclusions

We made a review on the recently
developed tools by these authors for  predicting the liquid water-hydrate equilibrium of the
water-hydrocarbon systems. A thermodynamic model based on equality of fugacity
concept was introduced for estimating the pure hydrocarbon hydrate former
solubility in pure water being in equilibrium with  gas hydrates. The model
employed the solid solution theory
of van der Waals-Platteeuw [20] for calculating the fugacity of water in the
hydrate phase while Henry’s law approach
and the activity coefficient method were used to calculate the pure hydrocarbon
hydrate former fugacity and water fugacity, respectively, in the liquid water
phase. The main advantages of this model are the availability of input data and the
simplicity of the calculations by ignoring the effect of pressure on the liquid
water-hydrate equilibrium. The reliability of  model was tested for three hydrocarbon hydrate
formers: methane, ethane, and propane. Acceptable agreements were achieved
between the results of this model and some selected experimental data from the
literature. The study showed a need for generating reliable experimental data
for the liquid water-hydrate equilibrium, especially for the systems containing
heavy hydrocarbon hydrate former. The deviations of the model predictions were
attributed to different factors, especially the quality of the experimental
data, correlations employed for the Langmuir constants and the vapor pressures
of the empty hydrate lattice, and finally expressions for the components
fugacities.Furthermore,
it was shown that the model can be extended to theCO2 + water
system by using an average value equal to
1.2 for the activity coefficient of water in theCO2 + water system,
which eliminates any need to use a model
for the activity coefficient of water. This value was already assumed equal
to unity for the single hydrocarbon hydrate former + water system.A feed-forward
artificial neural network algorithm with one hidden layer, which uses a modified
Levenberg-Marquardt optimization algorithm [38, 39], was developed
for estimating the solubility of single hydrocarbon hydrate former in pure
water being in equilibrium with gas hydrates. This algorithm has one output neuron
(logarithm of solubility of single hydrocarbon hydrate former in pure water
being in equilibrium with gas hydrates), two input neurons (temperature and
carbon number of hydrocarbon hydrate former, which represents the type of
hydrocarbon) and one neuron in the hidden layer taking advantage of a linear
activation function. The model was trained and developed using more
recent and reliable data reported in the literature and its predictions were
successfully compared with some independent experimental data (not used in
developing the artificial neural network algorithm) for three hydrate
formers: methane, ethane, and propane.



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*Source: 120481-2009-02-25.xml*