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i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 1 9 1 4e1 9 2 0

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Thermodynamic modeling of hydrate dissociation conditionsfor refrigerants R-134a, R-141b and R-152a

Fatemeh Nikbakht a, Amir A. Izadpanah a,**, Farshad Varaminian b,Amir H. Mohammadi c,*aDepartment of Chemical Engineering, Engineering Faculty, Persian Gulf University, Bushehr, IranbDepartment of Chemical Engineering, Oil and Gas, Semnan University, Semnan, IrancMINES ParisTech, CEP/TEP e Center Energetetique et Procedes, 35 Rue Saint Honore, 77305 Fontainebleau, France

a r t i c l e i n f o

Article history:

Received 8 March 2012

Received in revised form

25 June 2012

Accepted 5 July 2012

Available online 23 July 2012

Keywords:

Hydrate

Dissociation conditions

Refrigerant

Kihara potential function

Implicit least squares

* Corresponding author. Tel.: þ33 1 64 69 49** Corresponding author. Tel.: þ98 771 422216

E-mail addresses: Izadpanah@pgu.ac.ir (A0140-7007/$ e see front matter ª 2012 Elsevhttp://dx.doi.org/10.1016/j.ijrefrig.2012.07.004

a b s t r a c t

In this communication, a general model is presented for estimating hydrate dissociation

conditions for refrigerants R-134a, R-141b and R-152a which employs the cubic plus associ-

ation equation of state (CPA EoS) formodeling thefluid phases and vanderWaalsePlatteeuw

statistical model for the hydrate phase. The Kihara potential parameters for the latter

refrigerants are estimated by employing an implicit optimization scheme using minimiza-

tion of the chemical potential difference of water in the hydrate and in the liquid phase. The

minimizationwasperformedby applying genetic algorithm.Using thismodel, the agreement

between the experimental data and the model results is found acceptable.

ª 2012 Elsevier Ltd and IIR. All rights reserved.

Modelisation thermodynamique des conditions danslesquelles la dissociation des hydrates a lieu dans les casdes frigorigenes R-134a, R-141-b et R-152a

Mots cles : Hydrate ; Conditions de dissociation ; Frigorigene ; Fonction potentielle de Kihara ; Moindres carres implicites

70; fax: þ33 1 64 69 49 68.9..A. Izadpanah), amir-hossein.mohammadi@mines-paristech.fr (A.H. Mohammadi).ier Ltd and IIR. All rights reserved.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 1 9 1 4e1 9 2 0 1915

1. Introduction

Gas hydrates (or clathrate hydrates) are ice-like crystalline

compounds in which small molecules are entrapped in a cage

of water molecules under high pressure and low temperature

conditions (Sloan, 1990). Light hydrocarbons such as natural

gas components and some refrigerants can form hydrate

when come into contact with water under certain conditions

of temperature and pressure. Hydrates consist of three main

structures namely structure I (sI), sII and sH, respectively

(Sloan, 1990).

Gas hydrate appears like ice, but they may form at

temperatures well above the ice epoint. Since gas hydrate

dissociation is dramatically endothermic, it could be used in

cool storage and air conditioning applications (Tomlinson

et al., 1984; Tomlinson, 1982; Knebel, 1995; Tran et al., 1989;

Denkmann, 1985; Hensel et al., 1991). Refrigerant gas hydrates

are used as cooling storage systems, which have an appro-

priate melting temperature (277.15e280.15 K), a high specific

storage capacity (302.4e464 kJ kg�1) and a high heat transfer

coefficient (Douglas, 1990; Ternes, 1983). Watanabe et al.

(2004) and Ogawa et al. (2005) developed the refrigeration

technology based on hydrate formation and dissociation.

For cooling storage applications, the dissociation condi-

tions of refrigerant gas hydrates as well as suitable thermo-

dynamic models should be available and be employed. Liang

et al. (2001) published sets of equilibrium data for the

hydrate dissociation relevant to some refrigerants, such as

R-134a, R-141b and R-152a. Imai et al. (2005) reported hydrate

equilibrium conditions for Difluoromethane in the presence of

Cyclopentane or Tetra-n-butylammonium Bromide for usage

in the hydrate-based refrigeration technology. Li et al. (2006)

reported experimental data for the hydrate dissociation

conditions related to R-134a in the presence of copper

suspension nanoparticles and pure water. Because HCC,

HCFC, HFC and FC chemicals are known to have greenhouse

effects, these gases are target gases for emission reductions in

the Kyoto protocol. Therefore, some researchers attempted to

use CO2 hydrate for the phase change material for the

secondary refrigeration (Fournaison et al., 2004; Marinhas

et al., 2006). Because the dissociation enthalpy of sII hydrate

is much larger than sI hydrate and since the formation pres-

sure of sII is less than sI, Delahaye et al. (2006) used the sII

hydrate formed by CO2 and THF for the secondary refrigera-

tion system.

In the thermodynamic modeling of the refrigerant gas

hydrates, Liang et al. (2001) obtained Kihara potential

parameters for these refrigerants (R-134a, R-141b and R-152a)

based on Holder et al.’s (1980) model. Recently, Eslamimanesh

et al. (2011) developed a thermodynamic model for prediction

of the hydrate dissociation conditions of some refrigerants.

The cubic equations of state (i.e., SRK, PR) often could not

provide the accurately calculated volumetric property and the

phase equilibrium data of complex mixtures containing

hydrogen bonding compounds. This is because of the strong

hydrogen bonding forces, which cannot be well considered by

the corresponding terms of such EoS, especially when van der

Waals fluid mixing rules are used. By using the statistical

association fluid theory (SAFT) and some association EoS, this

problem can be solved. The Cubic-Plus-Association (CPA)

model is one of these applicable EoS (Kontogeorgis et al., 1996).

CPA can predict the properties of pure compounds, as well as

the mixtures of associating ones, such as water, alcohols,

glycols andmixtures of themwith hydrocarbons (Kontogeorgis

et al., 2006a,b; Folas et al., 2005). Also, CPA EoSwas successfully

used for the prediction of hydrate dissociation conditions in

absence of any aqueous phase (Youssef et al., 2009, 2010) and in

the presence of thermodynamic inhibitors such as methanol

and glycol (Haghighi et al., 2009a,b).

In their original work, van der Waals and Platteeuw (1959)

used the Lennard-Jones 6-12 pair potential. They applied the

Lennard-Jones potential more to monoatomic or spherical

molecules than to oblate or polar molecules. The inaccurate

prediction of this model leads to the fact that McKoy and

Sinanoglu (1963) suggested the Kihara core potential was

better for both large and non-spherical molecules. The Kihara

potential function is normally used,with the parameters fitted

to experimental hydrate dissociation data.

In this work, by using the cubic plus association equation

of state and van der Waals ePlatteeuw theory, hydrate

dissociation conditions for refrigerant R-134a, R-141b and

R-152a is modeled. Based on reference parameters for sI and

sII, reported by Sloan (1990), Kihara potential Parameters for

these materials are determined by an implicit least square

optimization scheme. In this method, the difference between

chemical potential of water in the hydrate and aqueous pha-

ses are calculated based on experimental hydrate dissociation

conditions and then minimized by using a genetic algorithm.

2. Thermodynamic modeling

2.1. Cubic e plus eassociation EoS (CPA)

The cubic-plus-association model is an equation of state

which combines the SRK (Soave-Redlich-Kwong) (Soave, 1972)

cubic equation of state and association term derived from

Wertheim theory (Wertheim, 1984a,b, 1986a,b; Huang and

Radosz, 1990) as applied in SAFT (Huang and Radosz, 1990).

The SRK model considers physical interaction between the

molecules and the association term takes into account the

specific siteesite interaction stem from hydrogen bonding

between similar molecules (self-association) and different

ones (cross-association).

The CPA EoS can be expressed in terms of pressure as

a summation of the SRK EoS and the contribution of associa-

tion term, as suggested by Michelsen and Hendriks (2001):

P ¼ R Ty� b

� ayðyþ bÞ �

R T2y

�1� y

vlngvy

�Xnci¼1

xi

XAi

�1� XAi

�(1)

where y is the molar volume, XAi is the fraction of A-sites on

molecule i that do not form bond with other active sites.

XAi, which is the key property in the association term

(Huang and Radosz, 1990), must satisfy Eq. (2)

XAi¼ 1

1þ 1y

Xncj¼1

xj

XBj

XBjDAiBj

(2)

Fig. 1 e Association schemes for water (Huang and Radosz,

1990).

Table 2 e Critical properties used in this work (Clarke andBishnoi, 2003).

Component u Tc(K) Pc (MPa)

Water 0.3440 647.13 22.055

R-134a 0.3256 374.20 4.055

R-141b 0.2055 482.73 4.386

R-152a 0.2557 386.89 4.444

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 1 9 1 4e1 9 2 01916

where Bj indicates the summation over all connectable

sites.

DAiBj , the association strength between site A onmolecule i

and site B on molecule j is given by Eq. (3):

DAiBj ¼ g�yrefm

��exp

�εAiBj

R T

�� 1

�bijb

AiBj (3)

εAiBjand bAiBj are the association energy and volume of inter-

action between site A on molecule i and site B on molecule j,

respectively. Also, g(ym)ref is the radial distribution function for

the reference fluid.

The expression for the radial distribution function has

been proposed by Kontogeorgis et al. (1999).

g�yrefm

� ¼ 11� 1:9h

; h ¼ b4ym

(4)

where h is the reduced fluid density.

The CPA EoS for pure associating fluid has five parameters.

Three of them (a0, b, c1) are related to SRK EoS and two of them

(εAiBj , bAiBj ) are related to association term. These five parame-

ters are normally determined by fitting experimental vapor

pressure and saturated liquiddensity data. For non-association

components the CPA EoS reduces to SRK equation of state.

The extension of the CPA EoS to mixtures, requires mixing

rules only for the parameters of the SRK part (Kontogeorgis

et al., 1999):

a ¼Xi

Xj

xixj

ffiffiffiffiffiffiffiffiaiaj

p �1� kij

�(5)

b ¼Xi

Xj

xixjbij (6)

bij ¼bi þ bj

2(7)

The binary interaction parameter (kij) is normally determined

using the solubility data of gases in liquids. Because the water

solubility data regarding the refrigerants used this work are

scarce, the zero value for (kij) is assumed.

2.2. Association schemes

As seen in Eq. (2), the association term of CPA depends on the

choice of association scheme, i.e., the number and the type of

Table 1 e CPA parameters for water (Kontogeorgis et al., 2006a

Component a0 (bar l2 mol-2) b (l mol�1)

Water 1.2277 0.014515

association sites for the associating compounds. Huang and

Radosz (1990) classified eight different association schemes.

The four-site 4C, association scheme is normally used forwater

in CPA EoS (Kontogeorgis et al., 1999). The bonding symmetry

means that all non-bonded site fractions are equal (Fig. 1).

The CPA parameters for water and critical properties used

in this work are shown in Tables 1 and 2, respectively.

2.3. Van der WaalsePlatteeuw model

In this model, the fugacity of water in the hydrate phase is

given by (Sloan, 1990):

fHw ¼ f bwexp

�� Dmb�H

w

R T

�(8)

where

Dmb�Hw ¼ mb

w � mHw ¼ R T

XNcavity

m¼1

nmln

0@1þ

XNH

j¼1

Cjmfj

1A (9)

In the above equation,NH is the number of hydrate formers,

Ncavity is the number of cavity types in the hydrate lattice, and

fj is the fugacity of the guest component in the vapor or liquid

phase, which is calculated by CPA equation of sate. The

quantity Dmb�Hw ¼ mb

w � mHw is the chemical potential difference

between the empty hydrate and the filled hydrate phase.

Van der Waals and Platteeuw (1959) used the Lenard-Jones

Devonshire theory (1938) and showed that the Langmuir

constant is given by (Sloan, 1990):

C ¼ 4pk T

ZN0

exp

�� uðrÞ

k T

�r2dr (10)

where u(r) is the spherically symmetric potential and r is the

radial distance from the center of the cavity.

The Kihara pair-potential functionmay be used to describe

the potential energy between guest and host molecules. This

potential is given as a function of separation distance by

(Sloan, 1990):

GðrÞ ¼8<:4ε

��s�

r� 2a

�12

��

s�

r� 2a

�6�r > 2a

N r < 2a(11)

An overall cell potential u(r) can be represented as:

) used in this work.

C1 εAB (bar l mol-1) b

0.67359 166.55 0.0692

Table 4 e The optimum values of the Kihara potentialparameters by using the implicit least squareoptimization scheme.

Component Hydratestructure(in largecavity)

ε

kðKÞ s � 10�10

(m)a � 10�10

(m)%AAD*in P

R-134a sII 235.19 2.7225 1.3113 1.6

R-141b sII 237.97 3.1721 0.8701 0.8

R-152a sI 202.95 2.9630 0.9135 1.8

* Average absolute deviation.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 1 9 1 4e1 9 2 0 1917

uðrÞ ¼ 2 Zε

�s12

R11r

d10 þ a

Rd11

� s6

R5r

d4 þ a

Rd5�

(12)

where

dN ¼ 1N

��1� r

R� a

R

��N

��1þ r

R� a

R

��N�(13)

whereN is equal to 4, 5, 10, or 11, Z is the coordination number

of the cavity, that is, the number of oxygen atoms at the

periphery of each cavity, and R is the radius of the cavity.

The fugacity of water in the empty hydrate lattice, f bw, can

be calculated with an equation similar to Eq. (8) from the

difference between the chemical potential of water in the

empty hydrate lattice and ice or pure liquid water. If

temperature is below the ice point, 273.15 K, then (Sloan,

1990):

f bw ¼ f Iwexp

�Dmb�I

w

R T

�(14)

If the temperature is above the ice point, then:

f bw ¼ f Lwexp

�Dmb�L

w

R T

�(15)

where Dmb�Iw ¼ mb

w � mIw, andDmb�L

w ¼ mbw � mL

w. The chemical-

potential differences are calculated from the equations given

by Holder et al. (1980):

Dmb�Iw

R T¼ Dm0

w

R T0�ZTT0

Dhb�Iw

R T2 dTþZPP0

Dyb�Iw

R TdP (16)

Dmb�Lw

R T¼ Dm0

w

R T0�ZTT0

Dhb�Lw

R T2 dTþZPP0

Dyb�Lw

R TdP� lnðawÞ (17)

At a temperature other than T0, the enthalpy difference is

evaluated by (Holder et al., 1980):

Dhb�I=Lw ¼ Dh0

w þZTT0

DcpdT (18)

Dcp ¼ Dc0p þ a0ðT� T0Þ (19)

The values for the parameters needed to calculate the

chemical potential of water in the hydrate phase are given in

Table 3. The definition of all parameters and variables of the

above equations is given in Nomenclature.

Table 3 e Thermodynamic reference properties for gashydrates (Sloan, 1990).

Structure I Structure II

Dm0wðliquidÞ ðJ=molÞ 1263.6 883.82

Dh0wðliquidÞ ðJ=molÞ �4623.27 �4987.27

Dy0wðliquidÞ ðcm3=molÞ 4.6 5.0

Dc0pðJ=mol:KÞ 38.12 38.12

a0 �0.141 �0.141

3. Results and discussion

In this work, the implicit least square minimization method

presented by Clark and Bishnoi (2003) is utilized to obtain

Kihara potential parameters for refrigerants R134a, R141b, and

R152a. One of the major advantages of this method is that

there is no further need to the reputational and time

consuming phase equilibrium calculations to estimate the

dissociation pressure of hydrates in a specific temperature. In

the presented method, a proportional objective function is

suggested to minimize the deviations, which are considered

as the differences between calculated fugacities or chemical

potential of components in separated phases. For example,

the following objective function (O.F.) is considered to obtain

the Kihara potential parameters of methane and ethane by

Clarke and Bishnoi (2003):

O:F: ¼XNi¼1

lnfHw;i � lnf Lw;i

2(20)

In this work, this equivalent function is used

O:F: ¼XNi¼1

DmMT�H

w;i � DmMT�Lw;i

RTi

!2

(21)

where N is the number of experimental data points. In Eq. (21)

the chemical potential difference is evaluated at each experimental

pressure and temperature. This objective function should be

minimized for calculating the optimized parameters. In this

Fig. 2 e Comparison between experimental data (Liang

et al., 2001) and the results in this work for hydrate

dissociation condition of R-134a.

50

150

250

350

450

272 276 280 284 288

P(kPa)

T(K)

Exp data

Calc data

Fig. 4 e Comparison between experimental data (Liang

et al., 2001) and the results in this work for hydrate

dissociation condition of R-152a.

0

10

20

30

40

272 274 276 278 280 282

P(kPa)

T(K)

Exp data

Calc data

Fig. 3 e Comparison between experimental data (Liang

et al., 2001) and the results in this work for hydrate

dissociation condition of R-141b.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 1 9 1 4e1 9 2 01918

work, a genetic algorithm is used for minimizing the objective

function, as mentioned earlier. By using FORTRAN program

and written code by Carroll (2003) for genetic algorithm, the

reasonable optimized Kihara potential parameters are ob-

tained. The results are reported in Table 4.

Also, Figs. 2e4 show the calculated hydrate dissociation

conditions using these parameters for R-134a, R-141b and

R-152a, respectively. As depicted in these figures, good

agreement is observed between calculated and experimental

data. It should be mentioned that the results with the opti-

mized parameters show that R-134a and R-141b occupy large

cavities of sII while R-152a occupies large cavities of sI.

4. Conclusion

In this work, the CPA EoS has been applied for modeling

aqueous and vapor phases, van der Waals ePlatteeuw model

has been employed for hydrate phase, and reference values

reported by Sloan (1990) as well as experimental data for

hydrate dissociation conditions above 273 K relevant to

refrigerants R-134a, R141b and R-152a have been considered in

order to simulate the hydrate dissociation conditions for the

latter refrigerants. This aim has been achieved by employing

an implicit optimization scheme, which minimizes the differ-

ence between chemical potential of water in the hydrate and in the

aqueous phases that calculated from experimental hydrate

dissociation data. The minimization has been performed by

genetic algorithm. Reasonable results have been obtained. It is

worth pointing out that the developed model is expected to

predict phase behavior of a blendof refrigerants although there

may be no or very limited corresponding experimental data.

Acknowledgment

The authors gratefully acknowledge the Vice Chancellor of

Research and Technology of the Persian Gulf University for

financial support (fund Number PGU/FE/12-1/1390/937).

Nomenclature

a energy term in the SRK EoS (bar L2 mol�2)

Ai site A in molecule i

b coevolume parameter (L mol�1)

Bj site B in molecule j

g radial distribution function

P pressure (bar or kPa)

R gas constant (bar Lmol�1 K�1)

R radius of cavity

T temperature (K)

TC critical temperature (K)

Tr reduced temperature

Vm molar volume (Lmol�1)

XAi fraction of A-sites of molecule i that are not

bonded

xi liquid mole fraction of component i

yi vapor mole fraction of component i

CP specific heat capacity coefficient

C Langmuir constant

Z coordination number of the cavity

O.F. objective function

k Boltzmann’s constant

fHw fugacity of water in the hydrate phase

fbw fugacity of water in the hypothetical empty

hydrate lattice

mbw chemical potential of water molecules in the

hypothetical hydrate phase

mHw chemical potential of watermolecules in the hydrate

phase

nm number of type m cavities per water molecule

Cjm Langmuir constant of component j in type i cavity

fj fugacity of a component j of guest molecule in the

hydrate phase

f I=Lw fugacity of water in the ice or liquid water phase

mbw chemical potential of water in the empty hydrate

phase

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 5 ( 2 0 1 2 ) 1 9 1 4e1 9 2 0 1919

mI=Lw chemical potential of liquid water (or ice)

T0 ice point of water (in absolute temperature)

Dm0w the chemical potential difference between the

empty hydrate phase and ice (or liquid water) at

the reference condition of T0 ¼ 273.15 K and zero

pressure.

Dhb�ðI=LÞw enthalpy differences between the empty hydrate

lattice and ice (or liquid water)

Dyb�ðI=LÞw volume enthalpy differences between the empty

hydrate lattice and ice (or liquid water)

Greek letters

b association volume parameter

D association strength

εAiBj association energy parameter (bar Lmol�1)

u acentric factor

r molar density (mol L�1)

s collision diameter

a radius of spherical molecular core

ε characteristic energy

u(r) spherically symmetric cell potential

h reduced fluid density.

r e f e r e n c e s

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