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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
Available o
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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: [email protected] (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), [email protected] (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.
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