Jce

Ra

b

c

a

ARRAA

KJJGT

1

tacsupoacttetc

�

0d

Fluid Phase Equilibria 306 (2011) 181–189

Contents lists available at ScienceDirect

Fluid Phase Equilibria

journa l homepage: www.e lsev ier .com/ locate / f lu id

oule–Thomson coefficients and Joule–Thomson inversion curves for pureompounds and binary systems predicted with the group contributionquation of state VTPR

ima Abbasa,b, Christian Ihmelsc, Sabine Endersb, Jürgen Gmehlinga,∗

Lehrstuhl für Technische Chemie, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, GermanyFachgebiet Thermodynamik und Thermische Verfahrenstechnik, Institut für Prozess- und Verfahrenstechnik, Technische Universität, Berlin, GermanyLaboratory for Thermophysical Properties (LTP GmbH), Marie-Curie-Str. 10, 26129 Oldenburg, Germany

r t i c l e i n f o

rticle history:eceived 28 April 2010eceived in revised form 29 March 2011ccepted 31 March 2011vailable online 8 April 2011

a b s t r a c t

In this work the accuracy of the prediction of Joule–Thomson coefficients for the gases CO2 and Arand the binary systems CO2–Ar and CH4–C2H6 was examined using the group contribution equationof state VTPR. Furthermore the experimental and correlated data of Joule–Thomson inversion curvesof a few compounds including carbon dioxide, nitrogen, benzene, toluene, methane, ethane, ethylene,propyne, and SF6 were compared with the results of the group contribution equation of state VTPR,

eywords:oule–Thomson coefficientoule–Thomson inversion curveroup contribution equation of state VTPRhrottling processes

the Soave–Redlich–Kwong (SRK), the Peng–Robinson (PR) and the Helmholtz equation of state (HEOS).Moreover, Joule–Thomson inversion curves for pure fluids, binary (CH4–C2H6, N2–CH4, CO2–CH4), andternary systems (CO2–CH4–N2, CH4–C2H6–N2, CO2–CH4–C2H6) were calculated with VTPR and comparedto the results of SRK, PR, HEOS and the molecular simulation results of Vrabec et al. It was found that thecalculated values for the Joule–Thomson coefficients and Joule–Thomson inversion curves are in goodagreement with the experimental findings.

. Introduction

The reliable knowledge of Joule–Thomson coefficients �JT andhe Joule–Thomson inversion curve are important for the designnd operation of throttling and refrigeration processes, e.g. forryogenic processes (e.g. air separation (Linde process)). One con-equence of the effect may be the cooling of natural gas in pipelinesnder pressure drop, which can lead to undesirable glaciations ofipelines The Joule–Thomson inversion curve is defined as the locusf the points where the Joule–Thomson coefficient becomes zerond separates the region of positive �JT from negative �JT. Thisurve is important in refrigeration and liquefaction processes sincehe sign of the Joule–Thomson coefficient determines whether theemperature of a real gas increases or decreases by isenthalpicxpansion. The Joule–Thomson coefficient is defined as the isen-halpic derivative of temperature T, with respect to pressure P atonstant enthalpy:

JT =(∂T

∂P

)h

(1)

∗ Corresponding author. Tel.: +49 441 798 3831; fax: +49 411 798 3330.E-mail address: gmehling@tech.chem.uni-oldenburg.de (J. Gmehling).URL: http://www.uni-oldenburg.de/tchemie (J. Gmehling).

378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.fluid.2011.03.028

© 2011 Elsevier B.V. All rights reserved.

When �JT is positive, throttling results in a temperaturedecrease. For negative values a temperature increase is observed. Inorder to express Eq. (1) in a more practical form the following pro-cedure may be followed. The total differential for the enthalpy of apure substance (which is a function of temperature and pressure)is defined as follows:

dh =(∂h

∂T

)P

dT +(∂h

∂P

)T

dP (2)

The molar heat capacity at constant pressure cp is defined as theisobaric derivative of the enthalpy, with respect to the temperatureT:

cp =(∂h

∂T

)P

(3)

From Eqs. (2) and (3) the Joule–Thomson coefficient can bedefined as follows:

�JT =(∂T

∂P

)h

=(

−(∂h/∂P)Tcp

)(4)

182 R. Abbas et al. / Fluid Phase Equil

0

40

80

120

160

200

10.80.60.40.20

P [b

ar]

x1, y1

(1) Ar(2) CO2

244.45 K233.32K

Fig. 1. Experimental and calculated vapor–liquid equilibria for Ar (1)–CO2 (2), (—)

VTPR, (—-) PR, (.......) SRK, experimental [34], � [35].

0

80

160

240

320

10.80.60.40.20

P [b

ar]

x1, y1

(1) N2(2) CO2

270 K

220 K

Fig. 2. Experimental and predicted vapor–liquid equilibria for N2 (1)–CO2 (2), (—)

V

tc

�tions was developed by Ahlers and Gmehling [7–11]. The groupcontribution equation of state VTPR was already successfully

TV

TV

TPR, (—-) PR, (.......) SRK, experimental [36], � [37].

For mixtures, the Joule–Thomson coefficient �JT is defined ashe derivative of the temperature T with respect to the pressure atonstant enthalpy h and constant composition x.

JT =(∂T

∂P

)=(−(∂h/∂P)T,x

cp

)(5)

h,x

able 1TPR group interaction parameters taken from DDB.

n/m anm (K) bnm cnm (K−1)

CO2/CH4 88.186 0 0CH2/CO2 941.82 −3.618 0.003429CH4/N2 69.493 0 0CH2/N2 162.63 0.39962 0CH2/CH4 177.41 −1.045 −0.000016

able 2TPR group interaction parameters fitted in this work.

n/m anm (K) bnm cnm (K−

CO2/N2 24.646 0.67499 0.00029CO2/Ar −215.08 1.5072 0

ibria 306 (2011) 181–189

The term (∂h/∂P)T can be defined with the help of thefundamental equation and the Maxwell relations as follows:(∂h

∂P

)T

= v − T(∂v∂T

)P

(6)

Since v = zRT/P, this equation can be also written as:(∂h

∂P

)T

= −RT2

P

(∂z

∂T

)P

(7)

where z is the compressibility factor. Substitution into Eq. (5) gives:

�JT = RT2

cpP

(∂z

∂T

)P

(8)

The curve connecting all state points for which the derivative:(∂z

∂T

)P

= 0 (9)

or the Joule–Thomson coefficient is 0 is called Joule–Thomsoninversion curve:

The reliable measurement of the Joule–Thomson inversioncurve is difficult because of the severe experimental condi-tions involved (up to 5 times critical temperature and 12 timescritical pressure). Therefore the accurate prediction of �JT andJoule–Thomson inversion curves for working fluids is of greatimportance. In various papers the results of cubic equations ofstate (EOS) for the prediction of Joule–Thomson inversion curvesfor pure fluids have been reported [1–3]. Particularly at high tem-peratures poor predictions for the Joule–Thomson inversion curveswere obtained using cubic equations of state. In the paper of Vrabecet al. [4], it was proved that molecular simulation delivers goodresults in predicting Joule–Thomson inversion curves for naturalgas and its pure compounds. Furthermore, Joule–Thomson inver-sion curves were calculated for pure fluids with molecular-basedmodels and presented in the work of Colina [5]. Results obtained bymolecular simulation were compared to the results of molecular-based models and cubic equations of state. Generally, it was shownthat molecular based methods are superior compared to classicalcubic EOS for the prediction of Joule–Thomson inversion curves.

The reliable knowledge of the behavior of supercritical fluidsor solutions is required to develop processes, such as super-critical extraction, the RESS (Rapid Expansion of SupercriticalSolution) technique for the production of amorphous fine pow-ders or thin films or gas-antisolvent crystallization [6]. A powerfulgroup contribution equation of state VTPR for engineering applica-

applied for the reliable prediction of vapor–liquid equilibria, excessenthalpies, azeotropic data, solid–liquid equilibria, gas solubilities.

amn (K) bmn cmn (K−1)

200.73 0 0−6.4843 −0.022 0.00098

4.484 0 051.147 −0.50797 0

−82.636 0.24879 0.0022776

1) amn (K) bmn cmn (K−1)

204 453.53 −1.6352 −0.00002551.13 −1.9153 0

R. Abbas et al. / Fluid Phase Equilibria 306 (2011) 181–189 183

-0.1

0.3

0.7

1.1

600450300150T [K]

P = 60 atmP = 100 atmP = 140 atmP = 200 atm

CO2μ J

T[K

/atm

]

Fig. 3. Experimental [22] and predicted Joule–Thomson coefficients using the groupcontribution equation of state VTPR for CO2.

0.0

0.2

0.4

0.6

0.8

550400250100T [K]

P = 60 atm

P = 100 atmP = 140 atm

P = 200 atm

Ar

μ JT[

K/a

tm]

Fc

Asa

tagntfc

2

botBPfs

P

av

0

0.4

0.8

1.2

1.6

2

450400350300250200

μ JT

[K/a

tm]

T[K]

P = 10 atm

(1) Ar(2) CO2

Fig. 5. Experimental [24] and predicted Joule–Thomson coefficients for Ar (1)–CO2

(2), (—) VTPR, (—-) PR, (.......) SRK, � yCO2 = 0.464, yCO2 = 0.754.

-2

0

2

4

6

8

10

400350300250

((μJT

,ca

lc.-μ

JT,

exp.)/μ

JT,

exp.)%

T [K]

VTPR

SRK

PR

P = 80 atm

ig. 4. Experimental [23] and predicted Joule–Thomson coefficients using the groupontribution equation of state VTPR for Ar.

lso other pure component and mixture properties such as den-ities, enthalpies, heat capacities, entropies, heats of vaporizationnd Joule–Thomson coefficients can directly be predicted.

The group contribution equation of state VTPR is a combina-ion of the volume-translated Peng–Robinson equation of statend the group-contribution method mod. UNIFAC (Do) [12–14]. Inroup-contribution methods it is assumed that the mixture doesot consist of molecules but of functional groups. The great advan-age of the group contribution concept is that the number of theunctional groups is much smaller than the number of possibleompounds.

. The group contribution equation of state VTPR

The Peng–Robinson (PR) equation of state was selected as theasis for the development of a new group contribution equationf state, because liquid densities for pure compounds and mix-ures are better described by the PR than the SRK equation of state.y implementing the volume translation, which is given for theeng–Robinson equation of state by Peneloux et al. [15], the resultsor liquid densities can be further improved. The VTPR equation oftate is given by the following equations:

= RT

(v + c − b)− a(T)

(v + c)(v + c + b) + b(v + c − b)(10)

(T) is the temperature dependent attractive parameter, b is the co-olume, c is the translation parameter, R is the general gas constant,

Fig. 6. Deviations between experimental and predicted Joule–Thomson coefficientsfor (Ar–CO2) at 80 atm and yCO2 = 0.464.

T is the absolute temperature, v is the molar volume and P is thepressure.

To improve the prediction for mixtures, especially asymmetricsystems, a modified gE mixing rule was developed by Chen et al.[16], where only the residual part of the excess Gibbs energy isused.

a

b=∑i

xi ·aiibii

+ gEres

−0.53087Pref = 1 atm (11)

gEres is the residual part of the mod. UNIFAC model. To calculate this

term, the following equation is used:

gEres = RT

∑i

xi ln �resi (12)

in which �resi

and xi are the activity coefficients and the molefractions in the liquid phase for compound i. The residual activitycoefficient is defined by the following equation (solution of groupsconcept):

ln�resi =

∑k

v(i)k

(ln �k −� (i)k

) (13)

with:

ln �k = Qk

⎡⎢⎢⎣1 − ln

(∑m

�m�mk

)−∑m

�m�mk∑�n�nm

⎤⎥⎥⎦ (14)

n

where v(i)k

is the number of subgroups k in compound i, � k and

� (i)k

are the group activity coefficient in the mixture respectively

184 R. Abbas et al. / Fluid Phase Equilibria 306 (2011) 181–189

0

0.5

1

1.5

2

2.5

600500400300200

μ JT

[K/M

Pa]

T[K]

CH4-C2H6

Fa

psg

t

�

X

ttt

phgt

0

4

8

12

6420

P rP r

Tr

Tr

N2

0

4

8

12

16

6420

CO2

Fig. 8. Experimental (+ [30], � [31]), correlated data (� [33]) and molecular sim-ulation together with predicted Joule–Thomson inversion curve (� [4]), (—) VTPR,

TP

TA

ig. 7. Experimental [25] and predicted Joule–Thomson coefficients for (CH4–C2H6)t xCH4 = 0.85, (—) VTPR, (—-) PR, (.......) SRK, � at 15 MPa, at 20 MPa.

ure compound i, Qk is the relative van der Waals surface area ofubgroup k and �nm describes the interactions between the mainroups n and m.

The group area fraction � and the group mole fraction Xm forhe group m are given by the following equations:

m = QmXm∑n

QnXn(15)

m =

∑j

v(i)m xj

∑j

∑n

v(i)n xj

(16)

In order to correctly describe the temperature dependence ofhe residual excess Gibbs energy used in the VTPR mixing rule,emperature-dependent group interaction parameters are used forhe large temperature range covered (often 250–300 K):

nm = exp

(−anm + bnmT + cnmT2

T

)(17)

The required temperature-dependent VTPR group interaction

arameters anm, bnm and cnm are simultaneously fitted to a compre-ensive database of reliable experimental vapor–liquid equilibria,as solubilities, solid–liquid equilibria of simple eutectic sys-ems, excess enthalpies and activity coefficient at infinite dilution

able 3R and SRK binary interaction parameters fitted in this work.

System oij

PR SRK

CO2/N2 −0.0136 −0.028CO2/CH4 0.06453 0.09827CO2/C2H6 0.134 0.14067CH4/N2 0.0389 0.0377CH4/C2H6 0.00445 0.000817N2/C2H6 0.0706 0.149CO2/Ar 0.1181 0.1216

able 4verage absolute relative deviations (AARDs) between experimental and predicted Joule–

Equation P = 10 atmT range (K) 233.15–383.15

P = 40T rang

xCO2 0.464 xCO2 0.754 xCO2 0

VTPR (AARD) % 2.68 3.55 1.38SRK (AARD) % 2.39 4.59 0.49PR (AARD) % 7.46 6.48 7.2

(––··) HEOS, (—-) PR, (.......) SRK.

(�∞). The required temperature-dependent VTPR group inter-action parameters of the compounds investigated are listed inTables 1 and 2.

The pure component parameters aii and bii can be determinedfrom the critical temperature Tc,i and critical pressure Pc,i.

2 2

aii(T) = 0.45724 ·R · T

c,i

Pc,i· ˛i(T) (18)

pij (K−1] qij (K−2]

PR SRK PR SRK

0 0 0 00.0001486 0 0 00 0 0 00 0 0 00 0 0 0

−0.000196 −0.00137 0 0.000003740 0 0 0

Thomson coefficients for Ar–CO2.

atme (K) 273.15–383.15

P = 80 atm T range (K)273.15–383.15

.464 xCO2 0.754 xCO2 0.464 xCO2 0.754

0.79 1.37 2.791.88 1.31 5.222.9 5.74 7.73

R. Abbas et al. / Fluid Phase Equilibria 306 (2011) 181–189 185

Table 5Average absolute relative deviations (AARDs) between experimental and predicted Joule–Thomson coefficients for a mixture of 0.85 CH4–0.15 C2H6.

Equation 0.85 CH4–0.15 C2H6

T range 250–350 (K)

P = 3 MPa P = 5 MPa P = 15 MPa P = 20 MPa

VTPR (AARD) % 1.52 1.48 4.17 5.23SRK (AARD) % 1.2 0.51 3.07 4.85PR (AARD) % 8.71 6.63 4.99 8.97

Table 6Average absolute relative deviations (AARDs) of the predicted Joule–Thomson inversion curves for selected pure substances from molecular simulation data.

Compound VTPR AARD % HEOS AARD % PR AARD % SRK AARD %

CO2 T range (K) 300–1050 9.67 8.75 12.43 18.76N2 T range (K) 125–500 6.23 4.06 18.56 14.83C2H6 T range (K) 275–1000 7.00 4.18 11.66 23.84CH4 T range (K) 178.3–891.3 5.42 4.86 19.3 14.58

0

4

8

12

16

6543210Tr

Tr

CH4

0

3

6

9

12

15

6420

P rP r

C2H6

FtH

˛

wawdt(

˛

I

˛

-75

-50

-25

0

25

950750550350150

((Pi,

calc

.-Pi,

MS)

/Pi,

MS)

%

T [K]

CH4

ig. 9. Experimental (* [32], correlated data �[33]) and molecular simulationogether with predicted Joule–Thomson inversion curve (� [4]), (—) VTPR, (––··)EOS, (—-) PR, (.......) SRK.

The ˛-function proposed by Twu et al. [17] is used:

i(T) = TNi ·(Mi−1)r,i exp[Li · (1 − TNi ·Mir,i )] (19)

here Tr,i is the reduced temperature for compound i, and Ni, Mi,nd Li the Twu–Bluck–Cunningham–Coon ˛-function parameters,hich have been determined by regression of the vapor pressureata of the pure components. If no experimental data are available,he ˛i(T) value can be calculated as function of the acentric factorωi) using the following generalized expressions [9,18]:

(0) (1) (0)

i(T) = ˛ +ωi · (˛ − ˛ ) (20)

f T < Tc,i:

(0) = T−0.1883273r,i

exp[0.1048767(1 − T2.1329765r,i )] (21)

Fig. 10. Deviations between molecular simulation and predicted Joule–Thomsoninversion curve, (—) VTPR, (––··) HEOS, (—-) PR, (.......) SRK.

˛(1) = T−0.6029386r,i

exp[0.5113343(1 − T2.2059312r,i )] (22)

If T > Tc,i:

˛(0) = T−0.792651r,i

exp[0.401219(1 − T−0.992615r,i

)] (23)

˛(1) = T−1.984711r,i

exp[0.024955(1 − T−9.98471r,i

)] (24)

The pure component parameter bii can be calculated from criti-cal data:

bii = 0.0778 · R · Tc,iPc,i

(25)

A quadratic mixing rule is used, to calculate the parameter b forthe mixture:

b =∑i

∑j

xi · xj · bij (26)

where the cross parameter bij is calculated using the followingcombination rule:

b3/4ij

=b3/4ii

+ b3/4jj

2(27)

The volume translation parameter c is defined by the linearmixing-rule:

c =∑

x · c (28)

i

i i

This parameter improves the prediction of the liquid densitiessignificantly. If experimental liquid densities at reduced temper-

186 R. Abbas et al. / Fluid Phase Equilibria 306 (2011) 181–189

0

4

8

12

16

6420

Benzene

0

3

6

9

12

15

543210

P rP r

Tr

Tr

Toluene

F(

ac

c

fp

c

Vomla

c

3

(nasP

o

0

4

8

12

16

6420

P rP r

P r

Tr

Tr

Tr

Ethylene

0

4

8

12

16

6420

Propyne

0

4

8

12

16

6420

SF6

ig. 11. (� [33]) correlated data and predicted Joule–Thomson inversion curve,—) VTPR, (––··) HEOS, (—-) PR, (.......) SRK.

ture Tr,i = 0.7 are available, the translation parameter of the pureompound is defined by the following equation:

i = vcalc,i − vexp,i (Tr,i = 0.7) (29)

When liquid density data are not available, ci can be estimatedrom the critical data and the compressibility factor at the criticaloint of compound i (zc,i) by using the following correlation:

i = −0.252R · Tc,iPc,i

(1.5448zc,i − 0.4024) (30)

Three pure component parameters (Tc, Pc, vc) are needed in theTPR equation, when the general ˛-function is used, as in SRKr PR (Tc, Pc, ω). In addition to the pure parameters, 2–6 binaryixture parameters are used in VTPR to describe the phase equi-

ibrium behavior and the excess properties in a wide temperaturend pressure range (see Tables 1 and 2).

In this work, Eq. (5) is used to determine the Joule–Thomsonoefficient for pure compounds and for mixtures using VTPR.

. Results and discussion

VTPR is a well established group contribution equation of stateGCEOS) for the prediction of phase equilibria and other thermody-amic properties. Figs. 1 and 2 show that the VTPR equation deliversbetter prediction of the vapor–liquid equilibrium behavior of the

ystems Ar–CO and N –CO at different temperatures than SRK or

2 2 2R [19,20].

The required parameters for PR and SRK for predicting the binaryr ternary systems are listed in Table 3. The binary temperature

Fig. 12. Experimental (x [38], correlated data � [33]) and predicted Joule–Thomsoninversion curve, (—) VTPR, (––··) HEOS, (—-) PR, (.......) SRK.

dependent interaction parameter kij for the quadratic mixing ruleof PR and SRK is given by the following equation:

kij = oij + pij.T + qij.T2 (31)

where oij = oji, pij = pji, qii = qjj.The required parameters are fitted simultaneously to all reliable

experimental vapor–liquid equilibria for these systems stored inthe Dortmund Data Bank (DDB) [21].

The Joule–Thomson coefficients for the pure compounds, i.e.CO2, N2 and Ar; and for mixtures CH4 + C2H6 and CO2 + Ar werecalculated using VTPR and compared with the published data.Fig. 3 shows the predicted Joule–Thomson coefficients using VTPRtogether with the experimental data for carbon dioxide at differ-ent pressures and temperatures, which were published by Roebucket al. [22]. It can be seen that a good agreement between the pre-

dicted and experimental data is obtained. In Fig. 4 the estimatedJoule–Thomson coefficients for Ar using VTPR are shown togetherwith the experimental data published in [23] and stored in the Dort-

R. Abbas et al. / Fluid Phase Equilibria 306 (2011) 181–189 187

0

20

40

60

80

100

12008004000

P [M

Pa]

T [K]

CO2-CH4

0

20

40

60

8006004002000

P [M

Pa]

T [K]

N2-CH4

0

20

40

60

80

12008004000

P [M

Pa]

T [K]

CH4-C2H6

F(

mf

da[paitaA

A

w

0

20

40

60

80

12009006003000

P [M

Pa]

T [K]

CO2-CH4-C2H6

0

20

40

60

9006003000

P [M

Pa]

T [K]

N2-CH4-C2H6

0

20

40

60

80

9006003000

P [M

Pa]

T [K]

CO2-CH4-N2

of VTPR, HEOS [26,29], SRK and PR equations of state for nitro-

ig. 13. (� [4]) molecular simulation and predicted Joule–Thomson inversion curve,—) VTPR, (––··) HEOS, (—-) PR, (.......) SRK.

und Data Bank (DDB) [21]. In general, good agreement is observedor the whole temperature and pressure range.

In Fig. 5 calculated Joule–Thomson coefficients for carbonioxide–argon mixtures at different compositions using VTPR, SRKnd PR are shown together with the available experimental values24]. Within the temperature range of 233.15–383.15 K at differentressures and compositions, the deviations between experimentalnd predicted Joule–Thomson coefficients for the Ar–CO2 systemncreases up to 3.77% for the VTPR in comparison up to 5.22% forhe SRK and up to 7.73% for the PR equation of state. The averagebsolute relative deviations (AARDs) are listed in Table 4, whereARD is defined as follows:

ARD = 1n

n∑i=1

∣∣∣∣�JTi,calc−�JTi,exp

�JTi,exp

∣∣∣∣ (32)

It can be seen that the lowest deviations are obtained by VTPR,hereas PR yields significantly higher deviations. Fig. 6 reflects

Fig. 14. (� [4]) molecular simulation and predicted Joule–Thomson inversion curve,(—) VTPR, (––··) HEOS, (—-) PR, (.......) SRK.

these findings for this system at 10 atm, 46.4 mol% CO2 at differenttemperatures.

In Fig. 7 the Joule–Thomson coefficients for CH4–C2H6 (with amethane composition of 85 mol%) predicted by the VTPR equation,SRK and PR are compared with the available data [25]. The AARDbetween the predicted and experimental data is listed in Table 5.

While SRK yields the lowest average absolute relative deviationfor �JT, VTPR delivers good results for this system.

Beside the prediction of�JT for pure compounds and binary mix-tures also the Joule–Thomson inversion curves are calculated withthe VTPR, HEOS [26–29], SRK and PR equation of state for purefluids, binary and ternary systems. Figs. 8 and 9 show a compar-ison of the available experimental data [30–32], NIST data [33] andthe molecular simulation data by Vrabec et al. [4] together withpredicted Joule–Thomson inversion curve calculated with the help

gen, carbon dioxide, methane and ethane. It can be seen that VTPRand HEOS can reproduce the experimental findings over the entiretemperature range. The predicted values from VTPR and HEOS are

188 R. Abbas et al. / Fluid Phase Equilibria 306 (2011) 181–189

Table 7Average absolute relative deviations (AARDs) of the predicted Joule–Thomson inversion curves for selected mixtures from molecular simulation data.

System VTPR AARD % HEOS AARD % PR AARD % SRK AARD %

CH4–C2H6 T range (K) 275–850 6.53 4.95 9.62 10.87CH4–CO2 T range (K) 275–775 4.20 2.61 10.45 19.12CH4–N2 T range (K) 200–550 5.82CH4–C2H6–N2 T range (K) 200–700 5.58CH4–C2H6–CO2 T range (K) 275–850 3.19

-80

-40

0

40

80

600400200

((Pi,

calc

.-Pi,

MS)

/Pi,

MS)

%

T [K]

N2-CH4

-40

-20

0

20

800500200

((Pi,

calc

.-Pi,

MS)

/Pi,

MS)

%

T [K]

CO2-CH4-C2H6

Fi

iseivstctJbts

samedm

sion curves.

ig. 15. Deviations between molecular simulation and predicted Joule–Thomsonnversion curves, (—) VTPR, (––··) HEOS, (—-) PR, (.......) SRK.

n good agreement with the available experimental and molecularimulation data. With the aim of evaluating the ability of the VTPRquation for prediction Joule–Thomson inversion curves, a compar-son between the molecular simulation data [4] and the predictedalues by VTPR, HEOS, SRK and PR are reported in Table 6 whichhows the average absolute relative deviations in the tempera-ure range covered by the molecular simulation data for nitrogen,arbon dioxide, methane and ethane. It can be recognized thathe VTPR results are similar to the results of HEOS in predictingoule–Thomson inversion curves with acceptable AARD and muchetter results than SRK and PR. For SRK in most cases high nega-ive deviations particularly at high temperatures are obtained, ashown in Fig. 10, for methane.

Figs. 11 and 12 show a comparison of the Joule–Thomson inver-ion curves for two aromatics compounds (benzene and toluene)nd for three other compounds (ethylene, propyne and SF6) esti-ated with VTPR, HEOS, SRK and PR with the available correlated

xperimental data [33]. Excellent agreement between the pre-icted and experimental data is achieved for all investigatedodels for benzene and toluene at low temperatures. At higher

5.10 19.22 18.066.68 14.51 21.784.94 8.03 8.61

temperature the models provide different results. Unfortunatelyexperimental data at high temperatures are not available.

In Figs. 13 and 14 comparisons between the predicted JT inver-sion curve using VTPR, HEOS, SRK and PR for six equimolar binaryand ternary systems are shown together with the molecular simu-lation data [4]. All studied equations deliver the correct shape at lowtemperatures; however significant deviations are again obtained athigh temperature. In Table 7 the mean absolute relative deviationsAARD are reported for the calculated Joule–Thomson inversioncurves for these mixtures.

AARD = 1n

n∑i=1

∣∣∣∣Pi,calc − Pi,MS

Pi,MS

∣∣∣∣ (33)

The results show that the lowest deviations are obtained forJoule–Thomson inversion curves predicted with the help of VTPRcompared to the other equations of state (HEOS, SRK, PR). AARDvalues with respect to the molecular simulation data (see Table 7)appear to suggest that the temperature-dependent group inter-action parameters for VTPR have a relevant influence on theprediction results of the JT inversion curves and yield better resultsthan HEOS. The deviation plots are presented in Fig. 15 for the tem-perature range covered by molecular simulation data. VTPR andHEOS agree quite well with these data, whereas SRK show poorresults with negative deviation at the high temperature branch ofthe Joule–Thomson inversion curve. The improved results for VTPRcompared to SRK and PR are mainly caused by the fact, that animproved ˛-function and at the time improved mixing rules forthe parameters a and b are used in VTPR.

4. Conclusion

Besides the excellent results of the VTPR group contributionequation of state for the prediction of vapor–liquid equilibria,excess enthalpies, azeotropic data, activity coefficients at infi-nite dilution, solid–liquid equilibria, gas solubilities, densities,enthalpies of vaporization, etc., the group contribution equation ofstate VTPR was examined in comparison to SRK and PR for the calcu-lation of Joule–Thomson coefficients of pure compounds CO2, N2, Arand the binary mixtures of CO2–Ar and CH4–C2H6. Good agreementis achieved between the predicted and the available experimentaldata using VTPR not only for the pure compounds but also for themixtures in the whole temperature and pressure range. Further-more the calculated results of VTPR for Joule–Thomson inversioncurves of CO2, N2, ethane, methane, benzene, toluene, ethylene,propyne, and SF6 are compared with available experimental andcorrelated data, and compared with the results of HEOS, SRK and PR.In all cases good agreement between the correlated experimentaland the calculated Joule–Thomson inversion curves equation wasfound using VTPR. Moreover, the prediction of binary and ternarysystems using VTPR, HEOS, SRK and PR is compared to molecularsimulation data. The comparison shows that VTPR is competitivewith most equations of state for predicting Joule–Thomson inver-

List of symbolsVTPR volume translated Peng–Robinson equation of state

Equil

PSHR�DAa

b

c

hxyTPvca

o

LgRXZQ

G��˛

v��

SimrrcePhTxkM

SE∞

[[[

[[[[

[

[

[[

[[[[[

[[[[[[[

[

[

[35] D. Koepke, R. Eggers, Chem. Eng. Technol. 79 (2007) 1235–1239.[36] T.A. Al-Sahhaf, A.J. Kidnay, E.D. Sloan, Ind. Eng. Chem. Fundam. 22 (1983)

372–380.[37] F.A. Somait, A.J. Kidnay, J. Chem. Eng. Data 23 (1978) 301–305.[38] S.R. de Groot, M. Geldermans, Physica (The Hague) 13 (1947) 538–542.

R. Abbas et al. / Fluid Phase

R Peng–Robinson equation of stateRK Soave–Redlich–Kwong equation of stateEOS Helmholtz equation of stateESS Rapid Expansion of Supercritical SolutionJT Joule–Thomson coefficientDB Dortmund Data BankARD average absolute relative deviation

attractive parameter of Peng–Robinson equation of stateand VTPR-equationvolumetric parameter of Peng–Robinson equation of stateand VTPR-equationtemperature-independent volume correction for theVTPR equationmolar enthalpyliquid phase mole fractionvapor phase mole fractionabsolute temperaturepressuremolar volume

p isobaric heat capacitynm, bnm, cnm temperature-dependent group interaction parame-

ters, p, q temperature dependent interaction parameters of PR and

SRK, M, N Twu–Bluck–Cunningham–Coon ˛-function parameters

Gibbs energy from mod. UNIFAC methodgeneral gas constantgroup mole fractioncompressibility factorrelative van der Waals surface area of a subgroup

reek lettersactivity coefficienttemperature-dependent function in the residual parttemperature-dependent function of the attractive param-eternumber of subgroupsgroup activity coefficientgroup area fraction

ubscripts, j components i and j

, n subgroup m, nreduced

es residual partalc calculatedxp experimental

isobaricisenthalpicisothermalat constant mole fractionsubgroup

S molecular simulation

uperscriptsexcessat infinite dilution

ibria 306 (2011) 181–189 189

Acknowledgments

The authors thank the Arbeitsgemeinschaft industriellerForschungsvereinigungen (AIF: 13885N) for the financial supportof this work. R. Abbas thanks the AL Baath University in Syria forthe fellowship.

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