Fe

BC

a

ARRA

KvVGP

1

ac

A

B

AcTclc

f

wf

0d

J. of Supercritical Fluids 55 (2010) 438–447

Contents lists available at ScienceDirect

The Journal of Supercritical Fluids

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

rom van der Waals to VTPR: The systematic improvement of the van der Waalsquation of state

astian Schmid, Jürgen Gmehling ∗

arl von Ossietzky Universität Oldenburg, Technische Chemie, D-26111 Oldenburg, Germany

r t i c l e i n f o

rticle history:eceived 30 August 2010eceived in revised form 13 October 2010ccepted 13 October 2010

eywords:an der Waals equation of stateTPRroup contribution equation of state

a b s t r a c t

Today the synthesis, design and optimization of the different processes in the oil and gas processing,pharmaceutical, food and environmental industry is carried out by solving the balance equations ofa mathematical model of the unit operation or the whole plant with the help of process simulators.The accuracy of the simulation mainly depends on the quality of the thermophysical properties used.Besides kinetic data for the chemical reaction, heat and mass transfer, pure component properties and inparticular a reliable knowledge of the phase equilibrium behavior of the system is required. Therefore,the development of powerful thermodynamic models was a very important step in the right direction.For the description of the required distribution coefficients or separation factors, reliable thermodynamic

E

rocess design models are required. For a long time, the application of g -models as well as group contribution methods

was most popular. However, equations of state show a lot of advantages compared to gE-models. Besidesthe phase equilibrium of subcritical compounds, they can handle supercritical compounds. At the sametime other important properties such as densities, heat capacities, and enthalpies of the pure compoundsand their mixtures are obtained. Starting from the cubic van der Waals equation of state, after importantimprovements today the volume translated Peng–Robinson group contribution equation can be applied

evelo

successfully for process d

. Introduction

Starting from the isofugacity condition, two differentpproaches can be applied. For vapor–liquid equilibria, onean start from the following approaches:

: xi�ifoi = yiϕVi P (1.1)

: xiϕLi = yiϕVi (1.2)

pproach A, also known as �–ϕ approach, is used for gE- or groupontribution models, such as UNIFAC [1] or modified UNIFAC [2].he reality of the liquid phase is taken into account using activityoefficients. As standard fugacity f o

i, usually the fugacity of the pure

iquid at system temperature and pressure is chosen, which can bealculated following Eq. (1.3)

oi (T, Psi ) = ϕLi Psi = ϕVi Psi ≡ ϕsi Psi (1.3)

here the fugacity coefficients ϕLi

resp. ϕVi

are replaced by theugacity coefficientϕs

iin the saturation state. At system pressure the

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

896-8446/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.supflu.2010.10.018

pment and design.© 2010 Elsevier B.V. All rights reserved.

influence of the pressure difference P − Psi

on the standard fugacitycan be taken into account with the help of the Poynting factor Poyi.

f oi (T, P) = ϕsi Psi Poyi (1.4)

The combination of Eqs. (1.1) and (1.4) lead to

xi�iϕsi Psi Poyi = yiϕVi P (1.5)

In the case of moderate pressures and only slightly associating com-pounds, the following approximation can be made:

ϕsi Poyi ≈ ϕVi (1.6)

which lead to a simplified relation (Eq. (1.7)).

xi�iPsi ≈ yiP (1.7)

The ϕ–ϕ approach B, uses fugacity coefficients to account for thereal behavior in both the liquid and the vapor phase. For the cal-culation an equation of state with reliable mixing rules has to beused to describe the pressure–volume–temperature behavior as afunction of composition of the liquid and the vapor phase.

In 1873 Johannes Diderik van der Waals presented the first cubicequation of state in his doctoral thesis. With his cubic equation ofstate for the first time effects like evaporation and condensation,the two phase region and critical phenomena could be describedwith only two parameters a and b. The application of equations of

percritical Fluids 55 (2010) 438–447 439

spte

tP

2c

a

P

Ttidpo

vTdwwf

obmd

2

P

Tp

B. Schmid, J. Gmehling / J. of Su

tate has a lot of advantages compared to gE-models. Besides theossibility to describe supercritical systems, at the same time otherhermophysical properties for process design, such as densities,nthalpies, and molar heat capacities, can be obtained.

The systematic improvements of the cubic van der Waals equa-ion of state led finally to VTPR, the universal volume translatedeng–Robinson group contribution equation of state.

. From the van der Waals equation of state to the groupontribution equation of state VTPR

The cubic van der Waals equation of state (Eq. (2.1)) consists ofrepulsive and an attractive part [3].

= R · Tv − b − a

v2(2.1)

he parameter a takes into account the attractive forces betweenhe different molecules and parameter b describes the closest pack-ng volume. Both parameters can directly be obtained from criticalata, since at the critical point the critical isotherm shows a saddleoint. This can be seen from Fig. 1, which shows the Pv – behaviorf a pure component.

The disadvantages of the van der Waals equation of state are theery poor results for the liquid densities and the vapor pressures.his can be seen from Fig. 2. In Fig. 2 the calculated liquid and vaporensities of methanol in the saturation state are shown togetherith the densities obtained by other equations of state togetherith the experimental data [4] for a reduced temperature range

rom 0.5 to 1.To improve the results, a large number of modified equations

f state were published. An important improvement was achievedy the equation of state published by Redlich and Kwong [5]. Theyodified the attractive term and introduced the first temperature

ependency of the attractive term.

.1. The Redlich–Kwong equation of state

= R · Tv − b − a

T0.5v(v + b)(2.2)

he Redlich–Kwong equation delivers improved results for vaporressures but still poor results for liquid densities.

Fig. 1. Pv-diagram of a pure compound.

Fig. 2. Experimental (�) and predicted densities for methanol in the saturation stateusing VTPR (—), Peng–Robinson ( ), Redlich–Kwong (—) and van der Waals (. . .).

2.2. The Soave–Redlich–Kwong equation of state

The introduction of a temperature dependent attractive param-eter a(T) realized in the Soave–Redlich–Kwong equation of state(Eq. (2.3)) was an important step in the right direction [6].

P = R · Tv − b − a(T)

v(v + b)(2.3)

Great improvement in the description of the vapor pressureswere obtained by Soave [6] who introduced a temperature depen-dency of the attractive parameter a with the help of a ˛-function,where the acentric factor ω is used as additional parameter. Onecan distinguish between generalized ˛-functions Eq. (2.4) and ˛-functions such as the Mathias–Copeman or Twu ˛-function, wherethe required parameters are fitted directly to experimental purecomponent vapor pressure data.

˛ii(T) = [1 + (0.48 + 1.574ωi − 0.176ω2i )(1 − T0.5

r,i )]2

(2.4)

Already with the introduction of the generalized˛-function, a gooddescription of the vapor pressures especially of hydrocarbons isobtained. But the description of the densities was only slightlyimproved compared to the Redlich–Kwong equation of state.

2.3. The Peng–Robinson equation of state

In 1976 Peng and Robinson suggested a slightly different attrac-tive part (Eq. (2.5)) with a modified generalized ˛-function (Eq.(2.6)) which leads to slightly improved results for liquid densities[7].

P = R · Tv − b − a(T)

v(v + b) + b(v − b)(2.5)

˛ii(T) = [1 + (0.37464 + 1.54226ωi − 0.26992ω2i )(1 − T0.5

r,i )]2

(2.6)

2.4. Mixing rules

For a long time the application of cubic equations of state waslimited to non-polar or slightly polar systems. This limitation was

mainly caused by the classical quadratic mixing rule used for theattractive parameter a (Eq. (2.7)) [8].

a =∑∑

xixjaij (2.7)

440 B. Schmid, J. Gmehling / J. of Supercrit

F3

wri

a

Har

TeSmts

ewrs[G

F3

ig. 3. Experimental and calculated VLE data for the system acetone(1)–water(2) at33.15 K – SRK + quadratic mixing rule.

here the binary parameter aij is calculated using a combinationule (Eq. (2.8)) with a binary parameter kij which is fitted to exper-mental data [9].

ij =√aiiajj(1 − kij) (2.8)

uron and Vidal [10] tried to combine the advantages of gE-modelsnd equations of state by the development of a so-called gE-mixingule (Eq. (2.9)) at a reference pressure Pref = ∞:

a

bRT=∑i

xiaiibiRT

+ gE∞/RT−0.6931

(2.9)

he improvements obtained in the description of vapor–liquidquilibria are exemplary shown in Figs. 3 and 4. While for theoave–Redlich–Kwong equation in combination with the classicixing rule very poor results are obtained for the polar system ace-

one (1)–water (2), the vapor–liquid equilibrium behavior of thisystem can be described reliably by using the gE-mixing rule.

The use of already published, directly fitted interaction param-ters of the implemented gE-models within the gE-mixing rules

ould save the time-consuming refitting procedure. But then the

eference pressure should be at pressures near atmospheric pres-ure. The first important step in this direction was done by Mollerup11]. He connected the excess Helmholtz energy aE and the excessibbs energy gE0 at zero pressure introducing the Flory–Huggins

ig. 4. Experimental and calculated VLE data for the system acetone(1)–water(2) at33.15 K – SRK + gE mixing rule UNIQUAC.

ical Fluids 55 (2010) 438–447

term∑

ixi ln b/bi:

aE

RT= gE0RT

− PvE

RT+∑i

xi lnb

bi(2.10)

The first order Modified Huron–Vidal (MHV1) mixing rule at thereference pressure Pref = 0 was derived by Michelsen [12] setting vE

equal to 0:

q1

(a

bRT−∑i

xiaiibiRT

)= gE0RT

+∑i

xi lnb

bi(2.11)

Where q1 depends on the equation of state.

2.5. The predictive Soave–Redlich–Kwong group contributionequation of state

The development of gE-mixing rules all of a sudden allowedthe combination of the Soave–Redlich–Kwong equation of statewith the gE-model (group contribution method) UNIFAC [1]. Theresult was the predictive Soave–Redlich–Kwong group contribu-tion equation of state (PSRK) which was published in 1991 byHolderbaum and Gmehling [13].

The resulting PSRK – gE-mixing rule for parameter a at a refer-ence state of Pref = 1.01325 bar is:

a

bRT=∑i

xiaiibiRT

+ gE0/RT +∑

ixi ln b/bi0.64663

(2.12)

PSRK allows using the already existing group interaction parame-ter matrix of the UNIFAC model [14]. Furthermore 30 gases havebeen added as additional main groups into the PSRK matrix toextend the range of applicability to processes, such as absorptionand supercritical extraction.

In PSRK the ˛-function proposed by Mathias and Copeman [15]was used to describe the vapor pressures as f(T) reliably. Whilefor reduced temperatures Tr < 1 the polynomial form (Eq. (2.13)) isused,

˛ii(T) = [1 + c1(1 − T0.5r,i ) + c2(1 − T0.5

r,i )2 + c3(1 − T0.5

r,i )3]2

(2.13)

for Tr > 1 the parameters c2 and c3 are set to zero. This leads to theSoave approach which is given in Eq. (2.14).

˛ii(T) = [1 + c1(1 − T0.5r,i )]

2(2.14)

The Mathias–Copeman parameters c1–c3 usually are fitted toexperimental vapor pressures. If no vapor pressures data are avail-able, the Mathias–Copeman equation is reduced to the Soaveapproach given in Eq. (2.14) and the parameter c1 can be estimatedfrom the acentric factor ω following Eq. (2.15).

c1 = 0.48 + 1.574ω − 1.76ω2 (2.15)

Today, PSRK is implemented in most process simulators and isused worldwide for a variety of applications. The great advantage ofthe PSRK model is that all of a sudden the UNIFAC group interactionparameters can be used at sub- as well as supercritical conditions.But nevertheless, the PSRK – model shows all the weaknesses ofthe SRK equation of state and the UNIFAC method. So for examplepoor results are obtained for the liquid densities of the pure com-pounds and the mixtures. Furthermore, the results for asymmetricsystems as well as the prediction of excess enthalpies and activitycoefficients at infinite dilution are not satisfying. Furthermore, the

Mathias–Copeman˛-function used shows a curvature, which is notphysically correct at high reduced temperature.

The next step in the development of a universal group con-tribution equation of state was the systematic removal of theseweaknesses. This was achieved by Ahlers and Gmehling. In 2001

percritical Fluids 55 (2010) 438–447 441

to

3

tcPmPt

P

3

eb

a

b

Ti

a

3

lTcp

˛

pctlpaaassfpif

3

elvm

c

Besides the change in the˛-function and the introduction of theconcept of volume translation, also improved mixing rules for theparameter a and b suggested by Chen et al. [21] were introduced.

B. Schmid, J. Gmehling / J. of Su

hey published the volume translated group contribution equationf state VTPR [16].

. The VTPR group contribution equation of state

The volume translated Peng–Robinson (VTPR) group con-ribution equation of state (Eq. (3.1)) is the result of aonsequent improvement of equations of state. It is based on theeng–Robinson equation of state [7], the group contribution modelod. UNIFAC (Do) [2] and the volume translation introduced by

eneloux et al. [17]. At the same time improved mixing rules forhe parameters a and b are used.

= RT

(v + c − b)− a(T)

(v + c)(v + c + b) + b(v + c − b)(3.1)

.1. Attractive parameter and co-volume

As for the van der Waals equation of state, the attractive param-ter aii and the co-volume bii for the pure components can directlye calculated from the critical data following Eq. (3.2) and (3.3).

ii(T) = ac,ii · ˛i(T) = 0.45724 ·R2 · T2

c,i

Pc,i· ˛i(T) (3.2)

ii = 0.0778 · R · Tc,iPc,i

(3.3)

he temperature dependency of the attractive parameter is takennto account by the Twu alpha function.

ii(T) = ac,i · ˛i(T) (3.4)

.2. ˛-Function

With the introduction of the Twu-˛-function [18], the prob-ems caused by the Mathias–Copeman˛-function were solved. Thewu-˛-function consists of two different forms. It depends on theonsidered state. For a reduced temperature Tr < 1, Twu et al. pro-osed Eq. (3.5).

i(T) = TNi ·(Mi−1)r,i · exp[Li · (1 − TNi ·Mir,i )] (3.5)

In this case, the parameters L, M and N are fitted to pure com-onent vapor pressures and sometimes additionally to liquid heatapacities, since liquid heat capacities deliver additional informa-ion about the slope of the vapor pressure curve especially atow temperatures, where the experimental determination of vaporressure data is difficult [19]. At supercritical conditions, a gener-lized form of the ˛-function is suggested, where the value of thelpha function is calculated with the help of the acentric factor ωnd generalized L, M and N parameters, which were fitted to gasolubilities of hydrogen and nitrogen in long chain alkanes. Aftereveral tests, we decided to use the subcritical form of the Twu ˛-unction at sub- as well as the supercritical conditions [20]. Thisrocedure leads to the consistency of the ˛-function and leads to

mproved results compared to the application of the generalizedorm in case of supercritical conditions.

.3. Volume translation

To improve the results for liquid densities calculated by cubicquations of state, Peneloux et al. [17] introduced a constant trans-ation parameter c. The c-parameter can be obtained from the

olumes calculated with the Peng–Robinson (PR) EOS and experi-ental volumes at a reduced temperature of 0.7 (Eq. (3.6)).

=∑i

xici with ci = vi,PR − vi,exp (3.6)

Fig. 5. Experimental and calculated densities as a function of temperature using thePeng–Robinson equation of state.

From a comparison of the results shown in Figs. 5 and 6 for differentcompounds the significant improvement for liquid densities calcu-lated by the VTPR EOS (Fig. 6) compared to the Peng–Robinson EOS(Fig. 5) can be recognized.

Still more impressive is a comparison with the results of the vander Waals equation of state. For methanol the results are shown inFig. 2 together with the results of other cubic equations of state.

As can be seen from Fig. 2, the experimental and predicted den-sities using VTPR near the critical point (Tr > 0.8) are not in a verygood agreement. Smaller deviations at high reduced temperaturescould be achieved introducing a temperature dependent transla-tion parameter. But the introduction of this temperature dependenttranslation parameter may result in a crossing of the isothermsat supercritical conditions. Therefore, a temperature independentvolume translation parameter was chosen, which does not showthese effects [13].

3.4. Improved mixing rules

Fig. 6. Experimental and calculated densities as a function of temperature using theVTPR equation of state.

442 B. Schmid, J. Gmehling / J. of Supercritical Fluids 55 (2010) 438–447

F sing Pa

T([

Ts

b

Tr

b

(mesraons

3

eppgop

F

Tlbnme

ig. 7. Experimental [4] and predicted vapor–liquid phase equilibrium behavior usymmetric system ethane (1)–dodecane (2).

he new VTPR gE-mixing rule for the attractive parameter a (Eq.3.7)) now only contains the residual part of the excess Gibbs energy2] at a reference pressure of 1.01325 bar.

a

b=∑i

xiaiibii

+ gEres−0.53087

(Pref = 1.01325 bar) (3.7)

he b parameter is calculated following the quadratic mixing rulehown in Eq. (3.8).

=∑i

∑j

xi · xj · bij (3.8)

he cross parameter bij is derived from a nonlinear combinationule with ¾ as exponent.

3/4ij

=b3/4ij

+ b3/4jj

2(3.9)

The exponent is adopted from the gE-model mod. UNIFACDo) where it was introduced to improve the results for asym-

etric systems. In Fig. 7 the predicted VLE data for the systemsthane–propane and ethane–dodecane using PSRK and VTPR arehown together with the experimental date. While similar goodesults are obtained for the system ethane–propane, only VTPR isble to predict the symmetric and the strong asymmetric systemver a wide temperature range, whereby it has to be mentioned thato group interaction parameters are required for alkane–alkaneystems.

.5. Group interaction parameters

While for alkane–alkane systems no group interaction param-ters are required, for other combinations group interactionarameters are fitted simultaneously to reliable experimentalhase equilibrium data like VLE, �∞, LLE, SLE of eutectic systems,as solubilities, azeotropic data and excess properties (hE, cEp). Thebjective function presented in Eq. (3.10) is minimized during thearameter fitting procedure.

= wVLE�VLE +wAZD�AZD +whE�hE +wcEP�cEP +wGLE�GLE

+w�∞��∞ +wSLE�SLE +wLLE�LLE = Minimum (3.10)

he contribution of the deviation between experimental and calcu-

ated values of the different data types to the objective function cane adjusted with the help of the weighting factors wi. VLE data oformal and low boiling components are suitable to provide infor-ation about the real behavior as a function of composition. Excess

nthalpies describe the temperature dependency of the activity

SRK (. . .) and VTPR (—) of the symmetric system ethane (1)–propane (2) and the

coefficient and can be used as supporting information at high tem-peratures. Gas solubilities as well as activity coefficients at infinitedilution provide the required information about the real behaviorin the dilute region. SLE data of eutectic systems can be used assupporting information at low temperatures. Finally, liquid–liquidequilibrium data are often the only available information for sys-tems with strong deviations from Raoult’s law.

In the VTPR model for the parameters the same temperaturefunction as in mod. UNIFAC (Do) is used (Eq. (3.11)). Depending onthe strength of the temperature dependence 2–6 group interactionparameters are fitted.

nm = exp

(−anm + bnmT + cnmT2

T

)(3.11)

Using the fitted group interaction parameters, the VTPR modelcan be applied to any kind of systems. In Fig. 8 the predicted resultsfor several n-alkane–alcohol systems using the VTPR group contri-bution equation of state and modified UNIFAC are shown togetherwith the experimental data.

It can be seen, that VTPR is able to predict the mixture proper-ties with the same quality as mod. UNIFAC (Do). But in contrast togE-models, equations of state can also handle systems with super-critical compounds, e.g. hexane–ethanol at 503.15 K. At the sametime other important thermophysical properties, such as densities,enthalpies etc., can be obtained.

In Fig. 9 the predicted results using the VTPR and PSRK modelare shown together with the experimental results for the binarysystem carbon dioxide(1)–cyclohexane(2).

It can be seen that besides vapor–liquid equilibria also reli-able results for excess enthalpies are predicted. Even the pressuredependence of these properties is described correctly.

The predicted VLE behavior using PSRK and VTPR for differentCO2–n-alkane systems is shown in Fig. 10 together with the exper-imental data. While the vapor–liquid equilibrium behavior of thesymmetric system CO2–propane and CO2–hexane is described reli-ably with both equations of state, only VTPR is able to describe thestrong asymmetric systems CO2–eicosane and CO2–octacosane.

4. Range of applicability

The applicability of the VTPR equation of state is not limited tothe reliable prediction of binary phase equilibria and excess prop-

erties. There are a lot of other fields of application for VTPR, forexample:

• the identification of separation problems, e.g. azeotropic pointsin multicomponent systems,

B. Schmid, J. Gmehling / J. of Supercritical Fluids 55 (2010) 438–447 443

and m

•••

••

••

•

•

•

F[

Fig. 8. Experimental and predicted properties using VTPR (—)

the construction of residual curves and borderlines,the design of separation columns,the selection of suitable solvents for separation processes, chem-ical processes,. . .the selection of working fluids for thermodynamic cycles,the consideration of the real behavior on the chemical equilib-rium conversion (K� , Kϕ),the prediction of flash points of flammable liquid mixtures,the prediction of the fate of a chemical in the environment, bioac-cumulation effects,. . .the prediction of various thermodynamic properties (h, �hv,s,. . .),

the reliable description of the diffusional mass transfer (using�ai,�fi,��i instead of�ci),. . .

ig. 9. VLE and excess enthalpies for the binary system CO2 (1)–ethane (2) calculated wit4].

od. UNFAC (Do) (. . .) for different n-alkane–alcohol systems.

4.1. Identification of separation problems

Because nearly 90% of the separation processes in the chemicalas well as in the oil and gas processing industry are distillationprocesses, the knowledge about azeotropic points is of a greatimportance. In Table 1, the predicted azeotropic points for the qua-ternary system carbon dioxide (1)–ethane (2)–hydrogen sulfide(3)–propane (4) at 266.5 K are listed and compared with the meanvalues stored in the Dortmund Data Bank [4].

As can be seen, the results obtained using the VTPR group con-tribution equation of state are in a very good agreement withthe experimental data stored in the Dortmund Data Bank. In all

cases azeotropic and non-azeotropic behavior is predicted cor-rectly. Unfortunately for the ternary and quaternary systems noinformation about azeotropic behavior was published.

h VTPR (—) and PSRK (. . .) together with the experimental data taken from the DDB

444 B. Schmid, J. Gmehling / J. of Supercritical Fluids 55 (2010) 438–447

fferen

4

abc

sF

4

spemGtra

TE

Fig. 10. Experimental and predicted vapor–liquid equilibrium behavior for di

.2. Construction of residual curves and borderlines

For distillation processes, besides the knowledge aboutzeotropic points, also information about the residual curves andorderlines are of interest to fully understand the separation pro-ess [22].

The predicted residual curves and the border line for the ternaryystem CO2–H2S–C2H6 with two binary azeotropes is shown inig. 11. As can be seen two distillation regions are obtained.

.3. Selection of suitable solvent for separation processes

The computer supported selection of a suitable entrainer for aeparation process is an important task in process development. Arocedure for the selection of suitable solvents for azeotropic orxtractive distillation with the help of predictive thermodynamic

odels or access to a large factual data bank was introduced bymehling and Möllmann [23]. Using this procedure for the separa-

ion of the azeotropic system CO2–ethane important for tertiary oilecovery higher alkanes are found by this procedure. For propanes entrainer the predicted result using VTPR are shown in Fig. 12.

able 1xperimental and calculated azeotropes using VTPR for the quaternary system carbon diox

System Predicted using VTPR

Type P (bar) y2,az y3,az

1–2 homPmax 33.36 0.31271–3 None1–4 None2–3 homPmax 20.59 0.9097 0.092–4 None3–4 homPmax 8.97 0.831–2–3 None1–2–4 None1–3–4 None2–3–4 None1–2–3–4 None

t sub- and supercritical CO2–n-alkane systems using PSRK (. . .) and VTPR (—).

As can be seen from Fig. 12 the addition of 80 mole% propaneleads to an increasing volatility of carbon dioxide which allowsseparating CO2–ethane by extractive distillation.

4.4. Prediction of the chemical equilibrium conversion

The equilibrium state of a reversible chemical reaction isdescribed by the equilibrium constant K, which can be calculatedusing standard thermodynamic properties.

K = Kx•K� (3.12)

Following Eq. (3.12), the real behavior has to be taken into accountto be able to calculate the chemical equilibrium conversion.

As example reaction, the acidic catalyzed cleavage reaction oftert-amyl-methyl-ether (TAME) is shown in Fig. 13. As catalyst,the macroreticular, sulfonic acid ion-exchange resin Amberlyst 36

from Rohm and Haas was used. The cleavage products are isoamy-lene (2-methyl-2-butene (2M2B), respectively, 2-methyl-1-butene(2M1B)) and methanol.

The experimentally determined and predicted chemical equilib-rium conversions using VTPR (—), modified UNIFAC (Do) (. . .) and

ide (1)–ethane (2)–hydrogen sulfide (3)–propane (4) at 266.5 K (n.a. = not available).

Experimental

Type P (bar) y2,az y3,az

homPmax 33.27 0.33NoneNone

03 homPmax 20.68 0.896 0.104None

01 homPmax n.a. 0.83n.a.n.a.n.a.n.a.n.a.

B. Schmid, J. Gmehling / J. of Supercritical Fluids 55 (2010) 438–447 445

Fig. 11. Border lines and residual curve for the ternary system CO2-ethane-H2S at266.5K derived from VTPR.

Fp

aa

sm

rarav

fid

Fma

least qualitative agreement is obtained. Furthermore the predictedLLE results for the binary system nitrogen–ethane as a functionof temperature are in good agreement with the experimentalfindings.

ig. 12. y–x-data for the system CO2 (1)–ethane (2) in the presence of 80 mole%ropane at 266.5 K and solvent free basis predicted with VTPR.

ssuming ideal behavior (—) for the TAME cleavage in the temper-ture range from 300 K to 370 K are shown in Fig. 14.

As can be seen, using the VTPR group contribution equation oftate leads to similar good results as the group contribution methodod. UNIFAC (Do).In Fig. 15, additionally the experimental [23] and predicted

esults for the chemical equilibrium conversion of the tert-myl-methyl-ether (TAME) cleavage reaction in the temperatureange from 300 K to 370 K in the presence of n-pentane (1:1)s inert solvent are shown together with the experimental

alues.

As can be concluded from the results shown above, modi-ed UNIFAC (Do) and the VTPR equation of state are able toescribe the equilibrium conversion of the TAME cleavage reli-

ig. 13. Reaction equation of the acidic catalyzed cleavage reaction of tert-amyl-ethyl-ether (TAME) to 2-methyl-2-butene (2M2B), 2-methyl-1-butene (2M1B)

nd methanol.

Fig. 14. Experimental (� [24], © [25], × [26]) and calculated results for the chemicalequilibrium conversion of tert-amyl-methyl-ether (TAME) in the temperature rangefrom 300 K to 370 K using VTPR (—), mod. UNIFAC (Do) (. . .) and assuming idealbehavior (—).

ably in the temperature range covered. It can be seen that thedilution effect, which following the LeChatelier principle shouldlead to a lower chemical equilibrium conversion is compensatedby the real behavior K� .

4.5. Prediction of miscibility gaps

Liquid–liquid equilibria can occur if a system shows strong devi-ation from Raoult’s law. In separation processes as for exampleextraction or hetero-azeotropic distillation processes the concen-tration differences in the two liquid phases are used for theseparation. For the description of the LLE behavior (distributioncoefficients, binodal curve) often gE-models, such as NRTL orUNIQUAC are applied. But of course also equations of state canbe applied. For the ternary system nitrogen-methane-ethane theresults of the VTPR group contribution equation of state are shownin Fig. 16 together with the experimental data. It can be seen that at

Fig. 15. Experimental (�) and calculated results for the chemical equilibrium con-version of tert-amyl-methyl-ether (TAME) in the temperature range from 300 Kto 370 K in the presence of n-pentane (1:1) as inert solvent using VTPR (—), mod.UNIFAC (Do) (. . .) and assuming ideal behavior (—).

446 B. Schmid, J. Gmehling / J. of Supercritical Fluids 55 (2010) 438–447

Fig. 16. Experimental [4] and predicted LLE data of the ternary

Ff

4

bdUoa

cttilefli

5

aons

ig. 17. Experimental (�) [4] and predicted solubilities of n-alkanes in water as aunction of temperature using VTPR (—) and mod. UNIFAC (Do) (. . .).

.6. Solubility of alkanes in water

For environmental technology the knowledge about the solu-ility of n-alkanes in water is of special interest, especially for theesign of crude oil processing plants and wastewater treatment.sing reliable group interaction parameters fitted to a huge amountf experimental data [4], the VTPR equation can be successfullypplied for the calculation of alkane solubilities in water.

As can be seen from Fig. 17, the solubility of n-alkanes in waterorrelated using the volume translated Peng–Robinson group con-ribution equation of state (solid lines) are in a good agreement withhe experimental data taken from the Dortmund Data Bank [4] andmproved to the results derived from the mod. UNIFAC (Do) corre-ation (dashed lines). For the comparison of both models with thexperimental findings, the respective group interaction parametersor VTPR as well as mod. UNIFAC (Do) were derived from a corre-ation of the same data base and are not identical with the groupnteraction parameters for the main groups H2O–CH2.

. Conclusion

Based on the doctoral thesis of Johannes Diderik van der Waals,lot of work was spent on the improvement of the cubic equationf state developed by him. Over the years, most of the weak-esses like the poor results for vapor pressures or liquid densities,trong polar or asymmetric systems could be removed. The pre-

system nitrogen–methane–ethane at 122 K using VTPR.

sented VTPR group contribution equation of state is the resultof the consequent improvement of cubic equations of state andcan be applied successfully for process development. No matterwhether the investigated system is a polar or non-polar multicom-ponent mixture containing supercritical compounds the volumetranslated Peng–Robinson group contribution equation of state canbe successfully applied. The prediction quality of VTPR is as goodas for modified UNIFAC (Do) and distinctly improved compared toPSRK. In contrast to modified UNIFAC (Do), VTPR at the same timeallows the prediction of densities, enthalpies, heat capacities andso on.

List of symbolsa cohesive energy parameter of cubic equations of stateaE excess Helmholtz energyanm group interaction parameterb co-volume parameter of cubic equations of statebnm group interaction parameterc translation parameter for the volume translated

Peng–Robinson equation of statecnm group interaction parametercp Isobaric heat capacityF objective functiong molar Gibbs energyh molar enthalpy�hv molar enthalpy of vaporization�hR molar enthalpy of reactionK equilibrium constantL,M,N Twu–Buck–Cunningham–Coon ˛-function parametersP pressureR universal gas constants molar entropyT absolute temperaturev molar volumew weighting factorx liquid phase mole fraction

Greek letters˛ temperature dependent a(T)-function� activity coefficient

ω acentric factorϕ fugacity coefficient temperature function� chemical potential� difference between two values

percrit

Scceir

SEim∞

AAGLSUVV

A

ip

R

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[25] L.K. Rihko, J.A. Linnekoski, A.O.I. Krause, Reaction equilibria in the synthesis of

B. Schmid, J. Gmehling / J. of Su

ubscriptscritical point

alc calculatedxp experimental,j component i,j

reduced

uperscriptsexcess

d ideal, n main groups m, n

infinite

bbreviationsZD azeotropic dataLE gas–liquid equilibriumLE liquid–liquid equilibriumLE solid–liquid equilibriumNIQUAC universal quasi chemical modelLE vapor–liquid equilibriumTPR volume translated Peng–Robinson group contribution

equation of state

cknowledgements

The authors would like to thank the “Arbeitsgemeinschaftndustrieller Forschungsvereinigungen” (AiF) for the financial sup-ort of this work (IGF-Project No. 15345N).

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