The IPRSV Equation of State-libre

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Fluid Phase Equilibria 330 (2012) 2435

Contents lists available at SciVerse ScienceDirect

Fluid Phase Equilibria

jou rna l homepage: www.elsevier .com/ locate / f lu id

The iPRSV equation ofstate

T.P. van der Stelt a,, N.R. Nannan b, P. Colonnaa

a Process andEnergyDepartment, Delft University of Technology, Leeghwaterstraat 44, 2628 CA Delft, TheNetherlandsb Mechanical Engineering Discipline, Antonde KomUniversity ofSuriname, Leysweg 86, PO Box9212, Paramaribo, Suriname

a r t i c l e i n f o

Article history:

Received 3 April 2012

Received in revised form 5 June 2012

Accepted 9 June 2012Available online 21 June 2012

Keywords:

Equation of state

PengRobinson

PRSV

-Function

Discontinuity

iPRSV

a b s t r a c t

The PengRobinson cubic equation ofstate with the StryjekVera modification (PRSV) is widely adopted

in scientific studies and engineering. However, it is affected by a discontinuityin allthe properties, which

is caused by a discontinuity of the -function. Aside of being non-physical, this discontinuity causesrobustness and accuracy issues in numerical simulations. The discontinuity in thermodynamic proper-

ties is eliminated here without affecting the overall accuracy of the model. In addition, the functional

form of(T) is optimized in such a way that itis not required to change the values ofthe fluid-dependent

parameters stored in the many available databases. The performance ofthe improved equation ofstate

(iPRSV) is assessed by comparing calculated properties with those obtained with the original PRSV equa-

tion ofstate, the Gasem et al. equation ofstate (PRG), which is alsocontinuous in temperature, a reference

multiparameter equation ofstate, and experimental data. It is shown that the accuracy ofthe new model

approaches the accuracy ofthe original equation ofstate and that it performs better than the PRG equa-

tion ofstate. The modified PRSV equation of state solves the issue ofthe artificial discontinuity in the

calculation ofproperties relevant to scientific and industrial applications, at the cost ofa small decrease

in overall accuracy.

2012 Elsevier B.V. All rights reserved.

1. Introduction

In order to obtain a better correlation of vapor pressures

for a wide variety of fluids, Stryjek and Vera [1,2] proposed to

use the Peng-Robinson [4] cubic equation of state (EoS), com-

plemented by the Soave [3] -function, but with a differenttemperature and acentric factor dependence. However, as a result,

the PengRobinson EoS with the StryjekVera modification (PRSV)

features a discontinuity in all the properties in correspondence of

the absolute critical temperature, Tc, of water and ofalcohols, and

at T=0.7 Tc for other fluids.

Over the last few decades, numerous modifications to the -function of Soave have been proposed, most of them with the

aim of obtaining a more accurate estimate of the pure-compound

vapor pressure. In particular, better performance has been sought

for reduced temperatures, Tr T/Tc, lower than 0.7, for substanceswith an acentric factor greater than 0.5, and for polar fluids

like alcohols. Some of the proposed modifications accomplish this

goal by introducing one or more component-dependent parame-

ters [1,2,5,6]. Other modifications involve changing the functional

form of in terms of either or Tr, or both. The -function depen-

dency can be either linear [7], exponential [8], quadratic [6], or a

combination of the aforementioned [5,9].

Corresponding author. Tel.: +31 15 2785412.

E-mail address: [email protected] (T.P. van der Stelt).

Because in most cases the proposed modifications are aimed

at improving only vaporpressure predictions, merely a handful

of researchers investigated the effect of their proposed modifi-

cation ofthe -function on the prediction ofall thermodynamicproperties, especially those dependent upon first or higher-order

derivatives of in the supercritical region.A number of thermodynamic models [1,7,9] suffer from the

reliance on the use of switching functions below and above the

critical temperature. Theseswitching functions can cause largedis-

continuities in the -function and its derivatives. Gasem et al. [8]addressed the problem ofswitching functions and proposed an

exponential and continuous -function. They determined the firstand second-order derivative of the -function with respect to the

temperature and compared values of heat capacities predicted by

theirmodel withexperimentaldata, for temperatures spanning the

range from Tr 0.5 up to values well above the critical point tem-perature for methane and nitrogen, andup to Tr = 1.14 for propane.

They obtained a significant improvement of the predicted heat

capacities with respect to the results from earlier models [3,5,7].

Neau et al. [10,11] analyzed in detail the influence of the functional

relation of the EoS and the first and second-order temperature

derivatives of the -function on the modeling of enthalpies andheat capacities for reduced temperatures as high as about 3.5. They

foundthat the second-order temperature derivative ofthe general-

ized models for of Twu et al. [7] and Bostonand Mathias [12] also

features abnormal extrema and inconsistent break points at the

critical temperature, due to the use ofdifferent sets of parameters

0378-3812/$ seefrontmatter 2012 Elsevier B.V. All rights reserved.

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T.P.vander Stelt etal./ FluidPhaseEquilibria330 (2012) 2435 25

Nomenclature

a attractive term of the PengRobinson EoS, Eq. (2)

A, B, C coefficients of the -function of the iPRSV EoS, Eq.(7)

D, E coefficients of the -function of the iPRSV EoS, Eq.(7)

b co-volume parameterofthe PengRobinsonEoS, Eq.

(3)c speed of soundC0, C1 coefficients for the ideal gas CPpolynomial, Eq. (4)

C2, C3 coefficients for the ideal gas CPpolynomial, Eq. (4)

CP isobaric heat capacity

Cv isochoric heat capacityh enthalpy

P pressure

R universal gas constant

s entropyT absolute temperature

v specific volume

Greek symbols

functionofreduced temperature andacentric factor,Eq. (4)

functionofreduced temperature andacentric factor,Eq. (5)

0 function ofthe acentric factor in the-function, Eq.(6)

1 pure compound parameter in the-function density acentric factor

Subscript

c critical

r reduced

Ref. EoS reference equations of state

tot total

Superscript

0 ideal gas state

below and above the critical temperature. They also pointed out

that the original-function ofSoave has a non-physical minimum,

but that this minimum is in the range of 2.3


3/12

26 T.P. van der Stelt et al. / Fluid Phase Equilibria330 (2012) 2435

Table 1

Thedefinitionof the-functionin theStryjekand Vera formulationofthe attractive

term in thePengRobinson equation of state.

(a) Water and alcohols: Tr < 1 = 0 + 1(1+

Tr)(0.7 Tr )

Tr 1 =0

(b) All other compounds: Tr 0.7 = 0 + 1(1 +

Tr )(0.7 Tr)

Tr>0.7 =0

Tr= T/ T

c

(0=0,

1=1)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 1. -Parameter of the PRSV CEoS for alcohols and w ater ( ) and all other

compounds ( ).

recommend 1 = 0, because there would be no advantage in usingEq. (5) in this region. The-function therefore introduces a discon-tinuity in (T) either at Tr =0.7 or at Tr = 1, and in thermodynamicproperties dependent upon and derivatives thereof.

The definition ofthe -function is summarized in Table 1. Fig. 1shows thevalue of for0 =0and1 =1asafunctionofthe reducedtemperature for both water and alcohols and all other compounds.

Figs. 2 and 3 demonstrate exemplary anomalies in the selected

thermodynamic properties caused by the mentioned discontinu-

ities. Fig. 2 depicts a line of constant isochoric heat capacity Cv

v[m3/kg]

P[M

Pa]

10-2

10-1

2

4

6

8

10

12

14

VLE

Fig. 2. Discontinuity ofthe PRSV model: Pv diagram ofmethanol displaying the

vaporliquid equilibrium region (VLE) and the non-physical discontinuity of an

exemplary iso-Cvline (Cv= 1.7 kJ/kg-K) ( ) crossing the critical isotherm

( ).

Table 2

Coefficients ofthe equation of theiPRSV EoS.

A=1.1

B= 0.25

C= 0.2

D= 1.2

E=0.01

calculated withthe PRSV model, togetherwith the criticalisotherm

and the saturated liquid and vapor lines in a Pv diagram formethanol. By following the iso-Cvline for increasing pressure anddecreasing specific volume, a non-physical discontinuity in the line

can be noted as it intersects the critical isotherm.

Fig. 3 shows a similar effect in the Ts diagram for methanol.

Together with the vaporliquid equilibrium region and the criti-

cal isotherm, exemplary isolines calculated with the PRSV model

are also shown. In order to illustrate the consequence of the

non-physical discontinuity, with reference to Fig. 3d, imagine the

expansion ofthe fluid through a nozzle starting from P=1.5MPa

and T= 240 C. As the pressure and temperature decrease, at the

critical temperature, an non-physical jump in the entropy value

occurs (from 0.119 to 0.1167 kJ/kg-K). Notice also, as an additional

example (Fig. 3c), that a state characterized by the same speed

of sound and entropy, features two values of temperature whichcontravenes the phase rule ofthermodynamics.

3. The iPRSV cubic equation of state

The improved PRSV EoS, iPRSV, is obtained by modifying the

equationfor the calculation ofthe-value, suchthat it is continuouswith the temperature, but by keeping the same parameters 0and1 in the functional form, with the same values. The -function inthe iPRSV thermodynamic model is therefore

= 0 + 1

[AD(Tr + B)]

2+ E+AD(Tr + B)

Tr + C. (7)

This functional form was obtained by matchingit to the original

PRSV formulation as close as possible, except for the discontinuity

(see Fig. 4). The implementation of the iPRSV in computer codes,relying on existing databases collecting the parameters for many

fluids, is quite straightforward. No refitting ofdata is necessary.

The coefficients A, B, C, D, and Eare presented in Table 2. The

derivatives of with respect to the temperature necessary forthe implementation of a complete thermodynamic model into a

computer program are given in Appendix A. The continuity of the

equation assures the continuity in the first and second derivative

of with respect to the temperature.1

In order to prevent a sign change in the first-order tempera-

ture derivative of, the new function follows as closely as possiblethe original PRSV formulation (b) in Table 1. With reference to

Fig. 5ad, it can be noted that, by varying coefficient E, the cur-

vature of(Tr) in correspondence ofTr = 0.7 can be changed. The

smaller the value ofE, the closer the values ofthe new function areto the original formulation.However, the smaller the value ofE, the

larger is the fluctuation of the second-order derivative withrespect

to the temperature. E comes therefore from a trade-off between

the counteracting need ofapproximating the original -value asclose as possible, andminimizing the variation ofthe second-order

temperature derivative.

Figs. 6 and 7 show a comparison between PRSV and iPRSV

related to the same exemplary diagrams reported in Figs. 2 and

3.

1 Notethat physics prescribesthat is a monotonefunction oftemperature,with-

out inflectionpoints, thereforeboth theoriginal formulation and theone proposed

here violate this constraint.


4/12


s[kJ/kg-K]

T

[C]

-2.5 -2 -1.5 -1 -0.5 0 0.50

50

100

150

200

250

VLE

ec db

(a) T s diagram of methanol displaying the critical isotherm

( ), the vapor-liquid equilibrium region (VLE), and the

non-physical discontinuities in some exemplary isolines (iso-h,

iso-c, isobar, isochor)( ). Enlargements of areas b, c, d,and e are shown in figures 3b, 3c, 3d, and 3e.

s [kJ/kg-K]

T

[C]

-1.6 -1.58 -1.56 -1.54 -1.52

230

235

240

245

250

h= -344.3 kJ/kg

h= -357.1 kJ/kg

h= -350 kJ/kg

(b) iso-h lines.

s[kJ/kg-K]

T

[C]

-0.22 -0.2 -0.18 -0.16

220

225

230

235

240

245

250

255

260

c= 346.4 m/s

c= 353.2 m/s

c= 350 m/s

(c) iso-c lines.

s[kJ/kg-K]

T

[C]

0.112 0.114 0.116 0.118 0.12239

239.2

239.4

239.6

239.8

240

P= 1.511 MPa

P= 1.5 MPa

P= 1.489 MPa

(d) isobars.

s[kJ/kg-K]

T

[C]

0.424 0.425 0.426 0.427 0.428239

239.2

239.4

239.6

239.8

240

v= 0.2493 m3/kg

v= 0.2507 m3/kg

v= 0.25 m3/kg

(e) isochors.

Fig. 3. Graphical representation ofthe non-physical discontinuities in some exemplary isolines in the Ts diagram ofmethanol calculatedwith theoriginal PRSV CEoS.

4. Performance of the iPRSV model

The attraction parameter in the PRSV EoS was proposed in

order to improve the accuracy of the calculation of the satu-

ration pressures. Firstly, in order to evaluate the performance

of the iPRSV model, the results of saturation pressure calcula-

tions are compared for fluids of different classes and molecular

complexity. Moreover, in order not to limit the evaluation to

the prediction of saturated properties, also PT data, spe-

cific enthalpies, and entropies are compared to values computed


5/12


Tr= T/ T

c

(0=0,

1=1)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Fig. 4. as function ofTr (with 0= 0 and 1= 1) for the original PRSV CEoS (alco-

hols and water , all other compounds: ), and the new iPRSV CEoS

( ).

with reference multiparameter equations of state. Such a more

extensive evaluation is limited to methanol and propane as exem-

plary fluids.

In order to obtain a complete thermodynamic model, the iPRSV

equation of state is complemented with a polynomial function for

the calculation ofthe ideal gas isobaric heat capacity, i.e.,

C0PR = C0 + C1T+ C2T

2 + C3T3,

v[m3/kg]

P[MPa]

10-2

10-1

2

4

6

8

10

12

14

VLE

Fig. 6. Pv diagram of methanol displaying the non-physical discontinuity of an

exemplary iso-Cv line (Cv=1.7 kJ/kg-K) calculated with the PRSV model () in cor-

respondence of the critical isotherm ( ), the same iso-Cv line calculated by the

iPRSV EoS ( ), and the vaporliquid equilibrium region (VLE).

where C0, C1, C2 and C3 are fluid-specific coefficients. Table 3 lists

the input data for the complete iPRSV model ofthe selected fluids.

The results of saturation pressure calculations performed with

the iPRSV EoS are compared with those obtained with the origi-

nal PRSV EoS, with thePRG EoS (PengRobinsonGasem), andwith

accurate measurements. The PRG EoS is a PR-type EoS implement-

ing the -function proposed by Gasem et al. [8]. The PRG EoS is

Tr= T/ T

c

(0=0,

1=1)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8(a)

Tr= T/ T

c

Tc2.d2/dT2(0=0,

1=1)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-6

-4

-2

0

2

4(b)

Tr= T/ T

c

(0=0,

1=1)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8(c)

Tr= T/ T

c

Tc2.d2/dT2(0=0,

1=

1)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-10

0

10

20

30

40

50(d)

Fig.5. Curvatureof the-functionoftheiPRSV EoSnear theintersectionwithx-axis(a andc) andits non-dimensional second-orderderivativewith respectto the temperature

T2

c (d

2

/dT2

) ( band d) f or E= 0.1 (aandb) and E=0.001 (c and d).


6/12


(b) iso-h lines. (c) iso-c lines.

(d ) isobars . (e) isochors.

(a) Tsdiagram of methanol displaying the critical isotherm

( ), the vapor-liquid equilibrium region (VLE), some ex-

non-physical discontinuities calculated with the PRSV model,

and the same, but continuous, isolines calculated by the iPRSV

model. Enlargeme nts of areas b, c, d, and e are shown in fig-

ures 7b, 7c, 7d, and 7e.

emplary isolines (iso-h,iso-c, isobar, isochor) displaying the

Fig. 7. Graphical representation of the non-physical discontinuities in some exemplary isolines in the Ts diagram of methanol calculated with the original PRSV CEoS

( ) together w ith t he c ontinuous i solines c alculated with t he i PRSV C EoS ( ).

included in this evaluation because its attractive parameter is a

continuous function of the temperature, much like in the iPRSV

model. It is therefore an alternative to the iPRSV EoS, if model

consistency is a concern. However, this thermodynamic model has

not been widely adopted in scientific and engineering applications

as testified by the lack of literature referring to the use ofthe PRG

model for practical purpose.

Fig. 8 shows charts displaying the percentage absolute devia-

tions (AD%) of the saturation pressures calculated by the iPRSV,

PRSV, and the PRG EoS with respect to experimental values.

The considered exemplary fluids are dodecane, methanol, water,

andMDM octamethyltrisiloxane,[(CH3)3SiO]2Si(CH3)2.TheiPRSV

model applied to dodecane (Fig. 8a) is somewhat less accurate in

predicting the saturated pressure than the PRSV for 0.65 0.7 and the deviation increases for increas-

ing temperature, while it is the most accurate of the three in

the temperature interval 0.55< Tr


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Tr= T/ T

c

|Psat,exp-P

sat,calc

|/P

sat,exp

*100%

0.4 0.5 0.6 0.7 0.80

0.5

1

1.5

2

2.5

3

3.5

4

(a) dodecane

Tr= T/ T

c

|Psat,exp-P

sat,calc

|/P

sat,exp

*100%

0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

(b) methanol

Tr= T/ T

c

|Psat,exp-P

sat,calc

|/P

sat,exp

*100%

0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

(c) water

Tr= T/ T

c

|Psat,exp-P

sat,calc

|/P

sat,exp

*100%

0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

(d) MDM

Fig. 8. Percentage absolute deviationbetween experimental data forthe vapor pressure (data markedaccepted in theDIPPR[33] database taken as a reference), and values

calculated with the iPRSV ( ), PRSV( ), and the PRG EoS ( ).

deviations are larger for all the models, with the PRG EoS being the

least accurate. It is noticeable that at very low pressure the accu-

racy of the measurements could be comparatively lower and the

percentage absolute deviation is inherently larger. Fig. 8b and c

show analogous trends for methanol and water. The lower accu-

racy of the iPRSV model for methanol and water for Tr >0.7 can be

expected,becausethe largestdifference betweenthe discontinuous

-function of the PRSV model and the continuous -functionof the iPRSV occurs in this temperature range (see Fig. 4). For

MDM and dodecane this effect is less pronounced. Furthermore,

Fig. 8d shows that the performance of the iPRSV model with

respect to PRSV, PRG, and experimental values in the case of

MDM cannot be clearly inferred. The PRG model is less accurate

for Tr > 0.75.

Table 3

Main fluid thermodynamic data for theiPRSV EoS ofsome fluids selected forthe evaluation ofits performance.Name Tc [K] Pc[MPa] 1 Ref er en ce Coef ficie nt s f or t he ide al gas CPpolynomial function

C1 C2 103 C3 10

6 C4 109 Reference

Inorganic

Ammonia 405.55 11.28952 0.25170 0.00100 [1] 27.31 23.83 17.07 11.85 [34]

Carbon dioxide 304.21 7.38243 0.22500 0.04285 [1] 19.80 73.44 56.02 17.15 [34]

Oxygen 154.77 5.090 0.02128 0.01512 [1] 28.11 3.680103 17.46 10.65 [34]

Water 647.286 22.08975 0.34380 0.06635 [1] 32.24 1.924 10.55 3.596 [34]

Alkanes

Propane 369.82 4.24953 0.15416 0.03161 [1] 4.224 306.3 158.6 32.15 [34]

Dodecane 658.2 1.82383 0.57508 0.05426 [1] 9.328 1.149 634.7 135.9 [34]

Ketones

Acetone 508.1 4.696 0.30667 0.00888 [1] 6.301 260.6 125.3 20.38 [34]

Alcohols

Methanol 512.58 8.09579 0.56533 0.16816 [1] 21.15 70.92 25.87 28.52 [34]Ethanol 513.92 6.148 0.64439 0.03374 [1] 9.014 214.1 83.90 1.373 [34]

2-Propanol 508.40 4.76425 0.66372 0.23264 [1] 32.43 188.5 64.06 92.61 [34]

1-Butanol 562.98 4.41266 0.59022 0.33431 [1] 3.266 418.0 224.2 46.85 [34]

1-Octanol 684.8 2.86 0.32420 0.82940 [1] 6.171 760.7 379.7 62.63 [34]

Ethers

Dimethyl e ther 400.1 5.240 0.18909 0.05717 [1] 17.02 179.1 52.34 1.918 [34]

Refrigerants

R134a 374.21 4.056 0.3259 0.0048 [39]a 16.7813 286.357 227.336 113.312 [39]

R245fa 427.2 3.640 0.3724 0.0060 [40,41]a 28.1594 335.454 144.213 0 b

Siloxanes

MDM 564.09 1.41516 0.5314 0.06195 a 97.3376 863.971 250.057 95.1544 [43]c

D5 619.15 1.16 0.6658 0.03885 a 90.9707 1564.41 1091.37 340.099 [43]c

a 1fitted to experimental data.b Coefficients fitted to ideal-gas heat capacity values obtained from a referencethermodynamic model implemented in a widely adopted computer code [42].c

C1, C2, C3, C4fitted to experimental data.


8/12


Table 4

Percentage average absolute deviations (AAD%) for the vapor pressures calculated

wit h t he PRSV, PRG and t he iPRSV E oS wit h r es pect t o the experimental values

marked as accepted in theDIPPR[33] database.

Fluid PRSV PRG iPRSV nPoints Range Tr

Ammonia 0.45 1.13 0.44 73 Tr> 0.48

Carbon dioxide 0.59 0.24 0.73 44 Tr> 0.71

Oxygen 0.35 1.08 0.36 74 Tr> 0.37

Water 0.12 7.62 0.72 51 Tr> 0.42

Propane 0.71 0.53 0.76 35 Tr> 0.35Dodecane 1.14 2.60 1.28 65 0.44 0.57

Ethanol 0.91 1.03 0.72 89 Tr> 0.57

2-Propanol 6.38 4.34 8.12 105 Tr> 0.37

1-Butanol 0.70 8.29 3.42 81 Tr> 0.52

1-Octanol 0.65 171 4.86 72 Tr> 0.42

Dimethyl ether 0.86 3.02 0.95 39 Tr> 0.44

R134a 0.40 0.85 0.40 151 Tr> 0.56

R245fa 0.92 0.49 0.95 32 Tr> 0.68

MDM 1.12 1.82 1.15 29 Tr> 0.46

D5 3.39 2.74 3.51 22 0.51


9/12


AD[%]

0

2

4

s[kJ/kg-K]

T[C

]

1 1.5 2 2.5 3

0

50

100

150

200

VLE

Fig. 10. Ts diagram ofpropane displaying the VLE region (calculated with a ref-

erence EoSfor propane [35]), several exemplary constant enthalpy lines calculated

with thereference EoS( ), the same lines of constant enthalpy calculated

with the iPRSV model (), and the percentage absolute deviation between data

calculated with thereference equation ofstate and the iPRSV model ().

[kg/m3]

10

-3

10

-2

10

-1

10

0

10

1

10

210-2

10-1

100

101

102

103

VLE

c

r

10-4

10-3

10-2

10-1

100

AD[%]

10-2

10-1

100

101

P[MPa]

Fig. 11. P diagram ofmethanol displaying theVLE region (calculated with a ref-

erence EoS for methanol [38]), several exemplary isotherms calculated with the

reference EoS ( ), thesame isothermscalculated withthe iPRSVmodel(),

and the percentage absolute deviation between data calculated with the reference

EoS and the iPRSV EoS ().

AD[%]

0

2

4

6

s[kJ/kg-K]

T[C]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

50

100

150

200

250

300

VLE

Fig.12. Ts diagram ofmethanoldisplaying the VLE region (calculatedwith a refer-

ence EoS for methanol [38]), several exemplary constant enthalpy lines calculated

with a reference EoS ( ), the same lines ofconstant enthalpy calculated

with the iPRSV model (), and the percentage absolute deviation between data

calculated with thereference EoSand theiPRSV EoS().

Table 5

Percentage absolute deviations (AAD%) in PTand Ths data ofthe PRSV, PRG

and the iPRSV EoS with respect to reference equations of state for propane [35]

and n-butane [36], and technical equations ofstatefor n-hexane,n-octane [37] and

methanol [38].

Fluid PT Ths

PRSV PRG iPRSV PRSV PRG iPRSV

Propane 2.46 2.46 2.47 0.69 0.68 0.73

n-Butane 40.0 39.9 40.0 0.35 0.41 0.33

n-Hexane 34.9 31.4 35.1 0.27 0.21 0.28

n-Octane 19.9 21.0 19.7 0.13 0.19 0.12

Methanol 9.60 9.74 9.56 0.64 1.16 0.91

Propanea 0.43 0.42 0.42

n-Butaneb 0.47 0.52 0.46

n-Hexanec 0.44 0.64 0.42

n-Octaned 0.45 1.16 0.40

Methanole 1.38 1.35 1.30

a Only r


10/12


Fig. 13. Convergingdiverging nozzle(a) and thecorresponding flow path in theTs plane (b).

position

P

[MPa]

-0.2 0 0.2 0.4 0.6 0.8 13.3

3.4

3.5

3.6

3.7

3.8

3.9

(a) pressure

position

T[C]

-0.2 0 0.2 0.4 0.6 0.8 1236

238

240

242

244

246

248

250

T

c

(b) temperature

position

Mach[-]

-0.2 0 0.2 0.4 0.6 0.8 10.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

(c) Mach number

number of iterations

log(r

esidualL

1-norm)

0 20000 40000 60000 80000 100000-12

-10

-8

-6

-4

-2

0

2

4

(d) residuals

Fig. 14. Visualisation ofthe simulation results ofa subsonic fluid flow ofa superheated methanol vapor through a nozzle using an explicit Euler solver. The charts display

the non-converged, discontinuous results ifthe PRSV CEoS ( ) is used forthe calculation of thethermodynamic properties, andthe convergedones that result from

adopting theiPRSV CEoS ( ) instead.

and the average absolute deviations (AAD%) are summarized in

Table 5. The differences ofthe PTdata and the Ths data of

propane between all models and the reference EoS are negligi-ble. For methanol, if the calculated liquid densities are taken into

account, the PRG model performs slightly worse. If they are not

taken into account, then iPRSV performs best, closely followed by

the PRGandthe PRSV. Thedifference indeviationsfor theThs cal-

culations is more substantial: the PRSV performs 0.27% point better

than the iPRSV model, and 0.52% point better than PRG in terms of

AAD.

Similar calculations as for propane and methanol were carried

out for alkanes with increasing sizes (n-butane, n-hexane, and n-

octane). Just the calculation results are summarized in Table 5.

In general the deviations are decreasing for increasing compo-

nent sizes. For PTdata in the vapor phase the deviations arequite similar for the PRSV and iPRSV model, but increasing for

PRG EoS.

5. Results and conclusions

The PRSV equation of state is widely used in computer pro-grams for scientific and industrial applications. It features a good

trade-off between accuracy and computational speed, and it pro-

vides the possibility ofextending the model to mixtures. The PRSV

thermodynamic model, however, features a numerical discontinu-

ity in all thermodynamic properties at the critical temperature Tcfor water andalcohols andat T= 0.7 Tcfor other fluids. The discon-

tinuity in the attractive termofthis cubic equation ofstate hasbeen

introduced in orderto improve the accuracyofcomputed saturated

properties. Such a discontinuity, besides being a non-physical phe-

nomenon, generates numerical problems in computer simulations

relying on thecomputationoffluidthermodynamic properties. This

issue is brieflytreated here using thecanonical example of the sim-

ulation ofa fluid flow through a convergingdiverging nozzle (see

Figs. 13 and 14).


11/12


The quasi-one dimensional inviscid flow through the noz-

zle is solved by means of the one-dimensional Euler equations

with source terms to mimic the convergingdiverging nozzle [44].

The conservation equations for mass, momentum and energy are

solved using a first-order finite volume scheme and an explicit

time integration. The convective fluxes are evaluated using the

approximate Riemann solver proposed by Liou [45]. Since the

approximate Riemann solver requires the evaluation of the speed

of sound, enthalpy and pressure, the solver is coupled to the PRSV

CEoS.

The conditions at the inlet and outlet ofthe nozzle (Fig. 13a) are

chosen in the superheated vapor phase slightly above the critical

temperature, such that the flow through the nozzle is acceler-

ated towards the throat and decelerated afterwards. A subsonic

flow develops, whereby at the throat the temperature has its

minimum, which is below the critical temperature. The flow

through the nozzle has therefore to cross the critical temperature

two times; slightlyupstream and slightlydownstream of the throat

location (Fig. 13b). The corresponding total conditions at the inlet

are T1,tot =253C and P1,tot = 4MPa and the static pressure at the

outlet is set to be P2= 3.85MPa.

The results in Fig. 14 show the pressure, temperature, Mach

number through the nozzle and the normalized L1-norm residu-

als of the energy equation for the simulations. The results obtainedwith the PRSV, containing the discontinuities in the CEoS, clearly

show non-smooth profiles for the pressure, temperature andMach

number at the location where the critical temperature is crossed.

These discontinuities prevent the solution to converge to steady

state as seen in Fig. 14d by means ofthe normalized L1 residual

of the energy equation as a function of the number of iteration

steps.

This paper presents a solution to this problem in the form

of a modification ofthe -function composing the temperature-

dependent attractive term of this van der Waals-type equation

of state. The continuous -function ofthe improved PRSV, iPRSV,equation of state closely resembles the original function apart

from the discontinuity. The requirement of a close match to the

original function was specified as a constraint during the devel-opment of the new function. The new formulation features the

additional and non-trivial advantage that the fluid parameters

of the original model, and 1 are unchanged. If a computer

program already implements the PRSV EoS, the database pro-

viding the fluid-specific input values need not to be modified,

and only changing few lines ofcode allows for a quick imple-

mentation. Fig. 14 shows that the iPRSV fluid model solves the

robustness issues affecting the PRSV EoS in fluid dynamics simu-

lations encompassing the critical point region. The solution for the

flow through the nozzle converges to machine precision (Fig. 14d)

in case the iPRSV CEoS is used, leading to smooth profiles for the

temperature, pressure and Mach number throughout the whole

nozzle.

The performance of the iPRSV model has been evaluated bycomparison with the original PRSV equation of state and with

the PRG equation of state. The latter is also consistent in the

entire thermodynamic space. Values computed with these cubic

equations of state have been compared, taking measurements of

saturated pressures and values calculated with reference equa-

tions of state as a reference for, respectively, saturated states,

and subcooled liquid and superheated vapor states for exemplary

fluids.

The accuracy of the iPRSV model is comparable to the

original PRSV model and better than the PRG equation of

state, thus solving the issue of the artificial discontinuity in

the calculation of properties relevant to scientific and indus-

trial applications, at the cost of a small decrease in overall

accuracy.

Acknowledgements

The authors would like to thank their colleague and friend Dr.

Rene Pecnikfor his help with andhis valuable comments about the

flow simulation calculations for the convergingdiverging nozzle.

Appendix A.

Derivatives ofthe equation ofthe iPRSV EoS with respect to

the temperature:

= 0 + 1(Tx + Ty)Tz

d

dT =

1(Tx + Ty)

Tc

1

2Tz

DTzTy

d2

dT2 =

1(Tx + Ty)

T2c

D

TyTz+

1

4T3z+

D2Tz

T3y(Tx Ty)

d3

dT3 =

31(Tx + Ty)

T3c

D

4TyT3z

+1

8T5zD2

T3y(TxTy)(

DTxTz

T2y+

1

2Tz)

with:

Tr =T

TcTx = AD(Tr + B)

Ty =

T2x + E

Tz=

Tr + C

A = 1.1B = 0.25C= 0.2D = 1.2E= 0.01

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