the iprsv equation of state-libre
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Fluid Phase Equilibria 330 (2012) 2435
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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
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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.
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T.P.vander Stelt etal./ FluidPhaseEquilibria330 (2012) 2435 27
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
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28 T.P. van der Stelt et al. / Fluid Phase Equilibria330 (2012) 2435
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).
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T.P.vander Stelt etal./ FluidPhaseEquilibria330 (2012) 2435 29
(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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30 T.P. van der Stelt et al. / Fluid Phase Equilibria330 (2012) 2435
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.
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T.P.vander Stelt etal./ FluidPhaseEquilibria330 (2012) 2435 31
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
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32 T.P. van der Stelt et al. / Fluid Phase Equilibria330 (2012) 2435
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
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T.P.vander Stelt etal./ FluidPhaseEquilibria330 (2012) 2435 33
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).
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34 T.P. van der Stelt et al. / Fluid Phase Equilibria330 (2012) 2435
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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http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1002/aic.12594http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1002/aic.12594