Paper ID: ETC2019-141 Proceedings of 13th European Conference on Turbomachinery Fluid dynamics & Thermodynamics ETC13, April 8-12, 2018; Lausanne, Switzerland

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COMPARISON OF THE CUBIC EQUATIONS OF STATE AND

DIFFERENT TRANSPORT PROPERTIES MODELS FOR ORC

TURBINES MODELING

A.A. Sebelev 1, M.V. Smirnov

2, N.I. Kuklina

1, K.L. Lapshin

2, A.S. Laskin

2

1Engineering center “Center of computer-aided engineering”, Peter the Great Saint-Petersburg

Polytechnic University, Saint-Petersburg, Russia

e-mail: a.sebelev.turbo@mail.ru 2Peter the Great Saint-Petersburg Polytechnic University, Saint-Petersburg, Russia

ABSTRACT

The present paper focuses on a comparison of cubic equations of state (EoS) with accurate

Span-Wagner (SW) EoS. Aungier Redlich-Kwong (ARK) EoS, Soave Redlich-Kwong (SRK)

EoS and standard Peng-Robinson (PR) EoS were chosen as the basic EoS implemented into

ANSYS CFX commercial solver. Furthermore, modeling of transport properties was also

considered. Rigid Non Interacting Sphere (NIS) and Interacting Sphere (IS) viscosity models

were chosen for a comparison with REFPROP data.

Analytically calculated density and viscosity were compared with the REFPROP values for

1 polar (ethanol) and 2 non-polar (R245fa, MDM) fluids at the first part of the investigation.

This part showed no significant difference between compared EoS for the non-polar fluids

and slightly lower accuracy of PR EoS for the polar fluid. A significant difference between NIS

and IS models was showed.

2D CFD simulations of VKI LS-89 cascade were performed for both MDM and MD4M

siloxanes at the second part. It was showed that a preliminary analysis of the fundamental

gasdynamic derivative behavior for isentropic expansion is needed for choosing of an

appropriate cubic EoS. The choice of NIS or IS transport models does not show a significant

effect on the obtained results.

KEYWORDS

ORC FLUIDS PROPERTIES, CUBIC EQUATIONS OF STATE, TRANSPORT

MODELS, CFX

NOMENCLATURE

a1 – a5 ideal gas specific heat coefficients α cascade discharge angle (deg.)

c velocity (m/s) ζ profile loss coefficient

Cax axial chord φ velocity ratio

Cf skin friction coefficient ρ density (kg/m3)

Г fundamental gasdynamic derivative μ dynamic viscosity (μPa·s)

H0 isentropic enthalpy drop (J/kg)

M molar mass (g/mol) Subscript

p pressure (Pa) * total parameters

T temperature (K) 0 parameters at the inlet

V molar volume (cm3/mol) 1 parameters at the outlet

x x coordinate (m) c critical point parameters

Z compressibility factor is isentropic

2

INTRODUCTION

Nowadays organic Rankine cycle (ORC) is one of the most popular technologies used for small

recovery power plants. However, an overall efficiency of ORC plants is basically below 25%. One

of the key factors for such a situation is an efficiency of ORC expander, which in many instances

varies from 65 to 75% due to the intricate physical phenomena of ORC fluids (Fu et al. (2015),

Weiß et al. (2018)). In order to tackle with this issue CFD approach is commonly used. Different

commercial and in-house software is in use for simulations of the whole cycle in general and the

expansion process in particular. All the software use different methods for the evaluation of

thermodynamic and transport properties of ORC fluids. These approaches could be divided into two

main groups: tabular specification of the fluid properties and using of real gas equations of state

(EoS).

It was shown by different researchers that tabular specification of the fluid properties is able to

provide a good accuracy of the simulation results (e.g., Hoffren et al. (2002), Wheeler and Ong

(2013)). Praneeth and Hickey (2018) showed that modern algorithms are very computationally

effective and may provide up to 2 times faster simulation than using traditional Peng-Robinson

EoS. Despite this lookup tables generation procedure itself has a significant computational cost and,

hence, this approach is not an objective of the current paper.

Real gas EoS, in its turn, could be divided into fluid individual (reference) EoS and universal

EoS (e.g., Span et al. (2001)). Among the variety of universal EoS the multiparameter EoS

demonstrate the best accuracy. Nowadays the most widely used multiparameter EoS is Span-

Wagner (SW) EoS, described by Span and Wagner (2003). However, its computational cost severely

limits a preliminary design process and optimization of ORC applications. Congedo et al. (2011)

showed that in case of dense-gas simulation through a plane cascade SW EoS has 36% higher

computational cost than Peng-Robinson-Stryjeck-Vera (PRSV) cubic EoS (Proust and Vera (1989)).

It was also shown that in the considered case the accuracy of PRSV EoS is in the range of 10%.

Harinck et al. (2010) presented a quantitative comparison of the simulations of sub- and

supercritical expansions of water steam, toluene and R245fa with using perfect ideal gas (PIG)

model, PRSV EoS and multiparameter EoS. It was shown that using of the PIG model leads to large

deviations up to 18 – 25% in density and other parameters. At the same time the simulations with

PRSV EoS demonstrated high accuracy even for supercritical expansions. Despite the advantages of

the PRSV EoS it is still not implemented into some commercial software.

Lujan et al. (2012) investigated the deviations between calculated thermodynamic properties

and NIST data for R245fa caused by using Soave Redlich-Kwong (SRK) EoS (Soave (1980)) and

Peng-Robinson (PR) EoS. The maximum deviations for SRK and PR EoS was in the range of 10%

for subcritical region.

Sauret and Gu (2014) performed an investigation of a R143a radial-inflow turbine with using

PR EoS. The choice of the EoS was motivated by the good balance between simplicity and

accuracy of PR EoS. Gori et al. (2015) performed a numerical investigation for the non-ideal

supersonic flow in nozzles. PRSV and SW EoS were used with the SU2 solver. Papes I. et al.

(2015) performed dynamic model for an efficiency evaluation of a screw expander. Aungier

Redlich-Kwong (ARK) EoS (Aungier (1995)) was used. Finally, Kolasinski et al. (2016) performed

an investigation of a rotary vane expander operated with R123. SRK EoS was used.

It is known that critical compressibility factor (Zcr) for SRK EoS is 0.333, for PR EoS – 0.307

(Ramdharee et al. (2013)). These values are very close to the Z values of the majority of ORC fluids

at real operating conditions. Consequently, using of the classical cubic EoS may cause the density

underprediction. Despite the sheer volume of the investigations of the cubic EoS accuracy there is

still no fully completed investigation which allows to establish their range of applicability. Thus, the

first scope of the present paper is to provide qualitative comparison of the cubic EoS (ARK, SRK,

PR) with SW multiparameter EoS for the different classes of ORC fluids at different conditions.

Another important issue is related to a modeling of ORC fluids transport properties. Cramer and

Park (1999) used Chung-Lee model (Chung et al. (1984)) for the transport properties prediction.

3

This is so-called Interacting sphere (IS) model in terms of ANSYS CFX solver. Cinnella and

Congedo (2008) also used Chung-Lee model for the transport properties modeling in the dense-gas

region. While it is obvious that non-interacting sphere (NIS) model might be invalid in the dense-

gas region, its applicability in the subcritical region is not well studied. It is also should be noticed

that there was not provided a qualitative comparison between NIS and IS models for the ORC

fluids. Thus, the second scope of the present paper is to provide the qualitative comparison of NIS

and IS models for the different classes of the ORC fluids at different conditions.

METHODOLOGY

At the first part analytically calculated density values for the selected EoS were compared with

the REFPROP values calculated with SW EoS. The comparison is provided for a wide range of the

thermodynamic parameters in superheated vapor region. The pressure range was set as

0.8 ≤ p/pc ≤ 1.5, the temperature range was set so as to fulfill 2 conditions:

minimal temperature is the saturated vapor temperature for the minimal pressure;

maximal temperature is defined with the maximal pressure and critical entropy.

The saturation curves were calculated with the reference SW EoS. Approximately, the

temperature range is 0.972 < T/Tc < 1.035.

This part implies only static comparison without modeling any complex gasdynamic

phenomena and allows to establish the range of pressure and temperature where the selected cubic

EoS provide acceptable deviation from SW EoS. The forms of ARK, SRK and PR cubic EoS are

widely described in literature (see Aungier (1995), Soave (1980)) and, hence, are omitted here.

The comparison of the transport models at the first part was provided in the same manner.

Analytically calculated viscosity values for NIS and IS models were compared with the REFPROP

values. The description of NIS and IS models is provided in literature (see Chung et al. (1984)) and

is also omitted. REFPROP uses different viscosity models for different fluids. For ethanol it is

SAFT-DFT/DMT model (Kiselev et al. (2005)), for R245fa – model described by Huber et al.

(2003). Viscosity of siloxanes is calculated in predictive mode with using form from Huber et al.

(2003). For the last case REFPROP vendor estimates viscosity deviations at a level of 5-10%. Thus,

for siloxanes the comparison with the REFPROP viscosity values does not provide secure data for

the NIS and IS models accuracy, but demonstrates how far are they applicable (or not) for the

viscosity calculation of complex ORC fluids.

At the second part there was performed the numerical simulation of the supercritical expansion

through VKI LS-89 cascade. Here the chosen EoS were compared to each other. The variation of

the transport models was also carried out.

Fluids Selection

The selection of the fluids is based on their applicability for ORC systems.

Kunte and Seume (2013) applied ethanol as the working fluid for the ORC automotive recovery

unit. Boretti (2012) used R245fa refrigerant for the same application. Siloxanes are a prospective

class of organic fluids for the ORC applications (see, e.g., Colonna et al. (2007)). MDM siloxane is

under investigation at TROVA shock tube at Milan polytechnic university (see, e.g., Spinelli et al.

(2016)). MD4M siloxane, as was showed by Guardone et al. (2014), has a BZT region in Clapeyron

diagram lower than its thermal stability limit and was chosen for LS-89 cascade simulation only.

Mixture working fluids were not considered. Thermodynamic properties of the chosen working

fluids are given in table 1.

Table 1: Thermodynamic properties of the chosen working fluids

M,

g/mol acentric factor

pc, MPa

Tc, K Vc,

cm3/mol

Tboil, K

Ideal gas specific heat capacity coefficients a1 a2, K^-1 a3, K^-2 a4, K^-3 a5, K^-4

Ethanol 84.04 0.259 3.492 346.3 193.600 225.86 2.577 2.3727E-02 4.8000E-06 -2.8240E-08 1.4390E-11

R245fa* 134.05 0.378 3.651 427.2 286.125 288.29 226.85 2.816E+00 3.3545E-01 -1.4421E-04 0

MDM**

236.53 0.529 1.415 564.1 921.282 425.66 9.4056 1.2051E-01 -6.8020E-05 1.5776E-08 0

MD4M***

458.99 0.825 0.877 653.2 1649.975 533.90 18.538 2.2507E-01 -1.3132E-04 3.1341E-08 0 *Harinck et al. (2010);

**Lai et al. (2011);

***Nannan et al. (2007)

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Simulation of the Supercritical Expansion through VKI LS-89 Cascade

VKI LS-89 planar cascade, widely described by Arts et al. (1990), was chosen as the test case

for the second part of the investigation. The numerical simulation was performed for MDM and

MD4M siloxanes only. The selected cubic EoS were combined with both NIS and IS transport

models. The initial conditions were chosen so as to fulfill 2 conditions:

start of the expansion process in the dense-gas region;

Z > Zcr.

The pressure ratio was taken from MUR49 test conditions, described by Arts et al. (1990). The

overview of the test conditions is presented in table 2. The diagram of the isentropic expansion

process is presented in figure 1a.

Table 2: The overview of the test conditions p0/pc p0, MPa p0/p1 T0/Tc T0, K Z0

MDM 1.10 1.5565 1.938

1.015 572.55 0.367

MD4M 1.08 0.9471 1.010 659.73 0.363

Figure 1: The diagram of the isentropic expansion process (a) and results of the grid

convergence study (b) (pressure distributions are presented for the pair PR+NIS, Cf

distributions – for the pair ARK+IS; C – coarse grid, M – medium grid, F – fine grid)

Solver settings

The numerical simulation was performed with using the commercial solver ANSYS CFX. The

second order so-called High resolution advection scheme was used. The k-ω SST turbulence model

with wall functions (y+ > 30 (High Reynolds)) and without wall functions (y+ < 5 (Low Reynolds))

was used. Medium turbulence intensity (5%) was specified at the inlet boundary. The pair total

parameters / static pressure was used as the boundary conditions. The following convergence

criteria were used:

drop of the RMS and maximum residuals not less than 104;

imbalances less than 0.5%;

fluctuation of velocities at the monitor points less than 1%.

Grid convergence study

The grid convergence study was performed for each combination EoS + Transport model +

HighRe / LowRe grid for MDM expansion process. The following H-grids were used: 150.9k nodes

(coarse), 284.8k nodes (medium), 776.2k nodes (fine). Figure 1b presents pressure and skin friction

coefficient distributions along the airfoil, table 3 presents convergence of the density and velocity

product at the cascade outlet. The control section is located 2 mm downstream the trailing edge. The

5

results for medium and fine grids are presented as the relative difference between sequent and

previous grids (medium to coarse, fine to medium).

Analysis of the table 3 shows that for all the combinations EoS + Transport model coarse grids

are sufficient: the difference in ρ1c1 between fine and coarse grids does not exceed 2%. Figure 1b

demonstrates very close distributions of the pressure and skin friction coefficient along the airfoil

for all the grids. However, detailed analysis of the solutions convergence shows that for the

considered grids consequent refinement leads to convergence deterioration. Comparison of the

results for HighRe and LowRe grids indicates negligible difference in ρ1c1 (less than 1.5%) and

expected discrepancy in the parameters along the airfoil where y+ < 30. Having regard to the above,

the results for medium LowRe grids were used for further analysis.

Table 3: Results of the grid convergence study

ρ1c1,

kg/(m2s)

MDM MD4M

HighRe LowRe LowRe

C M F C M F C M F

ARK+NIS 6976.15 +0.27% +0.19% 6997.44 -0.36% +0.22%

N/C

5381.15 +0.47%

ARK+IS 6979.68 +0.25% +0.02% 6962.00 +0.15% +0.09% 5382.26 +0.47%

SRK+NIS 6990.24 +0.63% +0.57% 7005.59 -0.44% +0.87% 5396.95 +0.49%

SRK+IS 6992.62 +0.59% +0.66% 7009.16 -0.44% +1.06% 5372.81 +0.37%

PR+NIS 7063.24 +0.65% +1.20% 7082.39 -0.44% +0.29% 5432.04 +0.63%

PR+IS 7064.40 +0.52% +0.50% 7080.64 -0.30% +0.60% 5435.71 +0.61% *C – coarse, M – medium, F – fine, N/C – not converged

DISCUSSION OF THE RESULTS

Analytical Comparison of the EoS and Transport Models

The results obtained for the direct comparison of the EoS and transport models are presented in

figures 2 – 4. The plots in figures 2, 3 represent percentage deviation between the results obtained

with selected cubic EoS and SW EoS. For the each fluid first plot in series represents the

compressibility factor values to establish the correlation between zones with Z < Zcr and density

error higher than 10%. White zones in each plot represent the two-phase region and the subcooled

liquid zone.

Figure 2: Density error plots. Ethanol: (a) Z-factor; (b) ARK density error; (c) SRK density

error; (d) PR density error. R245fa: (e) Z-factor; (f) ARK density error

6

Black zones in each plot represent the zones with the density deviation higher than 10%. The

viscosity deviation plots in figure 4 are performed at the same manner, but black zones represent the

zones with the viscosity deviation higher than 20%.

Analysis of the figures shows that in some cases even when Z > Zcr all the compared cubic EoS

may have the density error higher than 10%. However, the size of such regions is negligible and

maximum calculated deviations there do not exceed 25%.

Figure 2 demonstrates that for polar fluids, such as ethanol, PR EoS provides higher density

errors than SRK and ARK EoS. Nevertheless, ARK EoS is less stable while closing to the saturation

curve.

Figure 3: Density error plots. R245fa: (a) SRK density error; (b) PR density error. MDM:

(c) Z-factor; (d) ARK density error; (e) SRK density error; (f) PR density error

Figure 4: Viscosity deviation plots: (a) Ethanol, NIS; (b) Ethanol, IS; (c) R245fa, NIS;

(d) R245fa, IS; (e) MDM, NIS; (f) MDM, IS

For the non-polar fluids, such as R245fa and MDM, there is observed no significant difference

in the density errors. It should be noted that PR EoS demonstrates primarily positive density errors

7

while ARK and SRK EoS demonstrate only negative ones. To sum up, the maximal pressure for

each of the EoS, limited by the satisfactory deviations from SW EoS, is approximately 1.1pc. It

could be clearly seen that the temperature does not limits the applicability range. Moreover, in some

cases increasing of the temperature may help to get appropriate deviations at higher pressures.

The figure 4 shows that the range of 20% deviation in viscosity is strictly limited for both NIS

and IS transport models. It is surprising that the NIS model demonstrates smaller deviations in the

region 0.9 ≤ p/pc ≤ 1.15 where the IS model completely fails. Contrariwise, the IS model

demonstrates appropriate deviations while pressure is lower than 0.9pc where the NIS model fails.

These figures show that using of the NIS and IS transport models in the dense-gas region is severely

confined.

Numerical Simulation of VKI LS-89 cascade

The results of the LS-89 2D viscous simulations with selected cubic EoS are presented below.

The comparison of the transport models is presented in terms of velocity ratio (eq.1) and loss

coefficient (eq.2):

.2 0

1

H

cφ (1)

.1 2φ (2)

Obtained φ and ζ values for different combinations EoS + transport model are presented in table 4.

Analysis of the table shows that the results obtained with IS model does not significantly differ

from the NIS results. Thus, it can be concluded that both NIS and IS models provide the same

results for trans- and supercritical expansions. Therefore, further analysis is provided for EoS + NIS

combinations only.

Table 4: Obtained values of velocity ratio and loss coefficient for different combinations EoS

+ transport model

φ ζ α, deg. MDM MD4M

MDM MD4M MDM NIS IS NIS IS

ARK 0.9653 +0.05% 0.9819 -0.01% 0.0642 0.0358 17.37

SRK 0.9684 +0.00% 0.9893 -0.29% 0.0664 0.0212 17.33

PR 0.9466 +0.05% 0.9671 -0.01% 0.0998 0.0647 17.55

In general, the obtained distributions (fig.5, 6) for the MDM expansion are very close to each

over and to the results performed by Vitale et al. (2015) with using PRSV EoS. Comparison of the

pressure distributions (fig.6) shows inappreciable deviation of SRK values from ARK and PR ones

in MDM case. For MD4M case pressure distributions along the profile are almost identical for all

the EoS.

a) b) c) Figure 5: Mach number plots for the MDM case: (a) ARK+NIS; (b) SRK+NIS; (c) PR+NIS

8

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

0,98 0,99 1,00 1,01

Г

T/Tc

ARK

SRK

PR

SW

a) c)

d) e) f)

0,0

0,2

0,4

0,6

0,8

1,0

0,98 0,99 1,00 1,01

Г

T/Tc

ARK

SRK

PR

SW

b)

Figure 6: Fundamental gasdynamic derivative calculated for MDM (a) and MD4M (b)

isentropic expansion and airfoil distributions: pressure (MDM case) (c), pressure (MD4M

case) (d), density (MDM case) (e), density (MD4M case) (f)

Analysis of the obtained velocity ratios demonstrates extremely high values in MD4M case

simulated with ARK and SRK EoS and potentially too low value in MDM case with PR EoS.

According to Arts et al. (1990) for Reynolds and Mach numbers of interest the loss coefficient is

approximately 0.06 for the air medium. Despite the more complex shock wave structure compared

with air (fig.5) MDM and MD4M expansions are accompanied by significantly lower velocities.

Consequently, the levels of the loss coefficients may be the same.

Treatment of the fundamental gasdynamic derivative Г (see Colonna et al. (2009)) behavior

shows follows. Figures 6a, b depict Г values calculated for MDM and MD4M isentropic expansions

of interest with using each of the EoS, including reference SW EoS. It can be seen that in MDM

case ARK EoS provides closest to the reference EoS values of Г. Estimation of the cascade

discharge angle provided in table 4 allows to establish the nature of the obtained difference in loss

coefficients. As can be seen, higher difference in discharge angles correlates with higher difference

in loss coefficients. This means that the compared EoS provide different levels of the shock losses.

In addition to the difference in predicted density this provides obtained difference in loss

coefficients.

Contrariwise, in MD4M case ARK EoS completely fails since it predicts falling into BZT region

(Г < 0). SRK EoS predicts minimal value of Г close to zero. This causes significant underprediction

of the loss coefficient for both ARK and SRK EoS. At the same time PR EoS predicts Г > 0 during

the expansion and, hence, higher loss coefficient. To sum up, preliminary analysis of the

fundamental gasdynamic derivative behavior for isentropic expansion and its comparison with

distribution obtained with more accurate EoS may help to make correct choice of the cubic EoS.

9

CONCLUSIONS

The analytical comparison of ARK, SRK and PR EoS showed that the usage of these EoS

provides acceptable deviations from accurate SW EoS in the region p < 1.1pc. For the polar fluids

ARK and SRK EoS demonstrate higher deviations than PR EoS. For the non-polar fluids all the

investigated cubic EoS have similar deviations. Numerical simulations of MDM and MD4M

supercritical expansions showed that the condition Z >Zcr is not sufficient to determine the range of

applicability of the cubic EoS. Even though this condition is fulfilled the investigated EoS may

provide both overprediction and underprediction of the loss coefficient. In particular, it was

observed that in MDM case (non-BZT fluid) PR EoS overpredicts shock losses whereas in MD4M

case (BZT fluid) ARK / SRK EoS uderpredict the loss coefficient. Therefore, preliminary

comparison of the fundamental gasdynamic derivative behavior for isentropic expansion with

distribution obtained with more accurate EoS is a valuable instrument for a choosing of cubic EoS.

The comparison of NIS and IS transport models showed that the range of 20% deviation from

the REFPROP viscosity values is strictly limited for both NIS and IS transport models. Numerical

simulation of MDM and MD4M supercritical expansions showed that both NIS and IS models

provide the same end results.

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