# Corrigendum to: “Combining the entropy-scaling concept and cubic- or SAFT equations of state for modelling thermal conductivity of pure fluids”

Aghilas Dehlouz, Jean-Noël Jaubert, Guillaume Galliero, Marc Bonnissel, Romain Privat

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## To cite this version:

Aghilas Dehlouz, Jean-Noël Jaubert, Guillaume Galliero, Marc Bonnissel, Romain Privat. Corrigendum to: “Combining the entropy-scaling concept and cubic- or SAFT equations of state for modelling thermal conductivity of pure fluids”. International Journal of Heat and Mass Transfer, 2025, 236, pp.126121. 10.1016/j.ijheatmasstransfer.2024.126121. hal-04774958

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Corrigendum

Corrigendum to: “Combining the entropy-scaling concept and cubic- or
SAFT equations of state for modelling thermal conductivity of pure fluids”

<figure><img src="image_3.png" /></figure>

Aghilas Dehlouz<sup>a, b</sup>, Jean-Noël Jaubert<sup>a, *</sup>, Guillaume Galliero<sup>c</sup>, Marc Bonnissel<sup>b</sup>, Romain Privat<sup>a, *</sup>

The authors regret to report that, in their paper published in 2022
[1], they have identified an error in the calculation of the Tv-residual
entropy using the tc-PR equation of state. We sincerely apologize for
any caused inconvenience. For the sake of clarity, the mathematical
expression of the translated-consistent Peng-Robinson (tc-PR) [2] cubic
equation of state (EoS) and the corresponding correct formula for the
Tv-residual entropy are given below.

For a pure component, the tc-PR equation writes:

$$
P(T, v) = \frac{RT}{v + c_i - b_i} - \frac{a_i(T)}{(v + c_i)(v + c_i + b_i) + b_i(v + c_i - b_i)}
$$

$$
\begin{equation}
\begin{aligned}
& \text{with: } \left\{
\begin{array}{l}
a_i(T) = a_{c,i} \cdot \alpha_i(T_r); \quad T_r = \frac{T}{T_{c,i,\exp}} \\
\alpha_i(T_r) = T_r^{N_i(M_i-1)} \exp[L_i(1-T_r^{M_i}N_i)] \\
c_i = v_{i,liq}^{\text{sat},u-\text{PR}}(T_r=0.8) - v_{i,liq,\exp}^{\text{sat}}(T_r=0.8) \\
\eta_c = \left[1 + \sqrt[3]{4-2\sqrt{2}} + \sqrt[3]{4+2\sqrt{2}}\right]^{-1} \approx 0.25308 \\
a_{c,i} = \frac{40\eta_c + 8}{49 - 37\eta_c} \frac{R^2 T_{c,i,\exp}^2}{P_{c,i,\exp}} \approx 0.45724 \frac{R^2 T_{c,i,\exp}^2}{P_{c,i,\exp}} \\
b_i = \frac{\eta_c}{\eta_c + 3} \frac{RT_{c,i,\exp}}{P_{c,i,\exp}} \approx 0.07780 \frac{RT_{c,i,\exp}}{P_{c,i,\exp}}
\end{array}
\right.
\end{aligned}
\tag{C1}
\end{equation}
$$

Such a model is said translated [3,4] because a substance-dependent
volume-translation parameter, cᵢ, is introduced, in order that the

translated EoS exactly reproduces the experimental saturated liquid
volume at a reduced temperature of 0.8, [$v_{i,liq,exp}^{sat}(T_r = 0.8)$]:

$$
c_i = v_{i,liq}^{sat,u-CEoS}(T_r = 0.8) - v_{i,liq,exp}^{sat}(T_r = 0.8) \quad (C2)
$$

where $v_{i,liq}^{sat,u-CEoS}(T_r = 0.8)$ is the molar volume calculated with the original (untranslated) EoS at $T_r = 0.8$. In turn, *consistent* stands for the fact that the *tc-PR* EoS relies on an *α*-function that passes the consistency test proposed by Le Guennec et al. [5,6]

Starting from Eq. (C1), the mathematical expression of the Tv-
residual entropy can be straightforwardly determined by:

$$
\begin{align}
s_{Tv\text{-res}} &= \left[ \left. \frac{\partial}{\partial T} \int_{\infty}^{v} \left( P(T, v) - \frac{RT}{v} \right) dv \right]_{v} \nonumber \\
&= R \ln \left( \frac{v + c_i - b_i}{v} \right) - \frac{1}{b_i(r_1 - r_2)} \ln \left( \frac{v + c_i - b_i r_1}{v + c_i - b_i r_2} \right) \cdot \underbrace{a_{c,i}}_{\text{omitted in 2022}} \frac{\partial \alpha_i}{\partial T} \tag{C3}
\end{align}
$$

$$
\text{With: } \begin{cases} r_1 = -1 - \sqrt{2} \\ r_2 = -1 + \sqrt{2} \\ \frac{\partial \alpha_i}{\partial T} = \frac{\alpha_i N_i}{T} (M_i - 1 - L_i M_i T_r^{M_i N_i}) \end{cases}
$$

Unfortunately, in our paper from 2022, the a<sub>c,i</sub> quantity that appears
in Eq. (C3) was involuntary omitted in our software leading to erroneous
values of s<sub>Tv-res</sub>.

Despite this regrettable error, the general conclusions and observa-
tions reported in our original article remain valid. After correcting this

<footnote><sup>a</sup> Université de Lorraine, Ecole Nationale Supérieure des Industries Chimiques, Laboratoire Réactions et Génie des Procédés (UMR CNRS 7274), 1 rue Grandville, Nancy 54000, France</footnote>

<footnote><sup>b</sup> Gaztransport & Technigaz (GTT), 1 route de Versailles, Saint-Rémy-les-Chevreuses 78470, France</footnote>

<footnote><sup>c</sup> Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, LFCR UMR 5150, Avenue de l'Université, Pau 64000, France</footnote>

<footnote>DOI of original article: https://doi.org/10.1016/j.ijheatmasstransfer.2022.123286.</footnote>

<footnote>* Corresponding authors.</footnote>

<footnote>E-mail addresses: jean-noel.jaubert@univ-lorraine.fr (J.-N. Jaubert), romain.p Privat@univ-lorraine.fr (R. Privat).</footnote>
<figure><img src="image_4.png" /><figcaption>Fig. 5. Overview of all the experimental data in the $(ln\tilde{\lambda}, X_{ES})$ plane. The thermodynamic properties involved in the calculation were estimated with the I-PC-SAFT EoS in the left subplot and with the tc-PR EoS in the right subplot. Each color corresponds to a chemical family and each symbol corresponds to a given chemical specie.</figcaption></figure>

error, it was possible to conclude that the correction introduced at high temperatures for light molecules was no longer necessary when the tc-PR EoS was used. As a result, exactly the same formalism is now used for both the I-PC-SAFT and tc-PR EoS. This is a positive point. A more detailed analysis of results shows that by using component-specific parameters, the results remain excellent with a global average deviation around 4%. When chemical-family specific parameters are used, the overall mean deviation rises from 5.6% to 10.1%, becoming comparable to that obtained with the I-PC-SAFT EoS. This increase is due to a poorer correlation of hydrocarbons when using chemical-family parameters. By using universal parameters, the average deviation on our complete database increases slightly and the deviations obtained with the tc-PR EoS (around 12%) become comparable to those obtained with the I-PC-SAFT EoS (around 11%).

Clearly, the numerical values of the coefficients to be used in the developed correlations when the tc-PR EoS is considered have to be updated. All results related to the I-PC-SAFT EoS [7] are unaltered by our error.

**Corrigendum to the Supplementary Material:** all fitted coefficients related to the tc-PR EoS based model are now updated in the new version of the supplementary material that was built with the correct values of $s_{Tv-res}$.

**Corrigendum to section 2.3:** the section "Our paper devoted to viscosity underlined that numerical values of $s_{Tv-res}$ could differ significantly from one EoS to another. Although dividing the Tv-residual entropy by its value at the critical point strongly attenuates this difference, we found necessary to add a temperature-dependent term in the mathematical expression of the $X_{ES}$ variable (see Eq. (11)) when the tc-PR EoS is used. This patch, specific to the tc-PR EoS and only needed at very high temperatures ($4T_c < T < 8T_c$, where $T_c$ denotes the critical temperature of the fluid) was introduced to keep the same mathematical expression of the $X_{ES}$ variable (see Eq. (10)) regardless of the EoS used to estimate the density, residual isochoric heat capacity and residual entropy: it is indeed enough to select $f_{corr}(T) = 1$ in Eq. (11) when one works with the I-PC-SAFT EoS. To sum up, for heavy molecules for which the temperature domain $T > 4T_c$ is not of practical

interest, a null $\alpha$ value is selected in Eq. (11). Otherwise, for light molecules (having a low $T_c$) such as Ar, $N_2$ or Krypton for which experimental thermal conductivities are available at high reduced temperatures ($4T_c < T < 8T_c$) a non-zero coefficient was selected. The influence of this correction is illustrated with Argon in Fig. 16 of Appendix A. It is worth noting that as long as a given equation of state qualitatively reproduces the temperature and pressure dependence of the Tv-residual entropy regardless of the fluid conditions (as the I-PC-SAFT EoS does), it is not necessary to apply this patch", including (Eq. 11) of our original article, should be discarded since the concerned phenomenon is considerably reduced and becomes negligible when using the correct values of $s_{Tv-res}$. In other words, after correcting the Tv-residual entropy values, it was found that $f_{corr}(T) = 1$ was also suitable for the tc-PR EoS.

**Corrigendum to equation 16:** with the correct values of the residual entropy returned by the tc-PR EoS, the authors found it unnecessary to modify the $X_{ES}$ formulation for high temperatures and thus the same $X_{ES}$ as for the I-PC-SAFT EoS should be used for the tc-PR EoS. Therefore, (Eq. 16) should be corrected as follows.

$$ \left\{ \begin{aligned} & \ln \left( \frac{\lambda}{\lambda_{\text{ref}}^{\text{Rosenfeld, intDoF}}} \right)_{\text{calc}} = \left[ \left( \frac{a_1 + a_2 X_{\text{ES}}^{1/3}}{1 + e^{c_{\text{ES}}}} \right) + \left( \frac{b_1 + b_2 X_{\text{ES}}^{1/3}}{1 + e^{-c_{\text{ES}}}} \right) \right] X_{\text{ES}} + d \\ & \lambda_{\text{ref}}^{\text{Rosenfeld, intDoF}} = \rho_N^{2/3} k_b \sqrt{\frac{k_b T}{m_0}} \left[ 1 + \left( \frac{c_v^{\text{real}}}{R} \right) e^{-(\rho/\rho_c)} \right] \\ & X_{\text{ES}} = \left( -\frac{\tilde{s}_{\text{Tv-res}}}{\tilde{s}_{\text{Tv-res}}} \right) - \ln \left( \frac{\tilde{s}_{\text{Tv-res}}}{\tilde{s}_{\text{Tv-res}}} \right) \text{ for both I-PC-SAFT and tc-PR EoS} \end{aligned} \right. \quad (16) $$

**Corrigendum to the overall MAPEs obtained using the tc-PR EoS:** the overall MAPEs were 3.38%, 5.57% and 9.19% for the component-specific, the chemical-family specific and the universal parameters respectively. They are now 4.28%, 10.09% and 12.28% respectively after using the correct values of the Tv-residual entropy.

**Corrigendum to Appendix A:** the correction that was originally proposed to counteract some undesired behavior at high temperatures is
tc-PR - component-specific parameters

<figure><img src="image_5.png" /><figcaption>**Fig. 7.** Results of the combination of the component-specific parameters with the tc-PR EoS for modelling the thermal conductivity of n-alkanes, branched alkanes, cycloalkanes, alkenes, aromatics, esters, ethers and HFC-CFCs. Symbols represent the experimental data points; Solid lines represent the calculated values.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 9. Results of the combination of the component-specific parameters with the tc-PR EoS for modelling the thermal conductivity of alcohols, noble gases, aldehydes and other chemical species. Symbols represent the experimental data points; Solid lines represent the calculated values.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 10. Summary of MAPEs obtained with the I-PC-SAFT and the tc-PR EoS in combination with the component-specific parameters.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 12.** Results of the combination of the chemical-family specific parameters with the tc-PR EoS for modelling the thermal conductivity of paraffins and HFC-CFCs. Symbols represent the experimental data points; Solid lines represent the calculated values.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 13.** Summary of MAPEs obtained with the I-PC-SAFT and the tc-PR EoS in combination with the chemical-family specific parameters.</figcaption></figure>

now discarded due to the phenomenon becoming negligible while using the correct values of the Tv-residual entropy. Therefore, the appendix A which contains Fig. 16 should be discarded.

**Corrigendum** to figs 5, 7, 9, 10, 12, 13, 14, 15: after the correction of the Tv-residual entropy values, involved in the XES variable, the figures reporting the results of the tc-PR EoS are updated as follows.

**CRediT authorship contribution statement**

Aghilas Dehlouz: Conceptualization. Jean-Noël Jaubert: Writing –

review & editing. **Guillaume Galliero:** Supervision. **Marc Bonnissel:** Resources. **Romain Privat:** Writing – review & editing.

**Declaration of competing interest**

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
<figure><img src="image_10.png" /><figcaption>**Fig. 14.** Results of the combination of the universal parameters with the I-PC-SAFT and the tc-PR EoS for modelling the thermal conductivity. Each color corresponds to a chemical family and each symbol corresponds to a given chemical species. Solid lines represent the calculated values.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. 15.** Summary of MAPEs obtained with the I-PC-SAFT and the tc-PR EoS in combination with the universal parameters.</figcaption></figure>

## Acknowledgment

The authors would like to warmly thank Jarrell Richard ELLIOT, Professor at the university of Akron, for pointing out the error reported in this corrigendum.

## Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijheatmasstransfer.2024.126121.

## References

[1] A. Dehlouz, J.-N. Jaubert, G. Galliero, M. Bonnissel, R. Privat, Combining the entropy-scaling concept and cubic- or SAFT equations of state for modelling thermal conductivities of pure fluids, Int. J. Heat Mass Transf. 196 (2022) 123286, https://doi.org/10.1016/j.ijheatmasstransfer.2022.123286.

[2] Y. Le Guennec, R. Privat, J.-N. Jaubert, Development of the translated-consistent tc-PR and tc-RK cubic equations of state for a safe and accurate prediction of volumetric, energetic and saturation properties of pure compounds in the sub- and super-critical domains, Fluid Phase Equilib. 429 (2016) 301–312, https://doi.org/10.1016/j.fluid.2016.09.003.

[3] J.-N. Jaubert, R. Privat, Y. Le Guennec, L. Coniglio, Note on the properties altered by application of a Péneloux-type volume translation to an equation of state, Fluid Phase Equilib. 419 (2016) 88–95, https://doi.org/10.1016/j.fluid.2016.03.012.
[4] R. Privat, J.-N.; Le Jaubert, Y. Guennec, Incorporation of a volume translation in an equation of state for fluid mixtures: which combining rule? which effect on properties of mixing? Fluid Phase Equilib. 427 (2016) 414–420, https://doi.org/10.1016/j.fluid.2016.07.035.

[5] Y. Le Guennec, S. Lasala, R. Privat, J.-N. Jaubert, A consistency test for α-functions of cubic equations of state, Fluid Phase Equilib. 427 (2016) 513–538, https://doi.org/10.1016/j.fluid.2016.07.026.

[6] Y. Le Guennec, R. Privat, S. Lasala, J.-N. Jaubert, On the imperative need to use a consistent α-function for the prediction of pure-compound supercritical properties

with a cubic equation of state, Fluid Phase Equilib. 445 (2017) 45–53, https://doi.org/10.1016/j.fluid.2017.04.015.

[7] E. Moine, A. Pina-Martinez, J.-N. Jaubert, B. Sirjean, R Privat, I -PC-SAFT: an industrialized version of the volume-translated PC-SAFT equation of state for pure components, resulting from experience acquired all through the years on the parameterization of SAFT-type and cubic models, Ind. Eng. Chem. Res. 58 (45) (2019) 20815–20827, https://doi.org/10.1021/acs.iecr.9b04660.