Correspondence pubs.acs.org/IECR
Cite This: Ind. Eng. Chem. Res. 2019, 58, 5744−5745
Reply to Comment on “Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules” Joachim Gross*,† and Gabriele Sadowski‡ †
Institute of Thermodynamics and Thermal Process Engineering, University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart, Germany ‡ Laboratory of Thermodynamics, TU Dortmund University, Emil-Figge-Strasse 70, 44227 Dortmund, Germany
Ind. Eng. Chem. Res. 2019.58:5744-5745. Downloaded from pubs.acs.org by 5.189.205.200 on 04/14/19. For personal use only.
I
n a Comment, Canas-Marin et al.1 point out a misprint in our publication ref 2. We are aware of three misprints in that Article. Our discussions with scientists who implemented the equations of the PC-SAFT model showed that, fortunately, these misprints are rather easy to detect. The most subtle misprint is indeed the one commented by Canas-Marin et al. Identifying misprints is facilitated also by a (rather rudimentary) source code, that we made available on our Web sites as a reference for own implementations. We here take the opportunity to clarify three misprints in our publication,2 namely,
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equation of state. We detail the hard-sphere contribution to the overall residual chemical potential as an example, where ij ∂(ma ̃hsρ) yz 6 zz jj ̅ = (α0· ζ0, k + α1· ζ1, k + α2· ζ2, k + α3· ζ3, k) zzz jjj ∂ ρ π k {T , ρi ≠ k k
(3)
noting that the hard-sphere contribution, according to the notation of the original reference,2 comes with a prefactor m̅ . We define the shorthand notation π ζn , k = (∂ζn/∂ρk )T , ρi≠k = 6 mk dkn and abbreviations αn that, for
• Equation A.35 should have an additional term −(mk − 1) ln(ghs kk) on the right-hand side, as correctly pointed out by Canas-Marin et al. • The right-hand side of eq A.11 should be raised to the power of −1. It is then consistent with eqs 13 and 25. • The superscript of eq A.38 should be “disp” instead of “hs”, in agreement to the headline of the subsection.
computational efficiency, can be precomputed outside the loop over index k, as α0 = −ln(1 − ζ3)
α1 = 3
v , x)
kT
= a ̃res
ji
ij ζ yz 3ζ3 − 1 ζ − ζ2·(ζ2/ζ3)2 + ζ2jjj 2 zzz + 0 3 j ζ3 z (1 − ζ ) 1 − ζ3 (1 − ζ3) 3 k { 3 ij ζ yz − 2jjj 2 zzz ln(1 − ζ3) j ζ3 z (7) k {
j=1
α3 = 3
res y
zz zz ∂x zz k j {T , v , xi≠j
∑ xjjjjjj ∂a ̃
kT
ij ∂(a ̃resρ) yz zz = jjjj zz ∂ ρ k k {T , ρi≠k
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ζ1·ζ2
2
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: +49 (0)711 685 66105. Fax: +49 (0)711 685 66140.
(2)
ORCID
where ρ denotes the vector of substance densities ρi of all substances i. The variables of eq 1 and 2 are equivalent, with v−1 = ρ = ∑Ni ρi and with ρi = xiρ. Equation 2 leads to a rather compact and efficient implementation of the PC-SAFT © 2019 American Chemical Society
2
The other contributions, hard chain (hc), dispersion (disp), association (assoc),3−5 as well as quadrupolar−quadrupolar (qq),6 dipole−dipole (dd),7 and dipolar−quadrupolar (dq),8 are analogously obtained as simple derivatives of the Helmholtz energy density (ãρ) to substance density ρk. Also the modified hard-sphere contribution proposed by Paricaud,9 which suppresses artifacts at very high densities, is easily implemented.
(1)
where in this equation partial derivatives to xi are taken regardless of the summation relation 1 = ∑Nj xj, as if xi was a (unitless) mole-number. We use the notation of our previous work2 without detailed repetition of all quantities. We remind, however, that a ̃res denotes the dimensionless residual Helmholtz energy, a ̃res = Ares /(NkT ). In hindsight, we recommend a more obvious relation, namely, μkres (T , ρ)
(5)
(6)
ij ∂a ̃res yz zz + (Z − 1) + jjj j ∂xk zz k {T , v , xi≠k
N
−
ζ2 1 − ζ3
2 ij ζ2 yz ij ζ2 yz jij ζ1 zyz ζ2 j j z j z α2 = 3jj + jj zz + jj zz ln(1 − ζ3)zzz 2 jj 1 − ζ3 zz j ζ3 z (1 − ζ ) j ζ3 z 3 k { k { k {
CHEMICAL POTENTIAL We also take the opportunity to discuss equations for the chemical potential in a more general sense. In the original publication (eq A.33), we used the relation μkres (T ,
(4)
Joachim Gross: 0000-0001-8632-357X Published: March 28, 2019 5744
DOI: 10.1021/acs.iecr.9b01515 Ind. Eng. Chem. Res. 2019, 58, 5744−5745
Industrial & Engineering Chemistry Research
Correspondence
Gabriele Sadowski: 0000-0002-5038-9152 Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Cañas-Marín, W. A.; Gonzalez, D. L.; Hoyos, B. A. Comment on “Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2019, DOI: 10.1021/acs.iecr.8b06349. (2) Gross, J.; Sadowski, G. Perturbed-Chain SAFT: An Equation of State Based on a Perturbation Theory for Chain Molecules. Ind. Eng. Chem. Res. 2001, 40, 1244−1260. (3) Jackson, G.; Chapman, W. G.; Gubbins, K. E. Phase equilibria of associating fluids: Spherical molecules with multiple bonding sites. Mol. Phys. 1988, 65, 1−31. (4) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Phase equilibria of associating fluids: Chain molecules with multiple bonding sites. Mol. Phys. 1988, 65, 1057−1079. (5) Gross, J.; Sadowski, G. Application of the perturbed-chain SAFT equation of state to associating systems. Ind. Eng. Chem. Res. 2002, 41, 5510−5515. (6) Gross, J. An Equation-of-State Contribution for Polar Components: Quadrupolar Molecules. AIChE J. 2005, 51, 2556− 2568. (7) Gross, J.; Vrabec, J. An Equation-of-State Contribution for Polar Components: Dipolar Molecules. AIChE J. 2006, 52, 1194−1204. (8) Vrabec, J.; Gross, J. Vapor-Liquid Equilibria Simulation and an Equation of State Contribution for Dipole-Quadrupole Interactions. J. Phys. Chem. B 2008, 112, 51−60. (9) Paricaud, P. Extension of the BMCSL equation of state for hard spheres to the metastable disordered region: Application to the SAFT approach. J. Chem. Phys. 2015, 143, 044507.
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DOI: 10.1021/acs.iecr.9b01515 Ind. Eng. Chem. Res. 2019, 58, 5744−5745
