debugging of Ancillary.density_pT_Soave

[thorade]

For Pentane at p=1e-6 and T=578.275 the Soave algorithm goes to the supercritical branch, where three roots exist. According to Span, two of the three roots result in negative densities, but in this case this is not true.
Not a big problem, as this is used as starting value only, but it slows down the pT iteration.

Also, the RKS EoS should be able to correctly describe the liquid region, at least qualitatively.

[thorade]

see Span 2000, page 48-51
http://books.google.de/books?id=dd410GGw8wUC&pg=PA48

[thorade]

There is a section about solving cubic equations in
"Thermodynamics - Fundamentals for Applications" by O'Connell and Haile
(see Appendix C, page 620):
http://www.cambridge.org/9780521582063
http://books.google.de/books?id=1rOA5I6kQ7gC&&pg=PA620

[thorade]

Also see paper by Deiters 2014:
http://dx.doi.org/10.1021/ie4038664
And the WP articles on (solving) cubic functions:
https://en.wikipedia.org/wiki/Cubic_function

[thorade]

Also read the references cited in this blog post:
http://momentsingraphics.de/CubicRoots.html