PR /SRK equation of state - interaction parameters 1

[rotw]

Hello,

I am looking for the binary interaction coefficients (so called "K_ij") for either the SRK (Soave-Redlich-Kwong) or PR (Peng-Robinson) equation of state (EOS) - or both.
I am considering gases containing mixture of hydrocarbon compounds C1 to C6+ plus some non hydrocarbon(H2, H20, CO, CO2, H2S, N2, O2, and other rare gases). What I need is a "standard" table of those parameters for example - as published together with the original EOS technical papers.

Can someone points me please to the right direction (book, technical paper, website, etc.) ?

Thanks in advance.

 

[apetri]

you can find BIPs for PR / SRK with VDW mixing rule (one parameter) and standard alpha function in API TDB or Dechema , also several books as for example Chemical and Engineering Thermodynamics report these values,
note that different mixing rules or alpha functions can require different BIPs,
also many software applications adopt extended versions of these EOSs which require specific parameters,
finally, you may consider to use a data regression procedure to calculate BIPs not available in these lits.

 

[rotw]

Thanks this is helpful indeed.

 

[rotw]

Post deleted, as I found the problem which was the cubic equation solver (bogus). Thanks again for the link to API TDB!

 

[rotw]

While I am going through the development of the subject problem, I would like to ask for further direction regarding another aspect, though related to the topic.

With reference to the calculation of critical point for mixture, I found basically that two family of methods exist: Either the indirect approach which require the calculation of the whole phase diagram and then interpolate the critical point ( assuming it exists), or the direct method such as Heidemann and Khalil. However in both cases, "heavy mathematics" are involved while here - an implementation as straightforward as possible is preferred.

Does anyone know of a suitable procedure that would provide good estimate for critical point of mixture without the agonizing pain...?

As for the method that calculates pseudo-critical temperature and pressure solely based on compounds fraction and weighted average formula, it is extremely inaccurate for the application I am handling so it does not really help.

I am afraid there is no magic answer to my problem nor easy short cut...anyway, tentatively, I wanted to ask....
Thanks a lot

 

[apetri]

unfortunately I don't know simple ways to calculate the true critical point(s), if you adopt a EOS (such as Peng Robinson or Soave or the different variants) there is a large influence of Kij normally not included in simplified procedures,
yes, you can calculate the phase envelope, get an estimate of critical point(s) and adopt a specific procedure to improve the accuracy (solving Gibbs criteria for a critical point), for some examples see

http://www.prode.com/en/phaseenvelope.htm

as alternative, the method of Heidemann and Khalil doesn't require a good initial estimate but requires more calc's,
it depends from your application, for example, if you need only an estimate of critical point the methods discussed in chapter 5 of The Properties of Gases and Liquids may help.

 

[rotw]

Thanks apetri for your insight. Link is interesting, still reading through it.

Alternatively, out of my head I thought about the following approach...

Considering a cubic EOS, in principle the cubic equation has three real and distinctive roots once we cross the critical point and enter the bell shaped curve (in an ideal world). Thus if I screen starting from a relatively high pressure and go decreasing till the cubic equation gives three distinctive real roots, it should correspond to the critical point based on the said EOS. Understood it is highly inefficient to proceed in this way.

Except for complicated or exotic phase envelope situations (e.g. multiple critical points, etc) - would the method above be of some help? Would it work for example when the gas features retrograde condensation?

Consider this more for reasoning purpose / last resort before I am forced to admit that only a full blast Gibbs criteria / Michelsen method can be a proper path forward.

Thanks again

 

[apetri]

you know the critical points of pure components (working with Peng Robinson or Soave EOS) but for mixtures some iterative solution is required,

as far as I know roots can be difficult to track and there are several papers discussing the matter, my knowledge in this area is a bit outdated (graduated before 2000) and may be there are new developments,

Gundersen, Numerical aspects of the implementation of cubic equations of state in flash calculation routines

Mathias et al, Effective utilization of equations of state for thermodynamic properties in process simulation

Topliss et al, Computational Aspects of non cubic equation of state for phase equilibrium calculations

the papers of Michelsen about phase stability tests etc.

However, as said, The Properties of gases and liquids (chapter 5) reports several simplified procedures to estimate the critical point but expect some (may be large) deviation (from the true value calculated with a EOS, which may be not accurate, some papers report 5-20% errors),
why do you need a true Tc, Pc ?

 

[rotw]

Apetri,

You are right to say that roots can be difficult to track. A quick literature review tends to indicate that the number of roots is hard to correlate with the number of phases in multi-compound systems. I am trying to get some of the docs you mentioned just to understand more about the subject. The Properties of gases and liquids (chapter 5) is an excellent reference. Thanks for the references, this speeds up my progress.

Bottom line behind my query is that I want to avoid implementing a stability test. I currently use a thermodynamic test to discriminate whether gas is in liquid-like phase. Problem is, the test behave in the super-critical region as in liquid-like phase, which is a sort of a "false positive" for the case I am trying to discriminate.

Thinking is that if it would possible to discard the super-critical case by simply estimating Pc and Tc reliably enough, this should do the trick. It would not be a rigorous check but I guess still good enough. This the reason why I want to estimate Tc and Pc. Does this explanation indicate a little bit where I am coming from with my request?

 

[apetri]

stability test should not be too difficult to code but there are alternatives based on different properties (see the papers),
retrograde region and critical point area are good tests to verify if a code is reliable.

 

[rotw]

As an update...have implemented stability test and indeed this was the most reasonable way.

By the way I have a question...

I notice that the isentropic volume exponent deviates very sharply from isentropic temperature exponent and Cp/Cv exponent when entering / or approaching the two-phase region (gas-liquid). Just from the view point of liquid presence dimension, have I done something wrong when calculating the isentropic volume exponent or is this all plausible?

 

[apetri]

isentropic volume and temperature exponents can show large variations,
in case of doubts compare the values calculated by your code (calculated volume for a specified composition at t, p) with some reference software (to test a EOS such as Peng Robinson, Soave etc. I would suggest Nist Refprop or Prode Properties, both these applications can export values plus derivatives)

 

[rotw]

Ok Perfect.
I compared with Refprop (they call it "isentropic expansion"), values are in good agreement and it confirms the observation.
Thanks!

 

[rotw]

Apetri, all,

Could you please provide some guidance on the issue as follows

Do you happen to know any straightforward formula to obtain the isentropic coefficients (kt, kv) when knowing the compressibility Z (based upon SRK EOS)?

Any reference or link to a method would be appreciated.

Thanks

 

[apetri]

there are simplified correlations
but with a cubic EOS (SRK, PR etc.) better to adopt a rigorous derivation, see

https://en.wikipedia.org/wiki/Relations_between_heat_capacities

for that you need derivatives (as said applications as Nist Refprop or Prode Properties export derivatives for all models)
or you can simply adopt a numerical differentiation method...

a possible alternative could be based on corresponding states methodology (see Lee Kesler etc.), my copy of Chemical Engineering Thermodynamics (Daubert) includes the Fortran codes

 

[rotw]

Apetri, I noted your comments and references.
Also this is another one that may take certain time to sort out...will go through it.

And as usual thank you kindly for your orientation.