Calculation of isentropic exponent for real gases 3

[dtmeng]

Hi there,

I am doing some calculations on compression of co2 and therefore I need the isentropic volume exponent (kappa_p,v) and the isentropic temperature exponent (kappa_T,p) for a real gas.

These are defined as following (in "Thermodynamic Correlations, k-Exponents, Speed of Sound, and COP Data for Binary Refrigerant Mixtures" by Stegou-Sagia and Damanakis, University of Athens, Int. J. Thermodynamisc, Vol. 7 (No. 1), pp. 15-22, March 2004):

kappa_p,v = -v/p * c_p/c_v * (dp/dv)_T
kappa_T,p = 1 / (1-(p/c_p)*(dv/dT)_p)

(dp/dv)_T is the partial derivative of pressure with respect to specific volume at constant temperature.

My problem is that the everything I found about this subjects uses a Peng Robinson or a Redlich and Kwong EOS which is basically a classical van der Waals equation. Therefore it gives an analytical expression for the specific volume and can be differentiated so that you get an expression for the partial derivatives above.

But I cannot use the PR or RKS EOS for CO2 because of the accuracy so I'm using the LKP EOS which only gives an expression for Z that includes the compressibility factor of two fluids at a reduced state (the solution is calculated numerically as the expression for the reduced specific volume is not solveable analytically). This means no equation to differentiate.

Maybe I'm not seeing the wood for the trees but I have no idea how to calculate the isentropic exponents as I have no idea how to differentiate the EOS at a given state.

Any help is truly appreciated!

Greetings,
Daniel

PS: Please excuse any mistakes for English is not my native language.

 

[zdas04]

Daniel, I can't help much here, but I can say that your language skills are far better than most native-English speakers we see on this site. You have nothing to excuse in that category.

What is your accuracy issue with Peng-Robinson or Redlich-Kwong EOS? When I've used them and compared the results to the NIST REFPROP program I get excellent agreement. The agreement is not perfect, but when I put either EOS or the NIST data into an engineering calculation I get results that lead me to the same decision. I've never had to approach better "accuracy" than that.

David

 

[PaoloPemi]

possibly you need a tool which exports analytical derivatives, see for example Prode Properties (

http://www.prode.com)

or RefProp (

http://www.nist.gov),

Prode Properties includes Lee Kesler Plocker and exports several derivatives, for volume there is dV/dT (const P), dV/dP (const T) and others, RefProp provides similar data, you may need some general table to convert derivatives (this may be not easy as it requires some knowledge of thermodynamic laws)
hoping this helps.

 

[rossj]

as suggested by PaoloPemi have a look at Bridgman's tables, they are really useful to convert derivatives.
in my experience LKP is reasonably accurate for CO2.

 

[sailoday28]

What are the initial conditions prior to compression?
What are the approximate final conditions?

 

[hacksaw]

is the reluctance in forming the partial deriviatives numerically?

 

[dtmeng]

First of all, thanks to everybody who responded, gained some valuable input here.

My choice for LKP is based on a recommendation made by Lüdtke, "Process Centrifugal Compressors", 1st Ed., 2004. It just says that RKS is not recommended for CO2 in a pressure range up to 200 bar (about 2900 psi). But he says as well that the question is discussed controversially and for example Reid, Prausnitz and Sherwood recommend RKS for CO2. The compression process starts at ambient pressure and goes up 220 bar.

But I found a solution. In case somebody might encounters the same problem, here is a short description and the literature needed:

Expressions for the partial derivatives that use partial derivatives of the reduced quantities (e.g. (dp/dT)v equals (dp_r/dT_r)v_r*(p_c/T_c))and can therefore be expressed through the LKP equation can be found in Reid, Prausnitz, Sherwood; "The properties of gases and liquids, 3rd. Ed., 1977. I think you could differentiate the original equations by hand, but they are pretty nasty (in total they cover almost half a site in the book).
The formulas to calculate the heat capacities c_v and c_p are given by Plöcker (in his doctoral thesis from 1977). This is done by calculating the heat capacities for a perfect gas for the given temperature and then calculate the departure from the actual heat capacity to the perfect gas values.
The derivatives and the heat capacities have to be interpolated by means of the acentric factor (standard LKP procedure). From there the isentropic exponents can be calculated by the equations quoted above.

I've been comparing the results generated by this model to some charts of the isentropic exponents of propane for half a day now and it seems to fit properly.

But it's very good to know that tables like Bridgman's exist! Thanks for all the kind help. I'm amazed at how fast you get a competent answer around here. Hopefully I can contribute something in the future too.

Greetings,
Daniel

PS: All equations together can also be found in the book Lüdtke (above) - very helpful book!

 

[PaoloPemi]

so you did require cp/cv and, yes the Bridgman's tables could be useful... by the way you can quickly generate tables of values for cp, cv, speed of sound etc. with SRK,PR,LKP etc. with the mentioned Properties or REFPROP, see for example

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

 

[25362]

In this particular case why not use the data from:

http://webbook.nist.gov/cgi/fluid.cgi?ID=C124389&Action=Page