3 replies to this topic

[GS81Process]

I need to determine the isentropic expansion coefficient for a very non-ideal fluid using an Equation of State method, and I cannot assume that k=Cp/Cv.

I have the definition of the non-ideal isentropic expansion coefficient:

k= -v/P * (deltaP/deltav)_s = -v/P * (deltaP/deltav)_T * Cp/Cv

where v= specific volume
P= pressure
T= temperature
s= entropy
delta= partial derivative

I was able to differentiate the Peng-Robinson equation of state to find an expression for (deltaP/deltav)_T, but I am not sure how Cp and Cv are estimated. So, I am trying to figure out how a process simulation program estimates the constant pressure heat capacity (Cp) and constant volume heat capacity (Cv) for a non-ideal fluid.

I think that there might be a derivation that can be made from the definitions of Cp and Cv using differential calculus and Maxwell's relations to form relationships in terms of P,V, and T only.

Is anyone familiar with how a simulation program estimates Cp and Cv?

[PaoloPemi]

for solving partial derivatives Bridgeman's tables are quite useful...

[GS81Process]

I got in contact with Aspentech and I was able to determine how this calculation is done in HYSYS, it is roughly as follows:

- Calculate Cv= (deltaS/deltaT)_V​_. The trick is that HYSYS uses a small incremental temperature to perform the derivative. It makes T2= T1+0.01C. s1 and s2 (entropy values) are calculated from the entropy departure functions for the equation of state. P2 is selected so that V2 (molar volume) equals to v1.

- Cp can then be calculated using Cp-Cv= -T(deltaV/deltaT)P²/(deltaV/deltaP)_T

- The equation above can be derived from thermodynamic relationships and Maxwell's equations.

[PaoloPemi]

I use different tools (Prode Properties and Refprop) which include analytical derivatives for many properties as fugacities, enthalpy, entropy, volume vs. pressure, temperature and composition, these are easily available for SRK or PR models.
Once you have these derivatives use Bridgeman's tables to calculate all the derived properties, the procedure based on numerical differentiation which you have described may suffer from several problems (lack of convergence near critical boundaries, low accuracy etc.)