Real Gas Modeling

Overview

Since version 2022, Flow Simulator has used the compressibility factor, z, to adjust the ideal gas relationship for density. The compressibility factor improves the accuracy for fluids that are at pressures and temperatures where they do not behave like an ideal gas.

ρ = P z   R   T MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHbpGCcqGH9aqpdaWcaaWdaeaapeGaamiuaaWdaeaapeGaamOE aiaacckacaWGsbGaaiiOaiaadsfaaaaaaa@3EF8@

In version 2025, Flow Simulator has an improved accuracy option for fluids that do not behave as an ideal gas. This new approach uses modified exponents for the isentropic equations, reference 1. It still uses the compressibility factor as before. As before, only fluids using Coolprop can model the real gas effects.

From the Material Editor, select the Real Gas Gamma option.
Figure 1.


Equations

The two new “gammas” used for real gas:

γ P v = C p P   C v   β = γ i d e a l P   β MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHZoWzpaWaaSbaaSqaa8qacaWGqbGaamODaaWdaeqaaOWdbiab g2da9maalaaapaqaa8qacaWGdbWdamaaBaaaleaapeGaamiCaaWdae qaaaGcbaWdbiaadcfacaGGGcGaam4qa8aadaWgaaWcbaWdbiaadAha a8aabeaak8qacaGGGcGaeqOSdigaaiabg2da9maalaaapaqaa8qacq aHZoWzpaWaaSbaaSqaa8qacaWGPbGaamizaiaadwgacaWGHbGaamiB aaWdaeqaaaGcbaWdbiaadcfacaGGGcGaeqOSdigaaaaa@4FE0@
γ T v = 1 + v α C v   β MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHZoWzpaWaaSbaaSqaa8qacaWGubGaamODaaWdaeqaaOWdbiab g2da9iaaigdacqGHRaWkdaWcaaWdaeaapeGaamODaiabeg7aHbWdae aapeGaam4qa8aadaWgaaWcbaWdbiaadAhaa8aabeaak8qacaGGGcGa eqOSdigaaaaa@448E@
α = 1 v ∂ v ∂ T P = i s o b a r i c   e x p a n s i o n   c o e f f i c i e n t MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHXoqycqGH9aqpdaWcaaWdaeaapeGaaGymaaWdaeaapeGaamOD amaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIyRaamODaaWdaeaape GaeyOaIyRaamivaaaaaiaawIcacaGLPaaapaWaaSbaaSqaa8qacaWG qbaapaqabaaaaOWdbiabg2da9iaadMgacaWGZbGaam4Baiaadkgaca WGHbGaamOCaiaadMgacaWGJbGaaiiOaiaadwgacaWG4bGaamiCaiaa dggacaWGUbGaam4CaiaadMgacaWGVbGaamOBaiaacckacaWGJbGaam 4BaiaadwgacaWGMbGaamOzaiaadMgacaWGJbGaamyAaiaadwgacaWG UbGaamiDaaaa@601A@
β= −1 v ∂v ∂p T =isothermal compressibility factor MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHYoGycqGH9aqpdaWcaaWdaeaapeGaeyOeI0IaaGymaaWdaeaa peGaamODamaabmaapaqaa8qadaWcaaWdaeaapeGaeyOaIyRaamODaa WdaeaapeGaeyOaIyRaamiCaaaaaiaawIcacaGLPaaapaWaaSbaaSqa a8qacaWGubaapaqabaaaaOWdbiabg2da9iaadMgacaWGZbGaam4Bai aadshacaWGObGaamyzaiaadkhacaWGTbGaamyyaiaadYgacaGGGcGa am4yaiaad+gacaWGTbGaamiCaiaadkhacaWGLbGaam4Caiaadohaca WGPbGaamOyaiaadMgacaWGSbGaamyAaiaadshacaWG5bGaaiiOaiaa dAgacaWGHbGaam4yaiaadshacaWGVbGaamOCaaaa@6431@

For a fluid near the ideal gas conditions, the two new gammas are equivalent to the ideal gas gamma.

γ = γ P v = γ T v = C p C v   ,   for ideal gas conditions MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qacqaHZoWzcqGH9aqpcqaHZoWzpaWaaSbaaSqaa8qacaWGqbGaamOD aaWdaeqaaOWdbiabg2da9iabeo7aN9aadaWgaaWcbaWdbiaadsfaca WG2baapaqabaGcpeGaeyypa0ZaaSaaa8aabaWdbiaadoeapaWaaSba aSqaa8qacaWGWbaapaqabaaakeaapeGaam4qa8aadaWgaaWcbaWdbi aadAhaa8aabeaaaaGcpeGaaiiOaiaacYcacaGGGcGaaeOzaiaab+ga caqGYbGaaeiOaiaabMgacaqGKbGaaeyzaiaabggacaqGSbGaaeiOai aabEgacaqGHbGaae4CaiaabckacaqGJbGaae4Baiaab6gacaqGKbGa aeyAaiaabshacaqGPbGaae4Baiaab6gacaqGZbaaaa@611D@

The isentropic relations and the flow function equations can be rewritten using the new “gammas”.

P o P = 1 +   γ P v − 1 2 M 2   γ P v   γ P v − 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaamiua8aadaWgaaWcbaWdbiaad+gaa8aabeaa aOqaa8qacaWGqbaaaiabg2da9maabmaapaqaa8qacaaIXaGaey4kaS YaaSaaa8aabaWdbiaacckacqaHZoWzpaWaaSbaaSqaa8qacaWGqbGa amODaaWdaeqaaOWdbiabgkHiTiaaigdaa8aabaWdbiaaikdaaaGaam yta8aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaWdamaa CaaaleqabaWdbmaalaaapaqaa8qacaGGGcGaeq4SdC2damaaBaaame aapeGaamiuaiaadAhaa8aabeaaaSqaa8qacaGGGcGaeq4SdC2damaa BaaameaapeGaamiuaiaadAhaa8aabeaal8qacqGHsislcaaIXaaaaa aaaaa@538E@
T o T = 1 +   γ P v − 1 2 M 2   γ T v − 1   γ P v − 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGaamiva8aadaWgaaWcbaWdbiaad+gaa8aabeaa aOqaa8qacaWGubaaaiabg2da9maabmaapaqaa8qacaaIXaGaey4kaS YaaSaaa8aabaWdbiaacckacqaHZoWzpaWaaSbaaSqaa8qacaWGqbGa amODaaWdaeqaaOWdbiabgkHiTiaaigdaa8aabaWdbiaaikdaaaGaam yta8aadaahaaWcbeqaa8qacaaIYaaaaaGccaGLOaGaayzkaaWdamaa CaaaleqabaWdbmaalaaapaqaa8qacaGGGcGaeq4SdC2damaaBaaame aapeGaamivaiaadAhaa8aabeaal8qacqGHsislcaaIXaaapaqaa8qa caGGGcGaeq4SdC2damaaBaaameaapeGaamiuaiaadAhaa8aabeaal8 qacqGHsislcaaIXaaaaaaaaaa@5561@
m ˙ T o P o A = γ P v z o R   M 1 +   γ P v − 1 2 M 2 γ P v + 1 2 γ P v − 1 MathType@MTEF@5@5@+= feaahGart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 qadaWcaaWdaeaapeGabmyBa8aagaGaa8qadaGcaaWdaeaapeGaamiv a8aadaWgaaWcbaWdbiaad+gaa8aabeaaa8qabeaaaOWdaeaapeGaam iua8aadaWgaaWcbaWdbiaad+gaa8aabeaak8qacaWGbbaaaiabg2da 9maalaaapaqaa8qadaGcaaWdaeaapeWaaSaaa8aabaWdbiabeo7aN9 aadaWgaaWcbaWdbiaadcfacaWG2baapaqabaaakeaapeGaamOEa8aa daWgaaWcbaWdbiaad+gaa8aabeaak8qacaWGsbaaaaWcbeaakiaacc kacaWGnbaapaqaa8qadaqadaWdaeaapeGaaGymaiabgUcaRmaalaaa paqaa8qacaGGGcGaeq4SdC2damaaBaaaleaapeGaamiuaiaadAhaa8 aabeaak8qacqGHsislcaaIXaaapaqaa8qacaaIYaaaaiaad2eapaWa aWbaaSqabeaapeGaaGOmaaaaaOGaayjkaiaawMcaa8aadaahaaWcbe qaa8qadaWcaaWdaeaapeGaeq4SdC2damaaBaaameaapeGaamiuaiaa dAhaa8aabeaal8qacqGHRaWkcaaIXaaapaqaa8qacaaIYaWaaeWaa8 aabaWdbiabeo7aN9aadaWgaaadbaWdbiaadcfacaWG2baapaqabaWc peGaeyOeI0IaaGymaaGaayjkaiaawMcaaaaaaaaaaaaa@62E0@

For a fluid near the ideal gas conditions, these equations are equivalent to the ideal gas equations.

Output

The *.prop file has a column for these two new “gammas”. The “RG” at the end of “COOLPROPRG” in the output files also denotes the real gas gammas that were used in the analysis.
Figure 2.


References

  1. Nederstigt, P., Pecnik, R., “Generalised Isentropic Relations in Thermodynamics”, Energies 2023, 16, 2281.

    https://doi.org/10.3390/en16052281.