Materials

For each phase, an equation of state is required. The data for a given phase are gathered in a file. Every file must be in the folder ECOGEN/libEOS/ and must follow rules depending on the EOS. Five equations of state are implemented in ECOGEN:

  • Ideal Gas: For gaseous phase only.

  • Stiffened Gas: For condensed matter (liquid) in a pressure range where the compressible assumption is reasonable.

  • Noble-Abel Stiffened Gas: For condensed matter (liquid) subject to phase change.

  • Van der Waals and Polynomial: Experimental work on non-convex EOS therefore not detailed in the following.

Ideal Gas

It is a simple relation linking the pressure \(p\), the temperature \(T\) and the density \(\rho\) or the specific volume \(v=1/\rho\):

\[pv=rT\]

\(r\) is the universal gas constant divided by the molar weight of the gas (unit: \(J/(K.kg)\) (SI)). The Gibbs relation as well as Maxwell relations yield the definition of the internal energy:

\[e(p,v) = \frac{p}{\gamma-1} v + e_{ref}\]

One can easily obtain the entropy:

\[s(p,T) = c_v \ln{\left(\frac{T^{\gamma}}{p^{\gamma-1}}\right)} + s_{ref}\]

The following parameters are assumed constant and \(\gamma\), \(c_v\), \(e_{ref}\) and \(s_{ref}\) must be specified in the material file, as in the following example:

<?xml version = "1.0" encoding = "UTF-8" standalone = "yes"?>
<parametersEOS>
        <EOS fichier="IG_air.xml" type="IG"/>
        <parameters
                gamma="1.4"
                cv="800" cv_save="717.46"
                energyRef="0."
                entropyRef="0.">
        </parameters>
        <physicalParameters
                mu="1.85e-5"
                lambda="0.0262">
        </physicalParameters>
</parametersEOS>

Stiffened Gas

This equation of state was first presented in 1971 by Harlow & Amdsen [HA68]:

\[e = \left(\frac{p + \gamma p_{\infty}}{\gamma-1}\right) v + e_{ref}\]

This EOS takes into account the molecular attraction within condensed matter. This quite simple EOS is largely used in numerical simulation based on diffuse interface methods because it is able to reproduce shock relation as well as saturation curve for a liquid–vapor mixture when the phase transition is modeled. Hereafter, some useful relations for the specific volume, the enthalpy and the entropy:

\[\begin{split}v(p,T) &= \frac{ (\gamma-1) c_v T}{p + p_{\infty}} \\ h(T) &= \gamma c_v T + e_{ref} \\ s(p,T) &= c_v \ln{\left(\frac{T^{\gamma}}{\left(p+p_{\infty}\right)^{\gamma-1}} \right)} + s_{ref}\end{split}\]

A usefull description of these relations can be found in [LeMetayerMS04].

The ideal-gas law is recovered if \(p_{\infty}=0\). The following parameters are assumed constant \(\gamma\), \(p_{\infty}\), \(c_v\), \(e_{ref}\) and \(s_{ref}\). These parameters must be specified in the material file, as in the following example:

<?xml version = "1.0" encoding = "UTF-8" standalone = "yes"?>
<parametersEOS>
        <EOS fichier="SG_water.xml" type="SG"/>
        <parameters
                gamma="4.4"
                pInf="6.e8"
                cv="1000.0"
                energyRef="0."
                entropyRef="0.">
        </parameters>
        <physicalParameters
                mu="1.e-3"
                lambda="0.6">
        </physicalParameters>
</parametersEOS>

Noble-Abel Stiffened Gas

This equation of state is an improved version of the stiffened-gas equation of state in which a parameter of covolume b is introduced. This parameter allows to take into account the repulsive short distance effects linked to molecular motion. The Noble-Abel Stiffened-Gas (NASG) EOS was developed in 2016 by Le Métayer & Saurel [LMetayerS16] and it reads:

\[e(p,v) = \frac{p+\gamma p_{\infty}}{\gamma-1}(v-b) + e_{ref}\]

As the stiffened-gas EOS, the NASG EOS is able to take into account thermal agitation and attractive effects of condensed matter. This EOS is particularly interesting when phase transition is modeled because it is able to reproduce more accurately saturation curves of a liquid–vapor mixture than the traditionnal stiffened-gas EOS. Hereafter, some useful relations for the specific volume, the enthalpy and the entropy:

\[\begin{split}v(p,T) &= \frac{ (\gamma-1) c_v T}{p + p_{\infty}} + b \\ h(p,T) &= \gamma c_v T + + b p +e_{ref} \\ s(p,T) &= c_v \ln{\left(\frac{T^{\gamma}}{\left(p+p_{\infty}\right)^{\gamma-1}} \right)} + s_{ref}\end{split}\]

A complete description of the obtention of these relations is given in [LMetayerS16].

The stiffened-gas EOS is recovered if \(b=0\) and the ideal-gas law is recovered if \(b = p_{\infty} = 0\). The parameters \(\gamma\), \(p_{\infty}\), \(b\), \(c_v\), \(e_{ref}\) and \(s_{ref}\) are assumed constant. These parameters must be specified in the material file, as in the following example:

<?xml version = "1.0" encoding = "UTF-8" standalone = "yes"?>
<parametersEOS>
        <EOS fichier="NASG_dodLiq.xml" type="NASG"/>
        <parameters
                gamma="1.09"
                pInf="1.159e8"
                b="7.51e-4"
                cv="2393"
                energyRef="-794696."
                entropyRef="0.">
        </parameters>
        <physicalParameters
                mu="1.e-3"
                lambda="0.6">
        </physicalParameters>
</parametersEOS>