Stiffened Gas (SG) equations of state (EOS) calibration for compressible diphasic flow under mechanical equilibrium. Calibration of SG EOS parameters methods :

• Dynamic of a Shock wave
• Liquid and its vapor

The calibration dynamic of a shock wave is done with the dynamic adiatic experimental curve (shock speed function of the material speed) and the calibration liquid and its vapor is done with experimental saturation curves.

Le Métayer, O., Massoni, J., & Saurel, R. (2004). Elaborating equations of state of a liquid and its vapor for two-phase flow models.

This tool is fully compatible under Unix systems, however it is necessary to have the following utilities :

• Make
• Gnuplot

To install them through a package manager you can use the apt command :

$ sudo apt install pkg-name

With pkg-name the name of the package that you want to install.

The compilation is done with a Makefile. Under Unix systems it reads :

$ make 

To run the calibration tool you just need to type the following :

$ ./exe

If you want to delete all .o files from the compilation you can run :

$ make clean

In case you want to remove the executable exe:

$ make mrproper

To delete all the result files you can do :

$ make resClean

To calibrate your SG parameters you have to fill the input file located at SG-EOS_Calibration/input/input.txt. You have to provide integer numbers to select the calibration you want to use:

• (1) Dynamic of Shock wave
• (2) Liquid and its vapor

For liquid and its vapor calibration you can use two differents methods:

• (1) DM Calibration with two reference states and experimental saturated pressure curve
• (2) LSM Calibration with one reference state and a full-set of experimental curves at saturation

In the section Liquid and its vapor calibration you will have a complete description of these methods and the way to use them.

By default the shock wave calibration is done for liquid water and the liquid and its vapor calibration is configured for dodecane. To create a new calibration for a new fluid or under a new range of study you will need to modify the file according your desired method of calibration.

These files are located at the following path : SG-EOS_Calibration/input/xxx/ with xxx the method name of the calibration. Further description of these files is given in the next section.

According to the calibration method used you will also have to add experimental data files in the folder SG-EOS_Calibration/input/Shock/ and/or SG-EOS_Calibration/input/Liq-vap/.

Once the SG parameters are calibrated you will get their values in the shell environnement.

Theoritical curves can be found in the folder SG-EOS_Calibration/res/. If you want to plot theses curves and compare with the experimental ones you will have to go to the folder SG-EOS_Calibration/res/ and run the script ./runPlotShock.sh and/or ./runPlotLiqVap.sh. It might be possible that you don't have the rights to launch these scripts, thus you will have to do :

$ find . -type f -name "*.sh" -exec chmod +x {} \;

Scripts runPlotXXX.sh (XXX is the calibration used) display a preview of all curves in Gnuplot GUI and create in the same time plot files in eps format in the repository SG-EOS_Calibration/res/.

Under the folder SG-EOS_Calibration/input/lib/ you can find test cases for shock and liquid-vapor calibration. There are input files and the associated experimental data.

For shock wave calibration the test case is:

• Liquid water with experimental data extracted from S.P. Marsh, LASL Shock Hugoniot Data, University of California Press, Berkeley, CA, 1980

For liquid/vapor calibration the test cases are:

• Dodecane in the temperature range T ∈ [300-500 K]. Experimental data are from J. R. Simões-Moreira, “Adiabatic evaporation waves” Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, NY, 1994.
• Dodecane in the temperature range T ∈ [400-600 K]. Experimental data from the same source.
• Water and steam in the temperature range T ∈ [300-500 K]. Experimental data are extracted from NIST website.

The availibity of the test cases for liquid-vapor calibration according to the method used is summarized here:

To run a shock calibration you will have to fill the file SG-EOS_Calibration/input/Shock/Calib_shock.txt. You will have to give experimental values for the fluid at rest: the sound speed c₀ (m/s), the specific volume v₀ (m³/kg) and the pressure p₀ (Pa).

To calibrate SG parameters you will also need informations on dynamic adiabatic curves :

• if you already know the experimental proportionnal coefficient (written a) of dynamic adiabatic curve D = c₀ + a.u you can directly specify it. Therefore, you will to specify the shock speed interval of study in the section Interval of shock speed.
• if you don't know it yet, you have to put the value -1. to a and provide an experimental dynamic adiabatic file to let the tool determine it. This file must be located in SG-EOS_Calibration/input/Shock/AdiabDyn.txt. The first line of this file is not read, the first column is the material speed u (m/s) and the second column is the shock speed D (m/s).

To calibrate the SG parameters for liquid and its vapor you can use two differents methods:

• DM Calibration with two reference states and experimental saturated pressure curve
• LSM Calibration with one reference state and a full-set of experimental curves at saturation

DM calibration stands for Differential Method and LSM calibration stands for Least Square Method.

The Differential Method (DM) uses two reference states and experimental saturated pressure curve to calibrate SG parameters as it is done in Le Métayer, O., Massoni, J., & Saurel, R.. All the input files must be located under the following path SG-EOS_Calibration/input/Liq-vap/DM/, more details are given below.

To run a liquid/vapor DM calibration you have to fill the file SG-EOS_Calibration/input/Liq-vap/DM/Calib_liq-vap_DM.txt. In the section Compute Cp_k and q_k of this file you need to give two reference states: the state 0 and the state 1. At each of these reference states you have to give the temperature and the according enthalpies for the liquid and the gas. You must choose the states according to the use range of the EOS.

In the following section Compute pInf_k and gamma_k you have to give again two reference states. Of course, if you want you can use the same reference states than the previous section but this time at a given reference temperature you will also need to specify the specific volume and saturation pressure of each phase. Anyway, in this section you have the possibility to change the reference states, it's often used to get a better calibration for the liquid phase.

To allow the calibration you also must give an experimental saturation curve P_sat(T) at the path SG-EOS_Calibration/input/Liq-vap/DM/Psat_exp.txt. In this file, the first line is not read, the first column is temperature T (K) and the second column is the saturation pressure P_sat (Pa).

If you want to compare the theoric curves obtained with this calibration with experimental data you should provide experimental data in the folder SG-EOS_Calibration/input/Liq-vap/DM/Liq-vap/. For all these files the first line is not read and they are described below:

• phasic enthalpies hG_exp.txt for the Gas and hL_exp.txt for the Liquid. The first column is the temperature T (K) and the second one is the phasic enthalpy hG or hL (J/kg).
• the latent heat is defined throught the file Lv_exp. The first column is the temperature T (K) and the second one is the latent heat Lv (J/kg).
• phasic specific volumes vG_exp.txt for the Gas and vL_exp.txt for the Liquid. The first column is the temperature T (K) and the second one is the phasic specific volume vG or vL (m³/kg).
• the saturation pressure Psat_exp.txt. The first column is the temperature T (K) and the second one the saturation pressure P_sat (Pa).

Remark : in this method, each experimental data file can have a different number of points since they are only used to plot experimental curves (the pressure saturation curve is also used to calibrate as mentioned above).

The Least Square Method (LSM) is using one reference state and a full-set of saturation curves to determine the SG parameters. All the input files must be located under the following path SG-EOS_Calibration/input/Liq-vap/LSM/, more details are given below.

To run a liquid/vapor LSM calibration you have to fill the file SG-EOS_Calibration/input/Liq-vap/LSM/Calib_liq-vap_LSM.txt. In this file you have to specify the pressure p₀ (Pa), the density ρ₀ (kg/m³) and the sound speed c₀ (m/s) of the liquid reference state. You can also specify an interval of temperatures for enthalpies calibration to get a better fit and more accurate cp and q parameters. This interval must be within the range of temperatures given in experimental data (file expData.txt described below). If the user does not want to specify a sub-interval of temperatures or specify only one bound of the interval he has to put values -1. to (the) corresponding field(s).

You have to specify the experimental data at saturation in the file SG-EOS_Calibration/input/Liq-vap/LSM/expData.txt. The first line of this file is not read, there are 7 rows to fill (each one separated by a blank or a tab) as follows:

Each column is respectively the temperature T, the saturated pressure Psat, the specific volume of the gas vG and of the liquid vL, the enthalpy of the gas hG and of the liquid hL, and the latent heat Lv.

Remark: the decimal separator must be a dot.