/EOS/NASG

Block Format Keyword Describes the NASG (Noble-Abel-Stiffened-Gas) equation of state.

Format

(1)	(3)	(5)	(7)
/EOS/NASG/`mat_ID`/`unit_ID`
`eos_title`
`b`	$γ$	$P \infty$	`q`
`P`_sh	`P`₀	`C`_v	$ρ_{0}$

Definitions

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier. (Integer, maximum 10 digits)
`eos_title`	EOS title. (Character, maximum 100 characters)
`b`	Covolume. (Real)	$[\frac{m^{3}}{k g}]$
$γ$	Heat capacity ratio $γ = \frac{C_{p}}{C_{v}}$ . (Real)
$P \infty$	Stiffness parameter. (Real)	$[Pa]$
`q`	Heat bond. (Real)	$[\frac{J}{kg}]$
`P`_sh	Pressure shift. (Real)	$[Pa]$
`P`₀	Initial pressure. (Real)	$[Pa]$
`C`_v	Heat capacity at constant volume. (Real)	$[\frac{J}{kg \cdot K}]$
$ρ_{0}$	Reference density. Default = material density (Real)	$[\frac{kg}{m^{3}}]$

Example

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                  kg                   m                   s
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYDRO/7/1
WATER
#              RHO_I               RHO_0
              957.74                   0
#                 NU                PMIN
                   0                   0
/EOS/NASG/7/1
Noble-Abel-Stiffened-Gas EoS for WATER  (O.Le Metayer, R.Saurel)
#                  b               GAMMA               PSTAR                   Q      
             6.61E-4                1.19          7028.00E+5            -1177788 
#                Psh                  P0                  Cv                Rho0      
                 0.0            1.0453E5                3610              957.74
/EULER/MAT/7/1

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#enddata

Comments

NASG EOS (Noble-Abel-Stiffened-Gas equation of state) is based on the Stiffened-Gas and Noble-Abel equation of states.(1)
$(P + P_{\infty}) (v - b) = (γ - 1) C_{v} T$

Where,

$v$

Specific volume

$b$

Co-volume

$C_{v}$

Heat capacity at constant volume

$T$

Temperature

$γ = \frac{C_{p}}{C_{v}}$
This EOS summarizes in a simple formulation two main molecular effect:
- Agitation
- Attractive/Repulsive effects
The previous form $P = P (v, T)$ can be written in $P = P (μ, E)$ form.

Where, $µ = \frac{ρ}{ρ_{0}} - 1$ ; $E = \frac{E_{i n t}}{V_{0}}$ .

This gives $P (μ, E) = \frac{(γ - 1) (1 + μ) (E - ρ_{0} q)}{1 - b ρ_{0} (1 + μ)} - γ P_{\infty}$

Some comparisons with other EOS:

Noble-Able		NASG	Stiffened-Gas
$P (v, T)$	$P (v - b) = R T$	$(P + P_{\infty}) (v - b) = (γ - 1) C_{v} T$	$(P + P_{\infty}) v = (γ - 1) C_{v} T$
$P (μ, E)$	$P = \frac{(γ - 1) (1 + μ) E}{1 - b ρ_{0} (1 + μ)}$	$P = \frac{E - ρ_{0} q}{\frac{1}{1 + µ} - ρ_{0} b} (γ - 1) - γ P_{\infty}$	$P = (γ - 1) (1 + μ) E - γ P_{\infty}$
$c_{0}$	$c_{0} = \sqrt{\frac{γ P}{(1 - b ρ) ρ}}$	$c_{0} = \sqrt{\frac{γ (P + P_{\infty})}{(1 - b ρ) ρ}}$	$c_{0} = \sqrt{\frac{γ (P + P_{\infty})}{ρ}}$
${E_{0} \|}_{P = P_{0}, ρ = ρ_{0}}$	$\frac{P_{0} (1 - b ρ_{0})}{γ - 1}$	$\frac{(P_{0} + γ P_{\infty}) (1 - b ρ_{0})}{γ - 1} + q ρ_{0}$	$\frac{(P_{0} + γ P_{\infty})}{γ - 1}$

The Initial State is calculated from the input parameters:
$T_{0}$ from $v (P, T) = \frac{(γ - 1) C_{v} T}{P + P_{\infty}} + b$

Where,

$P = P_{0}$

$T = T_{0}$

$E_{0}$ from $e (P, T) = \frac{P + γ P_{\infty}}{γ - 1} (v - b) + q$

Where,

$P = P_{0}$

$v = \frac{1}{ρ_{0}}$

$E (P, T) = ρ_{0} e (P, T)$
Enthalpy can be calculated from:(2)
$h (P, T) = γ C_{v} T + b P + q_{\infty}$
The $P \infty$ parameter can be calculated with:(3)
$P_{\infty} = \frac{ρ_{0} c_{0}^{2} (1 - b ρ_{0})}{γ} - P_{0}$

Where, $c$ is the speed of sound in the material.
Equations of state are used by Radioss to compute the hydrodynamic pressure and are compatible with the material laws:
- /MAT/LAW3 (HYDPLA)
- /MAT/LAW4 (HYD_JCOOK)
- /MAT/LAW6 (HYDRO or HYD_VISC)
- /MAT/LAW10 (DPRAG1)
- /MAT/LAW12 (3D_COMP)
- /MAT/LAW49 (STEINB)
- /MAT/LAW102 (DPRAG2)
- /MAT/LAW103 (HENSEL-SPITTEL)

Table 1. Experimental Data for dodecane in kg, m, s units
	Liquid Phase	Vapor Phase
`C`_p	2608.0	2063.0
`C`_v	2393.0	2016.0
$γ$	1.09	1.02
$P \infty$	1159.0e+5	0.0
$b$	7.51e-4	0.0
$q$	−794696.0	−2685610.0

Reference state for the liquid phase: $ρ_{0} = 589.73 \frac{kg}{m^{3}}, P_{0} = 112800 Pa, c_{0} = 620.4 \frac{m}{s}$

Valid temperature range: [300 - 500 K]

Table 2. Experimental Data for water in kg, m, s units
	Liquid Phase	Vapor Phase
`C`_p	4285.0	1401.0
`C`_v	3610.0	955.0
$γ$	1.19	1.47
$P \infty$	7028.0e+5	0.0
$b$	6.61e-4	0.0
$q$	-1177788.0	2077616.0

Reference state for the liquid phase: $ρ_{0} = 957.74 \frac{kg}{m^{3}}, P_{0} = 104530 Pa, c_{0} = 1542 \frac{m}{s}$

Validity: T in [300 - 500 K]

¹ O Le Métayer, Richard Saurel, “The Noble-Abel Stiffened-Gas equation of state”, HAL Id: hal-01305974

² J.R. Simoes-Moreira, ”Adiabatic evaporation waves”, Ph.D. thesis, Rensselaer Polytechnic Institute, Troy, New-York (1994)

³ R. Oldenbourg, ”Properties of water and steam in SI-units”, Springer-Verlag Berlin Heidelberg, New-York (1989)