/EOS/NOBLE-ABEL

Block Format Keyword Describes the co-volume equation of state $P (v - b) = R T$ .

Format

(1)	(3)	(5)	(7)	(9)
/EOS/NOBLE-ABEL/`mat_ID`/`unit_ID`
`eos_title`
`b`	$γ$	`E`₀	`P`_sh	$ρ_{0}$

Definition

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`	Co-volume. (Real)	$[\frac{m^{3}}{k g}]$
$γ$	Heat Capacity ratio $γ = \frac{C_{p}}{C_{v}}$ . (Real)
`E`₀	Initial internal energy per unit reference volume. (Real)	$[\frac{J}{m^{3}}]$
`P`_sh	Pressure shift. (Real)	$[Pa]$
$ρ_{0}$	Reference density. Default = material density (Real)	$[\frac{k g}{m^{3}}]$

Example (Noble-Abel)

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/UNIT/1
unit for mat
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/HYDPLA/7
NOBLE_ABEL
#              RHO_I               RHO_0
             1.22e-6                   0
#                  E                  nu
                   0                   0
#                  a                   b                   n             eps_max           sigma_max
                1E30                   0                   0                   0                   0
#               Pmin                 Psh
                   0
/EOS/NOBLE-ABEL/7
NOBLE-ABEL EOS 
#                  b               GAMMA                  E0                 PSH                RHO0          
                1E-3                 1.4        0.2499999997                   0             1.22e-6
/ALE/MAT/7

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

Comments

A generalization of the ideal-gas thermal EOS, $P v = R T$ is the co-volume equation of state $P (v - b) = R T$
Where,

$v$

Specific volume

b

Covolume

R

Specific gas constant

T

Temperature

Previous form $P = P (v, T)$ , can be written in $P = P (μ, E)$ form.

Where,

$µ = \frac{ρ}{ρ_{0}} - 1$

$E = \frac{E_{i n t}}{V_{0}}$

$P (μ, E) = \frac{(γ - 1) (1 + μ) E}{1 - b ρ_{0} (1 + μ)}$

with $γ = \frac{C_{p}}{C_{v}}$
This EOS applies to dense gases at high pressure for which the volume occupied by the molecules themselves is no longer negligible.
Covolume b is usually in range $[0.9 \times 10^{- 3}, 1.1 \times 10^{- 3}]$ $[\frac{m^{3}}{k g}]$

Some comparison with ideal gas EOS:

	IDEAL-GAS	NOBLE-ABEL
$P (v, T)$	$P v = R T$	$P (v - b) = R T$
$P (μ, E)$	$(γ - 1) (1 + μ) E$	$\frac{(γ - 1) (1 + μ) E}{1 - b ρ_{0} (1 + μ)}$
Sound Speed $c$	$c = \sqrt{\frac{γ P}{ρ}}$	$c = \sqrt{\frac{γ P}{(1 - b ρ) ρ}}$
$E_{0} = E (0)$	$\frac{P_{0}}{γ - 1}$	$\frac{P_{0} (1 - b ρ_{0})}{γ - 1}$

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)