/EOS/IDEAL-GAS-VT

Describes the equation of state for ideal gas $P = ρ r T$ ; where, $c_{p}$ is temperature dependent and described with Janaf model: $c_{p} (T) = A_{0} + A_{1} T + A_{2} T^{2} + A_{3} T^{3} + A_{4} T^{4}$ (SI: $J . k g^{- 1} . K^{- 1}$ )

Format

(1)	(3)	(5)	(7)	(9)
/EOS/IDEAL-GAS/`mat_ID`/`unit_ID`
`eos_title`
$r$	`T`₀	`P`₀	`P`_sh	$ρ_{0}$
`A`₀	`A`₁	`A`₂	`A`₃	`A`₄

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)
$r$	Ideal gas constant. (Real)	$[J . k g^{- 1} . K^{- 1}]$
`T`₀	Initial temperature. If `P`₀ not defined, refer to Comment 3. (Real)	$[K]$
`P`₀	Initial pressure. If `T`₀ is not defined, refer to Comment 3. (Real)	$[Pa]$
`P`_sh	Pressure shift. (Real)	$[Pa]$
$ρ_{0}$	Reference density. Default = material density (Real)	$[\frac{kg}{m^{3}}]$
`A`₀	Specific heat capacity parameter. (Real)	$[\frac{J}{k g . K}]$
`A`₁	Specific heat capacity parameter. (Real)	$[\frac{J}{k g . K^{2}}]$
`A`₂	Specific heat capacity parameter. (Real)	$[\frac{J}{k g . K^{3}}]$
`A`₃	Specific heat capacity parameter. (Real)	$[\frac{J}{k g . K^{4}}]$
`A`₄	Specific heat capacity parameter. (Real)	$[\frac{J}{k g . K^{5}}]$

Example (Air)

#---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/LAW6/1/1
Air
#              RHO_I             
             1.22e-6     

#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/EOS/IDEAL-GAS-VT/1/1
Air
#                  r                  T0                  P0                 PSH                RHO0
                 287                   0                 0.1                   0             1.22e-6                  
#                 A0                  A1                  A2                  A3                  A4
              1004.5                   0                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/ALE/MAT/1

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

Comments

Ideal gas EOS is $P V = R T$
Where,

$V$

Volume

$R$

Universal ideal gas constant (SI: $J . k g^{- 1} m o l^{- 1}$ )

$T$

Temperature

It can also be written $P v = r T$ or $P = ρ r T$ .

Where,

$v$

Specific volume ( $v = 1 / ρ$ )

$r$

Gas constant (SI: $J . k g^{- 1} K^{- 1}$ ), $r = R / M$ ; where, $M$ is molecular mass
Alternative form of ideal gas is /EOS/IDEAL-GAS-VE (Volume Energy):(1)
$P (μ, E) = (γ - 1) (1 + μ) E$

Where,

$γ = \frac{c_{p}}{c_{v}}$

Is constant

$µ = \frac{ρ}{ρ_{0}} - 1$

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

In this case, equation of state /EOS/IDEAL-GAS-VT (Volume Temperature) is used when $γ$ is not constant, which is why the $c_{p} (T)$ function is required. $γ = \frac{c_{p}}{c_{v}} = \frac{c_{p}}{c_{p} - r}$ , since $c_{v} = c_{p} - r$ .

Consequently, when $γ$ is constant, defining $A_{0} = c_{p} = \frac{γ r}{γ - 1}$ , and $A_{1} = A_{2} = A_{3} = A_{4} = 0$ will provide the same equation of state as /EOS/IDEAL-GAS-VE formulation.
If T₀ is defined, then P₀ is computed:(2)
$P_{0} = r . ρ_{0} . T_{0}$

Otherwise, if P₀ is defined, then T₀ is computed:(3)
$T_{0} = P_{0} / r . ρ_{0}$
The $c_{p} (T)$ function is provided with Janaf model:(4)
$c_{p} (T) = A_{0} + A_{1} T + A_{2} T^{2} + A_{3} T^{3} + A_{4} T^{4}$

(SI: $J . k g^{- 1} . K^{- 1}$ )
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)