/EOS/IDEAL-GAS-VT

Block Format Keyword Describes the equation of state for ideal gas (volume - temperature).

Describes the equation of state for ideal gas $P=\rho rT$ ; where, ${c}_{p}$ is temperature dependent and described with Janaf model: ${c}_{p}\left(T\right)={A}_{0}+{A}_{1}T+{A}_{2}{T}^{2}+{A}_{3}{T}^{3}+{A}_{4}{T}^{4}$ (SI: $J.k{g}^{-1}.{K}^{-1}$ )

Format

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
/EOS/IDEAL-GAS/mat_ID/unit_ID
eos_title
$r$ T0 P0 Psh ${\rho }_{0}$
A0 A1 A2 A3 A4

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)

$r$ Ideal gas constant.

(Real)

$\left[J.k{g}^{-1}.{K}^{-1}\right]$
T0 Initial temperature. If P0 not defined, refer to Comment 3.

(Real)

$\left[\text{K}\right]$
P0 Initial pressure. If T0 is not defined, refer to Comment 3.

(Real)

$\left[\text{Pa}\right]$
Psh Pressure shift.

(Real)

$\left[\text{Pa}\right]$
${\rho }_{0}$ Reference density.

Default = material density (Real)

$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$
A0 Specific heat capacity parameter.

(Real)

$\left[\frac{\text{J}}{\text{k}\text{g}.\text{K}}\right]$
A1 Specific heat capacity parameter.

(Real)

$\left[\frac{\text{J}}{\text{k}\text{g}.{\text{K}}^{2}}\right]$
A2 Specific heat capacity parameter.

(Real)

$\left[\frac{\text{J}}{\text{k}\text{g}.{\text{K}}^{3}}\right]$
A3 Specific heat capacity parameter.

(Real)

$\left[\frac{\text{J}}{\text{k}\text{g}.{\text{K}}^{4}}\right]$
A4 Specific heat capacity parameter.

(Real)

$\left[\frac{\text{J}}{\text{k}\text{g}.{\text{K}}^{5}}\right]$

Example

#---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----|

1. Ideal gas EOS is $PV=RT$
Where,
$V$
Volume
$R$
Universal ideal gas constant (SI: $J.k{g}^{-1}mo{l}^{-1}$ )
$T$
Temperature

It can also be written $Pv=rT$ or $P=\rho rT$ .

Where,
$v$
Specific volume ( $v=1/\rho$ )
$r$
Gas constant (SI: $J.k{g}^{-1}{K}^{-1}$ ), $r=R/M$ ; where, $M$ is molecular mass
2. Alternative form of ideal gas is /EOS/IDEAL-GAS-VE (Volume Energy):(1)
$\mathrm{P}\left(\mu ,E\right)=\left(\gamma -1\right)\left(1+\mu \right)E$
Where,
$\gamma =\frac{{c}_{p}}{{c}_{v}}$
Is constant
$µ=\frac{\rho }{{\rho }_{0}}-1$
$E=\frac{{E}_{int}}{{V}_{0}}$
In this case, equation of state /EOS/IDEAL-GAS-VT (Volume Temperature) is used when $\gamma$ is not constant, which is why the ${c}_{p}\left(T\right)$ function is required. $\gamma =\frac{{c}_{p}}{{c}_{v}}=\frac{{c}_{p}}{{c}_{p}-r}$ , since ${c}_{v}={c}_{p}-r$ .

Consequently, when $\gamma$ is constant, defining ${A}_{0}={c}_{p}=\frac{\gamma r}{\gamma -1}$ , and ${A}_{1}={A}_{2}={A}_{3}={A}_{4}=0$ will provide the same equation of state as /EOS/IDEAL-GAS-VE formulation.

3. If T0 is defined, then P0 is computed:(2)
${P}_{0}=r.{\rho }_{0}.{T}_{0}$
Otherwise, if P0 is defined, then T0 is computed:(3)
${T}_{0}={P}_{0}/r.{\rho }_{0}$
4. The ${c}_{p}\left(T\right)$ function is provided with Janaf model:(4)
${c}_{p}\left(T\right)={A}_{0}+{A}_{1}T+{A}_{2}{T}^{2}+{A}_{3}{T}^{3}+{A}_{4}{T}^{4}$

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

5. 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)