# AIRBAG1

Uniform pressure is assumed inside the volume. Perfect gas law and adiabatic conditions are assumed. Injected mass (or mass flow rate) and temperature are defined as a time function using the injector property. A sensor can define the inflator starting time.

The deflation of the vent hole occurs after reaching a pressure ( ${P}_{def}$ ) or time ( ${t}_{def}$ ) criteria.

## Numerical Damping

Viscosity, $\mu$ can be used to reduce numerical oscillations.

If $\mu$ =1, a critical damping (shell mass and volume stiffness) is used. A viscous pressure, $q$ is computed as:

$q=-\frac{\mu }{A}\sqrt{\frac{PA\rho t}{V}}\frac{dV}{dt}$ if $\frac{dV}{dt}<0$

$q=0$ if $\frac{dV}{dt}>0$

Where,
$t$
Fabric thickness
$\rho$
Density of the fabric
$A$
Bag surface
The applied pressure is:(1)
$P-{P}_{ext}+q$

## Initial Conditions

To avoid initial disequilibrium and mathematical discontinuity for zero mass or zero volume, the following initial conditions are set at time zero (Iequil =0) or at the beginning of jetting (if Iequil =1).

• ${P}_{ext}={P}_{ini}$ external pressure
• ${T}_{0}={T}_{ini}$ initial temperature (295K by default)
• If the initial volume is less than ${10}^{-4}{A}^{3}{2}}$ , a constant small volume is added to obtain an initial volume: ${V}_{ini}={10}^{-4}{A}^{3}{2}}$
• Initial mass, energy and density are defined from the above values.

There is no need to define an injected mass at time zero.

## Gases Definition

1. Initial and injected gas is defined with /MAT/GAS. Four types of gas (MASS, MOLE, PREDEF, or CSTA) could be defined. Then the specific capacity per unit mass at constant pressure for the gas is:
• MASS type(2)
${C}_{p}=\left({C}_{pa}+{C}_{pb}\text{\hspace{0.17em}}T+{C}_{pc}\text{\hspace{0.17em}}{T}^{2}+{C}_{pd}\text{\hspace{0.17em}}{T}^{3}+\frac{{C}_{pe}}{{T}^{2}}+{C}_{pf}\text{\hspace{0.17em}}{T}^{4}\right)$
• MOLE type(3)
${C}_{p}=\frac{1}{MW}\left({C}_{pa}+{C}_{pb}\text{\hspace{0.17em}}T+{C}_{pc}\text{\hspace{0.17em}}{T}^{2}+{C}_{pd}\text{\hspace{0.17em}}{T}^{3}+\frac{{C}_{pe}}{{T}^{2}}\right)$

Where, $MW$ is the molecular weight of the gas.

• CSTA type

User input ${C}_{p}$ and ${C}_{V}$ with the unit of $\left[\frac{J}{kgK}\right]$ .

• PREDEF type

About 14 commonly used gases (N2, O2, Air, etc) predefined in Radioss.

2. Injected gas

Njet defines the number of injectors by monitored volume. The material of the injected gas is defined with /MAT/GAS. The injector properties (/PROP/INJECT1 or /PROP/INJECT2) define the injected mass curve defined fct_IDM and injected temperature curve defined fct_IDT.

Injected mass curve and injection temperature can be obtained:
• From the airbag manufacturer
• From a tank test

sens_ID is the sensor number to start injection.

3. Jetting effect Ijet is used only for /MONVOL/AIRBAG1 or /MONVOL/COMMU1
If Ijet ≠ 0, the jetting effect is modeled as an overpressure $\text{Δ}{P}_{jet}$ applied to elements of the bag.(4)
$\text{Δ}{P}_{jet}=\text{Δ}\mathrm{P}\left(t\right)\cdot \text{Δ}\mathrm{P}\left(\theta \right)\cdot \text{Δ}\mathrm{P}\left(\delta \right)\cdot \mathrm{max}\left(n·m,0\right)$

N1, N2, and N3 are defined based on the injector geometry (refer to the Radioss Starter Input Manual)

$\text{Δ}\mathrm{P}\left(t\right),\text{Δ}\mathrm{P}\left(\theta \right),\text{Δ}\mathrm{P}\left(\delta \right)$ are empirical functions provided by the user via $fct_I{D}_{Pt}$ , $fct_I{D}_{P\theta }$ , and $fct_I{D}_{P\delta }$

## Vent Hole Definition

Nvent defines the number of vent holes used.

$surf_I{D}_{v}$ is the surface identifier defining the vent hole

Avent is the vent area (if $surf_I{D}_{v}=0$ ) or a scale factor ( $surf_I{D}_{v}$ ≠ 0)

Bvent = 0 (if $surf_I{D}_{v}=0$ ) or a scale factor on the impacted surface ( $surf_I{D}_{v}$ ≠ 0)

Tstop is a stop time for venting

Tstart is the time at which leakage starts

$\text{Δ}{P}_{def}$ is the relative vent deflation pressure

$\text{Δ}t{P}_{def}$ is the time duration during which $\text{Δ}P>\text{Δ}{P}_{def}$

$fct_I{D}_{v}$ is the function identifier ${\mathrm{f}}_{P}\left(P-{P}_{ext}\right)$ for Chemkin model (Iform=2)(5)
If $fct_I{D}_{v}\ne 0$ , the outflow velocity, $v$ is defined by Chemkin as:(6)
$v=Fscal{e}_{v}{\mathrm{f}}_{v}\left(P-{P}_{ext}\right)$

Where, $Fscal{e}_{v}$ is the scale factor of the function $fct_I{D}_{v}$ .

and the outgoing mass is computed as:(7)
${\stackrel{˙}{m}}_{out}=\rho \cdot {A}_{vent}\cdot {\mathrm{f}}_{v}\left(P-{P}_{ext}\right)\cdot Fscal{e}_{v}$
Or, with the conservation of enthalpy between airbag and vent hole, adiabatic conditions and unshocked flow, it is then possible to express outgoing mass flow through vent holes as a function of ${P}_{ext}$ , $\rho$ , ${P}_{vent}$ , ${u}_{vent}$ and ${A}_{vent}$ .(8)
${\stackrel{˙}{m}}_{out}={\rho }_{vent}\cdot {A}_{vent}\cdot u=\rho {\left(\frac{{P}_{ext}}{P}\right)}^{\frac{1}{\gamma }}\cdot {A}_{vent}\cdot u$
In the case of supersonic outlet flow, the vent pressure, ${P}_{vent}$ is equal to external pressure, ${P}_{ext}$ for unshocked flow. For shocked flow, ${P}_{vent}$ is equal to critical pressure, ${P}_{crit}$ and velocity, $u$ is bounded to critical sound speed:(9)
${u}^{2}<\frac{2}{\gamma +1}{c}^{2}=\frac{2\gamma }{\gamma +1}\frac{P}{\rho }$
and(10)
${P}_{crit}=P{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}}$
(11)
${P}_{vent}=\mathrm{max}\text{ }\left({P}_{crit}\text{​},\text{ }\text{\hspace{0.17em}}{P}_{ext}\text{​}\right)$
The outgoing mass flow of gas $i$ is:(12)
${\stackrel{˙}{m}}_{out}{}^{\left(i\right)}=\frac{{V}^{\left(i\right)}}{V}{\stackrel{˙}{m}}_{out}$
Where, ${V}^{\left(i\right)}$ is the volume occupied by gas $i$ and satisfies:(13)
${V}^{\left(i\right)}=\frac{{n}^{\left(i\right)}}{n}V$

from $P{V}^{\left(i\right)}={n}^{\left(i\right)}RT$ and $PV=\left[\sum _{i}{n}^{\left(i\right)}\right]RT$ .

Then,(14)
${\stackrel{˙}{m}}_{out}{}^{\left(i\right)}=\frac{{n}^{\left(i\right)}}{\sum _{i}{n}^{\left(i\right)}}{\stackrel{˙}{m}}_{out}$

## Porosity

The isenthalpic model is also used for porosity. In this case, you can define the surface for outgoing flow:(15)
${A}_{eff}={C}_{ps}\cdot Are{a}_{ps}$
or, (16)
${A}_{eff}={\mathrm{C}}_{ps}\left(t\right)\cdot {\mathrm{Area}}_{ps}\left(P-{P}_{ext}\right)$
Where,
${\mathrm{C}}_{ps}\left(t\right)$
Function of fct_IDcps
${\mathrm{Area}}_{ps}\left(P-{P}_{ext}\right)$
Function of fct_IDaps

It is also possible to define closure of the porous surface when contacts occurs by defining the interface option Ibag=1.