# 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.

## 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 $\frac{dV}{dt}>0$

- $t$
- Fabric thickness
- $\rho $
- Density of the fabric
- $A$
- Bag surface

The applied pressure is:

## Initial Conditions

To avoid initial disequilibrium and mathematical discontinuity for zero mass or zero volume, the
following initial conditions are set at time zero (`I`_{equil} =0) or at the beginning of jetting (if `I`_{equil} =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}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ , a constant small volume is added to obtain an initial volume: ${V}_{ini}={10}^{-4}{A}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$
- 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

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

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

`PREDEF`typeAbout 14 commonly used gases (N2, O2, Air, and so on) predefined in Radioss.

- Injected gas
`N`_{jet}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_ID`_{M}and injected temperature curve defined`fct_ID`_{T}.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. - Jetting effect
`I`_{jet}is used only for /MONVOL/AIRBAG1 or /MONVOL/COMMU1If`I`_{jet}≠ 0, the jetting effect is modeled as an overpressure $\text{\Delta}{P}_{jet}$ applied to elements of the bag.$$\text{\Delta}{P}_{jet}=\text{\Delta}\mathrm{P}\left(t\right)\cdot \text{\Delta}\mathrm{P}\left(\theta \right)\cdot \text{\Delta}\mathrm{P}\left(\delta \right)\cdot \mathrm{max}\left(n\xb7m,0\right)$$N

_{1}, N_{2}, and N_{3}are defined based on the injector geometry (refer to the Radioss Starter Input Manual)$\text{\Delta}\mathrm{P}\left(t\right),\text{\Delta}\mathrm{P}\left(\theta \right),\text{\Delta}\mathrm{P}\left(\delta \right)$ 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

`N`_{vent}- Defines the number of vent holes used
- $surf\_I{D}_{v}$
- Surface identifier defining the vent hole
`A`_{vent}- Vent area (if $surf\_I{D}_{v}=0$ ) or a scale factor ( $surf\_I{D}_{v}$ ≠ 0)

`B`

_{vent}= 0 (if $surf\_I{D}_{v}=0$ ) or a scale factor on the impacted surface ( $surf\_I{D}_{v}$ ≠ 0)

`T`_{stop}- Stop time for venting
`T`_{start}- Time at which leakage starts
- $\text{\Delta}{P}_{def}$
- Relative vent deflation pressure
- $\text{\Delta}t{P}_{def}$
- Time duration during which $\text{\Delta}P>\text{\Delta}{P}_{def}$
- $fct\_I{D}_{v}$
- Function identifier
${\mathrm{f}}_{P}\left(P-{P}_{ext}\right)$
for Chemkin model (
`I`_{form}=2)

If $fct\_I{D}_{v}\ne 0$ , the outflow velocity, $v$ is defined by Chemkin as:

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

and the outgoing mass is computed as:

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 ${A}_{vent}$ .

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:

and

The outgoing mass flow of gas $i$ is:

Where, ${V}^{(i)}$ is the volume occupied by gas $i$ and satisfies:

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

Then,

## Porosity

The isenthalpic model is also used for porosity. In this case, you can define the surface for outgoing flow:

or,

- ${\mathrm{C}}_{ps}(t)$
- Function of
`fct_ID`_{cps} - ${\mathrm{Area}}_{ps}(P-{P}_{ext})$
- Function of
`fct_ID`_{aps}

It is also possible to define closure of the porous surface when contacts occurs by
defining the interface option `I`_{bag}=1.