# Venting Outgoing Mass Determination

Venting, or the expulsion of gas from the airbag, is assumed to be isenthalpic.

The flow is also assumed to be unshocked, coming from a large reservoir and through a small orifice with effective surface area, $T$ .

Conservation of enthalpy leads to velocity, $u$ , at the vent hole. The Bernouilli equation is then written as:

$\frac{\gamma }{\gamma -1}\frac{P}{\rho }=\frac{\gamma }{\gamma -1}\frac{{P}_{ext}}{{\rho }_{vent}}+\frac{{u}^{2}}{2}$ (vent hole)

$\frac{P}{{\rho }^{\gamma }}=\frac{{P}_{ext}}{{\rho }_{vent}{}^{\gamma }}$ (vent hole)

Therefore, the exit velocity is given by:

${u}^{2}=\frac{2\gamma }{\gamma -1}\frac{P}{\rho }\left(1-{\left(\frac{{P}_{ext}}{P}\right)}^{\frac{\gamma -1}{\gamma }}\right)$

with $\rho =\frac{\sum _{i}{m}^{\left(i\right)}}{V}$ the averaged density of the gas and $\gamma =\frac{\left[\sum _{i}{m}^{\left(i\right)}{c}_{p}{}^{\left(i\right)}\right]/\left[\sum _{i}{m}^{\left(i\right)}\right]}{\left[\sum _{i}{m}^{\left(i\right)}{c}_{v}{}^{\left(i\right)}\right]/\left[\sum _{i}{m}^{\left(i\right)}\right]}$ the fraction of massic averages of heat capacities at constant pressure and constant volume.

The mass flow rate is given by:

${\stackrel{˙}{m}}_{out}={\rho }_{vent}{A}_{vent}^{}u=\rho {\left(\frac{{P}_{ext}}{P}\right)}^{1/\gamma }{A}_{vent}u$

The energy flow rate is given by:

${\stackrel{˙}{E}}_{out}=\stackrel{˙}{m}\frac{E}{\rho V}={\left(\frac{{P}_{ext}}{P}\right)}^{1/\gamma }{A}_{vent}u\frac{E}{V}$

The total mass flow rate is given by:

$d{m}_{out}=\rho {\left(\frac{{P}_{ext}}{P}\right)}^{1/\gamma }{A}_{vent}u$

Where,
${A}_{vent}$
Vent hole surface.
The vent hole area or scale factor area, ${A}_{vent}$ , can be defined in two ways:
• a constant area taking into account a discharge coefficient
• a variable area equal to the area of a specified surface multiplied by a discharge coefficient.

## Supersonic Outlet Flow

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 $u$ is bounded to critical sound speed:

${u}^{2}<\frac{2}{\gamma +1}\text{\hspace{0.17em}}{c}^{2}=\frac{2*\gamma }{\gamma +1}\text{\hspace{0.17em}}\frac{P}{\rho }$

And,

${P}_{crit}=P{\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}}$

${P}_{vent}=\mathrm{max}\left({P}_{crit},{P}_{ext}\right)$

## Outgoing Mass per Gas

The mass flow of gas $i$ is $d{m}^{\left(i\right)}{}_{out}=\frac{{V}^{\left(i\right)}}{V}d{m}_{out}$ , where ${V}^{\left(i\right)}$ is the volume occupied by gas $i$ and satisfies:

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

It comes finally

$d{m}^{\left(i\right)}{}_{out}=\frac{{n}^{\left(i\right)}}{\sum _{i}{n}^{\left(i\right)}}d{m}_{out}$