Let
$T\left(t-\delta t\right)$
the temperature,
$P\left(t-\delta t\right)$
the pressure, and
$V\left(t-\delta t\right)$
the volume of the airbag at time
$t-\delta t$
, and
${\text{m}}^{(i)}$
the mass of gas
$i$
at time
$t-\delta t$
.
$T(t)$
,
$P(t)$
,
$V(t)$
are respectively temperature, pressure and volume of the
airbag at time
$t$
, and
${\text{m}}^{\text{(i)}}\text{}+\delta {\text{m}}_{\text{in}}^{\text{(i)}}\text{-}\delta {\text{m}}_{\text{out}}^{\text{(i)}}$
the mass of gas
$i$
at time
$t$
.

Using , the variation of total gas energy can be written as:

(1)
$$\text{\Delta}E=\left[{\displaystyle \sum _{i}({\text{m}}^{\text{(i)}}+\delta {m}^{(i)}{}_{\text{in}}-\delta {m}^{(i)}{}_{\text{out}})}\left({e}^{(i)}{}_{cold}+{\displaystyle \underset{{T}_{cold}}{\overset{T(t)}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)\right]-\left[{\displaystyle \sum _{i}{\text{m}}^{(i)}\left({e}^{(i)}{}_{cold}+{\displaystyle \underset{{T}_{cold}}{\overset{T(t-\delta t)}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]$$

which can be written as:

(2)
On the other hand, the basic energy equation

Thermodynamical Equations,

Equation 1 of the airbag and the expression of
enthalpy in

Thermodynamical Equations,

Equation 5 gives:

(3)
$$\begin{array}{l}\text{\Delta}E=\left[{\displaystyle \sum _{i}\delta {m}^{(i)}{}_{in}\left({e}^{(i)}{}_{cold}+\frac{R}{{M}^{(i)}}{T}^{(i)}{}_{in}+{\displaystyle \underset{{T}_{cold}}{\overset{{T}^{(i)}{}_{in}}{\int}}{c}_{v}{}^{(i)}\left(T\right)dT}\right)}\right]-\\ \text{\hspace{1em}}\text{\hspace{1em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[{\displaystyle \sum _{i}\delta {m}^{(i)}{}_{out}\left({e}^{(i)}{}_{cold}+\frac{R}{{M}^{(i)}}{T}^{(i)}{}_{out}+{\displaystyle \underset{{T}_{cold}}{\overset{{T}^{(i)}{}_{out}}{\int}}{c}_{v}{}^{(i)}\left(T\right)dT}\right)}\right]-\delta W\end{array}$$

Where,
$\delta {\text{m}}_{\text{in}}^{\text{(i)}}$
and
${T}^{(i)}{}_{in}$
are characteristics of the inflator and are considered as
input to the problem.
$\delta {\text{m}}_{\text{out}}^{\text{(i)}}$
and
${T}^{(i)}{}_{out}$
can be estimated from the velocity at vent hole
$u\left(t\right)$
.
$\delta W$
is the variation of the external work. This estimation will be
described hereafter.

It comes from

Equation 1 and

Equation 2:

(4)
$$\begin{array}{l}{\displaystyle \sum _{i}\left({m}^{(i)}+\delta {m}^{(i)}{}_{in}-\delta {m}^{(i)}{}_{out}\right){\displaystyle \underset{T(t-\delta t)}{\overset{T(t)}{\int}}{c}_{v}{}^{(i)}(T)dT}}=\\ \left[{\displaystyle \sum _{i}\delta {m}_{in}^{(i)}\left(\frac{R}{{M}^{(i)}}{T}^{(i)}{}_{in}+{\displaystyle \underset{T(t-\delta t)}{\overset{{T}_{in}^{(i)}}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]-\left[{\displaystyle \sum _{i}\delta {m}_{out}^{(i)}\left(\frac{R}{{M}^{(i)}}{T}_{out}^{(i)}+{\displaystyle \underset{T(t-\delta t)}{\overset{{T}_{out}^{(i)}}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]-\delta W\end{array}$$

The variation of the external work can be written as:

(5)
$$\delta W=\frac{\left(P(t)+P(t-\delta t)\right)}{2}(V(t)-V(t-\delta t))$$

Using

Thermodynamical Equations,

Equation 9, the last expression can be
written as:

(6)
$$\delta W=\frac{1}{2}\left(\frac{\left[{\displaystyle \sum _{i}\frac{{m}^{(i)}+\delta {m}^{(i)}{}_{in}-\delta {m}^{(i)}{}_{out}}{{M}^{(i)}}}\right]RT(t)}{V(t)}+\frac{\left[{\displaystyle \sum _{i}\frac{{m}^{(i)}}{{M}^{(i)}}}\right]RT(t-\delta t)}{V(t-\delta t)}\right)(V(t)-V(t-\delta t))$$

The last equation can be introduced to

Equation 4:

(7)
$$\begin{array}{l}\left[{\displaystyle \sum _{i}\left({m}^{(i)}+\delta {m}^{(i)}{}_{in}-\delta {m}^{(i)}{}_{out}\right){\displaystyle \underset{T(t-\delta t)}{\overset{T(t)}{\int}}{c}_{v}{}^{(i)}(T)dT}}\right]+\left[{\displaystyle \sum _{i}\frac{{m}^{(i)}+\delta {m}^{(i)}{}_{in}-\delta {m}^{(i)}{}_{out}}{{M}^{(i)}}}\right]RT(t)\frac{V(t)-V(t-\delta t)}{2V(t)}\\ =\left[{\displaystyle \sum _{i}\delta {m}^{(i)}{}_{in}\left(\frac{R}{{M}^{(i)}}{T}^{(i)}{}_{in}+{\displaystyle \underset{T(t-\delta t)}{\overset{{T}^{(i)}{}_{in}}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]-\left[{\displaystyle \sum _{i}\delta {m}^{(i)}{}_{out}\left(\frac{R}{{M}^{(i)}}{T}^{(i)}{}_{out}+{\displaystyle \underset{T(t-\delta t)}{\overset{{T}^{(i)}{}_{out}}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]-\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[{\displaystyle \sum _{i}\frac{{m}^{(i)}}{{M}^{(i)}}}\right]RT(t-\delta t)\frac{V(t)-V(t-\delta t)}{2V(t-\delta t)}\end{array}$$

The first order approximation

$\underset{T(t-\delta t)}{\overset{T(t)}{\int}}{c}_{v}{}^{(i)}(T)\approx {c}_{v}{}^{(i)}({T}_{|t-\delta t})}(T(t)-T(t-\delta t))$
for each gas, which allows rewrite

Equation 7 as:

(8)
$$\begin{array}{l}\left[{\displaystyle \sum _{i}\left({m}^{(i)}+\delta {m}^{(i)}{}_{in}-\delta {m}^{(i)}{}_{out}\right){c}_{v}{}^{(i)}({T}_{|t-\delta t})(T(t)-T(t-\delta t))}\right]+\\ \left[{\displaystyle \sum _{i}\frac{{m}^{(i)}+\delta {m}^{(i)}{}_{in}-\delta {m}^{(i)}{}_{out}}{{M}^{(i)}}}\right]RT(t)\frac{V(t)-V(t-\delta t)}{2V(t)}\\ =\left[{\displaystyle \sum _{i}\delta {m}^{(i)}{}_{in}\left(\frac{R}{{M}^{(i)}}{T}^{(i)}{}_{in}+{\displaystyle \underset{T(t-\delta t)}{\overset{{T}^{(i)}{}_{in}}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]-\left[{\displaystyle \sum _{i}\delta {m}^{(i)}{}_{out}\left(\frac{R}{{M}^{(i)}}{T}_{out}+{\displaystyle \underset{T(t-\delta t)}{\overset{{T}^{(i)}{}_{out}}{\int}}{c}_{v}{}^{(i)}(T)dT}\right)}\right]-\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[{\displaystyle \sum _{i}\frac{{m}^{(i)}}{{M}^{(i)}}}\right]RT(t-\delta t)\frac{V(t)-V(t-\delta t)}{2V(t-\delta t)}\end{array}$$

Which allows to determine the actual temperature
$T(t)$
. The actual pressure then computed from the equation of
perfect gas (Thermodynamical Equations, Equation 9).