Thermodynamical Equations

The basic energy equation of the airbag can be written as:
図 1.
d E a i r b a g = − P d V + d H i n = d H o u t
Where,
E MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
Internal energy
P MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
Pressure
V MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyraaaa@36C0@
Airbag volume
H i n MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGVbGaamyDaiaadshaaeqaaaaa@39D6@
Incoming enthalpy
H o u t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisamaaBa aaleaacaWGVbGaamyDaiaadshaaeqaaaaa@39D6@
Outgoing enthalpy
When the adiabatic condition is applied and assuming a perfect gas:
図 2.
P = ( γ − 1 ) E V

Where, γ is the gas constant. For air, γ = 1.4.

The two equations above allow the current airbag volume to be determined. The energy and pressure can then be found. To know the current airbag volume, derive energy and thus pressure.

Considering a gas such that the constant pressure and the constant volume heat capacities per mass unit (respectively, c p and c v ) vary in temperature T .

The following temperature dependency of the constant pressure heat capacity is assumed:
図 3.
c p = a + b T + c T 2

Where, a , b and c are the constants depending to characteristics of the gas.

The c p and c v satisfy the Mayer relation:
図 4.
c p (T)-c v (T) = R M
With R is the universal gas constant depending to the unit system ( R=8.3144 Jmo l −1 K −1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuaiabg2 da9iaaiIdacaGGUaGaaG4maiaaigdacaaI0aGaaGinaiaaysW7caWG kbGaamyBaiaad+gacaWGSbWaaWbaaSqabeaacqGHsislcaaIXaaaaO Gaam4samaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@45F2@ ):
図 5.
c v ( T ) = d e d T ( T ) h ( t ) = e ( t ) + P ρ
Where,
e
Specific energy
h
Specific enthalpy per mass unit of the gas at temperature T
You can then obtain:
図 6.
e ( T ) = e c o l d + ∫ T c o l d T c v ( T ) d T
and
図 7.
h ( T ) = e c o l d + ∫ T c o l d T c v ( T ) d T + R M T

Where, the lower index c o l d refers to the reference temperature T c o l d .

Now, assuming an ideal mixture of gas:
図 8.
P V = n R T
with n the total number of moles:
図 9.
n = ∑ i n ( i ) = ∑ i m ( i ) M ( i )
Where,
M ( i )
Mass of gas i
M ( i )
Molar weight of gas i
It follows:
図 10.
P = n R T V

With n = ∑ i m ( i ) M ( i ) .