/MONVOL/AIRBAG1

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Definitions

Comments

1. The airbag external surface should be built only from 4- and 3-noded shell elements. The airbag external surface cannot be defined with /SURF/SEG, or with /SURF/SURF, if a sub-surface is defined in /SURF/SEG.
2. The volume must be closed and the normals must be oriented outwards.
3. Abscissa scale factors are used to transform abscissa units in airbag functions, for example:(1)
   $$\text{F}(t\prime )={\text{f}}_t\left(\frac{t}{{\mathit{Ascale}}_t}\right)$$
   Where,

   $t$

   Time

   ${\mathrm{f}}_t$

   Function of fct_ID_t

   (2)

   $$\text{F}(P\prime )={\text{f}}_P\left(\frac{P}{{\mathit{Ascale}}_P}\right)$$
   Where,

   $P$

   Pressure

   ${\mathrm{f}}_P$

   Function of fct_ID_P

   The options are obsolete. Normally, the curve scaling parameters are used instead.
4. Pressure and temperature of external air and the initial pressure and temperature of air inside of airbag is set to P_ext. and T₀.
5. Initial thermodynamic equilibrium is written at time zero (I_equil =0) or at beginning of jetting (I_equil =1), based on the following equation with respect to the volume at time zero, or the volume at beginning of jetting:(3)
   $$P_{\mathit{ext}}V=R\frac{M_0}{M_i}{T}_0$$
   Where,

   $M_0$

   Mass of gas initially filling the airbag

   $M_{\mathrm{i}}$

   Molar mass of the gas initially filling the airbag

   $R$

   Gas constant depending on the units system given in /BEGIN card. For example in SI system:(4)

   $$R=8.314\frac{J}{\mathit{mole}\cdot K}$$

6. If jetting is used, an additional $\text{Δ}{P}_{jet}$ pressure is applied to each element of the airbag:(5)
   $$\Delta P_{\mathit{jet}}=\Delta \text{P}\left(t\right)\cdot \Delta \text{P}\left(\theta \right)\cdot \Delta \text{P}\left(\delta \right)\cdot \text{max}\left(\mathbf{n}·\mathbf{m},0\right)$$

   With m being the normalized vector between the projection of the center of the element upon segment (N₁ and N₃) and the center of the element; $\theta$ the angle between vectors MN₂ and m (in degrees), $\delta$ the distance between the center of the element and its projection upon segment (N₁ and N₃).

   The projection of a point upon segment (N₁ and N₃) is defined as the projection of the point in direction MN₂ upon the line (N₁ and N₃) if it lies inside the segment (N₁ and N₃). If this is not the case, the projection of the point upon segment (N₁ and N₃) is defined as the closest node N₁ or N₃.

   
   
   

   Figure 1. Dihedral Shape of the Jet

   with M between of N₁ and N₃

7. If node_ID₃ = 0, node_ID is set to node_ID₁ and the dihedral shape is reduced to a conical shape.
8. If fct_ID_v = 0: isenthalpic outflow is assumed, else Chemkin model is used and outflow velocity is:(6)
   $$v={\mathit{Fscale}}_v\cdot {\text{f}}_v\left(P-P_{\mathit{ext}}\right)$$
   Where, $f_v$ is the function of fct_ID_v.

   • Isenthalpic model

     Venting or the expulsion of gas from the volume 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, A.

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

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

     Applying the adiabatic conditions:

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

     Where, $P$ is the pressure of gas into the airbag and $\rho$ is the density of gas into the airbag.

     Therefore, the exit velocity is given by:(7)

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

     For supersonic flows the outlet velocity is determined as described in Supersonic Outlet Flow in the Theory Manual.

     The mass out flow rate is given by:(8)

     $${\dot{m}}_{\mathit{out}}={\rho }_{\mathit{vent}}·\mathit{vent}\_\mathit{holes}\_\mathit{surface}·u=\rho {\left(\frac{P_{\mathit{ext}}}{P}\right)}^{\frac{1}{\gamma }}·\mathit{vent}\_\mathit{holes}\_\mathit{surface}·u$$
     The energy flow rate is given by:(9)

     $${\dot{E}}_{\mathit{out}}={\dot{m}}_{\mathit{out}}\frac{E}{\rho V}={\left(\frac{P_{\mathit{ext}}}{P}\right)}^{\frac{1}{\gamma }}*\mathit{vent}\_\mathit{holes}\_\mathit{surface}*u\frac{E}{V}$$

     Where, $V$ is the airbag volume and $E$ is the internal energy of gas into the airbag.

   • Chemkin model(10)
     $${\dot{m}}_{\mathit{out}}=\rho \cdot \mathit{vent}\_\mathit{holes}\_\mathit{surface}\cdot {\text{f}}_v\left(P-P_{\mathit{ext}}\right)\cdot {\mathit{Fscale}}_v$$

     Where, $\rho$ is the density of the gas within the airbag and $f_v$ is the function of fct_ID_v.

9. Vent hole area is computed as:(11)
   $$vent\_holes\_area=A_{vent}\cdot A_{non\_impacted}\cdot {\mathrm{f}}_t\left(t\right)\cdot {\mathrm{f}}_P\left(P-P_{ext}\right)\cdot {\mathrm{f}}_A\left(\frac{A_{non\_impacted}}{A_0}\right)$$
   $$+B_{\mathit{vent}}\cdot A_{\mathit{impacted}}\cdot {\text{f}}_{t^{\prime }}\left(t\right)\cdot {\text{f}}_{P^{\prime }}\left(P-P_{\mathit{ext}}\right)\cdot {\text{f}}_{A^{\prime }}\left(\frac{A_{\mathit{impacted}}}{A_0}\right)$$
   With impacted surface:(12)

   $$A_{\mathit{impacted}}=\sum _{e\in S_{\mathit{vent}}}\frac{n_c\left(e\right)}{n\left(e\right)}{A}_e$$
   and non-impacted surface:(13)

   $$A_{\mathit{non}\_\mathit{impacted}}=\sum _{e\in S_{\mathit{vent}}}\left(1-\frac{n_c\left(e\right)}{n\left(e\right)}\right)A_e$$

   
   
   

   Figure 2.

   Where for each element e of the airbag materials $n_c\left(e\right)$ means the number of impacted nodes among the $n\left(e\right)$ nodes defining the element and $A_e$ is the area of element e.

   And,

   A₀ is the initial area of surface surf_ID_v

   ${\mathrm{f}}_t$ , ${\mathrm{f}}_P$ and ${\mathrm{f}}_A$ are functions of fct_ID_t, fct_ID_P and fct_ID_A

   ${\mathrm{f}}_{\mathrm{t\text{'}}}$ , ${\mathrm{f}}_{\mathrm{P\text{'}}}$ and ${\mathrm{f}}_{\mathrm{A\text{'}}}$ are functions of fct_ID_t', fct_ID_P' and fct_ID_A'

10. Functions fct_ID_t' and fct_ID_P' are assumed to be equal to 1, if they are not specified (null identifier).
11. If function fct_ID_A' is not specified, it is assumed as:(14)
    $${\text{f}}_{A^{\prime }}\left(A\right)=A$$
12. To account for contact blockage of vent holes and porous surface area, flag I_BAG must be set to 1 in the correspondent interfaces (Line 3 of interface TYPE7 or TYPE23). If not, the nodes impacted into the interface are not considered as impacted nodes in the previous formula for A_impacted and A_{non_impacted}.
13. When there is no sensor which activates gas injection, the vent hole membrane is deflated, if time T becomes greater than the T_start or if the pressure P exceeds P_def value longer than the time given in $\text{Δ}t{P}_{def}$ .
14. When at least one of the injectors is activated by the sensor, then the activation of venting and porosity options is controlled by I_ttf

    T_inj is the time of the first injector to be activated by the sensor.

    I_ttf = 0:

    	Venting, Porosity
    Activation	When $P>\text{Δ}{P}_{def}$ longer than the time $\text{Δ}t{P}_{def}$ , or $T>T_{start}$
    Deactivation	Tstop
    Time dependent functions	No shift

    I_ttf = 3:

    	Venting, Porosity
    Activation	When $T>T_{inj}$ and $P>\text{Δ}{P}_{def}$ longer than the time $\text{Δ}t{P}_{def}$ , or $T>T_{inj}+T_{start}$
    Deactivation	$T_{inj}+T_{stop}$
    Time dependent functions	Shifted by $T_{inj}+T_{start}$

    All other related curves are active when the corresponding venting, porosity or communication option is active.

    The variety of I_ttf values comes from historical reasons. Values I_ttf =1 and 2 are obsolete and should not be used. Usual values are I_ttf =0 (no shift) or I_ttf =3 (all relative options are shifted by T_inj).

15. Leakage by porosity formulations; the mass flow rate flowing out is computed as:
    • Iform_ps = 0 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\sqrt{2P\rho }{Q}^{\frac{1}{\gamma }}\sqrt{\frac{\gamma }{\gamma -1}\left[1-Q^{\frac{\gamma -1}{\gamma }}\right]}$ (Isentropic - Wang Nefske)

      with $Q=\frac{P_{\mathit{ext}}}{P}$

      and $A_{\mathit{eff}}=C_{\mathit{ps}}\cdot {\mathit{Area}}_{\mathit{ps}}$ or $A_{\mathit{eff}}={\text{C}}_{\mathit{ps}}(t)\cdot {\text{Area}}_{\mathit{ps}}(P-P_{\mathit{ext}})$

      ${\mathrm{C}}_{ps}(t)$ is the function of fct_ID_cps and ${\mathrm{Area}}_{ps}(P-P_{ext})$ is function of fct_ID_aps

      The effective venting area A_eff does not vary with different airbag fabric materials.

    • If Iform_ps > 0, the effective venting area A_eff is computed according to the input in the /LEAK/MAT input for fabric materials of TYPE19 or TYPE58.

      Iform_ps = 1 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\sqrt{2P\rho }{Q}^{\frac{1}{\gamma }}\sqrt{\frac{\gamma }{\gamma -1}\left[1-Q^{\frac{\gamma -1}{\gamma }}\right]}$ (Isentropic - Wang Nefske)

      Iform_ps = 2 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\rho v(P-P_{\mathit{ext}})$

      Where, $\upsilon$ is the outflow gas velocity (Chemkin).

      Iform_ps = 3 ${\dot{m}}_{\mathit{out}}=A_{\mathit{eff}}\sqrt{2\rho (P-P_{\mathit{ext}})}$ (Graefe)

16. If leakage blockage is activated, Iblockage=1, the effective venting area is modified as:(15)
    $$A_{\mathit{eff}}=A_{\mathit{non}\_\mathit{impacted}}$$

    A_{non_impacted} is non-impacted surface 11.

    The blockage will be active only if flag I_BAG is set to 1 in the concerned contact interfaces (line 3 of interface TYPE7 and TYPE23).

17. The lost heat flow is given by:(16)
    $$\dot{\text{Q}}\left(x,t\right)=H_{\mathit{conv}}\cdot \text{Area}\left(x,t\right)\cdot \left(\text{T}\left(x,t\right)-T_0\right)$$