2. Governing equations for airbag gas dynamics

An airbag inflator is, from a thermodynamic standpoint, a small, short-duration chemical reactor coupled to a rapidly expanding control volume. To predict how pressure, temperature, and mass evolve inside the cushion during the first 30–80 ms of deployment, we need a consistent set of governing equations: mass conservation, momentum (usually simplified), energy conservation, and an equation of state. This chapter establishes those equations in the control-volume form used by virtually all production airbag simulation codes, and shows how the inflator is coupled to the bag as a mass and energy source.

Control Volume Selection and Reference Conventions

We define a control volume (CV) as the gas region enclosed by the cushion fabric plus the inflator exit plane. The CV boundary is deformable: the fabric moves outward as the bag inflates, so the volume V(t) is a function of time. All fluxes are taken as positive into the CV unless stated otherwise. This sign convention is used consistently throughout the chapter and in the accompanying figure.

Two simplifications are standard for inflator-driven flows:

• The gas inside the bag is treated as a single lumped thermodynamic state — pressure p(t), temperature T(t), density ρ(t) — rather than a spatially resolved field. This is the uniform-pressure / lumped-parameter assumption.
• Kinetic energy of the gas bulk and gravitational potential energy are neglected relative to internal energy. This is valid because gas velocities inside the bag are typically below ~100 m/s and the deployment time is short.

Mass Conservation

The integral mass balance for the CV is

d(ρV)/dt = ṁ_in − ṁ_out

where ṁ_in is the mass flow rate entering from the inflator (and any secondary inlet), and ṁ_out is the mass flow rate leaving through vents, fabric porosity, or seam leakage. For a single-chamber driver-side bag with one inflator and one vent, this reduces to

d(ρV)/dt = ṁ_inflator(t) − ṁ_vent(t)

The inflator mass flow rate is the primary input from the gas generator. For a pyrotechnic inflator it is derived from the propellant burn rate; for a hybrid or stored-gas inflator it is derived from the orifice equation. In either case, ṁ_inflator(t) is supplied as a table or analytic function of time and, for hybrid systems, of bottle pressure.

Energy Conservation

Applying the first law to the deformable CV, neglecting kinetic and potential energy, gives

d(ρV u)/dt = ṁ_in h_in − ṁ_out h + Q̇ − Ẇ

Here u is specific internal energy, h is specific enthalpy, Q̇ is heat transfer to the gas (positive into the CV), and Ẇ is work done by the gas on the surroundings. The work term is the one that distinguishes an airbag from a rigid tank:

Ẇ = p dV/dt

Because the fabric moves, the gas does expansion work on the bag, which cools the gas and reduces pressure relative to a constant-volume vessel. This is why a bag that inflates freely reaches a lower peak pressure than the same mass of gas confined in a rigid tank.

For an ideal gas with constant specific heats, u = c_v T and h = c_p T, so the energy equation becomes

d(c_v ρV T)/dt = ṁ_in c_p T_in − ṁ_out c_p T − p dV/dt + Q̇

In most deployment simulations, Q̇ is small during the first tens of milliseconds and is either neglected or modeled with a simple convective coefficient. The dominant terms are the inflator enthalpy inflow and the expansion work.

Ideal Gas Assumptions and Their Limits

The ideal gas law closes the system:

p = ρ R T,  R = R_u / M

where R_u is the universal gas constant and M is the mixture molar mass. For airbag gases — typically a mixture of N₂, CO₂, H₂O, and small amounts of CO and NOₓ — the ideal gas assumption is reasonable at the temperatures and pressures of interest (roughly 300–1200 K, 1–3 bar absolute). Two caveats matter in practice:

• Variable composition. The inflator gas composition changes during the burn. If the molar mass or specific heat ratio γ varies significantly, use a mixture-averaged R and c_v rather than fixed values.
• High-pressure departures. Inside the inflator combustion chamber, pressures can exceed 100 bar, where real-gas effects are non-negligible. The bag CV, however, operates near ambient pressure, so the ideal gas law is adequate there.

Control Volume Formulation for Inflator-Driven Flow

Combining mass, energy, and the ideal gas law yields the working set of ODEs solved by the simulation:

d(ρV)/dt = ṁ_in − ṁ_out
d(c_v ρV T)/dt = ṁ_in c_p T_in − ṁ_out c_p T − p dV/dt + Q̇
p = ρ R T

These three equations are integrated in time together with a kinematic model for V(t) and dV/dt. The volume rate is obtained from the fabric geometry and the bag's unfolding kinematics, or from a coupled structural solver in FSI simulations. The vent mass flow is computed from the orifice equation using the instantaneous internal pressure and the ambient pressure:

ṁ_vent = C_d A_vent √(2ρ Δp)   for Δp > 0

where C_d is the discharge coefficient (typically 0.6–0.8 for fabric vents) and A_vent is the effective vent area.

Coupling to the Inflator Model

The bag CV equations require ṁ_in(t) and T_in(t) from the inflator. For a pyrotechnic inflator, these come from a burn model:

dm_prop/dt = A_b(r) · r_b(p) · ρ_prop
ṁ_in = dm_prop/dt · (1 + m_gas/m_prop)

where A_b is the burning surface area, r_b is the burn rate (often r_b = a p^n), and the second factor accounts for the mass of gas generated per unit mass of propellant. The inflator exit temperature T_in is the adiabatic flame temperature of the propellant, typically 1000–1500 K, reduced by heat loss to the inflator housing.

Summary of the Governing Set

These equations form the backbone of every lumped-parameter airbag deployment model. The next chapter extends them to multi-chamber bags and to the momentum equation needed when gas inertia inside the bag cannot be neglected.