2. Governing equations for inflator gas dynamics

An airbag inflator is, from a fluid-mechanics standpoint, a transient compressible-flow device. Within tens of milliseconds it converts stored chemical or compressed-gas energy into a high-temperature, high-pressure gas that must fill a cushion of roughly 60–120 liters. To predict the mass flow rate leaving the inflator, the pressure history inside the combustion chamber, and the temperature of the gas entering the bag, we need a consistent set of conservation laws. This section establishes those laws — mass, momentum, and energy — in the forms actually used for inflator modeling, and shows how they reduce to the working equations implemented in a lumped-parameter or 1-D simulation.

Control-volume formulation and sign conventions

All three conservation laws are written for a control volume (CV) that is fixed in space and encloses the region of interest — for example, the combustion chamber of a pyrotechnic inflator, or a short segment of the diffuser passage. The outward unit normal vector n points away from the CV. With this convention, mass, momentum, and energy fluxes are positive when they leave the CV, and a negative flux indicates inflow. This sign convention is used consistently throughout the course; any switch to an inflow-positive convention must be stated explicitly.

We denote density by ρ, velocity vector by u, pressure by p, specific internal energy by e, total specific energy by E = e + |u|²/2, and temperature by T. The gas is treated as a compressible, single-phase mixture with a temperature-dependent ratio of specific heats γ(T).

Conservation of mass

The integral mass balance states that the rate of change of mass inside the CV equals the net mass flux through its surface plus any mass source:

d/dt ∫_CV ρ dV  =  −∮_S ρ (u · n) dA  +  ṁ_source

For a pyrotechnic inflator, ṁ_source represents the gas generated by combustion of the propellant grain. In a lumped model the chamber is treated as a single well-mixed volume V_c, so the integral reduces to an ordinary differential equation:

d(ρ_c V_c)/dt = ṁ_gen − ṁ_out

where ṁ_gen is the propellant gas generation rate and ṁ_out is the mass flow through the exit orifice. This is the equation that governs the chamber pressure rise during the first few milliseconds of deployment.

Conservation of momentum

The momentum balance for the same CV is:

d/dt ∫_CV ρ u dV = −∮_S ρ u (u · n) dA − ∮_S p n dA + ∮_S τ · n dA + ∫_CV ρ g dV

The terms on the right are, in order: convective momentum flux, pressure force, viscous (shear) force, and body force. In inflator modeling the body force is negligible. For the chamber, momentum conservation is usually not solved directly; instead, the exit flow is modeled with an isentropic nozzle relation that already embeds the momentum balance. For the diffuser and the bag inlet, however, the 1-D momentum equation is retained:

∂(ρu)/∂t + ∂(ρu² + p)/∂x = −(f/D)(ρu²/2)

where f is the Darcy friction factor and D is the hydraulic diameter. This term captures the pressure loss through the diffuser ports and is essential for matching the measured bag-fill curve.

Conservation of energy

The energy equation, written for total energy, is:

d/dt ∫_CV ρE dV = −∮_S ρE (u · n) dA − ∮_S p (u · n) dA + Q̇ + Ẇ + ∫_CV ρ q̇ dV

Here Q̇ is heat transfer through the walls, Ẇ is shaft work (zero for an inflator), and q̇ is volumetric heat release from combustion. For the combustion chamber, the energy equation is often recast in terms of temperature using the caloric equation of state e = e(T):

m_c c_v dT_c/dt = ṁ_gen (h_gen − h_c) − ṁ_out (h_c − h_out) + Q̇_wall

This form makes the coupling explicit: the chamber temperature rises as hot propellant gas is generated, falls as gas leaves, and is moderated by heat loss to the metallic housing. The wall heat loss Q̇_wall is typically 5–15 % of the total energy release and cannot be ignored if the predicted peak pressure is to match test data.

Coupling through the equation of state

The three conservation laws are closed by an equation of state. For the high-temperature combustion products of a typical sodium-azide or guanidine-nitrate propellant, the ideal-gas law with a temperature-dependent γ is adequate up to about 2500 K:

p = ρ R T,   R = R_u / M_mix,   γ(T) = c_p(T)/c_v(T)

At higher temperatures, dissociation and condensed-phase products become significant, and a real-gas or equilibrium-chemistry model is required. The choice of equation of state directly affects the predicted mass flow rate, because the sonic velocity a = √(γRT) sets the choked-flow limit at the orifice.

From integral to discretized form

In a practical simulation the chamber is discretized into one or more control volumes, and the surface integrals become sums over faces. For a 1-D inflator model with N cells, the semi-discrete mass equation for cell i is:

d(ρ_i V_i)/dt = ṁ_{i−1/2} − ṁ_{i+1/2} + ṁ_gen,i

and the energy equation is advanced with the same face fluxes. A first-order upwind scheme is usually sufficient for the chamber, while the diffuser may require a second-order TVD scheme to resolve the expansion wave without excessive numerical diffusion. The time step is limited by the CFL condition Δt ≤ CFL · Δx / (|u| + a), with CFL ≈ 0.5 for explicit integration.

Worked example: chamber pressure during the first 5 ms

Consider a driver-side pyrotechnic inflator with a chamber volume of 45 cm³, an orifice area of 120 mm², and a propellant charge that generates 30 g of gas over 4 ms. Assume the gas has R = 320 J/(kg·K), γ = 1.25, and an initial chamber pressure of 1 atm at 300 K. Using the lumped mass and energy equations with a choked-flow exit model, the simulation predicts a peak chamber pressure of approximately 18–22 MPa at t ≈ 2.5 ms, followed by a rapid blowdown as the propellant is consumed. These values are illustrative and should be replaced by the actual propellant properties and geometry for a specific design.

The observable verification criteria are: (1) the peak pressure occurs before the propellant burn completes, (2) the pressure decay time constant is consistent with the orifice area and chamber volume, and (3) the integrated exit mass matches the total generated mass within 2 %. If the predicted peak pressure is too low, check the heat-loss term and the equation of state; if the decay is too slow, check the orifice discharge coefficient C_d, which is typically 0.85–0.95 for a sharp-edged port.

Practical notes on implementation

When implementing these equations, three details dominate the accuracy of the result. First, the discharge coefficient C_d at the orifice must be calibrated against a flow bench or a tank test; using C_d = 1.0 typically overpredicts the mass flow by 10–20 %. Second, the wall heat-transfer coefficient should be modeled with a correlation such as Dittus–Boelter for the turbulent combustion gas, not a constant value. Third, the time integration must be implicit or sub-cycled near the peak pressure, because the stiffness of the coupled mass–energy system can drive an explicit scheme unstable even when the CFL condition appears satisfied.

With the governing equations established, the next step in any inflator model is to select the constitutive relations — propellant burn rate, γ(T), and C_d — that close this system. Those choices determine whether the predicted pressure history matches the measured tank pressure curve within the tolerance required for occupant-restraint design.