4. Inflator internal ballistics

Internal ballistics describes what happens inside the inflator from the instant the initiator fires until the gas has fully vented into the airbag cushion. For a pyrotechnic or hybrid inflator, the entire useful event lasts roughly 20–80 ms, yet the pressure history in that window determines bag fill time, cushion peak pressure, and occupant injury metrics. Three coupled phenomena dominate: the rate at which gas is generated in the combustion chamber, the pressure that builds as a result, and the mass flow that escapes through the nozzle throat. This chapter develops the governing relations and shows how to evaluate them numerically.

Chamber Pressure Evolution

Treat the combustion chamber as a control volume with a single inlet (gas generation) and a single outlet (nozzle throat). Applying the ideal-gas equation of state and conservation of mass and energy, the chamber pressure P_c evolves according to a lumped-parameter ODE:

dP_c/dt = (γ - 1)/V_c · (ṁ_gen · H_g - ṁ_out · h_out) - γ · P_c/V_c · dV_c/dt

where γ is the ratio of specific heats, V_c the instantaneous chamber volume, ṁ_gen the propellant mass generation rate, H_g the heat of reaction per unit mass, ṁ_out the mass flow through the throat, and h_out the specific enthalpy of the exiting gas. For a fixed-volume chamber (typical of a hybrid inflator's combustion can), the last term vanishes. For a pyrotechnic inflator with a moving piston or rupturing burst disc, dV_c/dt must be retained.

In practice, ṁ_gen is not constant. Solid propellant burns according to Vieille's law:

ṁ_gen = ρ_p · A_b · a · P_c^n

with ρ_p the propellant density, A_b the burning surface area, and a, n empirical burn-rate coefficients. Because ṁ_gen depends on P_c and ṁ_out depends on P_c as well, the system is nonlinear and must be integrated numerically. A representative pyrotechnic inflator might use a = 0.05 (mm/s)/(MPaⁿ), n = 0.35, ρ_p = 1650 kg/m³, and an initial A_b of 2.0 × 10^-3 m². These values are illustrative and must be replaced with propellant-specific data for any real design.

Nozzle Throat Flow and Choked Conditions

Gas leaves the chamber through a convergent nozzle (or an orifice plate) whose minimum area is the throat, A_t. The mass flow rate depends on the ratio of chamber pressure to downstream (cushion) pressure, P_c/P_a. When this ratio exceeds a critical value, the flow chokes — the throat velocity reaches the local speed of sound and further reductions in downstream pressure cannot increase the mass flow.

The critical pressure ratio for an ideal gas is:

P_c/P_a ≥ ((γ + 1)/2)^(γ/(γ-1))

For γ = 1.3 (typical for combustion gases), the critical ratio is approximately 1.83. For γ = 1.4, it is about 1.89. In an airbag inflator, P_c can reach 20–60 MPa while the cushion starts at atmospheric pressure, so the ratio is far above critical and the throat is choked for essentially the entire discharge event.

Under choked conditions, the mass flow is:

ṁ_out = C_d · A_t · P_c · sqrt(γ/(R·T_c)) · (2/(γ+1))^((γ+1)/(2(γ-1)))

where C_d is the discharge coefficient (typically 0.85–0.95 for a well-machined convergent nozzle), R the specific gas constant, and T_c the chamber temperature. Note that ṁ_out is directly proportional to P_c and A_t, and inversely proportional to the square root of T_c. This linear dependence on P_c is what makes the chamber pressure equation stiff and requires small time steps in explicit integration.

Numerical Integration and Practical Observations

Because the chamber pressure equation is stiff, a simple explicit Euler scheme with a fixed time step will diverge unless the step is very small. A practical approach is to use an adaptive Runge–Kutta–Fehlberg (RKF45) integrator with a relative tolerance of 1 × 10^-4 and an initial step of 1 × 10^-6 s. The state vector is [P_c, T_c, m_prop, V_a], where m_prop is the remaining propellant mass and V_a is the cushion volume. At each step, the solver must:

1. Compute ṁ_gen from the current P_c and remaining A_b.
2. Evaluate the pressure ratio P_c/P_a and determine whether the throat is choked.
3. If choked, use the choked-flow equation; otherwise, use the subsonic isentropic relation.
4. Update P_c, T_c, and m_prop using the ODE derivatives.
5. Update the cushion pressure P_a from the accumulated mass and energy in the bag.

A typical simulation for a driver-side inflator might show P_c peaking at 35 MPa at t = 8 ms, with the throat choked from t = 1 ms to t = 25 ms. The cushion reaches 0.25 MPa (gauge) at t = 30 ms. These numbers are illustrative and depend strongly on propellant formulation, nozzle area, and cushion volume.

Design Implications

The choked-flow condition has a direct design consequence: once the throat is choked, the mass flow rate is controlled solely by P_c and A_t, not by the cushion pressure. This decouples the inflator from the bag to first order, which simplifies control but also means that any variation in A_t (due to manufacturing tolerance or thermal expansion) directly translates into a proportional variation in mass flow. A 5% variation in A_t produces roughly a 5% variation in the delivered gas mass, which can shift the cushion peak pressure by a similar percentage. For this reason, throat diameter is usually the tightest-toleranced dimension in the inflator.

Conversely, if the throat is not choked — for example, in a low-pressure hybrid inflator or during the late venting phase — the mass flow becomes sensitive to both P_c and P_a, and the coupling between inflator and cushion must be modeled explicitly. The transition between choked and subsonic flow is a common source of numerical difficulty and should be handled with a smooth blending function or a switch with hysteresis to avoid chattering.

When validating a model against test data, compare the chamber pressure trace first, then the cushion pressure trace. If the chamber pressure matches but the cushion pressure is low, the likely cause is an overestimated C_d or an underestimated A_t. If the chamber pressure decays too slowly, check the burn-rate exponent n and the initial burning area A_b. These diagnostics are far more informative than a single scalar error metric and should be part of any inflator model calibration workflow.