Fundamentals of Internal Ballistic Analysis & Modeling

A practical engineering introduction to combustion, pressure generation, thermodynamics and mechanical response in cartridge- and propellant-actuated devices.

Internal ballistic analysis is the study of processes occurring inside a pyrotechnically actuated mechanical device from the moment the pyrotechnic material, typically a solid propellant or mixture of propellants, is ignited until the projectile completes its stroke or exits the barrel of the device. In aerospace and defense applications these pyro-mechanical devices are commonly referred to as Cartridge and Propellant Actuated Devices (CAD/PAD).  CAD/PADs are frequently used in spacecraft applications for stage separation, booster jettison, service module jettison, parachute deployment, fairing separation and propulsion system fluid and gas flow control valves (pyrovalve). In defense and military applications, CAD/PADs are used in systems such as fighter jet crew egress systems, ejection seats, pylon jettison, and ballistic missile wing deployment systems, where rapid actuation is crucial.

CAD/PADs are powered by high pressure generated through rapid exothermic chemical reactions resulting from deflagration of propellants. In a constant volume vessel, the pressure primarily depends on the chemical and mechanical properties of the propellant such as grain geometry, which dictates the burning surface area progression, burn rate, isochoric flame temperature, molecular weight, particle size, and propellant mass. For stroking devices, the mechanical and structural aspects of the design govern the internal ballistics during combustion as the chamber volume expands. Key factors, including piston diameter, propelled mass, resisting and assisting external forces, friction, hydraulic and pneumatic dampers, and initial chamber volume, are critical components of CAD/PAD design and significantly influence the gas pressure profile throughout the stroke.

Modeling and analysis of a CAD/PAD involves simultaneously solving the governing equations of combustion and the kinematic equations of motion. While most custom ballistic analysis codes use traditional one-dimensional numerical solution methods to solve these equations with high accuracy, more complex mechanical designs with features such as non-linear external loads, linkages, locking mechanisms, shock absorbers, lubricants, and springs require Finite Element Analysis (FEA) or rigid body modeling tools to achieve accurate kinematic and structural solutions. However, preliminary estimates remain essential for initial design phases before detailed simulations.

1. Internal Ballistic Theory

Consider a generic pyrotechnic thruster as shown in Figure 1. The goal of ballistic analysis is to predict performance metrics (pressure, velocity, etc.) as function of time (or in some cases function of stroke) for a given set of ballistic parameters. Not all parameters are required for back-of-the-envelope estimate calculations but for a detailed numerical modeling, these parameters are essential and should be accurately defined.

Following descriptive set of differential equations must be solved to calculate the critical functions such as chamber pressure, chamber temperature, propellant weight requirement, stroke time, piston velocity, muzzle velocity, etc.

The conservation of energy for a propellant actuated stroking pyrotechnic device is represented by the following equation:

$$
\begin{flalign}
& c_v \frac{d(m_p T)}{dt}
= c_v T_0 \frac{dm_p}{dt}
– A_p p_c \frac{dx}{dt}
– \frac{d(HL)}{dt} \quad (1) &&
\end{flalign}
$$

The conservation of mass in the context of internal ballistics is defined as a function of the propellant burn rate and the burning surface area:

$$
\begin{flalign*}
& \frac{dm_p}{dt} = \rho S r \quad (2) &&
\end{flalign*}
$$

Propellant burn rate, r, is a function of the space-average pressure and is typically derived from the experimental data using the following equation:
$$
\begin{flalign*}
& r = B p^{\,n} \quad (3) &&
\end{flalign*}
$$

where B and n are empirical constants derived from the experimental data.

To define propellant grain geometry at any time during burn, a form function is required, which relates the mass of the propellant burned at any time to the distance burned through the propellant grain. For most propellant geometries, a cubic form function is sufficient to define the propellant geometry and the burn surfaces.

$$
\begin{flalign*}
& m_p = A_1 L + A_2 L^2 + A_3 L^3 \quad (4) &&
\end{flalign*}
$$

Where A₁, A₂ and A₃ are the form function constants, and the L is the distance burned through the grain.

For example, for an uninhibited cylindrical grain with length h, after burning through a distance L, the charge left unburned is

$$
\begin{flalign*}
& C – m_p
= \rho \cdot \text{(unburned volume)}
= \rho\,\frac{\pi}{4}(D – 2L)^{2}(h – 2L) &&
\end{flalign*}
$$

The initial unburned propellant mass for the cylindrical grain is

$$
\begin{flalign*}
& C = \rho\,\frac{\pi D^{2}}{4}\,h &&
\end{flalign*}
$$

Combining these two equations provides the A₁, A₂ and A₃ quadratic form function constants that can then be used in energy equation (1).

2. Noble-Abel Equation of State
The Noble-Abel non-ideal gas equation of state (EOS) is employed to model the behavior of high-density propellant gases at elevated temperatures. This EOS establishes the relationship between chamber pressure, volume, and temperature, incorporating a non-ideal gas correction factor known as the co-volume, η. This factor accounts for the finite volume occupied by gas molecules, which becomes significant under the conditions present in the chamber.

$$
\begin{flalign*}
& P_c \!\left[
V_c + A_p x
– \frac{C – m_p}{\rho}
– \eta m_p
\right]
= m_p F \frac{T}{T_0} \quad (5) &&
\end{flalign*}
$$

The chamber pressure, as derived from the equation of state, is then utilized to calculate the acceleration of the stroking piston by applying Newton’s second law:

$$
\begin{flalign*}
& M \frac{d^{2}x}{dt^{2}}
= P_c A_p – \sum F_{\text{ext}} \quad (6) &&
\end{flalign*}
$$

where F_ext is the total external loading, which includes friction, damping and any external resisting or assisting forces acting on the moving body.

3. Heat Loss
Due to the complexity of heat loss calculations, typically analysts use a “fudge factor” to take into account the total heat loss. If you prefer a realistic physics-based model and include true heat-loss in your numerical calculation, the following equation can be used.

$$
\begin{flalign*}
& \frac{d(HL)}{dt}
= K_D S_c P_c
\left( \frac{T}{T_0} – \theta_w \right)^{1/2}
\quad (7) &&
\end{flalign*}
$$

where

$$
\begin{flalign*}
& C_V T_0 = \frac{F}{\gamma – 1} &&
\end{flalign*}
$$

$$
\begin{flalign*}
& \gamma = \frac{C_P}{C_V} &&
\end{flalign*}
$$

Table below provides the full list of parameters used in these equations:

Most pyrotechnic designers and ballisticians use custom codes to solve these equations with high accuracy. Typically after an initial estimate and engineering tests, heat loss coefficient, burn rate coefficient for temperature extremes, and mechanical features such as friction coefficient are adjusted. Once the numerical model is fully anchored, analysts and development engineers proceed with a detailed parametric modeling and analysis to optimize the propellant charge weight, propellant geometry, hardware design features such as chamber volume and piston diameter.

Solution to these equations offers a ballistic description of a thruster performance, however there are more factors to be considered, when selecting the input parameters. Some of these factors are simply experienced based and usually verified through series of development tests.

4. Igniter Requirements

The igniter charge generally serves two functions. First, it must supply sufficient heat to the propellant surface to ensure ignition. Second, it must provide adequate hot gas to raise the chamber pressure to a level that will sustain propellant combustion. In addition, the igniter and propellant must be arranged so that the gas flow produced by the initial ignition spreads combustion throughout the entire charge.

Although typical ignition-mix weights are on the order of 5–10% of the total charge mass, the required igniter mass depends on the desired starting pressure \( P_{i} \). If a shot-start pressure \( P_{0} \) is required to overcome an internal or external lock or resisting load, it is ballistically advantageous to start with an ignition pressure greater than \( P_{0} \).

Some propellants require an initially high pressure to initiate and sustain burning. Therefore, it is important to estimate an adequate igniter charge mass for a reliable CAD design. The ignition-charge estimate can be obtained from the non-ideal gas equation of state below:

$$
\begin{flalign*}
& C_i
= \frac{P_i}{F_i\,\theta_i}
\left( V_c – \frac{C}{\rho} \right)
\quad (8) &&
\end{flalign*}
$$

where

F_i = Impetus of the ignition charge
_Pi = Required ignition pressure
ρ = main propellant charge density
\( \theta_{i} \)= Ignition temperature ratio (\( \theta_{i} \)~ 0.85-0.9 is usually a good value to use)
V_c  = initial volume
C = main charge wight

5. Additional Design Considerations

5.1 Operational Temperature Effects:
Most CAD must satisfy the performance requirements in the temperature range from -65°F to +200°F. This temperature range has significant effects on propellant burning rate, initial flame spread, and heat-loss as:

a) Burn Rate: Assume a change about 0.1% per degrees F. This assumption is a good estimate for initial design and should be adjusted with development testing

b) Flame Spread: Rapid ignition at high temperature, slower flame spread at lower temperature. This changes the effective propellant web and repressiveness.

c) Heat-Loss Rate: Slightly lower fractional heat loss at “hot” ambient conditions.

5.2 Ignition Effect:
The ignition system design must rapidly ignite the propellant while supplying significant pressure to sustain its combustion. Over-ignition can lead to propellant fracture at cold temperatures, which may induce performance failures. Under-ignition can lead to very slow initial flame spread which may also result in performance failure.

5.3 External Load Envelope:
Accurate modeling of external loads is critical because they directly determine the chamber pressure history and the resulting acceleration (see Eq. 6). For example, a mass that moves slower due to friction or other resistances produces a higher chamber pressure owing to a reduced rate of volume expansion. For ejection-seat catapults, propellant design must therefore be optimized so peak acceleration remains below the prescribed g-limit across a wide range of crew weights.

5.4 Heat Loss before and after burnout:
Burning of the propellant generates turbulent combustion gases in the chamber, which increase the convective heat-transfer coefficient between the gas and the thruster’s metal surfaces. After burning ends, the heat-loss factor \( K_{D} \)  typically decreases substantially — commonly by factors between about 3 and 10 (i.e., to roughly 1/3 – 1/10 of its in-burning value) — as turbulence decays and convective transfer subsides.

While rigorous solutions of the governing differential equations of internal ballistics are essential for an optimized CAD/PAD design, it is equally important to establish realistic bounds and first-order design parameters early in the development cycle—often as early as the proposal or concept phase. Typical CAD/PAD specifications for devices such as thrusters, actuators, or catapults define performance requirements in terms of a minimum end-of-stroke velocity and a maximum allowable thrust. These two requirements inherently compete with one another, necessitating careful optimization to satisfy both simultaneously.

Practical Ballistic Calculations