Practical Ballistic Calculations

A practical first-order approach to estimating pressure, work, function time, propellant web and charge weight for propellant-actuated devices.

Peak chamber pressure is rarely specified in end-item requirements. In practice, maximum pressure is constrained by the structural capability of the CAD housing, piston, and associated components. As such, pressure containment is generally treated as a design responsibility of the CAD developer rather than an externally imposed requirement from the end user.

Given velocity and thrust constraints, the first step in preliminary design is to determine the required average pressure over the stroke (not the peak pressure, yet) and the corresponding propellant energy needed to sustain that pressure. From this, the required propellant charge weight can be estimated, followed by selection of an appropriate propellant type based on grain geometry, burn rate, and relevant thermochemical properties.

In more complex designs, particularly thrusters or removers that must overcome a highly variable resisting load profile, a single propellant formulation may be insufficient. Systems requiring a rapid pressure rise to initiate motion or release an internal locking mechanism, followed by sustained pressure throughout the stroke, often benefit from the use of multiple propellant types. A common approach is to combine a fast-burning ball powder or flake type propellant to provide a sharp initial pressure rise with slower-burning cylindrical grains—perforated or unperforated—to maintain pressure during the remainder of the stroke. Equally critical is the physical arrangement of these propellants within the cartridge or gas generator, as loading order and spatial distribution directly influence ignition behavior, pressure evolution, and overall system performance.

1. Average Pressure and Work Relationships

An initial ballistic estimate typically begins by determining the average chamber pressure required to meet the specified performance. The average pressure over the full stroke may be expressed in terms of the mechanical work, W, performed on the piston.

The total work done on the piston is equal to the area under the thrust–stroke curve:

\begin{equation}
W = \int_{0}^{L_m} A\,P_c(x)\,dx
\tag{9}
\end{equation}

where $P_c$ is the chamber pressure and $A$ is the piston cross-sectional area.

This expression represents the summation of the instantaneous force on the piston, $A$$P_c$, multiplied by the differential stroke increment dx, integrated over the full stroke length.

The stroke-averaged pressure is therefore defined as

\begin{equation}
\bar{P}_c = \frac{1}{L_m}\int_{0}^{L_m} P_c(x)\,dx
\tag{10}
\end{equation}

Combining Equations (9) and (10), the average pressure may be written directly as

\begin{equation}
\bar{P}_c = \frac{W}{A\,L_m}
\tag{11}
\end{equation}

where $L_m$ is the total stroke required by the specification.

2. Work Requirement from End-Item Specifications

Given the end-user-defined moving mass M, constant resisting load $F_r$, stroke $L_m$, and end-of-stroke velocity $V_m$, the total work required is

\begin{equation}
W = \frac{1}{2} M V_m^2 + F_r L_m
\tag{12}
\end{equation}

Substituting Equation (12) into Equation (11) yields

\begin{equation}
\bar{P}_c = \frac{\frac{1}{2} M V_m^2 + F_r L_m}{A\,L_m}
\tag{13}
\end{equation}

Equation (13) provides the average operating pressure required to satisfy the end-user performance requirements.

3. Example Calculation

Consider a thruster accelerating a mass of 1200 lb (M=37.27 slugs) to an end-of-stroke velocity of 15 ft/s over a 6-inch stroke.
The system experiences a constant resisting load of 1000 lb, and packaging constraints limit the piston area to 2 in².

Using Equation (13), the required average pressure is

\[
\bar{P}_c = 4,693\ \text{psi}
\]

4. Piezometric Efficiency

A key ballistic parameter used in all propellant-actuated stroking devices—including guns, thrusters, catapults, cutters, pin pullers, and actuators—is the piezometric efficiency, η. This parameter is defined as the ratio of average pressure to peak pressure:

\begin{equation}
\eta = \frac{\bar{P}_c}{P_{\max}}
\tag{14}
\end{equation}

A constant-pressure system would yield η=1.0, which is not achievable in practical CAD/PAD designs. For well-optimized systems, a piezometric efficiency in the range η=0.4–0.6 is generally desirable.

For the example considered here, using η=0.5, the expected peak pressure is approximately

\[
P_{\max} \approx \frac{\bar{P}_c}{\eta} = 9{,}386\ \text{psi}
\]

Methods for Increasing Piezometric Efficiency

Several design strategies may be employed to increase η and thereby reduce peak pressure:

• Maximizing initial free volume
• Designing propellant burnout to occur near the end of stroke
• Using progressive propellant geometries (increasing burning surface)
• Initiating motion at a system start pressure close to the average pressure

5. Function Time Estimate

Function time (full-stroke time) is often a critical specification parameter. For preliminary estimates, assuming an average velocity equal to one-half the maximum velocity provides a reasonable first-order approximation:

\begin{equation}
t_m \approx\frac{L_m}{0.5\,V_m}
\tag{15}
\end{equation}

For the example system, this yields

\[
t_m = 67\ \text{ms}
\]

6. Propellant Web Calculations

The next step is to select an appropriate propellant geometry and burn rate. For granular propellants, burning occurs from both sides of the grain. The total propellant web burned, $w_o$, is therefore

\begin{equation}
w_0 = 2\,\bar{r}\,t_b
\tag{16}
\end{equation}

where $t_b$ is the burnout time and $\bar{r}$ is the average burn rate.
Using the average chamber pressure $\bar{P}_c$, the average burn rate is

\begin{equation}
\bar{r} = B\,\bar{P}_c^{\,n}
\tag{17}
\end{equation}

Substituting Equation (17) into Equation (16) yields

\begin{equation}
w_0 = 2\,B\,\bar{P}_c^{\,n}\,t_b
\tag{18}
\end{equation}

For the calculated pressure in this example, assuming an average burn rate of

\[
\bar{r} = 0.5\ \text{in/s}
\]

the required web thickness is

\[
w_0 = 0.067\ \text{in}
\]

7. Charge Weight Estimation

With estimates for average pressure and propellant web established, the next step is to determine the required propellant charge weight.

While accurate solutions require numerical integration of the full ballistic equations, a reasonable first-order estimate may be obtained using average values at end of stroke. Rather than integrating the full energy equation (Equation (1) from the Fundamentals of Internal Ballistics page) may be rewritten as

\[
C = \frac{\bar{P}_c}{3F}\left( V_c + A L_m \right)
+ \frac{\gamma – 1}{F}\left( (HL) + \bar{P}_c A L_m \right)
\quad (19)
\]

where the heat-loss energy HL is given by

\begin{equation}
(HL) = K_D\,\bar{P}_c\,S_c\,t_m
\tag{20}
\end{equation}

For this example, assume a composite propellant with following impetus (F) and $\gamma$

\[
F = 340{,}000\ \text{ft-lb/lb}, \qquad \gamma = 1.23
\]

From preliminary CAD layout,

\[
V_c = 10\ \text{in}^3, \qquad S_c = 80\ \text{in}^2
\]

A first-order heat-loss coefficient of $K_D$ = 0.7 is assumed.

Using Equation (19), the estimated required propellant charge is

\[
C \approx 15.1\ \text{g}
\]

8. Closing Remarks

For an initial, first-order estimate to satisfy the customer’s end-item performance requirements, the design may begin with an initial chamber volume of approximately 10 in³, a 6-inch piston stroke, a 2 in² piston head area, and a propellant exhibiting a linear burn rate of about 0.5 in/s at an average chamber pressure in the range of 4,500–5,000 psi.
These assumptions lead to an estimated propellant web thickness of 0.06–0.07 in and a required charge mass of approximately 15 g.

It should be noted that the approach described in this example represents only a very early back-of-the-envelope starting point intended to place the design within the correct performance regime. Ballistic testing is not only costly but also time-consuming, often resulting in significant schedule impacts. It is therefore essential to minimize reliance on testing through rigorous a priori design and sound ballistic analysis.