Solid Rocket Motors - Rocket Propulsion notes

Solid rocket motors store propellant as a solid “grain” containing both fuel and oxidizer. The grain burns from exposed surfaces, generating hot gas that is expelled through a nozzle.

Motor Configuration¶

Figure 1:Cross-section of a solid rocket motor showing: motor casing (pressure vessel), propellant grain with internal cavity (port), web thickness (distance from burning surface to casing), igniter at forward end, gas cavity, and nozzle. Section A-A shows a star-shaped grain cross-section with burning surface $A_b$ .

The propellant grain contains both fuel and oxidizer in a solid matrix.

Advantages of solid motors:

Simplicity — no pumps, valves, or plumbing
Compact — high propellant mass fraction
Storable — ready to fire on short notice (can be stored for years)

Disadvantages:

Lower $I_{sp}$ than liquid engines (typically 250–280 s)
No flexibility — cannot control $\dot{m}$ , cannot stop/restart
Mass flow rate depends on chamber conditions: $\dot{m} = f(P_c, T_i)$
Very sensitive to grain defects (cracks, voids, bubbles) which cause anomalous burning

Propellant Burning Rate¶

The burning rate $\dot{r}$ is the surface regression velocity---how fast the grain surface recedes normal to itself.

\dot{r} \quad \left[\frac{\text{mm}}{\text{s}}\right]

(1)

Figure 2:Diagram showing propellant grain with burning surface receding at rate $\dot{r}$ , chamber pressure $P_c$ , and total instantaneous burning area $A_b$ .

The mass generation rate from propellant burning is:

\boxed{\dot{m} = \dot{r} \, A_b \, \rho_p}

(2)

where:

$\dot{r}$ = burning rate (surface regression velocity)
$A_b$ = instantaneous burning surface area
$\rho_p$ = propellant density

Grain Geometry and Thrust Profiles¶

Since thrust is:

F = \dot{m} \, C_F \, c^*

(3)

and $C_F$ , $c^*$ are essentially constant for a given propellant and nozzle, the thrust profile depends on how $\dot{m}$ varies with time---which depends on $A_b(t)$ .

Figure 3:Common grain geometries and their thrust profiles showing cross-sections and corresponding $F$ or $P_c$ versus time curves.

Grain Types and Thrust Characteristics¶

Geometry	Cross-Section	Burn Pattern	Thrust Profile	$A_b(t)$
Tubular	Circular bore	Burns radially outward	Progressive (increasing)	Increases
Bone & Tube	Tube with slot	Combined surfaces	Neutral (constant)	~Constant
Star	Multi-pointed star	Complex internal	Approximately neutral	~Constant
Double Anchor	Anchor shape	Multiple surfaces	Regressive (decreasing)	Decreases

Burning Rate Law¶

The burning rate is an empirical function of chamber pressure:

Figure 4:Burning rate physics showing: boundary layer above grain surface, flame zone with heat release, and heat transfer back to the solid propellant causing surface regression at rate $\dot{r}$ . Chamber pressure $P_c$ affects flame properties and heat transfer.

An empirical relationship between burning rate and chamber pressure is:

\boxed{\dot{r} = a \, P_c^n}

(4)

where the pressure exponent $n$ is:

Not a function of $P_c$ or $T_i$
Measured experimentally for specific propellant combinations
Typically $n < 1.0$ (usually 0.3–0.7)

and the temperature coefficient $a$ :

Strong function of initial temperature $T_i$ : $\quad a = a(T_i)$
Not a function of pressure
Higher $T_i$ → higher $a$ → faster burning

Figure 5:Burning rate $\dot{r}$ versus chamber pressure $P_c$ on log-log axes for different initial grain temperatures. Higher initial temperature $T_i$ shifts the curve upward, giving higher burning rates at the same pressure. The slope of each line is the pressure exponent $n$ .

Equilibrium Chamber Pressure¶

Figure 6:Mass balance in a solid rocket motor: propellant mass is generated by grain burning ( $\dot{m}_p = \dot{r} A_b \rho_p$ ) and exits through the nozzle ( $\dot{m}_{out} = P_c A_t / c^*$ ). The motor casing acts as a pressure vessel containing the combustion gases.

Performing a mass balance:

\frac{dm}{dt} = \dot{m}_p - \dot{m}_{out}

(5)

where:

$\dot{m}_p = \dot{r} A_b \rho_p = a P_c^n A_b \rho_p$ (mass generation rate from burning)
$\dot{m}_{out} = \dfrac{P_c A_t}{c^*}$ (mass flow through nozzle)

At equilibrium (quasi-steady-state operation): $\dfrac{dm}{dt} \approx 0$ :

\dot{r} A_b \rho_p - \frac{P_c A_t}{c^*} = 0

(6)

Substituting the burning rate law:

a P_c^n A_b \rho_p - \frac{P_c A_t}{c^*} = 0

(7)

Solving for $P_c$ :

a P_c^n A_b \rho_p = \frac{P_c A_t}{c^*}

(8)

P_c^{n-1} = \frac{A_t}{c^* \rho_p a A_b}

(9)

\boxed{P_c = \left(\rho_p \, c^* \, a \, \frac{A_b}{A_t}\right)^{\frac{1}{1-n}}}

(10)

This is the equilibrium chamber pressure.

Klemmung (K-factor)¶

The ratio of burning area to throat area is called Klemmung ( $K$ ) factor:

K = \frac{A_b}{A_t} \gg 1

(11)

Typical values: $K \approx 100$

This large ratio means a small throat area can control the chamber pressure generated by a much larger burning surface.

Pressure-Time Behavior¶

Figure 7:Chamber pressure $P_c$ versus time showing the quasi-steady equilibrium. After ignition, pressure rises rapidly to equilibrium value, remains approximately constant during the quasi-steady burn phase (for neutral grain), then drops at burnout.