Solid Rocket Motors

Description

• A solid motor uses a cast propellant grain (oxidizer + binder + additives)
• Regression exposes fresh propellant
• Burning Area $A_b(t)$ and nozzle throat set chamber pressure $p_c(t)$ and thrust

Process

• Burning law (pressure-coupled):
  • $r=a{p}_c^n$ with $a,n$ from tests
  • erosive burning adds a velocity term if internal flow is high
• Mass generation:
  • ${\dot{m}}_p={\rho }_p{A}_{b}r$
• Nozzle choked flow:
  • ${\dot{m}}_n(p_c)$
  • In quasi-steady operation, ${\dot{m}}_p\approx {\dot{m}}_n$ closes $p_c$

Two Cases (end treatment)

Case A — end has protective coating (end-inhibited)

• Only side surfaces burn initially
• For a simple cylindrical bore ($r_i$ inner, $r_o$ outer, web $w=r_o-r_i$), area evolves:
  • $$A_b(t)\approx A_{\text{side}}(t)=2\pi L(r_i+r(t))$$
  • with $r(t)={\int }_0^t\mathrm{r}\left(\tau \right)\mathrm{d}\tau$
  • ends remain inactive until inhibition is removed (if ever)
• Tends to flatten $A_b(t)$ growth $\Rightarrow$ milder $p_c$ rise

Case B — end exposed (end-burning allowed)

• Ends contribute:
  • $$A_b=A_{\text{side}}+A_{\text{ends}}$$
  • giving larger and faster-changing area
• Can produce progressive $A_b(t)$ (and thrust) profiles if not balanced by nozzle sizing
• Real grains use complex inhibitors, slots, stars, finocyls to tailor $A_b(t)$ and thrust shaping

Burning Stability (qualitative)

• Pressure oscillations
  • couple to $r=a{p}_c^n$
  • Positive feedback risk if chamber acoustics feed back into flame zone
• Erosive burning
  • high internal velocities
  • can increase $r$ and destabilize startup
• Mitigations
  • tailored grain geometries, inhibitors, damping cavities, and acoustic/absorber features
  • strict control of $\Delta p_{\text{inj}}$ equivalents in igniter/manifolds