Solid propellants — fuel + oxidizer in polymer matrix

3.3.35 · Physics › Rocket Propulsion

WHAT is a solid propellant?

WHY apna khud ka oxidizer le jaana? Kyunki rocket ko vacuum mein kaam karna padta hai. Ek air-breathing engine (jet) atmosphere ke upar fail ho jaata hai. Rocket iska solution yeh nikaalta hai ki oxygen ko chemistry mein hi pack kar le.

WHY ek polymer matrix? Binder ek saath teen kaam karta hai:

1. Fine oxidizer crystals aur metal powder ko ek solid, castable, crack-free shape mein hold karta hai.
2. Structural material ki tarah kaam karta hai — grain ko apne weight aur acceleration ko survive karna hota hai.
3. Fuel ki tarah burn karta hai — HTPB (hydroxyl-terminated polybutadiene) ek hydrocarbon hai, isliye yeh bhi energy contribute karta hai.

HOW does it produce thrust? (derive from first principles)

Combustion solid ko chamber ke andar high pressure par hot gas mein convert kar deta hai. Gas nozzle se high speed par nikalta hai. Thrust mass ko peeche phenkne ki reaction hai.

Momentum conservation se shuru karte hain. Time $dt$ mein, motor mass $dm$ ko exhaust speed $v_e$ (rocket ke relative) par peeche eject karta hai. Gas ko diya gaya backward momentum $v_{e}dm$ hai. Newton ke third law se rocket ko equal forward momentum milta hai:

$F_{\text{thrust}}=\frac{dp}{dt}=v_e\frac{dm}{dt}=\dot{m}{v}_e$

Yeh step kyun? Force = rate of change of momentum. Rocket forward momentum exactly usi rate se gain karta hai jis rate se exhaust use backward carry karta hai.

Agar nozzle exit par exhaust pressure $p_e$ (area $A_e$) ambient pressure $p_a$ se alag ho, toh ek extra pressure term aata hai:

$\boxed{F=\dot{m}{v}_e+(p_e-p_a)A_e}$

Burn rate law (Saint-Robert / Vieille's law):

$r=a{p}_c^n$

$p_c$ = chamber pressure, $a$ = temperature-dependent constant, $n$ = pressure exponent. Yeh empirical hai lekin bahut zaroori hai: yeh kehta hai ki burn rate pressure ke saath badhta hai. Stability ke liye $n<1$ zaroori hai — warna pressure spike tezi se jalata hai → zyada gas → aur bhi zyada pressure → runaway explosion.

Grain geometry controls the thrust curve

Kyunki $\dot{m}\propto A_b$ hai, burning surface ka time ke saath shape thrust-vs-time curve ko shape karta hai:

• End-burner (cigarette burn): flat constant $A_b$ → lagbhag constant thrust, lambi duration.
• Internal cylindrical port: port chauda hone par $A_b$ badhta hai → thrust badhta hai (progressive).
• Star-shaped port: clever geometry $A_b$ ko roughly constant rakhti hai (neutral) → flat thrust.

Worked examples

Common mistakes (steel-manned)

Recall

Connections

• Tsiolkovsky Rocket Equation — jahan $v_e$ aur $\dot{m}$ final velocity mein feed hote hain.
• Liquid Propellants — contrast: throttleable, restartable, alag tanks.
• De Laval Nozzle — chamber gas ko supersonic $v_e$ par kaise accelerate kiya jaata hai.
• Specific Impulse — $I_{sp}=v_e/g_0$, efficiency figure of merit.
• Newton's Third Law — thrust ke peeche momentum principle.
• Combustion Chemistry — HTPB + AP + Al reaction energetics.

Concept Map