Grain geometry — BATES, star, wagon wheel; neutral - progressive - regressive burn

3.3.37 · Physics › Rocket Propulsion

HUM KYA DESCRIBE KAR RAHE HAIN?

BURNING AREA THRUST KO KYU CONTROL KARTA HAI? (scratch se derive karo)

Step 1 — Mass generation rate. Surface rate $r$ se recede karta hai (units m/s). Time $d t$ mein $A_{b}$ area pe $r d t$ thickness ki layer gas ban jaati hai. Uska mass: $d m = ρ_{p} A_{b} (r d t)$

Yeh step kyun? Volume burned = area × thickness receded; mass = density × volume. Pure geometry hai.

Toh mass generation rate: $\overset{m}{˙} = ρ_{p} A_{b} r$

Step 2 — Burn rate pressure pe depend karta hai (Vieille's law / Saint-Robert): $r = a p_{c}^{n}$

Kyun? Zyada chamber pressure $p_{c}$ solid mein heat zyada fast push karta hai, toh surface aur tezi se andar jaati hai. $a$ , $n$ propellant ke empirical constants hain ( $n$ usually $0.2$ – $0.5$ ).

Step 3 — Mass balance se chamber pressure. Steady operation mein, gas generated = gas leaving the nozzle throat ( $A_{t}$ ). Nozzle mass flow $\overset{m}{˙}_{o u t} \propto p_{c} A_{t}$ . Generation = exhaust set karo: $ρ_{p} A_{b} a p_{c}^{n} = \frac{p _{c} A _{t}}{c ^{*}}$ $p_{c}$ ke liye solve karo: $p_{c} = (\frac{ρ _{p} a c ^{*} A _{b}}{A _{t}})^{\frac{1}{1 - n}}$

Exponent $\frac{1}{1 - n}$ kyun? $p_{c}$ dono sides pe hai (left mein $r$ ke through), toh isolate karne par $(1 - n)$ power invert hoti hai. Isi liye $n < 1$ hona zaroori hai stability ke liye — warna pressure runaway ho jaata hai.

Step 4 — Thrust. $F = C_{F} p_{c} A_{t}$ Kyunki $A_{t}$ aur $C_{F}$ ≈ fixed hain, thrust $p_{c}$ ke saath track karta hai, jo $A_{b}$ ke saath track karta hai. Ratio define karo: $K = \frac{A _{b}}{A _{t}}$

HAR GEOMETRY $A_{b} (w)$ KO KAISE SHAPE KARTI HAI

Figure — Grain geometry — BATES, star, wagon wheel; neutral - progressive - regressive burn

1. BATES (cylindrical port, ends inhibited ya free)

Ek hollow cylinder. Bahar ki taraf jalta hai (port radius badhta hai, woh surface badhti hai) AUR ends se andar (annulus, axial length mein ghatta hai). Ek single unrestricted-end BATES segment ke liye, length $L$ , inner radius $R_{i}$ , outer $R_{o}$ , burning surfaces hain inner cylinder + do annular ends.

Maano web burned $= w$ . Inner radius $= R_{i} + w$ , length $= L - 2 w$ . $A_{b} (w) = port badhta hai 2 π (R_{i} + w) (L - 2 w) + ends ghatte hain 2 \cdot π [R_{o}^{2} - (R_{i} + w)^{2}]$

Yeh neutral trend kyun karta hai? Port term badhta hai jabki end term ghatta hai; sahi $L / D$ chunne se dono cancel ho jaate hain → near-neutral. Ek lamba, end-inhibited BATES (sirf inner surface jalta hai) purely progressive hota hai.

2. Star grain

Ek star-shaped port. Bahut saare points aur valleys ek bada initial perimeter dete hain jo, jalne par, round out hota hai — perimeter roughly constant rehta hai.

Neutral kyun? Shuruaat mein pointed "fingers" jal jaate hain (area girta) lekin valleys web mein khuljati hain (area badhta); ek achi tarah se designed star in dono ko balance karta hai → neutral. Case-bonded motor mein constant thrust ke liye classic choice hai (ends inhibited hain, toh end-burning ki chinta nahi).

3. Wagon wheel

Ek star jisme aur bhi gehri, patli spokes hain — ek extreme high-perimeter design.

Highly progressive/high-thrust-early kyun? Bahut bada initial burning perimeter ⇒ bahut bada initial $A_{b}$ ⇒ shuruaat mein badi thrust. Tab use karo jab tumhe high initial thrust / high mass flow chahiye (boost phase). Trade-off: patli spokes slivers chhodti hain aur regressive tail deti hain.

GEOMETRY SE BURN CLASS PADHNA

Common mistakes (steel-manned)

Flashcards

Fixed nozzle wale solid motor ka instantaneous thrust kya set karta hai?

Current burning surface area

A_{b}

(kyunki

\overset{m}{˙} = ρ_{p} A_{b} r

aur

p_{c} \propto A_{b}^{1/ (1 - n)}

Web burned

w

define karo.

Woh perpendicular distance jitni burning surface propellant ke andar ja chuki hai.

Saint-Robert / Vieille burn law kya hai?

r = a p_{c}^{n}

, burn rate vs chamber pressure,

n

typically 0.2–0.5.

n < 1

kyun hona chahiye?

Warna

p_{c} \propto A_{b}^{1/ (1 - n)}

blow up ho jaata hai / unstable ho jaata hai;

n < 1

pressure ko bounded aur self-regulating rakhta hai.

Neutral vs progressive vs regressive burn?

A_{b}

constant / increasing / decreasing over time → flat / rising / falling thrust.

Circular (BATES inner-only) port ka burn class?

Progressive — port perimeter bahar ki taraf recede hone par badhta hai.

Achi tarah se designed star grain ka burn class?

Neutral — jalne wale points aur khuljate valleys ek doosre ko balance karte hain taaki perimeter ≈ constant rahe.

Wagon wheel kyun use karte hain?

Bahut bada initial burning perimeter → bahut zyada initial thrust (boost phase); downside hai slivers/regressive tail.

End-burner ka burn class?

Neutral, low-thrust, long-duration (constant flat face area

π R^{2}

Case-bonded grain ke liye area aur port perimeter ka relation?

A_{b} (w) = P (w) \cdot L

(perimeter × grain length).

Geometry se thrust tak master chain?

A_{b} \to K = A_{b} / A_{t} \to p_{c} \propto K^{1/ (1 - n)} \to F = C_{F} p_{c} A_{t}

"Sliver" kya hota hai?

Web burnout ke baad bache hue propellant ke tukde jo regressively jalte hain, thrust ko tail off karte hue.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek birthday candle, lekin "flame" upar nahi hai — woh ek hole ki walls ke saath hai jo tumne candle mein kiya. Aag mombatti ko sideways khaati hai, hole se bahar har direction mein jaati hai. Jitni zyada wall abhi aag mein hai, utna zyada dhuaan (thrust) nikalta hai. Agar tum round hole karo, wall jalne par badi hoti jaati hai → zyada zyada aag → thrust badhta hai. Agar tum star-shaped hole karo, pointy bits jal jaate hain jabki dents khulte hain, aur sab balance ho jaata hai → steady aag. Agar tum wagon-wheel shape ka hole karo bahut saari patli spokes ke saath, shuruaat mein BAHUT zyada wall hoti hai → pehle bada whoosh, phir fade hota hai. Toh engineers mombatti nahi badlte — woh sirf hole ki shape badlte hain yeh decide karne ke liye ki rocket flat push kare, zyada-se-zyada, ya pehle bada phir soft.

Connections

Saint-Robert burn rate law — woh $r = a p_{c}^{n}$ jo timescale set karta hai.
Chamber pressure & throat area (Kn ratio) — $K = A_{b} / A_{t}$ $p_{c}$ drive karta hai.
Thrust coefficient and nozzle — $F = C_{F} p_{c} A_{t}$ .
Rocket equation & specific impulse — total impulse = $\int F d t$ = us curve ke neeche ka area jo tumne shape kiya.
Combustion instability — kyun $n < 1$ matter karta hai.
Case-bonded vs free-standing grains — determine karta hai ki ends jalte hain ya nahi.