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

3.3.37 · D2 · Physics › Rocket Propulsion › Grain geometry — BATES, star, wagon wheel; neutral - progres

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Step 1 — Ek flame jo sideway chalti hai

KYA HAI. Socho ek solid block rocket fuel ka jisme beech mein ek hollow tunnel hai. Hum tunnel ki andar wali wall ko aag lagate hain. Aag poore block ko ek saath nahi jala sakti — woh surface ko khaati hai, aur woh surface peeche hatti rehti hai, hamesha seedha fuel ke andar, apne aap ke perpendicular.

KYUN. Yeh "perpendicular retreat" woh ek fact hai jis par sab kuch tika hua hai. Ek candle ki flame candle ke neeche jaati hai; ek rocket grain ki flame har burning wall se ek saath bahar ki taraf jaati hai. Agar hum jaante hain ki wall kitni door gayi hai, toh hum jaante hain ki kitna fuel gas ban gaya.

PICTURE. Neeche, pale-yellow wall burning surface hai. Blue arrows dikhate hain ki woh ek chhoti si doori se bahar ki taraf kadam badha rahi hai. Woh doori — aag kitni gehri kat gayi — iska ek naam hai.

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Step 2 — "Surface peeche hatti hai" ko "gas banta hai" mein badalna

KYA HAI. Ek chhote se time $dt$ mein, poori burning surface — iski area $A_b$ kehlao — $rdt$ thickness ki ek patli layer ke andar kadam badhati hai. Solid ki woh patli shell gas ban jaati hai.

KYUN. Hum chahte hain gas ki mass per second, kyunki gas hi nozzle se nikalti hai aur thrust banati hai. Sabse saaf rasta pure geometry hai: volume = area × thickness, phir mass = density × volume.

PICTURE. Figure mein shaded rind woh layer hai jo ek pal mein jal jaati hai. Uska volume area aur uski (tiny) thickness ka product hai.

$$dm=\underset{\begin{array}{c}\text{fuel density}\\ {\text{kg per m}}^3\end{array}}{\underbrace{{\rho }_p}}\cdot \underset{\begin{array}{c}\text{burning}\\ {\text{area, m}}^2\end{array}}{\underbrace{A_b}}\cdot \underset{\begin{array}{c}\text{shell}\\ \text{thickness, m}\end{array}}{\underbrace{(rdt)}}$$

Dono sides ko $dt$ se divide karo rate paane ke liye:

$$\dot{m}=\underset{\text{density}}{\underbrace{{\rho }_p}}\underset{\text{area}}{\underbrace{A_b}}\underset{\text{speed}}{\underbrace{r}}$$

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Step 3 — Aag apne hi pressure par palti hai

KYA HAI. Burn speed $r$ koi fixed number nahi hai. Chamber ko zyada pressure $p_c$ par squeeze karo aur surface andar ki taraf tezi se khaane lagti hai. Yeh relationship, lab mein measure ki gayi, ek simple power law hai:

$$r=\underset{\begin{array}{c}\text{propellant}\\ \text{constant}\end{array}}{\underbrace{a}}\underset{\begin{array}{c}\text{pressure raised}\\ \text{to power }n\end{array}}{\underbrace{p_c^n}}$$

KYUN. Zyada pressure ka matlab hai hot gas solid ke khilaf zyada zor se dabayi jaati hai, usmein heat tezi se jaati hai, isliye solid tezi se vapourise hoti hai. Yeh Saint-Robert burn rate law hai. Hum iska istemal isliye karte hain kyunki Step 2 mein humne secretly $r$ use kiya bina yeh jaane ki kya set karta hai — aur $r$ us pressure par depend karta hai jo hum find karna chahte hain.

PICTURE. Neeche ka curve dikhata hai ki $r$ $p_c$ ke saath badhta hai. Exponent $n$ (typically $0.2$–$0.5$) steepness hai: chhota $n$ = ek gentle, well-behaved curve.

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Step 4 — Chamber na bhar jaye aur na khaali ho

KYA HAI. Gas burning surface par ban rahi hai aur gas nozzle ke sabse narrow point, throat (area $A_t$), se nikal rahi hai. Steady running mein yeh dono match karni chahiye: surface jo bhi banaye, throat nikaal de.

KYUN. Agar generation exhaust se zyada hoti, toh pressure hamesha badhta; agar exhaust generation se zyada hoti, toh aag bhukhi mar jaati. Balance hi pressure $p_c$ ko pin down karta hai.

Exhaust side (nozzle theory se, Kn ratio) kehti hai ki gas $p_c{A}_t$ ke proportional rate par nikalti hai:

$${\dot{m}}_{\text{out}}=\frac{p_c{A}_t}{c^{\ast }}$$

jahan $c^{\ast }$ ("c-star") characteristic velocity hai — burnt gases ki ek fixed property jo batati hai ki woh kitni efficiently bahar dhakeli jaati hain.

PICTURE. Ek see-saw: gas bani (left) versus gas expelled (right). Pressure us level par float karta hai jo dono ko balance karta hai.

Made = expelled set karo ($\dot{m}={\dot{m}}_{\text{out}}$), Step 3 se $r=a{p}_c^n$ plug karte hue:

$$\underset{\text{gas generated (Step 2 + 3)}}{\underbrace{{\rho }_p{A}_{b}a{p}_c^n}}=\underset{\text{gas expelled (Step 4)}}{\underbrace{\frac{p_c{A}_t}{c^{\ast }}}}$$

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Step 5 — Pressure solve karna, aur hidden amplifier

KYA HAI. Ab $p_c$ isolate karo. Yeh dono sides par hai — left par $p_c^n$ ke andar chhupa hua, right par ek plain $p_c$. $p_c$ ki powers gather karo:

$${\rho }_p{A}_{b}a{c}^{\ast }=\frac{p_c}{p_c^n}{A}_t=p_c^{1-n}{A}_t$$ $$\boxed{p_c={\left(\frac{{\rho }_{p}a{c}^{\ast }{A}_b}{A_t}\right)}^{\frac{1}{1-n}}}$$

Exponent $\frac{1}{1-n}$ kyun? Kyunki $p_c$ left par ek power ke roop mein tha, ise undo karne par $(1-n)$ flip hokar $\frac{1}{1-n}$ ban jaata hai. Yeh decoration nahi hai — yeh ek amplifier hai. $n=0.4$ ke saath, $\frac{1}{1-n}=1.67$, toh area mein $20\%$ rise $1.2^{1.67}\approx 1.36$ ban jaata hai — pressure mein $36\%$ rise.

$n<1$ kyun matter karta hai. Agar $n\ge 1$ toh exponent $\frac{1}{1-n}$ infinite ya negative ho jaata hai — pressure bhag jaata hai ya badly behave karta hai. $n<1$ motor ko self-regulating rakhta hai: pressure blip gas ko thoda tezi se nikaalti hai jitni bani thi, blip damp ho jaata hai. (Iske fail hone ke baare mein zyada jaankari Combustion instability mein.)

PICTURE. Do dials ek saath geared: chhoti $A_b$ dial thodi ghuma do, $p_c$ dial zyada ghoom jaati hai. Gear ratio $\frac{1}{1-n}$ hai.

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Step 6 — Pressure se Thrust tak

KYA HAI. Thrust woh hai jo throat-plus-nozzle pressure ko turn karta hai:

$$F=\underset{\begin{array}{c}\text{thrust}\\ \text{coefficient}\end{array}}{\underbrace{C_F}}\underset{\text{pressure}}{\underbrace{p_c}}\underset{\text{throat area}}{\underbrace{A_t}}$$

KYUN. $A_t$ aur $C_F$ (nozzle shape se set, dekho Thrust coefficient and nozzle) essentially fixed hardware hain. Toh ek matra mover $p_c$ hai, aur Step 5 se $p_c$ ke peeche ek matra mover $A_b$ hai. Sab chain karo:

$$A_b(w)\Rightarrow p_c\propto A_b^{\frac{1}{1-n}}\Rightarrow F\propto p_c$$

Isliye: thrust-vs-time curve ki shape $A_b(w)$ ki shape hai, amplified. Yahi wajah hai ki grain geometry design hai, chemistry nahi.

PICTURE. Poora pipeline, left se right: ek port shape ➜ ek area-vs-web curve ➜ ek pressure curve ➜ ek thrust curve, har ek pichhle ka amplified echo.

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Step 7 — Edge aur degenerate cases (kabhi skip mat karo)

KYA HAI & KYUN. Ek derivation jis par tum trust karo usse apni extremes survive karni chahiye. Teen yahan matter karte hain.

PICTURE. Teen mini-panels, har ek ek limiting shape.

• End-burner ($A_b=\pi R^2$, constant). Sirf ek flat circular face jalta hai. Area kabhi nahi badalta ⇒ exactly neutral, low thrust, bahut lamba burn (poori length khaani padti hai). Perfect sustainer.
• Burnout / sliver ($A_b\to$ small, phir $0$). Jab peeche hat rahi walls finally milti hain, bacha hua patla "slivers" ka fuel shrinking area ke saath jalta rehta hai ⇒ ek regressive tail. Toh "neutral" sirf main burn ke dauran ≈ constant hai, kabhi bhi bilkul end tak dead flat nahi.
• $n\to 1$ (forbidden). Jaise $n$ $1$ ki taraf badhta hai, $\frac{1}{1-n}\to \infty$: ek chhoti si area change pressure ko blast kar deti hai. Motor self-regulate nahi kar sakta. Yahi wajah hai ki real propellants $n\approx 0.2$–$0.5$ par rehte hain.

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Ek-picture summary

Poora page ek frame mein: ek receding wall gas banati hai ($\dot{m}={\rho }_p{A}_{b}r$), gas bani gas expelled ke barabar honi chahiye, woh balance pressure pin karta hai ($p_c\propto A_b^{1/(1-n)}$), aur thrust pressure par sawar hoti hai ($F=C_F{p}_c{A}_t$). Port ki shape bilkul left mein enter karti hai aur uska har wiggle echo hota hai — amplified — poori tarah thrust curve tak.

Recall Feynman retelling — simple shabdon mein bolo

Socho aag solid fuel mein se ek tunnel kha rahi hai. Aag kabhi jump nahi karti; woh bas seedha har wall mein ek steady crawl-speed se ghusati rehti hai. Ek heartbeat mein woh solid ki ek paper-thin shell ko gas mein badal deti hai — kitni gas banti hai yeh sirf burning surface kitni badi hai par depend karta hai, kyunki crawl-speed har jagah same hai. Woh gas ko kahin toh jaana hai: nozzle throat. Agar aag gas ko throat se tezi se nikalane ke liye tezi se banati hai, toh chamber pressure badhta hai — aur zyada pressure aag ko thoda tezi se crawl karwata hai, lekin throat bhi zyada tezi se thukta hai. Dono ek balance mein settle ho jaate hain, aur woh balance pressure fix karta hai. Yahan sneaky part hai: kyunki pressure burn speed mein wapas feed karta hai, burning-surface size mein ek chhoti si change pressure ko nudge nahi karti — woh use shove karti hai, $1/(1-n)$ ki power se. Akhir mein, thrust sirf pressure times ek fixed nozzle hai, toh thrust burning-surface-size ki kahani ka ek magnified copy hai. Aur burning-surface-size ki kahani ek cheez se set hoti hai jo tum design kar sakte ho: hole ki shape. Circle → hole bada hota hai → thrust badhta hai. Clever star → area steady rehta hai → flat thrust. Yahi grain geometry ki poori kala hai.

Recall Self-test

Thrust area ka magnified copy kyun hai, straight copy kyun nahi? ::: Kyunki pressure mass balance ke dono sides par appear karta hai; solve karne par $p_c\propto A_b^{1/(1-n)}$ milta hai, aur thrust $\propto p_c$. Exponent $1/(1-n)>1$ amplify karta hai. Web burned $w$ kya measure karta hai? ::: Woh perpendicular distance jitni burning surface solid ke andar peeche hat gayi hai. $n<1$ kyun hona chahiye? ::: Taaki $1/(1-n)$ finite aur positive rahe — motor self-regulate kare bhagane ki bajay. End-burner exactly neutral kyun hota hai? ::: Uski ek flat face ka constant area $\pi R^2$ hota hai jaise woh peeche hatti hai, toh $A_b$ kabhi nahi badalta.