Foundations — Over-expanded nozzle — oblique shocks in plume, efficiency loss

3.3.14 · D1 · Physics › Rocket Propulsion › Over-expanded nozzle — oblique shocks in plume, efficiency l

Yeh page yeh assume karta hai ki tum sirf itna jaante ho ki "hot gas tezi se bahar aati hai." Parent note parent topic mein use hone wala har letter, ratio, aur symbol yahaan banaya gaya hai, usi order mein jis par woh ek doosre par depend karte hain. Upar se neeche ek baar padho aur koi bhi symbol tumhe kabhi surprise nahi karega.

1. Pressure — $p$

Socho ek box of air hai. Tiny molecules har wall se bounce karte hain. Har bounce ek chhoti si shove hai; billions per second milke ek steady outward push banate hain. Zyada molecules pack karo, ya unhe itna heat karo ki woh harder bounce karein, toh $p$ badhta hai.

Topic ko $p$ ki zaroorat kyun hai? Kyunki poori kahaani do pressures ke beech ek contest hai:

• $p_e$ — nozzle exit par gas ka pressure (subscript $e$ = "exit").
• $p_a$ — bahar ki ambient hawa ka pressure (subscript $a$ = "ambient", rocket ke aas paas ka aasman).

Red arrows dekho: bahar ki hawa exhaust jet ke edge par andar ki taraf push karti hai. Exhaust apne pressure se bahar ki taraf push karta hai. Jo bhi zyada strong hoga, boundary uski hogi.

2. Exit ka area — $A_e$

Iski zaroorat kyun hai? Pressure force per area hai. Pressure ko actual push (force) mein badalne ke liye, tum usse area se multiply karte ho:

$\text{force}=p\times A$

Toh exit disc par kaam karne wala unbalanced pressure ek real force $(p_e-p_a)A_e$ deta hai. Yeh akela product decide karta hai ki pressure mismatch se kitne kilonewtons gain ya lose hote hain. Yaad rakho — yeh §7 mein wapas aayega.

3. Mass flow rate — $\dot{m}$

Socho tum exit par stopwatch lekar khade ho, har kilogram gas ko weigh karte ho jo stream karke jaati hai. Woh per-second weight $\dot{m}$ hai. Ek bada engine har second saikdon kg ugal sakta hai.

Kyun chahiye? Thrust fundamentally mass ko peeche tezi se phenka jaana hai. Tumhe jaanna chahiye kitna mass ($\dot{m}$) aur kitni tezi se (agla symbol) push compute karne ke liye.

4. Exit velocity — $v_e$

Nozzle ke diverging cone ka poora purpose $v_e$ ko jitna ho sake utna bada banana hai. Har second phenkay jaane wale mass ko us speed se multiply karo jis speed se pheka gaya hai, aur tumhe momentum thrust milega:

$\dot{m}{v}_e(\text{units: }\frac{\text{kg}}{\text{s}}\cdot \frac{\text{m}}{\text{s}}=\text{newtons})$

Yeh thrust ka "rocket isliye kick karta hai kyunki woh exhaust pheekta hai" waala part hai — hamesha positive, hamesha help karta hai.

5. Poora thrust equation — $F$

Ab ki $\dot{m}$, $v_e$, $p_e$, $p_a$, $A_e$ sab exist karte hain, hum parent note ka master formula assemble kar sakte hain:

• Pehla term: mass ko peeche phenka jaana. Hamesha help karta hai.
• Doosra term: exit disc par bachi hui pressure push. Help karta hai agar $p_e>p_a$, hurt karta hai agar $p_e<p_a$.

Over-expanded nozzle ke liye $p_e<p_a$, toh $(p_e-p_a)$ negative hai — doosra term subtract karta hai. Yeh pehla, sab se gentle tarika hai jismein over-expansion thrust churaati hai. Poori derivation dekho Thrust Equation mein.

6. Speed of sound $a$ aur Mach number — $M$

Is topic mein jo bhi violent hota hai woh isliye hota hai kyunki gas sound se tez move karti hai. Pehle humein sound speed ke liye ek symbol chahiye.

Ab hum flow ki speed ko us message speed se compare kar sakte hain.

Yeh itna important kyun hai? Sound ($a$ par travel karta hua) woh tarika hai jisse gas "message bhejti hai" ki aage pressure change ho rahi hai. Agar flow apne messages se tez hai ($M>1$), toh gas ko pehle se warn nahi kiya ja sakta ki smoothly mud jaaye ya slow ho jaaye. Adjustment abruptly honi chahiye — ek razor-thin wall mein jise shock kehte hain. Rocket nozzle ke exit par, $M_e$ typically $3$–$5$ hoti hai: deeply supersonic. Subscript $e$ = exit again, toh $M_e$ = "exit par Mach". Zyada jaankari Gas Dynamics mein.

Laal circle flow hai; yeh black sound-wave ripples se aage nikal jaati hai jo woh emit karta hai, toh ripples ek cone mein pile ho jaati hain uske peeche — yeh shock ki physical origin hai.

7. Shock wave — abrupt squeeze

Yahaan do flavours matter karte hain:

• Normal shock — shock surface flow ke perpendicular ($90°$ par) hoti hai. Maximum squeeze, maximum loss.
• Oblique shock — shock flow se ek angle par hoti hai. Sirf velocity ka woh part jo shock cross karta hai compress hota hai; baaki slide karta rehta hai. Gentler, kam loss.

Laal line oblique shock hai. Notice karo ki velocity arrow normal component (shock cross karta hai, compress hota hai) aur tangential component (slide karta hai, unchanged) mein split hoti hai. Yeh §8 ki key trick hai. Fundamentals Shock Wave Fundamentals mein.

8. Shock angles — $\beta$ aur $\theta$

Do Greek letters ek oblique shock ki geometry describe karte hain:

Over-expanded plume mein, ambient hawa exhaust ko andar ki taraf bend karti hai, toh $\theta$ inward turn hai aur $\beta$ nozzle lip se uthne wale shock ka angle hai. §7 se upstream Mach $M_1$ use karke (shock mein jaane wala flow), us Mach ka normal component hai

$M_{1n}=M_1\sin \beta$

Yahaan extra subscript $n$ ka matlab hai "normal component" — $M_1$ ka woh part jo shock line ke across point karta hai. $\sin \beta$ kyun? Kyunki $\sin \beta$ bilkul wahi fraction pick karta hai velocity ka jo shock line ke across point karta hai — sirf wahi part compress hota hai. Tangential part $\cos \beta$ use karta hai aur untouched rehta hai. (Agar $\beta =90°$, $\sin \beta =1$ aur hum normal shock recover karte hain — har case covered hai.)

9. Gas property — $\gamma$ (gamma)

Tumhe yahaan $\gamma$ derive nahi karna — bas yeh jaano ki yeh exhaust ki ek fixed property hai, jo har shock aur expansion formula mein plug ki jaati hai. Yeh set karta hai ki ek given squeeze ke liye pressure kitna badhta hai. Yeh parent ke oblique-shock relations mein aur Isentropic Flow ke isentropic formulas mein appear karta hai.

10. Stagnation pressure — $p_0$ aur efficiency $C_v$

Ek perfect (lossless) process $p_0$ ko constant rakhta hai. Ek shock permanently kuch $p_0$ destroy karta hai — woh lost fraction woh efficiency hai jo tum kabhi wapas nahi pa sakte. §7 ke shock subscripts use karke, hum $p_{01}$ likhte hain shock ke upstream stagnation pressure ke liye aur $p_{02}$ downstream ke liye, toh ek shock ke across survival fraction $p_{02}/p_{01}<1$ hai.

Ab efficiency. Hum do exit velocities compare karte hain:

• $v_{e,\text{ideal}}$ = woh velocity jo tum paate perfectly lossless (no-shock) expansion mein.
• $v_{e,\text{actual}}$ = woh velocity jo tum actually paate ho jab shocks kuch khaa leti hain.

Unka ratio velocity coefficient hai. Aur hum do stagnation pressures compare karte hain:

• $p_{0,\text{chamber}}$ = combustion chamber mein flow ki poori energy (kisi bhi loss se pehle).
• $p_{0,\text{actual}}$ = woh energy jo shocks ke baad exit tak survive karti hai.

Square root kahan se aata hai? Kinetic energy per kilogram $\frac{1}{2}{v}^2$ hai, aur flow ki usable energy $p_0$ se track hoti hai. Toh energy $\propto v^2\propto p_0$. Actual case ka ideal case se ratio lena:

$\frac{v_{e,\text{actual}}^2}{v_{e,\text{ideal}}^2}=\frac{p_{0,\text{actual}}}{p_{0,\text{chamber}}}⟹\frac{v_{e,\text{actual}}}{v_{e,\text{ideal}}}=\sqrt{\frac{p_{0,\text{actual}}}{p_{0,\text{chamber}}}}$

Square root isliye aata hai kyunki humne velocity squared se shuru kiya tha aur plain velocity ratio recover karne ke liye square undo karna pada. Efficiency Nozzle Area Ratio aur Altitude Compensation se link karti hai.

11. Area ratio — $\epsilon$

Divide karne se pehle, hume fraction ka bottom chahiye.

"$40:1$" jaisa likha ho toh matlab exit narrowest point se area mein 40 times wider hai. Current air pressure ke liye zyada bada $\epsilon$ exactly over-expansion ki definition hai, aur yahi wajah hai ki wahi engine jo space mein shine karta hai launch pad par over-expand karta hai. Dekho Nozzle Area Ratio aur Nozzle Flow Separation jab crush severe ho jaata hai.

Prerequisite map — yeh topic ko kaise feed karte hain

Map ko neeche se upar padho: upar ke boxes §1–§11 ke raw symbols hain; arrows dikhate hain kaun se symbols kahan merge hote hain, sab final topic box mein funnel ho kar.

• Left branch (§1–§5): pressure $p$, exit area $A_e$, mass flow $\dot{m}$, exit velocity $v_e$ sab thrust equation $F$ mein merge hote hain.
• Middle branch (§1): $p_e$ ko $p_a$ se compare karna pressure-match verdict deta hai, jo over-expansion ban jaata hai jab $p_e<p_a$.
• Right branch (§6–§9): Mach number $M$ (sound speed $a$ par built) shock waves ko possible banata hai; $\gamma$ ke saath yeh oblique-shock angles $\beta ,\theta$ set karte hain.
• Bottom join (§10): shocks stagnation pressure $p_0$ kaatti hain, jo efficiency $C_v$ set karta hai. Area ratio $\epsilon$ (§11) over-expansion ko feed karta hai. Sab topic mein land karta hai.

Equipment checklist

Khud ko test karo — sirf zor se jawab dene ke baad reveal karo.