Foundations — Altitude compensation methods — nozzle extension, aerospike

3.3.16 · D1 · Physics › Rocket Propulsion › Altitude compensation methods — nozzle extension, aerospike

Is page par parent note ke har symbol aur idea ko zero se build kiya jaayega, starting from wo cheezein jo ek 12-saal ka bachcha pehle se jaanta hai: pushing, squeezing, aur funnels. Yahan kuch bhi assume nahi kiya gaya ki aapne parent note pehle dekha hai — is note ko pehle padho topic note se.

1. Pressure $p$ — gas kitni zor se push karta hai

Socho ek box mein bouncing gas molecules hain, jaise neeche ke figure ke left mein draw kiya gaya hai. Har collision wall ke saath ek chhoti si shove hai; billions of shoves per second ko wall ke area par jod do aur tumhe ek steady outward push milti hai — woh steady push per unit area hi pressure hai.

Is page par humein teen alag-alag pressures chahiye, toh abhi unhe naam dete hain:

Neeche ka figure do panels mein hai: left panel woh bouncing molecules dikhata hai jo pressure banate hain, aur right panel ek nozzle dikhata hai jisme teeno pressures label kiye gaye hain aur atmosphere andar ki taraf squeeze kar raha hai.

Topic ko yeh kyun chahiye: poora subject $p_e$ (jo nozzle produce karta hai) ko $p_a$ (jo sky provide karti hai) se match karne ke baare mein hai. Jab woh match karte hain, jet "perfectly expanded" hoti hai. $p_a$ sea level par lagbhag $101\text{ kPa}$ se space mein almost $0$ tak girta hai, aur woh girta hua number hi poori problem hai.

Nozzle mismatch ke do tarike

Do ulte tarike hain jinmein exit pressure, ambient pressure ke barabar ho sakti hai, aur agli figure unhe side by side dikhati hai — abhi ise dekho, ek panel at a time.

In donoii extremes ke beech mein perfectly-expanded case baithta hai, $p_e=p_a$, jahan jet seedhi aur parallel nikaiti hai — yahi poore topic ka target hai.

2. Area $A$ — ek circular opening ka size

Neeche ka figure nozzle ko side view mein draw karta hai dono slices mark kiye hue — flow arrow ko left se right trace karo padhte waqt.

Nozzle ek hourglass ki tarah shaped hai jo side par rakha hua hai: gas rush karti hai andar, tight throat (red slice) se squeeze hoti hai, phir widening cone se exit (lavender slice) tak flare hoti hai. Throat woh jagah hai jahan flow sound ki speed tak pahunchi hai; uske baad gas tube ke widening ke saath accelerate hoti rehti hai.

Topic ko yeh kyun chahiye: in donoii areas ka ratio control karta hai ki gas kitni expand aur speed up hoti hai — jo humein is page ke sabse important symbol tak le jaata hai.

3. Expansion ratio $ϵ$ — funnel ka "flare"

Kyunki dono areas circles hain, $ϵ=(R_e/R_t{)}^2$ — radii ka squared ratio. Radius double karo, area quadruple ho jaata hai, $ϵ$ quadruple ho jaata hai.

Topic ko yeh kyun chahiye: har "altitude compensation" trick asliyat mein ek trick hai $ϵ$ change karne ki, ya uske benefits paane ki, jab rocket chadhta hai. Extendable nozzles literally $ϵ$ grow karti hain; aerospikes ek continuously-changing $ϵ$ fake karte hain.

4. Cone geometry — radius length ke saath kaise badhta hai

Nozzle flare (roughly) ek straight-walled cone hai. Do symbols use karte hain:

Jaise neeche figure mein dikhaya gaya hai, length $L$ ke cone ke neeche chalte hue, wall $L\tan \alpha$ se upar uthti hai. Toh exit radius hai: $R_e=R_t+L\tan \alpha$

Topic ko yeh kyun chahiye: aise hi ek extendable nozzle kaam karti hai — extra length $L$ slide out karne se $R_e$ bada hota hai, jo $ϵ$ bada karta hai. Parent note ka formula ${ϵ}_2={\left(1+\frac{L_2}{R_t}\tan \alpha \right)}^2$ bas $ϵ=(R_e/R_t{)}^2$ hai jisme yeh cone rule plug in kiya gaya hai.

5. Mach number $M$ — speed "sound-speeds" mein measure ki gayi

Hum khaas taur par $M_e$ ko naam dete hain, exit Mach number — jet munh se nikalne par kitni supersonic hai.

Topic ko yeh kyun chahiye: geometry ($ϵ$) aur speed ($M_e$) area–Mach relation se aapas mein locked hain — woh formula, jo aage define kiya jaayega, jo ek chosen flare ko ek definite exit speed mein convert karta hai.

Area–Mach relation — geometry speed set karti hai

Neeche ka figure is relation ko $\gamma =1.2$ ke liye plot karta hai: jitna zyada nozzle flare karo (bada $ϵ$, axis par upar jaao), utna zyada exit Mach number force hota hai (daayein jaao). Do dashed guide-lines follow karo: $ϵ=21$ ke paas land karta hai $M_e\approx 3.8$, aur $ϵ=84$ ke paas $M_e\approx 4.75$ — wahi numbers jo parent ka worked example use karta hai.

6. Gas properties: $\gamma$, $R$, $T$

Topic ko yeh kyun chahiye: ye teen fix karte hain ki ek diye gaye pressure drop se aap actually kitni speed nikalte ho. Yeh exhaust-velocity formula ke andar appear hote hain (agla section). Inhe yahan derive karne ki zaroorat nahi — bas inhe pehchano jab parent inhe likhta hai.

7. Exhaust par speed laana: $v_e$, $\dot{m}$, thrust $F$

Parent ka master equation hai: $F=\dot{m}{v}_e+(p_e-p_a)A_e$

Ise do pieces mein padho:

• $\dot{m}{v}_e$ — momentum push: mass ($\dot{m}$) ko fast ($v_e$) throw karna tumhe doosri taraf shove karta hai (Newton's third law).
• $(p_e-p_a)A_e$ — pressure push: agar jet sky se zyada pressure par nikle ($p_e>p_a$, under-expanded case) toh exit disc par leftover push hai; agar kam ho ($p_e<p_a$, over-expanded case) toh sky wapas push karti hai aur yeh term negative ho jaata hai.

Exhaust velocity khud (jo aapko parent ke boxed formula mein milegi) energy conservation se aati hai: $v_e=\sqrt{\frac{2\gamma }{\gamma -1}R{T}_0\left[1-{\left(\frac{p_e}{p_0}\right)}^{(\gamma -1)/\gamma }\right]}$ Notice karo ki yeh har symbol use karta hai jo humne build kiye: $\gamma$, $R$, $T_0$, $p_e$, $p_0$. Isi liye humne pehle unhe define kiya.

8. Specific impulse $I_{sp}$ — fuel efficiency score

Topic ko yeh kyun chahiye: yeh woh single number hai jo engineers nozzles compare karne ke liye quote karte hain. Altitude par better-matched nozzle zyada $v_e$ deti hai, isliye zyada $I_{sp}$ — altitude compensation ka payoff ek figure of merit mein.

Yeh foundations topic ko kaise feed karte hain

Dependency ek single chain mein chalti hai, neeche draw ki gayi hai. Geometry block ($A_t$, $A_e$, cone $\alpha$/$L$) akele expansion ratio $ϵ$ fix karta hai — yahan koi pressure nahi aata, $ϵ$ purely ek shape number hai. Woh $ϵ$ phir area–Mach relation ko feed karta hai $M_e$ set karne ke liye, jo $p_e$ set karta hai; exhaust velocity ke saath combine karo (gas properties aur chamber conditions se fed) aur thrust milti hai aur finally specific impulse. $p_e$ aur $p_a$ ke beech ka mismatch hi woh cheez hai jiske liye altitude compensation exist karta hai.

Equipment checklist

Right side cover karo aur dekho ki reveal karne se pehle har ek bata sakte ho.