Foundations — Optimum expansion — P_e = P_a for maximum thrust

3.3.13 · D1 · Physics › Rocket Propulsion › Optimum expansion — P_e = P_a for maximum thrust

Is page par assume kiya gaya hai ki aap kuch nahi jaante. parent note ka har letter neeche unpack kiya gaya hai, aur har ek usse pehle wale par build kiya gaya hai. Upar se neeche padho.

0. Char base measuring sticks

Kisi bhi physics symbol se pehle, hume woh rulers chahiye jinse har cheez measure ki jaati hai. Ye char internationally agreed base units hain — inhe aap neeche milenge.

Picture. Socho deewar par char labelled measuring tapes hain — ek kitna lamba, ek kitna bhaari, ek kitni ticks, ek kitna garam. Baad mein har symbol sirf in charon ka koi combination hai.

Topic ko inki zaroorat kyun hai. Area metres $\times$ metres hai, speed metres per second hai, mass flow kilograms per second hai, temperature kelvin mein hai. Agar ye rulers undefined hote, toh baad ke kisi bhi symbol ka koi matlab nahi hota.

1. Force $F$ aur newton

Hum force ko pehle define karte hain, kyunki pressure (agla section) isi se bana hai.

Picture. Ek $1\text{kg}$ ki water bottle ko smooth table par flick karo; tumhari flick ki strength newtons mein measure hoti hai.

Topic ko iski zaroorat kyun hai. Rocket jo kuch bhi karta hai woh sab force hai — pressures push karti hain, exhaust recoil karta hai. Aur thrust (poore topic ka point) ek force hai.

2. Area $A$ aur exit area $A_e$

Picture. Rocket bell ke andar seedha dekho. Bilkul end mein jo chamkila circle dikhta hai woh exit hai; uska size $A_e$ hai.

Topic ko iski zaroorat kyun hai. Badi surface par spread ki gayi push, usi push se alag hoti hai jo choti surface par ho. "Push-per-area" ko total force mein wapas convert karne ke liye area se multiply karte hain — yahi wajah hai ki baad mein introduce hone wale $(P_e-P_a)A_e$ term mein $A_e$ hai.

3. Pressure — invisible push

Ab jab force $F$ (section 1) aur area $A$ (section 2) dono define ho gaye hain, hum safely pressure ka formula likh sakte hain.

Picture. Socho ki laakhon chhote gas molecules ek box ke andar zing kar rahe hain, walls par drumming kar rahe hain. Har bounce ek chhota sa shove hai. Un sabhi shoves ko ek square metre wall par jod do — woh total push-per-area pressure hai.

Topic ko iski zaroorat kyun hai. Poori kahani do pressures ke beech ek contest hai: nozzle se $P_e$ pressure par nikli gas, aur $P_a$ pressure par bahar ki hawa. Sab kuch neeche inhi do numbers ko compare karta hai.

4. Topic ki do pressures: $P_e$ aur $P_a$

Picture. Nozzle rim par khade ho jaao. Andar se, hot gas bahar push karti hai ($P_e$). Bahar se, hawa andar push karti hai ($P_a$). Jo zyada push karta hai woh thodi extra ya missing thrust "jeet" leta hai.

Topic ko iski zaroorat kyun hai. Chapter ka poora result ek single line hai $P_e=P_a$. Aap woh line tab tak nahi padh sakte jab tak yeh nahi jaante ki ye do alag pushes ek hi jagah par hain.

5. Thrust — rocket ki forward force

Picture. Jab bhi aap kuch peeche phenkte ho, aap aage ki taraf recoil karte ho — jaise skateboard se uthna. Rocket gas peeche phenk ta hai; aage ki taraf recoil thrust hai.

Topic ko iski zaroorat kyun hai. Poore topic ka goal thrust ko jitna ho sake utna bada banana hai. "Optimum expansion" simply woh nozzle setting hai jo is forward force $F$ ko maximise karti hai.

6. Mass flow rate $\dot{m}$

Picture. Socho nozzle ke muh par ek checkpoint hai jo har second nikalne wali gas ki mass count karta hai, jaise paani pipe mein litres-per-second mein measure hota hai.

Topic ko iski zaroorat kyun hai. Phenki gayi gas se thrust hai (mass per second phenki gayi) $\times$ (kitni tezi se phenki gayi). Woh hai $\dot{m}{v}_e$ term — thrust ka sabse bada hissa.

7. Exhaust velocity $v_e$

Picture. Exit plane par ek akela gas molecule, speed se blurred, bell se hazaaron metres per second ki raftaar se bahar nikal raha hai.

Topic ko iski zaroorat kyun hai. Tezi exhaust = zyada recoil. Topic ka punchline yeh hai ki optimum expansion par, engine ki saari energy $v_e$ ko bada banane mein gayi hai, aur kuch bhi pressure mismatch par waste nahi hua.

8. Sab kuch jodna: thrust equation

Ab parent ki central formula mein har symbol define ho gaya hai, toh hum ise padh sakte hain:

Doosra term signs ke saath padho:

• Agar $P_e>P_a$: gas hawa se zyada push karti hai → term positive hai, thrust add karta hai.
• Agar $P_e=P_a$: pushes cancel ho jaati hain → term zero hai.
• Agar $P_e<P_a$: hawa zyada push karti hai → term negative hai, thrust chura leta hai.

Yahi wajah hai ki topic middle case dhundta hai. (Poora "zero kyun peak hai" Thrust equation derivation mein prove kiya gaya hai.)

9. Chamber conditions: $P_0$ aur $T_0$

Picture. Jalta hua fuel ka ek sealed furnace: bahut hot ($T_0$, kelvin mein), bahut high-pressure ($P_0$), barely moving. Yeh woh reservoir hai jis se nozzle draw karta hai.

Topic ko iski zaroorat kyun hai — $P_0,T_0$ kaise $P_e$ aur $v_e$ bante hain. Jaise hi gas widening nozzle se rush karta hai, woh thanda hota hai aur uska pressure drop hota hai jabki uski speed badhti hai — heat energy ko motion ke liye trade kiya jaata hai. Concretely, gas jitna zyada expand karta hai, utna hi $P_e$ girata hai aur $v_e$ badhta hai, ek fixed rule follow karte hue jise isentropic (constant-entropy) relations kehte hain. Words mein: high starting $P_0$ aur $T_0$ = convert karne ke liye ek bada "energy bank", toh ek well-expanded nozzle unhe large $v_e$ aur small $P_e$ mein badal deta hai. Tumhe yahan algebra ki zaroorat nahi hai; bas yeh chain yaad rakho $P_0,T_0\to \text{expansion}\to \text{lower }{P}_e,\text{higher }{v}_e$.

10. Do Greek helpers: $\gamma$ aur $\epsilon$

Picture. Nozzle sideways rakha hourglass hai: gas thin waist ($A^{\ast }$) se squeeze hoti hai aur phir wide exit ($A_e$) tak flare out hoti hai. Bada $\epsilon$ matlab ek lamba, dramatically flaring bell.

Topic ko iski zaroorat kyun hai. $\epsilon$ woh ek knob hai jo designer actually ghoomata hai. $\epsilon$ choose karna $P_e$, $v_e$, aur $A_e$ sabko ek saath fix kar deta hai (section 9 ke isentropic relations ke through). Toh "optimum thrust ke liye nozzle choose karo" ka matlab really hai "sahi $\epsilon$ choose karo". Geometry ke baare mein aur Nozzle expansion ratio mein.

11. "Dot-on-top" aur "d-something" notation

Humne pehle hi dot dekha ($\dot{m}$ = "per second"). Notation ka aakhri piece letter d hai jo $\frac{dF}{d{P}_e}$ mein use hota hai.

Picture. Page ke upar thrust $F$ aur across $P_e$ plot karo. Curve badhti hai, ek hilltop tak pahunchi hai, phir girti hai. Hill ke bilkul top par slope flat hota hai — woh flat point ($\frac{dF}{d{P}_e}=0$) maximum thrust hai, aur yeh exactly $P_e=P_a$ par land karta hai.

Topic ko iski zaroorat kyun hai. Mathematics mein "Maximum" = "flat slope". Is slope ko zero set karna woh tarika hai jisse parent note prove karta hai ki $P_e=P_a$ best hai sirf assert karne ki jagah.

Prerequisite map

Har arrow kehta hai "ise samajhne ke liye tumhe woh chahiye". Bottom node OPT parent topic hi hai.

Equipment checklist

Self-test: har question padho, apne dimag mein answer karo, phir ::: ke baad wala part reveal karo.