Foundations — Burn rate r = a·P^n — Vieille's law

3.3.36 · D1 · Physics › Rocket Propulsion › Burn rate r = a·P^n — Vieille's law

Is page par yeh assume kiya gaya hai ki aapko kuch nahi pata. $r=a{P}^n$ padhne se pehle aapko samajhna hoga ki har letter kya hai, woh kis picture ki taraf point karta hai, aur topic ko uski zaroorat kyun hai. Hum inhe dependency order mein build karte hain — har symbol ko use hone se pehle earn karna hoga.

0. Scene: hum dekh kya rahe hain?

Ek tube (rocket motor) ki picture banao jo ek rubbery solid fuel se bhari hui hai, jise grain kehte hain. Beech mein ek hole hai. Hum us hole ki surface ko jalate hain. Flame poore block ko uda nahi deti — woh solid ko dheere dheere chhabati hai, surface bahar ki taraf hatti rehti hai, jabki andar garam gas build up hoti hai aur peeche se bahar nikalti hai.

Vieille's law mein har cheez is picture ke kisi hisse se attached ek number hai. Aao har hisse ko naam dein.

1. Pressure $P$ — gas ka dhakka

Topic ko iski zaroorat kyun hai: $P$ Vieille's law mein cause hai. Sirf $P$ badlo aur burn rate badal jaati hai. Yeh woh ek knob hai jo nature ghumati hai.

Chamber pressure seedha is baat se juda hai ki gas kaise escape karti hai — dekhein Chamber Pressure and Nozzle Throat.

2. Burn rate $r$ — khane ki speed

Topic ko iski zaroorat kyun hai: $r$ effect hai — woh quantity jo poori law predict karti hai. $r$ ko burning surface area se multiply karo aur milega ki har second kitni gas ban rahi hai, jo thrust drive karta hai (Thrust Equation and Specific Impulse).

3. Exponents aur powers — $P^n$ padhna

$n$ se milne se pehle, ensure karo ki $P^n$ notation scary nahi lage.

Topic ko iski zaroorat kyun hai: actual burn rate double nahi hoti jab pressure double hota hai. Yeh zyada dheere badhti hai. $P^{0.35}$ jaisa fractional power exactly woh shape hai jo kehta hai "respond karta hai, lekin kamzori se." Koi aur simple function "badhta hai, lekin damped hai" ko itni saafai se capture nahi karta.

4. Pressure exponent $n$ — $r$ kitna $P$ ki sunta hai

Topic ko iski zaroorat kyun hai: $n$ safety ke liye sabse important number hai. Yeh decide karta hai ki ek random pressure bump self-correct hoga ya explosion mein bhag jaayega (woh stability story parent note mein build ki gayi hai). Engineers ko $n$ ki chinta sirf is liye hai: ise 1 se neeche rakho.

5. Burn-rate coefficient $a$ — chemistry-aur-temperature ka dial

Topic ko iski zaroorat kyun hai: $a$ burn ka scale set karta hai (absolute terms mein kitna tez), jabki $n$ shape set karta hai (pressure par kaise respond karta hai). Number nikalne ke liye dono chahiye.

6. Logarithm $\log$ — woh tool jo power law ko seedha karta hai

Vieille's law ek curve hai. Curves se constants padhna mushkil hota hai. Logarithm woh tool hai jo curve ko straight line mein badal deta hai.

Topic ko iski zaroorat kyun hai: aise constants measure hote hain. Kai pressures par ek strand jalao, $\log r$ ko $\log P$ ke against plot karo, best straight line kheencho — slope tumhe $n$ deta hai, intercept tumhe $\log a$ deta hai. Koi aur tool power-law ka exponent itni seedhai se nahi padhta.

7. Heat-balance ke characters (law ki yeh form kyun hai)

Parent note $r=a{P}^n$ ko heat balance se derive karta hai. Yahan supporting characters hain, har ek picture mein ek physical cheez.

Topic ko inki zaroorat kyun hai: yeh explain karte hain ki kyun pressure burn rate control karta hai. Zyada $P$ se $\delta$ shrink hota hai (flame paas aati hai), temperature gradient $\frac{T_f-T_s}{\delta }$ steeper hoti hai, heat tezi se conduct hoti hai wapas, aur surface tezi se hatti hai. Jab $\delta$ pressure ki power ke saath shrink hota hai, toh $r$ pressure ki power ke saath badhta hai — aur woh power $n$ hai.

Prerequisite map

Related deeper reading: Solid Rocket Motor Grain Geometry, Combustion Instability, aur parent Hinglish version.

Saare symbols ke saath ek chhota sa sanity calculation

Equipment checklist

Right side cover karo aur reveal karne se pehle jawab do.