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3.3.36 · HinglishRocket Propulsion

Burn rate r = a·P^n — Vieille's law

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3.3.36 · Physics › Rocket Propulsion

WHAT is being described?

WHY does pressure control burn rate?

Deriving the form from first principles

hum hat se nahi nikalte — yeh ek heat-balance argument se seedha aata hai.

Step 1 — Ek layer jalane ke liye zaruri heat. Kyun? par thande solid ko garam gas mein badalne ke liye, hume energy supply karni hogi. Unit surface area par, solid speed se feed hoti hai, toh mass flux ( = propellant density). Energy demand per area: jahan surface temperature hai. Yeh step kyun: mass flux specific heat temperature rise = power to heat the incoming solid.

Step 2 — Flame dwara deliver ki gayi heat. Kyun? Heat flame (at ) se surface tak ek flame standoff distance ke across conduct hoti hai: ( = gas conductivity). Kyun: Fourier's law — conduction temperature gradient.

Step 3 — Inhe balance karo. Steady burning mein, supply = demand:

\;\Rightarrow\; r = \frac{\lambda (T_f - T_s)}{\rho_p c_p (T_s - T_0)\,\delta}$$ **Step 4 — $\delta$ $P$ par kaise depend karta hai?** *Kyun power law aata hai:* Flame standoff diffusion aur reaction ke beech ek race se set hoti hai. Reaction rate $P^m$ ki tarah scale karti hai (bimolecular reactions concentration$^2\propto P^2$ ki tarah jaati hain, etc.), aur yeh $\delta \propto P^{-k}$ banata hai kisi $k>0$ ke liye. Ek power-law $\delta \propto P^{-n}$ substitute karne par sab constant $a$ mein aa jaata hai: $$r \propto \frac{1}{\delta}\propto P^{\,n}\quad\Longrightarrow\quad r = a\,P^{\,n}.$$ Yahi poora law hai: **yeh ek heat balance hai jahan flame-standoff distance pressure ki power ke saath shrink hoti hai.** > [!formula] Log-linearized form (kaise $a,n$ measure hote hain) > Dono sides ka $\log$ lo: > $$\log r = \log a + n\,\log P$$ > $\log r$ vs $\log P$ ka plot ek **straight line** hai: slope $= n$, intercept $= \log a$. Exactly isi tarah engineers strand-burner data se constants extract karte hain. ![[3.3.36-Burn-rate-r-=-a·P^n-—-Vieille's-law.png]] --- ## HOW does $n$ stability decide karta hai? (80/20 insight) > [!intuition] Kyun $n<1$ ek rocket ko alive rakhta hai > Chamber pressure khud burning se *produce* hota hai: zyada surface burn → zyada gas → zyada $P$. Generate hota mass $\propto r \propto P^n$. Nozzle throat se exhaust hota mass $\propto P^1$. > - Agar $n<1$: exhaust ($\propto P$) generation ($\propto P^n$) se **zyada fast** badhta hai jab $P$ badhta hai → ek pressure bump self-correct hota hai → **stable**. ✅ > - Agar $n>1$: ek bump generation ko exhaust se aage kar deta hai → runaway → **explosion**. ❌ > - $n=1$ knife-edge hai. > > Yeh single inequality, $n<1$, hi *woh* reason hai ki motors low-exponent propellants se design hote hain. Topic ka yeh 20% hai jo exam value ka 80% deta hai. --- ## Worked examples > [!example] Example 1 — Burn rate nikalo > Diya gaya hai $a = 5\ \text{mm/s}$ at $P=1\ \text{MPa}$ (yaani $r=a$ jab $P=1$), $n=0.35$. $P=7\ \text{MPa}$ par $r$ nikalo. > > $r = a P^n = 5 \times 7^{0.35}$. > *Kyun:* seedha law apply karo. $7^{0.35} = e^{0.35\ln 7}=e^{0.35(1.946)}=e^{0.681}=1.976$. > $$r = 5 \times 1.976 \approx 9.9\ \text{mm/s}.$$ > **Sense check:** pressure $7\times$ badha lekin $r$ sirf $\sim 2\times$ badha — chhota $n$ matlab weak sensitivity. ✔ > [!example] Example 2 — Do data points se $n$ nikalo > Measured: $r_1 = 4\ \text{mm/s}$ at $P_1=2\ \text{MPa}$; $r_2 = 6\ \text{mm/s}$ at $P_2=5\ \text{MPa}$. $n$ aur $a$ nikalo. > > *Kyun divide karein?* Divide karne se $a$ khatam ho jaata hai: > $$\frac{r_2}{r_1}=\left(\frac{P_2}{P_1}\right)^n \Rightarrow \frac{6}{4}=\left(\frac{5}{2}\right)^n.$$ > Logs lo: $n=\dfrac{\ln(1.5)}{\ln(2.5)}=\dfrac{0.405}{0.916}=0.442.$ > Phir $a = r_1/P_1^n = 4/2^{0.442}=4/1.358 = 2.95\ \text{mm/(s·MPa}^{n})$. > **Sense check:** $n<1$ → stable propellant. ✔ > [!example] Example 3 — Stability test > Propellant A ka $n=0.9$ hai; Propellant B ka $n=1.2$ hai. Dono par ek random pressure spike aati hai. Kaun sa motor bachega? > > *Kyun:* generation $\propto P^n$ ko exhaust $\propto P$ se compare karo. > - A: $n=0.9<1$ → exhaust generation se aage nikalta hai → damp out ho jaata hai → **bachega**. > - B: $n=1.2>1$ → generation runaway ho jaata hai → **pressure explosion**. --- ## Common mistakes > [!mistake] "Burn rate exhaust velocity hai." > **Kyun sahi lagta hai:** dono ki units speed hain aur dono "burning gas" describe karte hain. **Fix:** $r$ woh speed hai jis par *solid surface peeche hatti hai* (mm/s scale). Exhaust velocity woh gas hai jo nozzle se nikal rahi hai (km/s scale). Yeh $\sim10^6$ se differ karte hain aur alag sawaalon ke jawaab dete hain. > [!mistake] "Zyada $n$ better hai — pressure se zyada thrust boost." > **Kyun sahi lagta hai:** bada $n$ matlab burn rate strongly pressure ke saath respond karta hai, powerful lagta hai. **Fix:** wahi strong response **thermal runaway** karti hai jab $n\ge1$. Engineers *chahte hain* chhota $n$ (0.2–0.5) controllability ke liye. > [!mistake] "$a$ ek universal constant hai." > **Kyun sahi lagta hai:** yeh ek fixed material property ki tarah lagta hai. **Fix:** $a$ **initial grain temperature** par depend karta hai — ek cold-soaked motor aur ek garam wale ka alag $a$ hoga, toh ek hi rocket winter vs summer launch mein alag burn karta hai (temperature sensitivity $\sigma_p$). > [!mistake] $r$ vs $P$ ko linear plot par fit karna. > **Kyun sahi lagta hai:** slope ke liye straight line chahiye hoti hai. **Fix:** yeh sirf **log–log** mein linear hai. $\log r$ vs $\log P$ plot karo; slope $n$ hai, raw axes par nahi. --- ## #flashcards/physics Vieille's law formula ::: $r = a\,P^n$, jahan $r$ linear burn rate hai, $P$ chamber pressure, $a$ burn-rate coefficient, $n$ pressure exponent. $r$ physically kya represent karta hai? ::: Woh speed jis par burning propellant surface apne aap ke perpendicular peeche hatti hai (mm/s), exhaust gas speed NAHI. Exponent $n$ kya measure karta hai? ::: Burn rate kitni sensitively chamber pressure ke saath respond karti hai (combustion sensitivity). Kis plot par Vieille's law ek straight line hai, aur slope/intercept kya hain? ::: $\log r$ vs $\log P$: slope $=n$, intercept $=\log a$. Solid motor ke liye stability condition? ::: $n<1$ (exhaust $\propto P$ generation $\propto P^n$ se aage nikalta hai, toh pressure bumps self-correct hote hain). Physically kyun zyada pressure burn rate badhata hai? ::: Dense gas → flame zyada paas aur garm baith ti hai → steeper temperature gradient heat wapas zyada fast conduct karta hai → surface pehle ignition temperature par pahuncht hai. $a$ ke andar chemistry ke alawa kaun si physical quantity chhipi hai? ::: Initial grain temperature (temperature sensitivity). $r_1,P_1,r_2,P_2$ diye ho toh $n$ kaise nikaalein? ::: $n=\dfrac{\ln(r_2/r_1)}{\ln(P_2/P_1)}$. Usable propellants ke liye $n$ ki typical range? ::: Lagbhag $0.2$ se $0.5$ tak (stability ke liye 1 se kaafi kam). > [!recall]- Feynman: 12-saal ke bachche ko explain karo > Socho ek badi mombatti hai, lekin flame sirf upar baithne ki jagah wax mein *neeche eat karti jaati hai*. **Burn rate** yeh hai ki woh eating kitni fast hoti hai. Ab, agar mombatti par zyada phunko (zyada pressure/push), flame wax ke zyada paas aur garam ho jaati hai, toh zyada fast khati hai. Scientists ne ek simple rule dhoondha: burn speed = ek khaas number $a$, times push $P$ ek chhoti power $n$ tak raise kiya. Aur yeh safety trick hai: agar $n$ chhota ho (1 se kam), toh rocket ek hiccup ke baad khud ko shant kar leta hai. Agar $n$ bada ho, toh ek hiccup use zyada fast aur zyada fast khaane par majboor karta hai jab tak — boom. Toh rocket engineers hamesha chhote $n$ waali mombattiyaan (propellants) choose karte hain. > [!mnemonic] Yaad rakho > **"A Pen rate likhta hai"** → $r = a\,P^n$ ("A-P-n"). > Aur safety ke liye: **"n ek se neeche, motor bhaagna nahi seekhe"** ($n<1$ ⇒ stable). --- ## Connections - [[Solid Rocket Motor Grain Geometry]] — burning surface area $\times\,r$ mass flow set karta hai. - [[Chamber Pressure and Nozzle Throat]] — exhaust flux $\propto P$, stability balance ka doosra half. - [[Thrust Equation and Specific Impulse]] — $r$ mass flow $\dot m = \rho_p A_b r$ ko feed karta hai. - [[Fourier's Law of Heat Conduction]] — derivation mein conduction step. - [[Combustion Instability]] — jahan $n$-criterion sabse zyada matter karta hai. - [[Temperature Sensitivity of Propellants]] — kyun $a$ launch temperature ke saath vary karta hai. ## 🖼️ Concept Map ```mermaid flowchart TD P[Chamber pressure P] -->|controls| R[Burn rate r] V["Vieille's law r = a·P^n"] -->|defines| R A[Coefficient a] -->|depends on chemistry and temp| V N[Pressure exponent n] -->|combustion sensitivity| V P -->|packs molecules closer| FZ[Gas-phase flame zone] FZ -->|closer hotter flame| GRAD[Steeper temp gradient] GRAD -->|heat conducts back| HB[Heat balance] HB -->|supply = demand| R QN[Heat needed] -->|"ρp·r·cp·(Ts-T0)"| HB QS[Heat supplied] -->|"Fourier λ(Tf-Ts)/δ"| HB DELTA[Flame standoff δ] -->|"diffusion vs reaction P^m"| QS DELTA -->|yields power law| V ```