Burn rate r = a·P^n — Vieille's law

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WHAT is being described?

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WHY does pressure control burn rate?

Deriving the form $r = aP^n$ from first principles

We don't pull $aP^n$ out of a hat — it drops out of a heat-balance argument.

Step 1 — Heat needed to burn a layer. Why? To turn cold solid at $T_0$ into hot gas, we must supply energy. Per unit surface area, the solid feeds in at speed $r$, so mass flux $= \rho_p r$ ($\rho_p$ = propellant density). Energy demand per area: $\dot q_{\text{need}} = \rho_p\, r\, c_p (T_s - T_0)$ where $T_s$ is surface temperature. Why this step: mass flux $\times$ specific heat $\times$ temperature rise = power to heat the incoming solid.

Step 2 — Heat delivered by the flame. Why? Heat conducts from the flame (at $T_f$) down to the surface across a flame standoff distance $\delta$: $\dot q_{\text{supply}} \approx \lambda\,\frac{T_f - T_s}{\delta}$ ($\lambda$ = gas conductivity). Why: Fourier's law — conduction $\propto$ temperature gradient.

Step 3 — Balance them. In steady burning, supply = demand:

\;\Rightarrow\; r = \frac{\lambda (T_f - T_s)}{\rho_p c_p (T_s - T_0)\,\delta}$$ **Step 4 — How does $\delta$ depend on $P$?** *Why a power law appears:* The flame standoff is set by a race between diffusion and reaction. Reaction rate scales like $P^m$ (bimolecular reactions go as concentration$^2\propto P^2$, etc.), and this makes $\delta \propto P^{-k}$ for some $k>0$. Substituting a power-law $\delta \propto P^{-n}$ collapses everything constant into $a$: $$r \propto \frac{1}{\delta}\propto P^{\,n}\quad\Longrightarrow\quad r = a\,P^{\,n}.$$ That's the whole law: **it's a heat balance where the flame-standoff distance shrinks as a power of pressure.** > [!formula] Log-linearized form (how $a,n$ are measured) > Take $\log$ of both sides: > $$\log r = \log a + n\,\log P$$ > A plot of $\log r$ vs $\log P$ is a **straight line**: slope $= n$, intercept $= \log a$. This is exactly how engineers extract the constants from strand-burner data. ![[3.3.36-Burn-rate-r-=-a·P^n-—-Vieille's-law.png]] --- ## HOW does $n$ decide stability? (the 80/20 insight) > [!intuition] Why $n<1$ keeps a rocket alive > Chamber pressure is itself *produced* by burning: more surface burn → more gas → higher $P$. Mass generated $\propto r \propto P^n$. Mass exhausted through the nozzle throat $\propto P^1$. > - If $n<1$: exhaust ($\propto P$) grows **faster** than generation ($\propto P^n$) when $P$ rises → a pressure bump self-corrects → **stable**. ✅ > - If $n>1$: a bump makes generation outrun exhaust → runaway → **explosion**. ❌ > - $n=1$ is the knife-edge. > > This single inequality, $n<1$, is *the* reason motors are designed with low-exponent propellants. This is the 20% of the topic that gives 80% of the exam value. --- ## Worked examples > [!example] Example 1 — Find the burn rate > Given $a = 5\ \text{mm/s}$ at $P=1\ \text{MPa}$ (i.e. $r=a$ when $P=1$), $n=0.35$. Find $r$ at $P=7\ \text{MPa}$. > > $r = a P^n = 5 \times 7^{0.35}$. > *Why:* directly apply the law. $7^{0.35} = e^{0.35\ln 7}=e^{0.35(1.946)}=e^{0.681}=1.976$. > $$r = 5 \times 1.976 \approx 9.9\ \text{mm/s}.$$ > **Sense check:** pressure rose $7\times$ but $r$ only rose $\sim 2\times$ — small $n$ means weak sensitivity. ✔ > [!example] Example 2 — Extract $n$ from two data points > Measured: $r_1 = 4\ \text{mm/s}$ at $P_1=2\ \text{MPa}$; $r_2 = 6\ \text{mm/s}$ at $P_2=5\ \text{MPa}$. Find $n$ and $a$. > > *Why divide?* Dividing kills $a$: > $$\frac{r_2}{r_1}=\left(\frac{P_2}{P_1}\right)^n \Rightarrow \frac{6}{4}=\left(\frac{5}{2}\right)^n.$$ > Take logs: $n=\dfrac{\ln(1.5)}{\ln(2.5)}=\dfrac{0.405}{0.916}=0.442.$ > Then $a = r_1/P_1^n = 4/2^{0.442}=4/1.358 = 2.95\ \text{mm/(s·MPa}^{n})$. > **Sense check:** $n<1$ → stable propellant. ✔ > [!example] Example 3 — Stability test > Propellant A has $n=0.9$; Propellant B has $n=1.2$. A random pressure spike hits both. Which motor survives? > > *Why:* Compare generation $\propto P^n$ vs exhaust $\propto P$. > - A: $n=0.9<1$ → exhaust outpaces generation → damps out → **survives**. > - B: $n=1.2>1$ → generation runs away → **pressure explosion**. --- ## Common mistakes > [!mistake] "Burn rate is the exhaust velocity." > **Why it feels right:** both have units of speed and both describe "burning gas." **The fix:** $r$ is the speed the *solid surface recedes* (mm/s scale). Exhaust velocity is the gas leaving the nozzle (km/s scale). They differ by ~$10^6$ and answer different questions. > [!mistake] "Higher $n$ is better — more thrust boost from pressure." > **Why it feels right:** big $n$ means burn rate responds strongly to pressure, sounds powerful. **The fix:** that same strong response causes **thermal runaway** when $n\ge1$. Engineers *want* small $n$ (0.2–0.5) for controllability. > [!mistake] "$a$ is a universal constant." > **Why it feels right:** it looks like a fixed material property. **The fix:** $a$ depends on **initial grain temperature** — a cold-soaked motor and a hot one have different $a$, so the same rocket burns differently on a winter vs summer launch (temperature sensitivity $\sigma_p$). > [!mistake] Fitting $r$ vs $P$ on a linear plot. > **Why it feels right:** you want a straight line for slope. **The fix:** it's linear only in **log–log**. Plot $\log r$ vs $\log P$; slope is $n$, not on raw axes. --- ## #flashcards/physics Vieille's law formula ::: $r = a\,P^n$, where $r$ is linear burn rate, $P$ chamber pressure, $a$ burn-rate coefficient, $n$ pressure exponent. What does $r$ physically represent? ::: The speed at which the burning propellant surface recedes perpendicular to itself (mm/s), NOT the exhaust gas speed. What does the exponent $n$ measure? ::: How sensitively the burn rate responds to chamber pressure (combustion sensitivity). On what plot is Vieille's law a straight line, and what are slope/intercept? ::: $\log r$ vs $\log P$: slope $=n$, intercept $=\log a$. Stability condition for a solid motor? ::: $n<1$ (exhaust $\propto P$ outpaces generation $\propto P^n$, so pressure bumps self-correct). Why does higher pressure increase burn rate physically? ::: Denser gas → flame sits closer & hotter → steeper temperature gradient conducts heat back faster → surface reaches ignition sooner. What physical quantity is hidden inside $a$ besides chemistry? ::: The initial grain temperature (temperature sensitivity). Given $r_1,P_1,r_2,P_2$, how to get $n$? ::: $n=\dfrac{\ln(r_2/r_1)}{\ln(P_2/P_1)}$. Typical range of $n$ for usable propellants? ::: About $0.2$ to $0.5$ (well below 1 for stability). > [!recall]- Feynman: explain to a 12-year-old > Imagine a big candle, but instead of the flame just sitting on top, the flame slowly *eats down into the wax*. **Burn rate** is how fast that eating happens. Now, if you blow the candle harder (more pressure/push), the flame gets hotter and closer to the wax, so it eats faster. Scientists found a simple rule: burn speed = a special number $a$, times the push $P$ raised to a small power $n$. And here's the safety trick: if $n$ is small (less than 1), the rocket calms itself down after a hiccup. If $n$ is big, one hiccup makes it eat faster and faster until — boom. So rocket engineers always pick candles (propellants) with a small $n$. > [!mnemonic] Remember it > **"A Pen writes the rate"** → $r = a\,P^n$ ("A-P-n"). > And for safety: **"n below one, motor won't run away"** ($n<1$ ⇒ stable). --- ## Connections - [[Solid Rocket Motor Grain Geometry]] — burning surface area $\times\,r$ sets mass flow. - [[Chamber Pressure and Nozzle Throat]] — exhaust flux $\propto P$, the other half of the stability balance. - [[Thrust Equation and Specific Impulse]] — $r$ feeds mass flow $\dot m = \rho_p A_b r$. - [[Fourier's Law of Heat Conduction]] — the conduction step in the derivation. - [[Combustion Instability]] — where the $n$-criterion matters most. - [[Temperature Sensitivity of Propellants]] — why $a$ varies with launch temperature. ## 🖼️ 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 ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, solid rocket ka propellant ek baar mein nahi phatta — woh **layer by layer** jalta hai, jaise candle ki wax dheere-dheere niche khatam hoti hai. Jis speed se jalti hui surface piche hatti hai, usko hum **burn rate** $r$ bolte hain. Yaad rakhna: yeh gas ki nikalne ki speed NAHI hai, yeh solid ke ghisne ki speed hai (mm/s scale). > > Vieille's law kehta hai: $r = a\,P^n$. Matlab burn rate mostly **chamber pressure** $P$ pe depend karta hai. Physics kya hai? Zyada pressure → gas molecules paas-paas → flame surface ke kareeb aur zyada garam → heat wapas solid mein tezi se jaati hai → agli layer jaldi ignite hoti hai → surface tezi se ghista hai. $a$ propellant ki chemistry (aur initial temperature) batata hai, aur $n$ batata hai ki pressure pe response kitna strong hai. > > Sabse important cheez exam ke liye: **$n<1$ hona chahiye stability ke liye**. Kyun? Gas banti hai $\propto P^n$, aur nozzle se nikalti hai $\propto P$. Agar $n<1$ hai, toh pressure thoda badhte hi exhaust generation se aage nikal jaata hai aur system apne aap thanda ho jaata hai — safe. Lekin agar $n>1$, toh chhoti si pressure spike runaway ban jaati hai — motor phat sakta hai. Isliye engineers hamesha low $n$ (0.2–0.5) wale propellant choose karte hain. > > Ek practical trick: $a$ aur $n$ nikalne ke liye do data points lo aur ratio le lo — $a$ cancel ho jaata hai, aur $n = \ln(r_2/r_1)/\ln(P_2/P_1)$. Ya phir $\log r$ vs $\log P$ ka graph banao — woh straight line hoti hai jiska slope $n$ aur intercept $\log a$ hota hai. Bas itna samajh liya toh yeh topic pura clear hai! ![[audio/3.3.36-Burn-rate-r-=-a·P^n-—-Vieille's-law.mp3]]