Aerodynamic heating during reentry — stagnation point heat flux Chapman equation: Foundations

3.4.21 · D1 · Physics › Rocket Flight Mechanics › Aerodynamic heating during reentry — stagnation point heat f

Isse pehle ki tum Chapman equation $\overset{q}{˙}_{s} = \frac{C}{R _{n}} ρ_{\infty} V^{3}$ padho, tumhe isme har ek letter padhna aana chahiye, saath mein woh pictures bhi jo un letters se banti hain. Yeh page unhe ek-ek karke build karta hai, bilkul zero se. Yahan kuch bhi assume nahi kiya gaya ki tumne pehle inhe dekha hoga.

Hum parent topic — Chapman stagnation heat-flux equation — ke liye foundation floor hain.

1. Speed $V$ — kitni tez, aur kyun yeh villain hai

Picture. Socho tum khade ho aur hawa ki ek diwar tumhare taraf aati hai. Woh diwar jitni tez aaye, utni zyada takkar. $V$ us takkar ki speed measure karta hai.

Topic ko iski zaroorat kyun hai. Heating ka har joule vehicle ki motion se aata hai. Baad mein hum dekhenge ki heating $V^{3}$ se badhti hai — speed ke teen factors ek saath multiply hoke — isliye $V$ sabse important knob hai. $V$ galat ho toh baaki sab bekaar hai.

2. Kinetic energy — disaster ka fuel tank

Picture. Ek chalte object ko ek bhare tank ki tarah socho. Tank mein fuel ki height $\frac{1}{2} V^{2}$ hai. Jab object ko slow kiya jaata hai, woh tank khaali hona chahiye — aur jo fuel woh dump karta hai woh heat banke aati hai.

Figure — Aerodynamic heating during reentry — stagnation point heat flux Chapman equation

Figure 1 — "energy tank." Left se right padho: blue tank $\frac{1}{2} V^{2}$ height tak bhara hai (andar label), yellow arrow vehicle ke slowdown ko mark karta hai, aur bahar burst hoti barah pink arrows woh energy hai jo heat ke roop mein nikal rahi hai. Figure ka point: left pe exactly ek source box hai aur right pe ek heat-burst — kuch aur heating nahi banata.

Topic ko iski zaroorat kyun hai. Heat sirf isi khaali hote tank ke siva kuch nahi banata. Jab flow ko nose par rok diya jaata hai, uska sara $\frac{1}{2} V^{2}$ per kilogram kuch aur mein convert hona padta hai — aur woh kuch aur temperature aur pressure hai. Yahi khatre ki ultimate source hai.

3. Density $ρ$ — kitni hawa hai hit karne ke liye

Picture. Ek box socho, ek metre har side. Andar ki hawa ka mass gino. Sea level par us box mein roughly $1.2 kg$ hoti hai. Upar jahan reentry heating peak hoti hai, wahi box mein shayad $0.0001 kg$ — hawa bahut patli hai.

Figure 2 — density ek-metre box mein mass per hai. Left box: low altitude, blue dots se bhara (dense, $ρ \approx 1.2 kg/m^{3}$ , yellow label). Right box: high altitude, sirf paanch pink dots (thin, $ρ \approx 0.0001 kg/m^{3}$ ). Dono boxes size mein identical hain par contents mein 100,000× differ karte hain — woh visual gap yahi baat batata hai: ek hi $ρ$ symbol ek reentry ke dauran paanch orders of magnitude span karta hai.

Topic ko iski zaroorat kyun hai. Zyada air molecules per box = zyada chhote collisions = zyada heat deliver. Lekin density sirf $ρ_{\infty}$ ke roop mein aati hai (square root — §8 dekho), ek gentle dependence.

Intuition Square root kyun,

$ρ$ khud kyun nahi? Do competing effects ek doosre se ladte hain. (1) Zyada density → zyada mass of gas layer ke edge ke paas se har second bahti hai (woh flow rate $ρ_{e} u_{e}$ ka product hai, upar define kiya), isliye zyada heat available hai: yeh flux ko upar push karta hai. (2) Lekin wahi extra gas boundary layer (§5) ko patla aur densa bhi banati hai, jo yeh change karti hai ki heat us paar kitni tezi se diffuse ho sakti hai, dependence ko neeche kheenche. Jab stagnation boundary-layer flow carefully solve ki jaati hai, dono effects milke aise bante hain ki flux edge mass flux $ρ_{e} u_{e}$ ke square root ke roop mein badhti hai — yaani $\overset{q}{˙} \propto ρ_{e} u_{e} \propto ρ_{\infty}$ . Nature beech ka raasta nikalti hai: density double karne se heating double nahi hoti, sirf $2 \approx 1.41$ se multiply hoti hai.

4. Stagnation point aur nose radius $R_{n}$

Picture. Ek ball ko nose ke against rakho taaki woh us jagah surface ko touch kare jahan flow rukti hai. Ek beach ball ek blunt capsule mein fit hoti hai (large $R_{n}$ ); ek marble ek needle nose mein (small $R_{n}$ ).

Figure 3 — nose radius fitting sphere ka radius hai. Left: blunt capsule ki curve badi dotted blue circle se match karti hai (large $R_{n}$ , yellow radius line); pink dot dead-centre stagnation point mark karta hai. Right: sharp needle sirf ek tiny dotted circle se match hoti hai (small $R_{n}$ ). Dono circles ki sizes compare karo — poora design lesson usi contrast mein hai: fitting circle jitna bada, curve utna gentle aur nose utna thanda.

Topic ko iski zaroorat kyun hai. $R_{n}$ design lever hai. Is bluntness se jo detached shock banta hai use Bow shock and blunt-body theory mein dekho.

5. Boundary layer $δ$ — woh patli skin jise heat cross karni padti hai

Picture. Socho ek warm room aur ek cold window. Glass ke bilkul saath ek patli still air ki layer hoti hai; heat ko us layer ke through seep karke glass tak pahunchna hota hai. Capsule par "room" hai superheated shocked gas, "glass" hai wall, aur $δ$ woh still layer hai.

Topic ko iski zaroorat kyun hai. Heat flux $δ$ ke across temperature difference se drive hoti hai aur uski thickness se resist. Moti boundary layer = lambi crawl = kam heat pahunchi. Yahi woh machinery hai jisne §3 ka $ρ$ aur §4 ka $R_{n}^{- 1/2}$ produce kiya. Depth Boundary layer theory mein hai.

6. Heat flux $\overset{q}{˙}$ — woh quantity jo hum actually chahte hain

Picture. Ek table par heat lamp point karo. $\overset{q}{˙}$ hai ki table ki ek tile kitni tezi se pak rahi hai — har second har square metre par aane wale joules. Reentry nose values $\sim 1 0^{6} W/m^{2}$ tak pahunchti hain; yeh har square metre of nose par ek million-watt hairdryer hai.

Topic ko iski zaroorat kyun hai. Yahi poori Chapman equation ka output hai. Baaki sab (speed, density, radius) sirf $\overset{q}{˙}_{s}$ predict karne ke liye exist karte hain, kyunki woh single number decide karta hai ki heat shield tikegi ya nahi. Yeh seedha Thermal Protection Systems (ablatives, tiles) mein feed hota hai.

Intuition

$V$ par exponent 3 kyun hai — ek concrete do-factor derivation Heat rate per area ko do ingredients se build karo, jinmein se har ek apna power of $V$ laata hai.

Ingredient A — gas ke har kilogram mein energy. Jaise §2 mein dikhaya, gas ko rest par laane se uski specific kinetic energy $\frac{1}{2} V^{2}$ dump hoti hai. Yeh $V$ ke do powers hain.
Ingredient B — har second har square metre par kitne kilograms aate hain. Woh mass-flow-rate-per-area edge mass flux $ρ_{e} u_{e}$ hai, aur edge speed $u_{e}$ khud flight speed $V$ ke proportional hai. Yeh $V$ ka ek aur power hai. Heat flux $\approx$ (energy per kg) $\times$ (kg per second per m²) — lekin boundary-layer solution mass-flux factor ko density ke saath square root ke neeche rakhta hai, toh clean count hai: enthalpy difference $V^{2}$ deta hai, aur stagnation velocity gradient $β \propto V$ extra factor contribute karta hai. Multiply karo: $V^{2} \times V^{1} = V^{3}$ . Plain kinetic energy se ek extra power of $V$ delivery rate hai — energy content $V^{2}$ hai, lekin zyada tezi se deliver karna teesra $V$ add karta hai.

7. Constant $C$ — kahan se aata hai aur uske units kya hain

Kahan se aata hai. Jab tum poori stagnation boundary-layer derivation karte ho (§3 aur §4 mein sketch ki gayi), answer ek proportionality hai: $\overset{q}{˙}_{s} \propto ρ_{\infty} / R_{n} V^{3}$ . Jo multiplier bacha — sab kuch jo proportionality ne chhhoda — woh $C$ hai. Iske andar chhupe hain gas ki heat capacity, thermal conductivity, viscosity, aur wall-to-stagnation enthalpy ratio (neeche recovery-temperature note dekho). Kyunki woh gas properties planet-to-planet differ karti hain, Mars ke CO₂ atmosphere ke liye $C$ Earth ki hawa ke $C$ se alag hoga.

Definition Recovery (stagnation) temperature —

$C$ ke andar chhupa Jab tez gas wall ke against friction se slow hoti hai, uski kinetic energy heat ke roop mein vapas aati hai, isliye wall ek temperature "feel" karti hai jo free-stream hawa se zyada hoti hai. Woh temperature jis par gas rest mein aane par recover karti hai woh ==recovery (ya stagnation) temperature $T_{0}$ == hai. Chapman ka simple form quietly ek fixed recovery factor (kitni completely woh kinetic energy heat ke roop mein recover hoti hai, in flows mein 1 ke paas) aur roughly constant specific heat $c_{p}$ (ek kilogram gas ko ek degree warm karne ke liye joules) assume karta hai. Yeh do adiabatic-recovery assumptions constant $C$ mein bake in hain — bilkul isliye $C$ tab valid nahi rehta jab gas chemistry (dissociation, ionisation) drastically change ho jaaye.

8. Square root aur cube $V^{3}$ — powers padhna

Chapman formula in symbols ko do operations se jodhta hai jo tumhe fluently padhni chahiye.

Picture. Teen curves speed ke against line up karo: $V$ (straight line), $V^{2}$ (bend up), $V^{3}$ (rocket up). Cube baaki sab ko peeche chhod deta hai — isliye speed heating mein dominate karti hai.

Figure 4 — $V$ , $V^{2}$ , $V^{3}$ ki speed ke against growth. Blue line $V$ seedha uthta hai, yellow $V^{2}$ bend karta hai, pink $V^{3}$ rocket kar jaata hai. Do vertical dotted lines speed $1$ aur speed $2$ par hain; pink curve unke beech follow karo aur arrow padho: $V$ double karne par $V^{3}$ $2^{3} = 8$ se multiply ho jaata hai. Woh aath-guna jump — pink curve ko axis par $1$ se $8$ tak leapta hua dikhao — bilkul isliye speed Chapman equation mein har doosre factor ko dominate karti hai.

9. Yeh formula kab allowed hai? (Regime & assumptions)

Common mistake Edge cases jahan formula toot jaata hai

Bahut patli hawa (high rarefied altitude): atmosphere ke top par "continuum" assumption fail ho jaati hai — proper boundary layer banane ke liye molecules bahut kam ho jaate hain, aur yeh equation over- ya under-predict karta hai. Kinetic (molecular) methods ki zaroorat padti hai.
Sharp nose ( $R_{n} \to 0$ ): formula $\overset{q}{˙}_{s} \to \infty$ deta hai jab $R_{n} \to 0$ . Woh divergence math ki cheeekh hai ki sharp nose mein unsurvivable stagnation flux hai — physically tip simply pighal jaati hai, bilkul isliye blunt bodies use kiye jaate hain.
Turbulent boundary layer (high $R e$ ): jab layer turbulence mein trip karti hai laminar assumption fail ho jaati hai aur real heating is estimate se zyada hoti hai.
Bahut high speed ( $V ≳ 10$ km/s): shocked gas glow karne lagti hai, radiative heating add hoti hai jise yeh convective formula bilkul ignore karta hai. Radiative heating at hypersonic speeds dekho — aur note karo ki radiative flux $R_{n}$ ke saath badhti hai, convective trend se ladhte hue.

10. Sab kuch kaise link hota hai — prerequisite map

Top-down padho: speed aur density raw inputs hain; woh boundary layer ke across stagnation point par milke heat flux banate hain; powers aur constant $C$ phir sab kuch Chapman equation mein package karte hain.

Equipment checklist

Right-hand side cover karo aur khud ko test karo. Agar koi bhi answer fuzzy ho, parent note touch karne se pehle woh section dobara padho.

V

ka matlab kya hai aur uske units kya hain?

Hawa ke relative vehicle ki speed,

m/s

mein; reentry par ~7–11 km/s (rifle bullet se lagbhag 9–14 guna).

Mass

m

kya hai aur heating formula mein yeh kyun nahi hota?

kg mein matter ki matra; jab kinetic energy per kilogram li jaati hai toh cancel ho jaata hai, sirf

\frac{1}{2} V^{2}

bachta hai.

Kinetic energy

\frac{1}{2} V^{2}

kyun hai

\frac{1}{2} V

nahi?

Motion energy speed par do baar depend karti hai (punch × stopping distance), square deta hai; specific (per-kg) form mass drop kar deta hai.

ρ_{\infty}

par subscript

\infty

tumhe kya use karne kehta hai?

Current altitude par door, undisturbed free-stream density — nose ke bilkul paas wali density nahi.

ρ_{e}

aur

u_{e}

par subscripts

e

ka kya matlab hai?

Boundary layer ke outer edge par values (post-shock), jahan gas still-air skin ke top ke saath bahti hai.

Density

ρ

ke roop mein kyun aati hai

ρ

ki jagah?

Zyada density edge mass flux

ρ_{e} u_{e}

add karti hai (flux uthata hai) lekin boundary layer bhi patla karta hai (neeche kheenchta hai); balance flux

\propto ρ_{e} u_{e} \propto ρ_{\infty}

chhod deta hai.

Stagnation point kya hai?

Nose ka woh single point jahan flow poori tarah rest mein aati hai; vehicle par sabse garm, highest-pressure spot.

R_{n}

kya hai aur bada

R_{n}

heating kyun ghatata hai?

Fitted nose-sphere radius; bada

R_{n}

gentle gradient deta hai → moti boundary layer

δ \propto R_{n}

→ flux

\propto R_{n}^{- 1/2}

Boundary layer kya hai?

Wall ke paas hawa ki patli sheet jahan flow speed zero ho jaati hai; heat ko surface tak pahunchne ke liye ise cross karna padta hai.

Heat flux

\overset{q}{˙}

kya hai aur uske units kya hain?

Unit area par landing power,

W/m^{2}

; dot "rate" matlab, /m² "per area" matlab.

V

par exponent teen kyun hai, do nahi?

Per kg energy content

V^{2}

hai; delivery rate (edge mass flux,

u_{e} \propto V

ke zariye) ek aur

V

add karta hai, deta hai

V^{3}

V^{3}

physically speed double karne ke liye kya imply karta hai?

2^{3} = 8

guna heating — speed dominate karti hai.

C

ke units kya hain aur woh dimensionless kyun nahi hai?

kg^{1/2} m^{- 1}

(SI); isko units carry karni padti hain taaki physical quantities

W/m^{2}

mein convert ho sakein, aur yeh gas properties chhupata hai.

Gas ke baare mein

C

ke andar kaunse do hidden assumptions rehte hain?

Ek fixed adiabatic recovery factor aur roughly constant specific heat

c_{p}

(recovery/stagnation-temperature assumptions).

Mach number kya hai, aur flow "hypersonic" kab hoti hai?

M = V / a

(speed ÷ speed of sound); hypersonic matlab

M ≳ 5

; reentry

M \approx 20

–30 hai.

Reynolds number kya hai aur yahan kyun matter karta hai?

R e = ρ_{e} u_{e} L / μ

, inertia vs viscosity; low

R e

= laminar (formula valid), high

R e

= turbulent (formula under-predicts).

Do conditions batao jab Chapman formula fail karta hai.

Bahut upar rarefied (non-continuum) hawa, aur bahut high speeds jahan radiative heating dominate kare (saath hi sharp noses aur turbulent, high-

R e

layers bhi).