Aerodynamic heating — recovery temperature, heat flux: Foundations

3.1.28 · D1 · Physics › Compressible Flow & Aerodynamics › Aerodynamic heating — recovery temperature, heat flux

Hum parent note ki pehli equation padhne se pehle bhi, humein ek toolbox chahiye. Yeh page har symbol ko — bilkul zero se — ek aisi order mein banata hai jahan har naya idea sirf pehle se explain hue ideas par tikta hai.

1. Speed, aur "fast" ka measure — Mach number $M$

Topic ko yeh kyun chahiye. Slow flows mein hum heating ignore kar sakte hain. Lekin jaise hi $M$ badhta hai, hawa itni tezi se "raaste se hat" nahi paati, woh pile up ho jaati hai aur compress hoti hai — aur compression ka matlab hai heating. Toh $M$ woh single dial hai jo decide karta hai ki heating kitni violent hogi. Picture dekho: ek slow stream mein wahi wall thodi si garam hoti hai; ek fast stream mein pile-up, squashed hawa chamakti hai.

2. Temperature $T$ , aur iske do flavours: static vs stagnation

Topic ko yeh kyun chahiye. Ek vehicle apni surface par hawa ko rok leta hai, isliye wall ke paas ki hawa $T$ se zyada $T_{0}$ ke kareeb hoti hai. $T$ aur $T_{0}$ ke beech ka gap bilkul wahi heating budget hai. Subscript convention: $T_{e}$ ka matlab hai "wall ke paas ki thin layer ki edge par $T$ " — essentially woh free-stream static temperature jo vehicle dekhta hai.

3. Energy bookkeeping ka tool — enthalpy $h$ aur $c_{p}$

Yeh tool kyun aur plain energy kyun nahi? Aise flow ke liye jo bahar se heat na gain kare aur na turbine spin kare, woh quantity jo streamline ke along constant rehti hai woh enthalpy + kinetic energy hai, sirf internal energy nahi. Isliye parent likhta hai $c_{p} T + \frac{u ^{2}}{2} = c_{p} T_{0}$ — yeh do-jar accounting exactly hai. $h$ choose karna (aur $c_{p} T$ shortcut ke saath) messy energy balance ko ek-line temperature equation mein badal deta hai.

4. Gas ki personality — $γ$ aur $R$

Topic ko yeh kyun chahiye. Do identities sab kuch ek saath jodhti hain: $c_{p} = \frac{γ R}{γ - 1}, a^{2} = γ R T$ Pehli humein $c_{p}$ ko rewrite karne deti hai; doosri speed ko Mach number mein badal deti hai. Dono ko substitute karna hi woh process hai jis se parent $T_{0} = T (1 + \frac{u ^{2}}{2 c _{p} T})$ se clean $\frac{T _{0}}{T} = 1 + \frac{γ - 1}{2} M^{2}$ tak jaata hai. $γ$ aur $R$ ke bina hum $u$ ko $M$ se swap nahi kar sakte.

5. Thin warm skin — boundary layer

Topic ko yeh kyun chahiye. Boundary layer woh stage hai jahan recovery temperature aur heat flux khelti hain. Iske do rival processes — heat banna (dissipation) aur heat sideways leak hona (conduction) — wall temperature decide karte hain, jaisa ki agla symbol quantify karta hai.

6. Tug-of-war number — Prandtl number $P r$ aur recovery factor $r$

Topic ko yeh kyun chahiye. $r$ woh correction hai jo theoretical maximum $T_{0}$ ko real recovery temperature $T_{r}$ mein badalta hai jis par wall settle hoti hai. Dekho Prandtl number & thermal boundary layer.

7. Wall ki apni temperature $T_{w}$ aur heat flux $\overset{q}{˙}_{w}$

$T_{w} < T_{r}$ → heat wall ke andar jaati hai ( $\overset{q}{˙}_{w} > 0$ ).
$T_{w} = T_{r}$ → zero flux; yahi $T_{r}$ ko define karta hai ("adiabatic wall").
$T_{w} > T_{r}$ → wall heat gas ko de deti hai.

Driver $T_{r}$ kyun hai, $T_{0}$ ya $T_{e}$ kyun nahi. $T_{r}$ woh temperature hai jahan ek uncooled wall genuinely drift karti hai. Sirf $T_{r} - T_{w} = 0$ us wall ke liye zero heating deta hai, isliye $T_{r}$ honest thermal potential hai. Dekho Reynolds analogy & Stanton number ki engineers $h$ kaise nikaalte hain.

8. Prerequisite map

Ise upar se neeche padho: speeds aur temperatures $T_{0}$ banate hain; boundary layer aur $P r$ $r$ banate hain; woh dono $T_{r}$ banate hain; aur $T_{r}$ wall temperature ke saath heat flux banata hai.

Equipment checklist

Right side cover karo aur dekho ki kya aap reveal karne se pehle har cheez recite kar sakte ho.

Mach number

M

ka matlab

gas apni khud ki awaaz se kitni baar zyada tez chal rahi hai,

M = u / a

Static temperature

T

vs stagnation

T_{0}

T

= sirf molecular jiggling;

T_{0}

= woh temperature jab gas ko rokkar saari streaming motion heat mein badal di jaaye.

Enthalpy

h = c_{p} T

sahi bookkeeping variable kyun hai

kyunki

h + \frac{u ^{2}}{2}

adiabatic no-work flow mein conserved hota hai, ek-line temperature relation deta hai.

γ

aur

R

use karne wali do identities

c_{p} = γ R / (γ - 1)

aur

a^{2} = γ R T

; saath mein yeh speed ko Mach number mein convert karte hain.

No-slip condition

air speed wall par exactly zero tak girta hai, surface se chipak jaata hai.

Viscous dissipation

paas-paas ki air layers ek doosre se guzarti aur ragadti hain, boundary layer ke andar motion ko heat mein convert karte hue.

Prandtl number

P r = μ c_{p} / k

compare karta hai ki momentum kitni tezi se spread hota hai vs heat kitni tezi se spread hoti hai; air ke liye

\approx 0.71

r < 1

kyun

kyunki

P r < 1

, heat momentum se tezi se leak hoti hai, isliye wall poore

T_{0}

rise se kam recover karti hai.

Recovery factor values

laminar

r = P r \approx 0.84

; turbulent

r = P r^{1/3} \approx 0.89

Heat flux

\overset{q}{˙}_{w} = h (T_{r} - T_{w})

, aur iska driver

heat per second per square metre;

T_{r} - T_{w}

se driven kyunki

T_{r}

wahan hai jahan ek uncooled wall settle hoti hai.

Connections

Parent: Aerodynamic heating — jahan yeh saare symbols kaam aate hain.
Stagnation properties & isentropic relations — $T_{0} / T = 1 + \frac{γ - 1}{2} M^{2}$ ka ghar.
Boundary layers & viscous dissipation — woh stage jahan heat banti hai.
Prandtl number & thermal boundary layer — $r$ $P r$ par kyun depend karta hai.
Reynolds analogy & Stanton number — $h$ kaise estimate kiya jaata hai.
Hypersonic re-entry & thermal protection systems — extreme application.
Normal & oblique shock heating — woh $T_{e}$ set karta hai jo in formulas ko feed karta hai.