Choked flow — condition M = 1 at throat, maximum mass flow

3.1.8 · Physics › Compressible Flow & Aerodynamics

1. Choking kya hai? (WHAT)

Throat woh jagah hai jahan $A$ minimum hota hai. Yeh key fact hum prove karenge: sirf $M = 1$ par hi throat maximum possible mass flow carry kar sakta hai.

2. Building blocks (HOW hum ise set up karte hain)

Hum isentropic, steady, 1-D flow of a perfect gas use karte hain, ratio of specific heats $γ$ , gas constant $R$ . Stagnation (reservoir) conditions $p_{0}, T_{0}, ρ_{0}$ fixed hain.

$T_{0} / T$ step kyun? $h_{0} = h + \frac{1}{2} V^{2}$ ko $c_{p} T$ se divide karo: $\frac{T _{0}}{T} = 1 + \frac{V ^{2}}{2 c _{p} T}$ . Ab $\frac{V ^{2}}{2 c _{p} T} = \frac{M ^{2} γ R T}{2 c _{p} T} = \frac{M ^{2} γ R}{2 c _{p}} = \frac{γ - 1}{2} M^{2}$ . Done — pure algebra, koi memorization nahi.

3. Mass flow derive karna (DERIVATION FROM SCRATCH)

Area $A$ se mass flow: $\overset{m}{˙} = ρ A V$ . Har cheez ko $M$ aur stagnation quantities mein express karo.

Step 1 — $ρ$ , $V$ ko stagnation terms mein likho.

$ρ = ρ_{0} (1 + \frac{γ - 1}{2} M^{2})^{- 1/ (γ - 1)}$
$V = M a = M γ R T$ , aur $T = T_{0} (1 + \frac{γ - 1}{2} M^{2})^{- 1}$ , to $V = M γ R T_{0} (1 + \frac{γ - 1}{2} M^{2})^{- 1/2}$ .

Yeh step kyun? Hum $\overset{m}{˙}$ ko sirf $M$ ka function banana chahte hain (fixed $p_{0}, T_{0}$ ke liye) taaki hum dhundh sakein yeh kahan maximize hota hai.

Step 2 — combine karo. $ρ_{0} = p_{0} / (R T_{0})$ use karke:

$\overset{m}{˙} = \frac{A p _{0}}{T _{0}} \frac{γ}{R} \frac{M}{( 1 + \frac{γ - 1}{2} M ^{2} ) ^{\frac{γ + 1}{2 ( γ - 1 )}}}$

Yeh step kyun? Saari $M$ -dependence ab ek neat factor $f (M) = \frac{M}{( 1 + \frac{γ - 1}{2} M ^{2} ) ^{(γ + 1) /2 (γ - 1)}}$ mein aa gayi hai.

Step 3 — fixed throat area par $f (M)$ ko $M$ ke upar maximize karo. $\frac{df}{d M} = 0$ set karo. $ln f = ln M - \frac{γ + 1}{2 ( γ - 1 )} ln (1 + \frac{γ - 1}{2} M^{2})$ lekar aur differentiate karke:

$\frac{1}{M} - \frac{γ + 1}{2 ( γ - 1 )} \cdot \frac{( γ - 1 ) M}{1 + \frac{γ - 1}{2} M ^{2}} = 0$

$\frac{1}{M} = \frac{( γ + 1 ) M /2}{1 + \frac{γ - 1}{2} M ^{2}} \Rightarrow 1 + \frac{γ - 1}{2} M^{2} = \frac{γ + 1}{2} M^{2} \Rightarrow M^{2} = 1.$

4. Critical (sonic, "starred") conditions

$M = 1$ ko isentropic relations mein plug karo taaki critical ratios mile — choked hone par throat par conditions:

Figure — Choked flow — condition M = 1 at throat, maximum mass flow

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active-recall flashcards

#flashcards/physics

Choked nozzle ke throat par kaun sa Mach number hota hai?

M = 1

(sonic).

Choked hone ke baad mass flow kyon badhna band ho jaata hai?

Throat sonic hai; pressure signals

M = 1

se upstream nahi travel kar sakti, to throat lower back pressure "feel" nahi kar sakta.

Air ke liye critical pressure ratio

p^{*} / p_{0}

(

γ = 1.4

\approx 0.528

Air ke liye critical temperature ratio

T^{*} / T_{0}

\approx 0.833

(

= 2/ (γ + 1)

General formula

T^{*} / T_{0}

\frac{2}{γ + 1}

General formula

p^{*} / p_{0}

(\frac{2}{γ + 1})^{γ / (γ - 1)}

Kis

M

par

\overset{m}{˙}

per unit throat area maximize hoti hai?

M = 1

par (jahan

d \overset{m}{˙} / d M = 0

hota hai).

\overset{m}{˙}_{m a x}

p_{0}

T_{0}

A^{*}

ke saath kaise scale hoti hai?

\propto p_{0}

\propto A^{*}

\propto 1/ T_{0}

Choked-flow max mass-flow formula?

\overset{m}{˙}_{m a x} = A^{*} p_{0} γ / (R T_{0}) (2/ (γ + 1))^{(γ + 1) /2 (γ - 1)}

Supersonic exit ke liye kaisi nozzle shape chahiye?

Converging–diverging (de Laval).

Choked hone par, ambient tak remaining expansion kahan hoti hai?

Nozzle ke bahar (under-expanded jet).

Recall Feynman: 12-saal ke bacche ko explain karo

Socho tum toothpaste ko ek tube ke sabse patley hisse se squeeze kar rahe ho. Jaise jaise tum door wale end par zyada suction lagate ho, zyada nikalta hai — kuch der ke liye. Lekin jab patley spot ki toothpaste utni tez chal rahi ho jitni tez usme ek "message" travel kar sakta hai (speed of sound), tab woh patla spot tumhari sucking ko sunna band kar deta hai. To chahe tum kitna bhi kheencho, utni hi quantity nikalti rehti hai. Woh "sound message jitni tez" speed $M = 1$ hai, aur woh steady stuck quantity maximum mass flow hai.

Connections

Isentropic Flow Relations — $T_{0} / T$ , $p_{0} / p$ ratios supply karta hai jo yahan use hue.
Area–Mach Relation (A/A*) — kyun area wahan minimum hona chahiye jahan $M = 1$ ho.
Converging–Diverging (de Laval) Nozzle — choked throat ko supersonic exit tak extend karta hai.
Speed of Sound & Mach Number — kyun $M = 1$ upstream signaling block karta hai.
Normal Shock Waves — diverging section mein kya hota hai agar back pressure "galat" ho.
Stagnation (Total) Properties — $p_{0}, T_{0}, ρ_{0}$ ki definition.