Converging nozzle — subsonic flow, Mach 1 at exit

3.1.9 · Physics › Compressible Flow & Aerodynamics

WHAT ho raha hai (setup)

HOW gas behave karti hai jab hum $p_{b}$ lower karte hain:

$p_{b} = p_{0}$ : koi flow nahi.
$p_{b}$ thoda lower karo: flow shuru hota hai, exit subsonic hai, $p_{e} = p_{b}$ (exit back pressure "feel" karta hai).
Lower karte raho: exit accelerate karta hai. Ek special $p_{b} = p^{*}$ par, exit $M = 1$ hit karta hai. Choked.
$p_{b}$ aur lower karo: andar kuch nahi badalta! Exit $M = 1$ par rehta hai, $p_{e}$ $p^{*}$ par rehta hai. Extra expansion nozzle ke bahar hoti hai (flow ko "pata nahi" ki $p_{b}$ gira, kyunki signals Mach 1 par wapas travel nahi kar sakte).

Figure — Converging nozzle — subsonic flow, Mach 1 at exit

Choking condition scratch se derive karna

Hum isentropic (adiabatic + reversible) steady flow of a perfect gas use karte hain. Do conservation laws + thermodynamics.

Flashcards

Purely converging nozzle ke exit par maximum Mach number kya hai?

M = 1

(yeh choke ho jaata hai; supersonic nahi ja sakta).

Critical pressure ratio formula

p^{*} / p_{0}

(\frac{2}{γ + 1})^{γ / (γ - 1)}

Air ke liye numerical critical pressure ratio (

γ = 1.4

\approx 0.528

Choked flow define karo.

Exit

M = 1

reach karta hai aur mass flow maximize hoti hai; back pressure aur lower karne se

\overset{m}{˙}

nahi badh sakta.

Jab choked ho, exit pressure kya hota hai?

p_{e} = p^{*} = 0.528 p_{0}

(nozzle fix karta hai,

p_{b}

nahi).

Jab unchoked ho (subsonic exit), exit pressure kya hota hai?

p_{e} = p_{b}

(back pressure se match karta hai).

Converging duct mein flow

M = 1

kyun exceed nahi kar sakta?

Pressure signals sound speed par travel karti hain;

M = 1

par woh upstream propagate nahi kar sakti, isliye upstream flow ko speed up hone ke liye kaha nahi ja sakta — aur

\overset{m}{˙} (M)

M = 1

par peak karta hai.

Stagnation-to-static temperature relation?

T_{0} / T = 1 + \frac{γ - 1}{2} M^{2}

Master isentropic pressure relation?

p_{0} / p = (1 + \frac{γ - 1}{2} M^{2})^{γ / (γ - 1)}

T_{0} = 300

K (air) ke saath choke par, exit temperature?

T^{*} = \frac{2}{2.4} \times 300 = 250

Recall Feynman: 12-saal ke bache ko samjhao

Socho ek bheed ek narrow hoti corridor se guzar rahi hai. Jaise tighter hoti hai, log tez chalte hain. Lekin ek top speed hai: woh speed jis par "aage bado!" ki awaaz bheed mein wapas travel kar sakti hai — sound ki speed. Jab front line utni tez move kare, toh bahar se awaaz andar wapas nahi pahunch sakti, toh chahe aage ka room kitna bhi khali ho, har second utne hi log nikalte hain. Corridor "choked" ho gaya. Aur tez jaane ke liye tumhe corridor ko squeeze ke baad phir se chauda karna hoga — woh alag (de Laval) nozzle hai.

Connections

Isentropic Flow Relations — master equations jo har step mein use hui hain.
Speed of Sound — $a = γ R T$ define karta hai aur kyun $M = 1$ special hai.
Converging-Diverging (de Laval) Nozzle — actually supersonic jaane ke liye tumhe yahi chahiye.
Mass Flow Rate & Choking — $\overset{m}{˙} (M)$ maximization.
Stagnation Properties — reservoir conditions $p_{0}, T_{0}, ρ_{0}$ .
Normal Shock Waves — jets ki over-expansion/under-expansion aage kahan le jaati hai.

Converging nozzle — subsonic flow, Mach 1 at exit

WHAT ho raha hai (setup)

Choking condition scratch se derive karna

Worked examples

Common mistakes

Flashcards

Connections

Concept Map