Converging nozzle — subsonic flow, Mach 1 at exit

WHAT is going on (the setup)

HOW the gas behaves as we lower $p_b$ :

$p_b = p_0$ : no flow.
Lower $p_b$ a bit: flow starts, exit is subsonic, $p_e = p_b$ (exit "feels" the back pressure).
Keep lowering: exit accelerates. At a special $p_b = p^*$ , exit hits $M=1$ . Choked.
Lower $p_b$ further: nothing changes inside! Exit stays $M=1$ , $p_e$ stays at $p^*$ . The extra expansion happens outside the nozzle (the flow "doesn't know" $p_b$ dropped, because signals can't travel back in at Mach 1).

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

Deriving the choking condition from scratch

We use isentropic (adiabatic + reversible) steady flow of a perfect gas. Two conservation laws + thermodynamics.

Formula Step 1 — Energy conservation (stagnation temperature)

Steady adiabatic flow: enthalpy + kinetic energy is conserved along a streamline: $h + \tfrac{1}{2}V^2 = h_0 \quad\Rightarrow\quad c_p T + \tfrac12 V^2 = c_p T_0$ Why? No heat added, no shaft work, so total energy per unit mass is constant. $h_0$ is the value when $V=0$ (the reservoir).

Divide by $c_p T$ and use $V^2 = M^2 a^2 = M^2 \gamma R T$ and $c_p = \frac{\gamma R}{\gamma-1}$ : $\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2$ Why this step? We replaced velocity $V$ by Mach number $M$ using $a=\sqrt{\gamma RT}$ , the local speed of sound. Now everything is in terms of $M$ .

Worked examples

Common mistakes

Flashcards

What is the maximum Mach number at the exit of a purely converging nozzle?

M=1

(it chokes; it cannot go supersonic).

Critical pressure ratio formula

p^*/p_0

\left(\dfrac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}

Numerical critical pressure ratio for air (

\gamma=1.4

\approx 0.528

Define choked flow.

Exit reaches

M=1

and mass flow is maximized; lowering back pressure further cannot increase

\dot m

When choked, what is the exit pressure?

p_e = p^* = 0.528\,p_0

(fixed by nozzle, not by

p_b

When unchoked (subsonic exit), what is the exit pressure?

p_e = p_b

(matches back pressure).

Why can't the flow exceed

M=1

in a converging duct?

Pressure signals travel at sound speed; at

M=1

they can't propagate upstream, so the upstream flow can't be told to speed up — and

\dot m(M)

peaks at

M=1

Stagnation-to-static temperature relation?

T_0/T = 1+\frac{\gamma-1}{2}M^2

Master isentropic pressure relation?

p_0/p = \left(1+\frac{\gamma-1}{2}M^2\right)^{\gamma/(\gamma-1)}

At choke with

T_0=300

K (air), exit temperature?

T^*=\frac{2}{2.4}\times300=250

Recall Feynman: explain to a 12-year-old

Imagine a crowd squeezing through a narrowing corridor. As it gets tighter, people walk faster. But there's a top speed: the speed at which "move up!" shouts can travel back through the crowd — the speed of sound. Once the front line moves that fast, a shout from outside can't reach back in, so no matter how empty the room ahead is, the same number of people pour out each second. The corridor is "choked." To go even faster you'd need the corridor to widen again after the squeeze — that's a different (de Laval) nozzle.

Connections

Isentropic Flow Relations — the master equations used in every step.
Speed of Sound — defines $a=\sqrt{\gamma RT}$ and why $M=1$ is special.
Converging-Diverging (de Laval) Nozzle — what you need to actually go supersonic.
Mass Flow Rate & Choking — the $\dot m(M)$ maximization.
Stagnation Properties — reservoir conditions $p_0,T_0,\rho_0$ .
Normal Shock Waves — where over-expansion/under-expansion of jets leads next.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, converging nozzle ek aisi pipe hai jo aage jaake patli hoti jaati hai. Jaise highway pe lane kam ho jaayein toh gaadi tezz daudti hai, waise hi gas patli jagah pe accelerate karti hai. Lekin ek limit hai: exit pe gas zyada se zyada Mach 1 (sound ki speed) tak pahunch sakti hai — usse aage nahi. Converging-only nozzle me supersonic flow possible hi nahi.

Iske peeche reason simple hai: pressure ki "khabar" sound ki speed se travel karti hai. Jab exit pe flow exactly sound ki speed (M=1) ho jaata hai, tab downstream (bahar) ki koi bhi pressure change upstream tak signal nahi bhej paati. Isi ko hum choked flow kehte hain. Choke hone ke baad, chahe back pressure $p_b$ kitna bhi kam kar do, mass flow $\dot m$ aur nahi badhega — woh apni maximum value pe lock ho jaata hai.

Magic number yaad rakho: air ke liye critical pressure ratio $p^*/p_0 = 0.528$ . Matlab agar $p_b/p_0$ 0.528 se neeche chala gaya, toh nozzle choke ho gaya, exit Mach 1 fix, aur exit pressure $p_e = 0.528\,p_0$ ho jaata hai (back pressure ke barabar nahi!). Agar ratio 0.528 se upar hai, toh flow subsonic hai aur exit pressure simply $p_e = p_b$ ho jaata hai.

Yeh derive kaise hua? Bas do cheezein: energy conservation ( $c_pT + \frac12 V^2 = c_pT_0$ ) aur isentropic relation ( $p/\rho^\gamma$ = const). Inko mila ke master formula banta hai $p_0/p = (1+\frac{\gamma-1}{2}M^2)^{\gamma/(\gamma-1)}$ . Bas $M=1$ daal do, aur 0.528 aa jaata hai. Real life me yeh rocket nozzles, jet engines aur gas flow meters me directly use hota hai — choking ko samajhna must hai.

Converging nozzle — subsonic flow, Mach 1 at exit

WHAT is going on (the setup)

Deriving the choking condition from scratch

Worked examples

Common mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Compressible Flow & Aerodynamics

Connections