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

1. What is choking? (WHAT)

The throat is where $A$ is minimum. The key fact we'll prove: only at $M=1$ can the throat carry the maximum possible mass flow.

2. Building blocks (HOW we set it up)

We use isentropic, steady, 1-D flow of a perfect gas, ratio of specific heats $\gamma$, gas constant $R$. Stagnation (reservoir) conditions $p_0, T_0, \rho_0$ are fixed.

Why the $T_0/T$ step? Divide $h_0=h+\frac12V^2$ by $c_pT$: $\frac{T_0}{T}=1+\frac{V^2}{2c_pT}$. Now $\frac{V^2}{2c_pT}=\frac{M^2\gamma RT}{2c_pT}=\frac{M^2\gamma R}{2c_p}=\frac{\gamma-1}{2}M^2$. Done — pure algebra, no memorization.

3. Deriving the mass flow (DERIVATION FROM SCRATCH)

Mass flow through area $A$: $\dot{m}=\rho A V$. Express everything in $M$ and stagnation quantities.

Step 1 — write $\rho$, $V$ in stagnation terms.

• $\rho = \rho_0\left(1+\frac{\gamma-1}{2}M^2\right)^{-1/(\gamma-1)}$
• $V = Ma = M\sqrt{\gamma RT}$, and $T = T_0\left(1+\frac{\gamma-1}{2}M^2\right)^{-1}$, so $V = M\sqrt{\gamma R T_0}\left(1+\frac{\gamma-1}{2}M^2\right)^{-1/2}$.

Why this step? We want $\dot m$ as a function of $M$ alone (for fixed $p_0,T_0$) so we can find where it maxes out.

Step 2 — combine. Using $\rho_0 = p_0/(RT_0)$:

$\boxed{\;\dot{m} = \frac{A\,p_0}{\sqrt{T_0}}\sqrt{\frac{\gamma}{R}}\;\frac{M}{\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{\gamma+1}{2(\gamma-1)}}}\;}$

Why this step? All the $M$-dependence is now in one tidy factor $f(M)=\dfrac{M}{(1+\frac{\gamma-1}{2}M^2)^{(\gamma+1)/2(\gamma-1)}}$.

Step 3 — maximize $f(M)$ over $M$ at fixed throat area. Set $\dfrac{df}{dM}=0$. Taking $\ln f = \ln M - \frac{\gamma+1}{2(\gamma-1)}\ln(1+\frac{\gamma-1}{2}M^2)$ and differentiating:

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

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

4. Critical (sonic, "starred") conditions

Plug $M=1$ into the isentropic relations to get the critical ratios — the conditions at the throat when choked:

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active-recall flashcards

#flashcards/physics

Connections

• Isentropic Flow Relations — supplies the $T_0/T$, $p_0/p$ ratios used here.
• Area–Mach Relation (A/A*) — why area must be minimum where $M=1$.
• Converging–Diverging (de Laval) Nozzle — extends choked throat to supersonic exit.
• Speed of Sound & Mach Number — why $M=1$ blocks upstream signaling.
• Normal Shock Waves — what happens in the diverging section if back pressure is "wrong".
• Stagnation (Total) Properties — definition of $p_0, T_0, \rho_0$.

Concept Map

Hinglish (regional understanding)