Area-Mach number relation A - A - = f(M) — isentropic flow

3.1.6 · Physics › Compressible Flow & Aerodynamics

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$A^{\ast }$ actually hai kya?

Famous shape fact note karo (baad mein steel-man kiya gaya hai):

• Subsonic flow ($M<1$): speed up karne ke liye, area decrease honi chahiye (converge).
• Supersonic flow ($M>1$): speed up karne ke liye, area increase honi chahiye (diverge).
• Throat par ($A=A^{\ast }$, minimum area): $M=1$.

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First principles se derivation

Hum 1-D steady isentropic flow ke teen pillars use karte hain.

Pillar 1 — Mass conservation (continuity): $\dot{m}=\rho AV=\text{const}$

Pillar 2 — $A^{\ast }$ ki definition: $\dot{m}$ ko sonic point par evaluate karo jahan $\rho ={\rho }^{\ast },V=V^{\ast }=a^{\ast },A=A^{\ast }$: $\rho AV={\rho }^{\ast }{A}^{\ast }{a}^{\ast }$

Area ratio isolate karne ke liye rearrange karo: $\boxed{\frac{A}{A^{\ast }}=\frac{{\rho }^{\ast }{a}^{\ast }}{\rho V}}$

Pillar 3 — Isentropic + stagnation relations. Stagnation $T_0,{\rho }_0$ use karte hue:

\frac{\rho_0}{\rho}=\left(1+\frac{\gamma-1}{2}M^2\right)^{\frac{1}{\gamma-1}}$$ Ratio ko stagnation state ke through split karo: $$\frac{\rho^*}{\rho}=\frac{\rho^*/\rho_0}{\rho/\rho_0} =\frac{\left(1+\frac{\gamma-1}{2}\right)^{-\frac{1}{\gamma-1}}} {\left(1+\frac{\gamma-1}{2}M^2\right)^{-\frac{1}{\gamma-1}}}$$ **Speed** term ke liye, $V=Ma$ aur $a^*=1\cdot a^*$ likho, jahan $a=\sqrt{\gamma R T}$: $$\frac{a^*}{V}=\frac{a^*}{M a}=\frac{1}{M}\sqrt{\frac{T^*}{T}} =\frac{1}{M}\sqrt{\frac{T^*/T_0}{T/T_0}} =\frac{1}{M}\sqrt{\frac{\left(1+\frac{\gamma-1}{2}\right)^{-1}}{\left(1+\frac{\gamma-1}{2}M^2\right)^{-1}}}$$ > [!intuition] Yeh step kyun? > $V$ dono par depend karta hai — *yahan sound kitni fast chalti hai* ($a\propto\sqrt T$) > aur *kitne Mach* ($M$). Toh hum $a^*/V$ ko ek "temperature" piece (stagnation relations se > handle) aur ek explicit $1/M$ piece mein split karte hain. Sab kuch $M$ aur $\gamma$ par > reduce ho jaata hai. **$\dfrac{\rho^*}{\rho}\cdot\dfrac{a^*}{V}$ multiply karo.** Do $(1+\frac{\gamma-1}{2})$ constants combine ho jaate hain; exponents collect karne par $\left(-\frac{1}{\gamma-1}+\frac12\right)=-\frac{\gamma+1}{2(\gamma-1)}$ se clean result milta hai: > [!formula] Area–Mach relation > $$\left(\frac{A}{A^*}\right)^{2}=\frac{1}{M^{2}} > \left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{\gamma+1}{\gamma-1}}$$ > ya equivalently > $$\frac{A}{A^*}=\frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^{2}\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$$ > Key checks: $M=1$ par, $A/A^*=1$. Jab $M\to0$ ya $M\to\infty$, toh $A/A^*\to\infty$. ![[3.1.06-Area-Mach-number-relation-A-A-=-f(M)-—-isentropic-flow.png]] > [!intuition] Curve padhna (Dual coding) > Plot ek **U-shaped valley** hai jiska minimum $A/A^*=1$ hai $M=1$ par. Ek crucial baat > dhyan do: $A/A^* > 1$ ki har value curve ko **do** Mach numbers par hit karti hai — ek > subsonic, ek supersonic. **Sirf geometry decide nahi karti kaun sa branch hai.** Boundary/ > back-pressure conditions branch pick karti hain (subsonic everywhere, ya choked throat > ke baad supersonic). --- ## Worked examples > [!example] Example 1 — Area se Mach (pehle forecast karo) > Ek converging–diverging nozzle ka throat area $A^*=10\,\text{cm}^2$ hai aur exit area > $A_e=20\,\text{cm}^2$ hai, $\gamma=1.4$. Exit Mach number(s) nikalo. > > **Forecast:** $A_e/A^*=2$, comfortably $>1$ ⇒ DO solutions expect karo, ek $\approx 0.3$, > ek $\approx 2.2$. > > $2=\frac{1}{M}\left[\frac{1}{1.2}\left(1+0.2M^2\right)\right]^{3}$ numerically solve karo. > **Yeh step kyun?** $f(M)=A/A^*$ closed form mein invertible nahi hai, isliye iterate/tables > use karte hain. Roots: $M\approx 0.306$ (subsonic) aur $M\approx 2.197$ (supersonic). ✔ forecast se match karta hai. > [!example] Example 2 — $M$ se Properties > Supersonic root $M=2.197$ lo, $T_0=300\ \text{K}$, $p_0=500\ \text{kPa}$. > $$\frac{T}{T_0}=\frac{1}{1+0.2(2.197)^2}=\frac{1}{1.965}\Rightarrow T=152.7\ \text{K}$$ > $$\frac{p}{p_0}=\left(\frac{T}{T_0}\right)^{\gamma/(\gamma-1)}=(0.509)^{3.5}=0.0939 > \Rightarrow p=46.9\ \text{kPa}$$ > **Yeh step kyun?** Jab $M$ pata ho, toh *har* thermodynamic property isentropic stagnation > relations se milti hai — yahi $A/A^*$ se $M$ nikaalny ka faayda hai. > [!example] Example 3 — Choked throat se Mass flow > Reservoir $p_0=500$ kPa, $T_0=300$ K, $A^*=10\,\text{cm}^2=10^{-3}\,\text{m}^2$, air > ($R=287$, $\gamma=1.4$). Choked mass flow: > $$\dot m=\frac{p_0 A^*}{\sqrt{T_0}}\sqrt{\frac{\gamma}{R}} > \left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{2(\gamma-1)}}$$ > Yahan $\frac{2}{\gamma+1}=\frac{2}{2.4}=0.8333$ aur exponent $\frac{\gamma+1}{2(\gamma-1)}=3$ hai, > toh bracket factor $(0.8333)^{3}\approx 0.579$ hai: > $$=\frac{500000\cdot10^{-3}}{\sqrt{300}}\sqrt{\frac{1.4}{287}}\,(0.579) > \approx 1.16\ \text{kg/s}$$ > **Yeh step kyun?** Jab throat sonic ho ($A=A^*$), toh $\dot m$ reservoir conditions se > *fix* ho jaata hai — downstream pressure kam karne se zyada air pump nahi ho sakti (nozzle > **choked** ho jaata hai). --- ## Common mistakes (steel-manned) > [!mistake] "Choti area ka matlab hamesha fast flow hota hai." > **Kyun sahi lagta hai:** garden-hose / kitchen-tap ka experience — dabao, paani nikle. > Yeh **incompressible/subsonic** flow ke liye sach hai. > **Fix:** momentum + continuity combo deta hai $\frac{dA}{A}=(M^2-1)\frac{dV}{V}$. > $M>1$ ke liye sign flip ho jaata hai: supersonic flow accelerate karne ke liye area > **diverge** karni padti hai. > [!mistake] "$A/A^*$ ek unique Mach number deta hai." > **Kyun sahi lagta hai:** $f(M)$ ek function *lagta hai* isliye invert karna unique hona chahiye. > **Fix:** $f$ **U-shaped** hai, monotonic nahi. Har $A/A^*>1$ ka ek subsonic AUR ek > supersonic root hota hai; operating back-pressure decide karti hai kaun sa physically hota hai. > [!mistake] "$A^*$ ek real physical throat hai jo duct mein hamesha exist karti hai." > **Kyun sahi lagta hai:** CD nozzle mein throat *choked hone par* sonic hoti hai. > **Fix:** $A^*$ ek *reference* area hai. Purely subsonic flow ke liye actual minimum area > kabhi $M=1$ reach nahi kar sakti, phir bhi $A^*$ mathematically constant yardstick ke > roop mein exist karta hai. --- > [!recall]- Feynman: 12-saal ke bachche ko explain karo > Shaped pipe mein flow ho rahi air ek crowd jaisi hai jo ek hallway se guzar rahi hai jo > narrow aur wide hoti rehti hai. Sabse narrow spot **throat** hai. Ek magic speed hai > (speed of sound) jo crowd exactly throat par reach kar sakti hai. Formula ek chart hai: > bolo hall kitni wide hai us special narrow spot ke comparison mein, aur yeh batayega > log kitni fast chal rahe hain. Funny baat: same width ka matlab ho sakta hai ek *slow > polite crowd* ya ek *super-fast sprinting crowd* — tumhe pata hona chahiye ki woh throat > ke baad speed up hue ya nahi. > [!mnemonic] Yaad rakho > **"Sub Shrinks to Speed, Super Spreads to Speed; throat = ONE."** > Subsonic area shrink karke accelerate karta hai, Supersonic area spread karke accelerate > karta hai, aur sonic point ($M=1$) woh single minimum hai jahan $A/A^*=1$ hota hai. --- ## Active recall #flashcards/physics $A^*$ physically kya represent karta hai? ::: Woh (constant) area jis par flow exactly $M=1$ hogi same mass flow aur stagnation state ke liye — ek sonic reference yardstick. Area–Mach relation state karo. ::: $\frac{A}{A^*}=\frac1M\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$ $M=1$ par $A/A^*$ kya hota hai? ::: Exactly 1 (curve ka minimum). Har $A/A^*>1$ do Mach numbers kyun deta hai? ::: Function U-shaped hai (non-monotonic): ek subsonic root, ek supersonic root. Ise derive karne ke liye kaunse do conservation laws + relations use hote hain? ::: Continuity ($\rho A V=$const), $M=1$ par $A^*$ ki definition, aur $\rho/\rho_0$, $T/T_0$ ke liye isentropic stagnation relations. Supersonic flow ke liye accelerate karne ke liye area kaise change honi chahiye? ::: Area INCREASE (diverge) honi chahiye, kyunki $dA/A=(M^2-1)\,dV/V$. "Choked" ka matlab kya hai? ::: Throat $M=1$ reach karta hai; mass flow reservoir $p_0,T_0,A^*$ se fix ho jaati hai aur back pressure kam karne se increase nahi ho sakti. $M\to0$ aur $M\to\infty$ par $A/A^*$ ka limit kya hai? ::: Dono $\to\infty$. --- ## Connections - [[Isentropic Stagnation Relations]] — $\rho/\rho_0$, $T/T_0$, $p/p_0$ provide karta hai. - [[Continuity Equation 1-D Compressible Flow]] — derivation ka Pillar 1. - [[Area-Velocity Relation dA-A = (M^2-1) dV-V]] — shape behavior explain karta hai. - [[Choked Flow & Maximum Mass Flow]] — $M=1$ throat limit. - [[Converging-Diverging (de Laval) Nozzle]] — primary application. - [[Normal Shock Waves]] — kyun supersonic branch downstream persist nahi kar sakti. - [[Speed of Sound a = sqrt(gamma R T)]] — $M$ aur $a^*$ define karta hai. ## 🖼️ Concept Map ```mermaid flowchart TD MC[Mass conservation rho A V const] AS[Sonic reference area A*] AR[Area ratio A/A* = rho* a* / rho V] ISEN[Isentropic + stagnation relations] DR[rho*/rho from stagnation state] SP[a*/V split into 1/M and temperature] REL[A/A* = f of M and gamma] SUB[Subsonic M<1: converge to speed up] SUP[Supersonic M>1: diverge to speed up] THR[Throat A=A* gives M=1] PROPS[Local p, T, rho, V] MC -->|evaluate at sonic point| AS AS -->|rearrange| AR ISEN -->|gives rho*/rho| DR ISEN -->|gives a*/V| SP AR -->|substitute DR| REL DR --> REL SP --> REL REL -->|predicts| SUB REL -->|predicts| SUP REL -->|minimum area| THR REL -->|solve for M then| PROPS ```