Question bank — Converging nozzle — subsonic flow, Mach 1 at exit

False — ek converging duct subsonic gas ko sirf $M=1$ tak accelerate karta hai; mass-flow factor $\dot{m}(M)$ bilkul $M=1$ par peak karta hai, isliye $p_0$ badhane se flow scale up hoti hai lekin $M$ kabhi 1 se aage nahi jaata. Supersonic jaane ke liye area ko baad mein increase karna hoga (de Laval nozzle).

False — choked ka matlab hai $p_e=p^{\ast }=0.528p_0$, jo nozzle set karta hai, room nahi. $p^{\ast }$ se $p_b$ tak ka baaki expansion bahar expansion fans ke roop mein hota hai, isliye $p_e$ usually $p_b$ se upar hoti hai.

True — ek subsonic jet upstream pressure signals bhej sakta hai, isliye ye continuously adjust karta hai jab tak $p_e=p_b$ na ho jaye. Yahi wajah hai ki subsonic regime mein flow room ko "feel" karta hai.

False — $p_b/p_0=0.528$ se neeche flow choked hai aur $\dot{m}$ flat hai. Sonic exit kisi bhi pressure signal ko upstream travel karne se rokta hai, isliye reservoir kabhi nahi "sunta" ki $p_b$ giri.

$p_0$ badhao, toh mass flow badhti hai. True — $\dot{m}=A{p}_0\sqrt{\gamma /(R{T}_0)}\times (\text{factor at }M=1)$ seedha $p_0$ ke proportional hai. Choking Mach number ko cap karta hai, mass flow ko nahi; zyada $p_0$ ka matlab hai same sonic speed par denser gas, isliye zyada $\dot{m}$.

$\sqrt{\gamma R{T}_0}$ ke barabar hoti hai. False — exit speed local sound speed hai $a^{\ast }=\sqrt{\gamma R{T}^{\ast }}$, aur $T^{\ast }=0.833T_0<T_0$. Kyunki gas accelerate hote waqt thandi ho gayi, $a^{\ast }$ reservoir sound speed se chhoti hai.

False — kuch pile nahi hota; flow smooth aur steady hai. "Choking" ka matlab hai information channel cut ho jaata hai: $M=1$ par pressure waves flow ko outrun nahi kar sakti aur upstream travel nahi kar sakti, isliye downstream changes $\dot{m}$ ko influence karna band kar deti hain.

$\gamma =5/3$) ke liye critical ratio ab bhi 0.528 hai. False — 0.528 specifically $\gamma =1.4$ ke liye hai. $\gamma =5/3$ ko ${\left(\frac{2}{\gamma +1}\right)}^{\gamma /(\gamma -1)}$ mein plug karne par $\approx 0.487$ milta hai, isliye critical ratio gas par depend karta hai.

$p_b/p_0=0.30<0.528$, isliye exit pressure $0.30p_0$ hai." Error: exit pressure yahan $p_b$ nahi hai. Kyunki $0.30<0.528$ nozzle choked hai, isliye $p_e=0.528p_0$ hai, aur gas nozzle ke bahar baki expansion karta hai.

$M=1$ par hai, isliye gas speed of sound par nikalta hai jo reservoir mein measure hoti hai, $T_0=300$ K ke liye lagbhag 347 m/s." Error: $M=1$ local sound speed use karta hai thande exit par ($T^{\ast }=250$ K), jo $a^{\ast }=\sqrt{1.4\cdot 287\cdot 250}\approx 317$ m/s deta hai, reservoir value 347 m/s nahi.

$p_b$ kam karne ka kisi bhi cheez par zero effect hota hai." Error: nozzle ke andar aur $\dot{m}$ par zero effect hota hai, lekin ye bahar flow ko zaroor change karta hai — kam $p_b$ ka matlab hai under-expanded jet aur bhi zyada strongly downstream expand karta hai. Sirf internal state frozen hai.

$p_0/p_e=500/101$ set kiya aur isentropic relation solve ki." Error: ye tabhi kaam karta hai jab unchoked ho. Yahan $101/500=0.202<0.528$, isliye choked hai aur $M=1$ automatically hai; $p_e=p_b=101$ kPa ko relation mein use karne par ek fictitious supersonic $M$ milta hai jo converging duct reach nahi kar sakta.

$\dot{m}$ nahi badlega." Error: $\dot{m}\propto p_0/\sqrt{T_0}$, isliye $T_0$ badhane se $\dot{m}$ kam hoti hai (fixed $p_0$ par hotter gas kam dense hoti hai). Area akela lever nahi hai.

Error: internal nozzle flow (idealized) isentropic hai, lekin external under-expanded jet mein expansion fans aur possibly shocks hoti hain — nozzle ke bahar tum blindly isentropy assume nahi kar sakte (shock case ke liye Normal Shock Waves dekho).

$M=1$ kyun use karta hai, kisi aur Mach number kyun nahi? Kyunki $M=1$ wahan hai jahan pressure signals bilkul flow speed par travel karti hain — "downstream upstream se baat kar sakta hai" aur "nahi kar sakta" ke beech ki boundary. Yahan $\dot{m}(M)$ bhi peak karta hai, isliye ye converging duct ki natural physical ceiling hai.

zyada kyun hoti hai, kam kyun nahi? Nozzle $p_e$ ko $p^{\ast }=0.528p_0$ par lock karta hai kyunki sonic exit low $p_b$ ko andar relay nahi kar sakta. Isliye gas room ke relative over-pressured nikalta hai aur bahar expand karna finish karta hai — ise under-expanded jet kehte hain.

$V$ ko Mach number $M$ mein convert karne ki zarurat kyun hai? Kyunki special physics — sound-speed signalling, choking — $M=1$ par live karta hai, ek dimensionless threshold. $V=Ma$ likhna aur $a=\sqrt{\gamma RT}$ temperature ko story mein fold karta hai aur choking condition ko clean "$M=1$" banata hai, ek messy velocity value ki jagah.

Subsonic flow apni sound waves se slower hai, isliye pressure disturbances upstream travel karke reservoir ko inform kar sakti hain. $M=1$ par wo waves bilkul flow speed par move karti hain aur downstream sweep ho jaati hain — upstream side causally cut off ho jaata hai.

Steady subsonic flow ke liye, mass conservation ($\rho AV=\text{const}$) mein mild density change ke saath $A$ kam hone par $V$ force se badh jaata hai. Isentropic Flow Relations dikhata hai ki ye trend supersonic hone par reverse ho jaata hai — yahi wajah hai ki converging duct $M=1$ se aage nahi push kar sakta.

$T_0$ badhane se choked mass flow kyun kam hoti hai, halanki gas hotter aur faster hai? Fixed $p_0$ par hotter reservoir gas kam dense hoti hai, aur extra sonic speed compensate nahi karta: $\dot{m}\propto p_0/\sqrt{T_0}$. Density sound speed se zyada tezi se girta hai, isliye $\dot{m}$ drop karta hai.

$p_b>p_0$ (back pressure reservoir pressure se upar)? Koi forward push nahi hota — pressure gradient galat direction mein point karta hai. Gas reservoir mein ulta flow karne ka tendency rakhti hai, ya simply rukti hai; nozzle zero (ya reversed) flow deliver karta hai jab tak $p_b$ $p_0$ se neeche na gire.

$p_b=p_0$ exactly? Koi pressure difference nahi, isliye koi flow nahi: $V=0$, $M=0$, aur "exit" sirf reservoir hai. $\dot{m}=0$.

$p_b/p_0=0.528$ par (exact critical ratio)? Exit bas $M=1$ tak pahunchta hai aur $p_e=p_b=p^{\ast }$ simultaneously. Ye ek aisa single point hai jahan "unchoked, $p_e=p_b$" aur "choked, $p_e=p^{\ast }$" agree karte hain — dono regimes ke beech ki seam.

$p_b\to 0$ (vacuum mein exhaust karte hue)? Flow choked rehti hai $M=1$ par, $p_e=0.528p_0$ andar, aur $\dot{m}$ apni ceiling par rehti hai. Jet bahar aur bhi zyada violently expand karta hai, lekin nozzle ke andar kuch nahi badlta.

$p_b/p_0=1^{-}$ (1 se thoda neeche)? Ek tiny $M$ zero se thoda upar — flow barely trickle karta hai aur deeply subsonic hai, $p_e=p_b$ ke saath. Tum no-flow limit ko continuously approach karte ho.

$p_e<p^{\ast }$ ho sakta hai? Nahi — sirf converging duct ke andar exit $p^{\ast }=0.528p_0$ se neeche nahi ja sakta, kyunki iske liye $M>1$ chahiye hoga, jo geometry forbid karti hai. Sub-critical exit pressures sirf bahar ya diverging section mein hoti hain.

finite tank hai (infinite nahi), toh choked hone par time ke saath kya badlta hai? Jaise jaise gas drain hoti hai, $p_0$ (aur $T_0$) girta hai, isliye $\dot{m}\propto p_0/\sqrt{T_0}$ dhire dhire decrease hoti hai. Nozzle choked rehta hai jab tak $p_b/p_0\le 0.528$ hai, lekin "constant" mass flow tabhi constant hai jab $p_0$ constant ho.

Answers cover karo: kisi bhi $p_b/p_0$ ke liye tumhe instantly bolna chahiye (a) choked hai ya nahi, (b) $p_e$ kya hai, aur (c) exit par $M$ kya hai. Agar tum teeno kar sako, toh ye bank tumhare saath done hai.