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

3.1.8 · D5 · Physics › Compressible Flow & Aerodynamics › Choked flow — condition M = 1 at throat, maximum mass flow

True or false — justify

Choking ka matlab hamesha ye hota hai ki nozzle mein kahin supersonic flow hai.

False. Purely converging nozzle mein, choking ka matlab hai throat par

M = 1

aur upstream har jagah subsonic — nozzle ke andar kuch bhi supersonic nahi hai.

Ek baar converging nozzle choke ho jaye, to back pressure

p_{b}

kam karne se throat pressure bhi kam hoti hai.

False. Throat pressure

p^{*} = 0.528 p_{0}

par lock ho jaati hai (air ke liye). Lower

p_{b}

ke signals sonic throat ke past upstream travel nahi kar sakte, isliye throat us change ko "sun" hi nahi sakta.

Ek hotter reservoir (zyada

T_{0}

) choked throat se zyada mass push karta hai.

False.

\overset{m}{˙}_{m a x} \propto 1/ T_{0}

: hotter gas kam dense hoti hai, aur density ka loss sound speed mein

T

ki rise se zyada hota hai, isliye

\overset{m}{˙}

girti hai.

Reservoir pressure

p_{0}

double karne se choked mass flow bhi double ho jaati hai.

True.

\overset{m}{˙}_{m a x} \propto p_{0}

linearly (through

ρ_{0} = p_{0} / R T_{0}

), jab

T_{0}, A^{*}, γ, R

fixed hon.

Choking condition par,

M = 1

mass flow ko per unit throat area maximize karta hai.

True.

d \overset{m}{˙} / d M = 0

set karne par (

A

fixed ke saath)

M^{2} = 1

milta hai; woh stationary point flux function

f (M)

ka maximum hai.

Critical pressure ratio

0.528

sabhi gases ke liye ek universal constant hai.

False. Ye

(\frac{2}{γ + 1})^{γ / (γ - 1)}

hai, jo

γ

par depend karta hai.

0.528

specifically air ke liye hai (

γ = 1.4

); monatomic gases (

γ = 5/3

) ke liye alag value aati hai.

Jab choked ho aur bahut kam

p_{b}

par exhaust ho raha ho, to exit jet under-expanded hoti hai.

True. Exit pressure

p^{*} > p_{b}

par rehti hai, isliye gas nozzle ke bahar expand hoti rehti hai, aur shock diamonds ke saath ek under-expanded jet banati hai.

Choked flow isentropic nahi hoti kyunki wo "stuck" hai.

False. Nozzle ke andar throat tak flow isentropic rehti hai; choking ek condition hai, koi loss mechanism nahi. Irreversibility (shocks) downstream appear ho sakti hai, choking se nahi.

Why questions

Downstream pressure signals sonic throat mein wapas kyun nahi travel kar sakti?

Pressure disturbances local sound speed

a

par travel karti hain.

M = 1

par flow downstream

a

par move karti hai, isliye upstream-jaane wale signal ki net upstream speed zero hoti hai — wo throat par freeze ho jaata hai aur reservoir tak kabhi nahi pahunchta.

Maximum mass flow exactly

M = 1

par kyun hoti hai, kisi aur Mach number par kyun nahi?

Mass-flux function

f (M) = M / (1 + \frac{γ - 1}{2} M^{2})^{(γ + 1) /2 (γ - 1)}

ka apna stationary point sirf

M^{2} = 1

par hai; wahin

ρ V

per unit area sabse zyada hai, rising velocity aur falling density ke beech balance karke.

Nozzle choke hone ke baad throat pressure "lock" kyun ho jaati hai?

Kyunki throat

p_{b}

se isolated hai (koi upstream signal nahi), uski state sirf reservoir se fix hoti hai:

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

, downstream kisi bhi cheez se independent.

Supersonic flow ke liye throat ke baad diverging section kyun chahiye?

M = 1

hone ke baad, Area–Mach Relation (A/A*) dikhata hai ki aage acceleration (

M > 1

) ke liye increasing area chahiye. Sirf ek diverging duct ye supply kar sakta hai, isliye de Laval nozzles converging–diverging hote hain.

\overset{m}{˙}_{m a x}

p_{0}

ke proportional kyun hai lekin ratio

p^{*} / p_{0}

p_{0}

se independent kyun hai?

Critical ratio isentropic relation se

M = 1

par evaluate hoti hai, jisme sirf

γ

involve hota hai. Absolute

\overset{m}{˙}

reservoir density

ρ_{0} \propto p_{0}

ke saath scale karta hai, isliye

p_{0}

badhane se flow linearly badhti hai jabki ratio unchanged rehta hai.

Under-expanded jet ka extra expansion mass flow kyun nahi badhata?

Woh expansion nozzle ke bahar hoti hai, sonic throat ke downstream. Mass conservation

\overset{m}{˙}

ko throat par fix karta hai;

M = 1

ke downstream kuch bhi feed back karke use change nahi kar sakta.

Edge cases

Exactly critical back-pressure ratio

p_{b} / p_{0} = 0.528

(air) par

\overset{m}{˙}

ka kya hota hai?

Ye choking ki onset hai: throat abhi abhi

M = 1

reach karta hai aur

\overset{m}{˙}

abhi abhi apne maximum par pahunchti hai. Ye boundary hai — ek taraf subsonic-everywhere, doosri taraf choked-and-frozen.

Agar

p_{b} / p_{0}

0.528

se thoda zyada hai, to kya nozzle choked hai?

Nahi. Flow poori tarah subsonic hai, exit pressure

p_{b}

se match karti hai, aur

\overset{m}{˙}

abhi bhi back pressure par respond karti hai (jaise jaise kam karo, badhti hai).

Degenerate limit

p_{b} = p_{0}

(koi pressure difference nahi) mein mass flow kitni hogi?

Zero. Koi pressure drop nahi to koi driving force nahi,

M = 0

har jagah, aur

f (0) = 0

— flux curve ka far-left end, choked maximum se bilkul opposite.

Ek converging nozzle ke liye limit

p_{b} \to 0

(perfect vacuum downstream) mein

\overset{m}{˙}

kitni hai?

Phir bhi exactly

\overset{m}{˙}_{m a x}

— same as

p_{b} = 0.528 p_{0}

par. Nozzle

0.528

par choke hoti hai aur frozen rehti hai; vacuum sirf external jet ko zyada under-expanded banata hai, flow ko bada nahi karta.

Monatomic gas (

γ = 5/3

) versus air (

γ = 1.4

) ke liye, kya choking abhi bhi

M = 1

par hoti hai?

Haan —

M^{2} = 1

result

f (M)

ko kisi bhi

γ > 1

ke liye maximize karne se aata hai. Sirf critical ratios (

p^{*} / p_{0}

T^{*} / T_{0}

)

γ

ke saath change hote hain, sonic condition nahi.

Fully supersonic run karne wale de Laval nozzle mein, kya throat abhi bhi choke point hai?

Haan. Throat abhi bhi

M = 1

par baitha hai aur abhi bhi

\overset{m}{˙}_{m a x}

set karta hai. Diverging section exit Mach number change karta hai, lekin mass flow sonic throat se fix hoti hai.

Agar "throat" area actual minimum nahi hai (jaise upstream mein ek aur bhi narrow spot hai)?

Actual minimum area wo throat hai jo pehle choke hoti hai;

M = 1

wahin hota hai, aur wahi section — na ki geometric label —

\overset{m}{˙}_{m a x}

control karta hai.

Recall Self-test summary

Choked ::: throat par $M = 1$ ; given $p_{0}, T_{0}, A^{*}$ ke liye $\overset{m}{˙}$ apne maximum par. Throat pressure when choked (air) ::: $p^{*} = 0.528 p_{0}$ par locked, $p_{b}$ se independent. Effect of hotter reservoir ::: kam mass flow, kyunki $\overset{m}{˙}_{m a x} \propto 1/ T_{0}$ . Supersonic flow requires ::: converging–diverging (de Laval) nozzle, plain converging nahi.

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

True or false — justify

Spot the error

Why questions

Edge cases