Converging-diverging (de Laval) nozzle — subsonic, supersonic flow: Question bank

3.1.10 · D5 · Physics › Compressible Flow & Aerodynamics › Converging-diverging (de Laval) nozzle — subsonic, supersoni

Shuru karne se pehle symbols ka reminder (taaki koi line bina bataye term use na kare):

$A$ = tube ka ek point par cross-sectional area; $d A$ = downstream move karne par usmein chhota sa change.
$V$ = flow speed; $d V$ = speed mein chhota sa change.
$ρ$ = density; $a$ = local speed of sound; $T$ = local temperature.
$γ$ = ratio of specific heats $c_{p} / c_{v}$ gas ka (air ke liye lagbhag $1.4$ ); yeh $a = γ R T$ mein aur pressure/temperature ratios mein aata hai ( $R$ = specific gas constant).
$M = V / a$ = Mach number, flow speed ko local sound speed $a$ ki units mein measure kiya gaya.
$\overset{m}{˙} = ρ A V$ = har second mein flow hone wala mass. continuity ke zariye tube ke saath constant rehta hai.
Throat = sabse narrow point (area minimum, $d A = 0$ ).
Starred quantities ( $A^{*}, ρ^{*}, a^{*}, p^{*}, T^{*}$ ) = sonic condition $M = 1$ par values, yaani jab throat choked ho. Toh $A^{*}$ throat (sonic) area hai, $p^{*}$ throat pressure hai, aur aise hi aage.
$A_{e}$ = nozzle ka exit area (diverging section ke far end par opening); ratio $A_{e} / A^{*}$ exit ko throat se compare karta hai.
$p_{0}$ = stagnation (reservoir) pressure = tank mein gas ke move karne se pehle pressure, jahan $V = 0$ . Likewise $T_{0}$ stagnation temperature hai. Jaise-jaise gas accelerate hoti hai uski static pressure $p_{0}$ se neeche girती hai.
$p_{b}$ = back pressure = nozzle exit par environment ka pressure.

Kyunki yeh bank $(M^{2} - 1)$ ke sign aur hourglass geometry pe lean karta hai, reveals se pehle teen quick sketches sab kuch anchor kar dete hain:

Figure — Converging-diverging (de Laval) nozzle — subsonic, supersonic flow

True or false — justify

True or false: Tube ko narrow karna hamesha gas ko speed up karta hai.

False. Narrowing sirf subsonic flow ko speed up karta hai (

M < 1

M > 1

ke liye factor

(M^{2} - 1)

positive hai, isliye accelerate karne ke liye

d A > 0

chahiye — tube ko widen karna padta hai.

True or false: Properly running de Laval nozzle mein gas ki sabse fast speed throat par hoti hai.

False. Throat par exactly

M = 1

hota hai. Gas diverging section mein accelerate karti rehti hai, isliye sabse zyada speed exit par hoti hai.

True or false: Sonic flow (

M = 1

) nozzle mein kahin bhi ho sakta hai agar pressure sahi set karo.

False. Master equation force karta hai ki

M = 1

par

d A = 0

ho, isliye

M = 1

sirf local area minimum par hi baith sakta hai — yaani throat par.

True or false: Ek baar throat choked ho jaye, harder sucking (lower

p_{b}

) se zyada mass aata hai.

False.

M = 1

par throat pressure signals ko upstream travel karne se rokta hai, isliye

\overset{m}{˙}

apne maximum

\overset{m}{˙}_{ma x} = ρ^{*} A^{*} a^{*}

par lock ho jaata hai (density, area aur sound speed sab sonic throat par evaluate kiye gaye). Dekho Choked Flow & Mass Flow Limit.

True or false: Ek converging-only nozzle pressure drop kaafi bada ho toh supersonic exhaust produce kar sakta hai.

False. Uska sabse chhota area exit hai, jahan best case mein

M = 1

hota hai.

M > 1

cross karne ke liye diverging section chahiye (

d A > 0

), jo uske paas hai nahi.

True or false: Diye gaye diverging area ratio

A / A^{*} > 1

ke liye exactly ek hi possible flow hai.

False. Area-ratio relation ke do solutions hote hain: ek subsonic (decelerating) aur ek supersonic (accelerating). Back pressure decide karta hai ki Nature kaunsa choose karegi.

True or false: Area–velocity relation derive karte waqt assume kiya gaya tha ki friction aur heat loss chhote hain.

True. Yeh steady, 1-D, isentropic flow assume karta hai — koi friction nahi, koi heat transfer nahi. Isliye yeh Euler momentum equation use karta hai, koi viscous wali nahi.

True or false: Diverging section ke andar normal shock flow ko supersonic bana deta hai.

False. Normal shock supersonic se subsonic tak achanak jump hai; yeh flow ko abruptly decelerate karta hai, kabhi accelerate nahi karta.

Spot the error

Error: "

\frac{d A}{A} = (M^{2} - 1) \frac{d V}{V}

, toh

M = 1

par velocity change nahi ho sakti."

Error

d V

aur

d A

ko confuse karne mein hai.

M = 1

par equation deta hai

d A = 0

(area stationary hai), lekin

d V

phir bhi nonzero ho sakta hai — gas throat ke ek dum through accelerate karti rehti hai.

Error: "Supersonic jaane ke liye tube ko narrowing karte raho taaki kabhi rukne na pao."

M > 1

ke liye yeh galat sign hai. Continued narrowing (

d A < 0

) jab

M > 1

ho toh gas ko decelerate karega. Throat ke baad tube ko diverge karna padta hai.

Error: "Mass flow

\overset{m}{˙} = ρ A V

bina limit ke badhta rehta hai jaise-jaise hum

p_{b}

girate hain."

Ek baar throat

M = 1

reach kar le toh flow choked hai aur

\overset{m}{˙}

fix ho jaata hai. Formula sahi hai; yeh claim ki yeh badhta rehta hai throat par sonic gate ko ignore karta hai.

Error: "Density essentially constant hai, isliye area ko shrink karna hi gas ko har jagah speed up karta hai."

Yeh sirf low

M

ke liye sahi hai.

M > 1

ke liye density diverging section mein area ke badhne se tezi se girti hai; woh density collapse hi hai jo

V

ko upar force karta hai taaki

ρ A V

constant rahe.

Error: "Throat pressure reservoir pressure

p_{0}

ke barabar hoti hai."

Nahi —

p_{0}

stagnation (zero-speed) pressure hai. Choked throat par pressure girar ke

p^{*} \approx 0.528 p_{0}

ho jaata hai (

γ = 1.4

ke liye), kyunki gas ne speed pakad li hai aur uski static pressure gir gayi hai.

Error: "Design-matched nozzle ke exit par exit pressure reservoir pressure ke barabar honi chahiye."

Design condition par exit pressure back pressure

p_{b}

ke barabar hoti hai, jo

p_{0}

se kaafi neeche hai.

p_{0}

se match karna matlab hoga ki gas kabhi accelerate hi nahi hui.

Error: "Kyunki

a = γ R T

aur

T

downstream girti hai, sound speed badhti hai, isliye

M

automatically drop ho jaata hai."

Jaise gas accelerate aur cool hoti hai,

a

girti zaroor hai — lekin

V

bahut tezi se badhta hai, isliye

M = V / a

diverging section mein downstream badhta hai.

Why questions

(M^{2} - 1)

ka sign poora behaviour kyun palat deta hai?

Yeh

\frac{d V}{V}

ko multiply karta hai:

M < 1

ke liye yeh negative hai (accelerate karne ke liye area shrink chahiye),

M > 1

ke liye yeh positive hai (accelerate karne ke liye area badhna chahiye). Ek sign change poora subsonic/supersonic reversal hai.

M = 1

exactly throat par kyun rehna chahiye aur kahin nahi?

M = 1

par factor

(M^{2} - 1) = 0

hota hai, jo

d A = 0

force karta hai. Woh point jahan

d A = 0

hai aur area phir badhta hai woh area minimum hai — yahi throat ki definition hai.

Ek baar throat choked ho jaye toh downstream ka pressure signal reservoir tak kyun nahi pahunch sakta?

Pressure signals local sound speed

a

se gas ke relative travel karte hain. Throat par gas $a$ par move kar rahi hai (

M = 1

), isliye upstream jaane ki koshish karta signal ruk jaata hai — woh aage nahi nikal sakta. Dekho Speed of Sound in a Gas.

Rockets ko diverging bell kyun chahiye, sirf converging cone nahi?

Converging cone sirf apne exit par

M = 1

reach kar sakta hai. Thrust ke liye high exit velocity chahiye, isliye

M ≫ 1

chahiye, jiske liye supersonic acceleration ke liye diverging section ka hona zaroori hai taaki

d A > 0

rahe. Dekho Rocket Propulsion & Thrust.

Back pressure ko choking se neeche girane par throat ke upstream kya hota hai?

Converging section aur throat apne choked state mein already hain; sonic throat unhe isolate karta hai. Throat ke downstream ka flow (shocks, expansion) hi further

p_{b}

drop pe respond karta hai.

A / A^{*}

kabhi 1 se kam kyun nahi hota?

A^{*}

sonic (minimum) area hai. Relation ki minimum value exactly 1 hai

M = 1

par; koi bhi real area throat se

\geq

hai, isliye ratio dono branches ke liye

\geq 1

hai.

Edge cases

Edge case: Agar

p_{b}

exactly

p_{0}

ke barabar set kiya jaye toh kaunsa flow hoga?

Koi pressure difference nahi matlab koi driving force nahi — gas bilkul move nahi karti,

V = 0

\overset{m}{˙} = 0

. Nozzle idle hai.

Edge case: Converging-only nozzle apne exit par sabse zyada kitna

M

produce kar sakta hai?

Exactly

M = 1

(sonic), jo sirf tab reach hota hai jab woh choked ho. Yeh physically 1 se zyada nahi ho sakta kyunki

d A > 0

satisfy karne ke liye koi diverging area nahi hai.

Edge case: Agar

p_{b}

choking value se thoda sa neeche ho, toh supersonic flow kahan khatam hoti hai?

Gas throat ke thoda baad briefly supersonic hoti hai, phir diverging part mein ek normal shock khada ho jaata hai, jo use wapas subsonic kar deta hai exit se pehle.

p_{b}

girane se shock aur downstream chala jaata hai.

Edge case: Jaise

p_{b}

design point se kaafi neeche girti hai (under-expanded) toh

\overset{m}{˙}

ka kya hota hai?

Kuch nahi —

\overset{m}{˙}

\overset{m}{˙}_{ma x}

par fixed rehta hai kyunki throat abhi bhi choked hai. Sirf nozzle ke bahar ka plume badalta hai (expansion waves appear hoti hain).

Edge case: Limiting case

M \to 0

mein (lagbhag stationary gas), area–velocity relation kis cheez mein reduce ho jaata hai?

(M^{2} - 1) \to - 1

, jisse milta hai

\frac{d A}{A} \approx - \frac{d V}{V}

: speed badhane ke liye area proportionally shrink karo — plain incompressible venturi behaviour, jaisa classic nozzle theory predict karta hai.

Edge case: Agar diverging section bahut zyada lamba ho (bada

A_{e} / A^{*}

) diye gaye

p_{b}

ke liye, toh kya hoga?

Flow over-expand ho jaata hai: pressure

p_{b}

se neeche gir jaata hai, toh environment wapas push karta hai aur pressure ko

p_{b}

se match karne ke liye andar ya exit par ek shock (ya shock system) banta hai.

Recall Poore bank ke liye one-line memory hook

Sign of $(M^{2} - 1)$ sab kuch control karta hai ::: Mach 1 se neeche minus (speed badhane ke liye squeeze karo), Mach 1 se upar plus (speed badhane ke liye widen karo), zero sirf throat par.