De Laval nozzle geometry — conical, bell (Rao contour), 80% bell: Question bank

3.3.17 · D5 · Physics › Rocket Propulsion › De Laval nozzle geometry — conical, bell (Rao contour), 80%

Yeh parent topic ka misconception-hunting companion hai. Neeche har item ek one-line reveal hai: prompt padho, apna jawab zor se socho, phir check karo. Answers mein reasoning hoti hai, kabhi bare "yes/no" nahi — kyunki trap hamesha why mein hota hai.

Woh picture aur symbols jo hum use karte hain

Kisi bhi trap se pehle, hum drawing par har symbol build karte hain taaki kuch bhi assumed na ho. Figure 1 ek seedhi-deewar wali conical nozzle hai jo beech se kati hui hai; Figure 2 usi tarah ki ek bell nozzle hai taaki aap shapes ko side by side compare kar sako.

Figure — De Laval nozzle geometry — conical, bell (Rao contour), 80% bell

Figure 1 ko left se right padhte hue:

Axis (dashed gray) centerline hai — jis direction mein thrust push karta hai.
Sabse sankra point throat hai; uska radius $R_{t}$ (green) hai. Yahan gas ki speed of sound tak pahunchti hai.
Chauda opening exit hai; uska radius $R_{e}$ (orange) hai.
Throat se exit tak axis ke saath seedhi doori length $L$ (blue bracket) hai.
Deewar ek fixed half-angle $α$ (red) par uthti hai — yeh deewar aur axis ke beech ka angle hai. "Half" isliye kyunki cone ka poora opening angle $2 α$ hota hai.

Is triangle se seedhe do facts nikalte hain, aur hum unhe derive karte hain taaki koi formula hawa mein se na aaye:

Ab exit-flow angle $θ_{e}$ : yeh gas velocity aur axis ke beech ka angle hai jis waqt gas bahar nikalti hai. Cone mein gas deewar ko follow karti hai, toh $θ_{e} = α$ . Bell mein (Figure 2) deewar exit ke paas wapas mudi hui hai, toh gas almost seedhi nikalti hai aur $θ_{e}$ chhota hota hai — yahi curve ka poora point hai.

Divergence factor $λ$ kahan se aata hai

Humne claim kiya tha $λ = \frac{1 + c o s α}{2}$ ; yeh raha kyun, Figure 3 se jod ke.

Picture aur dono formulas earn karne ke baad, neeche ke traps spring karna safe hai.

True or false — justify karo

Half-angle

α = 0

wali conical nozzle mein

λ = 1

hoga

True —

λ = \frac{1 + c o s 0}{2} = 1

ke saath, lekin

α = 0

matlab ek seedhi tube jo kabhi expand hi nahi karti, toh supersonic exhaust produce karne ke liye yeh useless hai; ideal factor real hai, geometry degenerate hai.

Ek bell nozzle exactly

λ = 1

reach kar sakti hai

Practice mein False — ek real bell

λ \approx 0.98

–

0.995

deta hai kyunki finite length par kisi shock ke bina flow ko poori tarah axial karna impossible hai; sirf theoretically infinite ideal contour exactly

1

reach karta hai.

Ek 80% bell, full bell se 80% efficient hai

False — "80%" length ko refer karta hai (

L = 0.8 L_{cone}

), efficiency ko nahi; iska

λ \approx 0.985

full bell ke

\approx 0.99

ke lagbhag utna hi hai.

"Full Rao contour" (100% bell) apne equivalent cone jaisi length rakhta hai

True — "100% bell" ka matlab hai iska length

L

, same

ϵ

wale

1 5^{\circ}

cone ki length ke barabar hai; 80% bell woh reference length ka

0.8

hai, toh "%" hamesha reference cone ke against length measure karta hai, na ki kisi lambe bell ke against.

Cone half-angle

α

badhane se nozzle hamesha choti ho jaati hai

Length ke liye True lekin cost ke saath — bada

α

matlab

tan α

bada hota hai toh

L = R_{t} (ϵ - 1) / tan α

chhota hota hai, phir bhi

cos α

girta hai toh divergence loss badhta hai; length aur efficiency ulti disha mein khenchte hain.

Same expansion ratio wali do nozzles ki exit velocity ek jesi hogi

False — same

ϵ

ideal Mach number fix karta hai, lekin wall shape decide karta hai ki kitna divergence aur shocks ki wajah se loss hoga, toh same

ϵ

wala bell, cone se zyada usable axial velocity deliver karta hai.

Same

ϵ

aur efficiency wali conical nozzle se bell nozzle unconditionally halki hoti hai

Ek blanket rule ke roop mein False — bell ki choti wall mass bachati hai, lekin curved contour ko aksar mooti walls, stiffening rings, ya seedhe cone se zyada mahanga fabrication chahiye, toh net mass structural aur manufacturing choices par depend karta hai, sirf length par nahi.

Divergence factor

λ

gas properties par depend karta hai

Geometric factor ke liye False —

λ = \frac{1 + c o s α}{2}

sirf exit-flow angle par depend karta hai; gas properties

v_{e}

aur

\overset{m}{˙}

ko affect karti hain, jo

λ

ko multiply karte hain lekin use change nahi karte.

Error dhundho

"60% bell sabse best hai kyunki yeh sabse compact hai."

Error

4%

thrust loss (

λ \approx 0.96

) ko ignore karta hai; compactness akeli objective nahi hai — 80% bell standard hai kyunki yeh

\approx 98.5%

efficiency rakhta hai jabki cone se

20%

chota bhi hota hai.

"80% bell ka exit angle

0^{\circ}

hai, toh iska zero divergence loss hai."

Ek real 80% bell

θ_{e} \approx 5

–

1 0^{\circ}

par exit karta hai,

0^{\circ}

par nahi; flow almost axial hai, ek chhota residual loss deta hai — sirf theoretical full Rao contour exactly axial exit aim karta hai.

"Lambi nozzle → kam exit pressure → hamesha zyada thrust."

Sirf tab tak jab exit pressure ambient ke barabar ho; usse aage nozzle over-expands, internal shocks bante hain, aur thrust girta hai.

λ = \frac{1 + c o s α}{2}

ko seedha bell nozzle par uske exit angle use karke apply kar sakte hain."

Formula (Figure 3 dekho) uniform conical flow ke liye derive kiya gaya tha poore exit par; bell ka flow angle exit plane par vary karta hai, toh yeh ek approximation hai, exact result nahi — real

λ

actual contour ko integrate karke aata hai.

"Rao ka contour nozzle length minimize karta hai."

Rao ka calculus-of-variations problem fixed length ke liye thrust maximize karta hai, ulta nahi; length constraint hai, thrust optimize hone wali cheez hai.

"Kyunki bell contoured hai, manufacturing tolerance matter nahi karti."

Ulta — ek rough ya galat shaped bell wall friction add karti hai aur weak shocks create kar sakti hai, toh ek buri tarah bani bell ek smooth cone se bhi bura perform kar sakti hai; tolerance yahan zyada matter karti hai.

α = 1 5^{\circ}

wala cone apna

15%

thrust divergence mein kho deta hai."

Nahi — loss

1 - λ = 1 - \frac{1 + c o s 1 5 ^{\circ}}{2} \approx 1.7%

hai,

15%

nahi; angle percentage nahi hai, cosine averaging (Figure 3) loss ko bahut chhota bana deta hai.

Why questions

Bell nozzle exit ke paas flow ko axial kyun mod deta hai instead of use fan out hone dene ke?

Sideways momentum koi forward thrust produce nahi karta aur exit par cancel bhi ho sakta hai; deewar ko axis ki taraf wapas modna (Figure 2) us would-be-wasted momentum ko axial push mein convert karta hai,

λ

1

ki taraf le jaata hai.

Rao ka contour throat ke bilkul baad tezi se kyun expand karta hai?

Throat ke paas fast area growth pressure ko jaldi giraa deti hai jab gas abhi dense hai, zyaatar available energy pehle hi extract karke — phir deewar gently turn kar sakti hai end mein shocks avoid karne ke liye.

Bell ka final turning gradual kyun hona chahiye?

Ek sharp turn supersonic flow ko achanak compress karta, ek shock wave create karta jo kinetic energy ko heat aur disorder mein dump karta, bilkul wahi thrust kho deta jo contour bachane ki koshish kar raha hai.

Conical factor

cos α

ki jagah

cos θ

ka average kyun hai?

Exhaust

0

α

tak ke angles par poori conical surface se nikalta hai, toh aap us surface par axial fraction average karte ho (Figure 3); ring area

2 π R sin θ

se weight dene par mean factor

\frac{1 + c o s α}{2}

milta hai.

Vacuum-stage engines sea-level engines se lambi nozzles kyun use karte hain?

Vacuum mein over-expand karne ke liye koi ambient pressure nahi hai, toh bada

ϵ

(hence lambi nozzle) gas ko zyada velocity ke liye expand karta rehta hai.

Zyada expansion ratio akele better real thrust guarantee kyun nahi karta?

ϵ

ideal expansion set karta hai, lekin agar yeh current altitude ke liye over-expand karta hai, toh shocks aur flow separation aate hain aur net thrust girta hai; geometry ko sirf bade

ϵ

ke peeche nahi bhaagna chahiye, altitude aur wall shape dono match karni chahiye.

80% figure itne engines mein common kyun hai 75% ya 85% ki jagah?

Yeh diminishing-returns curve ke knee ke paas hai — roughly pehla

80%

length

\approx 98%

efficiency capture karta hai, toh uske baad aap tiny gains ke liye weight add karte ho; yeh ek practical sweet spot hai, koi physical constant nahi.

Edge cases

Jab cone half-angle

α \to 9 0^{\circ}

ho toh

λ

ka kya hoga?

cos 9 0^{\circ} = 0

toh

λ \to \frac{1}{2}

— ek nozzle jo gas sideways spray kare apna sirf aadha momentum axial rakhta hai; yeh dikhata hai formula smoothly degrade hota hai aur kabhi nonsense predict nahi karta.

Length formula

ϵ = 1

par kya deta hai?

L = R_{t} (1 - 1) / tan α = 0

— exit area throat area ke barabar hai, koi diverging section hi nahi hai, jo bilkul woh boundary hai converging-diverging aur plain throat ke beech.

Agar do designers alag

α

choose karein lekin same exit radius

R_{e}

demand karein?

Unhe same

R_{e} = R_{t} ϵ

milega lekin alag lengths (teekha

α

→ chhoti) aur alag

λ

(teekha → zyada lossy); same exit size ka matlab same performance nahi hai.

Kya yeh saari geometry tab bhi apply hoti hai agar throat mein flow choked na ho?

Nahi — yahan har formula assume karta hai ki throat sonic speed (choked) reach kar chuki hai toh diverging section supersonic chalta hai; bahut kam chamber pressure ya heavily throttled small engine mein flow subsonic reh sakta hai, jis case mein diverging cone gas ko ek ordinary diffuser ki tarah simply dhima karta hai aur bilkul bhi supersonic expansion produce nahi karta.

Jab nozzle itna over-expand ho ki flow deewar se separate ho jaaye, kya geometric

λ

tab bhi loss describe karta hai?

Nahi —

λ

assume karta hai ki gas nozzle ko bhar leti hai aur exit tak deewar ko follow karti hai; ek baar flow separate ho jaaye, effective exit angle aur area unpredictably change ho jaate hain, toh clean

\frac{1 + c o s α}{2}

picture ab valid nahi rehti.

Kya "0% bell" meaningful hai?

Nahi — zero length matlab koi diverging section nahi, toh gas kabhi supersonic exit conditions reach nahi karti; length fraction positive hona chahiye nozzle ke kaam karne ke liye.

Recall Harder application check

Ek designer cone half-angle $α$ ko half kar deta hai jabki $ϵ$ fixed rakhta hai. Length aur divergence loss ka kya hoga, aur woh dono par kyun nahi jeet sakte? ::: Length $L = R_{t} (ϵ - 1) / tan α$ roughly double ho jaati hai (chhota $tan α$ ), jabki divergence loss $1 - \frac{1 + c o s α}{2}$ zero ki taraf girta hai — aap efficiency length aur weight se khareedते ho, toh dono goals trade off karte hain aur koi single $α$ dono optimize nahi karta. Do engines $ϵ$ aur $λ$ share karte hain; ek 80% bell hai, ek cone. Kya bell guaranteed halki hai? ::: Guaranteed nahi — iski choti wall help karti hai, lekin mooti curved walls, stiffeners, ya mahanga fabrication saving ko mita sakti hai; length mass budget mein sirf ek term hai.

Woh picture aur symbols jo hum use karte hain

Divergence factor λ kahan se aata hai

True or false — justify karo

Error dhundho

Why questions

Edge cases

Divergence factor $λ$ kahan se aata hai