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

3.3.17 · Physics › Rocket Propulsion

Overview

De Laval nozzle hot combustion gases ko subsonic se supersonic speed tak accelerate karta hai. Basic converging-diverging shape physics se fixed hai, lekin diverging section geometry ko optimize kiya ja sakta hai. Teen primary geometries hain: conical, bell (Rao contour), aur 80% bell. Har ek performance (exhaust velocity), weight, aur manufacturing complexity ke beech balance banata hai.

Nozzle Geometry Kyun Matters Karta Hai

Conical nozzle simple hai lekin thrust waste karta hai kyunki gas angle pe exit karti hai. Bell nozzle flow ko wapas axial kar deta hai, wo loss recover karte hue. Sawaal ye hai: bell kitni lambi honi chahiye?

Conical Nozzle

Derivation: Divergence se Thrust Loss

Exhaust axis ke relative half-angle $α$ pe exit karti hai. Thrust direction mein momentum:

F_{thrust} = \overset{m}{˙} v_{e} cos α

Divergence correction factor hai:

λ = \frac{1 + cos α}{2}

Ye formula kyun? Conical surface pe momentum ka average karo. Angle $α$ pe uniform flow ke liye, cone ke upar $cos θ$ integrate karo:

λ = \frac{\int _{0}^{α} cos θ \cdot 2 π R sin θ d θ}{\int _{0}^{α} 2 π R sin θ d θ}

Simplify karo:

λ = \frac{\int _{0}^{α} cos θ sin θ d θ}{\int _{0}^{α} sin θ d θ} = \frac{[ \frac{1}{2} sin ^{2} θ ] _{0}^{α}}{[ - cos θ ] _{0}^{α}} = \frac{sin ^{2} α /2}{1 - cos α}

Identity $sin^{2} α = 1 - cos^{2} α = (1 - cos α) (1 + cos α)$ use karo:

λ = \frac{( 1 - cos α ) ( 1 + cos α )}{2 ( 1 - cos α )} = \frac{1 + cos α}{2}

Length: Expansion ratio $ϵ = A_{e} / A_{t}$ ke liye, throat se exit tak length:

L_{cone} = \frac{R _{t} ( ϵ - 1 )}{tan α}

jahan $R_{t}$ throat radius hai.

Dhundhna hai: Nozzle length aur thrust efficiency.

Solution:

Exit radius: $R_{e} = R_{t} ϵ = 1016 = 40$ cm
Length: $L = \frac{10 ( 4 - 1 )}{t a n 15°} = \frac{30}{0.2679} \approx 112 cm$
Efficiency: $λ = \frac{1 + c o s 15°}{2} = \frac{1 + 0.9659}{2} = 0.983$

Ye step kyun? Geometry ek simple cone hai. Slant height diverging section ki length deta hai. $cos α$ factor directly axial momentum fraction measure karta hai.

Bell Nozzle (Rao Optimum Contour)

Bell, Cone Se Behtar Kyun Hai

Bell nozzle ke teen sections hote hain:

Throat region — converging se diverging mein transition karta circular arc
Expansion region — flow expand karta parabolic/cubic curve
Exit region — curve axis ki taraf wapas mudta hai, flow ko axial banata hai

Curve is tarah design hota hai ki exit flow axis ke parallel ho ( $α_{e} = 0°$ ), jisse $λ \approx 1$ milta hai (koi divergence loss nahi).

Result ek smooth curve hai jo higher-order polynomials ya Bézier splines se describe hota hai.

Parabolic Approximation

Engineering ke liye, ek parabolic contour Rao ko approximate karta hai:

R (x) = R_{t} + a x b x^{2}

Coefficients $a, b$ fit kiye jaate hain taaki match ho:

Throat angle $θ_{n}$ (typically 20–30°)
Exit angle $θ_{e} = 0°$ (axial flow)
Expansion ratio $ϵ$

Length: Equivalent conical nozzle ki 80% length ke liye:

L_{bell} = 0.8 L_{cone}

Lekin efficiency $λ \approx 0.98 - 0.99$ (cone ke 0.983 ke comparison mein).

Bell nozzle (80% length):

Length: $L_{bell} = 0.8 \times 112 = 89.6$ cm
Efficiency: $λ_{bell} \approx 0.985$ (empirical data se)

Ye step kyun? Bell 20% chhoti hai lekin same ya better efficiency hai kyunki exit flow axial hai.

Thrust comparison ( $\overset{m}{˙} v_{e} = 100$ kN ideal ke liye):

Conical: $F = 100 \times 0.983 = 98.3$ kN
Bell: $F = 100 \times 0.985 = 98.5$ kN

Bell 0.2 kN zyada thrust deta hai aur 20% less length mein → 20% less weight.

80% Bell Nozzle

80% Kyun?

Trade-off analysis:

100% bell (full Rao contour): $λ \approx 0.99$ , lekin length = cone length (koi weight savings nahi)
60% bell: $λ \approx 0.96$ , bahut compact lekin 4% thrust loss
80% bell: $λ \approx 0.985$ , cone se 20% chhota, sweet spot

Geometry Details

80% bell contour:

Throat expansion angle $θ_{n} = 20 - 30°$ (rapid initial expansion)
Exit angle $θ_{e} = 5 - 10°$ (thodi non-axial, acceptable loss)
Wall profile: Parabolic ya cubic spline in angles ke saath fit kiya gaya

Solution:

Equivalent conical length ( $α = 15°$ ): $L_{cone} = \frac{8 ( 25 - 1 )}{t a n 15°} = \frac{8 \times 4}{0.2679} = 119.5 cm$
Bell length: $L_{bell} = 0.8 \times 119.5 = 95.6 cm$
Exit radius: $R_{e} = 825 = 40 cm$
Parabolic fit (simplified): $R (x) = 8 + 0.286 x + 0.0015 x^{2}$ jahan $x \in [0, 95.6]$ cm. Coefficients is liye choose kiye gaye ki $R (0) = 8$ , $R (95.6) = 40$ , $θ_{e} \approx 7°$ .

Ye step kyun? Parabola smooth expansion ensure karta hai bina shocks ke. Exit angle $θ_{e}$ itna chhota hai ki minimal divergence loss ho.

Comparison Table

Geometry	Length (relative)	Efficiency $λ$	Pros	Cons
Conical	100%	0.98	Banane mein simple	2% thrust loss, bhaari
Bell (100%)	100%	0.99	Max efficiency	Complex, koi weight savings nahi
80% Bell	80%	0.985	Achhi efficiency, 20% halka	Thoda mushkil manufacture karna
60% Bell	60%	0.96	Bahut compact	4% thrust loss

Common Mistakes

Fix: Tab tak sach hai jab tak exit pressure ambient se match kare. Over-expansion (exit pressure < ambient) shock waves create karta hai jo thrust reduce karte hain. Nozzle length altitude regime se match honi chahiye. Sea-level nozzles chhoti hoti hain; vacuum nozzles lambi hoti hain.

Fix: Chhote, saste upper stages ya experimental rockets ke liye conical theek hai. Manufacturing tolerance matter karta hai: rough walls wala poorly-made bell, smooth cone se zyada friction loss de sakta hai. Aur altitude compensation multi-stage rockets ke liye nozzle shape se zyada matter karta hai.

Fix: "Axial" ka matlab hai centerline ke parallel (0° angle), perpendicular nahi. "Radial" (perpendicular) aur "axial" (axis ke along) mein confusion hota hai. 80% bell mein, $θ_{e} = 7°$ matlab flow 7° off-axis hai, perpendicular nahi.

Feynman Explain-to-a-12-Year-Old

Recall Socho jaise garden hose squeeze kar rahe ho

Socho tumhare paas ek garden hose hai. Agar end squeeze karo, paani tezi se bahar aata hai, na? Kyunki tum same amount of paani ko chhote se hole se force kar rahe ho.

Rocket nozzle same kaam karta hai, lekin ulta: pehle narrow (throat) shuru hota hai, phir wide hota jaata hai (exit). Kyun? Kyunki gas already throat pe bahut tezi se chal rahi hai (sound se bhi tez!). Jab use zyada jagah do, aur tez ho jaati hai — jaise ek race car aur tez jaata hai jab road wide ho jaati hai.

Ab, trick ye hai: tum chahte ho gas seedhi peeche nikaale, side mein nahi. Cone imagine karo — gas angle pe bahar aati hai, kuch push sideways waste hota hai. Bell nozzle curve hota hai taaki gas end mein seedhi ho jaaye. Matlab zyada push!

Lekin bell kitni lambi honi chahiye? Agar bahut lambi ho, woh bhaari hai (rockets ke liye bura). Agar bahut chhoti ho, gas kaafi seedhi nahi ho paati. Engineers ne dhundha ki cone length ka 80% sweet spot hai: almost pura push milta hai (98.5%) lekin 20% weight bachta hai. Yehi 80% bell hai.

Connections

Converging-Diverging Nozzle Basics — De Laval nozzles kyun kaam karte hain
Expansion Ratio and Area-Mach Relation — $ϵ$ kaise $M_{e}$ determine karta hai
Nozzle Exit Pressure and Altitude Compensation — $p_{e}$ ko $p_{\infty}$ se match karna
Thrust Vectoring with Bell Nozzles — steering ke liye gimballing
Method of Characteristics for Nozzle Design — precise contour calculation
Manufacturing Tolerances in Nozzles — roughness $λ$ ko kaise affect karta hai

Active Recall

#flashcards/physics

What are the three main De Laval nozzle geometries? :: Conical, Bell (Rao contour), 80% bell

What is the divergence correction factor for a conical nozzle?

λ = \frac{1 + c o s α}{2}

, where

α

is the half-angle

Why does a bell nozzle have higher efficiency than a conical nozzle?

Bell flow ko exit pe axial kar deta hai (

θ_{e} \approx 0°

), divergence loss khatam ho jaata hai

What does "80% bell" mean?

Ek bell nozzle jiska length = equivalent conical nozzle length ka 80% hai

What is the typical efficiency of an 80% bell nozzle?

λ \approx 0.985

(ideal thrust ka 98.5%)

What are the two factors that determine nozzle performance?

Exit pressure (ambient se match hona chahiye) aur flow uniformity (axial exhaust)

For a conical nozzle with half-angle 15° and

ϵ = 16

, what is the thrust efficiency?

λ = \frac{1 + c o s 15°}{2} \approx 0.983

(98.3%)

Why is the 80% bell the "sweet spot"?

98.5% efficiency deta hai (cone ke 98% vs) lekin 20% chhota/halka. 80% ke baad diminishing returns

What is the Rao optimum contour?

Mathematically ideal bell nozzle shape jo diye gaye length ke liye maximum thrust deta hai, calculus of variations se nikala gaya

What happens if a nozzle is too long for its operating altitude?

Over-expansion: exit pressure < ambient, shock waves aur reduced thrust

Study with: Active recall (flashcards), derivation from scratch (all formulas), dual coding (diagram + equations), Feynman (ELI12 section), forecast-then-verify (prediction before examples), steel-man mistakes