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

Overview

The De Laval nozzle accelerates hot combustion gases from subsonic to supersonic speeds. While the basic converging-diverging shape is fixed by physics, the diverging section geometry can be optimized. Three primary geometries exist: conical, bell (Rao contour), and 80% bell. Each balances performance (exhaust velocity), weight, and manufacturing complexity.

Why Nozzle Geometry Matters

A conical nozzle is simple but wastes thrust because gas exits at angle. A bell nozzle curves the flow back to axial, recovering that loss. The question is: how long should the bell be?

Conical Nozzle

Derivation: Thrust Loss from Divergence

The exhaust exits at half-angle $\alpha$ to the axis. Momentum in the thrust direction:

The divergence correction factor is:

Why this formula? Average the momentum over the conical surface. For uniform flow at angle $\alpha$, integrate $\cos\theta$ over a cone:

Simplify:

Use identity $\sin^2\alpha = 1 - \cos^2\alpha = (1-\cos\alpha)(1+\cos\alpha)$:

Length: For expansion ratio $\epsilon = A_e / A_t$, length from throat to exit:

where $R_t$ is throat radius.

Find: Nozzle length and thrust efficiency.

Solution:

1. Exit radius: $R_e = R_t \sqrt{\epsilon} = 10\sqrt{16} = 40$ cm
2. Length: $L = \frac{10(4-1)}{\tan 15°} = \frac{30}{0.2679} \approx 112 \text{ cm}$
3. Efficiency: $\lambda = \frac{1 + \cos 15°}{2} = \frac{1 +0.9659}{2} = 0.983$

Why this step? The geometry is a simple cone. The slant height gives the diverging section length. The $\cos\alpha$ factor directly measures axial momentum fraction.

Bell Nozzle (Rao Optimum Contour)

Why Bell Beats Cone

A bell nozzle has three sections:

1. Throat region — circular arc transitioning from converging to diverging
2. Expansion region — parabolic/cubic curve expanding the flow
3. Exit region — curve bends back toward axis, making flow axial

The curve is designed so the exit flow is parallel to the axis ($\alpha_e = 0°$), giving $\lambda \approx 1$ (no divergence loss).

The result is a smooth curve described by higher-order polynomials or Bézier splines.

Parabolic Approximation

For engineering, a parabolic contour approximates Rao:

Coefficients $a, b$ are fitted to match:

• Throat angle $\theta_n$ (typically 20–30°)
• Exit angle $\theta_e = 0°$ (axial flow)
• Expansion ratio $\epsilon$

Length: For 80% the length of equivalent conical nozzle:

But efficiency $\lambda \approx 0.98\text{–}0.99$ (vs 0.983 for cone).

Bell nozzle (80% length):

• Length: $L_{\text{bell}} = 0.8 \times 112 = 89.6$ cm
• Efficiency: $\lambda_{\text{bell}} \approx 0.985$ (from empirical data)

Why this step? The bell is 20% shorter but has same or better efficiency because the exit flow is axial.

Thrust comparison (for $\dot{m} v_e = 100$ kN ideal):

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

Bell gives0.2 kN more thrust at20% less length → 20% less weight.

80% Bell Nozzle

Why 80%?

Trade-off analysis:

• 100% bell (full Rao contour): $\lambda \approx 0.99$, but length = cone length (no weight savings)
• 60% bell: $\lambda \approx 0.96$, very compact but 4% thrust loss
• 80% bell: $\lambda \approx 0.985$, 20% shorter than cone, sweet spot

Geometry Details

The 80% bell contour:

1. Throat expansion angle $\theta_n = 20\text{–}30°$ (rapid initial expansion)
2. Exit angle $\theta_e = 5\text{–}10°$ (slightly non-axial, acceptable loss)
3. Wall profile: Parabolic or cubic spline fitted to these angles

Solution:

1. Equivalent conical length ($\alpha = 15°$): $L_{\text{cone}} = \frac{8(\sqrt{25}-1)}{\tan 15°} = \frac{8 \times 4}{0.2679} = 119.5 \text{ cm}$

2. Bell length: $L_{\text{bell}} = 0.8 \times 119.5 = 95.6 \text{ cm}$

3. Exit radius: $R_e = 8\sqrt{25} = 40 \text{ cm}$

4. Parabolic fit (simplified): $R(x) = 8 + 0.286x + 0.0015x^2$ where $x \in [0, 95.6]$ cm. Coefficients chosen so $R(0) = 8$, $R(95.6) = 40$, $\theta_e \approx 7°$.

Why this step? The parabola ensures smooth expansion without shocks. The exit angle $\theta_e$ is small enough for minimal divergence loss.

Comparison Table

Common Mistakes

The fix: True only until exit pressure matches ambient. Over-expansion (exit pressure< ambient) causes shock waves that reduce thrust. Nozzle length must match the altitude regime. Sea-level nozzles are shorter; vacuum nozzles are longer.

The fix: For small, cheap upper stages or experimental rockets, conical is fine. Manufacturing tolerance matters: a poorly-made bell with rough walls can have more friction loss than a smooth cone. Also, altitude compensation matters more than nozzle shape for multi-stage rockets.

The fix: "Axial" means parallel to the centerline (0° angle), not perpendicular. Confusion between "radial" (perpendicular) and "axial" (along axis). In an80% bell, $\theta_e = 7°$ means the flow is 7° off-axis, not perpendicular.

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

A rocket nozzle does the same thing, but in reverse: it starts narrow (the throat), then gets wider (the exit). Why? Because the gas is already moving super fast at the throat (faster than sound!). When you give it more space, it speeds up even more — like a race car that goes faster when the road gets wider.

Now, here's the trick: you want the gas shooting straight back, not spraying sideways. Imagine a cone — the gas comes out at angle, some push is wasted sideways. A bell nozzle is curved so the gas bends back to straight at the end. That's more push!

But how long should the bell be? If it's too long, it's heavy (bad for rockets). If it's too short, the gas doesn't straighten out enough. Engineers found that 80% of the cone length is the sweet spot: you get almost all the push (98.5%) but save 20% of the weight. That's the 80% bell.

Connections

• Converging-Diverging Nozzle Basics — why De Laval nozzles work
• Expansion Ratio and Area-Mach Relation — how $\epsilon$ determines $M_e$
• Nozzle Exit Pressure and Altitude Compensation — matching $p_e$ to $p_{\infty}$
• Thrust Vectoring with Bell Nozzles — gimballing for steering
• Method of Characteristics for Nozzle Design — precise contour calculation
• Manufacturing Tolerances in Nozzles — how roughness affects $\lambda$

Active Recall

#flashcards/physics

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

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

Concept Map

Hinglish (regional understanding)