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

Before you can read a single formula in the parent note, you need the vocabulary. This page builds every symbol from nothing — a plain sentence, a picture, and the reason the topic can't do without it. Read top to bottom; each idea leans on the one before it.

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1. The nozzle shape itself: throat, exit, axis

Picture a wasp-waist: gas rushes in on the left, squeezes through the narrow throat in the middle, and flares out on the right.

• The throat is where the gas exactly reaches the speed of sound. Everything downstream of it is supersonic.
• The exit is the rim where gas finally leaves and joins the outside air.
• The axis is the imaginary straight line we want all the exhaust to travel along, because thrust is force along the axis.

Why the topic needs these: the whole discussion of "conical vs bell" is about the wall shape between throat and exit — so you must know exactly which two ends we are talking about.

Recall

Which point of the nozzle is where the flow first hits the speed of sound? ::: The throat (the narrowest neck).

For the deeper story of why narrowing then widening speeds gas up, see Converging-Diverging Nozzle Basics.

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2. Radius $R$ — how wide the pipe is

Two special radii appear constantly:

In the picture above, $R_t$ is the short red arrow at the waist, $R_e$ is the long red arrow at the mouth. A subscript is just a small label hanging off a letter to say which radius we mean — nothing more mysterious than that.

Recall

What does the subscript in $R_e$ tell you? ::: It labels which radius — here, the one at the exit.

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3. Area $A$ and the expansion ratio $ϵ$

The wall is a circle at every cross-section, so its area is $A=\pi R^2$ (area of a circle: $\pi$ times radius squared).

Because $A=\pi R^2$, the ratio of radii is the square root of the ratio of areas: $\frac{R_e}{R_t}=\sqrt{\frac{A_e}{A_t}}=\sqrt{ϵ}\Rightarrow R_e=R_t\sqrt{ϵ}$

That single line is used in every worked example in the parent note (e.g. $R_e=10\sqrt{16}=40$ cm). Now you know where the square root comes from: it's undoing the "squared" in the area of a circle.

Recall

If $ϵ=25$ and $R_t=8$ cm, what is $R_e$? ::: $R_e=8\sqrt{25}=8\times 5=40$ cm.

The link between $ϵ$ and the final gas speed is the subject of Expansion Ratio and Area-Mach Relation.

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4. Angles: the half-angle $\alpha$ and wall angles $\theta$

This is the heart of the whole topic, so we go slowly.

In the figure, the yellow wedge between the blue axis and the wall is $\alpha$. A bigger $\alpha$ means a fatter, more sharply flaring cone.

Recall

What exit angle ${\theta }_e$ wastes the least thrust, and why? ::: ${\theta }_e=0°$ — the flow leaves parallel to the axis, so all its momentum points straight back.

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5. The tools $\cos$, $\sin$, $\tan$ — reading a slanted line

The parent note uses $\cos \alpha$, $\sin \theta$, and $\tan \alpha$. Here is what each one asks, all on the same right triangle.

Draw the exhaust velocity as an arrow tilted at angle $\alpha$ from the axis. Drop it onto a right triangle:

• the side along the axis (the useful part) is the adjacent side,
• the side across the axis (the wasted part) is the opposite side,
• the arrow itself is the hypotenuse (the longest side, length = the full speed $v_e$).

Let's confirm the parent's number. With $R_t=10$, $ϵ=16$, $\alpha =15°$: $L=\frac{10(\sqrt{16}-1)}{\tan 15°}=\frac{10\times 3}{0.2679}\approx 112\text{ cm}✓$

Recall

Which ratio gives the fraction of exhaust speed that produces thrust? ::: $\cos \alpha$ (adjacent over hypotenuse = along-axis fraction).

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6. The efficiency score $\lambda$

The parent derives $\lambda =\frac{1+\cos \alpha }{2}$ by averaging $\cos \theta$ over the whole conical sheet of exhaust. You don't need the calculus yet — just read $\lambda$ as "the thrust that made it, as a fraction of the ideal."

Sanity checks across every case:

• $\alpha =0°$: $\lambda =\frac{1+1}{2}=1$ — a straight pipe wastes nothing. ✓
• $\alpha =15°$: $\lambda =\frac{1+0.9659}{2}=0.983$ — matches the parent. ✓
• $\alpha =90°$ (wall flat-out sideways, degenerate): $\lambda =\frac{1+0}{2}=0.5$ — half the momentum lost; the extreme limit. ✓

Recall

What does $\lambda =1$ physically mean? ::: No divergence loss — exhaust leaves perfectly along the axis.

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7. The flow-rate symbols: $\dot{m}$ and $v_e$

Recall

Why does $\dot{m}{v}_e$ have units of force? ::: Mass-per-second times velocity = momentum-per-second, and force is the rate of change of momentum.

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How the foundations feed the topic

Each box is a symbol you can now read. Together they let you follow every formula, table, and worked example in the parent note (De Laval nozzle geometry).

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Equipment checklist

Test yourself — cover the right side and answer aloud.

• Throat vs exit vs axis ::: Throat = narrow neck (sonic point); exit = wide mouth; axis = centre line we want thrust along.
• Meaning of $R_t$ and $R_e$ ::: Radii (axis-to-wall distance) at the throat and at the exit.
• Expansion ratio $ϵ$ ::: $A_e/A_t$ — how many times bigger the exit area is than the throat area.
• Get $R_e$ from $ϵ$ ::: $R_e=R_t\sqrt{ϵ}$ (square root undoes the squared in $A=\pi R^2$).
• Half-angle $\alpha$ ::: Tilt of a conical wall from the axis; full opening is $2\alpha$.
• ${\theta }_n$ vs ${\theta }_e$ ::: Steep throat-start angle vs gentle exit angle of a bell; ideal ${\theta }_e=0°$.
• $\cos \alpha$ meaning ::: Fraction of exhaust speed pointing forward (adjacent/hypotenuse).
• $\tan \alpha$ meaning ::: Steepness of the wall, sideways/along-axis — used for length.
• Efficiency $\lambda$ ::: Thrust-survived score, $0$ to $1$; $\lambda =\frac{1+\cos \alpha }{2}$ for a cone.
• $\dot{m}$ and $v_e$ ::: Mass leaving per second, and exit gas speed; their product is the ideal thrust force.