Vessel Volume Cylinder⚠ unverified
Mechanical / Pressure Vessels · Compute the internal volume of a cylindrical vessel
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| r | r | m | 1.0 | Internal radius |
| L | L | m | 1.0 | Cylinder length |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | V | m**3 | Internal volume, in cubic metres (m**3) |
The science & history
Understanding the Parameters
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Internal radius $r$ — enters as $r^2$, so capacity grows with the square of radius: a modest diameter increase adds volume quickly (and, note, also raises wall stress linearly — the capacity-vs-strength trade-off of vessel proportions).
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Length $L$ — the straight-shell length between the heads; volume is linear in $L$. This is the cylindrical volume only.
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Heads not included — a real vessel's total volume adds the two end heads (each a partial ellipsoid or hemisphere): a 2:1 ellipsoidal head adds $\tfrac{\pi}{12}D^3$, a hemisphere $\tfrac{2}{3}\pi r^3$. For a short vessel the heads are a significant fraction of the total.
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Uses — sizing for throughput/residence time, estimating fill mass, sizing relief valves, and gauging stored energy for hazard analysis.
Derivation (Approaching a Proof)
The volume of a right circular cylinder is its constant cross-sectional area times its length. The cross-section is a circle of radius $r$ with area
$$A = \pi r^2,$$
and sweeping it along the axis for length $L$ (integrating $A$ over $L$, with $A$ constant) gives
$$V = \int_0^L A\,dx = A\,L = \pi r^2 L.$$
Dimensional check. $V = \pi r^2 L = \text{m}^2\cdot\text{m} = \text{m}^3$ — a volume, as required ($\pi$ dimensionless).
History and Development
The cylinder-volume formula is elementary geometry (Archimedes), but in pressure-vessel practice it is the first line of a capacity calculation, to which the head volumes are added for the total. Combined with the wall thickness (Required Thickness Thin) and density it also feeds the vessel weight (Vessel Weight); combined with pressure it gives the stored pneumatic energy that governs the vessel's explosion hazard and relief-system sizing.
Related Concepts: Vessel Weight, Required Thickness Thin, Thin-Wall Hoop Stress, Pressure Vessel Design, Head Thickness Ellipsoidal
Notes: Cylindrical (shell) volume only — add head volumes for the total (2:1 head $\approx \tfrac{\pi}{12}D^3$; hemisphere $\tfrac23\pi r^3$). Capacity $\propto r^2$ (but wall stress $\propto r$). Feeds capacity, fill mass, and stored-energy/hazard estimates.