ASME F&D (Torispherical) Heads volume formula
Total volume, exact partial fill, derivation, worked example and a copy-ready fill table.
V ≈ πr²Lcyl + 2·0.0809·D³V(h) ≈ seg(r,h)·Lcyl + (0.1545·D/r)·πh²(3r−h)/3seg(r,h) = r²·acos((r−h)/r) − (r−h)·√(2rh−h²). All closed-form — spreadsheet-implementable with ACOS and SQRT.Derivation
True torispherical geometry (dish radius D, knuckle 0.06D, depth 0.1693D) has no tidy closed form, so we scale the spherical cap by an effective depth a_eff = 0.1545D chosen to match the standard 0.0809D³ full-head volume. Measured against exact numerical integration of the real profile, the worst error is 0.8% of head-pair volume — under 0.06% of a whole 500-gallon vessel. Both checks run in CI.
Worked example
Take D = 37.5 in, Lcyl = 94 in. Total capacity: 486.4 gal. At a stick reading of 15 in (40% of the 37.5 in maximum depth), the filled volume is 180.9 gal — 37.2% of capacity. Check it live on the calculator.
| Depth | Volume |
|---|---|
| 5% | 1.8% |
| 10% | 5% |
| 15% | 9.2% |
| 20% | 13.9% |
| 25% | 19.3% |
| 30% | 25% |
| 35% | 31% |
| 40% | 37.2% |
| 45% | 43.6% |
| 50% | 50% |
| 55% | 56.4% |
| 60% | 62.8% |
| 65% | 69% |
| 70% | 75% |
| 75% | 80.7% |
| 80% | 86.1% |
| 85% | 90.8% |
| 90% | 95% |
| 95% | 98.2% |
Need the full inch-by-inch table for specific dimensions? The dip chart generator prints it as a laminate-ready PDF.
FAQ
What is the formula for a asme f&d (torispherical) heads volume?
Total: V ≈ πr²Lcyl + 2·0.0809·D³. Partial fill at depth h: V(h) ≈ seg(r,h)·Lcyl + (0.1545·D/r)·πh²(3r−h)/3. Derivation and a worked example are above.
Is the fill-ratio table exact?
Yes — computed by the same engine as our calculators (a 500-gal-class vessel (shell ≈ 2.5×D)), verified by the automated test suite described on the methodology page.
Can I use this in a spreadsheet?
Yes — the formulas are closed-form (acos and sqrt only). Copy the 5%-step table for quick interpolation, or implement the formula directly for exact values.