Partially Filled Horizontal Tank Volume with Torispherical Heads

\[V=L\left(\frac{D^2}{4}\arccos\left(\frac{D-2h}{D}\right)-\left(\frac{D}{2}-h\right)\sqrt{Dh-h^2}\right)+KD^3\left(\frac{h}{D}\right)^2\left(3-2\frac{h}{D}\right)\]

Variables

Vliquid volume in the partially filled tank (m³)

Dinternal tank diameter (m)

Linternal cylindrical shell length (m)

hliquid depth measured from tank bottom (m)

Ktorispherical head volume coefficient

Description

What is this formula?

Partially Filled Horizontal Tank Volume with Torispherical Heads estimates the liquid volume inside a horizontal cylindrical tank equipped with two torispherical heads.

The formula combines the exact circular-segment volume of the cylindrical shell with an approximate partial-volume contribution from the two torispherical heads.

This is useful when the tank is partially filled and the available field measurements are the internal diameter, cylindrical shell length, liquid depth, and a torispherical head volume coefficient.

When to use it

Use this formula when the tank is horizontal, partially filled, and has two torispherical heads.

It is especially useful for approximate inventory calculations, process vessels, pressure vessels, petroleum tanks, chemical storage tanks, and industrial liquid level estimation.

Example

A horizontal tank with torispherical heads has:

D = 2.0 m

L = 10.0 m

h = 0.80 m

K = 0.1618

Formula:

V = L × [(D²/4) × acos((D−2h)/D) − ((D/2)−h) × √(Dh−h²)] + K × D³ × (h/D)² × [3 − 2(h/D)]

Substitution:

V = 10 × [(2²/4) × acos((2−2×0.8)/2) − ((2/2)−0.8) × √(2×0.8−0.8²)] + 0.1618 × 2³ × (0.8/2)² × [3 − 2×(0.8/2)]

Result:

V = 12.16 m³

The partially filled tank contains approximately 12.16 m³ of liquid.

Applications

- Torispherical head tank gauging

- Horizontal pressure vessel inventory

- Petroleum process vessels

- Chemical storage tanks

- Industrial liquid level calculations

- Approximate capacity and inventory estimation

Note

This formula is an engineering approximation intended for practical inventory estimation.

The cylindrical shell contribution is calculated using the exact circular-segment formula, but the torispherical head contribution is approximated using the coefficient K.

The coefficient K represents the full-volume contribution of both torispherical heads relative to D³.

For standard ASME F&D torispherical heads, a commonly used approximate value is K = 0.1618.

For non-standard