# Summary

The calculation of a horizontal vessels wetted area and volume is required for engineering tasks such fire studies and the determination of level alarms and control set points. However the calculation of these parameters is complicated by the geometry of the vessel, particularly the heads. This article details formulae for calculating the wetted area and volume of these vessels for various types of curved ends including: hemispherical, torispherical, semi-ellipsoidal and bumped ends.

# Definitions

\(A\) | : | Wetted Area |

\(D_{i}\) | : | Inside Diameter of Vessel |

\(D_{o}\) | : | Outside Diameter of Vessel |

\(h\) | : | Liquid level above vessel bottom |

\(L\) | : | Length of vessel, tan-line to tan-line |

\(L_{f}\) | : | Straight Flange |

\(R\) | : | Inside Vessel Radius |

\(R_{c}\) | : | Inside crown radius |

\(R_{k}\) | : | Inside knuckle radius |

\(t\) | : | Vessel Wall Thickness |

\(V_{p}\) | : | Partially Filled Liquid Volume |

\(V_{t}\) | : | Total Volume of head or vessel |

\(z\) | : | Inside Dish Depth |

\(\varepsilon\) | : | Eccentricity of elliptical heads |

# Introduction

The calculation of the liquid volume or wetted area of a partially filled horizontal vessel is best performed in parts, by calculating the value for the cylindrical section of the vessel and the heads of the vessel and then adding the areas or volumes together. Below we present the wetted area and partially filled volume for each type of head and the cylindrical section.

The partially filled volume is primarily used for the calculation of tank filling times and the setting of control set points, alarm levels and system trip points.

The wetted area is the area of contact between the liquid and the wall of the tank. This is primary used in fire studies of process and storage vessels to determine the emergency venting capacity required to protect the vessel.

The volume and wetted area of partially filled vertical vessels is covered separately.

# Hemispherical Heads - Horizontal Vessel

Hemispherical heads have a depth which is half their diameter. They have the highest design pressures out of all the head types and as such are typically the most expensive head type. The formula for calculating the wetted area and volume of one head are presented as follows.

## Wetted Area

\[ \displaystyle A = \pi h \frac{D_{i}}{2} \]

## Volume

\[ \displaystyle V_{p} = \frac{1}{6} \pi h^2 \left( 3R - h \right) \]

\[ \displaystyle V_{p} = D_{i}^{3}\frac{\pi}{12} \left(3\left(\frac{h}{D_{i}}\right)^{2}-2\left(\frac{h}{D_{i}}\right)^{3}\right) \]

# Semi-ellipsoidal or Elliptical Heads - Horizontal Vessel

The semi-ellipsoidal heads are shallower than the hemispherical heads and deeper than the torispherical heads and therefore have design pressures and expense lying between these two designs.

The most common variant of semi-ellipsoidal head is the 2:1 elliptical head which has a depth equal to 1/4 of the vessel diameter. The formula for calculating the wetted area and volume for one 2:1 semi-elliptical head are presented as follows.

## Wetted Area

For a 2:1 semi-ellipsoidal head ε is equal to 0.866, for other geometries the formula below may be used to calculate ε.

\begin{equation} \begin{split} A_{w} &= \frac{\pi D_{i}^{2}}{8} \left( \left(\frac{h}{D_{i}}-0.5 \right) B +1+\frac{1}{4 \varepsilon} ln \left( \frac{ 4 \varepsilon \left( \frac{h}{D_{i}}-0.5 \right) + B}{2- \sqrt{3}} \right) \right) \\ B &= \sqrt{1+12 \left( \frac{h}{D_{i}}-0.5 \right)^{2}} \\ \varepsilon &= \sqrt{1-\frac{4z^{2}}{D_{i}^{2}}} \end{split} \end{equation}The wetted area calculated using this method does not include the straight flange of the head. The length of the straight flange must be included in the calculation of the wetted area of the cylindrical section.

## Volume

\[ \displaystyle V_{p} = D_{i}^{3}C\frac{\pi}{12} \left(3\left(\frac{h}{D_{i}}\right)^{2}-2\left(\frac{h}{D_{i}}\right)^{3}\right)\]

Where,

for ASME 2:1 Elliptical heads:

\[ \displaystyle C = 1/2 \]

for DIN 28013 Semi ellipsoidal heads:

\[ \displaystyle C = 0.49951+0.10462\frac{t}{D_{o}}+2.3227\left(\frac{t}{D_{o}}\right)^{2} \]

The volume calculated does not include the straight flange of the head, only the curved section. The straight flange length must be included in the calculation of the volume of the cylindrical section.

# Torispherical Heads - Horizontal Vessel

Torispherical heads are the most economical and therefore is the most common head type used for process vessels. Torispherical heads are shallower and typically have lower design pressures than semi-elliptical heads. The formula for the calculation of the wetted area and volume of one partially filled torispherical head is presented as follows.

## Wetted Area

We can approximate the partially filled surface area of the torispherical head using the formula for elliptical heads. This approximation will over estimate the surface area because a torispherical head is flatter than a ellipsoidal head. This assumption is conservative for pool fire relieving calculations.

\begin{equation} \begin{split} A_{w} &= \frac{\pi D_{i}^{2}}{8} \left( \left(\frac{h}{D_{i}}-0.5 \right) B +1+\frac{1}{4 \varepsilon} ln \left( \frac{ 4 \varepsilon \left( \frac{h}{D_{i}}-0.5 \right) + B}{2- \sqrt{3}} \right) \right) \\ B &= \sqrt{1+12 \left( \frac{h}{D_{i}}-0.5 \right)^{2}} \\ \varepsilon &= \sqrt{1-\frac{4z^{2}}{D_{i}^{2}}} \end{split} \end{equation}The wetted area calculated using this method does not include the straight flange of the head. The length of the straight flange must be included in the calculation of the wetted area of the cylindrical section.

## Volume

\[ \displaystyle V_{p} = D_{i}^{3}C\frac{\pi}{12} \left(3\left(\frac{h}{D_{i}}\right)^{2}-2\left(\frac{h}{D_{i}}\right)^{3}\right)\]

Where,

for ASME Torispherical heads:

\[ \displaystyle C = 0.30939 + 1.7197 \frac{R_{k} - 0.06 D_{o}}{D_{i}} - 0.16116 \frac{t}{D_{o}} + 0.98997 \left( \frac{t}{D_{o}} \right)^{2} \]

for DIN 28011 Torispherical heads:

\[ \displaystyle C = 0.37802 + 0.05073 \frac{t}{D_{o}} + 1.3762 \left( \frac{t}{D_{o}} \right)^{2} \]

The volume calculated does not include the straight flange of the head, only the curved section. The straight flange length must be included in the calculation of the volume of the cylindrical section.

top# Bumped Heads - Horizontal Vessel

Bumped heads have the lowest cost but also the lowest design pressures, unlike torispherical or ellipsoidal heads they have no knuckle. They are typically used in atmospheric tanks, such as horizontal liquid fuel storage tanks or road tankers.

Here we present formulae for calculated the wetted area and volume for an arbitrary liquid level height in a single Bumped head.

## Wetted Area

We can approximate the partially filled surface area of the bumped head using the formula for elliptical heads. This approximation will over estimate the surface area, which is conservative for pool fire relieving calculations.

\begin{equation} \begin{split} A_{w} &= \frac{\pi D_{i}^{2}}{8} \left( \left(\frac{h}{D_{i}}-0.5 \right) B +1+\frac{1}{4 \varepsilon} ln \left( \frac{ 4 \varepsilon \left( \frac{h}{D_{i}}-0.5 \right) + B}{2- \sqrt{3}} \right) \right) \\ B &= \sqrt{1+12 \left( \frac{h}{D_{i}}-0.5 \right)^{2}} \\ \varepsilon &= \sqrt{1-\frac{4z^{2}}{D_{i}^{2}}} \end{split} \end{equation}## Volume

\[ \displaystyle V_{t} = \frac{1}{3} \pi z^{2} \left( 3R_{c}-z \right) \]

\[ \displaystyle V_{p} = \frac{3V_{t}}{4} \left(\frac{h}{R} \right)^{2} \left( 1-\frac{h}{3R} \right) \]

The partially filled volume equation is an approximation, but will give a reasonable accuracy for vessel volume calculations.

# Cylindrical Section - Horizontal Vessel

Here we present formulae for calculated the wetted area and volume for an arbitrary liquid level height in the cylindrical section of a horizontal drum.

## Wetted Area

\[ \displaystyle A_{p} = 2 L R cos^{-1} \left( \frac{R-h}{R} \right) \]

\[ \displaystyle A_{p} = L D_{i} cos^{-1} \left( 1- 2\frac{h}{D_{i}} \right) \]

## Volume

\[ \displaystyle V_{p} = L \left( R^{2} cos^{-1} \left( \frac{R-h}{R} \right) - (R-H) \sqrt{2Rh-h^{2}} \right) \]

\[ \displaystyle V_{p} = L D_{i}^2 \left( \frac{1}{4} cos^{-1} \left( 1-2\frac{h}{D_{i}} \right) - \left( \frac{1}{2}- \frac{h}{D_{i}} \right) \sqrt{ \frac{h}{D_{i}} - \left( \frac{h}{D_{i}} \right)^{2}} \right) \]

Where the vessel has torispherical or ellipsoidal heads the straight flange length of the head should be included in the cylindrical section length when calculating the volume or surface area.

# References

- B Wiencke, 2009,
*Computing the partial volume of pressure vessels* - R Doane, 2007,
*Accurate Wetted Areas for Partially Filled Vessels* - E Ludwing, 1997,
*Applied Process Design for Chemical and Petrochemical Plants (Volume 2)* - E Weisstein, 2013,
*Cylindrical Segment. From MathWorld*