Explanation of the calculation

The relation of filling level and filling volume in a horizontal or vertical vessel is quite complex. The main problem is that you can determine the volume of domed heads only by numeric integration.

Though the envelope curve respectively the envelope area of a domed head is educible with relatively simple equations:

See the head as a band of concentric circles with the radius r(z). z is the distance of one circle and the knuckle/skirt weld.

Variables:

r(z)          Radius starting at the axis of symmetry of the head, with one point on the inner surface

z              Distance of a point to the knuckle/skirt weld

R1           Crown radius

R2           Knuckle radius

Rx           Inner radius of head minus R2 (= distance knuckle center and axis of symmetry of the head)

These equations can be evaluated by numeric integration and you may determine the relation of filling level and filling volume in vertical and horizontal heads.

Please note;

Horizontal Tank

Vertical Tank

Elliptical head

Hemi-spherical head

Torispherical heads per DIN 28011

r₁ = d_a

r₂ = 0,1 · d_a

h₁ = 3,5 · s

h₂ = 0,1935 · d_a – 0,455 s

h₃ = h₁ + h₂

Semi-ellipsoidal heads per DIN 28013

r₁ = 0,8 · d_a

r₂ = 0,154 · d_a

h₁ = 3 · s

h₂ = 0,255 · d_a – 0,635 s

h₃ = h₁ + h₂

Domed heads

r₁ = d_a – 1,4 · d_a

r₂ = 30 – 50 mm bzw. nach Wunsch

h₁ = ≈ 20 – 30 mm bzw. nach Wunsch

Flat ends

r₂ = 30 – 50 mm bzw. nach Wunsch

h₁ = ≈ 20 – 30 mm bzw. nach Wunsch

Dished heads

r₁ = d_i

b = 40 – 70 mm

d_a = d_i + 2·b

Dished plates

r₁ = 0,8·d_a – 1,4·d_a

Tank Heads

Single-wall per DIN 6608 / 6616

Double-wall per DIN 6608-2 / 6606-2

r₁ = d_a

r₂ = ≈ 1/30 d_a

Variables: