The text is extracted from Towler G and Sinnott R (2013) Chemical Engineering Design: Principles,

Practice and Economics of Plant and Process Design, 2nd Ed., page 579-581.

Domed Ends Design of a Vessel

Design equations and charts for the various types of domed heads are given in the codes and standards
and should be used for detailed design. The codes and standards cover both unpierced and pierced heads.
Pierced heads are those with openings or connections. The head thickness must be increased to
compensate for the weakening effect of the holes where the opening or branch is not locally reinforced.

For convenience, simplified design equations are given in this section. These are suitable for the
preliminary sizing of unpierced heads and for heads with fully compensated openings or branches.

Hemispherical heads
For equal stress in the cylindrical section and hemispherical head of a vessel the thickness of the head
need only be half that of the cylinder; however, as the dilation of the two parts would then be different,
discontinuity stresses would be set up at the head and cylinder junction. For no difference in dilation
between the two parts (equal diametrical strain) it can be shown that for steels (Poisson’s ratio = 0.3) the
ratio of the hemispherical head thickness to cylinder thickness should be 7/17. However, the stress in the
head would then be greater than that in the cylindrical section, and the optimum thickness ratio is
normally taken as 0.6; see Brownell and Young (1959).

In the ASME BPV Code Sec. VIII D.1, the equation specified is the same as for a spherical shell:
𝑃𝑖 𝐷𝑖 Equation 14.17
𝑡=
4𝑆𝐸 − 0.4𝑃𝑖

Ellipsoidal heads
Most standard ellipsoidal heads are manufactured with a major and minor axis ratio of 2:1. For this ratio,
the following equation can be used to calculate the minimum thickness required (ASME BPV Code Sec.
VIII D.1 Part UG-32):
𝑃𝑖 𝐷𝑖 Equation 14.18
𝑡=
2𝑆𝐸 − 0.2𝑃𝑖

Torispherical heads
A torispherical shape is formed from part of a torus and part of a sphere (Figure
14.6). The shape is close to that of an ellipse but is easier and cheaper to
fabricate.

In Figure 14.6, Rk is the knuckle radius (the radius of the torus) and Rc the crown
radius (the radius of the sphere). The stress will be higher in the torus section
than the spherical section. There are two junctions in a torispherical end
closure: that between the cylindrical section and the head, and that at the
junction of the crown and the knuckle radii. The bending and shear stresses Figure 14.6 Torisphere
caused by the differential dilation that will occur at these points must be considered in the design of the
heads. The ASME BPV Code gives the design equation (Sec. VIII D.1 Part UG-32):
 0.885𝑃𝑖 𝑅𝑐
𝑡= Equation 14.19
𝑆𝐸 − 0.1𝑃𝑖
The ratio of the knuckle to crown radii should not be less than 0.06, to avoid buckling, and the crown
radius should not be greater than the diameter of the cylindrical section. Any consistent set of units can
be used with Equations 14.17 to 14.19. For formed heads (no welds or joints in the head) the joint
efficiency E is taken as 1.0.

Flat ends
Though the fabrication cost is low, flat ends are not a structurally efficient form, and very thick plates
would be required for high pressures or large diameters.

The design equations used to determine the thickness of flat ends are based on the analysis of stresses in
flat plates. The thickness required will depend on the degree of constraint at the plate periphery. The
ASME BPV Code specifies the minimum thickness as:

𝐶𝑃𝑖 Equation 14.16

𝑡 = 𝐷𝑒 √
𝑆𝐸

where 𝐶 = a design constant, dependent on the edge constraint

𝐷𝑒 = nominal plate diameter
𝑆 = maximum allowable stress
𝐸 = joint efficiency
*Any consistent set of units can be used.

Values for the design constant 𝐶 and the nominal plate diameter 𝐷𝑒 are given in the ASME BPV Code for
various arrangements of flat end closures (Sec. VIII D.1 Part UG-34).

Some typical values of the design constant and nominal diameter for the designs shown in Figure 14.4
are given below. For detailed design the ASME BPV Code should be consulted.

Figure 14.4 Flat-end closures: (a) flanged plate, (b)-(c) welded plate, (d)-(e) bolted cover.

(a) Flanged-only end, 𝐶 = 0.17 if corner radius is not more than 3𝑡, otherwise 𝐶 = 0.1; 𝐷𝑒 is equal to 𝐷𝑖 .
(b, c) Plates welded to the end of the shell with a fillet weld, angle of fillet 45° and weld depth 70% of the
thickness of the shell, 𝐶 = 0.33 𝑡/𝑡𝑠 , where ts is the shell thickness. 𝐷𝑒 = 𝐷𝑖 .
(d) Bolted cover with a full-face gasket, 𝐶 = 0.25 and 𝐷𝑒 is the bolt circle diameter (the diameter of a circle
connecting the centers of the bolt holes).
(e) Bolted end cover with a narrow-face gasket, 𝐶 = 0.3 and 𝐷𝑒 should be taken as the mean diameter of
the gasket.
Example

Select a design and estimate the thickness required for the component
part (i.e. domed head) of the vessel shown in the diagram. The vessel is to
operate at a pressure of 14 bar (absolute) and temperature of 260°C. The
material of construction will be plain carbon steel. Welds will be fully
radiographed. A corrosion allowance of 2 mm should be used.
Note: for flat head design, use a full face gasket.