Pressure Vessel Heads: 2:1 Ellipsoidal vs Torispherical vs Hemispherical

Quick Answer

For the same diameter, pressure and material, the head type sets the required thickness relative to the shell. Under ASME Section VIII Division 1 a hemispherical head needs about half the shell thickness, a 2:1 ellipsoidal head about the same as the shell, a European Korbbogen (crown radius 0.8D, knuckle 0.154D) about 5% more, a Klöpper (crown radius D, knuckle 0.1D) about 50% more, and an ASME flanged-and-dished torispherical head (crown radius D, knuckle 6% of D) about 75% more. Depth runs the other way: a hemispherical head is D/2 deep, a 2:1 ellipsoidal D/4, a Korbbogen about 0.25D, a Klöpper about 0.19D and an ASME F&D about 0.17D. That trade-off decides selection: 2:1 ellipsoidal heads are the default for pressure vessels and heat exchanger shells and channels; torispherical heads win at low pressure because they are shallow and cheap to form; hemispherical heads win at high pressure and large diameter, or in expensive alloys, where halving the thickness outweighs their extra forming and welding cost. Formed heads must be ordered by minimum thickness after forming, and ASME limits shape deviation to 1¼% of D outside and ⅝% of D inside the specified shape (UG-81).

The 2:1 ellipsoidal head in the photograph above belongs to a heat exchanger in production for a current Kasko Makine order. The bright ring at its rim is the machined weld bevel; the steel lugs welded to its outside are temporary handling points that will be cut off, ground and crack-tested before the exchanger is finished.

It looks like the simplest part of the vessel. It is not. A head has double curvature, a knuckle where the stress pattern changes sharply, a thickness that changes during forming, and a fit-up to the shell that has to meet tighter alignment limits than most people expect. And the choice of head type — made early, often by habit — can change the head's required thickness by a factor of three and a half.

What a Head Has to Do

A closed cylinder under internal pressure has a simple stress state: the hoop stress is twice the longitudinal stress. Close the end with a flat plate and the plate bends like a loaded floor, which is why flat closures are so thick. Close it with a curved shell and the pressure is carried mostly in membrane tension, which is far more efficient.

The more the head resembles a sphere, the lower its membrane stress. A sphere carries pressure with half the stress of a cylinder of the same diameter and thickness. That is why a hemispherical head can be half as thick as its shell.

The complication is the knuckle. A torispherical head is a shallow spherical crown joined to the cylinder by a tight toroidal knuckle. Where the curvature changes abruptly, the head bends, and the knuckle carries high local stresses — including compressive hoop stress, which on very thin large heads can cause the knuckle to buckle under internal pressure. A 2:1 ellipsoidal head spreads the transition over a smooth curve, which is why it performs so close to the shell at the same thickness.

The Main Head Types

2:1 ellipsoidal (semi-ellipsoidal) heads

The profile is half an ellipse with a major-to-minor axis ratio of 2:1, so the inside depth is D/4.

• ASME thickness: t = PD / (2SE − 0.2P) — essentially the same as the cylinder
• Depth: D/4, plus the straight flange
• Use: the default for pressure vessels, heat exchanger shells and channels, columns and reactors across moderate and high pressures
• Strengths: shell-equivalent thickness, smooth stress distribution, universally available, simple fit-up to a shell of the same thickness
• Nozzles: within a central circle of 80% of the shell diameter, the head can be treated as a sphere of radius 0.9D for opening reinforcement, which makes nozzles in the central region relatively economical

In practice most commercial "elliptical" heads are made as an approximation of the true ellipse from two radii — a crown radius and a knuckle radius — and the code accepts this where the shape tolerance is met.

Torispherical heads: ASME flanged and dished

A spherical crown of radius L joined to the shell by a knuckle of radius r.

• ASME standard proportions: crown radius L no greater than the outside diameter of the head skirt, knuckle radius at least 6% of the outside diameter and at least three times the head thickness
• ASME thickness for L = D and r = 6% of L: t = 0.885PL / (SE − 0.1P) — roughly 75% thicker than a 2:1 ellipsoidal head at the same pressure
• General torispherical rule: for other proportions, t = PLM / (2SE − 0.2P), with M = ¼(3 + √(L/r)). The tighter the knuckle relative to the crown, the larger M and the thicker the head
• Depth: about 0.17D for ASME standard proportions
• Use: low-pressure vessels, storage and process tanks, vacuum service with stiffening, and wherever a shallow head saves height

Torispherical heads: European Klöpper and Korbbogen

Two German standard shapes dominate European fabrication and are common wherever European engineering is specified:

The DIN standards define the radii on the outside diameter. The thickness comparison above applies the ASME torispherical rule to these proportions for a like-for-like comparison; the actual design is done to the code of construction, such as EN 13445 or AD 2000.

The practical reading: a Korbbogen behaves almost like a 2:1 ellipsoidal head, with similar depth and near shell-equivalent thickness, while a Klöpper is the economical low-to-moderate-pressure head, shallower but needing roughly half as much thickness again as the shell.

Hemispherical heads

Half a sphere, depth D/2.

• ASME thickness: t = PL / (2SE − 0.2P), where L is the inside radius — about half the shell thickness
• Use: high-pressure vessels, large-diameter vessels, spheres, and alloy or clad construction where every millimetre of thickness is expensive
• Fabrication: small hemispherical heads are pressed in one piece; large ones are assembled from pressed segments ("petals") and a crown cap, which adds welding, examination and fit-up work
• Joining: because the head is thinner than the shell, the thicker part is tapered at the joint — ASME requires a taper of at least 3:1 where the thickness difference exceeds a set limit

Conical heads and flat heads

Conical heads and cone transitions appear at the bottoms of hoppers, separators and vertical vessels and as reducers between shells of different diameter. The simple ASME cone rule applies up to a half-apex angle of 30°; steeper cones need a knuckle or more detailed analysis, and the cone-to-cylinder junction needs its own reinforcement check. See heat exchanger shells, cones and nozzles.

Flat heads — blind flanges, bolted channel covers, test blinds — carry pressure in bending and are far thicker than any dished head. Under ASME the basic rule is t = d √(CP/SE), with C depending on the attachment, and bolted covers adding a term for the bolting moment. For a 600 mm bolted cover at 20 bar with C = 0.3, the basic term alone gives about 40 mm before the bolting term — where a 2:1 head of the same diameter would need only a few millimetres. Flat covers are used where access matters more than weight: removable channel covers on TEMA A-type front ends, and blinds on nozzles. See also our guide to blind flanges and spectacle blinds.

A Worked Comparison

Same vessel, five head types. Inside diameter 1,800 mm — the size of the larger shells in the order shown in these photographs — design pressure 20 bar (2.0 MPa), SA-516 Grade 70 with an allowable stress of 138 MPa, joint efficiency 1.0, corrosion allowance 3 mm. Simplified ASME Division 1 calculation, for comparison only, not for design.

Three things to notice:

1. The 2:1 ellipsoidal and Korbbogen heads can be made from the same plate thickness as the shell — before forming allowance — which is why they are the defaults.
2. The ASME F&D needs about 10 mm more than the 2:1 head at 20 bar. At 3 bar the difference is a millimetre or two and the shallower, cheaper F&D wins. At 20 bar it no longer does.
3. The hemispherical head saves almost 7 mm, but it is twice as deep as the 2:1 head, uses a much larger blank or several pressed segments, and needs a tapered transition to the shell. In carbon steel at this pressure the 2:1 head is cheaper overall. In duplex, nickel alloy or clad construction the arithmetic changes quickly.

Forming Heads

Heads are formed from circular blanks — cut from a single plate or, for large diameters, welded from several plates — by one of three routes:

Cold pressing. The blank is pressed between dies. Fast and repeatable for standard sizes and moderate thicknesses.

Cold spinning and flanging. The crown is dished in a press, and the knuckle and straight flange are rolled on a flanging machine. This is how most torispherical and many ellipsoidal heads are made, and it handles large diameters without enormous dies.

Hot forming. Thick heads, and some materials, are heated to forming temperature and pressed hot. For materials supplied normalized, hot forming generally requires the head to be normalized again afterwards so that its properties match the certificate.

Thinning, and why heads are ordered by minimum thickness

Forming thins the plate — typically most in the knuckle of torispherical and ellipsoidal heads and in the crown of pressed hemispheres. The head maker therefore starts from a thicker blank so that the finished head meets a stated minimum thickness after forming everywhere. Ordering a head by nominal thickness invites a head that is under the required thickness at the knuckle.

Heat treatment after cold forming

Cold forming strains the outer fibres of the plate. ASME sets limits on that strain: carbon and low-alloy steels (UCS-79) and austenitic stainless steels (UHA-44) must be heat treated after forming when the calculated extreme-fibre elongation exceeds the code limit, with tighter triggers where, for example, impact testing is required or the plate is thick. This is settled with the head supplier at order, because it affects price, lead time and the condition the impact tests must represent.

Tolerances and Fit-Up

• Shape. ASME limits the deviation of the formed head from its specified shape to 1¼% of the inside diameter outward and ⅝% inward (UG-81), checked with templates.
• Circumference and roundness of the straight flange. These decide whether the head and shell meet within the code's alignment limits at the circumferential seam. A head that is a few millimetres too large in circumference produces an offset all the way round.
• Straight flange. The short cylindrical skirt keeps the head-to-shell weld away from the knuckle and gives room for fit-up and examination. Specify the length rather than accepting whatever the supplier's standard is.
• ID or OD basis. Heads may be dimensioned on the inside or outside diameter. A shell designed on the inside diameter and a head ordered on the outside diameter will not match, and this mistake still happens.
• Thickness map. Require ultrasonic thickness readings at the crown, knuckle and straight flange on the head certificate.

Nozzles in Heads

Nozzles are best placed in the central region of a 2:1 ellipsoidal head — inside the circle of 80% of the diameter, where the head behaves like a sphere of radius 0.9D. In the 1,800 mm example, that spherical-equivalent region needs 11.8 mm against 13.1 mm for the head as a whole before corrosion allowance, so less reinforcement is needed around openings there.

Openings in or near the knuckle are best avoided: the knuckle already carries the highest stresses, and reinforcement there is less effective and harder to analyse. On torispherical heads, keep openings within the spherical crown.

Head openings are machined to a controlled profile and bevel — large heads are clamped on a horizontal boring mill, as in the photograph below — so that the nozzle sits with an even weld gap and the full-penetration weld can be made and examined properly.

How to Choose

Common Specification Mistakes

1. Ordering by nominal thickness. The head arrives under the required thickness at the knuckle after forming.

   Prevention: Specify minimum thickness after forming and require an ultrasonic thickness map on the certificate.

2. ID and OD basis mixed up. The head does not match the shell circumference.

   Prevention: State the diameter basis — inside or outside — on the head order, and check the head circumference against the shell before fit-up.

3. Swapping head types on a revised drawing without recalculation. A Klöpper substituted for a 2:1 head needs roughly 50% more thickness at the same pressure.

   Prevention: Treat any change of head type as a design change, with a new thickness calculation.

4. Choosing a flanged-and-dished head at higher pressure because it is cheaper per head. The thickness penalty grows with pressure and soon outweighs the forming saving.

   Prevention: Compare cost on the finished head at the required thickness, not on catalogue price at nominal thickness.

5. No post-forming heat treatment agreed. The code requires it for the strain and material involved, and nobody planned it.

   Prevention: Settle forming method, calculated fibre elongation and heat treatment with the head supplier at order.

6. Impact tests that do not represent the final condition. A head hot-formed and renormalized is in a different condition from the plate that was tested.

   Prevention: Require impact testing, where needed, on material in the final heat-treated condition of the head.

7. Nozzles placed in the knuckle. Reinforcement is inefficient and stresses are already high.

   Prevention: Keep nozzles within the central 80% of a 2:1 head, or within the spherical crown of a torispherical head.

8. Short or unspecified straight flange. The head-to-shell weld sits too close to the knuckle and fit-up is difficult.

   Prevention: Specify straight flange length on the head drawing.

9. Temporary lugs left on, or removed without examination. Weld spots on the head become crack initiation sites.

   Prevention: Require removal, grinding flush and magnetic particle or dye penetrant examination of all temporary attachment locations.

Supply from Kasko Makine

Heads in complete equipment

Kasko Makine supplies heads as part of complete equipment — heat exchangers, industrial process columns and pressure vessels — with head type, forming method, heat treatment and thickness verification specified and documented as part of the vessel.

Head types

• 2:1 ellipsoidal heads
• Torispherical heads: ASME flanged and dished, Klöpper (DIN 28011) and Korbbogen (DIN 28013)
• Hemispherical heads, one-piece and segmented
• Conical and toriconical sections
• Flat covers, channel covers and test blinds

Materials

Carbon steels including low-temperature grades, alloy steels and austenitic stainless steels, with clad and weld-overlaid options for corrosive service.

Formed heads to drawing

Formed heads can be sourced to drawing for fabricators, with machined weld bevels, machined nozzle openings and the required documentation.

Certification

Material certificates to EN 10204 3.1 or 3.2, thickness-after-forming reports, shape tolerance checks, post-forming heat treatment records where applicable, and NDE reports for welded blanks.

Send your vessel or exchanger drawing and we will check the head selection and quote within five working days. Reach us at info@kaskomakine.com or WhatsApp +90 (537) 521 1399.

Continue Reading: Heat Exchanger Fabrication Series

• Heat Exchanger Manufacturing Process: Step by Step
• Heat Exchanger Shells, Cones and Nozzles
• Tubesheets, Baffles and Tube Bundles
• Carbon Steel Plate: ASTM A516 and A36

Frequently Asked Questions

Q: What is the difference between an ellipsoidal and a torispherical head?
A: A 2:1 ellipsoidal head has a smooth half-ellipse profile with a depth of D/4 and needs about the same thickness as the shell. A torispherical head is a shallow spherical crown joined to the shell by a tight knuckle, so it is shallower but carries high bending stress in the knuckle. An ASME flanged-and-dished head needs roughly 75% more thickness than a 2:1 ellipsoidal head at the same pressure.

Q: Why is a hemispherical head thinner than the shell?
A: A sphere carries internal pressure with half the membrane stress of a cylinder of the same diameter and thickness, so under ASME Section VIII Division 1 a hemispherical head needs about half the shell thickness. The trade-off is depth — D/2 instead of D/4 for a 2:1 head — plus larger blanks or welded segments and a tapered transition to the thicker shell.

Q: What is a Klöpper head and how does it compare with a Korbbogen head?
A: Both are German torispherical shapes. A Klöpper head (DIN 28011) has a crown radius equal to the diameter and a knuckle radius of 0.1D, a depth of about 0.19D, and needs around 50% more thickness than the shell. A Korbbogen head (DIN 28013) has a crown radius of 0.8D and a knuckle of 0.154D, is about 0.25D deep and performs almost like a 2:1 ellipsoidal head.

Q: What does minimum thickness after forming mean when ordering heads?
A: Forming thins the plate, especially in the knuckle of torispherical and ellipsoidal heads and in the crown of pressed hemispheres. The head maker starts from a thicker blank so the finished head meets a stated minimum thickness everywhere. Ordering by nominal thickness risks a head that falls below the required thickness after forming, so specify minimum thickness and ask for an ultrasonic thickness map.

Q: Where should nozzles be placed in a 2:1 ellipsoidal head?
A: Preferably within the central circle of 80% of the shell diameter. In that region ASME allows the head to be treated as a sphere of radius 0.9D for reinforcement, which reduces the reinforcement needed around openings. Openings in or near the knuckle should be avoided because the knuckle already carries the highest stresses and reinforcement there is less effective.

Q: Do cold-formed heads need heat treatment?
A: Sometimes. ASME Section VIII Division 1 requires heat treatment after cold forming when the calculated extreme-fibre elongation exceeds code limits, under UCS-79 for carbon and low-alloy steels and UHA-44 for austenitic stainless steels, with tighter triggers when impact testing is required or the plate is thick. The forming method and heat treatment should be agreed with the head supplier at order.

Q: What tolerance applies to a formed head?
A: ASME Section VIII Division 1 UG-81 limits the deviation of the formed inner surface from the specified shape to 1¼% of the inside diameter outward and ⅝% inward. In addition, the head's circumference and the roundness of its straight flange must allow the head-to-shell seam to meet the code's alignment limits, so they should be checked against the shell before fit-up.