pressure-vessel-heads

name	pressure-vessel-heads
description	Pressure vessel head design — hemispherical, ellipsoidal (2:1), torispherical (ASME, Kori-Svanfelt), flat plate head, conical transition, knuckle radius requirements, ASME VIII Div. 1 UG-32/UG-33 formulas, required thickness calculation, unstayed flat head (UG-34), external pressure head design, discontinuity stresses at head-shell junction, and head-to-shell weld category.
metadata	{"priority":7,"promptSignals":{"phrases":["pressure vessel head","ellipsoidal head","hemispherical head","torispherical head","flat head pressure vessel","ASME vessel head"],"minScore":3}}

Pressure Vessel Head Design — Complete Skill

Head Types and Applications

Selection Guide

Head Type	Relative Cost	Weight	Depth	Internal Pressure	External Pressure	Notes
Hemispherical	High	Moderate	D/2	Best (t = 1/2 cylinder)	Best	Optimal for high P
2:1 Ellipsoidal	Moderate	Moderate	D/4	Good	Good	Most common ASME
Standard flanged (ASME)	Low–Moderate	Moderate	Varies	Good	Moderate	Common for PV
Torispherical (Korbbogen)	Low	Low	~D/5	Adequate (E = 0.60–0.77)	Fair	European; dished
Conical (transition)	Low	Low	Varies	Adequate	Adequate	Equipment nozzle transitions
Flat plate	Very Low	High	Negligible	Poor (thick required)	Poor	Low P only; access

Design Pressure and Temperature

Design pressure P: maximum allowable working pressure (MAWP); from ASME UG-98 Design temperature: maximum operating T; determines allowable stress S from ASME IID tables Allowable stress S: from ASME Section IID (Tables 1A, 1B); function of material and temperature

ASME VIII Div. 1 Head Formulas

Hemispherical Head (UG-32(f))

Required thickness (internal pressure): t = P × L / (2SE - 0.2P) [ASME UG-32(f); L = inside radius of head = R; S = allowable stress; E = weld efficiency]

Simplified (where L = D_inside/2): t = P × D_i / (4SE - 0.4P) [equivalent form]

Thickness ratio vs. cylindrical shell: t_hemisphere / t_cylinder = 0.5 [hemisphere requires half the wall thickness of cylindrical shell; most efficient shape] Saves material for high-pressure vessels; common in nuclear pressure vessels

Minimum requirements: Joint efficiency E = 1.0 (full radiography); E = 0.85 (spot RT); E = 0.70 (no RT) Minimum thickness: 1.6 mm (ASME UG-16b); or per ASME UG-22 inspection

2:1 Semi-Ellipsoidal Head (UG-32(d))

Standard 2:1 ellipsoidal head: Major axis = D_inside/2; minor axis = D_inside/4 (depth = D_inside/4) Form factor K = 1.0 exactly for 2:1 ratio (simplifies formula)

ASME UG-32(d) formula: t = P × D / (2SE - 0.2P) [D = inside diameter; same form as cylinder but uses full D (not R)]

Note: this is same as cylindrical shell formula → ellipsoidal (2:1) head same thickness as shell Weld efficiency E = 1.0 for seamless heads (most common); E = 0.85 for welded (seamed) heads

Where not 2:1 ratio: General ellipsoidal (a:1 ratio where a ≠ 2): form factor K from ASME UG-32(d) table K > 1 for prolate (depth > D/4); K < 1 for oblate (flatter); t_required = K × t_2:1

2:1 ellipsoidal is the most common head for general industrial pressure vessels: Good balance of depth, weight, and fabrication cost; standard issue for many code-built vessels

Torispherical Head (UG-32(e)) — ASME "flanged and dished"

ASME standard torispherical head geometry: Crown radius L: typically L = D_inside (special provision in ASME UG-32(e)) Knuckle radius r: r ≥ 6% × D (minimum per ASME); typically r = 0.06 × D

ASME UG-32(e) formula: t = 0.885 × P × L / (SE - 0.1P) [for standard torispherical with L = D] Factor 0.885 reflects non-hemisphere geometry; less efficient than hemisphere

Weld efficiency E: Same as other heads; E = 1.0 preferred for seamless

Torispherical form factor M: M = (1/4) × (3 + √(L/r)) [general torispherical; not ASME std; from classical theory] t_general = P × L × M / (2SE - 0.2P)

European (Korbbogen) style: depth ≈ D/5; r/D ≈ 0.1–0.154; less deep than 2:1 ellipsoidal; cost-effective for large vessels

Minimum knuckle radius: r ≥ 3 × t_head (ASME UG-32 minimum) or 6% of D, whichever larger r < minimum → excessive localized bending stress at knuckle

Conical Head/Transition (UG-32(g) and Appendix F)

Geometry: Half-included angle α (apex angle/2); typically α ≤ 30° for without reinforcement

Required thickness (internal pressure, no stiffening): t = P × D / (2cos α × (SE - 0.6P)) [D = inside diameter at large end; α = half-angle] Valid for α ≤ 30°; for α > 30°: reinforcement ring required at junction

Junction reinforcement (ASME Appendix F): Transition from cylinder to cone creates discontinuity moment Area of reinforcement: A_req = H × D × tan α [H = computed thickness difference at junction] Provide fillet radius r_k ≥ t_shell at shell-cone junction

Conical bottom (hopper): Applied for chemical vessels, silos; α_hopper = 45–60° → thick cone required or stiffening rings

Flat Plate Head (UG-34) — Unstayed

ASME UG-34 formula: t = d × √(C × P / (S × E)) [d = inside diameter; C = shape factor from UG-34 Table; E = 1.0 for welded]

C factor: C = 0.20: flat head attached by full-penetration corner joint (most common, Category B weld) C = 0.25: flat head attached by fillet weld (less rigid connection) C = 0.13: flat head with small diameter (d < D); ring-type construction Lower C = more rigid attachment → lower required thickness

Full-faced gasket (flange and cover): For gasketed covers: pressure area term + bolt load in UG-34 Bolt-loaded covers: see ASME Appendix 2 (flange design)

Flat head thickness for typical vessel: t_flat = D × √(0.20 × P / (S × 1.0)) = D × √(0.20P/S) Much thicker than dished head: t_flat ≈ 3–5 × t_ellipsoidal for same pressure Cost penalty: use only for low-pressure (< 1 MPa) or special access requirements

External Pressure Head Design

Hemispherical and Ellipsoidal (ASME UG-33)

External pressure procedure (iterative):

1. Assume head thickness t; compute L/D and D/t ratios
2. Determine factor A from ASME Fig. G (external pressure chart)
3. From Fig. HA or HB (material-specific): determine factor B
4. Allowable external pressure: P_allow = B / (D_o/t)
5. Check P_allow ≥ P_external_design; if not, increase t and repeat

Hemispherical (simplified): t_hemi_external = 0.67 × t_cylinder_external [approximately; hemisphere more efficient than cylinder in buckling too]

Stiffening rings: For large-diameter low-pressure vessels under external pressure: add circumferential stiffening rings to cylinder; heads typically adequate without rings

Vacuum service: Design for 15 psig (1.034 bar) full vacuum for vessels that can be isolated and emptied Carbon steel vessels: minimum t from external pressure check (usually governs for large thin-wall vessels)

Head-to-Shell Weld

Weld Category

Category A (longitudinal seams): Longitudinal welds in shells, nozzles, and heads

Category B (circumferential seams): Head-to-shell junction weld: always Category B Category B allows lower weld efficiency E if not fully examined Full-penetration groove weld (butt joint) required for Category B > 1.6 mm thickness

Attachment details: Inside corner radius: r_inside ≥ minimum per UW-13 (usually t_head/3 minimum) Transition if head and shell have different thicknesses: ASME UW-13(b) taper or transition radius

Discontinuity Stresses at Head-Shell Junction

Bending and Membrane Forces

Structural discontinuity: Different radius of curvature between head and shell → different meridional strains → compatibility forces/moments at junction Result: localized bending stress superimposed on membrane stress

Stress concentration at junction: F = P × R / (2t) × f(geometry) [bending moment factor from shell theory] Peak stress at knuckle (torispherical): σ_peak = K × P × D / (4t) [K = 2–4 for typical designs; governs fatigue] ASME fatigue analysis: required for cyclic vessels per ASME VIII Div. 2

Shell theory (Flügge/Bijlaard): For accurate discontinuity stress: apply shell equations at junction M₀ = compatibility moment (per unit circumference) at junction σ_discontinuity = 6M₀/t² ≤ 3S (secondary stress limit, ASME approach)

Taper transition: If t_head ≠ t_shell: ASME UW-13(b) requires taper at 3:1 (length:thickness change) or fillet radius Reduces stress concentration at thickness mismatch

Worked Example

Design a head for: P = 1.5 MPa, D_inside = 800 mm, material SA-516 Gr. 70, T = 200°C, E = 1.0

S at 200°C (SA-516 Gr. 70): S = 137.8 MPa (from ASME IID Table 1A)

2:1 Ellipsoidal: t = P × D / (2SE - 0.2P) = 1.5 × 800 / (2 × 137.8 × 1.0 - 0.2 × 1.5) = 1,200 / 275.3 = 4.36 mm Round up to next standard: t = 6 mm (minimum per UG-16b + corrosion allowance) With 1.5 mm corrosion allowance: t_ordered = 6 + 1.5 = 7.5 mm → order 8 mm

Torispherical (ASME L = D): t = 0.885 × 1.5 × 800 / (137.8 × 1.0 - 0.1 × 1.5) = 1,062 / 137.65 = 7.71 mm Round up: t = 9 mm (with CA) [thicker than 2:1 ellipsoidal; less efficient]

Hemispherical: L = R = D/2 = 400 mm t = 1.5 × 400 / (2 × 137.8 × 1.0 - 0.2 × 1.5) = 600 / 275.3 = 2.18 mm → t = 4 mm (with CA) [Half the shell thickness; most efficient but expensive to fabricate]

Standards and References

Standard	Scope
ASME VIII Div. 1 UG-32	Internal pressure head design
ASME VIII Div. 1 UG-33	External pressure head design
ASME VIII Div. 1 UG-34	Unstayed flat heads and covers
ASME VIII Div. 1 UW-13	Weld requirements at head-shell junction
EN 13445-3	European unfired pressure vessel head formulas
BS PD 5500	British standard for pressure vessel heads

Output

Provide: head type selected (hemispherical/2:1 ellipsoidal/torispherical/flat) with justification (P, D, cost, depth constraint), design pressure P [MPa] and temperature T [°C], material (ASME designation; S at T [MPa]; from ASME IID), inside diameter D [mm], weld efficiency E (1.0/0.85/0.70; radiography requirement), ASME formula applied (UG-32 subsection), required head thickness t [mm] (calculated), corrosion allowance [mm] and ordered thickness [mm], minimum knuckle radius r [mm] (if torispherical; verify r ≥ 6% D), depth of head [mm], head-to-shell weld category (B; full penetration required?), external pressure check (if applicable: P_allow [MPa] from ASME charts vs. P_design), discontinuity stress estimate (K × P × D/(4t) [MPa] at knuckle vs. 3S limit), weight of head [kg] (π/2 × ρ × A × t for hemispherical; estimate for others), and applicable standard (ASME VIII UG-32/33/34, EN 13445-3).