Pressure vessel design context: for a cylindrical vessel intended to operate at high pressure (> 15 atm), which formed head profile is considered the most economical in terms of strength-to-thickness and material usage?

Introduction / Context:
Pressure vessel heads close the ends of cylindrical shells and must resist internal pressure safely. For high-pressure service, the selection of head geometry directly affects required thickness, weight, and cost. This question checks understanding of which head type is most economical when pressure exceeds about 15 atm, focusing on strength-to-thickness efficiency rather than fabrication convenience alone.

Given Data / Assumptions:

• Cylindrical pressure vessel operating at pressures greater than 15 atm.
• Comparison among common head shapes: hemispherical, ellipsoidal (2:1), torispherical/dished, conical, and flat.
• Economy refers to material efficiency for a given allowable stress (lower required thickness for the same diameter and pressure).

Concept / Approach:
Membrane theory shows that the hemispherical head develops uniform membrane stress with no knuckle stress intensification, thus requiring the least thickness among common head types for the same internal pressure and diameter. Lower thickness translates to lower material mass, which is the principal basis for calling it most economical for high-pressure duty, despite potentially higher fabrication complexity.

Step-by-Step Solution:

Verification / Alternative check:
Comparative design formulas show: t_hemisphere < t_ellipsoidal (2:1) < t_torispherical for the same P, D, allowable stress; flat heads have far greater thickness and are not used for high pressure.

Why Other Options Are Wrong:

• Dished (torispherical): needs greater thickness than hemispherical at the same pressure.
• Ellipsoidal (2:1): efficient but still thicker than hemispherical for equal conditions.
• Conical: requires reinforcement and often larger thickness near the knuckle region.
• Flat: impractical for high pressure due to excessive thickness requirement.

Common Pitfalls:
Confusing fabrication cost with material economy; assuming ellipsoidal is always most economical without considering very high-pressure regimes where thickness dominates cost.

Final Answer:
Hemispherical