Pressure-Vessel Mass Under Volume And Stress Limits

Introduction

A pressure vessel must hold a specified amount of fluid without exceeding the stress that its material can safely withstand. Increasing the wall thickness reduces stress but adds mass; changing the radius and cylindrical length changes both the capacity and the required amount of material.

We will model a thin-walled cylinder closed by two hemispherical ends. The design variables are the cylinder radius, cylindrical length, and wall thickness. The aim is to minimize modeled mass subject to a required capacity and an allowable-stress limit. Under the idealized model used here, the analysis has a subtle outcome: there is a finite lower bound on mass, but no design attains it.

Geometry and Assumptions

Let \(r>0\) be the vessel’s internal radius, \(L\geq 0\) the length of its cylindrical section, and \(t>0\) the uniform wall thickness. The vessel’s two hemispherical ends form one sphere. Thus the internal capacity is the sum of the cylindrical and spherical volumes.

The cylindrical wall has lateral area \(2\pi rL\), and the two hemispherical ends have the area of a sphere, \(4\pi r^2\). In a thin-wall model, the material volume is approximated by wall area times thickness. If the material density is \(\rho>0\), the modeled mass is therefore:

Let \(V_0>0\) be the required capacity, \(p>0\) the internal pressure, and \(S>0\) the allowable tensile stress. We use the standard thin-wall pressure-vessel stress model: the cylinder’s hoop stress is \(pr/t\); its longitudinal stress is \(pr/(2t)\); and the spherical ends’ membrane stress is \(pr/(2t)\). The hoop stress is the largest of these, so satisfying its limit also satisfies the other two in this model.

Use the Stress Limit to Choose Thickness

For fixed \(r\) and \(L\), mass increases with thickness because the surface area and density are positive. The hoop-stress condition is equivalent to \(t\geq pr/S\). Consequently, the least-mass thickness for those dimensions is the smallest allowable one:

Substituting this thickness into the mass model gives the least mass associated with each pair \((r,L)\):

Here and below, \(m(r,L)\) means the mass with the stress constraint active. Any thicker wall has greater mass for the same dimensions and cannot improve the objective.

Capacity Equality and Its Limitation

For a fixed radius, a longer cylinder has more capacity and more wall area. Let \(L_0\) be the length that gives capacity exactly \(V_0\) at radius \(r\). When \(L_0\ge0\) and a feasible design has \(L>L_0\), shortening it to \(L_0\) preserves the radius and thickness, keeps the required capacity, and strictly decreases mass; if \(L=L_0\), the mass is unchanged. Solving \(V(r,L)=V_0\) for \(L\) gives:

This length is physically allowed only when it is nonnegative. Equivalently, the radius must satisfy \(r\leq (3V_0/(4\pi))^{1/3}\). If \(r\) is larger, the hemispherical ends alone already exceed the required capacity; setting \(L=0\) cannot produce a capacity-equality design. Thus it is not valid simply to assume every feasible design can be shortened to capacity equality.

Deriving the Limiting Mass

When a capacity-equality design is possible, substitute its length into the stress-limited mass formula. The substitution simplifies the expression as follows:

For positive \(r\), this is greater than \(2\rho pV_0/S\), and it approaches that value as \(r\) approaches zero. But the capacity constraint then requires a very long cylinder. The limiting dimensions do not form a finite vessel.

We can establish the same lower bound for every feasible design, including ones whose hemispherical ends alone exceed the target capacity. First, the stress constraint gives \(t\geq pr/S\). Since area is positive, this yields a lower bound on mass. Then use the capacity constraint to rearrange that bound:

The strict final inequality follows from \(r>0\). It applies whether or not the vessel has excess capacity. To see that the lower bound can be approached, choose capacity-equality designs with positive radii tending to zero. For sufficiently small \(r\), the length \(V_0/(\pi r^2)-4r/3\) is positive, and the mass formula tends to \(2\rho pV_0/S\).

Worked Example: A Capacity-Equality Vessel

Worked Example: Comparing Two Feasible Radii

Worked Example: Approaching the Infimum

Worked Example: Excess Capacity Does Not Break the Bound

A Common Mistake

A tempting argument says, “If the vessel holds too much, shorten the cylinder until the capacity is exactly \(V_0\).” That step is not always possible. For example, when \(L=0\) and \(r=3\text{ m}\) in the preceding example, the ends already have capacity \(36\pi\text{ m}^3>100\text{ m}^3\). Shortening the cylindrical section cannot reduce the vessel below this capacity.

The correction is to use two complementary arguments. For designs where \(V_0/(\pi r^2)-4r/3\geq0\), shortening to the capacity-equality length can reduce mass. For all feasible designs, whether or not that length is nonnegative, the stress-and-capacity bound proves \(m>2\rho pV_0/S\). The bound is approached by a family of small-radius, long-cylinder designs, so it characterizes the infimum without incorrectly claiming that every design can be reduced to exact capacity.

What the Result Means for Design

The limiting result is a mathematical consequence of the assumptions, not a recommendation to build an indefinitely long vessel. As \(r\) decreases, the required thickness \(pr/S\) decreases in direct proportion, while the cylindrical length needed for fixed capacity grows approximately like \(1/r^2\). The model allows that length to grow without penalty, and it omits practical limits such as fabrication, transport, stability, connections, and minimum gauge thickness.

Adding any appropriate design restriction can change the optimization problem and may produce an attainable minimum. The first task in an applied optimization problem is therefore to identify the model’s constraints precisely. Here, the distinction between a minimum and an infimum is essential: the mass can be made arbitrarily close to its lower bound, but no finite positive radius reaches the bound.