Pressure Vessel Plate Cooler: Pressure vessel Design

Scaling

No matter what shape it takes, the minimum mass of a pressure vessel scales with the pressure and volume it contains and is inversely proportional to the strength to weight ratio of the construction material (minimum mass decreases as strength increases).

Scaling of stress in walls of vessel

Pressure vessels are held together against the gas pressure due to tensile forces within the walls of the container. The normal (tensile) stress in the walls of the container is proportional to the pressure and radius of the vessel and inversely proportional to the thickness of the walls. Therefore pressure vessels are designed to have a thickness proportional to the radius of tank and the pressure of the tank and inversely proportional to the maximum allowed normal stress of the particular material used in the walls of the container.

Because (for a given pressure) the thickness of the walls scales with the radius of the tank, the mass of a tank (which scales as the length times radius times thickness of the wall for a cylindrical tank) scales with the volume of the gas held (which scales as length times radius squared). The exact formula varies with the tank shape but depends on the density, ρ, and maximum allowable stress σ of the material in addition to the pressure P and volume V of the vessel. (See below for the exact equations for the stress in the walls.)

Spherical vessel

For a sphere, the minimum mass of a pressure vessel is

M = {3 \over 2} P V {\rho \over \sigma}

where:

$M$ is mass,
$P$ is the pressure difference from ambient (the gauge pressure),
$V$ is volume,
$\rho$ is the density of the pressure vessel material,
$\sigma$ is the maximum working stress that material can tolerate.

Other shapes besides a sphere have constants larger than 3/2 (infinite cylinders take 2), although some tanks, such as non-spherical wound composite tanks can approach this.

Cylindrical vessel with hemispherical ends

This is sometimes called a "bullet" for its shape, although in geometric terms it is a capsule.

For a cylinder with hemispherical ends,

M = 2 \pi R^2 (R + W) P {\rho \over \sigma}

where

R is the radius
W is the middle cylinder width only, and the overall width is W + 2R

Cylindrical vessel with semi-elliptical ends

In a vessel with an aspect ratio of middle cylinder width to radius of 2:1,

M = 6 \pi R^3 P {\rho \over \sigma}

Gas storage]

In looking at the first equation, the factor PV, in SI units, is in units of (pressurization) energy. For a stored gas, PV is proportional to the mass of gas at a given temperature, thus

M = {3 \over 2} nRT {\rho \over \sigma}

. (see gas law)

The other factors are constant for a given vessel shape and material. So we can see that there is no theoretical "efficiency of scale", in terms of the ratio of pressure vessel mass to pressurization energy, or of pressure vessel mass to stored gas mass. For storing gases, "tankage efficiency" is independent of pressure, at least for the same temperature.

So, for example, a typical design for a minimum mass tank to hold helium (as a pressurant gas) on a rocket would use a spherical chamber for a minimum shape constant, carbon fiber for best possible

\rho / \sigma

, and very cold helium for best possible

M / {pV}

Stress in thin-walled pressure vessels

Stress in a shallow-walled pressure vessel in the shape of a sphere is

\sigma_\theta = \sigma_{\rm long} = \frac{pr}{2t}

where

\sigma_\theta

is hoop stress, or stress in the circumferential direction,

\sigma_{long}

is stress in the longitudinal direction, p is internal gauge pressure, r is the inner radius of the sphere, and t is thickness of the cylinder wall. A vessel can be considered "shallow-walled" if the diameter is at least 10 times (sometimes cited as 20 times) greater than the wall depth.

Stress in the cylinder body of a pressure vessel.

Stress in a shallow-walled pressure vessel in the shape of a cylinder is

\sigma_\theta = \frac{pr}{t}

\sigma_{\rm long} = \frac{pr}{2t}

where:

$\sigma_\theta$ is hoop stress, or stress in the circumferential direction
$\sigma_{long}$ is stress in the longitudinal direction
p is internal gauge pressure
r is the inner radius of the cylinder
t is thickness of the cylinder wall.

Almost all pressure vessel design standards contain variations of these two formulas with additional empirical terms to account for wall thickness tolerances, quality control of welds and in-service corrosion allowances.

For example, the ASME Boiler and Pressure Vessel Code (BPVC) (UG-27) formulas are:

Spherical shells:

\sigma_\theta = \sigma_{\rm long} = \frac{p(r + 0.2t)}{2tE}

Cylindrical shells:

\sigma_\theta = \frac{p(r + 0.6t)}{tE}

\sigma_{\rm long} = \frac{p(r - 0.4t)}{2tE}

where E is the joint efficient, and all others variables as stated above.

The factor of safety is often included in these formulas as well, in the case of the ASME BPVC this term is included in the material stress value when solving for pressure or thickness.

Winding angle of carbon fibre vessels

Wound infinite cylindrical shapes optimally take a winding angle of 54.7 degrees, as this gives the necessary twice the strength in the circumferential direction to the longitudinal.

Pressure Vessel Plate Cooler

Friday, 27 March 2015

Pressure vessel Design