GB2086823A - Pressure Vessel End Wall Contour - Google Patents

Abstract

A surface of revolution end closure for a pressure vessel, having minimum ratio of minor axis to major axis length, includes a wall substantially contoured according to the equation <IMAGE> where X=x/a (a being the length of the semi-major axis) and Y=y/a, and x, y are Cartesian co-ordinates in a meridional plane. For large-diameter thin-walled pressure vessels such as lighter-than- air lifting bodies.

Description

SPECIFICATION Pressure Vessels The present invention relates to large diameter axi-symmetric thin walled pressure vessels of the type used for lighter than air lifting devices.

This type of vessel, containing a gas at a low differential pressure, has a relatively thin wall which usually acts as a membrane. It is, therefore, important that compressive stresses, which would cause wrinkling, folding or geometrical changes, are avoided.

A common configuration for a pressure vessel, which is comparatively easy to design and construct, is a cylinder with end closures shaped as surfaces of revolution. The end closures are conventionally spherical, ellipsoidal, torispherical or duo-torispherical. The latter three of these closures have lower ratios of minor axis to major axis than has the spherical end closure, and there is frequently a requirement for this ratio to be as small as possible. Absolute minimisation of the ratio is restricted by the need to avoid compressive stresses.

According to the present invention a pressure vessel has an end closure including a wall substantially contoured according to a surface of revolution defined by the equation

where X=x/a (a being the length of the semi major axis) and Y=y/a.

In a preferred embodiment of the invention the end closure has a spherical end cap.

Details of the invention will now be described with reference to the accompanying drawings, of which: Figure 1 illustrates a segment of an axi-symmetrical wall, and Figure 2 is an elevation of a section of half of an end closure according to the invention.

A segment 10 of a wall of an axi-symmetric pressure vessel (Figure 1) is situated at that part 11 of the wall cut by a vertical cylinder of radius r. The wall is assumed to be sufficiently thin for stresses therein to be given adequately by membrane theory.

Vertical equilibrium of this part of the wall is given by the equation 7Gr2p=27GrNg sin 0 (1) where p is the differential pressure, No is the meridional load per unit length of the cut 11, and sJ is the angle a tangent to the meridian makes with the horizontal.

Similarly equilibrium of the segment 10 normal to the surface yields.

No No p=- + (2) r, r2 where N0 is the hoop load/unit length and r1, r2 are the principal radii of curvature.

Considering meridional curves where Ne=O (3) so that there is no possibility of buckling under internal pressure, and substituting (3) into (1) and (2) gives r-2r1 sin j=O (4) Introducing Cartesian co-ordinate for the meridional curves, so that r=x,

tan =y' equation (4) becomes

and writing t=z' (7) equation (6) becomes dt x =2t(1+t2) (8) dx which can be integrated to give

where a is the semi-major axis of the meridional curve.

Finally, if X=x/a, Y=y/a (10) equation (9) (with Y positive, say) becomes

which can be integrated to give

Half of a pressure vessel end closure having a wall 12 following this contour is shown by the full line in Figure 2.

The ratio of semi minor axis, b, to the semi major axis a in this contour can be expressed in terms of the r-function:

This is significantly lower than the ratio for ellipsoidal, torispherical or duo-torispherical shells, which can be shown to be 0.707, 0.750 and.691 respectively, and is the theoretical lowest value consistent with the avoidance of compressive stresses.

However, the radius of curvature r,, as given by equations (5), (10) and (12), is a2 a2 2x (14) and substituting into equations (2) and (3) gives pa2 No= (15) 2x indicating that N and r, approaches infinity as x approaches zero. However, this conclusion is based on the use of membrane theory, which is inaccurate in a region where the curvature vanishes. In practice a curvature will be introduced by the pressure itself acting on the flexural properties of the wall 12.

The load N, although in practice finite, will be difficult to estimate. For design purposes, therefore, the end cap can be made spherical, as illustrated by the dotted line 1 2a in Figure 2.

If R is the radius of the spherical end cap R=x cosec 0 (16) =a2/x Therefore, if the maximum allowable value of No is pa (the value of No in a cylinder of radius (a) it can be shown that R=2a (17) and the spherical end cap covers the region Ixl < wa (18) The optimum ratio b/a of equation 13 then becomes b/a0.620 (19) which is still appreciably lower than the ratios for other designs.

It will be appreciated that the wall 12, 1 2a as described with reference to Figure 2 will in practice form the end cap of a pressure vessel, the main body of the vessel being of, for example, cylindrical section.

Claims (1)

1. Claims

   1. A pressure vessel having an end closure which includes a wall substa~ntiaWly contoured according to a surface of revolution defined by the equation

   where X=x/a (a being the length of the semi-major axis) and Y=y/a, and x, y are Cartesian co-ordinates in a meridional plane.