Ellipsoidal dished head

The ellipsoidal dished head with a major to minor axis ratio of 2 1 is popular for economic reasons, even though the theory for thin-walled vessels predicts that the head of this shape should have twice the thickness of a hemispherical head where the major and minor axes are equal. Such an ellipsoidal head used for vessels under internal pressure has the same thickness as the cylindrical shell if the same allowable stresses and joint efficiencies are applied to both parts. The 1962 ASME Code Section VIII, Division 1 gives the following equation for the thin-walled ellipsoidal dished heads with a 2 1 major to minor axis ratio  

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In addition to the hemispherical and ellipsoidal heads, the torispherical head has been used extensively for closures on a large variety of cylindrical vessels. It is shaped, not as an ellipsoid, but by the use of two radii. The crown radius is the radius of dish for the spherical crown which constitutes the major portion of the head, and the knuckle radius, sometimes referred to as the comer radius, is the radius joining the spherical crown to the cylindrical shell. Heads of this type require less forming than ellipsoidal dished heads, so forming costs are lower. 

Because of its lower forming cost and smaller depth, the ellipsoidal dished head is often used in preference to the hemispherical dished head. Also ellipsoidal dished heads are more readily available in a variety of sizes. The designer must consider the hemispherical and ellipsoidal dished sizes available from the head supplier. He must also include the costs for material, forming and shipping and must consider other factors such as delivery dates before he can make the best selection. 

In years past, the ellipsoidal dished head was so much more available than the hemispherical dished head that the hemispherical head was limited primarily to small-diameter vessels. But vessel closure manufacturers have increased the nimiber of dies stocked for hemispherical head production and these are now becoming more widely used. If it were not for the cost of forming, the engineer would usually choose the hemispherical closure for the top head of a tall vertical vessel since it has the least weight and the lowest discontinuity stresses and is therefore the strongest. 

The torispherical head has more local stresses from discontinuities than the hemispherical or ellipsoidal heads due to the discontinuity in shape at the junction of the two radii and to the discontinuity at the junction of the knuckle with the straight flange. In fact, stressed beyond its elastic limit, the torispherical head tends to become more like the ellipsoidal dished head in shape, demonstrating that the ellipsoidal dished head has a better natural geometry. 

In general, the saving realized is not great enough to warrant the use of torispherical instead of ellipsoidal dished heads in vessels as expensive as most tall vertical towers. Use hemispherical dished heads whenever possible, because there is very little stress discontinuity in the junction of these heads with the shell, as compared to the ellipsoidal dished or the torispherical dished heads. 

The volumes of heads must be calculated separately and added to the volume of the cylindrical portion of the tank. The four types of heads most frequently used are the standard dished head, torispherical or ASME head, ellipsoidal head, and hemispherical head. Dimensions and volumes for all four of these types are given in Luken.s Spun Heads, Lnkens Inc., Coatesville, Pennsylvania. Approximate volumes can also be calculated by the formulas in Table 10-65. Consistent units must be used in these formulas. 

Although spherical vessels have a limited process application, the majority of pressure vessels are made with cylindrical shells. The heads may be flat if they are suitably buttressed, but preferably they are some curved shape. The more common types of heads are illustrated on Figure 18.16. Formulas for wall thicknesses are in Table 18.3. Other data relating to heads and shells are collected in Table 18.5. Included are the full volume V0 and surface S as well as the volume fraction V/V0 corresponding to a fractional depth H/D in a horizontal vessel. Figure 18.17 graphs this last relationship. For ellipsoidal and dished heads the formulas for V/V0 are not exact but are within 2% over the whole range. 

If the hemispherical dished head is not feasible, the ellipsoidal dished should be preferred over the torispherical dished head for tall vertical vessels, since the cost of the top closure is only a small part of the total cost of the vessel. For a battery of horizontal storage tanks or small vertical vessels, the torispherical dished head may be the more economical choice. Closures with conical, tori-conical, flanged and dished (not ASME) or flat shapes have no place in the design of tall vertical towers unless the process requires such special shapes for some particular reason. 

Head types fall into one of three general categories hemispherical, torispherical, and ellipsoidal. Hemispherical heads are analyzed as spheres and were covered in the previous section. Torispherical (also known as flanged and dished heads) and ellipsoidal head formulas for stress are outlined in the following form. 

Volume Relationships. A cylindrical vessel closed at both ends with elliptical dished heads has a volume equal to the volume of the cylindrical section plus twice the volume contained in one of the heads. The volume contained in a head can be expressed in terms of a cylinder of e( uiva-lent volume having the same inside diameter as the cylindrical section of the heafl. Figure 5.2 is a cross section of an ellipsoidal head having a 2 1 major-to-minor-axis ratio. 

K = factor for ellipsoidal heads as determined from Eq. 9.6 L = spherical crown radius of flanged and dished heads L = effective length of cylindrical shell L, = effective length of conical section... 

ASME Design Equations for Ellipsoidal and Flanged and Dished Heads... 

ASME DESIGN EQUATIONS FOR ELLIPSOIDAL AND FLANGED AND DISHED HEADS... 

Dished or Basket Heads consist of a spherical segment normally dished to a radius equal to the inside diameter of the tank cylinder (or within a range of 6 inches plus or minus) and connected to the straight cylindrical flange by a knuckle whose inside radius is usually not less than 6 per cent of the inside diameter of the cylinder nor less than 3 times the thickness of the head plate. Basket heads closely approximate hemi-ellipsoidal heads. 

Hemispherical, ellipsoidal and torispherical heads are collectively referred to as domed heads. They are formed by pressing or spinning large diameters are fabricated from formed sections. Torispherical heads are often referred to as dished ends. 

Standard torispherical heads (dished ends) are the most commonly used end closure for vessels up to operating pressures of 15 bar. They can be used for higher pressures, but above 10 bar their cost should be compared with that of an equivalent ellipsoidal head. Above 15 bar an ellipsoidal head will usually prove to be the most economical closure to use. 

Figure 18.17. Types of heads for cylindrical pressure vessels, (a) Flat flanged KR = knuckle radius, SF= straight flange, (b) Torispherical (dished), (c) Ellipsoidal, (d) Spherical, (c) Conical, without knuckle, (f) Conical, with knuckle, (g) Nonstandard, one of many possible types in use.

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