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PRESSURE CYLINDERS 39.10
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CLASSICAL STRESS AND DEFORMATION ANALYSIS
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For equilibrium, the resistive force must be equal to the force due to pressure.Therefore, 2 di pi = dit h 4 h = pidi 4t or h = piri 2t (39.23)
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Equation (39.23) gives the relevant maximum stress in a spherical shell. The expression for hoop stress for a thin h t cylindrical shell and that for a thin spherdi ical shell are similar. This simple dePi h duction is of great importance in the design of pressure vessels because the thickness requirement for a spherical vessel of the same material strength and FIGURE 39.3 Spherical shell under internal thickness-to-diameter ratio is only onepressure. half that required for a cylindrical shell. The change in internal volume V of a spherical shell can be evaluated easily from consideration of volumetric strain and the original volume. Volumetric strain is equal to the sum of three equal and mutually perpendicular strains. The change in internal volume due to internal pressure is given by 3p d V = i i (1 )Vo (39.24) 4Et 39.4.5 Vessels Subjected to Fluid Pressure During the process of pressurization of a vessel, the fluid used as the medium changes in volume as the pressure is increased, and this must be taken into account when determining the amount of fluid which must be pumped into a cylinder in order to increase the pressure level in the vessel by a specified amount. The cylinder is considered initially full of fluid at atmospheric pressure. The necessary change in volume of the pressurizing fluid is given by Vf = where K = bulk modulus of fluid p = pressure Vo = original volume The additional amount of fluid necessary to raise the pressure must take up the change in volume given by Eq. (39.25) together with the increase in internal volume of the cylinder. The amount of additional fluid Va required to raise the cylinder pressure by p is given by Va = pdi pV [5 4 ]V + 4Et K (39.26) pVo K (39.25)
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Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright 2004 The McGraw-Hill Companies. All rights reserved. Any use is subject to the Terms of Use as given at the website.
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It can be shown by similar analysis that the additional volume of fluid Va required to pressurize a spherical vessel is given by Va = 3pdi pV (1 )V + 4Et K (39.27)
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39.4.6 Cylindrical Vessel with Hemispherical Ends One of the most commonly used configurations of pressure vessels is a cylindrical vessel with hemispherical ends, as shown in Fig. 39.4. The wall thickness of the cylindrical and hemispherical parts may be different. This is often necessary because the hoop stress in the cylinder is twice that in a sphere of the same inside diameter and wall thickness. The internal diameters of both parts are generally considered equal. In order that there should be no distortion or mismatch of hoop stress at the junction, the hoop stresses for the cylindrical part and the hemispherical part must be equal at the end junctions. Therefore, pdi pdi [2 ] = [1 ] 4Etc 4Ets (39.28)
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where tc = thickness of the cylinder, and ts = thickness of the hemisphere. Simplification of Eq. (39.28) gives ts (1 v) = tc (2 v) 39.4.7 Effects of Joints and End Plates in Vessel Fabrication The preceding sections have assumed homogeneous materials and uniform material properties of the components. The effects of the end plates and joints will be the reduction of strength of the components due to characteristic fabrication techniques, such as riveted joints, welding, etc. To some extent, this reduction is taken into account by using a parameter, joint efficiency, in the equation of the stresses. For a thin cylindrical vessel as depicted in Fig. 39.5 the actual hoop and longitudinal stresses are given by the following equations: h = pdi 2t l (39.30) (39.29)
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