2.1 Introduction to pressure vessels
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1 2.1 Introduction to pressure vessels Pressure vessels in the form of cylinders and tanks are used for storing variety of liquids and gasses at different temperatures and pressures. Some of the substances stored can be lethal to human beings if exposed and some can be highly explosive. Bursting of pressure vessels due to improper design can prove fatal to human life and property. It is imperative that a designer should have a comprehensive understanding of the principles of designing such pressure vessels based on national and international standards. 2.2 Stresses in thin cylinders If the wall thickness is less than about 7% of the inner diameter then the cylinder may be treated as a thin one. Thin walled cylinders are used as boiler shells, pressure tanks, pipes and in other low pressure processing equipments. A thin cylinder is also defined as one in which the thickness of the metal is less than 1/20 of the diameter of the cylinder. In thin cylinders, it can be assumed that the variation of stress within the metal is negligible, and that the mean diameter, d m is approximately equal to the internal diameter, d i. In general three types of stresses are developed in pressure cylinders viz. circumferential or hoop stress, longitudinal stress in closed end cylinders and radial stresses. These stresses are demonstrated in figure 2.1. Figure 2.1 Radial stress in thin cylindrical shells can be neglected as the radial pressure is not generally high and that the radial pressure acts on a larger area. The internal pressure, p tends to increase the diameter of the cylinder and this produces a hoop or circumferential stress (tensile). If the stress becomes excessive, failure in the form of a longitudinal burst would occur.
2 Consider the half cylinder shown. Force due to internal pressure, p i is balanced by the force due to hoop stress, σ t i.e. hoop stress x area = pressure x projected area Longitudinal stress in a cylinder: σ t x 2 L t = p x L x d i σ t = (p d i ) / 2 t Figure 1.2 The internal pressure p also produces a tensile stress in the longitudinal direction as shown in figure 1.2. The force P acting on an area (πd 2 i / 4) is balanced by longitudinal stress σ t acting over an approximate area πd i t. σ t x πd i t = p (πd 2 i / 4) σ t = pd i / 4t Since hoop stress is twice longitudinal stress, the cylinder would fail by tearing along a line parallel to the axis, rather than on a section perpendicular to the axis. The equation for hoop stress is therefore used to determine the cylinder thickness. Pressure vessels are generally manufactured from curved sheets joined by welding. Mostly V butt welded joints are used. The riveted joints may also be used. Since the plates are weakened at the joint due to the rivet holes, the plate thickness should be enhanced by taking into account the joint efficiency. Allowance is made for this by dividing the thickness obtained in hoop stress equation by efficiency (i.e. tearing and shearing efficiency) of the joint. A typical welded construction of a pressure vessel is shown in figure 2.3 and riveted construction is shown in figure 2.4.
3 Figure 2.3. Welded construction of a pressure vessel Figure 2.4. Rivetted construction of a pressure vessel 2.3 Stress induced in a spherical shell: A sphere is the most favorably stressed shaped for a vessel requiring minimum wall thickness. It is used for extremely high pressure options. It is used in space vehicles and missiles for the storage of liquefied gasses at lower pressures but with light weight thin walls. Spheres also have the greatest buckling resistance. Spherical vessels are used as pressure carrying structures and as living space in most deep submerged vehicles for oceanography. The stress induced in a spherical vessel is as shown in figure 2.5 and is given by: σ t = pd i / 4t Figure 2.5 Stress in a spherical pressure vessel
4 2.4 EXAMPLE ON THIN CYLINDERS E1. An air receiver consisting of a cylinder closed by hemispherical ends is shown in Figure below. It has a storage capacity of 0.25 m 3 and an operating internal pressure of 5 MPa. It is made of plain carbon steel 10C4 with an ultimate tensile strength of 340 MPa. Factor of safety to be used is 4. Neglecting the effect of welded joints, determine the dimensions of the receiver. Volume of the vessel V= π d 2 i L/4 + πd 3 i/6 Substituting L=2d i and simplifying; d i = (3V/2π) ⅓ = m or 500 mm L=2di = 500 X 2 =1000 mm. Allowable stress= 340/4 =85 MPa. Thickness of the cylinder = (neglecting the effect of welded joint) t= pd i /2σ t t= 14.7 mm or 15 mm. Thickness of the hemispherical Ends: t= pd i /4σ t t=7.35 mm or 7.5 mm. Problems for practice: EP1. A seamless pipe 800 mm in diameter contains air at a pressure of 2 MPa. If the permissible stress of the pipe material is 100 MPa, find the minimum thickness of the pipe. EP2. A cylindrical air receiver for a compressor is 2m in internal diameter and made of plate 15 mm thick. If the hoop stress is not to exceed 90 MPa and longitudinal stress is not to exceed 60 MPa find the safe air pressure. EP3. A cylindrical shell of 2.2m internal diameter is constructed of mild steel plate. The shell is subjected to an internal pressure of 0.8 MPa. Determine the thickness of the shell plate by adopting a factor of safety of 6. The ultimate tensile strength of steel is 470 MPa. The efficiency of the longitudinal joint may be taken as 78%.
5 2.5 THICK CYLINDERS If the wall thickness is more than about 7% of the inner diameter then the cylinder may be treated as a Thick Cylinder. Difference in Treatment between Thin and Thick Cylinders In thin cylinders the hope stress is assumed to be constant across the thickness of the cylinder wall. In thin cylinders there is no pressure gradient across the wall. In thin cylinders, radial stress is neglected (while it is of significant magnitude in thick cylinders). In thick cylinders none of these assumptions can be used and the variation of hoop and radial stress will be as shown in figure 2.6. Figure 2.6 Variation of stresses in thin and thick cylinders A thick cylinder subjected to both internal and external pressure is shown in figure 2.7. Figure 2.7 Thick cylinder subjected to both internal and external pressure Figure 2.8 Radial and Tangential stress (hoop stress) distribution in a thick cylinder subjected to internal pressure only
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