Thick-walled vessels

The principal stresses (see Section 13.3.1) acting at a point in the wall of a vessel, due to a pressure load, are shown in Figure 13.1. If the wall is thin, the radial stress comparison with the other stresses, and the longitudinal and circumferential stresses o and <72 can be taken as constant over the wall thickness. In a thick wall, the magnitude of the radial stress will be significant, and the circumferential stress will vary across the wall. The majority of the vessels used in the chemical and allied industries are classified as thin-walled vessels. Thick-walled vessels are used for high pressures, and are discussed in Section 13.15. 

The equation given in the British Standard PD 5500 differs slightly from equation 13.40, as it is derived from the formula for thick-walled vessels see Section 13.15. 

This is the hoop stress for thin-waUed vessels, i.e., t < r/4. For thick-walled vessels, see Ashby, p. 396 [1]) The longitudinal stress can be obtained from a similar shell balance (see Figure 8.7)... 

Fig. 4.3-1. Basic features of the thick-walled vessels (adopted from Lewin et. al. [1]). A welded, with cover and closed-end B welded, with two covers...

This test is also based on the determination of the range distance of a heavy projectile. The explosive is suspended in a thick-walled vessel, and an accurately fitting cap of the vessels is projected. This apparatus is stronger, and the weight of the charge may be made as large as 500 g. 

A mixture of 100 g. (0.63 mole) of dry malonic ester and 70 g. (0.66 mole) of benzaldehyde is placed in a 200-ml. thick-walled vessel, and 2 g. of piperidine is added gradually. The reaction vessel is stoppered securely and allowed to stand for 2 days. The reaction mixture is then heated on a water bath for 12 hours. Ether is added, and the resulting solution is washed with dilute aqueous acid and then with water. The ethereal solution is dried over anhydrous sodium sulfate and distilled. There is obtained a 70% yield of ethyl benzalmalonate boiling at 185-186°/11 mm. and melting at 27-27.5°. 

The calculations described in this chapter apply only to cylindrical vessels whose lining is thin relative to the radius of the vessel. Thick-walled vessels, and vessels of other shapes such as rectangular or spherical, will require considerably more complex mathematical analysis which is beyond the scope of this handbook. In such cases, exact formulae are difficult or impossible to obtain, and the designer must resort to computer programs for performing the complex calculations. 

Treat 20 grammes of benzaldehyde in a stoppered cylinder or a thick-walled vessel with a cold solution of 18 grammes of potassium hydroxide in 12 grammes of water, and shake until a permanent emulsion is formed the mixture is then allowed to stand over night. The vessel is closed by a cork, and not a glass stopper, since at times a glass stopper becomes so firmly fastened that it can be removed only with great difficulty. To the crys-1B. 14, 2394. 

The classic approach to pressure vessel design differentiates between a thin-walled and a thick-walled vessel. 

The formulas for thin-walled pressure vessels are first-order equations and are easier to rearrange and solve for minimum thickness and maximum stress values. The thick-walled vessel formulas provide the most accurate value for the stresses in the pressure vessel wall, but solving the thin-walled equations provides comparatively accurate results and is, therefore, quite useful for preliminary design estimates. 

A comparison of the equations developed above is shown in Fig. 1. Each of the four formulas, referred to as Thin-1, Thin-2, ASME, and Thick-walled, were used to evaluate the stress on the walls of the same hypothetical pressure vessel (with a radius of 2 in. and a pressure of 1000 psi) at various wall thicknesses between 1/8 and 7/8 in. The resultant stresses for the first three formulas were then normalized to the calculated thick-walled vessel stress and plotted to demonstrate the relative accuracy of the various methods. 

The safe design of a pressure vessel takes into account the strength of the material and the stresses that are imposed on it by internal pressure and exterior forces. The approach to satisfactory design can be best understood with an appreciation of the equations of a thin-walled vessel. Expansion of these concepts to the implications of a thick-walled vessel will lead to the code rules written by the ASME. 

Much information is available on the deformation and fatigue behavior of simple thick-walled cylinders [10-17], but it must be remembered that most process reactors will not be a simple hollow cylinder. Components such as connectors, threads and sleeves, windows, and removable closures make a complete analytical solution for a high-pressure system design problem quite involved. Useful design criteria for thick-walled vessels can be derived, however, under the assumption that the material of which the vessel is made is isotropic and that the cylinder is long (more than five diameters) and initially free from stress. The radial and tangential stresses in the walls are then only functions of the radius coordinate (r) and the internal pressure. Given the outer-to-inner wall radius ratio as o/i = w, and the yield point (To) of the material, the yield pressure (py) is... 

Figure 2.1-1 Thick walled vessels (a) double walled cylinder with interference fit d (b) single walled, autofrettaged. (o = outer radius, i = inner radius, c = interference diameter).

DESCRIBE why thermal shock is a major concern in reactor systems when rapidly heating or cooling a thick-walled vessel. 

Thermal stresses are a major concern in reactor systems due to the magnitude of the stresses involved. With rapid heating (or cooling) of a thick-walled vessel such as the reactor pressure vessel, one part of the wall may try to expand (or contract) while the adjacent section, which has not yet been exposed to the temperature change, tries to restrain it. Thus, both sections are under stress. Figure 1 illustrates what takes place. 

A vessel is considered to be thick-walled or thin-walled based on comparing the thickness of the vessel wall to the radius of the vessel. If the thickness of the vessel wall is less than about 1 percent of the vessel s radius, it is usually considered a thin-walled vessel. If the thickness of the vessel wall is more than 5 percent to 10 percent of the vessel s radius, it is considered a thick-walled vessel. Whether a vessel with wall thickness between 1 percent and 5 percent of radius is considered thin-walled or thick-walled depends on the exact design, construction, and application of the vessel. 

Shock experienced by a thick-walled vessel due to the combined stresses from a rapid temperature and/or pressure change. 

Solution The stresses existing in the walls of a cylindrical vessel under pressure can be reduced to (1) the tangential or hoop stress o, (2) the radial stress Oj, and (3) the longitudinal stress Oj. For a cylinder under internal pressure, and Oj are tensile stresses while o, is a compressive stress. vessel failure. For a thick-walled vessel (i.e., where r /rj = R > 1.2) ... 

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