Thick-walled cylinder equations

Chamberlain [114] derived an alternate model for transverse expansion of unidirectional composites using thick-walled cylinder equations for the case of a fiber embedded in a cylindrical matrix section. The radial displacements on the outside of the cylinder were related to in the radial direction. The expression for a is given by the following equation ... 

Bursting tests have been carried out on neatly a hundred thick-walled cylinders made of carbon, low alloy, and stainless steels, together with some nonferrous materials. The diameter ratio of the cylinders varied from 1.75 to 5.86, and some tests were carried out at 660°C. An analysis of the results (19) showed that 90% of the cylinders burst within 15% of the value given by equation 17. 

For a thick-walled cylinder, the rate of conduction of heat through lagging is given by equation 9.21 ... 

If the pressure in a thick-walled cylinder is raised beyond the yield pressure pei according to the equation (9), the yield will spread through the wall until it reaches the outer diameter [10]. For a perfectly elastic-plastic material the ultimate pressure for complete plastic deformation of the thick wall pCOmpi-pi, also called collapse pressure, can be calculated by equation (4.3-10). As the ductile materials used for high pressure equipment generally demonstrate strain... 

As a final result of the explanations about strengthening measures the admissible static internal pressure for thick-walled cylinders is compared in Fig. 4.3-7 for different design strategies according to the equations (4.3-9), (4.3-10), (4.3-12) and (4.3-13) and the explained assumptions and optimisations. In the case of the monobloc (A), the two-piece shrink fit and the autofrettaged cylinders the maximum stress at the inner diameter stays within the elastic limit (00.2). Comparatively much larger is the admissible pressure when complete plastic yielding occurs as shown for the collapse pressure (pCOmpi pi. = Pcoii D). 

Equation (10.12) can be used to calculate the flow of heat through a thick-walled cylinder. It can be put in a more convenient form by expressing the rate of flow of heat as... 

Derive the equation for steady-state heat transfer through a spherical shell of inner radius and outer radius rj. Arrange the result for easy comparison with the solution for a thick-walled cylinder. 

There are many important elastic problems that involve circular symmetry. Of interest here is the solution for a thick-walled cylinder under the action of internal and external pressures and respectively, as shown in Fig. 4.15. For this problem, the symmetry is such that the stresses will not depend on 6. Hence and all dx/d 0 terms vanish, which allows Eq. (4.28) to be reduced to the ordinary differential equation. 

Thick walled cylinders can be designed to either ASME Section VIII, Divisions 1, 2 or 3. However, what is different in each of the Divisions is the allowable stress. Using the same allowable stress in the equations for the three divisions will yield approximately the same results. An example has been provided to illustrate this. 

This chapter will focus on developing the equations, assumptions and procedures one must use to solve two and three dimensional viscoelastic boundary value problems. The problem of an elastic thick walled cylinder will be used as a vehicle to demonstrate how to obtain the solution of a more difficult reinforced viscoelastic thick walled cylinder. In the process, we first demonstrate how the elasticity solution is developed and then apply the correspondence principle to transform the solution to the viscoelastic domain. Several extensions to this problem will be discussed and additional practice is provided in the homework problems at the end of the chapter. 

Determine the stresses in a thick wall cylinder similar the one of Fig. 9.1 using the integral equation solution given by Eq. 9.58. Use the assumption of a step input in pressure as well as elastic response in bulk and Maxwellian in shear as in the earlier example. Compare your solution to that obtained using the correspondence principle. 

In many applications in the process industry, the need arises to calculate the heat transfer through a thick-walled cylinder, as in pipes. Consider a hollow cylinder with an inside radius r where the temperatme is Tj, and an outside radius rj, where the temperature is T2, having a length L measmed in meters. Heat is flowing radially from the inside surface to the outside. Equation 6.4, as given earlier, is readily applied, with distance dr instead of dx. 

The second method is the Lame approach (Kashani and Young, 2008), which is based on displacement differential equations and is applicable to any cylindrical vessel with any diameter-to-wall-thickness ratio. The Lame method is often referred to as the solution for thick wall cylindrical pressure vessels. Equations for the hoop stress and radial stress in a thick-walled cylinder were developed by Lame in the early nineteenth century (Timoshenko and Goodier, 1969) ... 

Using the equations for stress in thick-walled cylinders from the theory of elasticity, or from a design textbook derive Equation 7.42. Hint Use Hooke s law in the form... 

Elastic Behavior. In the following discussion of the equations relevant to the design of thick-walled hoUow cylinders, it should be assumed that the material of which the cylinder is made is isotropic and that the cylinder is long and initially free from stress. It may be shown (1,2) that if a cylinder of inner radius, and outer radius, is subjected to a uniform internal pressure, the principal stresses in the radial and tangential directions, and <7, at any radius r, such that > r > are given by... 

Preparation of Potassium Hydroxide by the Electrolysis of a Potassium Chloride Solution. Assemble an electrolyzer (see Fig. 130, p. 231). Place small cylinder 2 (8 cm in height and 4 cm in diameter) made from uncalcined clay into 0.5-litre thick-walled beaker 1. Pour a saturated potassium chloride solution into both vessels so that the level of the liquid in them will be the same. Add a few drops of phenolphthalein to the electrolyte. Use carbon rod 4 as the anode and thick iron wire 3 as the cathode. Secure both electrodes with corks in the electrolyzer lid. A d-c source at 10 V is needed for the experiment. After assembling the electrolyzer, switch on the current. What happens in the anode and cathode compartments Write the equations of the reactions. What substances can form in the absence of a diaphragm ... 

The calculation of the strength resp. the admissible internal pressure varies with the wall-thickness thick-walled hollow cylinders are calculated by neglecting the radial stress (equal to the pressure) which is small compared to the tangential. On the other side the thick-walled hollow cylinders are calculated with the Lame equations (1833). 

The structure of this formula can quickly be related to the thin-walled pressure vessel cylinder equation. Using the equation that calculates the stress at the center of the vessel wall, ux = P R + 0.5t)/t, and rearranging to solve for the thickness, results m. t = PR/ ux — 0.5P. The addition of the weld joint efficiency, E, and changing the coefficient before P to 0.6 results in the ASME code formula, t = PR/ SE — 0.6P), which they feel best represents the minimum wall thickness required to contain an internal pressure, P, in a cylindrical vessel having a radius, R, and made of a material with an allowable stress, S. 

No exact solutions for Rac are available when the walls are not finlike and L is finite. But for circular cylinder cavities with thick walls and for LID —> =, Ostroumov [216] showed that Fig. 4.30 is valid, provided k%, in Eq. 4.83 is equated to km, where kw, (the equivalent finlike wall conductivity of a very thick wall material—see Table 4.6, entry 7) is defined by... 

The state of stress in a cylinder subjected to an internal pressure has been shown to be equivalent to a simple shear stress, T, which varies across the wall thickness in accordance with equation 5 together with a superimposed uniform (triaxial) tensile stress (6). 

In some cases, particularly for the radial flow of heat through a thick pipe wall or cylinder, the area for heat transfer is a function of position. Thus the area for transfer applicable to each of the three media could differ and may be A, A2 and A3. Equation 9.3 then becomes ... 

The minimum wall thickness required to resist the hydrostatic pressure can be calculated from the equations for the membrane stresses in thin cylinders (Section 13.3.4) ... 

A 4-in schedule 40 pipe has an outside diameter of 4.5 in, a wall thickness of 0.237 in, and an inside diameter of 4.026 in. From Table 12-3 the tensile strength SM for stainless 316 is 85,000 psi. Equation 12-4 for cylinders is used to compute the pressure necessary to rupture this pipe ... 

Over the years, vapour adsorption and condensation in porous materials continue to attract a great deal of attention because of (i) the fundamental physics of low-dimension systems due to confinement and (ii) the practical applications in the field of porous solids characterisation. Particularly, the specific surface area, as in the well-known BET model [I], is obtained from an adsorbed amount of fluid that is assumed to cover uniformly the pore wall of the porous material. From a more fundamental viewpoint, the interest in studying the thickness of the adsorbed film as a function of the pressure (i.e. t = f (P/Po) the so-called t-plot) is linked to the effort in describing the capillary condensation phenomenon i.e. the gas-Fadsorbed film to liquid transition of the confined fluid. Indeed, microscopic and mesoscopic approaches underline the importance of the stability of such a film on the thermodynamical equilibrium of the confined fluid [2-3], In simple pore geometry (slit or cylinder), numerous simulation works and theoretical studies (mainly Density Functional Theory) have shown that the (equilibrium) pressure for the gas/liquid phase transition in pores greater than 8 nm is correctly predicted by the Kelvin equation provided the pore radius Ro is replaced by the core radius of the gas phase i.e. (Ro -1) [4]. Thirty year ago, Saam and Cole [5] proposed that the capillary condensation transition is driven by the instability of the adsorbed film at the surface of an infinite... 

Table 2.7 Equations for the calculation of the solidifying time t for cylindrical and spherical layers of thickness s, see Fig. 2.39. R always indicates the radius of the cylinder or sphere at the side where the solidified layer develops. AR is the thickness of the wall of the cylinder or sphere.

The value of Sg can be determined in a manner somewhat similar to that proposed in the part on foundation bolts. There, the overturning load was calculated as a function of the periphery of the foundation bolt circle by means of equations (11-30) and (11-33). The bolt circle was assumed to be a hollow cylinder the wall thickness being infinitely small, as compared with the diameter. 

An analytic solution has been found for representing the rate of ablative melting of a solid cylinder pressed against a horizontal wall maintened at (7 ). In steady state, a liquid layer of constant thickness is formed between the hot surface and the rod, with a radial flow of liquid. The resolution of the equation of liquid flow associated with that of energy balance between the two surfaces allows to derive the following relationship ... 

Unsteady-state conduction in a cylinder. In deriving the numerical equations for unsteady-state conduction in a flat slab, the cross-sectional area was constant throughout. In a cylinder it changes radially. To derive the equation for a cylinder. Fig. 5.4-3 is used where the cylinder is divided into concentric hollow cylinders whose walls are Ax m thick. Assuming a cylinder 1 m long and making a heat balance on the slab at point n, the rate of heat in — rate of heat out = rate of heat accumulation. 

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