Conduction, thermal hollow cylinders

Derive equations that describe the temperature profiles for a plane wall, long hollow cylinder, and hollow sphere. Assume constant thermal conductivity, and temperature at the walls as Tt and 7 2. 

J 8 An electrical conductor 2 cm in diameter fits closely inside a long hollow cylinder having an inside diameter of 2 cm and an outside diameter of 6 cm. The material from which the cylinder is madehas a temperature-dependent thermal conductivity (Fig.2P-15). The outer surface of the cylinder is maintained at 1600 C, and the electrical conductor dissipates 500 W per m of length. Compute the temperature at the inside surface of the hollow cylinder. 

HEAT pW THROUGH A CYLINDER. Consider the hollow cylinder represented by Figf 10 3. The inside radius of the cylinder is r.-, the outside radius is r , and the length of the cylinder is L. The thermal conductivity of the material of which the cylinder is made is k. The temperature of the outside surface is %, and that of the inside surface is 7. It is desired to calculate the rate of heat Sow outward for this case. 

In Eqs. (4.77) and (4.78) A is the thermal conductivity of the material of the hollow cylinder, a the coefficient for convective heat transfer, and Bi Biot s number on the inner (i) and outer (a) surfaces of the hollow cylinder, respectively, with... 

Figure 12, Thermal conductivity of packed beds of hollow cylinders at various temperatures according to Ref. and Equation 23. (a) Ceramic- nitrogen (h) steelnfiitrogen.

This assumes that the power density R and the thermal conductivity k are uniform and the temperature at r = ft is taken as the reference zero. For solid cylinders the logarithmic term becomes zero and the distribution is parabolic. When such a temperature distribution is imposed on a finite cylinder it distorts so that the flat end faces bulge out and the curved surface bends outwards, the rim being displaced further than the center belt. An exaggeration of this shape is shown in Fig. 2 for sections of a solid cylinder and a hollow cylinder where a jb = 0.2. In this case most of the rim displacement takes place within 0.1 of the cylinder length from the end. For a cylinder with a length to diameter ratio of 1 the difference between the radial displacements of the rim and belt is approximately half... 

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