Cylinder and Sphere Key Problem

Let the rate of energy per unit volume generated in a solid cylinder or a solid sphere be u (r) = u , the radius and the thermal conductivity of the cylinder or the sphere be R and k(T) = kr (alternate notations ur and kr are used for convenience in the following formulation). Under steady conditions, the total energy generated in the cylinder or sphere is transferred, with a heat transfer coefficient h, to an ambient at temperature Too. This cylinder could be one of the fuel rods of a reactor core, or one of the elements of an electric heater, and the cylinder or sphere could be a bare, homogeneous reactor core. We wish to determine the radial temperature distribution. 

Following the five steps of formulation, first we consider the differential system (Step 1) shown in Fig. 2.21(a). The first law of thermodynamics (Step 2), Eq. (1.16) interpreted in terms of Fig. 2.21(b), yields 

Note variable Ar is kept inside the derivative. Fourier s law of conduction (Step 3),.  

The alternate condition in Eq. (2.90), T (0) = Finite, turns out to be more convenient to use than dT(0)/dr = 0 in cases involving curvature. Note that mathematical solutions that would lead to an infinite temperature at the center are not physically meaningful. 

For a distributed energy generation and a variable thermal conductivity, integrating Eq. (2.88) twice results in 

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