Question

Consider a homogeneous spherical piece of radioactive material of radius r0 = 0,04 m that is generating heat at a constant rate of g = 4 x 10⁷ W/m³. The heat generated is dissipated to the environment steadily. The outer surface of the sphere is maintained at a uniform temperature of 80 0C and the thermal conductivity of the sphere is k = 15 W/m. 0C. Assuming steady one-dimensional heat transfer express the differential equation and the boundary conditions for heat conduction through the sphere.

Answer 1

The differential equation for steady-state heat conduction through a sphere is derived using the heat equation in spherical coordinates with boundary conditions accounting for symmetry at the center and a set temperature at the surface.

Step-by-step explanation:

To derive the differential equation for heat conduction through a homogeneous spherical piece of radioactive material, we use the heat equation in spherical coordinates assuming steady-state conditions and one-dimensional radial heat transfer. This equation can be expressed as:

dT/dr = (1/r^2) * (d/dr) (r^2 * (-k) * dT/dr) + g/k

where T is the temperature, r is the radial coordinate, k is the thermal conductivity, and g is the volumetric heat generation rate.

The boundary conditions for this problem are:

1. At r = 0, the symmetry requires that (d/dr)(-k * dT/dr) = 0, meaning the radial derivative of the heat flux is zero.
2. At r = r0, the temperature is maintained at 80°C, so T(r0) = 80°C.

We can then solve this differential equation with the given boundary conditions to find the temperature distribution within the sphere.