(Spherical HDE with heat generation) Thermal ablation is used to treat cancerous tissue by heating it to a high enough temperature. A recent technique is to implant small metallic spheres (thermoseeds) at precise location within the cancer tumor. The metallic spheres generate volumetrically uniform heat rate when subjected to an oscillating magnetic field, while the surrounding tumor does not have a heat generation. The generated heat is conducted from the spheres into the surrounding tumor tissue. We are interested in find the ID temperature field associated with a single thermoseed placed in an infinite medium of tissue. Interactions with neighboring thermoseeds can be neglected. Assume the thermoseed has thermal conductivity $k_{ts}$, radius $r_{ts}$, and volumetric heat generation rate of $Q$. The temperature far away from the thermoseed is the body temperature $T_b$. The tumor tissue has a thermal conductivity of $k_t$.
Part A: Determine the steady state radial temperature variation $T(r)$ in the thermoseed, and in the surrounding tissue.
Part B: Determine the location and magnitude of the maximum temperature in the tissue.
Part C: Assuming $r_{ts} = 1$ mm, $k_{ts} = 10$ W/m-K, $T_b = 37^\circ$C, $k_t = 0.5$ W/m-K, and that 1 W of heat generation is uniformly distributed throughout the thermoseed, determine the maximum temperature from Part B, and sketch the temperature variation.
