Homework # $1$

Due: Monday October $21,2024$

Question $1$: Please provide a step$-$by$-$step derivation of the general heat diffusion equation in cylindrical coordinates. Hint: your starting point is the conservation of energy equation. Please do not forget to provide necessary schematic.

Question $2$: A spherical particle of radius $r_1$ experiences uniform thermal generation at a rate of $q.$ The particle is encapsulated by a spherical shell of outside radius $r_2$ that is cooled by ambient air. Thermal conductivities of particle and shell are $k_1$ & $k_2,$ respectively. Schematic is attached below. Hint: remember to look up the formulas for volume and surface area of the sphere. Surface area of sphere is the surface area of a hemisphere multiplied by $2.$

By applying the conservation of energy principle to spherical control volume A$,$ which is placed at an arbitrary location within the sphere determine a relationship between the temperature gradient $d\frac{T}{d}r$ and the local radius $r,$ for $0r{r}_1.$ Assume steady state conditions.

By applying the conservation of energy principle to spherical control volume B $,$ which is placed at an arbitrary location within the spherical shell, find the relationship between $d\frac{T}{d}r$ and r for $r_{1}r{r}_2.$ Assume steady state conditions.