$1\mathrm{.}5.$ Distributed Thermal Model for Thermocouple

If the plastic bead covering the tip of the thermocouple described in Problem $1\mathrm{.}4$ is quite large, and since plastic usually sustains a very low value of thermal conductivity, then the simple lumped model solution becomes quite inaccurate. To improve the model, we need to account for thermal conductivity in the $($assumed$)$ spherical shape of the plastic bead.

$($a$)$ Assuming the bead is a perfect sphere, contacted everywhere by external fluid of temperature $T_f,$ perform a shell balance on an element of volume $4r^{2}r$ and show that

$C_p\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{t}}=k\frac{1}{r^2}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r^2\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}})$

$($b$)$ Perform an elemental heat balance at the surface of the sphere and deduce

$-k\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}}=h(T-T_f),at,r=R$