On a gloomy night, Prof. Sirkegiannis lights a candle and watches how it melts. The candle with length $L$ is cylindrical and burning in his large office where temperature can be taken constant, Ta$.$ The convective heat transfer coefficient $($h$)$ of air in the office can also be taken constant. The top surface of the candle has always the melting temperature $($due to burning of the candlewick and wax$)$ Tm$.$ The bottom temperature is at Ta since the candle is long. He considers the heat transfer $ln2D$ for this question at the steadystate case that can be expressed by the following model equation.

$\frac{1}{r}\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{r}}(r\frac{\text{d}\text{e}\text{l}\text{T}}{\text{d}\text{e}\text{l}\text{r}})+\frac{de{l}^{2}T}{del{z}^2}=0$

a$)$ Briefly explain the physical meanings of each term in the model equation and what terms he ignored for us in the energy conservation equation?

More clue was given thanks to his ingenuity that integration of the above equation in radial direction yields

$-\frac{hR}{k}(T-T_a)+\frac{R^2}{2}\frac{d^{2}T}{d{z}^2}=0$

b$)$ Introduce $=\frac{z}{L}$ and $=\frac{T-T_a}{T_m-T_a},$ to show that the above equation reduces to

$\frac{d^2}{d{}^2}=K^2$

c$)$ Solve the last ODE to obtain the temperature profile T$($z$)$ for Sirkegiannis's candle.