Question

Two chunks of metal with heat capacities $C_1$ and $C_2$ are interconnected by a rod of length $l$ and cross-sectional area A and fairly low conductivity k. The whole system is thermally insulated from the environment. At a moment t = 0, the temperature difference between two chunks of metal equals $(\Delta T{)}_0$. Assuming the heat capacity of the rod to be negligible, find the temperature difference between the chunks as a function of time.

Answer 1

Given: Two chunks of metal with heat capacities $C_1$ and $C_2$ are interconnected by a rod of length $l$ and cross-sectional area A and fairly low conductivity $k$. The whole system is thermally insulated from the environment. At a moment $t=0$, the temperature difference between two chunks of metal equals $(\Delta T{)}_0$. Assuming the heat capacity of the rod to be negligible,

To find the temperature difference between the chunks as a function of time.

Solution:

Suppose the chunks have temperature $T_1,T_2$ at $t$ and $T_1-d{T}_1,T_2+d{T}_2$ at $dt+t$.

Then, $C_{1}d{T}_1=C_{2}d{T}_2=\frac{kS}{l}(T_1-T_2)dt$

Thus, $d\Delta T=\frac{-kS}{l}(\frac{1}{C_1}+\frac{1}{C_2})\Delta Tdt$ where $\{\Delta T=T_1-T_2\}$

Hence, $\Delta T={\left(\Delta T\right)}_0^{e^{\frac{-t}{\tau }}}$ where $\{\frac{1}{\tau }=\frac{kS}{l}(\frac{1}{C_1}+\frac{1}{C_2})\}$

is the required temperature difference between the chunks as a function of time.