Question

Four identical rods AB, CD, CF and DE are joined as shown in figure (28-E8). The length, cross-sectional area and thermal conductivity of each rod are l, A and K respectively. The ends A, E and F are maintained at temperature T₁, T₂ and T₃ respectively. Assuming no loss of heat to the atmosphere, find the temperature at B.

Answer 1

Let the temperature at junction B be T.
Let q₁, q₂ and q₃ be the heat currents, i.e. rate of flow of heat per unit time in AB, BCE and BDF, respectively.
From the diagram, we can see that
q₁ = q₂ + q₃
The rate of flow of heat is givem by
$q=\frac{KA∆T}{l}$
Using this tn the above equation, we get

$$\begin{gathered} \frac{KA\left(T_1-T\right)}{l}=\frac{KA\left(T-T_3\right)}{{\displaystyle \frac{3l}{2}}}+\frac{KA\left(T-T_2\right)}{{\displaystyle \frac{3l}{2}}} \\ \Rightarrow T_{\mathit{1}}\mathit{-}T\mathit{}\mathit{=}\mathit{}\frac{\mathit{2}\mathit{}\left(\mathit{T}\mathit{-}{\mathit{T}}_{\mathit{3}}\right)}{\mathit{3}}\mathit{+}\frac{\mathit{2}\mathit{}\left(\mathit{T}\mathit{-}{\mathit{T}}_{\mathit{2}}\right)}{\mathit{3}} \\ \mathit{\Rightarrow }3\left({\mathit{T}}_{\mathit{1}}\mathit{-}\mathit{T}\right)\mathit{=}\mathit{}2T\mathit{-}2T_{\mathit{3}}\mathit{+}2T\mathit{-}2T_{\mathit{2}} \\ \mathit{\Rightarrow }T\mathit{}\mathit{=}\mathit{}\frac{\mathit{-}\mathit{3}{\mathit{T}}_{\mathit{1}}\mathit{+}\mathit{2}\mathit{}\left({\mathit{T}}_{\mathit{2}}\mathit{+}{\mathit{T}}_{\mathit{3}}\right)}{\mathit{7}} \end{gathered}$$