Solved Problem on Heat Transfer
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A metal rod of constant cross-section and length L has its ends maintained at constant temperatures t1 and t2. Determine the temperature at the midpoint of the rod when heat flows through it at a steady state. The side surfaces of the bar are thermally insulated.
Problem data:
- Temperature at the ends of the bar: t1 e t2;
- Bar Length: L.
The problem tells us that heat flows in a steady state, that is to say, that the heat flux through a
cross-section of the bar is constant.
The length e of the bar between the ends is e=L−0=L, and the length between the end at temperature t1 and any cross-section is e=x−0=x (Figure 1).
The length e of the bar between the ends is e=L−0=L, and the length between the end at temperature t1 and any cross-section is e=x−0=x (Figure 1).
Solution
The heat flux is given by
\[ \bbox[#99CCFF,10px]
{\phi =kA\frac{\left(t_{1}-t_{2}\right)}{e}}
\]
Since the flux φ is constant, the amount of heat that passes through the ends, maintained at temperatures
t1 and t2, is equal to the amount of heat that passes through the end at
temperature t1 and any section at temperature tx
\[
\begin{gather}
\phi=kA\frac{\left(t_{1}-t_{2}\right)}{L}=kA\frac{\left(t_{1}-t{_x}\right)}{x}\\
\frac{t_{1}-t_{2}}{L}=\frac{t_{1}-t_{x}}{x}\\
t_{1}-t_{x}=x\frac{\left(t_{1}-t_{2}\right)}{L}\\
t_{x}=t_{1}-x\frac{\left(t_{1}-t_{2}\right)}{L}
\end{gather}
\]
Generally, this expression gives the temperature at any point x of the bar, in particular, we want
the temperature at the midpoint, where
\( x=\frac{1}{2}L \),
substituting this value
\[
\begin{gather}
t_{\frac{11}{2}}=t_{1}-\frac{1}{2}L\frac{\left(t_{1}-t_{2}\right)}{L}\\
t_{\frac{11}{2}}=t_{1}-\frac{t_{1}}{2}+\frac{t_{2}}{2}\\
t_{\frac{11}{2}}=\frac{2t_{1}-t_{1}}{2}+\frac{t_{2}}{2}\\
t_{\frac{11}{2}}=\frac{t_{1}}{2}+\frac{t_{2}}{2}
\end{gather}
\]
\[ \bbox[#FFCCCC,10px]
{t_{\frac{1}{2}}=\frac{t_{1}+t_{2}}{2}}
\]
The temperature at the midpoint will be the average of the temperatures at the ends.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .