Consider three cases of 1D heat conduction with the same heat tra
Consider three cases of 1D heat conduction with the same heat transfer rate ‘q’ as shown in the figure. These three cases are three walls with values of temperature on the both sides of it. Find out the correct relation of thermal conductivity.
A. <span class="math-tex">\({k_1} = {k_3} < {k_2}\)</span>
B. <span class="math-tex">\({k_2} = {k_3} < {k_1}\)</span>
C. <span class="math-tex">\({k_1} < {k_3} < {k_2}\)</span>
D. <span class="math-tex">\({k_1} = {k_3} > {k_2}\)</span>
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Right Answer is: A
SOLUTION
Concept:
Using Fourier law of conduction
\(q = - k\frac{{dT}}{{dx}}\)
As, |q| is constant in all cases
\({\left| {k\frac{{dT}}{{dx}}} \right|_1} = {\left| {k\frac{{dT}}{{dx}}} \right|_2} = {\left| {k\frac{{dT}}{{dx}}} \right|_3}\)
Let’s find out \(\left( {\frac{{dT}}{{dx}}} \right)\) of all cases,
\({\left| {\frac{{dT}}{{dx}}} \right|_1} = \frac{{35 - 30}}{{0.5}} = 10\)
\({\left| {\frac{{dT}}{{dx}}} \right|_2} = \frac{{35 - 32}}{{0.4}} = 7.5\)
\({\left| {\frac{{dT}}{{dx}}} \right|_3} = \frac{{35 - 28}}{{0.7}} = 10\)
\(\therefore {\left| {\frac{{dT}}{{dx}}} \right|_1} = {\left| {\frac{{dT}}{{dx}}} \right|_3} > {\left| {\frac{{dT}}{{dx}}} \right|_2}\)
∴ k1 = k3 < k2