Question

For steady state one-dimensional heat conduction through a plane wall with constant thermal conductivity and no internal heat generation, the temperature distribution within the wall will be:
1. hyperbolic
2. ​elliptic
3. linear
4. non-linear

Answer 1

Correct Answer - Option 3 : linear

Explanation:

The temperature distribution for various types of geometry with constant thermal conductivity and no internal heat generation is given in the table below:

Geometry	Temperature Profile	Nature	Heat conduction equation
Plane wall	\(\frac{{T - {T_1}}}{{{T_2} - {T_1}}} = \frac{x}{L}\)	Linear	\(Q = \frac{{\left( {{T_1} - {T_2}} \right)}}{{\frac{L}{{kA}}}}\)
Hollow cylinder	\(\frac{{T - {T_1}}}{{{T_2} - {T_1}}} = \frac{{ln\left( {\frac{r}{{{r_1}}}} \right)}}{{ln\left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}\)	Logarithmic	\(Q = \frac{{\left( {{T_1} - {T_2}} \right)}}{{\frac{{In\left( {\frac{{{r_2}}}{{{r_1}}}} \right)}}{{k\left( {2\pi L} \right)}}}}\)
Hollow sphere	\(\frac{{T - {T_1}}}{{{T_2} - {T_1}}} = \frac{{\frac{1}{{{r_1}}} - \frac{1}{r}}}{{\frac{1}{{{r_1}}} - \frac{1}{{{r_2}}}}}\)	Hyperbolic	\(Q = \frac{{\left( {{T_1} - {T_2}} \right)}}{{\frac{{\left( {{r_2} - {r_1}} \right)}}{{4\pi {r_1}{r_2}k}}}}\)