A solid aluminum bar forms a thermal "bridge" that carries energy from a hot plate to…

MATHEMATICS · Multi-part · CBSE · Mathematics · Area · Area

Problem

A solid aluminum bar forms a thermal "bridge" that carries energy from a hot plate to a cold reservoir. The bar’s length is fixed at

0.050\,\text{m}

, but space constraints require its rectangular cross-section to have a constant perimeter of

0.20\,\text{m}

. The temperature difference between the two ends of the bar is held constant during operation. (a) Express the cross-sectional area of the bar in terms of its width

w

and height

h

, and write an equation that relates

w

and

h

using the fixed-perimeter condition. (b) Using algebraic reasoning (no calculus required), show that the cross-sectional area is maximized when the bar’s cross-section is a square. State the corresponding optimal dimensions. (c) Calculate the maximum possible conduction rate through the bar in terms of the aluminum’s thermal conductivity

k_{\text{Al}}

and the imposed temperature difference

\Delta T

. Express your answer as a multiple of

k_{\text{Al}}\Delta T

. (d) Compare the conduction rate of the optimized square bar to that of a bar having the same perimeter but dimensions

w = 0.020\,\text{m}

and

h = 0.080\,\text{m}

. State the ratio of the two rates and interpret its physical meaning.

Related theory

Browse the problem library