Two rods of equal cross-sectional area

PHYSICS · Multi-part · A Level · Physics · Further mechanics and thermal physics · Further mechanics and thermal physics

Problem

Two rods of equal cross-sectional area, one of brass (thermal conductivity

k_\text{b}

) and the other of steel (thermal conductivity

k_\text{s}

), are welded end-to-end to form a composite rod of total length

L

. The brass segment occupies a fraction

f

of the total length, so its length is

fL

and the steel segment’s length is

(1-f)L

. The free ends of the composite rod are kept at fixed temperatures that differ by

\Delta T

. Assume no energy is lost through the sides of the rod and that steady-state conditions have been reached. (a) Derive an expression for the steady-state energy transfer rate

R

through the composite rod in terms of

k_\text{b}

k_\text{s}

A

L

f

, and

\Delta T

. (b) Show algebraically that when

k_\text{b}\gg k_\text{s}

, the overall rate

R

is limited primarily by the steel segment. (c) For

k_\text{b}=110\,\text{W}\!\cdot\!\text{m}^{-1}\!\cdot\!^{\circ}\text{C}^{-1}

k_\text{s}=50\,\text{W}\!\cdot\!\text{m}^{-1}\!\cdot\!^{\circ}\text{C}^{-1}

A=4.0\times10^{-4}\,\text{m}^2

L=0.40\,\text{m}

f=0.30

, and

\Delta T=75\,^{\circ}\text{C}

, compute

R

numerically. (d) Suppose the brass fraction

f

is doubled while all other quantities remain fixed. Determine the percentage change in

R

and interpret the physical significance of your result.

Related theory

Browse the problem library