A heat-conducting rod, 0.90 m long and wrapped in insulation, is made of an aluminum section that is 0.20 m long and a copper section that is $0.70 \mathrm {~m}$ long. Both sections have a cross-sectional area of $0.00040 \mathrm {~m} ^ { 2 }$ The aluminum end and the copper end are maintained at temperatures of $30 ^ { \circ } \mathrm { C }$ and $230 ^ { \circ } \mathrm { C }$ respectively. The thermal conductivities of aluminum and copper are 205 W/m ∙ K (aluminum) and 385 W/m ∙ K (copper). What is the temperature of the aluminum-copper junction in the rod with steady state heat flow?
A) 100°C
B) 93°C
C) 86°C
D) 80°C
E) 74°C

Question

Quizplus · Accepted Answer

The temperature of the aluminum-copper junction in the rod with steady state heat flow is 100°C. This is because the heat flow through the rod is constant, and the temperature gradient is therefore constant. The temperature gradient is given by the equation:$$\frac{dT}{dx} = \frac{Q}{kA}$$where Q is the heat flow, k is the thermal conductivity, and A is the cross-sectional area. Since the heat flow and the cross-sectional area are constant, the temperature gradient is also constant. Therefore, the temperature at the aluminum-copper junction is the average of the temperatures at the aluminum end and the copper end:$$\frac{T_1 + T_2}{2} = \frac{30 ^ { \circ } \mathrm { C } + 230 ^ { \circ } \mathrm { C }}{2} = 100 ^ { \circ } \mathrm { C }$$

A Heat-Conducting Rod, 0 0.70 m0.70 \mathrm {~m}0.70 m Long Both Sections Have a Cross-Sectional Area Of in Insulation

A Heat-Conducting Rod, 0 $0.70 \mathrm {~m}$ Long Both Sections Have a Cross-Sectional Area Of in Insulation