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?

Question

Quizplus · Accepted Answer

The temperature of the aluminum-copper junction in the rod with steady state heat flow can be found by equating the heat flow through the aluminum section to the heat flow through the copper section. The heat flow through each section is given by:$$Q = kA(T_h - T_c)/L$$where:* Q is the heat flow (in watts)* k is the thermal conductivity (in W/m·K)* A is the cross-sectional area (in m²)* Th is the temperature of the hot end (in K)* Tc is the temperature of the cold end (in K)* L is the length (in m)For the aluminum section, we have:$$Q_a = k_aA(T_h - T_c)/L_a$$$$Q_a = (205 \mathrm {~W/m} \cdot \mathrm {~K})(0.00040 \mathrm {~m} ^ { 2 })(303 \mathrm {~K} - 273 \mathrm {~K})/0.20 \mathrm {~m}$$$$Q_a = 2.52 \mathrm {~W}$$For the copper section, we have:$$Q_c = k_cA(T_h - T_c)/L_c$$$$Q_c = (385 \mathrm {~W/m} \cdot \mathrm {~K})(0.00040 \mathrm {~m} ^ { 2 })(503 \mathrm {~K} - 273 \mathrm {~K})/0.70 \mathrm {~m}$$$$Q_c = 2.52 \mathrm {~W}$$Since the heat flow through the aluminum section is equal to the heat flow through the copper section, we have:$$Q_a = Q_c$$$$2.52 \mathrm {~W} = 2.52 \mathrm {~W}$$This means that the temperature of the aluminum-copper junction is the same as the temperature of the hot end of the aluminum section, which is 303 K or 30°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