A heat conducting rod, 1.60 m long and wrapped in insulation, is made of an aluminum section that is 0.90 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 $170 ^ { \circ } \mathrm { C }$ respectively. The thermal conductivities of aluminum and copper are 205 W/m ? K (aluminum)and 385 W/m ? K (copper). At what rate is heat conducted in the rod under steady state conditions?
A)9.0 W
B)7.9 W
C)10 W
D)11 W
E)12 W

Question

Quizplus · Accepted Answer

Explanation:The rate of heat conduction in the rod is given by:$$Q = \frac{kA(T_h - T_c)}{L}$$where:* Q is the rate of heat conduction (in W)* k is the thermal conductivity (in W/m·K)* A is the cross-sectional area (in m²)* L is the length (in m)* $T_h$ is the temperature of the hot end (in K)* $T_c$ is the temperature of the cold end (in K)For the aluminum section, we have:$$Q_a = \frac{k_aA(T_h - T_c)}{L_a}$$$$Q_a = \frac{(205 \mathrm{W/m·K})(0.00040 \mathrm{m}^2)(170 ^ { \circ } \mathrm { C } - 30 ^ { \circ } \mathrm { C })}{0.90 \mathrm{m}}$$$$Q_a = 7.9 \mathrm{W}$$For the copper section, we have:$$Q_c = \frac{k_cA(T_h - T_c)}{L_c}$$$$Q_c = \frac{(385 \mathrm{W/m·K})(0.00040 \mathrm{m}^2)(170 ^ { \circ } \mathrm { C } - 30 ^ { \circ } \mathrm { C })}{0.70 \mathrm{m}}$$$$Q_c = 11 \mathrm{W}$$The total rate of heat conduction in the rod is:$$Q = Q_a + Q_c = 7.9 \mathrm{W} + 11 \mathrm{W} = 19 \mathrm{W}$$Therefore, the correct answer is A) 9.0 W.

A Heat Conducting Rod, 1 0.70 m0.70 \mathrm {~m}0.70 m Long Both Sections Have a Cross-Sectional Area Of Wrapped

A Heat Conducting Rod, 1 $0.70 \mathrm {~m}$ Long Both Sections Have a Cross-Sectional Area Of Wrapped