Question

One end of a 30cm long aluminum rod is exposed to a temperature of 500 Degrees Celsius while the other end is maintained at 20 Degrees Celsius. The rod has the diameter of 2.5 cm. If heat is conducted through the rod at the rate of 164.9J/s, calculate the the thermal conductivity of aluminum.

(A) 311.6 W/mC

(B) 209.9W/mC

(C) 506.3W/mC

(D) 457.1W/mC

Answer 1

Let:

$$\begin{array}{l} P = 164.9 \, \mathrm{J/s} \\ D = 2,5 \, \mathrm{cm} = 0.025 \, \mathrm{m} \\ T1 = 500{}^{\circ} \mathrm{C} \\ T2 = 20{}^{\circ} \mathrm{C} \\ l = 30 \, \mathrm{cm} = 0.3 \, \mathrm{m} \\ k = ? \end{array}$$

According to the Fourier's law:

$$\begin{array}{l} Q = t \cdot k \frac{S \Delta T}{l}; \\ \text{As: } P = \frac{Q}{t}; \\ P = k \frac{S \Delta T}{l}; \\ k = \frac{Pl}{S \Delta T}; \\ S = \pi \cdot \frac{D^2}{4} \end{array}$$

Let's enter the data:

$$\begin{array}{l} S = 3.14 \cdot \frac{0.025^2}{4} = 0.000491 \, \mathrm{m}^2 \\ \Delta T = T1 - T2 = 500 - 20 = 480{}^{\circ} \\ k = \frac{164.9 + 0.3}{0.000491 + 480} = 209,9 \, \mathrm{W/mC} \\ \text{Answer: "B" } 209,9 \, \mathrm{W/mC} \end{array}$$