Question

An aluminum pan whose thermal conductivity is \(237 \mathrm{W} / \mathrm{m} \cdot^{\circ} \mathrm{C}\) has a flat bottom whose diameter is \(20 \mathrm{cm}\) and thickness \(0.6 \mathrm{cm} .\) Heat is transferred steadily to boiling water in the pan through its bottom at a rate of 700 W. If the inner surface of the bottom of the pan is \(105^{\circ} \mathrm{C}\), determine the temperature of the outer surface of the bottom of the pan.

Short answer

Answer: The temperature of the outer surface of the bottom of the pan is approximately 101.16°C.

Step-by-step solution

List the given data

Here is the given data from the problem: - Thermal conductivity of the aluminum pan (K) = 237 W/m°C - Bottom diameter = 20 cm - Bottom thickness (L) = 0.6 cm - Heat transfer rate (Q) = 700 W - Temperature of the inner surface (T_inner) = 105°C

Convert measurements to SI units

To make calculations easier, let's convert the bottom diameter and thickness to meters: Bottom diameter = 20 cm = 0.2 m Bottom thickness (L) = 0.6 cm = 0.006 m

Find the area of the bottom

We can find the area of the bottom of the pan (A) with the following formula using the bottom diameter: A = π(D/2)^2 A = π(0.2/2)^2 A = π(0.1)^2 = 0.0314 m^2

Calculate the temperature difference

We can use the formula for heat transfer Q to calculate the temperature difference between the inner and outer surface of the pan's bottom: \(Q = K \cdot A \cdot \frac{T_{inner} - T_{outer}}{L}\) First, rearrange the equation to solve for the temperature difference: \((T_{inner} - T_{outer}) = \frac{Q \cdot L}{K \cdot A}\) Now, plug in the given values: \((T_{inner} - T_{outer}) = \frac{700 W \cdot 0.006m}{237 W/m °C \cdot 0.0314 m^2}\) \((T_{inner} - T_{outer}) = 3.84°C\)

Calculate the temperature of the outer surface

To find the temperature of the outer surface (T_outer), we can simply subtract the temperature difference from T_inner: T_outer = T_inner - (T_innter - T_outer) T_outer = 105°C - 3.84°C = 101.16°C The temperature of the outer surface of the bottom of the pan is approximately 101.16°C.