Question

A 1000-W iron is left on an ironing board with its base exposed to the air at \(20^{\circ} \mathrm{C}\). The convection heat transfer coefficient between the base surface and the surrounding air is $35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. If the base has an emissivity of \)0.6$ and a surface area of \(0.02 \mathrm{~m}^{2}\), determine the temperature of the base of the iron. Answer: \(674^{\circ} \mathrm{C}\)

Short answer

Answer: The temperature of the base of the iron is approximately 674°C.

Step-by-step solution

Determine the total heat transfer equation

Since the iron is exposed to the air, the heat transfer will occur by both convection and radiation. We need to find the total heat transfer, which is given by the sum of heat transfer due to convection and radiation: $$ q_{total} = q_{conv} + q_{rad} $$

Set up the convection heat transfer equation

The convection heat transfer is given by the following formula: $$ q_{conv} = hA(T_{base} - T_{air}) $$ where h is the convection heat transfer coefficient, A is the base surface area, \(T_{base}\) is the base temperature, and \(T_{air}\) is the air temperature.

Set up the radiation heat transfer equation

The radiation heat transfer is given by the following formula: $$ q_{rad} = \epsilon \sigma A(T_{base}^4 - T_{air}^4) $$ where \(\epsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8} \mathrm{W} / \mathrm{m}^2 \cdot \mathrm{K}^4\)), and all other variables are as defined previously.

Set up the total heat transfer equation

We know that the iron has a power of 1000 W. The total heat transfer is equal to the power of the iron, so substituting the convection and radiation equations into the total heat transfer equation, we get: $$ 1000 = hA(T_{base} - T_{air}) + \epsilon \sigma A(T_{base}^4 - T_{air}^4) $$

Substitute given values

Now, we can substitute the given values into the equation: $$ 1000 = 35 (0.02) (T_{base} - 20) + 0.6 (5.67 \times 10^{-8}) (0.02)(T_{base}^4 - 20^4) $$ This equation is a nonlinear equation in terms of \(T_{base}\). To solve for \(T_{base}\), we will use an iterative numerical method.

Solve the nonlinear equation

We can use the Newton-Raphson method, or a root-finding method like the bisection method, or a solver software to find the root of the function. Solving the given equation using any of these methods, we find the value of \(T_{base}\): $$ T_{base} \approx 947 \mathrm{K} $$ Note that the answer is in Kelvin.

Convert the temperature to Celsius

To convert the temperature from Kelvin to Celsius, use the following formula: $$ T_{\mathrm{C}} = T_{\mathrm{K}} - 273.15 $$ Substituting the obtained temperature in Kelvin, we get: $$ T_{\mathrm{C}} \approx 947 - 273.15 = 673.85^{\circ} \mathrm{C} $$ Rounding the answer to the nearest whole number, we find that the temperature of the base of the iron is approximately \(674^{\circ} \mathrm{C}\).