Question

Two surfaces, one highly polished and the other heavily oxidized, are found to be emitting the same amount of energy per unit area. The highly polished surface has an emissivity of \(0.1\) at \(1070^{\circ} \mathrm{C}\), while the emissivity of the heavily oxidized surface is \(0.78\). Determine the temperature of the heavily oxidized surface.

Short answer

Answer: The temperature of the heavily oxidized surface is approximately 557.08°C.

Step-by-step solution

Understand the Stefan-Boltzmann Law

The Stefan-Boltzmann Law relates the radiant emittance, emissivity, and temperature of an object. The law is given by the following formula: \[E = e \sigma T^4\] where: \(E\) = radiant emittance (energy per unit area) \(e\) = emissivity of the surface (between 0 and 1) \(\sigma\) = Stefan-Boltzmann constant (\(5.67 \times 10^{-8} W m^{-2} K^{-4}\)) \(T\) = temperature (in Kelvin)

Know given values

We are given the following values: Emissivity of the highly polished surface, \(e_1 = 0.1\) Temperature of the highly polished surface, \(T_1 = 1070^{\circ} \mathrm{C}\) Emissivity of the heavily oxidized surface, \(e_2 = 0.78\) We need to find the temperature of the heavily oxidized surface, \(T_2\).

Convert given temperature to Kelvin

To work with the Stefan-Boltzmann Law, we need the temperature in Kelvin. To convert the given Celsius temperature to Kelvin, we use the following formula: \[T(K) = T(^{\circ} \mathrm{C}) + 273.15\] So, \(T_1 = 1070 + 273.15 = 1343.15 \mathrm{K}\)

Use the Stefan-Boltzmann Law

Since both the surfaces are emitting the same amount of energy per unit area, we can write: \[E_1 = E_2\] which means: \(e_1 \sigma T_1^4 = e_2 \sigma T_2^4\)

Solve for the temperature of the heavily oxidized surface

We can now solve for \(T_2\): \[\frac{e_1}{e_2} = \frac{T_2^4}{T_1^4}\] Plug in the given values and solve for \(T_2\): \[\frac{0.1}{0.78} = \frac{T_2^4}{(1343.15)^4}\] \(T_2^4 = (1343.15)^4 \times \frac{0.1}{0.78}\) \(T_2 = \sqrt[4]{(1343.15)^4 \times \frac{0.1}{0.78}}\) \(T_2 \approx 830.23 \mathrm{K}\)

Convert temperature back to Celsius

Finally, convert the temperature back to Celsius using the formula: \[T(^{\circ} \mathrm{C}) = T(K) - 273.15\] \(T_2 \approx 830.23 - 273.15 \approx 557.08^{\circ} \mathrm{C}\) The temperature of the heavily oxidized surface is approximately \(557.08^{\circ} \mathrm{C}\).