So, I have some problems understanding Kirchoff's Radiation law.

My textbook, Transport Processes and Separation Process Principles, by Geankoplis, states that at the same temperature T1 the emissivity and absorptivity of a surface is equal, which holds for any black or non black solid surface.

In a problem from my professor it is given that : The sun radiate a flat surface with 1000 W/m2. The absorptivity of the plate is 0.9 and the emissivity is 0.1. The air temperature is 20 C and the heat transfer coefficient is 15 W/Km2. Calculate the surface temperature at equilibrium if the bottom is isolated.

My question is: how is it possible that the emissivity and absorptivity in this case is not equal, which contradicts Kirchoff's law?

• That's because Kirchoff's only applies to equilibrium. – FGSUZ May 13 '18 at 20:50
• But we are told to calculate the surface temperature AT equilibrium. shouldn't Kirchoff's law hold? – Jmei May 13 '18 at 21:00
• @FGSUZ Kirchhoff's law can be "derived" using equilibrium and the second law of thermodynamics, but absorptivity and emissivity are properties of a surface that apply generally. – Pieter May 14 '18 at 6:19

Emissivity and absoprtivity are both functions of wavelength. The plate may absorb 90 % of sunlight ($\lambda \approx 0.5 \mu$m) and have an emissivity and absorptivity of 0.1 in the thermal infrared $(\lambda \approx 10 \mu$m).

These numbers are a bit unlikely, it is usually the other way around.

• What implications to my problem does that have? – Jmei May 13 '18 at 21:10
• I explained how the numbers could be different. Calculating is up to you. The plate will radiate in the thermal infrared. – Pieter May 13 '18 at 21:17

The absorptivity quoted is an average over all the wavelengths of light incident on the body from the Sun. This light may have an average wavelength of $\lambda \approx 0.5 \mu \text{m}$ or so which presumably has an absorbtivity of 0.9 in your case.