# Heating of a metallic 'gray body' by radiation

I am interested in a realistic model for calculating the heating of a metallic body by solar radiation.

Assumption (0) is vacuum, so neither conduction nor convection.

Assumption (1) is integration over frequency; using this we get for absorbed and emitted power = heat $$Q_\pm$$ per time

$$\dot{Q}_+ = \alpha_T \, g \, A \, u$$

$$\dot{Q}_- = -\epsilon_T \, \sigma \, A \, T^4$$

with incident radiation power $$u$$ per area $$A$$, a geometric factor $$g$$ and a temperature-dependend absorption coefficient $$\alpha_T$$; then a temperature-dependend emission coefficient $$\epsilon_T$$ and the Stefan-Boltzmann constant $$\sigma$$.

Assumption (2) would be Kirchhoff's law in integrated form, i.e. $$\alpha_T = \epsilon_T$$. I will use this at the very end.

Assumption (3) is Newton's cooling law, assuming a homogeneous temperature, which should make sense for metals with high conductivity.

For the temperature dependence one finds

$$\dot{T} + (mc)^{-1}\,A \left[\epsilon_T \, \sigma \, T^4 - \alpha_T \, g \, u \right] = 0$$

Using Kirchhoff's law the target temperature can be calculated as

$$\sigma \, T_\infty^4 - g \, u = 0$$

However, I have doubts about the assumptions. Can you please comment on integration over frequency and Kirchhoff's law, especially for the relevant frequency range of solar radiation? Can it be used to calculate the target temperature, i.e. assuming equilibrium? Or can it be used even earlier to calculate the time dependence?

In addition, the calculation of temperature $$T(t)$$ as a function of time $$t$$ requires a specific model for the temperature dependence of the absorption $$\alpha_T$$ and emission coefficient $$\epsilon_T$$. What would be a valid model for metals?

First, absorption and emission are to sides of the same coin, so $$\alpha_T = \epsilon_T$$.

Second, different metals have different emissivities, and those emissivities are dependent on wavelength. They're also dependent on surface treatment and (if the piece was manufactured outside a perfect vacuum) aging.

So you can't solve the problems for generic "metal"*.

You can demonstrate this to yourself in almost any culture that has computers by pulling a few coins out of your pocket (or, in a few years, visiting a museum). Copper is red-brown, silver is, well, silvery, nickle is shiny-white as is aluminum, iron is gray-white, etc. The reason each different metal has a different color is exactly because their absorption/emissivity varies across the visible spectrum.

So for any given metal, you need to look it up, or determine it by experiment. I'm sure there's handbooks out there. I'm equally sure (because I've tried to find this stuff) that they're hard to find because of the narrowness of your question.

Definitely in any sort of reactive atmosphere, and at least for the first time in a vacuum, if you're heating the metal hot enough there may be changes in the surface emissivity, just to keep you on your toes (steel, for instance, develops a dark oxide layer in air when it's red hot). So you need to account for that.

* To preserve your sanity, you may try using a generic, constant emissivity number. I guarantee nothing, except that red-hot aluminum looks like a silvery liquid at room temperature, so you can assume that it's emissivity is pretty low compared to steel which glows red when it's red hot.

• So would it make sense to rewrite the ansatz as $\dot{Q}_+ \to \int d\lambda \, \alpha_T(\lambda) \, u_+(\lambda)$, and $\dot{Q}_-$ in a similar way?
– TomS
May 19, 2023 at 14:59
• Yes, that would make sense. May 19, 2023 at 19:13