Estimating fraction of radiant energy absorbed by a metal

Question

I have a couple of texts on thermodynamics and radiant energy but am finding it difficult to figure out from these how energy absorption and reflection work.

The area of interest is heating ferrous metal alloys to working temperatures, which would be from about 300F to 2500F.

For these temperatures, most radiant energy seems to be emitted in the near infrared spectrum, but as the temperature increases, a greater and greater proportion is emitted in the visible spectrum.

Question 1: A mirror will reflect nearly all energy in the visible spectrum, and I understand it can reflect infrared energy as well. Is the reflection of infrared energy from a mirror as complete as it is for energy in the visible spectrum? If not, what amount is not reflected for a given near infrared wavelength?

Question 2: Let's imagine we have a piece of metal hottest at the surface at some radiant temperature, like 2000F, but its core is cooler, say 500F, so there is a temperature gradient. A mirror is placed right next to the metal bar causing the radiant energy from the bar to be reflected back towards it. (Assume there is a vacuum or some other means of preventing external convection, so that heat loss from the bar is purely via radiation.) Presumably, nearly all of the reflected energy will be re-absorbed back into the metal, because it has a non-reflective surface. The loss will be only that energy which either does not hit the mirror, or which the mirror reflects through the gap between the bar and the mirror. Clearly, the smaller the gap, the less the loss. Therefore, there will be sort of gradient system where heat from the surface of the bar goes in two directions: towards the mirror, or migrates by conduction towards the core. The portion towards the mirror will return (with some small gap loss), and be re-absorbed, so the heat will be kind of bouncing back and forth between the mirror and the bar. So we have two rates: the rate of conduction to the core, and the rate of loss to the gap between the bar and the mirror. How can I mathematically model these two rates as a differential equation and thereby compute the rate of net conduction to the core?

Answer 1

Today one uses a multi-physics simulation program. To reflect IR requires special mirrors; gold is often used, but not at these temperatures.

You can use the black body formula for a rough estimate of the spectral output; then find materials that can take the heat, and also reflect efficiently.