- #1
cc94
- 19
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Homework Statement
This isn't a homework question but something I'm working on that I thought should be simple. Two disks (area ##A## and thickness ##d##) are joined together and placed under a radiation heater in vacuum, so that one side of the top disk is heated with a constant power. Assume heat is only lost out of the two-disk system by radiation. 1. Is there a temperature gradient in either of the materials? 2. What is the steady-state temperature of the bottom disk/bottom surface?
Homework Equations
For conduction between the two materials, I found this equation on Wikipedia: https://en.wikipedia.org/wiki/Thermal_contact_conductance#Thermal_boundary_conductance.
For radiation, we have the Stefan-Boltzmann law,
[tex] P = \epsilon\sigma AT^4 [/tex]
The Attempt at a Solution
I'm not sure what all I'm supposed to consider between these two bodies. My thinking is, first, we have some amount of heat ##Q_{in}## coming in. The top disk absorbs this as
[tex]Q_{abs} = \epsilon_{top} Q_{in}[/tex]
Now here's where I was confused. I know the top disk emits radiation out of the top face, but do I also include radiation emitted from the back face in contact with the bottom disk? Or does the conduction from Fourier's law actually include the fact that the disk is radiating? If we don't include a radiation term, then we have two equations for the top and bottom disks:
[tex]\epsilon_{top} Q_{in} = \epsilon_{top}\sigma AT_{top}^4 + \frac{T_{top}-T_{bot}}{d/(k_{top}A) + 1/(hA) + d/(k_{bot}A)}[/tex]
[tex]\frac{T_{top}-T_{bot}}{d/(k_{top}A) + 1/(hA) + d/(k_{bot}A)} = \epsilon_{bot}\sigma AT_{bot}^4[/tex]
where ##k## is the thermal conductivity and ##1/h## is the thermal contact resistance. If we include radiation, I should just add those terms to each equation. But then I have two more unknowns since I think there's 2 different temperatures at the interface between the disks.