1. The problem statement, all variables and given/known data Let's say you have a 3m long copper pipe, 3mm in thickness with a diameter of 170mm. You fix one end at 1K and insulate it to prevent conduction or convection between the air and the pipe itself. There is still radiation. Assume that the inside of the pipe has no effect (radiative, conductive or otherwise). emissivity is 0.1, find the steady state temperature at the other end of the pipe. The atmosphere is at 300K. We can assume the problem is 1 dimensional and approximate that the temperature will be constant across the 3 mm thickness. 2. Relevant equations [tex]\alpha*d^2 u/dx^2 = du/dt = 0 [/tex](steady state) [tex]Q = mCdT[/tex] [tex]P = \epsilon \sigma A*T^4[/tex] 3. The attempt at a solution The problem I'm having with solving this scenario is that the differential equation becomes something I've never seen before. There is heat emitted through radiation at every dx due to the temperature at that x coordinate, which, coupled with the conduction across the pipe will then be equal to the heat gained from the input radiation. The differential equation turns out to look like: [tex]d^2 T(x)/dx^2 = C - \epsilon \sigma A*T(x)^4[/tex] where [tex]C = \epsilon \sigma A*T(outer)^4 [/tex] I don't have a second boundary condition, and this turns into a DE that I wouldn't wish on my worst enemies. I've considered approximating it by saying that T_copper << T_atm so T^4 effect from copper is insignificant, which makes the equation a much more manageable: [tex]d^2 T(x)/dx^2 = C[/tex] But I would prefer not to make this assumption. Does anyone have any suggestions? I realize if the pipe was long enough I would just be able to say that the other end is fixed at 300K equal to the surroundings, but I would need to show that 3m is enough distance from the infinite 1K source to use this assumption.