iterative model, conducted heat is a problem 2

[someguy79]

I'm trying to model a metal box sitting on the ground. It's 3" thick steel, painted white, and not insulated. It sits in the sun on a hot day with no wind and comes to steady state. Inside the box is a small heater.

The goal is to estimate the internal air temperature.

I've been trying to model the box by developing a heat balance on its external surface.

I know the solar radiation heat input (including incident angles, direct, reflected and diffuse components), as well as background radiation heat input (from ambient temperature, and absorbtivity).

Convection is assumed to be natural convection driven by difference between surface temperature and ambient temperature.

Net heat conducted (through all walls of box) heat is equal to the heater power inside the box.

The difficulty comes because I do not know surface temperature(s) of the box.

Solve this I've been trying to create an iterative model to get a solution. I'm finding that when I vastly increase thermal resistance to conduction (as if insulated) I can get the model to converge and give reasonable results. Unfortunately, even when I work the model to converge at smaller and smaller thermal resistances, at some point it fails and diverges.

I also tried looking at the box as a lumped capacitance (isothermal walls) but I end up with conduction equations that don't work.

Is there another strategy I should be looking at?

 

[gruntguru]

If I can assume that the internal air is mixed well, then the internal air temperature (T1a) is very near to the internal wall temperture (Tis). It's not a perfect assumption, but it shouldn't be too bad.
I would question this assumption. With conduction resistance very low, the internal surface temps will be more related to the exterior temps and convection to the internal walls will dominate.

 

[someguy79]

Thank you all for your help. I've made quite a lot of progress with this model. It's converging and giving reasonable results. The internal temperature is showing around 150°F, and the heat flows are balancing.

IRstuff,
Thanks for the clarification on my emissivity and absorbtivity. The solar absorbtivity I have is now only applied to the solar heating term. The emissivity is now applied to the background radiation term, and the radiative emission term for the box. I'm keeping the background temperature as the local ambient (Tamb) as this is more conservative for this calculation. It will produces a higher internal temperature.

prex,
I can't assume a convection coefficient as suggested. The client objected to assuming a convection coefficient before. it must be calculated. Because convection coefficient is highly dependent on temperature difference, (and several other key pieces of data are not known) I can't do this as a simple and direct calculation.

gruntguru,
You are correct, that the assumption is no good for cases where conduction resistance is low. I've put in several more steps in the calculation to go through calculating internal wall temperatures (Tis) and convection coefficients inside.