Heat and Mass Transfer: Thermal Radiation Circuit

[jdawg]

Homework Statement

1.) A piece of bread (ε = 0.85) is being toasted in a toaster, as shown below. The heating element of the toaster consists of a wire 2 mm in diameter that is wound to approximate cylinders spaced 2 cm apart. This element acts as a blackbody. The housing for the toaster is a well-insulated piece of sheet metal (ε = 0.5). For the sake of time, assume no air flows through the assembly. You may also assume the configuration shown below repeats over a long distance.

a.) Sketch the radiation circuit for this arrangement and label all relevant resistances.

b.) Calculate the relevant resistances for the radiation circuit above. You may approximate the bread and metal surface areas with the separation, s, and the element surface area with the circumference.

c.) Set up the equations necessary to estimate the temperature distribution in the bread. Start with the full version of the conduction heat equation, state the relevant assumptions and simplify accordingly, and define any appropriate boundary and initial conditions.

d.) Comment on how this analysis would change if air flowed through the toaster. In particular, address how the radiation circuit would change and why it would change.

Homework Equations

The Attempt at a Solution

So at the moment I'm stuck at part A. I attached my attempt at the solution so far. I guess I'm not sure if its appropriate to treat all of the cylinders as one surface as opposed to having some sort of elaborate parallel thermal circuit trying to account for all four cylinders shown.

Sorry for the awful picture quality, my scanner is acting up so I had to take a picture with my phone.

 

[Babadag]

Because I am only an electric engineer I would be curious to understand the problem.
The relevant equations are missing.
If radiosity [or M-excitance] J=εσT^4+ρE [W/m^2] where:
E=dΦ/dA; Φ=power of the source and ρ=part of the E of the source
how do you define the resistance?