# Derive differential equation that describes temperature

1. Feb 17, 2016

### Shackleford

I'm taking an engineering heat transfer course as an elective.

1. The problem statement, all variables and given/known data

Copper tubing is joined to a solar collector plate of thickness t, and the working fluid maintains the temperature of the plate above the tubes at To. There is a uniform net radiation heat flux q”rad to the top surface of the plate, while the bottom surface is well insulated. The top surface is also exposed to a fluid at T that provides for a uniform convection coefficient h.

(a) Derive the differential equation that governs the temperature distribution T(x) in the plate.
(b) Obtain a solution to the differential equation for appropriate boundary conditions.

2. Relevant equations

3. The attempt at a solution

I want to first analyze a differential control volume.

Ac = yt
As = ydx

qcond + qrad = qconv + qcond, x + dr

-kAcdT/dx + εσ[T4(x) - T4] ydx = h[T(x) - T] ydx + -kAcdT/dx -kd/dx(AcdT/dx)

Of course, t is constant and we're assuming that temperature does not vary with the y-coordinate. However, I wanted to start with an actual volume and see where y factors out. Am I on the right track?

Last edited: Feb 17, 2016
2. Feb 22, 2016