Derive differential equation that describes temperature

[Shackleford]

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

1. Homework Statement

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.

Homework Equations

Conduction, convection, and radiation

The Attempt at a Solution

I want to first analyze a differential control volume.

A_c = yt
A_s = ydx

q_cond + q_rad = q_conv + q_{cond, x + dr}

-kA_cdT/dx + εσ[T⁴(x) - T⁴_∞] ydx = h[T(x) - T_∞] ydx + -kA_cdT/dx -kd/dx(A_cdT/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?

 

[KataKoniK]

Dear student,

Thank you for sharing your attempt at solving this problem. It seems like you are on the right track and have a good understanding of the different heat transfer mechanisms involved in this system.

To derive the differential equation that governs the temperature distribution T(x) in the plate, we can use the energy balance equation for a control volume as follows:

Σ(in) = Σ(out) + Σ(generation)

Assuming steady-state conditions and neglecting any internal heat generation, we can write the energy balance equation as:

qcond + qrad = qconv

Where qcond is the conduction heat transfer, qrad is the radiation heat transfer, and qconv is the convection heat transfer.

To solve for the temperature distribution, we can use the Fourier's law of heat conduction for the conduction heat transfer, the Stefan-Boltzmann law for the radiation heat transfer, and the Newton's law of cooling for the convection heat transfer. This will result in a second-order differential equation with appropriate boundary conditions that can be solved using standard methods.

I would recommend carefully reviewing the equations you have used and making sure that all the terms are accounted for and properly balanced. Also, it might be helpful to draw a control volume diagram to better visualize the problem and the different heat transfer mechanisms involved.

I hope this helps and good luck with your studies!