Heat Transfer, Finite difference, Curved geometry

[MechEngrStdnt]

Homework Statement

Homework Equations

I could really use a push on how to approach this problem. My primary problem is it asks for the heat flux into the page, which makes no sense to me as that is the z direction and this is in the x/y plane. If anyone could explain this problem and maybe give me a push in the right direction I would really appreciate it!

The Attempt at a Solution

 

[Stephen Tashi]

My primary problem is it asks for the heat flux into the page,

It asks for "heat transfer per unit depth into the page". I interpret the diagram as a cross section of a 3D object. The greater the depth into the the page of the object, the more heat (per unit time) the 3D object can transfer between the two adiabatic surfaces because the surfaces have an area that depends on the depth of the object into the page.

The flux involves the transfer of heat between the surfaces "across the page".

 

[Chestermiller]

They want you to assume that the temperature is constant along all radial lines in the curved section, and is also constant along all horizontal lines in the rectangular section. So, in the curved section, the temperature gradient is $\frac{1}{r}\frac{dT}{d\phi}$ and, in the rectangular section, the temperature gradient is $\frac{dT}{dy}$. The temperatures are to match at the interfaces of the subdomains. So, in the curved section, the heat flow rate is given by $$Q=-w\frac{dT}{d\phi}\int_{r_0}^{r_0+\Delta r}{\frac{dr}{r}}$$where w is the depth into the page. I leave it up to you to do the rectangular section, and to combine the two sections.