A long cylinder (surface 1) with a diameter of 0.05 m sits inside of a long tube; the outer tube is made up of two parts, creating 3 total surfaces. The plate at the bottom (surface 2) has a width of 0.14142 m (it's the diameter of the outside tube divided by the square root of two). The top part (surface 3) is $\frac{1}{4}$ of a tube with a diameter is 0.2 m, and sits concentric with the central cylinder such that the view factor from surface 1 to surface 3 is 0.75 ($F_{13}=0.75$). The length of the three components is not needed, but you may use an arbitrary length "L" in your solutions as necessary, though it should always cancel out.
Cross-sectional view of the three surface; Length "L" is into the page.
Problem 3, Part a
Determine all the view factors, and place them in a matrix ($F_{11}$ and $F_{13}$ have been completed for you); each row represents the surface it is from. Each column represents the surface it is to. Show your work in the submitted.pdf.
$F_{xx}$
1
2
3
1
0
0.75
2
3
Problem 3, part b
Surface 2, the flat plate, is insulated on the bottom, and a known heat generation ($\dot{q}_2$) occurs within it. It loses heat to surfaces 1 and 3 via radiation to remain steady-state. All three surface are blackbodies ($\epsilon = 1$). If the thickness of the plate ($t_2$), and the surface temperatures of the other surfaces ($T_1$ and $T_3$) are known, what is the surface temperature of surface 2 ($T_2$)? You may assume that the view factors from part a (and the surface dimensions given in part a) are known. Solve symbolically; for full credit, show as $T_2 = \dots$.
Problem 3, part c
Assuming that $T_2$ (the surface temperature) is now known, and thermal conductivity of the plate ($k_2$) is known, determine the temperature profile through the plate from 0 to $t_2$. Assume $x = 0$ at the bottom where it's insulated, and that at that location, $dT/dx = 0$ (since it is insulated). What is the maximum temperature in the plate?