Solve the following problem:
Q: The top surface of a plate, including its grooves, is maintained at a uniform temperature of T, = 200°C. The lower surface is at T2 = 20°C, the thermal conductivity is 15 W/m · K, and the groove spacing is 0.16 m.
(a) Using a finite-difference method with a mesh size of Δx = Δy = 40 mm, calculate the unknown nodal temperatures and the heat transfer rate per width of groove spacing (w) and per unit length normal to the page.
(b) With a mesh size of Δx = Δy = 10mm, repeat the foregoing calculations, determining the temperature field and the heat rate. Also, consider conditions for which the bottom surface is not at a uniform temperature T2 but is exposed to a fluid at T8 = 20°C. With ?x = ?y = 10mm, determine the temperature field and heat rate for values of h = 5, 200, and 1000 W/m2 · K, as well as for h=8.
