The top surface of a plate, including its grooves, is maintained at a uniform temperature of T, = 200C. The lower surface is at T2

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 ∆x = ∆y = 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 ∆x = ∆y = 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 T∞ = 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 → ∞.

w/4 w/2 w/2- -T2