Does Bi<0.1 really suggest a negligible temperature gradient? A straight fin of uniform circular cross section is fabricated from a material with a constant thermal conductivity. The fin has a radius of 20 mm and a length of 100 mm. The convection heat transfer coefficient is 500 W/m²-K with an ambient fluid temperature of T=30 °C. The base of the fin is maintained at T,=100 °C, while the fin tip is heavily insulated. Temperature gradients in the circumferential direction of the fin are negligible. Using the finite-volume technique with volumes that are 2 mm on a side, estimate the temperature distribution within the fin for a fin with a thermal conductivity of (a) 50 W/m-K and (b) 0.5 W/m-K. 1. For both cases, plot the temperature along the fin axis on the centerline of the fin and on the outer surface of the fin. 2. For both cases, also calculate the fin heat transfer rate, qin- 3. Compare the fin heat transfer rate that you determined via FVM with the one- dimensional, analytical solution from Table 3.4. 4. Comment on the validity of the Biot number as a guide regarding the importance of temperature gradients.

This question requires the use of the conduction charts given in the appendix at the end of this paper. A wooden sphere of diameter 5 cm is initially at a uniform temperature of 21 °C. Its surface temperature is suddenly raised to 83 °C. The thermal conductivity and thermal diffusivity of the wood are 0.15 W/m K and 0.82 x 10-7 m2/s, respectively. a) Calculate the temperature 2 cm beneath the surface of the sphere 30 min after the sudden increase in temperature. b) Calculate the energy gained by the sphere during this time.

Temp base is 156 not 456 as in the diagram

Please solve quickly