Reference no: EM13379210

Problem 1:

Heat is being generated uniformly throughout a sphere at volumetric rate 4. The radius of the sphere is R and the sphere is made of a material having thermal conductivity k. The surface of the sphere is exposed to a convection environment with fluid temperature T_x and convection heat transfer coefficient h. Assume one-dimensional steady conduction in the radial direction and do the following:

a) Use the equation for V²T in spherical coordinates and the heat diffusion equation to produce the differential equation that models the conduction in the sphere.

b) Find the general form of the solution for T(r) answer : T(r) = ^{47-2 C} + 6kr

c) Find the temperature at the center of the sphere ( 4R²/4R answer : T(0) =__ +___ +T 6k 3h

Problem 2

A 6-mm thick steel plate will be heated from 300°C to 600°C by exposing both sides of the plate to 700°C air having h = 100 W/m²- K . The density, heat capacity, and thermal conductivity of the steel are, respectively, 7900 kg/m" , 640 J/kg • K , and

30 W/m K

a) Verify that heating the steel plate may be treated as a lumped-capacity process.

b) Find how long it will take to heat the plate.

Problem 3, Heat Transfer, Spring 2014

The surface of a long 1-mm diameter aluminum wire is exposed to 15°C ambient air having h = 25 W/m²- K . The temperature of the wire is initially the same as the temperature of the ambient air. The wire has an electrical resistance per unit length of

= 0.034 K)/m and at time t = 0 a 20 ampere current begins flowing through the wire.

a) Verify that heating the wire may be treated as a lumped-capacity process.

b) Find the steady-state temperature of the wire.

c) Find how long it takes for the temperature of the wire to reach 100°C.