ii.

18.5 Cast-iron cannonballs used in the War of 1812 were occasionally heated for some extended time so that, when fired at houses or ships, they would set them afire. If one of these the socalled "hot shot" were at a uniform temperature of 2000F, how long after being exposed to air at 0F with an outside convective heat-transfer coefficient of 16But/hft2F, would be required for the surface temperature to drop to 600F ? What would be the center temperature at this time? The ball diameter is 6 in. The following properties of cast iron may be used: kcp=23Btu/hftF=0.10Btu/lbmF=460lbm/ft3. i. Write down the governing differential equation along with the appropriate initial and boundary conditions. (12 points) Define dimensionless temperature =ToTT(r,t)T, and dimensionless radius R=Rr to convert the governing equation to a dimensionless form. Here R is the radius of the sphere. iii. Convert the boundary and initial condition to the dimensionless form. iv. Write down the solution to the governing equation in (ii) subject to the initial and boundary conditions in (iii). v. Show that the solution in (iv) satisfies the governing equation and the boundary conditions. vi. Calculate the time taken for the surface of the orange to reach 0C ? vii. At the time you calculated in (vi), what is the temperature of the center of the orange? viii. Derive an expression for the average temperature of the orange