E5.20 Wall of Finite Thickness - Heated on One Side

Example 5.20: A thick steel wall is initially at 100 °F. One of its surfaces is suddenly changed to 1000 °F. Determine the temperature of the material 4 in below its hot surface after 4 hours. Determine heat flux at that time. Use the following properties of the steel:

Specific heat, c = 0.12 Btu/(lb·°F),

thermal conductivity k = 24 Btu/(h·ft·°F), density, ρ = 488 lb/ft³

Solution: This is a situation of a wall of infinite thickness heated on one side. The solution is given in terms of an error function of argument χ

Where x is the distance from the surface, and θ is the time lapsed, and α is thermal diffusivity given by

Now we can see that θ = 4 h, and x = 4 in = 0.333 ft, so argument χ can be found to be

And error function of χ can be read from the standard tables as 0.146. Initial temperature of the body, T_o, is 100 °F. The surface temperature, T_s, is 1000 °F. The temperature of the body at a depth of 4 in from the surface after 4 h can be found to be

And heat flux after 4 hours is