Unsteady State Heat Transfer (Conduction): Another area of heat transfer is unsteady state transfer. Gurnie-Lurie has developed a set of charts for different geometries of the solids. The cases considered are wall of infinite thickness heated on one side, wall of finite thickness heated on both sides, wall of finite thickness heated by a fluid with a contact resistance.

Wall of infinite thickness is heated on one side: In this situation, the temperature of the material at a distance x from the surface at θ hours is given by Table 5.11 Equation 1, where T_o is initial temperature, T_s is surface temperature, χ is a dimensionless length, α is thermal diffusivity, and q is flux passing the surface at any time θ.

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