E5.22 Finite and Semi-Finite Shapes Heated with Contact Resistance

Example 5.22: A steel circular rod having diameter 8 in, length 12 ft, is initially at a temperature of 400 °F. It is immersed in a bath having a constant temperature of 200 °F. What is the temperature 2 in below the surface after 15 minutes if the following data apply.

Free convection coefficient from oil, h = 50 Btu/(h·ft²·°F).

Thermal conductivity, k = 25 Btu/(h·ft·°F);

Thermal diffusivity, α = 0.425 ft²/h.

Solution: This is a situation where unsteady state heat transfer occurs through a cylindrical object with a contact resistance.

Characteristic length of the cylinder is the diameter of the cylinder, L = 8 in = 0.667 ft

Time lapsed, θ = 15 min = 0.25 h. And the parameter values can be calculated as

We can find the dimensionless temperature for cylindrical object from Gurney-Lurie Chart, Fig 18.13 as Y = 0.31.

The initial temperature, T_o of the body is 200 °F, and the surface temperature, T_s of the body is 400 °F. Plugging these values, we can find the temperature at the center of the cylinder as