A stainless steel cylindrical billet [k = 14.4 W/(m K), ? = 3.9 × 10–6 m2/s] is heated to 593°C preparatory to a forming process. If the minimum temperature permissible for forming is 482°C, how long can the billet be exposed to air at 38°C if the average heat transfer coefficient is 85 W/(m2 K)? The shape of the billet is shown in the sketch.

Question

GIVEN

• A stainless steel cylindrical billet exposed to air

• Thermal conductivity (k) = 14.4 W/(m K)

• Thermal diffusivity (?) = 3.9 × 10–6 m2/s

• Initial temperature (To) = 593°C

• The minimum temperature permissible for forming is 482°C

• Air temperature (T?) = 38°C

• Average heat transfer coefficient ( )ch = 85 W/(m2 K)

FIND

• How long may the billet be exposed to the air?

ASSUMPTIONS

• End effects are negligible

• Constant heat transfer coefficient

• Conduction in the radial direction only

• Uniform thermal properties

SKETCH

Answer 1

The Biot number is calculated to determine if internal resistance is significant



Therefore, internal resistance is important, and approximate solution is used.

The approximate solution is used for the problem. The approach will be as follows:

1. Apply (Tr,,t – T? )/(To – T?) and the Biot number to find the Fourier number

2. Use the Fourier number to find the time it takes for the surface to cool to the given minimum surface

temperature.

For infinite cylinder with Bi=0.3 we have



for r= r0, we have



The surface temperature must not fall below 482°C



Solving for the time