Consider a heat treatment process in which steel rods with a 10 cm diameter at an initial temperature of 600°C are inserted into an oil bath at 25°C. Assuming a convection coefficient of 400 W/(m2 K) between the oil and the rod, estimate how long it takes for the centerline of the rod to cool to 60°C. If the steel rod, which is 1 m long, is withdrawn from the bath at this point in the process, i.e., when the centerline temperature has reached 60°C, estimate the rate at which heat must be extracted from the oil to maintain its temperature constant during the process of treating 25 rods per hour.

Question

GIVEN

• Heat treatment of steel rod inserted in oil bath

• Rod outside diameter = 0.1 m

• Initial temperature (T0) = 600°C

• Oil bath temperature (T?) = 25°C

• The average heat transfer coefficient ch = 400 W/(m2 K)

FIND

• Time it takes for centerline of the rod to cool to 600C.

• Rate at which heat must be extracted from oil to maintain its temperature constant during the process of treating 25 rods per hour.

ASSUMPTIONS

• Radial conduction only in billet

• Uniform and constant properties

SKETCH

Answer 1

(a) The Biot number for the steel rod based on r0 is





Using approximate solutions, for r=0 we have



For Bi=0.466 for infinite cylinder we have



Solving for the time



The surface temperature is needed to find the surface heat flux. Thus using approximate solution

for r=r0 we have T (0, T 0



J 0(0.915) =0.8012



can be used to calculate the heat transferred from one ball during the cooling time



J,(0.915) =0.41103



the rate of heat that must be extracted to maintain oil temperature constant during treatment of 25 rods per hour is