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rod, estimate how long it takes for the centerline of the rod to cool to 60°C. If the steel rod, which is 1 m long, is withdrawn from the bath at this point in the process, i.e., when the centerline temperature has reached 60°C, estimate the rate at which heat must be extracted from the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.
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
(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
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A mercury-in-glass thermometer at 40°C (OD = 1 cm) is inserted through duct wall into a 3 m/s air stream at 66°C. This can be modelled as cylinder in cross-flow, as shown in figure. Estimate the heat transfer coefficient between the air and the thermometer.
GIVEN
• Thermometer in an air stream
• Thermometer temperature (Ts) = 40°C
• Thermometer outside diameter (D) = 1 cm = 0.01 m
• Air velocity (V) = 3 m/s
• Air temperature (Tb) = 66°C
FIND
• The heat transfer coefficient ch
ASSUMPTIONS
• Steady state
• Turbulence in the free stream approaching the thermometer is low
• Effect of the duct walls in negligible
SKETCH
PROPERTIES AND CONSTANTS
From Appendix 2, Table 28, for dry air at the bulk temperature of 66°C
Thermal conductivity (k) = 0.0282 W/(m K) Kinematic viscosity (?) = 20*10-6 m2/s Prandtl number (Pr) = 0.71 At the thermometer surface temperature of 40°C, the Prandtl number (Prs) = 0.71
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