A long copper cylinder 0.6-m in diameter and initially at a uniform temperature of 38°C is placed in a water bath at 93°C. Assuming that the heat transfer coefficient between the copper and the water is 1248 W/(m2 K), calculate the time required to heat the center of the cylinder to 66°C. As a first approximation, neglect the temperature gradient within the cylinder, then repeat your calculation without this simplifying assumption and compare your results.

Question

GIVEN

• A long copper cylinder is placed in a water bath

• Diameter of cylinder (D) = 0.6 m

• Initial temperature (To) = 38°C

• Water bath temperature (T?) = 93°C

• The heat transfer coefficient ( ) ch = 1248 W/(m2 K)

FIND

Calculate the time required to heat the center of the cylinder to 66°C assuming

(a) Negligible temperature gradient within the cylinder

(b) Without this simplification, then

(c) Compare your results

ASSUMPTIONS

• Neglect end effects

• Radial conduction only

SKETCH

Answer 1

(a) For a negligible temperature gradient within the cylinder, the temperature-time history is given by



Solving for the time



(b) The approximate solution method can be used to take the temperature gradient within the cylinder into account.



for infinite cylinder, we have



For Bi=0.95 for Infinite cylinder, we have



Solving for the time