A solid lead cylinder 0.6-m in diameter and 0.6-m long, initially at a uniform temperature of 121°C, is dropped into a 21°C liquid bath in which the heat transfer coefficient h cis 1135 W/(m2 K). Plot the temperature-time history of the center of this cylinder and compare it with the time histories of a 0.6 m-diameter, infinitely long lead cylinder and a lead slab 0.6 m-thick.

Question

GIVEN

• A solid lead cylinder dropped into a liquid bath

• Cylinder diameter (D) = 0.6 m

• Cylinder (L) = 0.6 m

• Initial uniform temperature (To) = 121°C

• Liquid bath temperature (T?) = 21°C

• Heat transfer coefficient h c= 1135 W/(m2 K)

FIND

(a) Plot the temperature-time history of the cylinder center (b) Compare it with the time history of a 0.6 m diameter, infinitely long lead cylinder (c) Compare it with the time history of a lead slab 0.6 m thick

ASSUMPTIONS

• Two dimensional conduction within the cylinder

• Constant and uniform properties

• Constant liquid bath temperature

SKETCH

Answer 1

The Biot number based on radius is



Therefore, internal resistance is significant.

(a) This two-dimensional system required a product solution. From the product solution is



Since the length of the cylinder is the same as its diameter, the Biot number based on length is the same as that based on radius



The temperature of the center of the cylinder (x = 0, r = 0) is determined by calculating the Fourier number for that time, finding P(0) on finding C(0) on Figure 3.10, and applying



(b) The center temperature for a long cylinder is



(c) The center temperature for a slab is



The temperature-time histories of these three cases are tabulated and plotted below