An aluminum sheet, 0.4-m-tall, 1-m-long, and 0.002-m-thick is to be cooled from an initial temperature of 150°C to 50°C by immersing it suddenly in water at 20°C. The sheet is suspended from two wires at the upper corners as shown in the sketch.

Question

(a) Determine the initial and the final rate of heat transfer from the plate.

(b) Estimate the time required.

(Hint: Note that in laminar natural convection, h ? ?T 0.25)

GIVEN

• A vertical aluminum sheet in water

• Plate dimensions: height (H) = 0.4 m, length (L) = 1 m, thickness (s) = 0.002 m

• Initial plate temperature (Tsi) = 150°C

• Water temperature (T?) = 20°C

• Final plate temperature (Tsf) = 50°C

FIND

(a) The initial and final heat transfer rates

(b) The time required

ASSUMPTIONS

• Constant and uniform water temperature

SKETCH



PROPERTIES AND CONSTANTS

for aluminum at the mean temperature of 100°C

Thermal conductivity (ka1) = 238 W/(m K)

Density (?) = 2702 kg/m3

Specific heat (c) = 896 J/(kg J)

Answer 1

(a) The Grashof number based on the height of the plate is

Initial



Final



(Turbulent)

The average heat transfer coefficient from a vertical plate with a turbulent boundary layer is



Initial



Final



The rate of convective heat transfer from the plate is



Initial



Final



(b) The initial Biot number for the aluminum sheet is



Therefore, the internal thermal resistance of the aluminum sheet is negligible during the entire cool down and the temperature-time history of the sheet is



Solving for the time



Using the average of the initial and final heat transfer coefficients