A 2.5 × 2.5 m steel sheet 1.5-mm-thick is removed from an annealing oven at a uniform temperature of 425°C and placed in a large room at 20°C in a horizontal position. (a) Calculate the rate of heat transfer from the steel sheet immediately after its removal from the furnace, considering both radiation and convection. (b) Determine the time required for the steel sheet to cool to a temperature of 60°C. Hint: This will require numerical integration.

Question

GIVEN

• Horizontal steel sheet in air

• Sheet dimensions = 2.5 m × 2.5 m × 0.0015 m

• Sheet initial temperature (Tsi) = 425°C = 698 K

• Air temperature (T?) = 20°C = 293 K FIND

(a) The initial rate of heat transfer (q) (b) The time required for the sheet to cool to 60°C (333 K)

ASSUMPTIONS

• The room behaves as a black body at T?

• The steel sheet behaves as a black body (? = 1.0)

• Heat transfer takes place from both top and bottom of the sheet

• The steel is 1% carbon steel

• Heat transfer from the edges of the plate is negligible

SKETCH



PROPERTIES AND CONSTANTS

The Stephan-Boltzmann constant (?) = 5.67 × 10–8 W/(m2 K4)

for 1% carbon steel

Specific heat (cs) = 473 J/(kg K)

Density (?s) = 7801 kg/m3

for dry air at the mean temperature of 496 K (223°C)

Thermal expansion coefficient (?) = 0.00203 1/K

Thermal conductivity (k) = 0.0384 W/(m K)

Kinematic viscosity (?) = 38.7 × 10–6 m2/s

Prandtl number (Pr) = 0.71

Answer 1

(a) The characteristic length for this geometry is



The Grashof and Rayleigh numbers are



The average Nusselt number on the bottom of the plate is given by



The average Nusselt number on the top of the plate is



The total rate of heat transfer is the sum of the convective and radiative components



(b) As the plate cools, the rate of heat transfer will decrease. The cooling time will be estimated by calculating a new sheet temperature and heat transfer every time period.

The Biot number for the sheet is



This is slightly above 0.1. For a first order approximation, we can neglect the thermal resistance in the plate. For the first 20 second interval



Solving for temperature after 20 sec



This temperature is then used to calculate new transfer coefficients and heat transfer rates as shown below. This procedure is followed until the temperature of the plate is 333 K.







Interpolating between 200 and 300 seconds

The time required to reach 333 K is approximately 290 seconds = 4.8 min.