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# This is a transient convection problem that utilizes dimensionless numbers for the solution.

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A 10 cm thick pure gold plate is initially at 85ËšC. Both top and bottom boundary surfaces are exposed to an ambient gas at 15ËšC with a heat transfer coefficient of h=400 W/C-m^s Calculate the center temperature at t = 6 minutes after the start of cooling.

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A 10 cm thick pure gold plate is initially at 85ËšC. Both top and bottom boundary surfaces are exposed to an ambient gas at 15ËšC with a heat transfer coefficient of Calculate the center temperature at t = 6 minutes after the start of cooling.

Solution:

The physical constants for gold can be found by referring to any general text on heat transfer. These are found to be:

and

For a plate, the transient temperature charts can be utilized. The chart to be used is the transient temperature chart for a slab with thickness 2L subjected to convection at both boundary surfaces. To utilize this chart correctly, the thickness is set equal to 2L:

2L = 10 cm.

Rewriting,

.

Calculating the dimensionless time parameter, or Fourier number, Ï„:

Calculating the reciprocal of the Biot number to determine the appropriate curve on the chart:

Reading off the value from the chart using the calculated dimensionless parameters above yields:

Thus the center temperature is at t = 6 minutes after the start of cooling is:

=15ËšC + (85ËšC - 15ËšC)(0.34)=38.8ËšC

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