A copper sheet of thickness 2.09 mm is bonded to a steel sheet of thickness 1.35 mm. The outside surface of the copper sheet is held at a temperature of 100.0°C and the steel sheet at 26.3°C.
a) Determine the temperature (in °C) of the copper-steel interface?
b) How much heat is conducted through 1.00 m^2 of the combined sheets per second?
Use the following values:
k_steel=220 W/m.K
k_copper= 386 W/m.K
To determine the temperature at the copper-steel interface and the heat conducted through the combined sheets, we can use the concept of heat conduction and thermal resistance.
a) Temperature at the copper-steel interface:
The heat flow through the combined sheets can be modeled as heat flowing through two resistances in series: the copper sheet and the steel sheet. The temperature drop across each sheet can be determined using the formula:
ΔT = (Q * L) / (k * A)
where:
ΔT is the temperature difference across the sheet,
Q is the heat flow rate,
L is the thickness of the sheet,
k is the thermal conductivity of the material, and
A is the area of the sheet.
Let's assume that the temperature at the copper-steel interface is T.
Therefore, the temperature drop across the copper sheet (from the outside to the interface) is (100 - T) °C, and the temperature drop across the steel sheet (from the interface to the inside) is (T - 26.3) °C.
Applying the thermal resistance concept, the temperature drop across each sheet is equal to the resistances multiplied by the heat flow rate:
[(100 - T) °C] = ((Q / k_copper) * 2.09 mm * 0.001 m) / 1 m^2
[(T - 26.3) °C] = ((Q / k_steel) * 1.35 mm * 0.001 m) / 1 m^2
Since the temperature at the interface is the same for both materials, we can set the temperature drops equal to each other:
(100 - T) °C = (T - 26.3) °C
Solving for T:
T = (100 + 26.3) / 2
Therefore, the temperature at the copper-steel interface is T = 63.15 °C.
b) Heat conducted through 1.00 m^2 of the combined sheets per second:
The heat flow rate (Q) can be determined using the formula:
Q = (k_copper * A * ΔT) / (L_copper + L_steel)
where:
Q is the heat flow rate,
A is the area of the sheets,
ΔT is the temperature difference, and
L_copper and L_steel are the thicknesses of the copper and steel sheets, respectively.
Substituting the known values:
Q = (386 W/m.K * 1 m^2 * (100 - 63.15) °C) / (2.09 mm * 0.001 m + 1.35 mm * 0.001 m)
Evaluating this expression will give you the heat conducted through 1.00 m^2 of the combined sheets per second.