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 of the copper-steel interface, we can use the concept of heat conduction and the formula for thermal resistance in series:
R_total = R_copper + R_steel
The thermal resistance (R) is given by the equation:
R = (thickness) / (thermal conductivity * area)
For this problem, let's calculate the thermal resistances for copper (R_copper) and steel (R_steel) separately.
a) Calculate the temperature of the copper-steel interface:
1. Calculate the thermal resistance of copper (R_copper):
R_copper = (thickness of copper) / (thermal conductivity of copper * area)
= (2.09 mm) / (386 W/m.K * 1 m^2)
= 0.0054 K/W
2. Calculate the thermal resistance of steel (R_steel):
R_steel = (thickness of steel) / (thermal conductivity of steel * area)
= (1.35 mm) / (220 W/m.K * 1 m^2)
= 0.0061 K/W
3. Calculate the total thermal resistance:
R_total = R_copper + R_steel
= 0.0054 K/W + 0.0061 K/W
= 0.0115 K/W
4. Calculate the temperature difference across the two sheets:
ΔT = (temperature of copper - temperature of steel)
= (100.0°C - 26.3°C)
= 73.7°C
5. Calculate the temperature of the copper-steel interface:
Temperature of copper-steel interface = (temperature of copper) - (R_total * ΔT)
= 100.0°C - (0.0115 K/W * 73.7°C)
= 99.16°C
Therefore, the temperature of the copper-steel interface is approximately 99.16°C.
b) Calculate the heat conducted through 1.00 m^2 of the combined sheets per second:
1. Calculate the heat flow rate (Q) using Fourier's Law of heat conduction:
Q = (thermal conductivity * area * ΔT) / (total thermal resistance)
= (386 W/m.K * 1 m^2 * 73.7°C) / (0.0115 K/W)
= 249949.56 W
Therefore, the heat conducted through 1.00 m^2 of the combined sheets per second is approximately 249,949.56 W or 249.95 kW.