A copper sheet of thickness 2.49 mm is bonded to a steel sheet of thickness 1.51 mm. The outside surface of the copper sheet is held at a temperature of 100.0°C and the steel sheet at 24.1°C.
a) Determine the temperature (in °C) of the copper-steel interface.
b) How much heat is conducted through 1.00 m2 of the combined sheets per second?
To determine the temperature of the copper-steel interface, we can use the concept of thermal equilibrium. At thermal equilibrium, the heat flows from a higher temperature region to a lower temperature region until both regions reach the same temperature.
a) To find the temperature at the copper-steel interface, we can assume that the heat flows only in the thickness direction of the sheets. Therefore, the heat flowing through both sheets must be equal to each other.
The rate of heat transfer (Q) can be calculated using the formula:
Q = k * A * ΔT / d
Where:
- Q is the rate of heat transfer
- k is the thermal conductivity of the material
- A is the cross-sectional area through which heat is conducted
- ΔT is the temperature difference
- d is the thickness of the material
Since the heat flowing through both sheets must be equal to each other, we have:
(Q_copper) = (Q_steel)
k_copper * A * (T_copper - T_interface) / d_copper = k_steel * A * (T_interface - T_steel) / d_steel
Simplifying the equation by canceling out A (since it's the same for both sides) and rearranging the terms, we get:
k_copper * (T_copper - T_interface) / d_copper = k_steel * (T_interface - T_steel) / d_steel
Now we can substitute the values given:
k_copper = thermal conductivity of copper (assumed constant) = 401 W/(m·K)
k_steel = thermal conductivity of steel (assumed constant) = 50 W/(m·K)
T_copper = temperature of the copper sheet = 100.0°C
T_steel = temperature of the steel sheet = 24.1°C
d_copper = thickness of the copper sheet = 2.49 mm = 0.00249 m
d_steel = thickness of the steel sheet = 1.51 mm = 0.00151 m
Plugging in the values, we get:
401 * (100.0 - T_interface) / 0.00249 = 50 * (T_interface - 24.1) / 0.00151
Now, we can solve this equation to find the temperature (T_interface) at the copper-steel interface.
b) To calculate the amount of heat conducted through 1.00 m2 of the combined sheets per second, we can use the formula:
Q = k * A * ΔT / d
Where:
- Q is the rate of heat transfer
- k is the thermal conductivity of the material
- A is the cross-sectional area through which heat is conducted
- ΔT is the temperature difference
- d is the thickness of the material
Since we already have the values for thermal conductivity, temperature difference, and thickness, we can substitute them into the formula and calculate the rate of heat transfer (Q).