Consider a composite wall made up of three layers in series. The first layer is 2 cm thick with thermal conductivity k1 = 10 W/m·K, the second layer is 5 cm thick with thermal conductivity k2 = 20 W/m·K, and the third layer is 3 cm thick with thermal conductivity k3 = 15 W/m·K. If the temperatures at the two ends of the wall are 100°C and 50°C, calculate the rate of heat transfer through the wall.
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To calculate the rate of heat transfer through the composite wall, we need to determine the thermal resistance of each layer and then sum them up.
The thermal resistance of each layer is given by:
R = Thickness / (Thermal conductivity)
For the first layer:
R1 = 0.02 m / 10 W/m·K = 0.002 K·m²/W
For the second layer:
R2 = 0.05 m / 20 W/m·K = 0.0025 K·m²/W
For the third layer:
R3 = 0.03 m / 15 W/m·K = 0.002 K·m²/W
To calculate the total thermal resistance, we sum up the individual thermal resistances:
R_total = R1 + R2 + R3 = 0.002 K·m²/W + 0.0025 K·m²/W + 0.002 K·m²/W = 0.0065 K·m²/W
The rate of heat transfer through the wall is given by:
Q = (T1 - T2) / R_total
Where T1 and T2 are the temperatures at the two ends of the wall.
Given that T1 = 100°C = 100 + 273.15 K = 373.15 K
And T2 = 50°C = 50 + 273.15 K = 323.15 K
We can now calculate the rate of heat transfer:
Q = (373.15 K - 323.15 K) / 0.0065 K·m²/W
= 50 K / 0.0065 K·m²/W
= 7692.31 W
Therefore, the rate of heat transfer through the composite wall is 7692.31 W.
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