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 can use the formula for steady-state heat conduction:
Q = (kAΔT)/L
Where:
Q is the rate of heat transfer
k is the thermal conductivity
A is the cross-sectional area
ΔT is the temperature difference
L is the thickness of the layer
Let's calculate the rate of heat transfer for each layer and then sum them up.
First layer:
k1 = 10 W/m·K
A1 = A (assuming the cross-sectional area is the same for all layers)
ΔT1 = (100°C - 50°C) = 50°C
L1 = 2 cm = 0.02 m
Q1 = (k1 * A1 * ΔT1) / L1
Q1 = (10 * A * (50)) / 0.02
Q1 = 2500A W
Second layer:
k2 = 20 W/m·K
A2 = A
ΔT2 = 50°C
L2 = 5 cm = 0.05 m
Q2 = (k2 * A2 * ΔT2) / L2
Q2 = (20 * A * 50) / 0.05
Q2 = 20000A W
Third layer:
k3 = 15 W/m·K
A3 = A
ΔT3 = 50°C
L3 = 3 cm = 0.03 m
Q3 = (k3 * A3 * ΔT3) / L3
Q3 = (15 * A * 50) / 0.03
Q3 = 25000A W
The rate of heat transfer through the composite wall is the sum of the rates for each layer:
Q_total = Q1 + Q2 + Q3
Q_total = 2500A + 20000A + 25000A
Q_total = 47500A W
Therefore, the rate of heat transfer through the wall is 47500A watts, where A is the cross-sectional area of the wall.
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