Heat Transfer
A metal rod of length 1.2 meters and thermal conductivity 40 W/(m·K) is heated at one end and maintained at a constant temperature of 200°C. The other end of the rod is kept at a constant temperature of 50°C. The cross-sectional area of the rod is 0.5 m². Given that the rate of heat transfer through the rod is 80 W, find the thermal gradient across the rod.
The rate of heat transfer through a material can be calculated using the formula:
Q = k * A * ΔT / L
where:
Q = Rate of heat transfer (W) k = Thermal conductivity of the material (W/(m·K)) A = Cross-sectional area of the material (m²) ΔT = Temperature difference across the material (K) L = Length of the material (m)
We are given: k = 40 W/(m·K) A = 0.5 m² ΔT = (200 - 50)°C = 150 K L = 1.2 m Q = 80 W
Let's substitute the given values into the formula and solve for ΔT:
80 = 40 * 0.5 * ΔT / 1.2
Multiplying both sides of the equation by 1.2:
80 * 1.2 = 0.5 * ΔT * 40
96 = 20ΔT
Dividing both sides of the equation by 20:
ΔT = 4.8 K
Therefore, the thermal gradient across the rod is 4.8 K/m.