Question:

A metal rod of length 2.0 m and cross-sectional area 0.05 m^2 conducts heat with a thermal conductivity of 80 W/(m·K). The hot end of the rod is in contact with a heat source at 200°C, while the cold end is in contact with a heat sink at 20°C. Calculate the rate of heat transfer through the rod.

Given:

Length of the rod (L) = 2.0 m Cross-sectional area of the rod (A) = 0.05 m^2 Thermal conductivity of the rod (k) = 80 W/(m·K) Temperature of the hot end (T₁) = 200°C Temperature of the cold end (T₂) = 20°C

Assumptions:

• The rod is made of a homogeneous material.
• Heat transfer occurs only through conduction.
• The thermal conductivity of the rod remains constant throughout.

Answer:

The rate of heat transfer through conduction can be determined using the formula:

Where: Q = Heat transfer rate (in watts, W) k = Thermal conductivity (in W/(m·K)) A = Cross-sectional area of the rod (in square meters, m^2) ΔT = Temperature difference between the hot and cold ends (in Kelvin, K) L = Length of the rod (in meters, m)

Given values: k = 80 W/(m·K) A = 0.05 m^2 ΔT = (200 + 273) K - (20 + 273) K = 453 K (converting temperatures to Kelvin) L = 2.0 m

Substituting the given values into the formula:

Calculating further:

Therefore, the rate of heat transfer through the rod is 453 W.