Question:

The diagram below shows a metal rod of length L and cross-sectional area A. The temperature at one end of the rod is maintained at T1, while the other end is maintained at T2. The thermal conductivity of the material of the rod is given by k.

a) Derive the expression for the rate of heat transfer (Q) through the rod in terms of the temperature difference, thermal conductivity, and other relevant parameters.

b) A copper rod of length 1.5 m, cross-sectional area 0.015 m², and thermal conductivity 400 W/m·K has a temperature difference of 45°C across its ends. Calculate the rate of heat transfer through the rod.

Answer:

a) The rate of heat transfer (Q) through the rod can be calculated using the formula:

where:

• Q is the rate of heat transfer (in watts, W)
• k is the thermal conductivity of the material (in watts per meter per Kelvin, W/m·K)
• A is the cross-sectional area of the rod (in square meters, m²)
• T2 is the temperature at one end of the rod (in Kelvin, K)
• T1 is the temperature at the other end of the rod (in Kelvin, K)
• L is the length of the rod (in meters, m)

b) Given:

• L = 1.5 m
• A = 0.015 m²
• k = 400 W/m·K
• T2 - T1 = 45°C = 45 K (Temperature difference is given in Celsius)

Substituting the given values into the formula, we get:

Simplifying the equation, we have:

Therefore, the rate of heat transfer through the copper rod is 900 W.

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