Question: A metal rod of length 2 meters is heated on one end and kept at a constant temperature of 100°C. The other end of the rod is kept at 0°C. The thermal conductivity of the metal is 50 W/(m·K).
a) Calculate the rate of heat transfer through the rod.
b) Determine the temperature of the rod at a distance of 0.5 meters from the heated end.
Answer: a) The rate of heat transfer through the rod can be calculated using Fourier's Law of heat conduction:
q = -k * A * (dT/dx)
where:
q is the rate of heat transfer (W or J/s)k is the thermal conductivity of the metal (W/(m·K))A is the cross-sectional area of the rod (m²)dT/dx is the temperature gradient along the length of the rod (K/m)As the rod is cylindrical, its cross-sectional area A can be given by:
A = π * r²
where r is the radius of the rod (m).
We need to find the temperature gradient dT/dx. Since the rod is held at a constant temperature of 100°C on one end and 0°C on the other, the change in temperature along its length is given by:
ΔT = T₂ - T₁ = 100°C - 0°C = 100°C
The change in position along the length of the rod is:
Δx = x₂ - x₁ = 0m - 2m = -2m
Therefore, the temperature gradient dT/dx can be calculated as:
dT/dx = ΔT / Δx = (100°C) / (-2m) = -50°C/m
Substitute the given values into Fourier's Law to calculate the rate of heat transfer:
q = -k * A * (dT/dx) = -50 W/(m·K) * (π * r²) * (-50°C/m)
= 2500πr² W/m
b) To determine the temperature of the rod at a distance of 0.5 meters from the heated end, we can use the heat conduction formula:
q = -k * A * (dT/dx)
Rearrange the equation to solve for the temperature gradient dT/dx as:
dT/dx = q / (-k * A)
The change in temperature ΔT at a distance Δx from the heated end is given by:
ΔT / Δx = dT / dx
Substituting the given values into the formula:
(-50°C/m) = (ΔT) / (0.5m)
Solving for ΔT:
ΔT = -50°C/m * 0.5m = -25°C
Thus, the temperature of the rod at a distance of 0.5 meters from the heated end is:
100°C - 25°C = 75°C
Therefore, the temperature of the rod at a distance of 0.5 meters from the heated end is 75°C.
Remember to provide the final answer, i.e., "The temperature of the rod at a distance of 0.5 meters from the heated end is 75°C."