Question

A 12 -m-long and 8-cm-diameter hot-water pipe of a district heating system is buried in the soil \(80 \mathrm{~cm}\) below the ground surface. The outer surface temperature of the pipe is \(60^{\circ} \mathrm{C}\). Taking the surface temperature of the earth to be \(2^{\circ} \mathrm{C}\) and the thermal conductivity of the soil at that location to be $0.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, determine the rate of heat loss from the pipe.

Short answer

Answer: The rate of heat loss from the pipe is approximately 1278.40 watts.

Step-by-step solution

Convert given values to appropriate units

Convert the diameter and depth of the pipe to meters: 8 cm = 0.08 m 80 cm = 0.8 m

Calculate the outer radius of the pipe

Calculate the outer radius \(r_1\) by dividing the diameter by 2: \(r_1 = \frac{0.08 \ \text{m}}{2} = 0.04 \ \text{m}\)

Calculate the distance from the center of the pipe to the ground surface

Add the depth of the pipe and the outer radius to obtain \(r_2\): \(r_2 = 0.8 \ \text{m} + 0.04 \ \text{m} = 0.84 \ \text{m}\)

Apply the formula to find the heat loss rate

Use the given values and the ones calculated previously to find the rate of heat loss (\(q\)) using the formula mentioned above: $$ q = \frac{2 \pi (0.9 \ \text{W/m·K}) (12 \ \text{m})(60^{\circ} \mathrm{C} - 2^{\circ} \mathrm{C})}{\ln \frac{0.84 \ \text{m}}{0.04 \ \text{m}}} $$

Calculate the heat loss rate

Solve the equation to obtain the heat loss rate \(q\): $$ q \approx 1278.40 \ \text{W} $$ The rate of heat loss from the pipe is approximately 1278.40 watts.