Question

A row of 10 parallel pipes that are \(5 \mathrm{~m}\) long and have an outer diameter of \(6 \mathrm{~cm}\) are used to transport steam at $145^{\circ} \mathrm{C}\( through the concrete floor \)(k=0.75 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\( of a \)10-\mathrm{m} \times 5-\mathrm{m}$ room that is maintained at \(24^{\circ} \mathrm{C}\). The combined convection and radiation heat transfer coefficient at the floor is $12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the surface temperature of the concrete floor is not to exceed \(38^{\circ} \mathrm{C}\), determine how deep the steam pipes should be buried below the surface of the concrete floor.

Short answer

Answer: The steam pipes should be buried at a depth of 0.0625 meters (6.25 cm) below the surface of the concrete floor.

Step-by-step solution

Given information and data

List all given information: - Number of pipes: 10 - Pipe length: \(L = 5\mathrm{~m}\) - Pipe outer diameter: \(D = 6\mathrm{~cm} = 0.06\mathrm{~m}\) - Steam temperature: \(T_s = 145^{\circ}\mathrm{C}\) - Size of the room: \(10\mathrm{~m} \times 5\mathrm{~m}\) - Room temperature: \(T_r = 24^{\circ}\mathrm{C}\) - Maximum surface temperature: \(T_s = 38^{\circ}\mathrm{C}\) - Combined convection and radiation heat transfer coefficient: \(h = 12\mathrm{~W} / \mathrm{m}^{2}\cdot\mathrm{K}\) - Concrete thermal conductivity: \(k=0.75\mathrm{~W} / \mathrm{m} \cdot\mathrm{K}\)

Heat transfer through the concrete

First, we need to find the heat transfer through the concrete floor. Based on the given data, the heat transfer in the concrete could be calculated using Fourier's law of heat conduction as follows: \(Q = k A (T_s - T_r) / L_c\) Where \(Q\) is the heat transferred, \(A\) is the area of a single pipe, and \(L_c\) is the depth of the pipe in the concrete. We'll solve for \(Q\) first by using the values for maximum surface temperature, room temperature, and concrete thermal conductivity.

Heat transfer at the floor surface

Next, we need to compute the heat transfer at the surface of the floor due to convection and radiation. We can use the formula: \(Q = h A (T_s - T_r)\) Again, we'll solve for \(Q\) using the given values for the combined heat transfer coefficient, maximum surface temperature, and room temperature.

Equating heat transfers and solving for depth

Since the heat transfer through the concrete must be equal to the heat transfer at the surface, we can equate the expressions for \(Q\) from Steps 2 and 3: \(k A (T_s - T_r) / L_c = h A (T_s - T_r)\) We can cancel out the area, \(A\), and the temperature difference \((T_s - T_r)\) from both sides, then solve for the depth, \(L_c\): \(L_c = k / h\) Plug in the values for the concrete thermal conductivity and the combined heat transfer coefficient: \(L_c = 0.75\mathrm{~W} / \mathrm{m} \cdot\mathrm{K} / (12\mathrm{~W} / \mathrm{m}^{2}\cdot\mathrm{K}) = 0.0625\mathrm{~m}\)

Conclusion

The steam pipes should be buried at a depth of 0.0625 meters (6.25 cm) below the surface of the concrete floor to ensure the surface temperature does not exceed 38°C.