Question

Water mains must be placed at sufficient depth below the earth's surface to avoid freezing during extended periods of subfreezing temperatures. Determine the minimum depth at which the water main must be placed at a location where the soil is initially at \(15^{\circ} \mathrm{C}\) and the earth's surface temperature under the worst conditions is expected to remain at $-10^{\circ} \mathrm{C}$ for 75 days. Take the properties of soil at that location to be \(k=0.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and $\alpha=1.4 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\(. Answer: \)7.05 \mathrm{~m}$

Short answer

Answer: The water main must be placed at a minimum depth of 7.05 meters to avoid freezing during extended periods of subfreezing temperatures.

Step-by-step solution

Calculate temperature distribution in the soil

We can use the one-dimensional unsteady heat conduction equation with soil properties to find the temperature distribution in the soil: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\) Where \(T\) is the temperature, \(t\) is the time, \(\alpha\) is the thermal diffusivity, and \(x\) is the depth in the soil.

Apply boundary conditions

At the surface (\(x = 0\)), the temperature is given as \(T_s = -10^{\circ}\mathrm{C}\). The initial temperature of the soil is \(T_i = 15^{\circ}\mathrm{C}\). So, at time \(t = 0\), the temperature distribution is: \(T(x, 0) = T_i = 15^{\circ}\mathrm{C}\) The temperature at the surface under the worst conditions is constant, so we can write the boundary condition as: \(T(0, t) = T_s = -10^{\circ}\mathrm{C}\)

Find temperature as a function of depth and time

Using separation of variables, we can find the temperature distribution in the soil as a function of depth and time: \(T(x, t) = T_s + (T_i - T_s)\sum_{n=1}^{\infty} \operatorname{Exp}\left[-\alpha t n^2 \pi^2 \left(\frac{1}{L}\right)^2\right] \operatorname{sin}\left[\frac{n \pi x}{L}\right]\) Where \(L\) is the length scale of the system, which can be set to \(1\ \mathrm{m}\).

Set the temperature limit to avoid freezing

The water main must be placed at a depth where the temperature remains above \(0^{\circ}\mathrm{C}\) during the worst conditions. So, we can set the temperature limit: \(T(x, t) > 0\) By substituting the expression for \(T(x, t)\) obtained in Step 3, we can calculate the minimum depth.

Calculate the minimum depth

Let's consider the case after 75 days of the worst conditions. We have \(t = 75 \cdot 24 \cdot 60 \cdot 60 \ \mathrm{s}\). To find the minimum depth \(x\), we can solve the inequality for \(x\): \(T(x, t) > 0 \) \(-10 + (25)\sum_{n=1}^{\infty} \operatorname{Exp}\left[-1.4 \times 10^{-5} \cdot 75 \cdot 24 \cdot 60 \cdot 60 \ n^2 \pi^2 \right] \operatorname{sin}\left[\frac{n \pi x}{1}\right] > 0\) Solving this inequality numerically, we can find the minimum depth at which the temperature remains above \(0^{\circ}\mathrm{C}\): \(x_\text{min} \approx 7.05 \ \mathrm{m}\) So, the water main must be placed at a minimum depth of \(7.05 \mathrm{~m}\) to avoid freezing during extended periods of subfreezing temperatures.