Question

In areas where the air temperature remains below \(0^{\circ} \mathrm{C}\) for prolonged periods of time, the freezing of water in underground pipes is a major concern. Fortunately, the soil remains relatively warm during those periods, and it takes weeks for the subfreezing temperatures to reach the water mains in the ground. Thus, the soil effectively serves as an insulation to protect the water from the freezing atmospheric temperatures in winter. The ground at a particular location is covered with snowpack at $-8^{\circ} \mathrm{C}$ for a continuous period of 60 days, and the average soil properties at that location are $k=0.35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( and \)\alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$. Assuming an initial uniform temperature of \(8^{\circ} \mathrm{C}\) for the ground, determine the minimum burial depth to prevent the water pipes from freezing.

Short answer

Answer: The minimum burial depth required is approximately 2.44 meters.

Step-by-step solution

Convert time to seconds

First, we need to convert the time given in days to seconds. We have 60 days of continuous freezing temperature, so we will convert this time period into seconds: Time (t) = 60 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute = 5,184,000 seconds

Calculate parameter `L`

Calculate the parameter L using the formula: \(L = 2 \sqrt{\alpha t}\) where \(t\) is the time in seconds and \(\alpha\) is the thermal diffusivity. \(L = 2 \sqrt{(0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s})(5,184,000 \mathrm{~s})} = 3.58 \mathrm{~m}\)

Calculate burial depth using temperature at `L`

Now we'll calculate the burial depth using the temperature at `L`. To do this, we'll use the formula: \(\frac{T(x) - T_{surf}}{T_{initial} - T_{surf}} = erf(\frac{x}{2\sqrt{\alpha t}})\) where - \(T(x)\) is the temperature at burial depth (we want it to be \(0^\circ \mathrm{C}\)), - \(T_{surf}\) is the surface temperature (-8°C), - \(T_{initial}\) is the initial ground temperature (8°C), - \(x\) is the burial depth we want to find, - \(\alpha\) is the thermal diffusivity (0.15 × 10⁻⁶ m²/s), - \(t\) is the time in seconds (5,184,000 s), - erf is the error function. Thus, solving the equation for \(x\), we get: \(\frac{ 0 - (- 8)}{8 - (- 8)} = erf\big(\frac{x}{2\sqrt{(0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s})(5,184,000 \mathrm{~s})}}\big)\)

Solve for burial depth `x`

Now, we'll solve the equation for \(x\). We have: \(\frac{1}{2} = erf\big(\frac{x}{3.58 \mathrm{~m}}\big)\) Inverting the error function, we find the value of its argument: \(erf^{-1}(0.5) = \frac{x}{3.58 \mathrm{~m}}\) Now, solve for \(x\): \(x = 3.58 \mathrm{~m} \times erf^{-1}(0.5) = 2.44 \mathrm{~m}\) So the minimum burial depth to prevent the water pipes from freezing is approximately 2.44 meters.