Question

The water main in the cities 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 a period of 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 minimum depth at which the water main must be placed is 7.05 meters.

Step-by-step solution

Understand the given data

We are given the following information: - Initial soil temperature: \(T_i = 15°C\) - Surface temperature: \(T_s = -10°C\) - Duration of subfreezing temperatures: \(t = 75\) days - Soil thermal conductivity: \(k = 0.7 \frac{W}{m \cdot K}\) - Soil thermal diffusivity: \(\alpha = 1.4 \times 10^{-5} \frac{m^2}{s}\) We need to find the critical depth \(x\) where the temperature drops to 0°C after 75 days.

Convert the time from days to seconds

To work with the given values of thermal diffusivity (\(\alpha\)), convert the time from days to seconds. \(t = 75 \text{ days} \times 24 \frac{\text{hours}}{\text{day}} \times 60 \frac{\text{minutes}}{\text{hour}} \times 60 \frac{\text{seconds}}{\text{minute}} = 6480000 \text{ seconds}\)

Use the 1D unsteady heat conduction equation to find temperature distribution in the soil

The temperature distribution in a semi-infinite medium is given by the following equation: $$T(x,t) = T_s + (T_i - T_s) \text{erfc} \left(\frac{x}{2 \sqrt{\alpha t}}\right)$$ Where \(T(x,t)\) is the temperature at a depth \(x\) and time \(t\), and \(\text{erfc}\) is the complementary error function. We need to find the critical depth where \(T(x, t) = 0°C\). Set \(T(x, t) = 0\) and solve for \(x\).

Solve for the critical depth

\(0 = -10 + (15 - (-10)) \text{ \text{erfc} }\left(\frac{x}{2 \sqrt{1.4 \times 10^{-5} \cdot 6480000}}\right)\) Simplify the expression: %% $$ (20) \text{ \text{erfc} }\left(\frac{x}{2 \sqrt{1.4 \times 10^{-5} \cdot 6480000}}\right) = 10 $$ %% So, the complementary error function value is: $$\text{erfc} \left(\frac{x}{2 \sqrt{1.4 \times 10^{-5} \cdot 6480000}}\right) = 0.5$$ Now, we need to find the inverse of the erfc function: $$\frac{x}{2 \sqrt{1.4 \times 10^{-5} \cdot 6480000}} = \text{erfc}^{-1}(0.5)$$ Using a calculator or software that has an inverse erfc function, we find: $$\text{erfc}^{-1}(0.5) = 0.4769$$ Now we solve for \(x\): %% x = 2 \sqrt{1.4 \times 10^{-5} \cdot 6480000} \times 0.4769 $$ Calculate the value of \(x\): \(x = 2 \times \sqrt{1.4 \times 10^{-5} \times 6480000} \times 0.4769 = 7.05 \text{ meters}\) So, the minimum depth at which the water main must be placed at the location is 7.05 meters.

Key concepts

Heat Transfer Fundamentals

Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and exchange of thermal energy between physical systems. Fundamental modes of heat transfer include conduction, convection, and radiation. Heat conduction, or thermal conduction, is the transfer of heat from one part of a body to another with which it is in contact, without appreciable movement of the body as a whole.

In the exercise regarding the prevention of water pipes from freezing, we're focusing on the concept of unsteady heat conduction, which occurs when the temperature distribution within a material changes over time. The temperature change at a specific point within the material depends on the thermal properties of the material and the conditions at the boundaries. The exercise requires us to apply knowledge of unsteady heat conduction to determine how deep water pipes need to be buried to avoid freezing.

Thermal Conductivity

Thermal conductivity is a property of materials that indicates their ability to conduct heat. It is denoted by the symbol \(k\) and typically measured in watts per meter per kelvin (\(W/m \times K\)). A high thermal conductivity means that the material can transfer heat effectively, while a low thermal conductivity indicates poor heat transfer ability.

In the textbook exercise, we use the soil's thermal conductivity value to calculate how heat would move through the soil over time. The depth of the water main is directly influenced by the soil's ability to transfer heat to or from the pipe. A factor to consider is that different materials will have different thermal conductivity values, therefore changing the value of \(k\) could affect the required depth of the water pipes to resist freezing temperatures.

Temperature Distribution

Temperature distribution refers to how temperature varies within a material. When considering unsteady heat conduction, temperature distribution changes with both location and time. An understanding of temperature distribution within a material is critical to solving a wide array of thermal problems.

In the exercise we are focusing on, we explore temperature distribution within soil over a period of 75 days where the surface temperature is maintained at a constant \(-10^\circ C\). To calculate the distribution, we apply the 1D unsteady heat conduction equation which includes the thermal diffusivity of the soil (\(\alpha\)) and the initial and boundary temperatures. The calculated temperature distribution helps to determine the minimum depth for the water main to prevent it from reaching the freezing point. It's important to realize that the temperature distribution is not uniform across the soil depth; it changes with the specific soil properties and environmental conditions.