Question

The average rate at which heat flows out through the surface of the Earth in North America is \(54 \mathrm{~mW} / \mathrm{m}^{2}\) and the average thermal conductivity of the near surface rocks is \(2.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). Assuming a surface temperature of \(10^{\circ} \mathrm{C}\), what should be the temperature at a depth of \(33 \mathrm{~km}\) (near the base of the crust)? Ignore the heat generated by radioactive elements; the curvature of the Earth can also be ignored.

Short answer

The temperature at a depth of 33 km is approximately 742°C.

Step-by-step solution

Identify given variables

From the exercise, given variables include: heat flow rate (q) = 54 mW/m² or 0.054 W/m², thermal conductivity (k) = 2.5 W/m.K, surface temperature (Ts) = 10 °C or 283.15K (converting to Kelvin), and depth (d) = 33 km or 33000 m.

Rearrange the formula to solve for temperature at depth

The formula \(q=kA\frac{\Delta T}{d}\) can be rearranged as \(\Delta T = \frac{qd}{kA}\). Since it's the temperature at a depth of 33 km that is needed, denote that temperature as T1. Now, the equation becomes \(\Delta T = T1 - Ts\), where Ts is the surface temperature. Substitute \(\Delta T\) in the initial formula with \(T1 - Ts\), then the formula becomes \(T1 = Ts + \frac{qd}{kA}\)

Substitution and calculation

Substitute given values into the formula: \(T1 = 283.15 + \frac{0.054\times33000}{2.5\times1}\) The area (A), often denoted in equations, doesn't need to be accounted for in this scenario since the heat per unit area was given. Calculate the expression above to get the value for T1 in Kelvin.

Convert the result back to Celsius

The previous step yields the temperature in Kelvin. Convert this back to Celsius for the final result (remember, to convert Kelvin to Celsius, you subtract 273.15 from the Kelvin temperature.

Key concepts

Heat Flow Rate

The heat flow rate, often denoted as 'q', is a measure of the amount of heat energy that moves through a material per unit time and area. It is usually expressed in units of watts per square meter (W/m²). In the context of geothermal energy and Earth sciences, it describes the rate at which heat escapes from the Earth's interior to its surface.

For instance, in the provided exercise, the heat flow rate is given as 54 milliwatts per square meter (mW/m²) for North America. This seemingly small number is crucial for calculating the temperature difference caused by the heat escaping through the ground. Together with thermal conductivity, this rate helps to determine the temperature at different depths below the Earth's surface, assuming that other factors like heat generation from radioactive decay are negligible.

To enhance understanding, imagine you have a heat source under a pan. The rate at which heat is transferred from that source to the pan's surface is analogous to the heat flow rate from the Earth's core to its surface. This concept is vital in many practical applications, such as understanding the Earth's temperature distribution, designing geothermal energy systems, and even in creating efficient insulation materials.

Temperature Gradient

A temperature gradient is the rate of temperature change with respect to distance in a particular direction. It is a vector quantity, which means it has both magnitude and direction, and is expressed in units like degrees Celsius per meter (°C/m) or Kelvin per meter (K/m). In geology and environmental science, gradient indicates how temperature increases or decreases with changes in elevation or depth.

Within the Earth, temperature gradient plays a role in a variety of processes, including geothermal heat transfer. For example, the exercise mentions the task of finding the temperature at a depth of 33 kilometers, given the surface temperature. To do this, we use the heat flow rate and thermal conductivity to determine the temperature gradient, and then use it to extrapolate the subsurface temperature.

Understanding temperature gradients is essential for several reasons. It aids in predicting how heat will distribute in a given material over time and also helps in estimating the effects of temperature changes on Earth's climate, engineering structures, and biological ecosystems. It's a fundamental concept in meteorology, oceanography, geology, and many other fields where heat distribution is essential.

Geothermal Gradient

Geothermal gradient is the specific type of temperature gradient that represents how the temperature changes with depth below the Earth's surface. It is a critical concept in the fields of geology and geophysics. Understanding the geothermal gradient is key for determining the earth's inner structure, studying plate tectonics, and is especially important for exploring and harnessing geothermal energy resources.

In the context of our textbook exercise, the geothermal gradient is used to calculate the expected temperature at around 33 kilometers beneath the Earth's crust. The geothermal gradient can vary depending on geographical location and geological conditions, such as the presence of volcanic activity or hot springs. For instance, the geothermal gradient in regions with active volcanic processes might be much higher, leading to hotter temperatures at shallower depths.

When considering improvements in educational content about this topic, it is beneficial to incorporate visual aids such as graphs or diagrams showing the relationship between depth and temperature. Such visuals aid in reinforcing the abstract concept of geothermal gradient, making it easier for students to grasp how temperature naturally increases as one delves deeper into the Earth.