Question

Assume that the radioactive elements in the Earth are uniformly distributed through a nearsurface layer. The surface heat flow is \(70 \mathrm{~mW} \mathrm{~m}^{-2}\), and there is no heat flow into the base of the layer. If \(k=4 \mathrm{~W} \mathrm{~m}^{-1} \mathrm{~K}^{-1}, T_{0}=0^{\circ} \mathrm{C},\) and the temperature at the base of the layer is \(1200^{\circ} \mathrm{C}\), determine the thickness of the layer and the volumetric heat production.

Short answer

The thickness of the layer is approximately 68.57 km, and the volumetric heat production is about \(1.02 \times 10^{-6} \text{ W/m}^3\).

Step-by-step solution

Understand the Given Data

We have the following information:- Surface heat flow, \( q = 70 \text{ mW/m}^2 \).- Thermal conductivity, \( k = 4 \text{ W/mK} \).- Temperature at the surface, \( T_0 = 0^{\circ} \text{C} \).- Temperature at the base of the layer, \( T_b = 1200^{\circ} \text{C} \).- There is no heat flow from below, meaning all heat is generated within the layer itself.

Set Up the Heat Flow Equation

Using Fourier's law of heat conduction, the heat flow \( q \) is given by:\[q = -k \frac{dT}{dz}\]where \( \frac{dT}{dz} \) is the temperature gradient. Since the heat flow is constant across the layer, we can express the temperature gradient as:\[\frac{dT}{dz} = \frac{T_b - T_0}{L}\]where \( L \) is the thickness of the layer.

Solve for Layer Thickness

Substitute the given values into the equation from Step 2:\[70 \times 10^{-3} = 4 \times \frac{1200 - 0}{L}\]Rearrange to solve for \( L \):\[L = \frac{4 \times 1200}{70 \times 10^{-3}}\]Calculate \( L \) to find the thickness of the layer:\[L \approx \frac{4800}{0.07} = 68571.43 \text{ m} = 68.57 \text{ km}\]

Determine Volumetric Heat Production

The total heat at the surface comes from the volumetric heat production \( A \) in the layer. Therefore:\[A \cdot L = q\]Solving for \( A \), we have:\[A = \frac{q}{L} = \frac{70 \times 10^{-3}}{68571.43}\]Calculate \( A \):\[A \approx 1.02 \times 10^{-6} \text{ W/m}^3\]

Key concepts

Fourier's Law of Heat Conduction

Fourier's Law of Heat Conduction is a fundamental principle that describes how heat energy transfers through a material. This law is especially significant when studying geothermal heat flow, as it helps us understand how heat moves from the Earth's interior to the surface. The law is mathematically expressed as:

• \[ q = -k \frac{dT}{dz} \]

In this formula:

• \( q \) is the heat flow rate per unit area (measured in watts per square meter, W/m²).
• \( k \) stands for thermal conductivity, reflecting how easily heat flows through a material (measured in watts per meter per kelvin, W/mK).
• \( \frac{dT}{dz} \) represents the temperature gradient, or how temperature changes with depth.
• The negative sign indicates heat flows from warmer to cooler regions.

The temperature gradient is calculated by dividing the temperature difference across a material by its thickness. This formula helps us determine the heat flow when the thermal conductivity and temperature gradient are known.
By applying Fourier's law in geothermal studies, scientists can estimate the thickness of Earth's layers and the rate at which heat is produced and escapes to the surface.

Volumetric Heat Production

Volumetric heat production refers to the rate at which heat is generated within a specific volume of Earth's crust, often due to radioactive decay of elements such as uranium, thorium, and potassium. It is usually expressed in watts per cubic meter (W/m³).
In the context of geothermal heat flow, knowing the volumetric heat production is crucial as it directly correlates to the surface heat flow. This can be determined by evaluating the heat flow per unit area and the thickness of the crustal layer.

• Total surface heat flow is the combination of internally generated heat and any heat flowing in from deeper layers.
• In this exercise, it's assumed that there's no heat coming from below the layer, implying all heat felt at the surface is due to internal generation.
• The volumetric heat production \( A \) is calculated by using the surface heat flow \( q \) and the layer thickness \( L \) through the formula: \[ A = \frac{q}{L} \]

Understanding volumetric heat production helps in modeling Earth's thermal state and the rate of geological processes driven by internal heat.

Thermal Conductivity

Thermal conductivity is a property of materials that indicates how efficiently heat energy is transferred through a material due to a temperature difference. In geothermal contexts, this is an essential parameter, as different layers of the Earth have varying conductivities.

• Higher thermal conductivity means heat flows easily through the material without much resistance.
• Conversely, lower thermal conductivity implies the material is a poor conductor of heat, potentially acting as an insulating layer.
• In our exercise, the given \( k = 4 \text{ W/mK} \) was used to understand how heat from below moves through the Earth's layer to reach the surface.

Thermal conductivity importantly affects the temperature gradient. As seen from Fourier's law, it helps determine how steep or gradual the temperature profile will be across a layer.
Accurate knowledge of the thermal conductivity of Earth materials helps scientists build models defining heat movement and distribution within Earth’s crust. It affects energy resource management and predicts geothermal potential and efficiency.