Question

The total rate at which power is used by humans worldwide is approximately 15 TW (terawatts). The solar flux averaged over the sunlit half of Earth is \(680 \mathrm{~W} / \mathrm{m}^{2}\) (assuming no clouds). The area of Earth's disc as seen from the Sun is \(1.28 \times 10^{14} \mathrm{~m}^{2}\). The surface area of Earth is approximately 197,000,000 square miles. How much of Earth's surface would we need to cover with solar energy collectors to power the planet for use by all humans? Assume that the solar energy collectors can convert only \(10 \%\) of the available sunlight into useful power.

Short answer

Approximately 4.32% of Earth's surface would need to be covered with solar energy collectors to power the entire planet for use by all humans.

Step-by-step solution

Calculate Total Solar Power Received by Earth

First, we need to find the total amount of solar power received by the Earth. We have the average solar flux over the sunlit half of Earth (680 W/m²) and the area of Earth's disc as seen from the Sun (1.28 x 10^14 m²). Total solar power received by Earth = Average solar flux x Area of Earth's disc = 680 W/m² x 1.28 x 10^14 m² = 8.704 x 10^16 W

Calculate Power Received by Solar Collectors

Now, we need to calculate the amount of power that can be collected by solar energy collectors. They can convert only 10% of the available sunlight into useful power. Therefore, the power received by the solar energy collectors is 10% of the total power received by Earth. Power received by solar collectors = 10% of 8.704 x 10^16 W = 0.10 x 8.704 x 10^16 W = 8.704 x 10^15 W

Calculate the Required Power Generated by Solar Collectors

We are given that the total rate at which power is used by humans worldwide is 15 TW. Thus, we need the solar energy collectors to generate this amount of power. 1 TW = 10^12 W, so 15 TW = 15 x 10^12 W

Calculate the Required Area of Solar Collectors

Now let's determine the area required to generate the power needed by all humans. We have the power received by solar collectors and the required power. Area required = (Power required by humans) / (Power received by solar collectors per square meter) Area required = (15 x 10^12 W) / (680 W/m²) = 2.20588 x 10^10 m²

Convert Square Meters to Square Miles

We need to convert the area in square meters to square miles. 1 square meter is equal to approximately 3.86102 x 10^-7 square miles. Area required in square miles = 2.20588 x 10^10 m² x 3.86102 x 10^-7 mi/m² ≈ 8,521,621 square miles

Calculate the Percentage of Earth's Surface

Finally, let's compute the percentage of Earth's surface that needs to be covered by solar panels. Percentage = (Required area of solar panels) / (Total surface area of Earth) x 100 Percentage = (8,521,621 square miles) / (197,000,000 square miles) x 100 ≈ 4.32 % Approximately 4.32% of Earth's surface would need to be covered with solar energy collectors to power the entire planet for use by all humans.

Key concepts

Solar Flux

Solar Flux is an important measure in understanding how much solar energy hits a certain area on the Earth's surface. In simpler terms, it indicates the power or energy received per square meter. This gives us an idea of how intense the sunlight is in any particular region. Average solar flux changes depending on time of day, geographical location, and weather conditions. However, for calculation purposes here, we're using an average value of 680 W/m² (watts per square meter) when the sun is directly overhead and there are no clouds.

Solar flux is crucial in determining how much energy can be harvested from the sun. It is calculated by multiplying the average solar flux by the area receiving sunlight. For Earth, this area is represented by its disc facing the sun. By understanding solar flux, we can estimate the potential of solar power generation on a global scale.

Energy Conversion Efficiency

Energy Conversion Efficiency refers to the percentage of solar energy that a solar panel system converts into usable electricity. Not all sunlight that hits a solar panel is converted into electricity due to various technological and material limitations. In many cases, state-of-the-art solar panels have efficiencies around 15-20%. However, for this example, we're assuming an efficiency of just 10%.

This means that only 10% of the 680 W/m², as stated in the solar flux value, is converted to electrical power that can be used. The rest is lost as heat or reflected back. Higher efficiency rates reduce the area required for the panels but have other costs and limitations.

To calculate how much solar energy can realistically be used, you take the total amount of energy measured by solar flux and multiply it by the conversion efficiency. Understanding the efficiency of solar panels allows us to better plan and optimize the area needed and technology to harness solar energy effectively.

Global Energy Consumption

Global Energy Consumption is the total amount of energy used by all humans and is typically measured in terawatts (TW). In our exercise, this value is about 15 TW. A terawatt equals one trillion watts. This energy supports everything from lighting our homes, powering industries, to running transportation systems.

This consumption rate is a crucial figure because it sets the benchmark of how much energy the world needs, and it assists in determining the scale of renewable energy sources required to meet those needs. If we know the global energy consumption, we can calculate the percentage of Earth's surface that must be covered in solar panels, given their efficiency, to meet these energy demands.

Understanding global energy consumption helps us realize the importance of scaling renewable energy sources such as solar power. This allows for designing better policies and infrastructure to shift towards a more sustainable energy future. By matching the consumption with solar energy potential, we can better address and plan for the world’s energy demands.