Question

The total rate at which power is used by humans world wide 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 sun light into useful power.

Short answer

To power the planet for use by all humans with solar energy collectors at 10% efficiency, approximately 85.14 square miles of Earth's surface needs to be covered with solar energy collectors.

Step-by-step solution

Find the power collected from the sun

First of all, we need to find out how much power is collected from the sun per square meter. We can get this value from the given solar flux. Solar flux equals the power collected per unit area, which is given as 680 W/m².

Calculate the power transformed into useful power

Now, we need to find how much of the solar energy is converted into useful power. This can be found by multiplying the solar energy collected with the efficiency of the solar energy collectors. The efficiency is given as 10%, which means 0.1 in decimal form. Therefore, the useful power per square meter can be calculated as follows: Useful_Power = Solar_Flux × Efficiency Useful_Power = 680 W/m² × 0.1 = 68 W/m²

Calculate the required area to fulfill the power consumption

Now we know the useful power per square meter. To find the area required to fulfill the total power consumption of humans worldwide, which is 15 TW or 15,000,000,000,000 W, we need to divide the total power consumed by the useful power per square meter: Required_Area = Total_Power / Useful_Power Required_Area = 15,000,000,000,000 W / 68 W/m² = 220,588,235 m²

Convert area to square miles

Finally, we need to convert the calculated area to square miles. To do this, we just need to multiply the area in square meters by the conversion factor to get the area in square miles. The conversion factor is 1 square meter equals 3.86102 x 10⁻⁷ square miles. Required_Area_sq_miles = 220,588,235 m² × 3.86102 × 10⁻⁷ = 85.14189183 square miles Therefore, to power the planet for use by all humans with solar energy collectors at 10% efficiency, we need to cover approximately 85.14 square miles of Earth's surface with solar energy collectors.

Key concepts

Solar Flux

Solar flux is a measure of the power per unit area received from the Sun in the form of electromagnetic radiation. In the provided exercise, solar flux is given as 680 W/m², which is an average value for the sunlit side of Earth. This quantity is a crucial factor in determining the potential of solar energy harvesting. Since the Earth is constantly rotating, solar flux values vary throughout the day and across different geographical locations.

Solar collectors are designed to capture this energy and convert it into electricity or heat. However, the total solar power a collector can harness is directly proportional to the solar flux and the collector's area. Therefore, understanding solar flux is essential for planning and optimizing the layout of solar energy systems for maximum efficiency and energy capture.

Power Consumption

Power consumption refers to the amount of electrical energy used by various devices, systems, and processes within a certain time period. It is often measured in watts (W), kilowatts (kW), or terawatts (TW) for large-scale consumption. In our exercise, the total rate of power used worldwide is approximately 15 TW.

Calculating the power consumption helps us understand the scale of energy demand that must be met to sustain human activities. It is a starting point to determine how much energy we need to generate, particularly from renewable sources like solar power, to replace or supplement conventional energy resources.

Renewable Energy

Renewable energy comes from natural sources or processes that are replenished constantly, such as sunlight, wind, rain, tides, waves, and geothermal heat. Unlike fossil fuels, these sources are sustainable over the long term and have a much lower impact on the environment.

Solar energy is a prominent form of renewable energy due to its vast availability and potential to meet human energy demands. By converting sunlight into electrical power, we can reduce our dependence on non-renewable sources and minimize greenhouse gas emissions. Educating people about the significance of renewable energy leads to increased adoption and advancements in technology, making it a keystone in the transition toward a cleaner energy future.

Energy Efficiency

Energy efficiency involves using less energy to perform the same task or produce the same result, thereby eliminating energy waste and reducing power consumption. In the context of our exercise, the solar energy collectors are only 10% efficient, meaning they convert only 10% of the sunlight they capture into usable energy.

This low efficiency requires us to use more area for solar collectors to meet the world's energy needs. Increasing the efficiency of solar panels is a critical research area. Even small improvements can significantly reduce the required space for panels and the resource investment needed for manufacturing and installation. High energy efficiency is not only cost-effective but also environmentally beneficial, leading to reduced carbon footprints and better resource management.