Question

If typical insolation is \(200 \mathrm{~W} / \mathrm{m}^{2}\), how much land area would be needed for a \(15 \%\) efficient flat PV array supplying an average of \(10 \mathrm{~kW}\) of power-which is the U.S. individual share? If arranged in a square, how large is the side-length of this array? Compare its size or area to something familiar.

Short answer

To supply an average power of 10 kW to a community, given a flat photovoltaic (PV) array of 15% efficiency and 200 W/m² insolation, the land area needed for the PV array is 333.33 m². The side length of the square array is approximately 18.25 meters, which is similar in size to a small public basketball court.

Step-by-step solution

Convert required power output to watts

Given that the average power supply required is \(10 \mathrm{~kW}\), we need to convert it to watts. Since there are 1000 watts in 1 kilowatt, we will multiply 10 kilowatts by 1000. So, we have: \(10 \mathrm{~kW} \times 1000 = 10000 \mathrm{~W}\) The required power output in watts is \(10000 \mathrm{~W}\).

Calculate the actual power output of the PV array

We are given that the flat PV array is \(15 \%\) efficient. We now find the actual output power of the array by dividing the required output power by the efficiency of the array: \(Actual \, Power \, Output = \cfrac{Required \, Power \, Output}{Efficiency} = \cfrac{10000 \mathrm{~W}}{0.15} = 66666.67 \mathrm{~W}\) The actual power output of the PV array is \(66666.67 \mathrm{~W}\).

Calculate the land area for the required PV array

We are given that typical insolation is \(200 \mathrm{~W} / \mathrm{m}^{2}\). To find the land area needed, we can divide the actual power output by the insolation value: \(Land \, Area = \cfrac{Actual \, Power \, Output}{Insolation} = \cfrac{66666.67 \mathrm{~W}}{200 \mathrm{~W} / \mathrm{m}^{2}} = 333.33 \mathrm{m}^{2}\) Thus, the land area needed for the required PV array is \(333.33 \mathrm{m}^{2}\).

Calculate the side length of the square array

To calculate the side length of the square PV array, we can take the square root of the land area calculated in Step 3: \(Side \, Length = \sqrt{Land \, Area} = \sqrt{333.33 \mathrm{m}^{2}} = 18.25 \mathrm{m}\) (approximately) The side length of the square PV array is approximately \(18.25 \mathrm{m}\).

Compare the size or area of the PV array to something familiar

The area of the PV array required is \(333.33 \mathrm{m}^{2}\), which is equivalent to the area of a square with a side length of approximately 18.25 meters or roughly the size of a small public basketball court.

Key concepts

Photovoltaic (PV) Efficiency

Photovoltaic (PV) efficiency is a measure of how effectively a solar panel converts sunlight into electrical energy. It's expressed as a percentage, representing the ratio of the electric power output compared to the solar energy input. In other words, if we have a solar panel with an efficiency of 15%, it means that it can convert 15% of the sunlight hitting it into electricity.

Understanding PV efficiency is crucial because it directly impacts the size of the array needed to produce a certain amount of electrical power. Higher efficiency panels can generate more power in a smaller area. For practical usage, a homeowner or business owner needs to know how many panels they'll need to meet their power demands, which will hinge on the efficiency of the panels they choose. To improve conceptual understanding in this area, one can compare the efficiencies of different panel technologies or investigate how factors like temperature and shade affect the efficiency in real-world conditions.

Insolation Calculation

Insolation, a portmanteau of 'incoming solar radiation,' refers to the amount of solar energy received by a specific area over time. Typically, it is measured in watts per square meter (\( \text{W/m}^2 \)). To calculate the energy produced by a solar panel or an array of panels, one needs to know the typical insolation as well as the area over which the solar panels are spread.

For example, with an insolation of 200 W/m², it means that each square meter of the solar panel is receiving 200 watts of solar energy. By integrating insolation with PV efficiency, one can determine the power output of a solar panel and subsequently the area needed for installation to meet a specific energy requirement. This aids students in clearly understanding the relationship between solar energy inputs, efficiency of conversion, and output power.

Renewable Energy Mathematics

Renewable energy mathematics is about using numerical methods to plan and predict the performance of renewable energy systems like solar panels. This involves calculations of efficiency, insolation, and other factors such as economic costs and energy storage capacities. With renewable sources becoming increasingly significant, mathematics allows us to address critical questions like 'How much energy can be produced?' 'What is the best system size for a particular area?' or 'How will different conditions affect power production?'

By calculating the energy needs of a typical US individual and the land area needed for PV arrays to meet those needs, students apply renewable energy mathematics to real-life scenarios. They learn to translate theoretical efficiency rates into practical spatial requirements, and this hands-on approach helps solidify their understanding. Furthermore, comparing the PV array's size to familiar objects helps contextualize the abstract numbers and fosters a more intuitive grasp of the scale of such installations.