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 \(10,000 \mathrm{~W}\) for every person in the U.S. (population 330 million). If arranged in a square, how large is the side-length of this array? Draw it on top of a state of your choice, to scale.

Short answer

Answer: The required land area is 110 billion m² and the side length of the square array is approximately 331,662.48 m (or 331.66 km).

Step-by-step solution

Calculate the total power needed for the entire U.S. population

First, find the total power needed for the entire U.S. population by multiplying the average power per person (10,000 W) by the population (330 million): Total power needed = Power per person × Population Total power needed = 10,000 W × 330,000,000 Total power needed = 3,300,000,000,000 W (3.3 TW)

Calculate the power generated per square meter by PV array

Next, find the power generated per square meter by multiplying the typical insolation (200 W/m²) by the efficiency of PV array (15%): Power generated per square meter = Insolation × Efficiency Power generated per square meter = 200 W/m² × 0.15 Power generated per square meter = 30 W/m²

Calculate the required land area

Now, find the required land area by dividing the total power needed by the power generated per square meter: Required land area = Total power needed ÷ Power generated per square meter Required land area = 3,300,000,000,000 W ÷ 30 W/m² Required land area = 110,000,000,000 m² (110 billion m²)

Calculate the side length of the square array

To find the side length of the square array, take the square root of the required land area: Side length of square array = √Required land area Side length of square array = √110,000,000,000 m² Side length of square array = 331,662.48 m (approx.) At this point, we have calculated the required land area and the side length of the square array. To draw it on top of a state of your choice, please use any mapping software or a ruler and a map to scale. It's important to note that the side length of the square array is very large (331.66 km), so it may occupy a significant portion of a state.

Key concepts

Insolation

Insolation refers to the amount of solar radiation that reaches a given area. It's an important factor in calculating the potential solar energy that can be harnessed by photovoltaic (PV) panels. The term, which is short for 'incoming solar radiation,' is typically measured in watts per square meter (\( \text{W/m}^2 \)).

For the purpose of our exercise, we consider typical insolation to be 200 W/m². It's worth noting that insolation varies by location, time of day, season, and atmospheric conditions. This variability can greatly influence the efficiency and design of a solar power installation.

Understanding insolation helps in estimating the amount of solar energy that could be converted into electricity. Our exercise involves calculating the land area needed for a flat PV array supplying 10,000 W per person in the U.S., where insolation plays a pivotal role in determining the total efficiency of the solar energy system.

Photovoltaic (PV) Efficiency

Photovoltaic efficiency is a measure of how effectively a solar panel can convert sunlight into usable electricity. It is expressed as a percentage and represents the ratio of the electrical power output compared to the solar energy input. A solar panel with an efficiency of 15% converts 15% of the solar energy striking it into electricity.

High-efficiency PV panels produce more electricity in a smaller area, but they tend to be more expensive. The efficiency of solar panels has been steadily increasing due to technological advancements, but practical limits do exist. Factors that affect panel efficiency include the type of materials used in solar cells, the configuration of the cells, the quality of manufacturing, and even local weather conditions.

In our exercise, a 15% efficient PV array exemplifies a common efficiency level for commercially available solar panels. When calculating the needed land area, we multiplied the typical insolation by the panel's efficiency to determine the power generated per square meter, highlighting the direct impact of PV panel efficiency on the scale of a solar installation.

Renewable Energy Mathematics

Renewable energy mathematics involves the calculations and equations used to design and evaluate renewable energy systems. It's a vital part of planning for sustainability and energy independence. In the context of our solar power land area calculation, the mathematics centers around determining the necessary land to supply a given power demand using solar energy.

This exercise underscores the importance of mathematical proficiency in the renewable energy sector. To calculate the total power requirement for the U.S. population, we multiplied the average power needed per person by the population. We then used efficiency and insolation values to figure out how much power can be produced per square meter of solar panels. Finally, we divided the total power requirement by the power generated per square meter to find the needed land area.

Peaceful exercises like this allow students to understand how mathematical concepts translate into realistic renewable energy solutions. These calculations pave the way for us to envision and create large-scale solar power capabilities that have tangible impacts on how societies generate and consume energy.