Problem 36 - College Physics

Question

Solar power. A solar power plant is to be built with an aver- age power output capacity of 2500 \(\mathrm{MW}\) in a location where the average power from the sun's radiation is 200 \(\mathrm{W} / \mathrm{m}^{2}\) at the earth's surface. What land area \(\left(\mathrm{in} \mathrm{km}^{2}\) and \(\mathrm{mi}^{2}\right)\) must the solar collectors occupy if they are (a) photocells with 42\(\%\) efficiency. (b) mirrors that generate steam for a turbine- generator unit with an overall efficiency of 21\(\%\) ?

Step-by-Step Solution

Answer

For photocells, 29.76 km² or 11.49 mi²; for mirrors, 59.52 km² or 22.98 mi².

Step 1: Calculate Required Energy Output from Photocells

To find the land area needed, first determine the total energy output required from the solar collectors. Since the efficiency of the photocells is 42%, the energy required from the sun is more than 2500 MW. The actual energy output needed can be calculated using the formula: \[ \text{Energy from Sun} = \frac{\text{Power Output}}{\text{Efficiency}} \] For photocells: \[ \text{Energy from Sun} = \frac{2500 \text{ MW}}{0.42} \approx 5952.38 \text{ MW} \] This is the required energy from the sun to achieve 2500 MW output with photocells.

Step 2: Calculate Required Land Area for Photocells

Next, calculate the land area needed when the average power from the sun is 200 W/m², using the expression: \[ \text{Area} = \frac{\text{Energy from Sun}}{\text{Sun's Energy per Unit Area}} \] So for photocells: \[ \text{Area} = \frac{5952.38 \times 10^6 \text{ W}}{200 \text{ W/m}^2} = 29761900 \text{ m}^2 \] Convert this area into km²: \[ \text{Area in km}^2 = 29761900 \text{ m}^2 \times \frac{1 \text{ km}^2}{10^6 \text{ m}^2} = 29.76 \text{ km}^2 \] And into mi²: \[ \text{Area in mi}^2 = 29.76 \text{ km}^2 \times \frac{0.3861 \text{ mi}^2}{1 \text{ km}^2} \approx 11.49 \text{ mi}^2 \] So, the land area required for the photocells is approximately 29.76 km² or 11.49 mi².

Step 3: Calculate Required Energy Output from Mirrors

Now repeat the calculation for mirrors that generate steam for a turbine, with an efficiency of 21%. \[ \text{Energy from Sun} = \frac{2500 \text{ MW}}{0.21} \approx 11904.76 \text{ MW} \] Thus, the energy required from the sun to achieve 2500 MW output with mirrors.

Step 4: Calculate Required Land Area for Mirrors

Calculate the land area when using mirrors with 21% efficiency. \[ \text{Area} = \frac{11904.76 \times 10^6 \text{ W}}{200 \text{ W/m}^2} = 59523800 \text{ m}^2 \] Convert this area into km²: \[ \text{Area in km}^2 = 59523800 \text{ m}^2 \times \frac{1 \text{ km}^2}{10^6 \text{ m}^2} = 59.52 \text{ km}^2 \] And into mi²: \[ \text{Area in mi}^2 = 59.52 \text{ km}^2 \times \frac{0.3861 \text{ mi}^2}{1 \text{ km}^2} \approx 22.98 \text{ mi}^2 \] Thus, the land area required for mirrors is approximately 59.52 km² or 22.98 mi².

Key Concepts

Energy Efficiency

Energy efficiency is a vital concept in solar power calculations. It refers to the percentage of solar energy radiated towards the solar collectors that is successfully converted into usable power. In our exercise, we explore two types of collectors: photocells and mirrors for steam generation. The photocells have an efficiency of 42%, meaning 42% of the received solar energy is converted into electricity, while the remaining 58% is lost. Mirrors, on the other hand, use an efficiency of 21%, which indicates that they convert 21% of solar energy into machinable energy.

To find the required energy from solar power, we use the formula: \[ \text{Energy from Sun} = \frac{\text{Power Output}}{\text{Efficiency}} \] Thus, higher efficiency means we need less sunlight to meet the same power output of 2500 MW, resulting in a smaller area for solar collectors.

Land Area Calculation

Calculating the land area for solar collectors involves understanding the energy required from the sun and the power available per unit area. The energy needed from the sun is divided by the solar energy available per square meter to find the total area needed. This is done using the following expression:
\[ \text{Area} = \frac{\text{Energy from Sun}}{\text{Sun's Energy per Unit Area}} \]
For this exercise, the average solar radiation at the Earth’s surface is 200 W/m². We calculated areas for photocells and mirrors as follows:

• For photocells, an area of 29.76 km² or 11.49 mi² is required
• For mirrors, a larger area of 59.52 km² or 22.98 mi² is needed

The difference in required land area is due to the differing efficiencies of these technologies.

Solar Energy Conversion

Solar energy conversion is the process by which sunlight is transformed into electricity or usable energy. This exercise showcases two conversion methods: photovoltaic cells and solar thermal power (mirrors for steam generation).

• Photocells (Photovoltaics): These use semiconductors that generate electricity when exposed to sunlight. They directly convert solar energy into electrical power.
• Solar Thermal (Mirrors): Mirrors focus sunlight to heat a fluid, creating steam that drives a turbine generator. This indirect conversion works well for large-scale power generation but has lower efficiency compared to photocells.

Both methods have distinct applications with their pros and cons, depending on energy demand, geographic location, and available resources.

Power Generation

Power generation refers to the process of producing electricity from various energy sources. In solar power plants, it involves converting sunlight into electricity through solar collectors. The exercise focuses on achieving a power output of 2500 MW, which dictates the setup required, whether it be using highly efficient photocells or less efficient mirrors.

Understanding power generation in this context involves:

• Recognizing the capacity needs: 2500 MW in this case
• Deciding the technology based on the resources and area available
• Calculating the necessary land area based on the efficiency rates of the chosen technology

Each technique and setup influences the overall generation capacity, costs, and ecological impact of the plant, highlighting the importance of careful planning.