Question

If solar panels are placed in the Mojave Desert in California that generate 100 megawatts on a 1.3-square-mile site with an average of \(7.50\) hours of productive daylight each day, how many tons of coal are equivalent to the energy produced in one day by the solar panels? (One ton of coal will produce \(26.6\) gigajoules of energy, 1 Watt \(=1 \mathrm{~J} / \mathrm{s}\).)

Short answer

3000 tons of coal are equivalent to the energy produced by the solar panels in one day.

Step-by-step solution

Calculate Daily Energy Output from Solar Panels

First, we need to calculate the total amount of energy generated by the solar panels in one day. The formula to calculate the daily energy output in joules (J) is Energy (in joules) = Power (in watts) * Time (in seconds). Since we have the power in megawatts (MW), we convert it to watts by multiplying by 10^6 (1 MW = 10^6 W), and we convert hours to seconds by multiplying by 3600 (1 hour = 3600 seconds). Thus, the daily energy output is 100 x 10^6 W * 7.5 hours * 3600 seconds/hour.

Calculate the Energy Yield of One Ton of Coal

Next, we convert the energy produced by one ton of coal from gigajoules into joules for comparison. One gigajoule is equivalent to 10^9 joules. Therefore, one ton of coal produces 26.6 x 10^9 J of energy.

Determine the Equivalent Tons of Coal

To find out how many tons of coal are equivalent to the energy produced by the solar panels, we divide the daily energy output by the solar panels (calculated in step 1) by the energy yield of one ton of coal (calculated in step 2).

Perform the Calculation

Now perform the actual calculation using the numbers from the previous steps. This will give us the number of tons of coal equivalent to the energy produced by the solar panels in one day.

Key concepts

Solar Energy

Solar energy harnesses the power of the sun, a renewable and abundant source of energy. Unlike fossil fuels, solar power is inexhaustible and emits no greenhouse gases during generation, making it a key player in sustainable energy solutions. Photovoltaic cells in solar panels convert sunlight directly into electricity through a process known as the photovoltaic effect. Each solar panel consists of many solar cells, which work together to produce a significant amount of energy.

In the exercise, solar panels located in the Mojave Desert utilize the region's ample sunshine, converting the solar irradiance into electrical energy. The power generation capability of such a system is measured in watts (W), or in this case, megawatts (MW). Such large-scale solar installations can have a substantial impact on reducing reliance on traditional energy sources, like coal, and contribute to a lower carbon footprint.

Coal Energy Equivalence

Understanding the equivalence between coal energy and other forms of energy is crucial for comparing the environmental impact and the sustainability of different energy sources. Coal, a fossil fuel, has a high energy content; one ton of coal can produce a substantial amount of energy, typically expressed in gigajoules (GJ).

In our exercise, we establish that one ton of coal can yield 26.6 GJ of energy. To equate solar energy output to coal energy, we must know the amount of energy a typical unit of coal produces. This allows us to estimate how much coal would be needed to generate the same amount of energy as the solar panels, providing a tangible comparison of the two energy sources. It's important to remember that coal combustion has environmental consequences, including greenhouse gas emissions, which aren't associated with solar energy generation.

Energy Output Calculation

Calculating energy output is a fundamental step to understand how much energy a system produces over a given period. In our scenario, the energy output of solar panels is determined by multiplying the power capacity (in watts) by the number of seconds of productive daylight received. The product gives us the total energy in joules (J) generated in a day.

For the solar array in the Mojave Desert, this calculation informs us how much energy is produced with 100 MW capacity over 7.5 hours of productive daylight. It's important in such calculations to ensure unit consistency, converting hours to seconds and megawatts to watts when necessary. This precise calculation allows us to compare different energy sources like solar and coal on an even footing.

Unit Conversion

Unit conversion is essential in energy conversion calculations since it ensures comparisons are accurate and meaningful. As energy can be expressed in multiple units such as joules, watt-hours, kilowatt-hours, and gigajoules, converting these units properly is crucial for clear communication and comparison.

In our given exercise, there are several conversions required: megawatts to watts, hours to seconds, and gigajoules to joules. These conversions are necessary to align the units of coal energy output with the units of solar energy output. Ensuring that we're working with the same units across different forms of energy makes the calculation coherent and the result reliable. Remember, accurate unit conversion is the basis for comparisons in energy output between diverse energy sources.