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Chapter 7: Problem 52

As an energy-efficiency consultant, you're asked to assess a pumped-storage facility. Its reservoir sits \(140 \mathrm{m}\) above its generating station and holds \(8.5 \times 10^{9} \mathrm{kg}\) of water. The power plant generates 330 MW of electric power while draining the reservoir over an 8.0 -h period. Its efficiency is the percentage of the stored potential energy that gets converted to electricity. What efficiency do you report?

Short Answer

Expert verified
The efficiency of the power plant is calculated by substituting the calculated values for total potential energy and total generated energy into the formula for efficiency. As we are dealing with huge quantities, final calculations should be done with conscientious manner to avoid errors. Just substitute and calculate to obtain the final answer in percentage.

Step by step solution

01

Calculate the total potential energy stored in the reservoir

The total gravitational potential energy (PE) of the water in the reservoir could be calculated using the equation: PE= m * g * h, where m is the mass of the water, g is the acceleration due to gravity (approximately \(9.8 ms^{-2}\)), and h is the height above the ground. Substituting the given values, we find PE = \(8.5 * 10^{9} kg * 9.8 ms^{-2} * 140m\)
02

Calculate the total energy converted to electricity

The power plant drains the reservoir over an 8-hour period, during which it generates 330MW of power. To calculate the total energy produced, we multiply the power output by the time over which it was produced. But first, we need to convert 8 hours into seconds: \(8hr * 60min/hr * 60s/min = 28800 s\).The total energy (E) generated by the plant is E = Power output * Time = \(330 MW * 28800 s\).Remember to convert MW to W: \(1 MW = 10^6 W\). So, E = \(330 * 10^6 W * 28800s\)
03

Compute the efficiency

The efficiency (η) of the power plant can be calculated by dividing the energy output by the total potential energy and multiplying the result by 100 to turn it into a percentage. Therefore, η = \(\frac{E}{PE} * 100\%\) where E is the total energy generated and PE is the potential energy of the water. Substituting the values obtained in steps 1 and 2 we find the efficiency of the plant.

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