Problem 97 - Physics for Scientists and Engineers with Modern Physics

Question

Some electric power companies use water to store energy. Water is pumped by reversible turbine pumps from a low reservoir to a high reservoir. To store the energy produced in 1.0 hour by a 180 -MW electric power plant, how many cubic meters of water will have to be pumped from the lower to the upper reservoir? Assume the upper reservoir is \(380 \mathrm{~m}\) above the lower one, and we can neglect the small change in depths of each. Water has a mass of \(1.00 \times 10^{3} \mathrm{~kg}\) for every \(1.0 \mathrm{~m}^{3}\)

Step-by-Step Solution

Answer

Approximately 1,715,595 cubic meters of water must be pumped.

Step 1: Calculate the Energy Produced in One Hour

First, we need to determine the total amount of energy produced by the 180 MW power plant in one hour. Since power is the rate of energy production, we use the given power rating and time duration to find the energy. Power is given as \(P = 180 \text{ MW}\). Since 1 MW = \(10^6\text{ W}\), we have \(P = 180 \times 10^6 \text{ W}\).Time is given as \(t = 1 \text{ hour} = 3600 \text{ seconds}\).Thus, the energy \(E\) in joules is:\[E = P \times t = 180 \times 10^6 \times 3600\]

Step 2: Use Gravitational Potential Energy Formula

The energy needed to lift the water to the higher reservoir is stored as gravitational potential energy (GPE). The formula for gravitational potential energy is:\[GPE = mgh\]where \(m\) is the mass of the water in kilograms, \(g\) is the acceleration due to gravity (approximately \(9.81\text{ m/s}^2\)), and \(h\) is the height difference between reservoirs (given as \(380\text{ m}\)).

Step 3: Relate Energy Produced to Mass of Water

Since the energy produced is used to lift the water, the gravitational potential energy is equal to the energy produced. Thus,\[180 \times 10^6 \times 3600 = m \times 9.81 \times 380\]Solve this equation for \(m\), the mass of water.

Step 4: Solve for Mass \(m\)

Rearrange the equation from Step 3:\[m = \frac{180 \times 10^6 \times 3600}{9.81 \times 380}\]This calculation will give the mass of the water in kilograms.

Step 5: Convert Mass to Volume

Since water has a density of \(1000\text{ kg/m}^3\), the volume \(V\) of water is given by:\[V = \frac{m}{1000}\]Use the mass obtained in Step 4 to calculate the volume in cubic meters.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy (GPE) is an essential concept in understanding how energy can be stored using height differences. When an object is lifted to a certain height, it gains gravitational potential energy. This energy is essentially stored within the object due to Earth's gravity. The formula to calculate GPE is given by:\[GPE = mgh\]The variables here stand for:

• \(m\) - mass of the object in kilograms (kg)
• \(g\) - acceleration due to gravity (approximated as \(9.81 \text{ m/s}^2\) on Earth)
• \(h\) - height in meters (m) above the reference point, such as the ground level or a lower reservoir in a hydropower system

In our exercise, GPE is utilized to determine how much energy is needed to pump water to a higher elevation. As water is pumped upwards, it stores energy as GPE, ready to be converted back into other forms when the water is released back down. This makes pumped storage a practical method for energy storage and management.Understanding GPE is crucial because it is the cornerstone of many energy systems that rely on height differences, such as hydroelectric power plants and pumped storage systems.

Energy Conversion

Energy conversion is the process of changing energy from one form to another. In the context of pumped storage hydropower, energy conversion plays a crucial role. This method of energy storage involves converting electrical energy into gravitational potential energy. Here's how it works:

• During times of low electricity demand, excess electrical energy generated by power plants is used to pump water from a lower reservoir to a higher reservoir.
• This conversion from electric energy to GPE allows for energy storage.
• When energy demand peaks, the stored GPE in the elevated water is converted back to electrical energy by allowing the water to flow back down through turbines, generating electricity.

The beauty of this system is the ability to shift energy usage, efficiently balancing the supply and demand of electricity on a grid. It also highlights a critical point about energy conservation: energy is never lost but rather transformed from one type to another, showcasing the physics law of conservation of energy. By converting electric energy to GPE and back, pumped storage systems exemplify this principle on a large scale.

Electric Power Generation

Electric power generation is the process of producing electrical energy from various forms of energy sources. In traditional power plants, energy conversion is typically from mechanical energy derived from burning fuels or through movement, such as in wind or hydropower systems, into electricity. Pumped storage hydropower acts as both a generator and a storage system:

• When water is released from the higher to the lower reservoir, the falling water drives turbines, converting GPE back into mechanical energy.
• The turbines are connected to generators, which transform this mechanical energy into electrical energy, ready for use in the power grid.
• This process effectively allows for time-shifting energy production—storing it when excess is available and generating it when there is higher demand.

The efficiency and reliability of electric power generation through pumped storage is essential for modern energy systems. It provides a stable energy supply, supports grid flexibility, and plays a vital role in integrated energy management systems. Understanding electric power generation within pumped storage highlights how energy transformation facilitates meeting variable electricity demand.