Question

The biggest hydroelectric installation in the U.S. is the Grand Coulee dam on the Columbia River. The enormous flow rate reaches its maximum at \(4,300 \mathrm{~m}^{3} / \mathrm{s}\), and the dam (reservoir) height is \(168 \mathrm{~m}\). At an efficiency of \(90 \%\), at what rate is this dam capable of producing hydroelectric power (in \(\mathrm{GW}^{40}\) )? Don't forget the density of water and that \(g \approx 10 \mathrm{~m} / \mathrm{s}^{2}\).

Short answer

Answer: The hydroelectric power produced by the Grand Coulee dam is approximately 6.458 GW.

Step-by-step solution

(Step 1: Write down the given values)

We are given the following values: - Flow rate, \(Q = 4,300\) m³/s - Dam height, \(h = 168\) m - Efficiency, \(\eta = 90\% = 0.9\) - Gravity constant, \(g \approx 10\) m/s² - Water density, \(\rho \approx 1,000\) kg/m³ (Note: This value is not given in the problem, but it is a common value for water density.)

(Step 2: Use the hydroelectric power formula)

We will now use the formula for hydroelectric power, which is: \(P = \eta \rho Q g h\) Plug in the given values: \(P = 0.9 \times 1,000 \times 4,300 \times 10 \times 168\)

(Step 3: Compute the power)

Multiply the values together: \(P = 0.9 \times 1,000 \times 4,300 \times 10 \times 168 = 6,458,040,000~\text{W}\)

(Step 4: Convert the power to GW)

The result is in watts. To convert it to gigawatts (GW), divide by \(10^9\): \(P = \frac{6,458,040,000~\text{W}}{10^9} = 6.458~\text{GW}\) So the Grand Coulee dam is capable of producing hydroelectric power at a rate of approximately \(6.458~\mathrm{GW}\).

Key concepts

Hydroelectric Energy

Hydroelectric energy is a form of renewable energy produced by converting the kinetic and potential energy of water into electrical energy through the use of turbines and generators. The fundamental principle of hydroelectric power is based on the water cycle, where water is constantly moved through the atmosphere and Earth due to the energy from the sun.

When it comes to hydroelectric installations, like the Grand Coulee Dam, the water stored in the reservoir represents potential energy due to its height, which is then released as kinetic energy when it flows down through the dam. The turbine blades are turned by the moving water, which subsequently drives the generators to produce electricity. The efficiency of energy conversion is a key factor in the amount of power that a hydroelectric plant can produce. The provided example of 90% efficiency is notably high, meaning that the system effectively converts 90% of the mechanical energy into electrical energy, with only 10% lost due to factors such as friction and conversion losses.

Physical Principles in Energy Production

The production of energy, especially in hydroelectric power, is heavily reliant on the physical principles of mechanics. These principles govern the transfer of energy from one form to another. In the context of the Grand Coulee Dam, the significant physical principles include the following:

• Conservation of energy - asserts that energy can neither be created nor destroyed, only transformed from one form to another. The dam converts the potential energy of water into kinetic energy and eventually into electrical energy.
• Gravitational potential energy - the water at the top of the dam has potential energy that is related to its height above the bottom of the dam. This energy is given by the formula \(E_p = mgh\), where \(m\) is mass, \(g\) is the acceleration due to gravity, and \(h\) is the height.
• Kinetic energy - as the water falls, it gains speed and thus kinetic energy, which can be expressed by \(E_k = \frac{1}{2}mv^2\), where \(m\) is mass and \(v\) is the velocity of the water.

The efficiency of this energy transformation is crucial in determining how much electrical energy can be produced by the hydroelectric plant. The Grand Coulee Dam's ability to generate power efficiently is due, in part, to the engineering that maximizes these physical principles.

Renewable Energy Resources

Renewable energy resources are those which can be used repeatedly and do not deplete over time. Such resources include solar, wind, geothermal, and of course, hydroelectric power. These resources are often contrasted with non-renewable resources like coal, oil, and natural gas, which can run out and are responsible for a significant amount of environmental pollution.

Hydroelectric power is particularly compelling as a renewable resource because it leverages the natural water cycle, and there is no direct emission of greenhouse gases during electricity generation. Facilities like the Grand Coulee Dam harness flowing water without depleting it, making the operation sustainable in the long term. However, there are environmental and social impacts associated with large dams, such as disruption of aquatic ecosystems and displacement of communities, which need to be carefully managed. The idea of sustainability is also about minimizing negative impacts while maximizing energy output.

Mechanical Work and Power

In physics, mechanical work and power are measures of energy transfer. Work is done when a force is applied to an object, and the object moves through a distance.

Mechanical power, on the other hand, is the rate at which work is done or energy is transferred. It is calculated as work done per unit time. In the case of the Grand Coulee Dam, the power produced can be calculated using the equation \(P = \eta \rho Q g h\), where \(P\) is the power, \(\eta\) is the efficiency of the turbine and generator, \(\rho\) is the water density, \(Q\) is the volumetric flow rate of water, \(g\) is the acceleration due to gravity, and \(h\) is the height from which the water falls. This equation encapsulates how hydroelectric plants convert the movement of water into electrical power that can be used by consumers. The example solution shows how these concepts are applied in a practical context to find the hydroelectric power output of the Grand Coulee Dam in gigawatts.