Question

One can use turbines to exploit the energy contained in ocean currents, just like one can do it for wind. If the maximum amount of power is \(918.8 \mathrm{~kW}\), which can be extracted from an ocean current with a turbine of rotor diameter \(25.5 \mathrm{~m}\), what is the speed of the ocean current? (Hint 1: The density of seawater is \(1024 \mathrm{~kg} / \mathrm{m}^{3}\). Hint 2 . The Betz limit applies to any fluid, including seawater.).

Short answer

Answer: The speed of the ocean current is approximately 1.991 m/s.

Step-by-step solution

Write down the known values and formula for kinetic power of a fluid

We are given the following information: - Maximum power: \(P = 918.8 \mathrm{~kW}\) - Rotor diameter: \(D = 25.5 \mathrm{~m}\) - Density of seawater: \(\rho = 1024 \mathrm{~kg} / \mathrm{m}^{3}\) - Betz limit: \(\eta = 0.593\) First, convert the power to watts: \(P = 918.8 \mathrm{~kW} \times 1000 = 918800 \mathrm{~W}\) The formula for the kinetic power of a fluid is: \(P = \frac{1}{2}\rho AV^3\eta\) where \(A\) is the rotor area and \(V\) is the speed of the ocean current.

Calculate the rotor area

Since the rotor is circular, we can calculate its area using the formula for the area of a circle: \(A = \pi(\frac{D}{2})^2\) Now, plug in the given value for the rotor diameter: \(A = \pi(\frac{25.5 \mathrm{~m}}{2})^2 \approx 510.68 \mathrm{~m}^2\)

Solve the formula for the speed of the ocean current

We can now rearrange the equation to solve for \(V\): \(V = \sqrt[3]{\frac{2P}{\rho A\eta}}\) Substitute the known values into the equation: \(V = \sqrt[3]{\frac{2 \cdot 918800 \mathrm{~W}}{1024 \mathrm{~kg} / \mathrm{m}^{3} \cdot 510.68 \mathrm{~m}^{2} \cdot 0.593}}\) Now, calculate the value of \(V\): \(V \approx 1.991 \mathrm{~m} / \mathrm{s}\) The speed of the ocean current is approximately \(1.991 \mathrm{~m} / \mathrm{s}\).

Key concepts

Kinetic Power of Fluid

The energy present in moving fluids, like water or air, can be harnessed to produce power. This is essentially what is done in hydro and wind power systems.

The kinetic power of a fluid is dependent on three main factors: the fluid's density \( \rho \), the area it flows through \( A \), and the velocity of the flow \( V \). The kinetic energy is converted into mechanical power by devices such as turbines. The specific formula used to describe this is \( P = \frac{1}{2}\rho AV^3 \), showing how the power varies with the cube of the velocity, emphasizing why even small increases in fluid speed can lead to significant increases in power.

Understanding this relationship allows us to calculate the potential energy that can be extracted from a fluid system, such as ocean currents, if we know these variables.

Betz Limit

The Betz limit, named after the German physicist Albert Betz, suggests that only 59.3% of the kinetic energy in a fluid stream can theoretically be harnessed using a turbine. This concept is crucial not only in wind energy extraction but also applies to ocean currents. No matter how efficient a turbine is, it cannot convert more than this fixed percentage of the fluid's kinetic energy into mechanical energy.

This limit is expressed with the symbolic value \( \eta \), where \( \eta = 0.593 \). When calculating the potential power output of a turbine in a fluid medium, the Betz limit is incorporated as a factor to ensure realistic and attainable energy projections.

Rotor Area Calculation

The rotor area of a turbine is critical as it directly influences how much fluid energy the turbine can interact with and thus extract power from. For a circular rotor, the area can be calculated using the equation for the area of a circle: \( A = \pi(\frac{D}{2})^2 \), where \( D \) is the diameter of the rotor.

This is key because, as per the power formula mentioned earlier \( P = \frac{1}{2}\rho AV^3 \), a larger rotor area translates to the interaction with a greater volume of the fluid each second, leading to the potential for more energy extraction. This understanding assists in designing turbines that are appropriately sized for the intended energy output.

Fluid Dynamics

Fluid dynamics is the study of the flow and forces of fluids, which can be both liquids and gases. It involves understanding how fluids move and how they react to various influences, like obstacles, changes in pressure, and gravitational forces.

In the context of energy extraction, fluid dynamics helps us to predict and optimize the interaction between the fluid—such as ocean currents or wind—and the turbines that convert the fluid's kinetic energy into electricity. It is the comprehensive analysis of these dynamics that enables the creation of turbines and other such devices that are efficient, reliable, and effective in harvesting energy from fluid flows.