Question

One can use turbines to exploit the energy contained in ocean currents, just like one can do it for wind. What is the maximum amount of power that can be extracted from a current flowing with a speed of \(1.35 \mathrm{~m} / \mathrm{s}\), if one uses a turbine with rotor diameter of \(24.5 \mathrm{~m}\) ? (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

The maximum amount of power that can be extracted from the ocean current using a turbine with a rotor diameter of 24.5 m and a current speed of 1.35 m/s is approximately 290,807 W.

Step-by-step solution

Calculate the area of the rotor

We are given the diameter of the rotor as \(D = 24.5 \mathrm{~m}\). To find the area swept by the rotor, we use the formula for the area of a circle: \(A = \pi r^2\) where \(r\) is the radius of the rotor. We can find the radius by dividing the diameter by 2: \(r = \frac{D}{2} = \frac{24.5 \mathrm{~m}}{2} = 12.25 \mathrm{~m}\). Now, we can calculate the area: \(A = \pi (12.25 \mathrm{~m})^2 = 471.9 \mathrm{~m}^2\)

Calculate the kinetic energy flow rate

Now that we know the area swept by the rotor, we can find the kinetic energy flow rate passing through this area. We use the formula: \(KE = \frac{1}{2} \rho A v^3\) where \(\rho\) is the density of seawater (\(1024 \mathrm{~kg} / \mathrm{m}^{3}\)), \(A\) is the rotor area calculated in step 1, and \(v\) is the speed of the current (\(1.35 \mathrm{~m} / \mathrm{s}\)). Plugging in the values, we get: \(KE = \frac{1}{2} \times 1024 \mathrm{~kg} / \mathrm{m}^3 \times 471.9 \mathrm{~m}^2 \times (1.35 \mathrm{~m} / \mathrm{s})^3 = 487895.5 \mathrm{~W}\)

Apply the Betz limit

To find the maximum amount of power that can be extracted from the ocean current, we apply the Betz limit, which states that only about 59.3% (approximately \(\frac{16}{27}\)) of the kinetic energy can be extracted. So, we multiply the kinetic energy flow rate calculated in step 2 by the Betz limit: \(P_{max} = \frac{16}{27} \times 487895.5 \mathrm{~W} = 290806.9 \mathrm{~W}\) Therefore, the maximum amount of power that can be extracted from the ocean current using a turbine with a rotor diameter of \(24.5 \mathrm{~m}\) and a current speed of \(1.35 \mathrm{~m} / \mathrm{s}\) is approximately \(290807 \mathrm{~W}\).