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 \(571.8 \mathrm{~kW}\), which can be extracted from a current flowing with a speed of \(1.57 \mathrm{~m} / \mathrm{s}\), what is the rotor diameter of the turbine? (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 approximate diameter of the turbine rotor is 39.72 meters.

Step-by-step solution

Write down the formula for power

We can write down the formula for the power extracted from the ocean current as follows: $$P = \frac{1}{2} \rho A v^3 C_p$$ where, \(P\) = power (W), \(\rho\) =density of fluid (kg/m³), \(v\) = velocity of fluid (m/s), \(A\) = swept area of the turbine (m²), \(C_p\) = power coefficient (dimensionless). In this case, we know the value of \(P\), \(\rho\), and \(v\). Also, the Betz limit states that the maximum power coefficient that can be achieved is \(C_p=0.593\). We aim to find the swept area \(A\).

Rearrange the formula for A

We want to find the swept area of the turbine, so we need to rearrange the formula for \(A\): $$A = \frac{2P}{\rho v^3 C_p}$$

Substitute the given values and find A

Now, substitute the values of \(P\), \(\rho\), \(v\), and \(C_p = 0.593\) to find the area: $$A = \frac{2(571.8 \times 10^3)}{(1024)(1.57^3)(0.593)}$$ Calculating this expression gives us: $$A \approx 1240 \mathrm{~m}^2$$

Find the diameter using the Area

The swept area of the turbine is given by: $$A = \pi R^2$$ where \(R\) is the radius of the turbine rotor. To find the diameter, we need to first find the radius: $$R = \sqrt{\frac{A}{\pi}}$$ Substituting the value of \(A\) we found above: $$R = \sqrt{\frac{1240}{\pi}}$$ Calculating this expression gives us: $$R \approx 19.86 \mathrm{~m}$$ Since the diameter \(D\) is twice the radius \(R\): $$D = 2R = 2 \times 19.86$$ This gives us the rotor diameter as: $$D \approx 39.72 \mathrm{~m}$$ So, the diameter of the turbine rotor is approximately 39.72 meters.