Question

If placed in a steady current of \(1 \mathrm{~m} / \mathrm{s}\), and at a generation efficiency of \(40 \%\), how large would an ocean current rotor be (diameter) to satisfy the \(10,000 \mathrm{~W}\) demand of the average American? Put this scale in the context of some familiar object or space.

Short answer

Answer: The approximate diameter of the ocean current rotor is 7.87 meters, which is roughly the length of a school bus or around half the width of a basketball court.

Step-by-step solution

Identify given parameters

In this problem, we are given: - Power output demand (P) = 10,000 W - Efficiency (η) = 40% - Steady current speed (v) = 1 m/s

Calculate actual power generation

The efficiency of the system is given, so we need to calculate the actual power generation (Pg) of the rotor before losses due to efficiency. We can use the following formula: Pg = P ÷ η Where η needs to be converted to its decimal value, which is 0.40. Pg = 10,000 W ÷ 0.40 Pg = 25,000 W

Use power formula to find rotor area

The general formula for the power generated from an ocean current is given by: Pg = 0.5 × ρ × A × v³ Where: - Pg is the power generated (25,000 W) - ρ is the density of seawater (approximately 1025 kg/m³) - A is the rotor area (m²), which we want to find - v is the steady current speed (1 m/s) First, rearrange the formula to solve for A: A = Pg ÷ (0.5 × ρ × v³) Next, plug in the given values and calculate A: A ≈ 25,000 ÷ (0.5 × 1025 × 1³) A ≈ 48.78 m²

Calculate rotor diameter

Since the rotor is circular, we can relate the rotor area to the diameter of the rotor using the area of a circle formula: A = π × (D/2)² Where: - A is the rotor area (48.78 m²) - D is the diameter of the rotor (which we want to find) Rearrange the formula to solve for D: D = 2 × √(A ÷ π) Next, plug in the given values and calculate D: D ≈ 2 × √(48.78 ÷ 3.1416) D ≈ 7.87 m

Compare diameter to familiar objects

The diameter of the ocean current rotor is found to be approximately 7.87 meters. To give context, this is roughly the length of a school bus or around half the width of a basketball court. This comparison can help us visualize the scale of the ocean current rotor.

Key concepts

Ocean Current Turbines

Ocean current turbines are innovative devices designed to harness energy from the natural movement of water beneath the ocean's surface. These turbines operate in a way similar to wind turbines but utilize the steady and predictable currents found underwater.
The primary advantage of these turbines is their ability to offer a consistent and predictable source of renewable energy. Ocean currents usually change much slower than winds, providing steady energy flow.
These turbines consist of rotor blades that capture the kinetic energy of flowing water. As the water moves, it rotates these blades, converting kinetic energy into mechanical energy. This mechanical energy is then transformed into electricity.
Ocean current technology is especially beneficial in coastal areas where water currents are significant, providing an opportunity to replace or supplement traditional power sources, contributing to a reduction in carbon emissions.

Energy Efficiency

Energy efficiency is a key concept in renewable energy systems, as it determines how much of the energy captured by a system can be converted into usable power. For the ocean current turbine, energy efficiency measures how effectively it converts the kinetic energy of ocean currents into electrical energy.
In our exercise, the efficiency of the turbine is given as 40%. This means that only 40% of the kinetic energy is actually converted into electrical energy that can be used. The rest is lost, typically due to mechanical friction, resistance in the electrical system, or leakage.
To calculate the necessary power input of the turbine, it is essential to consider these losses. More precise and efficient turbine designs seek to boost this percentage, thereby maximizing the electrical output while using the same amount of kinetic energy.

Power Generation Calculations

Power generation calculations are crucial to determining the feasibility and size of renewable energy systems, such as ocean current turbines. These calculations help determine the area and size of the turbine needed to achieve the desired power output.
Initially, we find the actual power requirement (before considering losses from efficiency) by dividing the desired power output by the efficiency ( η). For example, if the efficiency is 40%, the calculation involves dividing by 0.40, resulting in the requirement of a higher initial power input.
The power captured from ocean currents is determined using the formula: \[ Pg = 0.5 \times \rho \times A \times v^3 \]
- \( Pg \) is the power generated - \( \rho \) is seawater density (1025 kg/m³) - \( A \) is the turbine's area - \( v \) is the current speed
Solving for the area \( A \) and subsequently the rotor diameter involves manipulating this formula, which provides a practical dimension for constructing an efficient energy system. Such calculations inform the design and scalability of ocean current turbines in sustainably meeting energy demands.