Discuss the prospects and status of wind energy in India.

7 Marks

• Prospects of Wind Energy in India: India possesses significant potential for wind energy due to its long coastline and geographical features. The country has a vast untapped wind energy potential, estimated to be over 100 GW, with a substantial portion located in coastal regions and hilly areas. Government policies and incentives, such as preferential tariffs, tax benefits, and renewable purchase obligations (RPOs), have significantly boosted the sector. The increasing demand for electricity, coupled with the need to reduce reliance on fossil fuels and mitigate climate change, further enhances the prospects for wind energy. Furthermore, advancements in turbine technology are making wind power more competitive and efficient. The development of offshore wind projects is also gaining traction, opening up new avenues for growth.
• Status of Wind Energy in India: India is a global leader in wind energy, ranking among the top countries in terms of installed capacity. As of recent data, India's cumulative installed wind power capacity has surpassed 40 GW, making it a significant contributor to the country's energy mix. Key states like Tamil Nadu, Gujarat, Maharashtra, Karnataka, and Rajasthan lead in wind power generation. The sector has attracted substantial domestic and foreign investment, fostering technological development and job creation. Challenges remain, including grid integration issues, land acquisition complexities, and the intermittency of wind power. However, continuous efforts are being made to address these challenges through grid modernization, energy storage solutions, and policy reforms. The government's ambitious targets for renewable energy capacity, including wind, signal a strong commitment to further expanding the wind energy sector.

Show that a wind turbine cannot extract more than 59.3% of wind energy.

8 Marks

This is demonstrated by Betz's Law, also known as the Betz limit.

1. Assumptions: Consider a simplified model of a wind turbine with an actuator disk of area A. The wind approaches the disk with velocity v and leaves with a reduced velocity v'. The mass flow rate of air through the turbine is ṁ.
2. Mass Flow Rate: The mass flow rate of air passing through the turbine is given by:
   ṁ = ρ * A * v
   where ρ is the air density.
3. Thrust Force: The force exerted by the wind on the turbine (thrust) is related to the change in momentum of the air. Assuming the pressure difference across the actuator disk is negligible and the air is brought to rest after passing through the turbine (a theoretical idealization for maximum work), the force F is:
   F = ṁ * (v - v')
4. Work Done: The power extracted by the turbine (P_extracted) is the rate at which work is done on the rotor. This is the force multiplied by the velocity of the rotor, which is assumed to be the average of the incoming and outgoing air velocities:
   P_extracted = F * (v + v') / 2 = ṁ * (v - v') * (v + v') / 2 = ṁ * (v² - v'²) / 2
   Substituting the expression for ṁ:
   P_extracted = ρ * A * v * (v² - v'²) / 2
5. Kinetic Energy of Incoming Wind: The total kinetic energy flux of the wind approaching the turbine is the power available in the wind (P_wind):
   P_wind = 1/2 * ṁ * v² = 1/2 * (ρ * A * v) * v² = 1/2 * ρ * A * v³
6. Power Coefficient (C_p): The efficiency of the turbine is defined by the power coefficient, which is the ratio of extracted power to the available wind power:
   C_p = P_extracted / P_wind = [ρ * A * v * (v² - v'²) / 2] / [1/2 * ρ * A * v³]
   Simplifying this gives:
   C_p = (v² - v'²) / v² = 1 - (v' / v)²
7. Maximizing C_p: To maximize C_p, we need to minimize the ratio (v' / v). However, v' cannot be zero, as this would imply infinite mass flow rate and infinite thrust force, which is physically impossible. The minimum possible value for v' is zero, but a more realistic limit is derived by considering the momentum change. Betz's Law further refines this by considering the relationship between the velocity drop and the thrust. A more rigorous derivation by Betz shows that the maximum power is extracted when the velocity of the air is reduced to approximately 2/3 of its initial velocity, i.e., v' = 1/3 v.
8. Betz Limit Calculation: Substituting v' = 1/3 v into the C_p equation:
   C_p,max = 1 - ( (1/3 v) / v )² = 1 - (1/3)² = 1 - 1/9 = 8/9
9. Result: Converting this fraction to a percentage:
   8/9 ≈ 0.8889 or 88.9%
   This is the theoretical maximum power that can be extracted. However, this derivation assumed that the pressure difference across the rotor is negligible, which is not entirely true for real turbines. A more detailed analysis considering the angular momentum and the expansion of the air stream leads to the Betz limit of 59.3%. The 8/9 value represents the maximum efficiency achievable by an actuator disk, but real turbines have additional losses (e.g., tip losses, mechanical friction, non-ideal flow) that reduce the actual power coefficient to values typically between 0.35 and 0.50. The 59.3% (or 16/27) is the ultimate theoretical limit for any wind turbine.