• The Sun is the primary source of most energy on Earth, either directly or indirectly.
• Solar energy powers photosynthesis, drives the hydrological cycle (enabling hydroelectric power), and causes wind by uneven heating.
• Fossil fuels are ancient solar energy stored in organic matter.
• Exceptions to solar origin include nuclear and geothermal energy.

Solar cells convert sunlight directly into electrical energy by utilizing the photoelectric effect. Made from semiconductors such as silicon, they rely on a p-n junction to separate charge carriers. When photons with energy $E_{\text{photon}} \geq E_g$ (the bandgap energy) strike the semiconductor, electrons are excited from the valence band to the conduction band, producing electron-hole pairs. The built-in electric field at the p-n junction separates these charges, generating current.

Working Principle

1. p-type semiconductor: Rich in holes (positive carriers).
2. n-type semiconductor: Rich in electrons (negative carriers).
3. Photon absorption excites electrons, enabling current flow through an external circuit.

1. Open-circuit voltage, $V_{oc}$: Voltage when no current flows (open circuit).
2. Short-circuit current, $I_{sc}$: Current when the circuit is closed with zero resistance.
3. Maximum power point, $P_{\text{max}}$: The maximum power output on the current-voltage (I-V) curve.
4. Fill Factor (FF): Quality measure of solar cell performance:

where $V_{mp}$ and $I_{mp}$ are voltage and current at maximum power point.

5. Efficiency, $\eta$: Ratio of electrical power output to incident solar power:

where $P_r$ is the incident solar radiation power.

• Solar irradiance of $1000\, \mathrm{W/m^2}$
• Air Mass (AM) 1.5 spectrum (solar zenith angle approximately 48.19°)
• Temperature of 25°C (298 K)

Air Mass (AM) quantifies the path length sunlight travels through the atmosphere relative to the shortest path at the zenith:

Given:

• Solar cell surface area: $100\, \mathrm{cm}^2 = 1 \times 10^{-2}\, \mathrm{m}^2$
• Wavelength $\lambda = 760\, \mathrm{nm}$
• Open circuit voltage: $V_{oc} = 0.67\, \mathrm{V}$
• Temperature: $T = 300\, \mathrm{K}$
• Assuming 100% quantum efficiency (one photon produces one electron)

Efficiency of a Solar Cell

The efficiency ($\eta$) of a solar cell is the ratio of the useful electrical power output to the solar power input received by the cell, measured under standard illumination conditions at the maximum power point.

Formula:

where:

• Pmax = Maximum electrical power delivered by the solar cell (at the maximum power point)
• Pr = Power of the solar radiation incident on the solar cell (input power)

Limiting Factors:

1. Light absorption: Not all sunlight is absorbed effectively by the cell.
2. Charge separation: Efficiency of separating the electron-hole pairs generated by light.
3. Charge transport: Efficient movement of charge carriers without recombination losses.
4. Cell temperature: Higher temperatures generally reduce cell efficiency.
5. Internal resistance: Resistive losses inside the cell reduce power output.

Wind is the movement of atmospheric air caused primarily by the uneven absorption and redistribution of solar radiation on Earth's surface. This results from differences in heat capacity of land and water, creating spatial and temporal variations in temperature, which induce pressure gradients in the atmosphere. Air moves from zones of high atmospheric pressure to low pressure, generating wind currents. This is the fundamental driving mechanism for all wind phenomena, including local effects such as sea breezes during daytime and land breezes at night.

Sea and Land Breezes: During the day, land heats faster than the sea, causing air over land to rise and create a low-pressure zone, while cooler air from the sea (high pressure) moves inland (sea breeze). At night, the process reverses because land cools faster than the sea, causing air to flow from land to sea (land breeze).

Conversion of Wind Energy to Usable Forms

To harness wind energy, a wind turbine is used to convert the kinetic energy of moving air into mechanical rotational energy, which can then be converted into electrical energy or used mechanically (e.g., pumping water or grinding).

Wind Turbines

Wind turbines consist mainly of:

1. Blades: Capture wind energy and convert it to rotational motion.
2. Hub: Central part connecting blades to the shaft.
3. Main Shaft: Transfers rotational motion from blades to the gearbox or generator.
4. Gearbox: Increases rotational speed from low RPM of blades to higher RPM needed for electrical generators.
5. Generator: Converts mechanical rotational energy into electrical energy.
6. Nacelle: Housing that contains gearbox, generator, and controls.
7. Tower: Supports the turbine at a height to access stronger, less turbulent wind.

Figure: Components of a Wind Turbine for Electricity Generation

Principles of Operation of Wind Turbine Blades

There are two primary aerodynamic principles by which wind turbines operate:

1. Drag-based turbines: The wind exerts a force on the blades in the direction of the airflow, pushing them to rotate. These tend to be less efficient.
2. Lift-based turbines: The blades are shaped like aerofoils (airfoils). When wind passes over the blade, a difference in air velocity between the upper and lower surfaces generates a pressure difference, producing a lift force perpendicular to the wind direction, which causes the rotor to spin more efficiently.

Figure: Aerofoil Blade Creating Lift Force

Mathematical Model of Wind Power

The kinetic energy $E$ contained in moving air mass of mass $m$ with velocity $v$ is:

$E = \frac{1}{2} m v^2$

For air flowing continuously through a cross-sectional area $A$ at velocity $v$, the mass flow rate $\frac{dm}{dt}$ is given by:

$\frac{dm}{dt} = \rho A v$

where $\rho$ is the density of air (approximately $1.225 \, \text{kg/m}^3$ at sea level and 15°C). The power $P$, which is the rate of kinetic energy transfer by the wind, is:

This shows the crucial fact that wind power is proportional to the cube of wind speed, which implies small increases in wind speed lead to large increases in available power.

Extractable Power and Betz Limit

Not all wind power can be extracted. According to Betz's Law, the maximum theoretical efficiency (power coefficient $C_p$) of a wind turbine is about 59.3%:

$C_{p, max} = 0.593$

Thus, the practical extractable power from wind is:

where $C_p$ is the power coefficient dependent on turbine design and operating conditions. Typical modern turbines achieve $C_p$ values between 0.35 and 0.45.

Tip Speed Ratio (TSR)

An important design parameter of wind turbines is the Tip Speed Ratio (TSR), defined as the ratio of the blade tip speed to the wind speed:

where:

• $\omega$ = angular velocity of the rotor (rad/s)
• $r$ = radius of the turbine blades (m)
• $v$ = wind speed (m/s)

The TSR affects the aerodynamic efficiency and noise. If TSR is too low, the rotor moves slowly, and much wind passes unused. If TSR is too high, blades cause excessive drag and stall. Optimizing TSR is crucial for maximizing power extraction and turbine longevity.

Extractable Power from Wind

Problem: Calculate the power extractable by a wind turbine with blade length $r = 52\, \text{m}$, wind speed $v = 12\, \text{m/s}$, air density $\rho = 1.23\, \text{kg/m}^3$, and power coefficient $C_p = 0.4$.

Solution: The swept area $A$ of the turbine is the area of a circle:

The extractable power $P_{ext}$ is:

Calculate:

Tip Speed Ratio and Power Efficiency

A turbine with rotor diameter $d = 0.9\, \text{m}$ rotates at frequency $f = 0.198\, \text{Hz}$, with wind speed $v = 15\, \text{m/s}$ and rated power output $P_{rated} = 0.13\, \text{MW}$.

a. Calculate the tip speed to wind speed ratio (TSR):

b. Calculate power conversion efficiency $C_p$: Power available in wind at 15 m/s:

Assuming $A = \pi r^2 = \pi \times (0.45)^2 = 0.636\, \text{m}^2$ and $\rho = 1.23\, \text{kg/m}^3$:

Power conversion efficiency:

Note: This is unrealistic; possibly the rated output power or other assumptions need revisiting, or the rated power may be peak instantaneous power at higher wind speeds or a larger swept area. This illustrates the importance of consistent data.

1. Wind energy conversion efficiency is limited by aerodynamic and mechanical factors (Betz Limit).
2. Site selection depends on average wind speed, terrain roughness, and turbine height to maximize energy yield.
3. Wind speed variations, turbulence, and maintenance affect actual power output.
4. Advancements include variable pitch blades, yaw mechanisms, and grid integration for optimized power generation.