1.1 Solar Photovoltaics

Solar radiation can be converted directly into electricity using the photoelectric effect in semiconductor devices called solar cells. A solar cell is essentially a p-n junction diode operated in reverse bias.

When photons with energy greater than the semiconductor's band gap energy ($E_g$) strike the junction, electrons are excited from the valence band to the conduction band, creating electron-hole pairs. The built-in electric field at the junction separates these charges, allowing current to flow in an external circuit.

The current-voltage relationship for a solar cell is:

$I = I_0 \left[ e^{\frac{qV}{kT}} - 1 \right] - I_L$

where:

• $I$ = output current
• $I_0$ = dark saturation current (reverse bias leakage)
• $I_L$ = photo-generated current
• $V$ = cell voltage
• $q$ = electron charge ($1.6 \times 10^{-19}$ C)
• $k$ = Boltzmann's constant ($1.38 \times 10^{-23}$ J/K)
• $T$ = absolute temperature (K)

1.2 Open-Circuit Voltage and Short-Circuit Current

• Open-circuit voltage ($V_{oc}$): The voltage across the cell when no load is connected (current = 0). Typical silicon cells give $V_{oc}$ of 0.5–0.7 V.
• Short-circuit current ($I_{sc}$): The current when voltage = 0 (terminals shorted).
• Maximum power point ($V_{mp}, I_{mp}$): The point on the I-V curve where power output is maximum.

1.3 Fill Factor and Efficiency

The fill factor (FF) measures solar cell quality:

$FF = \frac{P_{max}}{I_{sc}V_{oc}} = \frac{I_{mp}V_{mp}}{I_{sc}V_{oc}}$

Cell efficiency under standard test conditions (1000 W/m², AM 1.5 spectrum, 25°C):

$\eta = \frac{P_{max}}{P_r}$

1.4 Temperature Effects

Solar panel efficiency decreases as temperature rises above 25°C. The temperature coefficient (typically –0.4%/°C) quantifies this loss. For every 1°C increase, a panel loses about 0.4% of its rated power.

Worked Example 1: A 300 W solar panel is rated at 25°C with a temperature coefficient of –0.4%/°C. Calculate the output at 35°C.

Solution:

• Temperature increase = 10°C
• Power loss = 10 × 0.4% = 4%
• Actual output = 300 W × (1 – 0.04) = 300 × 0.96 = 288 W

2.1 Wind Power Extraction

Wind is atmospheric air in motion caused by uneven solar heating of Earth's surface. Wind turbines convert the kinetic energy of wind into rotational mechanical energy, then electrical energy via a generator.

The power in wind passing through area $A$ at speed $v$ is:

$P = \frac{1}{2}\rho Av^3$

where $\rho$ is air density (≈1.23 kg/m³ at sea level).

2.2 Betz Limit

No wind turbine can capture all wind kinetic energy. The theoretical maximum efficiency is 59.3% (the Betz Limit). Accounting for the power coefficient $C_p$:

$P_{ext} = \frac{1}{2}\rho Av^3 C_p$

Where $C_p \leq 0.593$ for real turbines.

2.3 Tip Speed Ratio (TSR)

The tip speed ratio is crucial for efficient design:

$TSR = \frac{\omega r}{v} = \frac{\text{blade tip speed}}{\text{wind speed}}$

Optimal TSR values depend on turbine design (typically 6–8 for three-blade turbines).

2.4 Factors Affecting Wind Power Output

1. Wind speed: Power ∝ v³ (doubling speed increases power 8×)
2. Air density: Higher density → more power
3. Rotor swept area: Longer blades sweep more area
4. Tower height: Higher towers access stronger, steadier winds
5. Turbine efficiency: Limited by Betz limit and mechanical losses

Worked Example 2: A wind turbine has blade length 52 m. Wind speed is 12 m/s, air density 1.23 kg/m³, and $C_p$ = 0.4. Calculate extractable power.

Solution:

• Swept area: $A = \pi r^2 = \pi(52)^2 = 8495$ m²
• $P_{ext} = \frac{1}{2} \rho A v^3 C_p$
• $P_{ext} = \frac{1}{2} × 1.23 × 8495 × (12)^3 × 0.4$
• $P_{ext} = 0.5 × 1.23 × 8495 × 1728 × 0.4$
• $P_{ext} ≈$ 3.61 MW

3.1 Overview

Hydropower converts the gravitational potential energy of water stored in reservoirs into electricity. The energy conversion follows: potential → kinetic → mechanical → electrical.

3.2 Energy Conversion Stages

Stage 1: Reservoir Storage Water at height $h$ has potential energy: $E_p = mgh$

Stage 2: Water Flow Through Penstocks Potential energy converts to kinetic energy: $E_k = \frac{1}{2}mv^2$

Bernoulli's principle describes the energy conservation: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

Stage 3: Turbine Rotation Water strikes turbine blades, converting kinetic energy to rotational mechanical energy: $E_{rot} = \frac{1}{2}I\omega^2$

Stage 4: Electricity Generation Faraday's law of electromagnetic induction: $E = -N\frac{d\phi_B}{dt}$

Stage 5: Voltage Transformation Transformers step up voltage for efficient transmission: $\frac{V_s}{V_p} = \frac{N_s}{N_p}$

3.3 Types of Hydro Turbines

• Pelton turbines: High-head, low-flow conditions
• Francis turbines: Medium-head, medium-flow
• Kaplan turbines: Low-head, high-flow conditions

4.1 Overview

Geothermal energy originates from Earth's interior—from radioactive decay and residual heat from Earth's formation. When underground rocks reach 700–1300°C, magma forms and heats nearby aquifers.

4.2 Extraction for Electricity

Three main types of geothermal power plants:

1. Dry steam plants: Direct steam from fractures drives turbines
2. Flash steam plants: High-pressure hot water (over 200°C) flashes to steam
3. Binary cycle plants: Hot water transfers heat to organic fluid with lower boiling point

4.3 Heat Transfer Calculations

Heat absorbed by water: $Q = mc_p\Delta T$

Energy to convert water to steam: $Q = mL$

Worked Example 3: Water at 90°C flows from a geothermal source at 50 kg/s and cools to 40°C. Calculate heat extracted ($c_v$ = 4200 J/kg·K).

Solution:

• $\Delta T = 90 - 40 = 50$°C
• Heat rate = $\dot{m}c_v\Delta T$
• $= 50 × 4200 × 50$
• $= 10.5 × 10^6$ W = 10.5 MW

In Tanzania, understanding solar and wind energy is directly relevant to everyday life. For instance, many households in rural areas like Singida and Dodoma now use solar panels (typically 100–300 W) to charge phones and power LED lights, reducing dependence on expensive diesel generators. A 100 W solar panel under Tanzanian sunlight can charge a mobile phone (5–10 W requirement) for several hours daily, saving approximately TSh 5,000–10,000 per month that would otherwise be spent on charging services at local shops.