Renewable Energy Technology (ME-604 (C)) - Unit 2 Short Notes

Unit 2: Renewable Energy Technologies - Short Notes

I. Introduction to Renewable Energy

A. Fundamentals

• Renewable Energy (RE): Energy derived from natural sources that are replenished on a human timescale (e.g., solar, wind, biomass, hydro, geothermal, ocean).

• Need for Adoption:

  • Depleting fossil fuel reserves.

  • Environmental pollution and climate change mitigation.

  • Energy security and diversification.

  • Rural electrification and sustainable development.

• Classification: Based on source and form: Solar (thermal, PV), Wind, Biomass, Hydropower, Geothermal, Ocean (tidal, wave, OTEC), Fuel Cells.

B. Environmental and Climate Context

• Greenhouse Effect: Natural process where greenhouse gases (GH₂O, CO₂, CH₄) trap infrared radiation, warming the Earth. Enhanced by anthropogenic emissions.

• Impact of Fossil Fuels: Primary source of CO₂ emissions → global warming → climate change (sea-level rise, extreme weather).

• Sensible Heat: Heat exchanged that causes a temperature change (ΔQ = m·c·ΔT).

• Latent Heat: Heat exchanged during a phase change without temperature change (e.g., evaporation, condensation).

C. Energy Storage Management

• Role: Mitigates intermittency of RE sources (solar, wind), shifts energy from time of generation to time of use, provides grid stability.

• Management Strategies: Peak shaving, load leveling, backup power. Technologies: Batteries (Li-ion), pumped hydro, flywheels, thermal storage.

[!TIP] Common exam question: Link fossil fuels → CO₂ → greenhouse effect → global warming. Be ready to define sensible/latent heat with examples.

II. Solar Energy

A. Solar Radiation and Geometry

• Measurement:

  • Instruments: Pyranometer (global radiation), Pyrheliometer (direct radiation), Pyrometer (diffuse radiation).

  • Units: W/m², kWh/m²/day.

• Key Angles:

  • Declination (δ): Angle between Sun-Earth line and equatorial plane. Varies ±23.45° annually.

  • Hour Angle (ω): Angular displacement of Sun from local solar noon (15° per hour).

  • Altitude Angle (α): Angle between Sun's rays and horizontal plane.

  • Incidence Angle (θ): Angle between Sun's rays and normal to the surface.

  • Solar Time: Based on apparent solar motion; differs from clock time by equation of time and longitude correction.

• Energy-Wavelength Relation: Energy of a photon, \( E = \frac{hc}{\lambda} \). For λ = 1 μm, \( E = \frac{(6.626×10^{-34})(3×10^8)}{1×10^{-6}} = 1.986×10^{-19} \, \text{J} \). Convert to eV (1 eV = 1.602×10⁻¹⁹ J) → \boxed{E \approx 1.24 , \text{eV}}.

• Radiation on Tilted Surface: \( I_T = I_b \cos\theta + I_d \frac{1+\cos\beta}{2} + I \rho_g \frac{1-\cos\beta}{2} \), where β = tilt angle, ρ_g = ground reflectance.

[!TIP] Solar geometry calculations are frequent. Practice finding θ for given δ, ω, β, latitude (φ). Remember: cosθ = sinδ sinφ cosβ - sinδ cosφ sinβ cosγ + cosδ cosφ cosβ cosω + cosδ sinφ sinβ cosγ cosω + cosδ sinβ sinω sinγ (γ = azimuth).

B. Solar Thermal Systems

• Collector Classification:

  • By Concentration: Non-concentrating (flat plate), Concentrating (parabolic trough, dish, tower).

  • By Circulation: Passive (thermosyphon), Active (pump-driven).

• Flat Plate Collector (FPC):

  • Construction: Absorber plate (selective coating), glass cover (insulation, greenhouse effect), insulation (back/sides), casing, header/riser tubes.

  • Working: Solar radiation passes through glass, absorbed by plate → heats fluid in tubes → natural convection (passive) or pump (active) circulation.

  • Efficiency: \( \eta = \frac{Q_u}{A_c I_T} = F_R \left[ \tau \alpha - U_L \frac{(T_i - T_a)}{I_T} \right] \), where F_R = heat removal factor.

• Systems:

  • Thermosyphon: Passive, no pump. Density-driven flow. Requires tank above collector.

  • Forced Circulation: Active, uses pump. Controlled by differential thermostat.

• Applications: Water heating, space heating, drying (agricultural, industrial), industrial process heat (<250°C).

C. Solar Photovoltaic (PV) Systems

• Principle: Photovoltaic Effect → generation of voltage/current when light strikes a p-n junction. Photons with energy > bandgap (E_g) excite electrons from valence to conduction band.

• Solar Cell Technologies:

  | Type | Material | Efficiency | Features | | :--- | :--- | :--- | :--- | | Mono-Si | Single crystal Si | 15-22% | High cost, high efficiency, uniform blue color | | Poly-Si | Multi-crystal Si | 13-18% | Lower cost, lower efficiency, speckled blue | | a-Si | Amorphous Si | 6-9% | Low cost, flexible, higher temp coefficient | | CdTe | Cadmium Telluride | 10-17% | Low-cost thin-film, toxic Cd | | CIGS | Cu(In,Ga)Se₂ | 12-16% | High potential, flexible substrates | | GaAs | Gallium Arsenide | 25-30% | Very high efficiency, expensive, space/CPV |

• Fabrication: Cells (0.5-1V, 3-8A) connected in series (for voltage) and parallel (for current) → laminated into modules (encapsulation with EVA, glass, backsheet).

• PV System Components:

  • PV Array (modules)

  • Charge Controller (protects battery from overcharge/discharge)

  • Battery Bank (storage, e.g., lead-acid, Li-ion)

  • Inverter (DC-AC conversion)

  • Mounting structure, wiring, protection devices.

• Maximum Power Point Tracking (MPPT): Algorithm to operate PV at Maximum Power Point (MPP) where dP/dV=0. Perturb and Observe (P&O):

  1. Measure initial V, I → calculate P.

  2. Perturb (increase/decrease) voltage by ΔV.

  3. Measure new P. If P increases, continue perturbation in same direction; if decreases, reverse direction.

  4. Repeat. Simple but oscillates around MPP under rapid irradiance change.

• Limitations of SPV:

  • Low efficiency (15-22% commercial).

  • Intermittency (day/night, weather).

  • High initial cost (though decreasing).

  • Storage dependency for off-grid.

  • Degradation (0.5-1%/year).

  • Land requirement for large plants.

• Applications:

  • Standalone: Remote homes, telecommunication, water pumping.

  • Grid-Connected: Rooftop, solar farms (with/without net metering).

  • PV Generation Systems: Large-scale solar parks with inverters, transformers, grid interface.

[!TIP] P&O algorithm is a favorite. Draw flowchart: Measure (V,I) → Calc P → Perturb V → Measure new P → Compare → Decide direction. Know why it oscillates. For tidal/wind, remember power ∝ v³.

III. Wind Energy

A. Wind Resource and Power

• Wind Regimes: Characterized by Weibull Distribution: \( f(v) = \frac{k}{c} \left( \frac{v}{c} \right)^{k-1} e^{-(v/c)^k} \), where k = shape, c = scale parameter.

• Energy Estimation:

  • Mean wind speed: \( \bar{v} = c \Gamma(1 + \frac{1}{k}) \).

  • Wind Power Density (WPD): \( P_d = \frac{1}{2} \rho \bar{v}^3 \) (W/m²). More accurately, \( P_d = \frac{1}{2} \rho \int_0^\infty v^3 f(v) dv \).

• Power from Wind Turbine (Betz's Limit):

  1. Mass flow rate: \( \dot{m} = \rho A v \) (A = swept area, v = upstream wind speed).

  2. Kinetic energy/sec (power) in wind: \( P_{wind} = \frac{1}{2} \dot{m} v^2 = \frac{1}{2} \rho A v^3 \).

  3. Power extracted: \( P = \frac{1}{2} \dot{m} (v_1^2 - v_2^2) \), where v₁ = upstream, v₂ = downstream.

  4. Using momentum theory (actuator disk), \( v_2 = v(1 - 2a) \), where a = axial induction factor.

  5. \( P = 2 \rho A v^2 a (1 - a) \). Maximize w.r.t a: \( \frac{dP}{da} = 0 \Rightarrow a = \frac{1}{3} \).

  6. \boxed{P_{max} = \frac{16}{27} \cdot \frac{1}{2} \rho A v^3 = \frac{8}{17} \rho A v^3} ).

  7. Power Coefficient: \( C_p = \frac{P}{P_{wind}} \leq C_{p,max} = \frac{16}{27} \approx 0.593 \) (Betz Limit).

• Power Curve: Shows P vs. v.

  • Cut-in speed (v_c): ~3-4 m/s (turbine starts).

  • Rated speed (v_r): ~12-15 m/s (rated power reached).

  • Cut-out speed (v_f): ~25 m/s (shut down for safety).

B. Wind Energy Conversion Systems (WECS)

• Types:

  • HAWT (Horizontal Axis): Rotor shaft parallel to ground. Most common. Needs yaw mechanism. Blades like airplane wings (aerofoils).

  • VAWT (Vertical Axis): Rotor shaft perpendicular to ground.

    • Darrieus: "Egg-beater", lift-based, high efficiency, needs external start.

    • Savonius: Drag-based, low efficiency, self-starting, robust (low wind).

• Aerofoil:

  • Definition: Cross-sectional shape of a blade designed to generate lift.

  • Types: Symmetrical (zero lift at 0° AoA), Cambered (positive lift at 0° AoA).

  • Forces: Lift (perpendicular to relative wind, primary for HAWT), Drag (parallel, opposes motion).

• Main Components:

  • Rotor Blades (capture energy)

  • Nacelle (housing)

  • Gearbox (increases speed, often omitted in direct-drive)

  • Generator (converts mechanical to electrical)

  • Tower (height ↑ wind speed)

  • Yaw mechanism (faces HAWT into wind)

  • Control systems (pitch, brake).

C. Challenges and Hybrid Systems

• Limitations/Prohibitions:

  • Intermittency: Variable wind → grid stability issues.

  • Grid Integration: Need for transmission lines, reactive power support, frequency regulation.

  • Visual & Noise Impact: Public acceptance.

  • Wildlife: Bird/bat mortality.

  • Cost: Capital intensive, though LCOE competitive.

  • Site Availability: Need high, consistent wind resources (coastal, hilltops).

• Wind-Diesel Hybrid: Combines wind turbines with diesel generators in remote grids.

  • Configuration: Wind → rectifier → DC bus → inverter → AC grid; Diesel gen → AC grid. Often with battery buffer.

  • Operation: Wind used as primary source; diesel supplements during low wind or peak demand.

  • Advantages: Reduces diesel fuel consumption/cost, lowers emissions, improves reliability in remote areas.

[!TIP] Derivation of Betz's limit is crucial. Practice step-by-step: mass flow, energy extraction, momentum theory (v₂ = v(1-2a)), maximize P(a). Know why C_p ≤ 0.593. For hybrid systems, emphasize role of battery/buffer.

IV. Biomass Energy

A. Biomass Fundamentals

• Definition: Organic matter from plants/animals (living or recently dead) used as fuel.

• Sources: Agricultural residues (straw, bagasse), energy crops (sugarcane, switchgrass), municipal solid waste (MSW), animal waste (dung), forestry residues.

• Usefulness:

  • Renewable (via photosynthesis).

  • Carbon-neutral (CO₂ released ≈ CO₂ absorbed during growth).

  • Waste management solution.

  • Rural employment/development.

• Photosynthetic Pathways:

  • C3 Plants: (Rice, wheat, potatoes). Calvin cycle only. Fix CO₂ into 3-carbon compound. Efficient in cool, moist conditions. Photorespiration loss → lower efficiency (~0.5-1% solar to biomass).

  • C4 Plants: (Sugarcane, maize, sorghum). Additional C4 cycle in mesophyll cells concentrates CO₂ → minimizes photorespiration. Higher efficiency (~1-2%), better in hot, sunny conditions.

B. Biomass Conversion Processes

• Biochemical:

  • Anaerobic Digestion (AD): Microbial breakdown in absence of O₂.

    • Stages: 1. Hydrolysis (complex polymers → sugars/amino acids), 2. Acidogenesis (→ acids, alcohols, H₂, CO₂), 3. Acetogenesis (→ acetic acid, H₂, CO₂), 4. Methanogenesis (→ CH₄, CO₂).

    • Biogas Composition: ~55-65% CH₄, 35-45% CO₂, traces H₂S, H₂O.

• Thermochemical:

  • Gasification: Partial oxidation at high T (700-1000°C) → producer gas (CO, H₂, CH₄, CO₂, N₂).

    • Types: Updraft (simple, high tar), Downdraft (low tar, common), Crossdraft, Fluidized bed (good mixing, high efficiency).

  • Pyrolysis: Thermal decomposition in absence of O₂ → bio-oil, char, syngas. Fast pyrolysis → max bio-oil.

  • Combustion: Complete oxidation → heat (for steam turbine).

C. Biomass Energy Systems

• Anaerobic Digesters:

  • Batch: Fill, digest, empty. Simple, uneven gas.

  • Continuous: Constant feed/output, steady gas.

  • Fixed Dome (Chinese): Brick/cement, gas holder fixed. Low cost, masonry skill needed.

  • Floating Drum (Indian): Movable steel drum as gas holder. Easy to see gas volume, maintenance.

  • Plug Flow: Long, narrow tank (like a sausage). For dung/water mixes.

• Biogas Plant Design:

  • Components: Inlet (feed), Digester tank (anaerobic zone), Outlet (slurry), Gas holder/storage, Piping.

  • Sizing: Based on retention time (15-50 days for dung) and loading rate (kg VS/m³/day). Volume = (daily feed × retention time).

• Power from Gasification:

  • System: Biomass → Gasifier → Producer Gas → Cooling (remove tar/particulates) → Filtering → Gas Engine → Generator → Electricity.

  • Cleanup Critical: Tar and particulates damage engines.

[!TIP] AD stages must be in order: Hydrolysis → Acidogenesis → Acetogenesis → Methanogenesis. Gasifier types: Downdraft most common for engines (low tar). Biogas plant: Fixed dome vs. floating drum – know construction and pros/cons.

V. Hydropower

A. Hydropower Systems

• Classification by Capacity:

  | Type | Capacity | Head | Flow | Application | | :--- | :--- | :--- | :--- | :--- | | Micro | < 100 kW | Low | Low | Single home/village | | Mini | 100 kW – 1 MW | Medium | Medium | Small community | | Small | 1 – 25 MW | Variable | Variable | Mini-grid, small utility |

• Main Components (Small Hydro):

  1. Intake: Diverts water, screens debris.

  2. Penstock: Pressurized pipe (head loss critical).

  3. Turbine: Converts hydraulic to mechanical energy.

  4. Generator: Converts mechanical to electrical.

  5. Tailrace: Returns water to river.

  6. Control Systems: Governor, valves, instrumentation.

B. Hydraulic Turbines

• Classification & Working:

  | Type | Principle | Head (m) | Flow | Example Application | | :--- | :--- | :--- | :--- | :--- | | Impulse (Pelton) | Jet impacts buckets, kinetic → mechanical | High (>300) | Low | High-head, low-flow | | Reaction (Francis) | Pressure & velocity change in runner, enclosed | Medium (30-300) | Medium | Most common, medium-head | | Reaction (Kaplan/Propeller) | Adjustable blades, axial flow | Low (<30) | High | Low-head, high-flow rivers |

• Selection Criteria:

  • Net Head (H): Available head after losses.

  • Design Flow (Q): Available discharge.

  • Specific Speed (N_s): \( N_s = \frac{N \sqrt{P}}{H^{5/4}} \) (imperial) or metric equivalent. Indicates turbine type suitability.

  • Efficiency, Cost, Site constraints (cavitation, size).

• Speed & Voltage Regulation:

  • Governor: Mechanical/hydraulic device controlling wicket gates/nozzles to maintain constant speed (frequency) under varying load.

  • Excitation System: Controls generator field current → regulates output voltage.

  • Grid Synchronization: Turbine-generator must match grid frequency, voltage, phase before connection.

C. (Covered in Hybrid Systems section of Wind)

VI. Geothermal Energy

A. Geothermal Resources

• Types of Deposits:

  • Vapor-dominated (Dry Steam): Steam-filled fractures (e.g., The Geysers, USA). Direct use.

  • Liquid-dominated (Wet Steam/Hot Water): Hot water + steam. Most common. Requires separation.

  • Hot Dry Rock (HDR): Hot impermeable rock. Needs Enhanced Geothermal Systems (EGS) – inject water to create fractures.

  • Geopressured: Hot brine under high pressure. Contains methane.

• Site Selection Criteria:

  • High geothermal gradient (>30°C/km).

  • Permeable reservoir (fractures, porosity).

  • High temperature (>150°C for power).

  • Adequate water (recharge or injection).

  • Accessibility, environmental sensitivity, proximity to grid.

B. Geothermal Power Generation

• Processes:

  1. Dry Steam: Direct use of geothermal steam → turbine → condenser. Simplest.

  2. Flash Steam: High-pressure hot water → throttled (flashed) to lower pressure → mixture of steam + water → steam separated → turbine → condenser. Single-flash (one stage), Double-flash (two stages for higher efficiency).

  3. Binary Cycle: Geothermal fluid (low temp, 100-180°C) heats secondary working fluid (low boiling point: isobutane, pentane) in heat exchanger → vapor drives turbine → condenser → fluid recycled. Allows use of lower T resources. \boxed{\text{Organic Rankine Cycle (ORC)}}.

• Thermodynamics: Modification of Rankine cycle. Key components: vaporizer (heat exchanger), turbine, condenser, pump.

• Environmental Benefits:

  • Low GHG emissions (mostly non-condensable gases like CO₂, H₂S – can be reinjected).

  • Small land footprint per MW.

  • Baseload capability: Runs 24/7, high capacity factor (>90%).

  • Minimal visual impact (small plants).

[!TIP] Distinguish flash vs. binary. Flash uses geothermal steam directly; binary uses heat exchanger with secondary fluid. Binary allows lower T resources. HDR/EGS is future tech but not commercial yet.

VII. Ocean Energy

A. Tidal Energy

• Principle: Harness kinetic (tidal streams) or potential (tidal range) energy from gravitational pull of Moon/Sun.

• Configurations:

  • Tidal Barrage: Dam across estuary. Basin fills/empties with tide → water flows through turbines in dam.

  • Single Basin: One basin. Generates during flood (filling) or ebb (emptying), not both (unless reversible turbines).

  • Double Basin: Two basins at different phases → continuous generation.

  • Tidal Stream Generators (TSG): Underwater "wind turbines" in fast currents (no dam).

• Energy Calculation (Single Basin, Ebb Generation):

  • Potential energy in basin at high tide: \( E_{high} = \frac{1}{2} \rho g A H_1^2 \)

  • Potential energy at low tide (when turbine stops at head H₂): \( E_{low} = \frac{1}{2} \rho g A H_2^2 \)

  • Extractable Energy: \( E = \eta \cdot \frac{1}{2} \rho g A (H_1^2 - H_2^2) \), where η = turbine-generator efficiency.

  • Note: H₁ = tidal range, H₂ = minimum operating head. For filling process, similar formula with H₁ and H₂ swapped.

  • Example: A=30×10⁶ m², H=12 m, H₂=3 m, η=0.73. \( H_1 = 12\, \text{m}, H_2 = 12-3=9\, \text{m} \) (if emptying from full to 3m head). \( E = 0.73 \times \frac{1}{2} \times 1025 \times 9.81 \times 30\times10^6 \times (12^2 - 9^2) \) → calculate in J, convert to kWh (1 kWh = 3.6×10⁶ J).

B. Wave Energy

• Significant Wave Height (H₁/₃): Average height of the highest one-third of waves in a wave spectrum. Statistically represents sea state.

• Conversion Devices (Brief):

  • Oscillating Water Column (OWC): Wave drives air column → air turbine.

  • Point Absorber: Buoy moves with waves → drives generator.

  • Attenuator: Long, multi-segment device (like Pelamis) flexes with wave → hydraulic pump.

C. Ocean Thermal Energy Conversion (OTEC)

• Principle: Exploit temperature difference (ΔT) between warm surface water (~25-30°C) and cold deep water (~5-10°C). Requires ΔT > 20°C (tropical oceans).

• Types:

  • Open Cycle: Warm seawater → flash evaporator → low-pressure steam → turbine → condenser (cold water) → condensed freshwater (byproduct).

  • Closed Cycle: Warm seawater heats working fluid (e.g., ammonia) in evaporator → vapor → turbine → condenser (cold seawater) → fluid recycled.

  • Hybrid: Combines aspects of both.

• Challenges: Very low thermodynamic efficiency (3-4%) due to small ΔT, huge infrastructure (pipes), biofouling, high capital cost.

[!TIP] Tidal energy formula is \boxed{E = \eta \frac{1}{2} \rho g A (H_1^2 - H_2^2)}. Be careful: H₁ and H₂ are heads, not necessarily the full tidal range if turbine stops early. OTEC efficiency low because ΔT small (Carnot efficiency ∝ ΔT).

VIII. Fuel Cells

A. Fuel Cell Fundamentals

• Working Principle: Electrochemical device. Fuel (H₂, CH₄, etc.) + Oxidant (O₂ from air) → Electricity + Water + Heat. No combustion.

  • Anode: Fuel oxidized (H₂ → 2H⁺ + 2e⁻).

  • Cathode: Oxidant reduced (½O₂ + 2H⁺ + 2e⁻ → H₂O).

  • Electrolyte: Ion-conducting membrane (selective).

• Classification by Electrolyte:

  | Type | Electrolyte | Temp (°C) | Fuel | Efficiency | Applications | | :--- | :--- | :--- | :--- | :--- | :--- | | PEMFC | Polymer (Nafion) | 60-80 | Pure H₂ | 40-60% | Transport, portable, backup | | SOFC | Solid Oxide (ZrO₂) | 800-1000 | H₂, CO, CH₄ | 50-60% | Stationary, large-scale | | MCFC | Molten Carbonate | 650 | H₂, CO, CH₄ | 50-60% | Stationary, utility | | AFC | Alkaline (KOH) | 60-90 | Pure H₂, O₂ | 40-60% | Space (Apollo), specialty | | PAFC | Phosphoric Acid | 200 | Reformed fuels | 40% | Stationary, early commercial |

• Comparison: Temp ↑ → fuel flexibility ↑, efficiency ↑, but start-up time ↑, materials challenge ↑. PEMFC: quick start, good for vehicles. SOFC/MCFC: high efficiency, fuel flexible, slow start (stationary).

B. Fuel Cell Systems

• Main Components:

  1. Fuel Processor/Reformer: Converts hydrocarbon fuel (NG, methanol) to H₂-rich gas (steam reforming, partial oxidation).

  2. Fuel Cell Stack: Series of cells → desired voltage.

  3. Power Conditioner: DC-DC converter (for voltage), inverter (DC-AC).

  4. Heat Recovery System: Uses waste heat (cogeneration/CHP) → overall efficiency 70-90%.

• Advantages:

  • High efficiency (40-60% electrical, >80% with CHP).

  • Very low emissions (water, heat, trace NOx).

  • Modular, scalable.

  • Quiet, vibration-free.

  • Quick refueling (vs. battery charging).

[!TIP] Know electrolyte → temp → fuel → application mapping. PEMFC = vehicles (H₂ fuel cell cars). SOFC = stationary power (high temp, fuel flexible). AFC = space (pure H₂/O₂). Reformer needed for non-H₂ fuels.

IX. Grid Integration and Economics

A. Electricity Tariffs

• Flat Rate: Fixed charge per unit (kWh) regardless of time.

• Block Rate (Increasing/Decreasing Block): Different rates for different consumption blocks (e.g., first 100 kWh @ ₹3, next @ ₹5).

• Time-of-Use (TOU): Different rates for different time periods (peak, off-peak, shoulder). Encourages load shifting.

• Demand Charges: Charge based on maximum power (kW) drawn in a billing period (common for commercial/industrial).

• Feed-in Tariff (FiT): Rate paid by utility to renewable generator for power fed into grid. Often above retail rate to incentivize RE.

B. Integration Challenges

• Intermittency Management: Solar/wind variability → need for flexible resources: storage, flexible generation (gas turbines), demand response.

• Grid Stability:

  • Frequency: Balance of generation-load. Inertia from rotating machines (turbines) helps; inverter-based RE has low inertia → need synthetic inertia/grid-forming inverters.

  • Voltage: Reactive power support. Traditional generators provide VARs; PV/wind need power electronics (STATCOM, inverters) for voltage control.

• Need for Storage/Backup: To firm capacity, provide ancillary services.

• Grid Codes: Technical requirements for RE plants (fault ride-through, power factor, ramp rates).

[!TIP] FiT vs. Net Metering: FiT = sell all generation at fixed rate; Net Metering = offset own consumption, export at retail rate (or lower). Intermittency → storage/grid flexibility. Inertia issue with renewables is key.

X. Cross-Cutting Calculations and Examples

A. Solar Geometry Calculations

• Solar Time: \( t_{sol} = t_{clock} + \frac{4(L_{std} - L_{loc})}{60} + EOT \), where L_std = standard meridian, L_loc = local longitude, EOT = equation of time (approx: \( EOT = 9.87 \sin(2B) - 7.53 \cos(B) - 1.5 \sin(B) \), \( B = \frac{360}{365}(N-81) \), N = day number).

• Incidence Angle on Tilted Surface (facing south, γ=0): \( \cos\theta = \sin\delta \sin\phi \cos\beta - \sin\delta \cos\phi \sin\beta + \cos\delta \cos\phi \cos\beta \cos\omega + \cos\delta \sin\phi \sin\beta \cos\omega + \cos\delta \sin\beta \sin\omega \).

  • Simplified for south-facing: \( \cos\theta = \cos\theta_z \cos\beta + \sin\theta_z \sin\beta \cos(\gamma - \alpha) \), where θ_z = zenith angle.

• Optimal Tilt (β): For max annual insolation, β ≈ φ. For max winter, β = φ + 10-15°; for max summer, β = φ - 10-15°.

B. Tidal Energy Calculation

• Single Basin (Ebb/Fill): \( E = \eta \frac{1}{2} \rho g A (H_1^2 - H_2^2) \).

  • H₁ = head at start (e.g., high tide = 12m).

  • H₂ = head when turbine stops (e.g., 3m).

  • η = overall efficiency (turbine + generator).

  • ρ = seawater density (~1025 kg/m³), g = 9.81 m/s².

  • Always check: Is it filling or emptying? Which head is larger? (H₁ > H₂).

C. Wind Power Derivation

• Step-by-step (Betz):

  1. \( \dot{m} = \rho A v \) (mass flow through rotor).

  2. \( P_{wind} = \frac{1}{2} \dot{m} v^2 = \frac{1}{2} \rho A v^3 \).

  3. \( P = \frac{1}{2} \dot{m} (v_1^2 - v_2^2) = \frac{1}{2} \rho A v (v_1^2 - v_2^2) \).

  4. Continuity: \( v_1 A = v_2 A_2 = v A \) → \( v_2 = v(1 - 2a) \), where a = axial induction factor.

  5. Substitute: \( P = \frac{1}{2} \rho A v [v^2 - v^2(1-2a)^2] = 2 \rho A v^3 a (1-a) \).

  6. Max: \( \frac{dP}{da} = 2 \rho A v^3 (1 - 2a) = 0 \Rightarrow a = \frac{1}{3} \).

  7. \( P_{max} = 2 \rho A v^3 (\frac{1}{3})(\frac{2}{3}) = \frac{4}{9} \rho A v^3 \).

  8. \( C_{p,max} = \frac{P_{max}}{P_{wind}} = \frac{4/9}{1/2} = \frac{8}{27} \approx 0.593 \).

D. Photovoltaic Energy Conversion

• Photon Energy: \( E_{photon} = \frac{hc}{\lambda} \). h = 6.626×10⁻³⁴ J·s, c = 3×10⁸ m/s.

  • For λ = 1 μm = 10⁻⁶ m: \( E = \frac{(6.626e-34)(3e8)}{1e-6} = 1.9878e-19 \, \text{J} \).

  • Convert: \( 1 \, \text{eV} = 1.602e-19 \, \text{J} \) → \( E = \frac{1.9878e-19}{1.602e-19} \approx 1.24 \, \text{eV} \). \boxed{E \approx 1.24 , \text{eV}}.

• Bandgap (E_g): Minimum energy needed to excite electron. Only photons with E > E_g generate electron-hole pairs.

• Shockley-Queisser Limit: Maximum theoretical efficiency of a single-junction solar cell under standard conditions (~33.7% for Si, E_g ~1.1 eV). Losses: below-gap photons (thermalization), above-gap photons (thermal relaxation), radiative recombination, fill factor.

[!TIP] Tidal: Identify H₁ (larger head) and H₂ (smaller head). Wind: Derivation must show a = 1/3 for max. PV: E(eV) = 1240 / λ(nm). So λ=1000 nm → 1.24 eV. Shockley-Queisser is theoretical max, real cells lower due to practical losses.

Final Exam Strategy: Focus on derivations (Betz, tidal energy), definitions (C3/C4, aerofoil, fuel cell types), diagrams (flat plate collector, PV cell, WECS types, tidal barrage, fuel cell schematic), and numerical problems (solar geometry, tidal, wind power). Always box final formulas.