Comprehensive Notes on Hydroelectric Power, Wave Energy Conversion, and Energy Densities

Environmental Impacts of Hydroelectric Reservoirs and Greenhouse Gas Accounting

• Submerged Biomass Preservation and Decomposition:

  • Forests submerged beneath artificial reservoirs can remain structurally intact for extensive durations. Historical aerial and underwater imagery indicates that tree trunks flooded approximately 70 years prior remain standing, with the exception of foliage and finer organic matter.

  • The long-term underwater decay of flooded plant matter results in sustained greenhouse gas emissions, fundamentally altering environmental life-cycle assessments.

• Comparative Emissions in Canadian Hydroelectricity:

  • Greenhouse gas accounting methodologies indicate that Hydro-Québec electricity produces approximately five times higher emissions per unit of energy than British Columbia (BC) hydropower.

  • This discrepancy is attributed primarily to the accounting and decomposition dynamics of submerged forests located within large northern reservoirs.

Hydroelectric Dam Mechanics and Fundamental Power Equations

• Electromechanical Energy Conversion:

  • Water impoundment drives massive hydraulic turbines mechanically coupled to large electromagnetic generators (also referred to as alternators or magnetic motors).

  • Operation requires high hydraulic pressure, derived from the hydrostatic head created by the gravitational weight of the impounded water column.

• System Boundary Selection and Governing Equations:

  • An internal pressure analysis models the hydrostatic pressure directly acting on the penstock and turbine runner.

  • By positioning the thermodynamic control volume boundary at the reservoir surface and the tailrace exit, internal pressure terms can be eliminated, directly converting gravitational potential energy into electrical output.

  • The theoretical mechanical power output ($P$) is governed by:         $P = \dot{m} g \Delta h$         where:

    • $\dot{m}$ is the mass flow rate of water ($\text{kg}/\text{s}$

    • $g$ is the acceleration due to gravity ($9.81\,\text{m}/\text{s}^2$

    • $\Delta h$ is the net hydraulic head or vertical elevation drop across the dam ($\text{m}$

• Hydraulic Mass Flow Rate Formulations:

  • At the turbine or conduit level, the mass flow rate depends on internal pipe geometry and fluid velocity:         $\dot{m} = \rho_{\text{water}} A_{\text{pipe}} v_{\text{water}}$         where $\rho_{\text{water}}$ is the density of water ($1000\,\text{kg}/\text{m}^3$), $A_{\text{pipe}}$ is the cross-sectional flow area ($\text{m}^2$), and $v_{\text{water}}$ is the flow velocity ($\text{m}/\text{s}$

  • From a macro-scale watershed sustainability perspective, the maximum steady-state mass flow rate ($\dot{m}_{\text{sust}}$) is constrained by annual hydrological precipitation and drainage area:         $\dot{m}_{\text{sust}} = \rho_{\text{water}} \times (\text{Annual Rainfall}) \times (\text{Catchment Area})$

• Reservoir Depletion Constraints:

  • Exceeding the sustainable mass flow rate lowers the reservoir level, reducing hydraulic head and total energy capacity.

  • Real-World Case Study: Hoover Dam in the United States has experienced severe drops in water surface elevation due to excessive regional water consumption and upstream agricultural diversion along the Colorado River valley, directly starving the power generation infrastructure.

Hydrological Modeling and the Daniel-Johnson Dam Case Study

• Hydrological and Structural Specifications:

  • Facility: Daniel-Johnson Dam (Manic-5), Québec.

  • Hydraulic Head ($\Delta h$): $214\,\text{m}$ (also noted as $212\,\text{m}$; calculations use $214\,\text{m}$

  • Installed Electrical Capacity: $2.6\,\text{GW} = 2{,}600\,\text{MW}$

  • Catchment Area ($A_{\text{catch}}$: Approximately $30{,}000\,\text{km}^2$

  • Reservoir Surface Area ($A_{\text{res}}$: Approximately $2{,}000\,\text{km}^2$ (yielding an approximate 15:1 to 10:1 ratio of catchment area to lake surface area).

  • Average Reservoir Depth: $85\,\text{m}$

  • Total Reservoir Water Volume: $142\,\text{km}^3$

  • Regional Precipitation: Annual rainfall in this region of Québec averages approximately $1\,\text{m}/\text{year}$

• Mass Flow Rate Calculation:

  • Conversion of one year to seconds:         $1\,\text{year} = 365\,\text{days} \times 24\,\text{hours} \times 60\,\text{minutes} \times 60\,\text{seconds} \approx 3.15 \times 10^7\,\text{s}$

  • Annual mass inflow:         $\text{Mass} = 1000\,\text{kg}/\text{m}^3 \times 1\,\text{m}/\text{year} \times (30{,}000 \times 10^6\,\text{m}^2) = 3 \times 10^{13}\,\text{kg}$

  • Average sustainable mass flow rate:         $\dot{m} = \frac{3 \times 10^{13}\,\text{kg}}{3.15 \times 10^7\,\text{s}} \approx 10^6\,\text{kg}/\text{s} = 10^6\,\text{L}/\text{s}$

• Theoretical Power Output Determination:

  • Applying the potential energy conversion equation:         $P = (10^6\,\text{kg}/\text{s}) \times (9.81\,\text{m}/\text{s}^2) \times (214\,\text{m})$         $P \approx 2.14 \times 10^9\,\text{W} = 2{,}140\,\text{MW}$

  • The theoretical estimate of $2{,}140\,\text{MW}$ closely tracks the actual dam rating of $2{,}600\,\text{MW}$.

• Sensitivity to Climate Change:

  • Because electrical generation scales linearly with precipitation, shifting rainfall patterns directly threaten hydroelectric security. Decreased precipitation reduces available throughput, while extreme precipitation events introduce structural flooding and spillway discharge challenges.

Area-Normalized Power Density Metrics in Hydroelectricity

• Reservoir-Basis Power Density:

  • Defined as the electrical generation capacity divided by the flooded reservoir surface area ($A_{\text{res}}$

  • Calculation for Daniel-Johnson Dam:         $\text{Power Density}_{\text{res}} = \frac{2{,}140\,\text{MW}}{2{,}000\,\text{km}^2} = \frac{2{,}140 \times 10^6\,\text{W}}{2{,}000 \times 10^6\,\text{m}^2} \approx 1\,\text{W}/\text{m}^2$

  • This provides a benchmark for evaluating direct land alteration against other energy technologies (such as solar or wind power).

  • Reservoirs provide secondary utility beyond power generation, including water storage, recreational zones (e.g., camping, boating), and downstream flood prevention.

• Catchment-Basis Power Density:

  • Calculated across the broader hydrological watershed boundary ($A_{\text{catch}}$

  • Calculation for Daniel-Johnson Dam:         $\text{Power Density}_{\text{catch}} = \frac{2{,}000\,\text{MW}}{30{,}000\,\text{km}^2} = \frac{2}{30}\,\text{W}/\text{m}^2 \approx 0.067\,\text{W}/\text{m}^2$

  • This metric reflects the total geographic territory required to gather precipitation and funnels into calculations estimating total national hydroelectric limits.

Global Hydroelectric Resource Limits and Turbine Efficiencies

• Turbine Thermodynamics and Conversion Efficiencies:

  • Unlike thermal power plants constrained by Carnot efficiency limits, hydraulic turbines convert mechanical work directly without an intermediate thermal reservoir; their efficiency is theoretically bound only by friction and fluid irreversibilities.

  • Kaplan Turbines: Characterized by adjustable propeller blades, designed for low-head applications ($\Delta h < 70\,\text{m}$

  • Francis Turbines: Reaction turbines designed for medium to high-head configurations ($\Delta h$ up to approximately $600\,\text{m}$

  • Modern hydraulic turbines routinely operate at efficiencies around $90\%$

• Global Macro-Scale Limits:

  • Hydroelectricity accounts for approximately $60\%$

  • Total worldwide theoretical hydroelectric potential is estimated at approximately $2\,\text{TW}$

  • More than half of this resource ($> 1\,\text{TW}$) is already developed.

  • Relative to total global primary energy consumption (approximately $12\,\text{TW}$), full development of every viable river on Earth would be insufficient to meet global energy demand.

• Hydrological Accounting Complexities:

  • National potential cannot be calculated purely from average rainfall multiplied by land surface area. Non-mountainous regions (e.g., the Prairie provinces) lack necessary hydraulic head.

  • Hydrological assessments must deduct significant losses due to ground infiltration and evapotranspiration from vegetation.

Wave Energy Physics and Working Principles of Wave Energy Converters

• Kinematic Nature of Ocean Waves:

  • Visual observation of breaking waves at shorelines creates the illusion of net forward water mass transport.

  • In open-ocean deep water, fluid particles undergo closed, reciprocal orbital (vertical and lateral) motion with zero net forward displacement over a complete period.

  • The wave is a propagating energy packet rather than bulk fluid translation.

• Forms of Harnessable Energy:

  • Wave Energy Converters (WECs) ride the wave motion rather than capturing dynamic fluid impact forces.

  • The extracted energy is predominantly gravitational potential energy associated with the continuous elevation and descent of fluid parcels, rather than fluid kinetic energy.

• Classification of Wave Energy Converters:

  • Point Absorbers / Tethered Buoys: Floating surface buoys attached via taut or sprung tethers to seafloor anchors. The relative heave motion drives internal generators, linear alternators, or fluid pumps.

  • Attenuators / Hinged Articulated Structures (e.g., Pelamis): Multi-segment floating cylinders oriented parallel to wave propagation. Wave profiles cause bending moments at flexible joints, which actuate hydraulic rams driving electric generators.

  • Oscillating Water Columns (OWC): Enclosed coastal or offshore chambers open to the sea below. Rising and falling internal water surfaces compress and decompress trapped air columns, driving bidirectional air turbines (such as Wells turbines).

  • Salter's Duck (Nodding Duck): Cam-shaped floating bodies comprising a buoyant forward section and a heavily ballasted body. As waves pass, the buoyant nose rocks relative to the fixed ballast shaft, converting rotational shaft motion into power.

Mathematical Derivation of Wave Power Density per Unit Crest Length

• Wave Parameterization:

  • Wavelength: $\lambda$ ($\text{m}$

  • Wave Period: $T$ ($\text{s}$

  • Apparent Wave Propagation Speed: $v_w = \frac{\lambda}{T}$

  • Wave Amplitude: $h$ ($\text{m}$), defined as half the total peak-to-trough vertical distance ($2h$

  • Wavefront Width (crest length perpendicular to travel): $L$ ($\text{m}$

• Derivation of Displaced Mass ($m$):

  • Approximating the sinusoidal wave crest above the mean equilibrium water level as a symmetrical triangle:

    • Crest height: $h$

    • Base of crest along propagation axis: $\frac{\lambda}{2}$

  • Cross-sectional area of water mass elevated above equilibrium:         $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times \left(\frac{\lambda}{2}\right) \times h = \frac{1}{4} h \lambda$

  • Total volumetric displacement across width $L$:         $V = \frac{1}{4} h \lambda L$

  • Mass of elevated water:         $m = \rho_{\text{water}} V = \frac{1}{4} \rho_{\text{water}} h \lambda L$

• Effective Center-of-Mass Vertical Drop ($\Delta h$):

  • While the topmost surface water drops by $2h$ from peak to trough, water at the equilibrium node experiences zero vertical movement.

  • The vertical displacement between the center of mass of the crest and the center of mass of the trough is exactly equal to the wave amplitude:         $\Delta h = h$

• Power Derivation per Unit Wavefront Length ($P/L$):

  • Total gravitational potential energy delivered over one wave cycle period ($T$):         $P = \frac{m g \Delta h}{T} = \frac{\left(\frac{1}{4} \rho_{\text{water}} h \lambda L\right) g h}{T}$

  • Substituting the wave propagation velocity relation ($v_w = \frac{\lambda}{T}$):         $P = \frac{1}{4} \rho_{\text{water}} g h^2 v_w L$

  • Normalizing per unit of wavefront crest length ($L$):         $\frac{P}{L} = \frac{1}{4} \rho_{\text{water}} g h^2 v_w$

• Numerical Estimations:

  • Constants: $\rho_{\text{water}} = 1000\,\text{kg}/\text{m}^3$, $g \approx 10\,\text{m}/\text{s}^2$, nominal wave amplitude $h = 1\,\text{m}$

  • Low-Velocity Sea State ($v_w = 1\,\text{m}/\text{s}$):         $\frac{P}{L} = \frac{1}{4} \times 1000 \times 10 \times (1)^2 \times 1 = 2{,}500\,\text{W}/\text{m} = 2.5\,\text{kW}/\text{m}$

  • High-Velocity Sea State ($v_w = 10\,\text{m}/\text{s}$):         $\frac{P}{L} = \frac{1}{4} \times 1000 \times 10 \times (1)^2 \times 10 = 25{,}000\,\text{W}/\text{m} = 25\,\text{kW}/\text{m}$

  • High-Energy Storm Regimes: Waves with $h = 2\text{--}3\,\text{m}$ and $v_w = 10\,\text{m}/\text{s}$ reach linear power densities up to $60\,\text{kW}/\text{m}$.

Global Wave Energy Potential and Coastline Geometry

• The Rationale for Linear Density ($\text{kW}/\text{m}$):

  • Unlike solar or wind power, which capture omnidirectional flux distributed across land area ($\text{W}/\text{m}^2$), waves travel in unidirectional wavefronts.

  • Operating converters extract energy directly from the passing wave, causing downstream waves to exit with reduced amplitude and diminished speed.

  • Placing converters in series directly behind one another along the propagation line yields negligible output; installations must align linearly along the incident wavefront normal.

• Coastline Fractal Paradox:

  • Measuring physical coastline perimeter exhibits fractal behavior, where measured length scales toward infinity as measurement resolution decreases toward microscopic sand grains.

  • For wave energy resource estimation, the relevant length parameter is the projected line normal to the predominant ambient open-ocean wave vectors, rather than the convoluted geographic shoreline.

• MacKay Regional Wave Assessments:

  • European Average: Standard open-ocean resource of approximately $10\,\text{kW}/\text{m}$, delivering an equivalent potential of roughly $2\,\text{kWh}/\text{person}/\text{day}$.

  • United Kingdom Coastal Model:

    • Coastline boundary length: $1{,}000\,\text{km} = 10^6\,\text{m}$

    • National population: $60{,}000{,}000$

    • Per capita frontage allocation: $\frac{10^6\,\text{m}}{60 \times 10^6\,\text{people}} = \frac{1}{60}\,\text{m}/\text{person}$

    • Assuming an offshore energetic sea state of $40\,\text{kW}/\text{m}$ over a 24-hour cycle:             $\text{Per Capita Energy} = \left(\frac{1}{60}\,\text{m}\right) \times (40\,\text{kW}/\text{m}) \times (24\,\text{h}) = 16\,\text{kWh}/\text{day}/\text{person}$

Questions & Discussion: Wave Resonance, Energy Scales, and Exam Review

• Wave Amplitude Amplification via Internal Mechanics:

  • Question: Can engineered resonant mechanisms (such as internal tethered spring systems designed to accelerate and triple buoy stroke height, as trialed in Portugal) overcome base wave power limitations?

  • Response: Mechanical resonance cannot exceed thermodynamic or physical limits. The total ambient energy carried by a wave parcel is finite. Engineering systems alter internal impedance and resonance matching to maximize hydrodynamic extraction efficiency toward theoretical limits, but they cannot create excess wave energy beyond the baseline ambient flux.

• Course Logistics and Examination Review:

  • Grading Protocol: Initial grading and rubric assignments are conducted by the teaching assistant (Varzee). Inquiries regarding grading deductions must be submitted to Varzee first for evaluation before formal instructor review.

• Chemical vs. Gravitational Potential Energy Comparison (The Soda Bottle Problem):

  • Problem Statement: Determine the height to which a single bottle of soda containing $150\,\text{food calories}$ must be lifted such that its gravitational potential energy equals its internal chemical potential energy.

  • Thermodynamic Unit Error: The most prevalent error is confusing dietary "food calories" with thermodynamic gram-calories. One food calorie is one kilocalorie ($1\,\text{kcal} = 1{,}000\,\text{cal} \approx 4{,}184\,\text{J}$

  • Energy Calculation:         $E = 150\,\text{kcal} \times 4{,}184\,\text{J}/\text{kcal} \approx 6.28 \times 10^5\,\text{J}$

  • Calculated Height Drop/Rise: Assuming a bottle mass around $0.355\text{--}0.5\,\text{kg}$, the equivalent gravitational height is approximately $200\,\text{km}$

  • Comparative Scale: An altitude of $200\,\text{km}$ reaches the thermosphere and the lower boundary of Low Earth Orbit (LEO, typically $200\text{--}500\,\text{km}$).

  • Equivalent Free-Fall Velocity: Dropping this mass in the absence of aerodynamic drag converts this energy to kinetic velocity, yielding terminal speeds between Mach 6 and Mach 7 (approximately $2\,\text{km}/\text{s}$).