13.1 Hydroelectric, Geothermal & Renewable Energy Power Plants

13.1 Hydroelectric, Geothermal & Renewable Energy Power Plants

Renewable energy power plants harvest energy from natural, replenishable hydrologic, geothermal, solar, atmospheric, and biological cycles. In the Philippine licensure examination for Mechanical Engineers (MELE), mastery of energy conversion thermodynamics, hydraulic head equations, turbine specific speeds, flash steam thermodynamics, and wind power aerodynamics is critical.

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1. Hydroelectric Power Plants

Hydroelectric power conversion converts the gravitational potential energy of elevated water into rotational mechanical energy via hydraulic turbines, which in turn drive electrical generators.

Hydraulic Heads and Energy Losses

• Gross Head ($H_g$): The vertical elevation difference between the headwater surface reservoir level and the tailrace surface water level.
• Friction Head Loss ($h_f$): Hydraulic energy loss per unit weight of fluid due to pipe wall friction and fitting resistance in the penstock, calculated using the Darcy-Weisbach equation: $h_f = f \cdot \frac{L}{D} \cdot \frac{V^2}{2g}$
• Net Head or Effective Head ($H$): The actual head available at the turbine inlet nozzle/entrance: $H = H_g - h_f$

Power Calculations

• Water Power (Hydraulic Power Input, $P_w$): $P_w = \gamma \cdot Q \cdot H = \rho \cdot g \cdot Q \cdot H$ Where $\rho = 1000\text{ kg/m}^3$, $g = 9.81\text{ m/s}^2$, $\gamma = 9.81\text{ kN/m}^3$, $Q$ is volumetric flow rate ($\text{m}^3/\text{s}$), and $H$ is net head ($\text{m}$).
• Turbine Shaft Power Output ($P_t$): $P_t = P_w \cdot \eta_h = \rho g Q H \eta_h$ Where $\eta_h$ is the hydraulic efficiency of the turbine.
• Electrical Generator Power Output ($P_e$): $P_e = P_w \cdot \eta_{overall} = \rho g Q H (\eta_h \cdot \eta_m \cdot \eta_g)$ Where $\eta_m$ is mechanical efficiency and $\eta_g$ is generator electrical efficiency.

Hydraulic Turbine Classifications

• Pelton Wheel: Uses high-velocity free jets impinging on double-hemispherical buckets. Needle valves regulate flow rate without altering jet velocity.
• Francis Turbine: A mixed-flow reaction turbine enclosed in a spiral volute casing with adjustable guide vanes (wicket gates) directing water inward onto runner blades.
• Kaplan Turbine: An axial-flow propeller turbine featuring automatically adjustable runner blades and guide vanes to maintain high efficiency across partial load operations.

Specific Speed ($N_s$)

Specific speed represents the speed in rpm at which a geometrically similar turbine runner would operate to produce unit power under unit net head: $N_s = \frac{N \sqrt{P}}{H^{5/4}}$ (Note: In SI/Metric units, $N$ is in rpm, $P$ is in kW, and $H$ is in meters. In US customary units, $P$ is in HP and $H$ is in feet).

Function of the Draft Tube

The draft tube is a gradually expanding divergent pipe connected between the reaction turbine runner exit and the tailrace. Its functions are:

1. Allows the turbine to be set above the tailrace level without losing net head.
2. Recovers a major portion of the kinetic energy ($\frac{V_2^2}{2g}$) of water exiting the runner by converting it into static pressure head before discharging into the tailrace.

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2. Geothermal Power Plants

Geothermal power plants convert hydrothermal energy stored within subterranean magma reservoirs into electricity. The Philippines is one of the world's top producers of geothermal power, with major fields located in Tiwi (Albay), Leyte, Mak-Ban (Laguna/Batangas), Tongonan, and Palinpinon.

Geothermal Power Conversion Cycles

1. Dry Steam Plants: Geothermal wells yield superheated or dry saturated steam ($> 150^\circ\text{C}$) containing no liquid water. Steam is piped directly to condensing steam turbines.
2. Single-Flash Steam Plants: Deep geothermal wells deliver high-pressure liquid-dominated brine. When pressure is reduced in a separator (flash vessel), a fraction flashes into saturated vapor: $x = \frac{h_{fluid,well} - h_{f,flash}}{h_{fg,flash}}$ Vapor powers the turbine, while liquid brine is directed to reinjection wells.
3. Double-Flash Steam Plants: High-pressure liquid discharge from the primary flash separator is flashed a second time at a lower pressure, generating additional low-pressure steam to feed a secondary turbine stage. This increases energy extraction efficiency by 15% to 20% over single-flash designs.
4. Binary Cycle (Organic Rankine Cycle - ORC): Used for lower temperature reservoirs ($90^\circ\text{C} \le T \le 150^\circ\text{C}$). Geothermal brine passes through a heat exchanger, evaporating an organic working fluid with a low boiling point (e.g., isobutane, n-pentane, R-134a) operating in a closed Rankine cycle.

Non-Condensable Gas (NCG) Removal and Reinjection

• NCG Extraction: Geothermal steam contains 1% to 5% by weight of non-condensable gases ($\text{CO}_2$, $\text{H}_2\text{S}$, $\text{NH}_3$, $\text{CH}_4$). If allowed to accumulate in the condenser, NCGs raise backpressure, drastically reducing turbine power output. Steam jet ejectors or liquid-ring vacuum pumps continuously purge NCGs from the condenser.
• Reinjection Wells: Spent geothermal brine and condensate are pumped back into the periphery of the deep geothermal reservoir to maintain reservoir hydraulic pressure, prevent land subsidence, and avoid surface disposal of toxic dissolved minerals (e.g., arsenic, boron, silica).

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3. Solar, Wind & Biomass Systems

Wind Power Aerodynamics & Betz Limit

The theoretical kinetic power available in an atmospheric wind stream crossing swept area $A$ at velocity $V$ is: $P_{wind} = \frac{1}{2} \cdot \rho_{air} \cdot A \cdot V^3$ Where $\rho_{air} \approx 1.225\text{ kg/m}^3$ at standard conditions ($15^\circ\text{C}, 101.325\text{ kPa}$).

• Betz Limit ($C_p$): According to 1-D momentum theory across an actuator disk, the maximum theoretical power coefficient that a wind turbine can extract from unconstrained wind is: $C_{p,max} = \frac{16}{27} \approx 59.3\%$
• Actual Extracted Power: $P_{elec} = \frac{1}{2} \rho_{air} A V^3 C_p \cdot \eta_{mech} \cdot \eta_{gen}$

Solar and Biomass Systems

• Solar Photovoltaics (PV): Efficiency $\eta = \frac{P_{max}}{G \cdot A_{panel}}$, where $G$ is solar irradiance (W/m$^2$). Concentrated Solar Power (CSP) utilizes parabolic troughs or central receiver towers to drive high-temperature Rankine steam cycles.
• Biomass Systems: Direct combustion of agricultural residues (bagasse in Philippine sugar mills, coconut shells, wood chips) in boiler furnaces to generate high-pressure steam.

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Worked Hydroelectric Plant Calculation

Problem: A hydroelectric plant in Mindanao operates under a gross elevation head of $H_g = 185\text{ m}$ with penstock friction head loss $h_f = 10\text{ m}$. The volumetric water flow rate is $Q = 15\text{ m}^3/\text{s}$. The turbine hydraulic efficiency is $\eta_h = 92%$, mechanical efficiency is $\eta_m = 96%$, and generator efficiency is $\eta_g = 97%$. Determine (a) net head, (b) total electrical power output, and (c) metric specific speed if the turbine rotates at $450\text{ rpm}$.

Step-by-Step Solution:

1. Calculate Net Head ($H$): $H = H_g - h_f = 185\text{ m} - 10\text{ m} = 175\text{ m}$

2. Calculate Water Power Input ($P_w$): $P_w = \rho g Q H = 1000\text{ kg/m}^3 \times 9.81\text{ m/s}^2 \times 15\text{ m}^3/\text{s} \times 175\text{ m} = 25,751,250\text{ W} = 25.751\text{ MW}$

3. Calculate Total Combined Efficiency ($\eta_{overall}$): $\eta_{overall} = \eta_h \times \eta_m \times \eta_g = 0.92 \times 0.96 \times 0.97 = 0.856685 \text{ (85.67\%)}$

4. Calculate Electrical Power Output ($P_e$): $P_e = P_w \times \eta_{overall} = 25.75125\text{ MW} \times 0.856685 = 22.061\text{ MW}$

5. Calculate Turbine Mechanical Shaft Power ($P_{kW}$): $P_{kW} = P_w \times \eta_h = 25751.25\text{ kW} \times 0.92 = 23,691.15\text{ kW}$

6. Calculate Metric Specific Speed ($N_s$): $N_s = \frac{N \sqrt{P_{kW}}}{H^{5/4}} = \frac{450 \times \sqrt{23691.15}}{(175)^{1.25}} = \frac{450 \times 153.9193}{638.6517} = 108.45\text{ rpm}\cdot\text{kW}^{1/2}/\text{m}^{5/4}$ Conclusion: Since $N_s = 108.45$ falls within the range $50 \le N_s \le 300$, a Francis reaction turbine is the correct selection.