💧 Hydro Power Plant: Fundamentals

"India's first hydro project: Darjeeling, 1897"
Water stored at high elevation flows through penstock → drives turbine + alternator → electrical energy.

Energy Conversion Chain:

$$ \text{Potential Energy }(mgh) \;\rightarrow\; \tfrac{1}{2}mv^2\;\text{(Kinetic)} \;\rightarrow\; \text{Electrical Energy} $$

Water stored at height → Flows through penstock → Drives turbine + alternator

Power in Hydroelectric Plants:

$$ \boxed{P = 9.81\times10^{-3}\times W\,Q\,H\,\eta \;\;\text{kW}} $$

$$ \boxed{P = \frac{0.736}{75}\,W\,Q\,H\,\eta \;\;\text{kW}} $$

$$ \boxed{P = \frac{WQH\eta}{75} \;\;\text{HP}} $$

Where:

• $W$ = Specific weight of water = $9.81\times10^3\,\text{N/m}^3 \approx 1000\,\text{kg/m}^3$
• $Q$ = Rate of flow (m³/s) • $H$ = Head / height of fall (m) • $\eta$ = Overall efficiency

Unit Conversions:

$$ 1\,\text{HP} = 0.746\,\text{kW} = 550\,\text{ft} \cdot \text{lb/s} $$

Factors Before Constructing:

1.Capital cost of plant
2.Capital cost of erecting & maintaining transmission line
3.Cost of energy generation should be minimum

Elements of Hydroelectric Plants:

🏔️ Storage Reservoir 🧱 Dam 🌊 Forebay ⚡ Alternator 🌐 Spillway 🚪 Intake 🔧 Valves & Gates 🗑️ Trash Racks 🏞️ Tail Race 📐 Draft Tubes 🔵 Penstock 🔮 Surge Tank 🌀 Water Turbine

⚡ Hydroelectric Power Plant

How moving water generates electricity — animated diagram

Water Flow

Surge Tank

Turbine (Left Bay)

Drive Shaft

Generator (Right Bay)

Transformer

Electricity

① Reservoir → ② Penstock → ③ Surge Tank → ④ Turbine ―→ ⑤ Generator → ⑥ Transformer → ⑦ Grid

🔄 Energy Conversion

PE (mgh)

KE (½mv²)

Mech

Elec ⚡

🧮 Hydro Power Calculator

Q (m³/s): 50 H (m): 100 η (%): 85

            P = — kW
          

🌊 Classification by Water Flow Regulation

Three Types Based on Pondage/Regulation:

1. Run-off River (No Pondage)

▸ Low capacity & seasonal
▸ Water taken directly from river
▸ No storage possible
▸ Used only when water available
▸ Mainly for irrigation

2. Run-off River (With Pondage)

▸ Pondage = storage at plant
▸ Copes with hourly–weekly fluctuations
▸ Base load OR peak load
▸ Depends on stream flow

3. Reservoir Plants

▸ Water stored in big reservoir
▸ Full control of water flow
▸ Base or peak load as needed
▸ Maximum flexibility

Classification by Water Head:

LOW Head ($H < 30\,\text{m}$)

Created near dam — no surge tank
Turbines: Francis, Propeller, Kaplan

MEDIUM Head ($30 < H < 300\,\text{m}$)

Forebay → penstock → turbine
Forebay acts as reservoir + surge tank
Turbines: Francis, Propeller, Kaplan

HIGH Head ($H > 300\,\text{m}$)

Steep valley dammed for storage
Surge tank before valve house
Reduces water hammer severity
Turbine: Pelton Wheel

Water Hammer Pressure:

$$ \boxed{\Delta P = \rho \cdot a \cdot \Delta v} $$

$a$ = wave velocity $= \sqrt{K/\rho}$, $K$ = bulk modulus of water

Surge tank reduces $\Delta P$ by absorbing pressure waves

🔄 Plant Type Comparison

Seasonal only — No storage

📊 Head Classification

Head H (m): 100

LOW
<30m

MED
30-300m

HIGH
>300m

            MEDIUM Head → Francis / Propeller / Kaplan
          

💥 Water Hammer Demo

ρ (kg/m³): a — wave speed (m/s): Δv (m/s):

⚡ Base Load vs Peak Load Plants

🔵 Base Load Plants

▪Take load on base portion of load curve

▪Usually large capacity

▪Work on nearly constant load

▪Operate at HIGH load factor

▪Unit cost of energy: LOW

▪Ex: Coal, Nuclear, Run-of-river

$$ LF_{base} = \frac{P_{avg}}{P_{max}} > 0.8 $$

🔴 Peak Load Plants

▪Supply peak — top of load curve

▪Smaller capacity, quick start-up

▪Run-off with pondage = peak load

▪Reservoir plants = peak load possible

▪Unit cost of energy: HIGH

▪Ex: Gas turbines, Pumped storage, Hydro

$$ LF_{peak} = \frac{P_{avg}}{P_{max}} < 0.3 $$

Full Comparison:

Parameter	Base Load	Peak Load
Load curve part	Bottom (constant)	Top (variable)
Load factor	High (>0.8)	Low (<0.3)
Capacity	Large	Smaller
Operation	Nearly continuous	Only during peaks
Unit cost	Low	High
Examples	Coal, Nuclear, Run-off	Gas, Pumped storage
Hydro type	Run-off (no pondage)	Pondage / Reservoir

Pumped Storage Plant:

Off-peak (night): Pump water UP to reservoir (motor mode — consumes cheap base load energy)
Peak (day): Release water DOWN through turbine (generator mode — produces expensive peak energy)

$$ \eta_{round} = \eta_{pump} \times \eta_{turbine} \approx 70\text{-}85\% $$

📈 24-Hour Load Duration Curve

🔄 Pumped Storage Cycle

📊 Load Factor Calculator

Avg Load (MW): Max Demand (MW):

🔄 Pumped Storage Efficiency

η_pump (%): η_turbine (%):

🌀 Water Turbine Classification

Classification by Type of Flow:

1. Axial flow — parallel to rotation axis

2. Inward radial flow — radially inward

3. Tangential/peripheral — tangent to runner

4. Mixed flow — radial inlet, axial outlet

Major Turbine Types:

⚙️ Pelton Wheel (Impulse)

Tangential/peripheral flow | High head (>300 m)
Water jets hit cup-shaped buckets. All pressure drop in nozzle (not runner).

$$ u_{opt} = \frac{v_{jet}}{2} \qquad \eta_{hyd} = \frac{2u(v_j-u)(1+\cos\beta)}{v_j^2} $$

🌀 Francis Turbine (Reaction)

Mixed flow (inward radial + axial) | Medium head (30-300 m)
Water enters radially, exits axially. Pressure drops across runner.

🔄 Kaplan Turbine (Reaction)

Axial flow (propeller type) | Low head (<30 m)
Adjustable runner blades. Best for large discharge, low head. High efficiency over wide range.

Specific Speed:

$$ \boxed{N_s = \frac{N\sqrt{P}}{H^{5/4}}} $$

$N_s$ = specific speed, $N$ = rpm, $P$ = power (HP), $H$ = net head (m)

Pelton: Ns = 10-50 Francis: Ns = 50-300 Kaplan: Ns = 300-900

Unit Quantities (Turbine Similarity):

Unit Speed

$$ N_u = \frac{N}{\sqrt{H}} $$

Unit Discharge

$$ Q_u = \frac{Q}{D^2\sqrt{H}} $$

Unit Power

$$ P_u = \frac{P}{D^2 H^{3/2}} $$

🌀 Turbine Comparison

📊 Head → Turbine Selector

Kaplan

Francis

Pelton

0m30m300m1000m

🔢 Specific Speed Calculator

N (rpm): P (HP): H (m):

📉 Pelton Efficiency vs u/vⱼ

📊 Hydro Plant Complete Reference

Formula Quick Reference (Click to flip):

💧Hydro Power (kW)

$P = 9.81 \times 10^{-3} WQH\eta$

🐴Hydro Power (HP)

$P = \frac{WQH\eta}{75}$

💥Water Hammer

$\Delta P = \rho \cdot a \cdot \Delta v$

⚙️Specific Speed

$N_s = \frac{N\sqrt{P}}{H^{5/4}}$

📊Load Factor

$LF = \frac{P_{avg}}{P_{max}}$

🔄Pumped Storage η

$\eta_r = \eta_p \times \eta_t$

⚡Pelton Optimal

$u_{opt} = v_{j}/2$

🌊Unit Speed

$N_u = \frac{N}{\sqrt{H}}$

Head Classification Summary:

Classification	Head Range	Turbine Type	Ns Range
Low Head	< 30 m	Kaplan / Francis	300 – 900
Medium Head	30 – 300 m	Francis	50 – 300
High Head	> 300 m	Pelton	10 – 50

Plant Type Summary:

Run-off

No Pondage — Seasonal

Pondage

Base or Peak Load

Reservoir

Full Control — Any Load

Pumped

Store & Release Energy

🧮 Master Hydro Calculator

Q (m³/s): H (m): η (%): N (rpm):

🧠 Plant Type Quiz

Click "Next Question" to start!

⚙️ Turbine Action Types & Full Comparison

Impulse: All pressure converted to KE in nozzle → jet drives wheel. Runner at atmospheric pressure.
Reaction: Water pressure + velocity both act on runner. Pressure drops across runner.

Impulse Turbine:

$$ \text{Entire pressure} \xrightarrow{\text{nozzle}} \text{Kinetic Energy (jet)} \rightarrow \text{Drives wheel} $$ $$ P_{\text{static, runner}} = 0 \quad \text{(atmospheric)} $$

Reaction Turbine:

$$ \text{Water pressure + velocity} \rightarrow \text{Combined action on runner} $$ $$ \text{Pressure drops across runner blades} $$

Four Turbine Comparison:

Pelton Wheel	Francis Turbine	Kaplan Turbine	Propeller
Impulse type	Reaction type, inward mixed flow	Reaction type, governing like Francis	Axial flow reaction, fixed blades
High head, low flow, tangential	Medium head & flow	Water strikes axially	Low head, large flow
Elliptical buckets on periphery	Velocity + pressure difference	High Ns → low head, large flow	η = 92% at full load only
Horizontal shaft mostly	η ≈ 92% full load; H or V	η ≈ 90% at all loads	Runner can reverse as pump
Not suitable H < 200 m	Higher speed than Pelton	Low cost runner & alternator	No blade adjustment while running
η_mech up to 90%	η ≈ 92%	η ≈ 90% (all loads)	η = 92% (full load)

Key Efficiencies:

$$ \eta_{Pelton,max} \approx 90\% \qquad \eta_{Francis,full\,load} \approx 92\% $$ $$ \eta_{Kaplan,all\,loads} \approx 90\% \qquad \eta_{Propeller,full\,load} = 92\% $$

⭐ Kaplan Advantage:

Efficiency stays ≈ 90% over WIDE load range (adjustable blades). Runner can reverse → pump operation.

⚙️ Four-Panel Turbine Comparison

🏔️ Pelton Wheel

🌀 Francis Turbine

🔄 Kaplan Turbine

⚡ Propeller Turbine

📊 Efficiency vs Load Comparison

💧 Impulse vs Reaction Pressure Drop

IMPULSE

Full drop in nozzle

100% → 0% at nozzle exit

REACTION

Gradual drop through runner

Gradual: nozzle → runner → exit

🏔️ Head Range Selector: 150 m

Francis Turbine — Ns: 50–300 — Medium Head

📍 Site Selection & Hydro Plant Merits

Selection of Site for Hydroelectric Plants:

1. Availability of water → Water storage

2. Water head → Distance from load centre

3. Effective transportation → Land availability

4. Water pollution can cause corrosion of plant

5. Large catchment area → Large reservoir area

✅ Merits of Hydro Power Plant:

1. No fuel required

2. High reliability, cheap operation

3. Fast run-up & synchronization

4. Load varies quickly & rapidly

5. Accurate governing: f = const

6. No stand-by losses

7. Robust, longer life

8. η not a function of age

9. Neat and clean

10. Also serves irrigation

❌ Demerits of Hydro Power Plant:

1. Requires larger area

2. Firm capacity is low

3. Construction cost is high

4. Long TL required (hilly areas)

5. Dry season affects supply

Hydro vs Thermal Quick Comparison:

💧 HYDRO

✦ No fuel, clean, long life

✦ High capital cost, remote

🔥 THERMAL

✦ Near load centre, controllable

✦ Fuel cost, pollution, ash

📊 Site Selection Radar Chart

Water
Head
Transport
Land
Catchment

Site Score: 70% — Good

⚖️ Merits vs Demerits Balance

MERITS (10)

⚖️

Green wins!

DEMERITS (5)

🕐 Plant Life Comparison

Hydro

50-100 yrs

Thermal

25-30 yrs

"Efficiency unchanged with age" — Hydro advantage

📏 Specific Speed & Types of Head

Specific Speed (Dimensionless Quantity):

$$ \boxed{N_s = \frac{N\sqrt{P_o}}{H_{net}^{5/4}}} $$

$N_s$ = Specific speed · $N$ = Actual speed (rpm) · $H_{net}$ = Net head (m) · $P_o$ = Output power (HP)

$$ \text{Speed of geometrically similar turbine under unit head, developing unit power} $$

Specific Speed Ranges:

10–50

Pelton

50–300

Francis

300–900

Kaplan

Types of Head:

Gross Head ($H_g$)

Total head available

Net Head ($H_{net}$)

Head used for power

Loss Head ($H_f$)

Friction losses

$$ \boxed{H_{net} = H_g - H_f} $$

Frictional Head Loss ($H_f$):

DARCY-WEISBACH

$$ \boxed{H_f = \frac{fLV^2}{2gD}} $$

FANNING'S EQUATION

$$ \boxed{H_f = \frac{4f'LV^2}{2gD}} $$

$f$ = Darcy friction factor · $f'$ = Fanning factor · $L$ = Penstock length (m)
$V$ = Water velocity (m/s) · $D$ = Penstock diameter (m) · $g$ = 9.81 m/s²

$$ f_{Darcy} = 4 \times f_{Fanning} $$

⚠️ Always check which friction factor is used!

Net Head for Power:

$$ P = \rho g Q H_{net} \eta $$ $$ H_{net} = H_g - \frac{fLV^2}{2gD} \qquad V = \frac{Q}{\pi D^2/4} $$

🏔️ Head Diagram

🧮 Darcy-Weisbach Hf Calculator

f (friction): L (m): V (m/s): D (m): H_g (m): g (m/s²):

⚡ Specific Speed N_s Calculator

N (rpm): P_o (HP): H_net (m):

🔄 Darcy vs Fanning Friction Factor

DARCY f

0.020

FANNING f'

0.005

f_Darcy = 4 × f_Fanning — Same H_f result!