Hydraulic Machines — Pumps & Turbines

1. Classification of Hydraulic Machines

1.1 Impulse vs Reaction Turbines

2. Euler’s Turbomachinery Equation

Euler’s turbomachinery equation is derived from the angular momentum theorem and gives the theoretical (ideal) work done by or on the fluid per unit mass in any rotodynamic machine.

2.1 Velocity Triangles

At any blade row, three velocities form a triangle:

3. Centrifugal Pump — Working, Heads & Efficiency

3.1 Working Principle

A centrifugal pump consists of a rotating impeller inside a casing (volute). Fluid enters axially at the eye (centre) of the impeller and is thrown radially outward by centrifugal action. The kinetic energy imparted by the impeller is converted to pressure energy in the volute casing (which acts as a diffuser).

3.2 Head Definitions

3.3 Pump Efficiency

3.4 Priming

A centrifugal pump must be primed (filled with liquid) before starting because it cannot develop sufficient suction to lift liquid from a dry state — it can only pressurize liquid already present. Self-priming pumps have special features to handle this automatically.

4. Pump Characteristic Curves & System Curve

4.1 Pump Characteristic Curves

The pump manufacturer provides characteristic curves at a fixed speed N, showing how H, efficiency η, and power P vary with discharge Q:

• H–Q curve: Head decreases as discharge increases (centrifugal pump). At Q = 0: H = H₀ (shut-off head). As Q increases, H decreases.
• η–Q curve: Efficiency peaks at the best efficiency point (BEP) and falls on both sides.
• P–Q curve: Power increases with Q for centrifugal pumps — motor must be sized for maximum expected flow.

4.2 System Curve

The system curve represents the head required to push flow through the pipe network at various flow rates:

5. Pumps in Series and Parallel

6. Reciprocating Pump

A reciprocating pump uses a piston or plunger moving back and forth in a cylinder to displace fluid. Unlike centrifugal pumps, it is a positive displacement pump — it delivers a fixed volume per cycle regardless of the head.

6.1 Air Vessel

An air vessel is a closed chamber connected to the suction or delivery pipe of a reciprocating pump. It acts as an accumulator — absorbing the pulsating flow from the pump and delivering a steady stream to the pipe. Benefits: reduces acceleration head, allows higher pump speeds, prevents separation in suction pipe, and saves power.

7. Specific Speed — Pumps & Turbines

Specific speed is a dimensionless (or dimensional, in common practice) parameter that characterises the geometry and type of a hydraulic machine operating at its best efficiency point. It allows comparison and selection of machine types independently of size.

7.1 Specific Speed for Pumps

7.2 Specific Speed for Turbines

7.3 Physical Meaning of Specific Speed

Specific speed is the speed at which a geometrically similar turbine (or pump) of unit size would run while developing unit power (or delivering unit discharge) under unit head. It is the same for all geometrically similar machines regardless of size — it characterises the shape (geometry) of the runner.

8. Affinity Laws (Similarity Laws) for Pumps

The affinity laws relate the performance of a pump at different speeds (or impeller sizes) using dimensional analysis and dynamic similarity.

These laws allow manufacturers to predict performance at different operating speeds without testing at every speed, and allow engineers to scale model test results to prototype dimensions.

9. Pelton Wheel (Impulse Turbine)

The Pelton wheel is an impulse turbine in which one or more high-velocity water jets strike double-cupped (bucket-shaped) vanes attached to the rim of a wheel. All the available head is converted to kinetic energy in the jet before striking the buckets — there is no pressure change across the runner.

9.1 Velocity Triangle Analysis

9.2 Design Features

• Jet deflector: Deflects jet away from buckets during load rejection to prevent overspeed.
• Spear (needle) valve: Regulates jet area (and hence flow) smoothly — flow control without surge.
• Number of jets: Single jet standard; multi-jet (up to 6) for high power — jets equally spaced around the wheel.
• Jet ratio (m): m = D/(d_jet) — typically m = 12 to 25. Higher m → lower N_s → more suitable for higher heads.

10. Francis Turbine (Reaction Turbine)

The Francis turbine is an inward-flow reaction turbine — water enters radially inward through adjustable guide vanes (wicket gates) and exits axially. Both pressure energy and kinetic energy decrease across the runner. The turbine is enclosed in a spiral casing (scroll casing) that maintains uniform velocity distribution around the runner periphery.

10.1 Draft Tube

A draft tube is a diverging conduit connecting the turbine runner exit to the tailwater. It serves two functions:

• Recovers kinetic energy: The expanding tube decelerates the exit flow, recovering velocity head as pressure head — this increases the effective net head across the turbine.
• Allows runner installation above tailwater: The draft tube creates a sub-atmospheric pressure at the runner exit (suction effect), allowing the turbine to be set above the tailwater level without losing the head between runner exit and tailwater.

11. Kaplan Turbine (Propeller Turbine)

The Kaplan turbine is an axial-flow reaction turbine with adjustable runner blades (like a ship’s propeller). Water flows axially through the runner. The blade angle can be adjusted as guide vane opening changes, maintaining high efficiency over a wide range of flows — making it ideal for run-of-river plants with highly variable flow.

12. Turbine Comparison & Selection

13. Cavitation & NPSH

13.1 Cavitation Mechanism

Cavitation occurs when the local pressure in a flowing liquid drops to or below the vapour pressure p_v at that temperature. Vapour bubbles form and then collapse violently when they move into higher-pressure regions, generating intense local shock waves, noise, vibration, surface pitting (erosion), and loss of efficiency.

13.2 Thoma’s Cavitation Number (σ)

13.3 Net Positive Suction Head (NPSH) for Pumps

13.4 Effects of Altitude and Temperature on NPSH

• Higher altitude: p_atm decreases → NPSH_available decreases → maximum suction lift decreases.
• Higher water temperature: p_v increases → NPSH_available decreases → pump must be placed closer to (or below) the sump.

14. Worked Examples (GATE CE Level)

Example 1 — Centrifugal Pump Power and Efficiency (GATE CE type)

Problem: A centrifugal pump delivers 0.06 m³/s of water against a total head of 25 m. The shaft power input is 18 kW. Find (a) the overall efficiency and (b) the power delivered to water.

Answer: P_water = 14.72 kW; η = 81.75%

Example 2 — Pelton Wheel Efficiency (GATE CE 2021 type)

Problem: A Pelton wheel has a jet velocity V₁ = 45 m/s and bucket velocity u = 20 m/s. The side leakage angle β₂ (deflection from straight) is 165° so the exit angle for the relative velocity is β = 15° from horizontal. Friction factor k = 0.9. Find the hydraulic efficiency.

Answer: Hydraulic efficiency η_h = 92.3%

Example 3 — Specific Speed of a Turbine (GATE CE type)

Problem: A turbine runs at 300 rpm under a head of 80 m and produces 5000 kW. Find the specific speed and identify the turbine type.

Answer: N_s = 88.7 → Francis turbine

Example 4 — Affinity Laws: Speed Change (GATE CE type)

Problem: A centrifugal pump operates at 1200 rpm delivering 0.05 m³/s against a head of 30 m, consuming 18 kW. If the speed is increased to 1500 rpm (same impeller), find the new discharge, head, and power.

Answer: Q₂ = 62.5 L/s; H₂ = 46.9 m; P₂ = 35.2 kW

Example 5 — NPSH and Maximum Suction Head

Problem: A centrifugal pump is to be installed to lift water at 25°C (p_v = 3.17 kPa absolute) from a sump. Atmospheric pressure = 101.3 kPa. NPSH_required = 4.0 m. Head loss in suction pipe = 0.5 m. Find the maximum permissible suction head.

Answer: Maximum suction head = 5.5 m. Pump must be installed within 5.5 m above the sump water surface.

Example 6 — Reciprocating Pump Discharge (GATE CE type)

Problem: A double-acting reciprocating pump has a piston diameter of 200 mm and a piston rod diameter of 40 mm. The stroke length is 300 mm and the pump speed is 60 rpm. Volumetric efficiency = 95%. Find the actual discharge.

Answer: Q_act = 17.55 litres/s

15. Common Mistakes

Mistake 1 — Confusing Specific Speed Formulas for Pumps vs Turbines

Error: Using N_s = NQ^1/2/H^3/4 (pump formula) for a turbine, or N_s = N√P/H^5/4 (turbine formula) for a pump.
Root Cause: Pumps are characterised by discharge Q (output), turbines by power P (output) — different quantities are used because they represent different performance outputs.
Fix: Pump specific speed uses Q (what the pump delivers). Turbine specific speed uses P (what the turbine produces). The H exponent differs (3/4 for pump, 5/4 for turbine). Always identify the machine type first.

Mistake 2 — Using the Affinity Law H ∝ N Instead of H ∝ N²

Error: Writing H₂ = H₁ × (N₂/N₁) instead of H₂ = H₁ × (N₂/N₁)².
Root Cause: Mixing up the three affinity laws. Q scales linearly with N, but H scales with N² and P scales with N³.
Fix: Memorise as a sequence: Q ∝ N¹; H ∝ N²; P ∝ N³. The exponent increases by one each time: 1, 2, 3.

Mistake 3 — Forgetting That u = V₁/2 Gives Maximum Pelton Wheel Efficiency

Error: Assuming the Pelton wheel runs at its most efficient when bucket speed equals jet speed (u = V₁).
Root Cause: At u = V₁, the relative velocity is zero — no work is done at all. Maximum efficiency occurs when u = V₁/2 (half jet speed) — this is derived by differentiating the hydraulic efficiency expression with respect to u.
Fix: Optimal condition: u/V₁ = 0.5 → u = V₁/2. At this condition, η_h,max = (1 + k cosβ)/2.

Mistake 4 — Including the Draft Tube in the Net Head Calculation Incorrectly

Error: Not accounting for draft tube suction head H_s in the net head for a Francis turbine, computing H_net = headwater level – tailwater level without the draft tube correction.
Root Cause: The draft tube allows the runner to be set above tailwater while still utilising the full gross head. The net head = gross head + draft tube recovery – draft tube losses. For preliminary calculations, H_net ≈ gross head is acceptable, but detailed design must account for draft tube efficiency.
Fix: For GATE problems, use H = gross head (difference between headwater and tailwater levels) unless the problem specifically asks about draft tube effects. When turbine setting height H_s is given, check cavitation using Thoma’s σ.

Mistake 5 — Applying NPSH Incorrectly When Pump is Below Sump

Error: Using a positive H_s in the NPSH formula when the pump is installed below the sump (flooded suction), leading to an artificially reduced NPSH_available.
Root Cause: Sign convention for H_s: H_s is positive when the pump is above the sump (suction lift) and negative when the pump is below the sump (flooded suction — reduces cavitation risk). Many students always use a positive H_s.
Fix: NPSH_available = H_atm – H_v – H_s – h_f,s. If pump is below sump: H_s is negative → NPSH_available increases (flooded suction improves cavitation safety). If pump is above sump: H_s is positive → NPSH_available decreases.

16. Frequently Asked Questions

Q1. Why does a centrifugal pump need priming but a reciprocating pump does not?

A centrifugal pump develops head by centrifugal action — it imparts angular momentum to the fluid. For this to work, the impeller must be surrounded by a liquid (not air), because the density difference between air and water means the centrifugal pump cannot generate enough pressure difference to push air out of the suction pipe and draw water up from the sump. If the pump runs dry (air-filled), it develops negligible head and no suction is created. In contrast, a reciprocating pump is a positive displacement device — the piston physically compresses the fluid in the cylinder and forces it out regardless of its density. Even with air in the cylinder, each stroke displaces a volume and creates pressure, so reciprocating pumps are self-priming by nature. They can pump air initially and gradually evacuate the suction line, drawing water up. Self-priming centrifugal pumps incorporate a recirculation system that retains a reservoir of liquid to prime the impeller automatically before operation begins.

Q2. What determines the choice between a Francis and a Kaplan turbine for a low-head hydropower project?

The primary factor is specific speed N_s — which itself is determined by the head and desired speed. For heads below about 40–50 m, Kaplan turbines have higher specific speed and higher efficiency at part load than Francis turbines, making them the preferred choice for run-of-river plants where flow varies significantly throughout the year. The key advantage of the Kaplan turbine is its double regulation: both the guide vanes (wicket gates) and the runner blade angle are adjusted simultaneously, maintaining near-peak efficiency even when operating at 50–60% of design flow. A Francis turbine has fixed runner geometry — its efficiency drops sharply at part load. In India, run-of-river projects on Himalayan tributaries (where monsoon and dry-season flows differ by a factor of 5 to 10) typically use Kaplan turbines for this reason. Francis turbines dominate storage-type hydro schemes (dams with reservoirs) where flow can be regulated to keep the turbine near its design point.

Q3. How does the concept of specific speed relate to geometrically similar turbine families?

Specific speed N_s = N√P/H^5/4 is the same for all geometrically similar turbines (those with the same shape of flow passages, blade angles, and proportions) regardless of their actual size — it is a shape parameter, not a size parameter. This means that all Pelton wheels with single jets belong to a narrow band of N_s values (4–30), all Francis turbines to their band (51–255), and all Kaplan turbines to their band (255–860), regardless of whether they are 1 MW or 500 MW machines. The physical reason is that N_s is derived from dimensional analysis of the governing physics (flow rate, speed, head, and power) — once the N_s value is fixed, the ratios of all key dimensions (impeller diameter to setting diameter, blade angles, volute shape) are determined by geometric similarity. This is why turbine manufacturers use model tests at convenient small scales (typically 1:5 to 1:20) and scale up results using dimensional similarity — the efficiency and flow characteristics of the prototype are predicted from the model’s N_s and appropriate scale correction formulas.

Q4. What is the engineering significance of the Thoma cavitation number σ in turbine design?

Thoma’s cavitation number σ = (H_atm – H_s – H_v)/H quantifies the margin of safety against cavitation in a turbine installation. The numerator represents the available pressure head above vapour pressure at the runner, and the denominator normalises this by the total operating head. The critical value σ_c is determined from model tests for each turbine design — if the installed σ falls below σ_c, cavitation begins on the runner blades, causing pitting and vibration. Since σ increases as H_s decreases (runner set lower relative to tailwater), the Thoma number dictates the maximum permissible setting height of the turbine above tailwater. For a given turbine design and head H, the setting height limit is H_s ≤ H_atm – H_v – σ_cH. In practice, high-head Francis turbines have low σ_c and can be set well above tailwater, while low-head Kaplan turbines have high σ_c and are typically set below tailwater level (H_s negative, requiring a deeper powerhouse excavation) to prevent cavitation.