Turbomachinery Principles

Introduction

Turbomachines transfer energy continuously between a flowing fluid and a rotating or stationary blade system. Pumps and compressors add energy to fluids, whereas hydraulic turbines and gas turbines extract energy from fluids. Fans, blowers, wind turbines, turbochargers, propellers, and aircraft engines are further examples.

Unlike positive-displacement machines, turbomachines operate through the dynamic interaction between fluid velocity and moving blades. Their analysis therefore depends on angular momentum, velocity triangles, blade geometry, losses, and similarity laws. These principles allow engineers to predict torque, power, head, efficiency, operating range, and scale effects.

Learning Objectives

By the end of this chapter, the learner should be able to:

• apply the angular-momentum equation to turbomachines;
• derive and use Euler’s turbomachinery equation;
• construct and interpret velocity triangles;
• determine blade forces, torque, power, and hydraulic head;
• explain degree of reaction and its importance;
• calculate specific speed and apply similarity laws;
• interpret efficiency definitions and characteristic curves;
• identify major loss mechanisms and practical design limitations.

1. Classification and Fundamental Terminology

A turbomachine may be classified in several ways:

The rotor is the rotating blade assembly. The stator is stationary and may accelerate, decelerate, or redirect the flow. The runner is the rotor of a hydraulic turbine. The impeller is the rotating element of a pump or compressor.

The absolute fluid velocity is V, the blade velocity is U, and the velocity relative to the blade is W. At radius r and rotational speed ω,

where D is diameter in metres and N is rotational speed in revolutions per minute. The velocity triangle is the vector relationship

For a machine with predominantly axial flow, V_f denotes the axial or flow component and V_w denotes the tangential, whirl, or circumferential component. The blade angle β is usually measured from the tangential direction for rotor-relative velocity, while α is the absolute-flow angle measured from the tangential direction.

2. Angular-Momentum Theory

Consider steady flow through a rotor. The moment of momentum equation about the shaft axis gives the torque exerted by the rotor on the fluid:

Here T is torque on the fluid in N·m, ṁ is mass flow rate in kg/s, r is radius in m, and V_w is whirl velocity in m/s. Subscripts 1 and 2 denote rotor inlet and outlet. The sign convention must be stated: positive whirl is taken in the direction of rotor rotation.

The shaft power transferred to the fluid is

Thus, the specific work is

This is Euler’s turbomachinery equation in its most general rotor form. It applies to steady, one-dimensional mean-flow analysis when changes in radius and velocity distribution are represented by mean values. It assumes that shaft work is the principal energy transfer and neglects heat transfer and changes in potential energy. Losses do not invalidate the equation; they affect the actual fluid state and efficiency.

For a pump or compressor, positive w represents work supplied to the fluid. For a turbine, the sign is interpreted in the opposite direction, and the work extracted from the fluid is commonly written as

The corresponding ideal Euler head for a pump is

where H_E is in metres of fluid and g is gravitational acceleration, approximately 9.81 m/s².

Physical interpretation

Energy transfer requires a change in angular momentum. If the fluid enters and leaves a rotor with no change in rV_w, the ideal torque is zero. A pump impeller normally gives the fluid positive whirl. A turbine rotor removes whirl from the fluid, producing shaft torque.

3. Velocity Triangles

Velocity triangles are graphical or analytical representations of the velocity relationship at rotor inlet and outlet. For a rotor rotating at U, the absolute velocity V is resolved into a whirl component V_w and a flow component V_f. The relative velocity W is the velocity observed by an observer rotating with the blade.

For the common convention in which α and β are measured from the tangential direction,

and

Similarly, for relative velocity,

Because the blade has no velocity in the flow direction, the flow component of W equals that of V for a simple axial-flow analysis. The triangle is constructed from

although the equivalent vector equation V = U + W is usually easier to interpret.

For an axial rotor with constant blade speed U and constant flow velocity V_f, the inlet and outlet whirl velocities determine the work. If V_{w1}=0, a frequent pump approximation, then

For radial machines, U₁ and U₂ differ because blade speed changes with radius. Inward-flow turbines may have U₂ < U₁, while centrifugal pumps generally have U₂ > U₁.

Blade-speed and flow relations

For a rotor of diameter D and speed N,

The volume flow rate is

where A is the flow area. For an annular axial passage,

approximately, where D_m is mean diameter and b is blade height. For a radial impeller, the ideal outlet area is approximately

before accounting for blade thickness and blockage.

4. Blade Force, Torque, and Power

The tangential force exerted on the fluid is related to torque by

and

More directly, the shaft torque is

The axial force is associated with the change in axial momentum:

The force on the blades is equal and opposite to the force exerted by the blades on the fluid. Bearing and casing loads depend on both tangential and axial forces. Accurate structural design must also consider pressure forces, unsteady blade-passing loads, vibration, and hydraulic thrust.

5. Euler Equation and Energy Conversion

The steady-flow energy equation for a rotor, neglecting heat transfer and potential-energy change, may be written as

where h₀ is stagnation enthalpy. For an incompressible pump, this ideal specific work corresponds to Euler head. For a compressible compressor, the enthalpy rise is more appropriate than a constant-density head.

Euler’s equation is an ideal energy-transfer relation. It does not by itself give the actual pressure rise. Slip, incidence, friction, leakage, mixing, and mechanical losses reduce performance. In a real pump,

where H_p is the head actually imparted to the fluid and η_h is hydraulic efficiency. Definitions vary slightly by manufacturer, so the reference stations must always be specified.

Worked Example 1: Euler head and power

A centrifugal pump impeller has inlet diameter 0.16 m and outlet diameter 0.32 m and rotates at 1,450 rpm. The outlet whirl velocity is 18 m/s, and inlet whirl is negligible. The outlet flow rate is 0.12 m³/s, and water density is 1,000 kg/m³. Determine the ideal Euler head and ideal power transfer.

First,

The ideal specific work is

Thus,

The mass flow rate is

Therefore, ideal power transferred to the water is

The result assumes uniform mean velocities, no slip, and negligible inlet whirl.

6. Degree of Reaction

The degree of reaction indicates how much of the static enthalpy or pressure change in a stage occurs in the rotor. It is defined generally as

For a turbine stage,

with signs written according to the selected station convention. In a pump or compressor, the corresponding definition uses the static enthalpy rise in the rotor divided by the total stage static enthalpy rise.

An impulse stage has approximately R = 0: the principal pressure drop occurs in stationary nozzles, and rotor passages mainly turn the flow. A 50-percent-reaction axial turbine stage has approximately equal static enthalpy drops in stator and rotor. Reaction stages require careful control of blade-passage pressure gradients and usually have greater sensitivity to tip leakage and sealing.

For an ideal axial turbine with constant blade speed and constant axial velocity, an often-used velocity-triangle relation is

This special formula depends on the stated axial-flow assumptions and angle convention; it should not be applied indiscriminately to radial machines or nonuniform-flow stages.

7. Impulse and Reaction Machines

In an impulse turbine, fluid pressure falls mainly in a nozzle or stator. The rotor receives a high-velocity jet and extracts momentum. The ideal rotor work depends strongly on the change in whirl velocity. In a reaction turbine, pressure changes occur in both stator and rotor, and the rotor passages act partly as nozzles.

The distinction is useful but real machines may have intermediate behavior. A centrifugal pump impeller is a reaction device because pressure rises while fluid passes through the rotating passages. A Pelton wheel is primarily an impulse turbine, whereas Francis and Kaplan turbines are reaction turbines.

8. Specific Speed and Machine Selection

Specific speed is a dimensionless or convention-dependent index used to compare geometrically similar machines and guide preliminary selection. For pumps, a common SI form is

where N is in revolutions per second, Q is in m³/s, and H is in m. If N is entered in rpm, the numerical value belongs to a different convention and must not be compared with the SI form without conversion.

For turbines, one common unit-dependent form is

where P is power and H is effective head. Because turbine specific-speed definitions vary substantially among countries and manufacturers, the units and definition must always accompany the value.

Low pump specific speed generally corresponds to radial-flow impellers, medium values to mixed-flow machines, and high values to axial-flow machines. For turbines, low specific speed is associated broadly with Pelton-type machines, intermediate values with Francis machines, and high values with Kaplan or propeller turbines. These are selection trends, not rigid boundaries.

9. Similarity Laws

Geometrically similar turbomachines are compared using dimensionless groups. The principal pump affinity laws, for fixed fluid and approximately constant efficiency, are

Thus, between two similar operating conditions,

These relations assume geometric similarity, dynamically similar flow, comparable roughness and Reynolds-number effects, and operation in corresponding regions of the characteristic curve. They are not exact when viscosity, cavitation, leakage, or efficiency changes become important.

For constant diameter, doubling speed ideally doubles flow, increases head fourfold, and increases power eightfold. This explains why overspeed can rapidly overload a motor and cause excessive pressure or cavitation.

10. Efficiencies

A turbomachine’s overall performance is represented by several efficiencies.

Hydraulic or fluid efficiency compares useful fluid energy change with Euler energy transfer. For a pump,

Mechanical efficiency compares shaft power transferred to the fluid or extracted from the fluid with the power at the shaft, depending on machine type.

Volumetric efficiency accounts for leakage. For a pump it may be expressed approximately as useful discharge divided by impeller circulation flow, although exact definitions vary.

Overall efficiency is useful output divided by input. For a pump,

where ρ is fluid density, Q is discharge, H is developed head, and P_shaft is shaft input power. The numerator is hydraulic power in watts.

For a turbine,

when the available fluid power is represented by ρgQH. If efficiencies are separated,

provided the component definitions are mutually consistent.

11. Characteristic Curves and Operating Point

A pump characteristic curve plots head H, efficiency η, and often power P against discharge Q at fixed speed. System resistance is commonly represented by

where H_static is the static elevation or pressure requirement and K represents combined flow resistance under a consistent unit system.

The operating point is the intersection of the pump curve and system curve. Increasing system resistance moves the operating point toward lower flow and often higher head on a centrifugal-pump curve. Throttling changes the system curve; changing speed shifts the pump curve according to affinity laws. The best-efficiency point, or BEP, is the region of lowest hydraulic and mechanical stress for sustained operation.

For turbines, characteristic curves commonly relate power, efficiency, and discharge to head and speed. Governing systems alter flow or nozzle opening to match load while maintaining acceptable speed and efficiency.

12. Loss Mechanisms

The principal losses are:

1. Blade friction: boundary-layer shear on blade surfaces and end walls.
2. Incidence loss: shock or separation caused by mismatch between incoming flow angle and blade inlet angle.
3. Passage loss: secondary flow, separation, and turbulent mixing inside blade channels.
4. Tip-clearance leakage: fluid bypasses blade tips, reducing useful circulation and generating vortices.
5. Disc and windage loss: rotating surfaces shear the surrounding fluid.
6. Volute, diffuser, and casing loss: excessive diffusion or abrupt area changes cause separation.
7. Recirculation loss: operation far from the design flow produces inlet or outlet recirculation.
8. Mechanical loss: bearings, seals, couplings, and lubrication systems consume power.
9. Cavitation: local pressure falls below vapour pressure, producing vapour cavities and possible erosion.

Loss reduction requires suitable blade angles, controlled diffusion, adequate surface finish, tight but safe clearances, proper alignment, sufficient inlet pressure, and operation near the design region.

Cavitation criterion

The available net positive suction head is

The pump requires an NPSH value supplied by the manufacturer. Safe operation requires NPSH_A to exceed NPSH_R by an appropriate margin. NPSH values are heads of the pumped liquid and must be calculated using consistent absolute-pressure and density data.

13. Worked Example 2: Similarity and power

A geometrically similar pump operates at 1,200 rpm with Q = 0.08 m³/s, H = 24 m, and shaft power 25 kW. It is operated at 1,500 rpm with the same impeller diameter and approximately unchanged efficiency. Estimate the new performance.

The speed ratio is 1,500/1,200 = 1.25. Therefore,

The large increase in power demonstrates why speed changes must be checked against motor capacity, pressure limits, vibration, and cavitation margin. The calculation is an ideal similarity estimate, not a substitute for a manufacturer’s revised performance curve.

14. Applied Case: Selection and Diagnosis of a Cooling-Water Pump

Background

An industrial machine shop uses a centrifugal pump to circulate cooling water through heat exchangers. The original duty is 0.10 m³/s at 32 m total head. The motor consumes 45 kW, and the pump operates near its best-efficiency point.

Problem

Production is increased, and the required flow becomes 0.12 m³/s. Operators partially close a discharge valve, but the pump develops vibration, the motor current fluctuates, and the measured flow remains below target.

Analysis

The valve increases system resistance and moves the operating point away from the original design point. The resulting lower flow may cause discharge recirculation, while the increased suction losses reduce NPSH_A. The symptoms are consistent with operation outside the preferred range and possible cavitation or hydraulic instability.

A preliminary speed estimate using the constant-diameter affinity relation gives

The corresponding ideal head would become

or approximately 46.1 m, and ideal power would increase by

This increase is too large for the existing 45-kW motor if efficiency remains unchanged. The affinity estimate therefore indicates that simply increasing speed is not an acceptable solution.

Approach and solution

Engineers first measure suction pressure, discharge pressure, flow, speed, vibration, and motor power. They verify the system curve, clean the suction strainer, inspect impeller wear, and compare NPSH_A with NPSH_R. A larger pump or a parallel pump is then evaluated using a system-level life-cycle analysis. The selected solution must satisfy the required duty near its BEP, maintain adequate cavitation margin, and remain within motor and piping limits.

Outcome

A parallel pump is selected for peak production, while the original pump serves normal demand. Variable-speed control is used within a restricted range for intermediate loads. The pumps operate closer to their efficient regions, vibration falls, and the system retains redundancy.

Learning points

• A valve can change the operating point but cannot create pump head.
• Affinity laws are powerful screening tools but not complete design methods.
• Motor power, NPSH, vibration, and efficiency must be assessed together.
• Operation near BEP generally improves reliability and energy performance.

15. Design Practice, Safety, and Contemporary Developments

Good turbomachine design begins with a clear duty specification: fluid properties, flow range, pressure or head, temperature, speed, transient conditions, allowable noise, and maintenance requirements. Engineers should use manufacturer curves based on the correct fluid and speed, specify measurement uncertainty, and validate field performance.

Safety issues include rotating guards, pressure containment, overspeed protection, hot or toxic fluids, electrical isolation, and safe lockout procedures. Cavitation damage and fatigue can create sudden failures, so condition monitoring is part of responsible operation.

Modern machines increasingly use computational fluid dynamics, three-dimensional blade optimization, laser and pressure measurements, digital twins, and condition-monitoring algorithms. Artificial intelligence can detect deviations in vibration, power, and pressure signals, but it does not replace physical understanding or calibrated sensors. Models should be validated against measured data, and automated decisions should include alarm limits and human oversight.

Environmental performance is also important. Improving efficiency reduces electrical consumption, greenhouse-gas emissions, and operating cost. Material selection must consider erosion, corrosion, recyclability, and the consequences of leakage. Ethical engineering requires conservative treatment of uncertainty rather than reporting an ideal calculated performance as guaranteed field performance.

Chapter Summary

Turbomachinery transfers energy through changes in fluid angular momentum. The central relation is Euler’s equation, w = U₂V_w2 − U₁V_w1, which links rotor speed, whirl velocity, torque, and specific work. Velocity triangles resolve absolute, relative, flow, and whirl velocities and provide the basis for blade-angle and performance analysis. Blade forces follow from linear and angular momentum equations.

Degree of reaction describes the proportion of stage static enthalpy change occurring in the rotor and distinguishes impulse-like from reaction behavior. Specific speed assists preliminary machine selection, while affinity laws permit estimates of flow, head, and power for similar machines. Actual performance is reduced by hydraulic, volumetric, mechanical, leakage, incidence, friction, and cavitation losses. Characteristic curves and system curves determine the operating point, and reliable design requires attention to BEP, NPSH, motor capacity, vibration, safety, and validated performance data.

Key Terms

Angular momentum: Moment of linear momentum about the machine axis.

BEP: Best-efficiency point at which a machine operates with maximum or near-maximum efficiency.

Blade speed: Tangential speed of a rotor blade, U = ωr.

Cavitation: Formation and collapse of vapour cavities when local pressure falls below vapour pressure.

Degree of reaction: Fraction of stage static enthalpy change occurring in the rotor.

Euler head: Ideal head predicted from the change in angular momentum.

Incidence loss: Loss caused by mismatch between incoming flow direction and blade inlet angle.

Impulse machine: Machine in which most pressure change occurs in stationary nozzles or diffusers.

NPSH: Net positive suction head, a measure of suction pressure margin above vapour pressure.

Reaction machine: Machine in which pressure changes occur significantly within the rotor.

Rotor: Rotating blade assembly.

Specific speed: Similarity index used to compare machine type and operating characteristics.

Slip: Difference between ideal and actual fluid exit whirl caused mainly by finite blade number and flow effects.

Stator: Stationary blade or nozzle row.

System curve: Relationship between required head and flow for a piping system.

Tip-clearance loss: Loss caused by fluid leakage across the gap between blade tip and casing.

Velocity triangle: Vector representation of absolute, blade, and relative velocities.

Volumetric efficiency: Measure accounting for leakage or discrepancy between circulation flow and useful flow.

Whirl velocity: Tangential component of absolute fluid velocity.

Review Questions

Short Answer Questions

1. State Euler’s turbomachinery equation and explain the meaning of each term.
2. What is the physical significance of whirl velocity?
3. Distinguish between an impulse stage and a reaction stage.
4. Define specific speed and state one use in machine selection.
5. List four important loss mechanisms in turbomachines.

Long Answer Questions

1. Derive the rotor torque equation from the angular-momentum principle and relate it to shaft power.
2. Explain how inlet and outlet velocity triangles are constructed for an axial-flow rotor.
3. Discuss degree of reaction, including impulse and 50-percent-reaction stages.
4. Explain pump affinity laws and their assumptions and limitations.
5. Describe how pump and system characteristic curves determine the operating point.

Analytical / Application Questions

1. A pump operates at 1,450 rpm with Q = 0.06 m³/s and H = 20 m. Estimate Q, H, and relative power at 1,740 rpm for the same diameter, assuming similarity and unchanged efficiency.
2. A rotor has ṁ = 12 kg/s, r₁ = 0.20 m, r₂ = 0.30 m, V_{w1} = 4 m/s, and V_{w2} = 28 m/s. Calculate the torque using the angular-momentum equation and state the sign interpretation.
3. A centrifugal pump shows rising vibration, reduced flow, and noisy operation after a suction-side valve is partly closed. Analyse the likely mechanisms and recommend measurements and corrective actions.

Multiple Choice Questions with Answers

Question 1

Which equation gives the ideal specific work transferred by a rotor?

A. w = U₂V_w2 − U₁V_w1

B. w = ρgQH

C. w = Q/A

D. w = U/V

Correct Answer: A. w = U₂V_w2 − U₁V_w1

Question 2

A pump impeller has outlet diameter 0.32 m and speed 1,450 rpm. Its outlet blade speed is approximately:

A. 7.73 m/s

B. 24.29 m/s

C. 45.57 m/s

D. 96.00 m/s

Correct Answer: B. 24.29 m/s

Question 3

If a similar pump operates at 1.25 times its original speed with unchanged diameter, the ideal head ratio is:

A. 1.25

B. 1.50

C. 1.5625

D. 1.9531

Correct Answer: C. 1.5625

Question 4

In an impulse turbine stage, the principal pressure drop occurs in the:

A. Rotor bearings

B. Stationary nozzle or stator

C. Shaft coupling

D. Tailwater channel

Correct Answer: B. Stationary nozzle or stator

Question 5

The tangential component of absolute velocity is called:

A. Flow velocity

B. Relative velocity

C. Whirl velocity

D. Slip velocity

Correct Answer: C. Whirl velocity

Question 6

For a pump, overall efficiency is commonly expressed as:

A. Shaft power divided by hydraulic power

B. Hydraulic power divided by shaft power

C. Head divided by discharge only

D. Torque divided by speed only

Correct Answer: B. Hydraulic power divided by shaft power

Question 7

The operating point of a centrifugal pump is found from the intersection of the:

A. Blade and shaft curves

B. Pump and system curves

C. Pressure and temperature curves

D. Efficiency and viscosity curves

Correct Answer: B. Pump and system curves

Question 8

Which condition most directly promotes cavitation at a pump suction?

A. Higher absolute suction pressure

B. Lower liquid vapour pressure

C. Local pressure falling below vapour pressure

D. Reduced fluid velocity everywhere

Correct Answer: C. Local pressure falling below vapour pressure

Question 9

For geometrically similar pumps at constant speed, the ideal flow-rate relation is:

A. Q ∝ D

B. Q ∝ D²

C. Q ∝ D³

D. Q ∝ D⁵

Correct Answer: C. Q ∝ D³

Question 10

A degree of reaction of approximately zero characterizes a stage in which:

A. The rotor has no blades

B. The main pressure change occurs in stationary passages

C. The machine has zero flow

D. All energy transfer is mechanical friction

Correct Answer: B. The main pressure change occurs in stationary passages