Fundamentals of Turbomachinery and Governing Laws

What are the fundamentals of Fluid Machinery?


What is fluid machinery?

  • A fluid machinery (turbo-machinery) is a mechanical device that converts the energy stored by a fluid into mechanical energy or vice versa
  • The energy stored by a working fluid may be in the form of potential, kinetic, and intermolecular energy
  • The mechanical energy is usually transmitted by a rotating shaft.
  • Hydraulic Machines: Machines use mainly water, for most of the practical applications
  • In this article, the basic working principle of fluid machinery and energy transfer is discussed

  • The application of fluid machinery is show below :

What is the difference between fluid machinery and turbomachinery?

  • Fluid machinery is the device used to transfer energy from fluid to mechanical devices or vice versa. There is an interaction between fluid and machine for power generation or industrial applications. Turbomachinery is part of fluid machinery focussing on mechanical machines.
  • Turbo-machinery is defined as a device that transfers fluid energy between a rotor (a moving wheel with blades)  and a fluid.  It involves rotating parts called as rotors. The rotor can be an impeller for pumps and a turbine for power generation. 


Classification Turbo-machinery

• Turbo-machines are classified into two types of power generation or power absorption 

  • Example of Turbo-machinery: steam turbine in  power generation

  • Example of Turbo-machinery for power absorption
Turbo-machines  based on  Directions of fluid flow 

Parts of the Axial Turbine Stage

  • The major components of the axial flow turbine are given as below:
    • The stator comprises  a series of stationary blades that acts as guide vane for incoming flows
    •  The rotor comprises  a series of moving  blades and provides work/power to the shaft
  •   Application of Axial Flow Turbine: thermal  power plant and gas turbine of (aircraft engine)
  • Major parts of axial stage turbine: The following figure shows part of the turbine and blades for a single-stage axial turbine

  • Parts of Axial stage turbine


Classification Based on Direction of Energy Conversion

The fluid machines may be classified under different categories as follows:

Power Producing Device

  • The device in which the energy stored in a fluid is converted in the form of mechanical energy of a rotating member is called as a turbine
  • The energy stored in the fluid may in kinetic, potential, or intermolecular energy

Power Absorbing Device

  • The machines which transfer mechanical energy from moving parts to a fluid by increasing its pressure or velocity are known as power absorbing fluid machine such as fans, compressors, pumps, or blowers


Power Absorbing Turbo-machinery

Compressor Types

  • Positive displacement can have different types

 Axial Fans or blowers 

  • Different types of fans/blowers is shown below
  • Types of blowers that are popular in many industries and domestic applications like cleaning and cooling purposes

Water Pump

  • The water pump consists of a pump casing, an electric motor to run the impeller
  • Water is sucked at the inlet at low pressure and it exhausted with high velocity at the outlet
  • Parts of the Water pump are shown below

Classification on the Principle of Operation and Fluid type

Positive Displacement Machines

  • whose functioning depends essentially on the change of volume of a certain amount of fluid within the machine
  • The positive displacement means that there is a physical displacement of the boundary of the closed system containing a certain mass of fluid. This principle has been used in practice with the reciprocating motion of a piston within, a cylinder during the continuous flow of fluid across the system
  • Therefore, the word reciprocating engine is also used for positive displacement machines which produce mechanical energy using a reciprocating piston is called a reciprocating pump or reciprocating compressor

Classification of Positive Displacement Pump

Rotodynamic Machines

  • The machine whose functioning depends basically on the principle of fluid dynamics is called a Rotodynamic machines
  • They are different from positive displacement machines by relative motion between the moving part of the machine and fluid
  • The rotating part of the machine usually comprises a number of vanes or blades, called as rotors or impellers. The non-moving (fixed part) is termed a stator
  • Depending upon the major direction of the fluid flow path in the rotor, the machine can be called a radial flow or axial flow machine
  • Relative motion: Radial flow or Axial flow machine:
    • As per thermodynamic laws,  the work is done by the fluid on the rotor of turbines, but  the rotor does the work on the fluid of the pump or compressor using external prime movers
    • In the rotor of the radial flow machine, the main direction of fluid flow is radial but in the axial flow machine, the fluid path is axial
    • For radial flow turbines, all fluid flows toward the center of the rotor. In contrast, for pumps and compressors, the fluid flows away from the center of the machine. Therefore, radial flow turbines can be referred to as radially inward flow machines
    •  Radial flow pumps, the fluid flows are for radially outward flow machines.
    • Examples of radial flow machines:  Francis turbines,  centrifugal pumps, compressors
    • Examples of axial flow machines: are Kaplan turbines and axial flow compressors.
    • However, in some machines the flow can be partly radial and partly axial such a machine is called a mixed-flow machine

Based on the Type of Fluid Used

  • Depending on availability, fluid machines use either liquid (water) or gas as the working fluid
  • A machine such as hydraulic pump transfers the mechanical energy of the rotor to the energy of the liquid. Gas turbines use high-temperature and pressure dry steam (H2O) in gas phase. A compressor or a fan or a blower uses air as working fluid.
  • A compressor is a machine where main purpose is to increase the static pressure of a gas. Therefore, the mechanical energy stored by the fluid is mainly in the form of pressure energy. On the other hand, fans and blowers utilize the mechanical energy of the rotor to increase mostly the kinetic energy of the fluid. In these machines, the change in static pressure is relatively small compared to pumps.
  • Water or Hydraulic Turbines: For all practical cases, water is used as working fluid in turbines which is known as water turbines or hydraulic turbines.
  • Steam or Gas Turbine: turbines using gases as a working fluid in practical fields are usually known to as steam turbines, gas turbines, and air turbines depending upon whether they use steam, gas (the mixture of air and products of burnt fuel in the air), or air.


Representation of A Single-stage Axial Turbine

 A single stage Turbine 

  • A simple turbine consist of one stator and rotor arranged on one single shaft
  • Due to fluid force, the torque is generated to rotate the shaft connected with electrical power generator

A Simple Turbine_ Exploded View

Cascade view of the axial turbine

  • The view of the turbine is taken by the plane passing through the axis of the turbine

 Meridional  view of axial flow turbine

  • The view of the turbine is taken by the radial plane

Multistage Axial Turbine

  •  A series of stages form a multistage turbine
  • The energy (work) transfer in a single stage is limited by the blade speed
  • If more energy transfer per unit mass is required, then the number of stages is added one after other

 Multi-stage steam turbine for thermal power plant

  • Schematic of multi-stage steam turbine
  • Turbine power-house
  • Multi-stage Steam turbine without casing

Parts of the Radial Turbine

  • Application of radial turbine: Gas turbine, turbocharger, and process industry   
  • Components of radial turbine
Components of the radial turbine

Representation of Flow through the Pump

Sectional View of Pump

  • The pump is easy to understand with two sectional views of the pumps
  • The impeller is an essential part of the pump which rotates to create the desired pressure of water

Cascade view of Pump

  •  The cascade view of the pump is shown below with a 3D view
  • Cascade view of a centrifugal pump

 Aero-foil Theory 

What is an aerofoil?

  • Aero-foil is a  curved structure designed to give the most desirable ratio of lift to drag in flight
  • It is used as the basic form of the wings, fins, windmills,s and planes of most aircraft are defined as an airfoil.
  • The major role of an airfoil is to produce the maximum lift.  Lift is used to produce torque for power generation. There is drag force which is undesirable.

Forces acting on Aerofoil

  • Both lift and drag are perpendicular to each other
  • Lift is the component of fluid force such that the force is perpendicular to the direction of motion, and drag is the component parallel to the direction of motion but opposite fluid flows.

aerodynamic_force -aerofoil-lift-drag

Fundamental Laws used in Turbomachinery

 Second Law of Thermodynamics

  • The second law of thermodynamics is devised based on a physical understanding with universal experience related to heat and energy interconversion.
  • Heat always moves from hotter objects (source – high energy level ) of to colder objects  (sink-low energy level) unless external energy is supplied to reverse the direction of heat or energy flow. Another example is that water flows under gravity.

Second law of thermodynamics-Heat Engine


  • The Clasius Inequality


  • For a reversible cyclic process

  • Entropy changes of a state for an isentropic process when the process is reversible and adiabatic

  • We can re‐write the above definition using the first law of thermodynamics

Bernoulli’s Equation

  • Writing an energy balance for a flow without heat transfer or power generation

  • Applying Bernoulli’s equation for infinitesimal (differential) control volume: dh + cdc + gdz = 0

where enthalpy is dh = vdp=dp/ρ When the process is isentropic: Tds =dh-vdp


Euler’s motion equation:

  • Integrating this equation into stream direction, the Bernoulli equation is obtained

  • When the flow is incompressible, density does not change, thus the equation becomes

Where Po is the stagnation pressure

  • For hydraulic turbo-machines, head is defined as

H= Z + p/(ρg)

  • If the pressure and density change are small than the stagnation pressures at inlet and outlet conditions are equal to each other. This is applied to compressible isentropic processes.

Compressible flow relations

  • For a perfect gas, the Mach # can be written as,

  • For a perfect gas, the Mach # can be written as, . Here a is the speed of sound, R, T and γ are universal gas constant, temperature in (K), and specific heat ratio, respectively
  • For the Mach number greater than 0.3, the flow is considered as compressible, therefore fluid density is not constant
  • With the stagnation enthalpy definition, for a compressible fluid:

  • Relation between static and stagnation temperatures

  • Relation between static and stagnation pressures

  • Stagnation temperature – pressure relation between two arbitrary points

  • Capacity (non‐dimensional flow rate)

  • Relative stagnation properties and Mach

Thermodynamic Graphs

  • Relation of static‐relative‐stagnation temperatures on a T‐s diagram

  • Temperature – gas properties relation

Efficiency definitions used in Turbomachinery

Overall efficiency

  • Isentropic – hydraulic efficiency:

Mechanical efficiency

Steam and Gas Turbines

The adiabatic total‐to‐total efficiency:

  • When inlet‐exit velocity changes are small

Expansion and compression Process

  • Enthalpy – entropy relation for turbines and compressors

Total‐to‐static efficiency

  • The definition of The following efficiency is used when the kinetic energy is not utilized and entirely wasted. Here, the exit condition corresponds to ideal‐ static exit conditions are utilized (h2s)

  • Hydraulic turbines

Polytrophic Efficiency

  • Polytrophic efficiency is defined to show the differential pressure effect on the overall efficiency, resulting in an efficiency value higher than the isentropic efficiency

Working principle of Rotodynamic Machine

  • In this section, the basic principle of rotodynamic machines and the performance of different types of fluid machines are discussed
  • The major part of a rotodynamic machine is the rotor which consists of several vanes or blades
  • The relative motion exists between the rotor blades/vanes and the fluid. The fluid enters over a blade with a certain velocity component and hence of momentum in the direction tangential to the rotor. While fluid flows over the rotor blades, its tangential velocity, and momentum change compared to incoming fluids.
  • The rate of change of tangential momentum changes corresponds to a tangential force on the rotor
  • In a turbine, the tangential momentum of the fluid is drawn to cause rotational effect. Therefore, work is done by the working fluid to the moving rotor. However, in the case of pumps or compressors, there is an increase in the tangential momentum of the fluid by supplying external energy source. Therefore work is absorbed by the working fluid from the moving rotor.

Basic Equation of Energy Transfer in Rotodynamic Machines

  • The basic equations of mass, momentum and energy of fluid dynamics is applied to a fluid element flowing a rotor using the Newton’s Laws of Motion. These equations is same for all rotodynamic machines
  • Here we shall discuss the momentum theorem applied to a fluid element while flowing through moving or fixed blades or vanes.
  • The following represents diagrammatically a rotor of a generalized fluid machine, with 0 – 0 the axis of rotation and the angular velocity.
  • The fluid enters initially the rotor at the first point 1, then it passes through the rotor and discharged at the end point 2. The points 1 and 2 are at radii and from the center of the rotor, and the directions of fluid velocities at 1 and 2 may be at any arbitrary angles.

Assumptions considered for the analysis of energy transfer

  • Steady-state flow
  • The mass flow rate is constant across any section. There is no storage or depletion of fluid in the rotor
  • The heat and work interactions between the rotor and its surroundings take place at a constant rate
  • Uniform velocity over any area normal to the flow direction. This means that the velocity vector at any point is representative of the total flow over a finite area. There is no leakage and the entire fluid undergoes the same conditions

Details of Velocity Triangles

  • The velocity (V) at any point can be resolved into three mutually perpendicular components (Va, Vf, Vw)
  • is the axial component of velocity as it is directed parallel to the axis of rotation
  • is the radial component which is directed radially through the axis to the rotation
  • is the tangential component which is directed at right angles to the radial direction and the tangent to the rotor at that part
  • Changes in the axial velocity components at the inlet and outlet of the rotor result in a change in the axial momentum and give rise to an axial force, which is taken by a thrust bearing of the stationary rotor casing. The rate of change in radial velocity results in the change of rotor momentum in the radial direction.
  • However, for an axisymmetric flow, this does not result in any net radial force on the rotor. In case of a nonuniform flow distribution over the periphery of the rotor in practice, a change in momentum in the radial direction may result in a net radial force which is carried as a journal load
  • The tangential component only has an effect on the angular motion of the rotor. In consideration of the entire fluid body within the rotor as a control volume, we can write from the moment of momentum theorem


Inlet and outlet Velocity Triangle of Steam Turbine

Euler’s Turbo-machine Equations

  •  The velocity diagram for the pump is given below


Torque Generated on Rotor

  • The torque exerted by the rotor on the moving fluid


where m is the mass flow rate of fluid through the rotor. The subscripts 1 and 2 denote values at the inlet and outlet of the rotor respectively. The rate of energy transfer to the fluid is then given by


  • Where is the angular velocity of the rotor and which represents the linear velocity of the rotor
  • Therefore and are the linear velocities of the rotor at points 2 (outlet ) and 1 (inlet) respectively The Eq, (2) is known as Euler’s equation in relation to fluid machines. The Eq. (2) can be written in terms of head gained ‘H’ by the fluid as

  • As per the usual sign convention in the fluid machines, the head delivered by the working fluid to the rotor (impeller) is considered to be positive or vice-versa. Therefore the above is written with a change in the sign on the right-hand  side according to the sign convention as

  • These equations are applicable irrespective of changes in fluid density or components of velocity in other directions
  • Moreover, the shape of the path taken by the fluid in moving from the inlet to the outlet is of no consequence. The expression involves only the inlet and outlet conditions. A rotor, the moving part of a fluid machine, usually consists of a number of vanes or blades mounted on a circular disc.
  • Velocity triangles are important to understand the inlet and outlet of a rotor.
  • The inlet and outlet portions of a rotor vane may change as per applications
  •  velocities at the inlet and outlet in the vector diagrams have  two velocity triangles, where the velocity of fluid is relative to the rotor

Angle Made by Absolute Velocities:

  • The  angles made by the directions of the absolute velocities at the inlet and outlet respectively with the tangential direction,

Angle Made by Relative Velocities:

  • the angles made by the relative velocities with the tangential direction. The angles must match with vane or blade angles at the inlet and outlet respectively for a smooth. Note that there is shockless entry and exit of the fluid to avoid undesirable losses.
  • Now we shall apply a simple geometrical relation as follows: From the inlet velocity triangle,

or (5)

  • similarly from the outlet velocity triangle.

Using the expressions of and we get H (Work head, i.e. energy per unit weight of fluid, transferred between the fluid and the rotor as) as

  • The above equation is an important form of Euler’s equation for fluid machines as it has the three components of energy transfer. These components illustrate the nature of the energy transfer
  • The first term in the equation is a change in absolute kinetic energy or dynamic head of the working fluid while flowing through the rotor
  • The second term of is represents the change in fluid energy because of the movement of the rotating fluid from one radius of rotation to another.

Scope of CFD Modeling 

  • CFD modeling has been used widely in industries for the design and performance evaluation of turbo-machinery
  • First, the configuration of the turbine or pump must be selected for CFD simulation.
  • Steps for CFD simulation for turbomachinery
    • Geometry: create a 3D model based on the blade profile and performance curve for a selected turbo-machinery
    • Meshing: create the mesh model using the meshing platform
    • Simulation: select CFD solvers for  turbo-machinery using suitable CFD models
    • Post-processing
  • ANSYS has separate modules for geometry and meshing of turbo-machinery



  • Fluid machinery or Turbomachinery works on the second law of thermodynamics
  • They are classified based on power absorption or generation, the direction of fluid flows
  • The moving part is a rotor which is a common part in all turbo-machinery
  • CFD modeling will help for the design and optimization of performance


  • All essential books of turbo-machinery are on the page references 
  • There are several books focussing on thermodynamics, fluid mechanics and CFD modeling of turbo-machinery



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