33927 - Machines M

Course Unit Page


This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Affordable and clean energy

Academic Year 2022/2023

Learning outcomes

The Student improve in the knowledge of Turbines (Steam and Hydraulic) and Compressors.

Course contents


Compressible flow turbomachines. Static and total quantities. Equations of fluid motion in moving channels. The speed of sound and flow regimes. Compressible fluid flow equations. Stodola Ellipse. The chocking. Maximum mass flow rate and flow parameter.

Link between transversal area and flow in a channel depending on the regime of motion: Hugoniot equation. Converging channel and converging-diverging channel. Mass flow rate and flow parameter as function of the ratio between downstream pressure and upstream total pressure.

Stator losses, representation of physical states in the enthalpy diagram. Enthalpy diagram for converging channel and converging-diverging channel.

Euler equation and equation with the differences of kinetic energy for a rotor row. Degree of reaction. Total to total and total to static efficiencies, representation of physical states in the enthalpy diagram. Rotor losses.

The axial turbines reaction stage: velocity triangles. Normal stage. Maximum work and velocity triangles. Total to total efficiency definition and maximum point.

The impulse stage: the velocity triangles, the maximum work, physical states on the enthalpy diagram. Total to static efficiency. comparison between reaction stage and impulse stage losses.

De Laval Turbine scheme and operating principle. De Laval Turbine limitations on the enthalpy change.

Curtis Wheel Turbine scheme and operating principle. The velocity triangles. Maximum work evaluation for a Curtis Wheel Turbine. Total to static efficiency. Comparison with the impulse turbine efficiency.

The Multiple impulse stage turbine scheme and operating principle. Total to static efficiency and recovery factor.

The reaction turbine scheme and operating principle. The role of balancing drum. Limits on input and output volumetric flow of a reaction turbine. Analytic expression of the input and output volumetric flow. The mixed and double flow turbines.



Dynamic compressors. Operational differences between axial and centrifugal compressors. Limits on the flow rate that can be disposed of by an axial compressor. Principle of conservation of flow rate in cylindrical coordinates. Solenoidal motion field. Rotation of the fluid element in cylindrical coordinates. Vorticity and rotation. Solenoidal field of motion: the velocity potential and its properties. Irrotational motion field: the vector potential and its properties: orthogonality with the iso velocity potential lines. Current lines and tangent line coinciding with the direction of the velocity. Mass flow rate between two iso lines. Complex variable functions, Cauchy Reiemann conditions and holomorphic functions. The complex potential and the complex velocity. The conformal transformation: maintenance of the angles between the tangents to the curves in a plane and the corresponding tangents to the curves in the transformed plane. Complex potential in the point source case and in the free vortex case. The overall potential in the point source and in the free vortex. The axial compressor: operating principle and speed triangles in the case of reaction degree equal to 0.5. Compressor blades and NACA profiles: classification. Construction of a 4-digit NACA profile. Strength on an isolated profile. Lift, lift and pressure coefficient for a profile. Blasius formula for an isolated profile. Correlation between lift and circulation for an isolated profile. Th. of Kutta-Joukowsky. Kutta condition. Determination of the coefficient of lift and drag for a profile placed in the row in the case of no losses. Determination of the coefficient of lift and drag for a profile placed in the row in the event of leaks. Analytical expression for the coefficient of lift and drag for a mobile profile. Degree of reaction, flow coefficient, load coefficient and its dependence on the lift coefficient. Link between load coefficient and degree of reaction. Speed triangles at the compressor inlet and outlet and their degree of reaction. Diagram h, s and characteristic curves of an axial compressor. Three-dimensional flow problem in an axial compressor: momentum balance in cylindrical coordinates. Conservation of energy in cylindrical coordinates and Cracco's formula. Application of Cracco's formula to the three-dimensional problem of flow. Variation of the tangential speed with the radius. Design with the free vortex design method (work, degree of reaction angles alpha and beta). Circumferential speed trend with Constant reaction design and Exponential design. Comparison between the three methods.



Pelton Turbine. Scheme and operating principle. Distributor outlet velocity and flow rate evaluation. Inlet and outlet rotor velocity triangles. Maximum work, total and hydraulic efficiencies. Manual and oleo-dynamic regulating Doble nozzles. Minimum and maximum blades number. Off-design and characteristic curves: flow rate as function of rotational speed and efficiency as function of flow rate.

Mechanical, kinematical, dynamical and geometrical similarity: hydraulic similarity. Specific speed. Specific diameter and non-dimensional impeller speed. Multi-jet Pelton turbines, scheme and operating principle.

Francis Turbine. Scheme and operating principle. Balance of the axial forces on the turbine rotor and design solutions. Degree of reaction. Inlet and outlet rotor velocity triangles. Maximum work. Fink distributor effect on inlet rotor velocity triangles. Importance of the discharge duct in a Francis turbine plant and description of the outlet pressure trend as a function of design parameters at the duct outlet. Achievable work with and without the discharge duct. Pressure at the Francis discharge section. The cavitation problem and Thomanumber definition. Description of the sudden off-design problems and solutions. Off-design and characteristic curves. Rotor shape improvements as a function of specific speed in (high flow rate and low head).

Propeller and Kaplan Turbine. Scheme and operating principle. Inlet and outlet rotor velocity triangles from hub to tip of the impeller. Off-design and characteristic curves. Kaplan Turbine, operating principle and load regulation curves.


Three books:

- Turbomacchine termiche motrici, Vol.1 - M. Bianchi, A. Peretto - Ed. Pitagora

-Compressori assiali e centrifughi, Vol.2 - A. Peretto. Ed. Pitagora

-Turbomacchine Idrauliche Motrici, Vol.3 - , F. Melino, A. Peretto - Ed. Pitagora

Teaching methods

Lessons developed in the classroom with electronic pad

Assessment methods

Required diagrams

(the student must know how to draw the diagram in a realistic manner):

  • T,s diagram of water with isobaric curves inside and outside of the liquid-vapor region
  • H,s diagram of water with isobaric and isothermal curves inside and outside of the liquid-vapor region

Required drawings:

  • De Laval Turbine

  •  Curtis Wheel

  • Multiple stage impulse Turbine

  • Reaction Turbine

  •  Pelton Turbine, oleo-dynamic and mechanical nozzle

  • Francis Turbine

  • Propeller and Kaplan Turbine 

Teaching tools

Parts of Compressors and Turbines given and explained to the students in the classroom during the lessons.

Office hours

See the website of Antonio Peretto