Eulers Equation for Turbomachines P M V Subbarao

Euler’s Equation for Turbomachines P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Scientific Method for Creation of A Better Energy Harvester. .

Applications of Euler’s Equation • • • Euler equation applies to all kinds of turbomachines. Wind turbines ---- Pumps --- Gas turbines. Axial flow machines to Radial flow to Mixed flow. Energy extraction machines – Energy consuming machines. For a turbine the sign of Equation is negative. For pumps, fans, blowers and compressors it is positive.

Euler’s View of Vectors : A Cylindrical Coordinate System

Order of Velocity Vectors • The real flow through any turbomachine is three dimensional. • Axial, Radial and Circumferential. • Axi-symmetry: Inter blade row space. • Axi-symmetry assumes an average value to represent the state of working fluid in the blade-to-blade plane.

The Reference Plane of Euler Equation : Meridional Plane The momentum balance is considered in a plane constructed through axis of rotation and radial axis. This Plane is called Meridional Plane.

Euler’s Vision of 2 DVector

Euler’s Version of Newton’s Second Law • In turbo-machines much useful information is obtained by employing Newton’s second law in the form where it applies to the moments of forces. • This form is of central importance in the analysis of the energy transfer process in turbo-machines.

Euler’s Statement for Second Law For a system of mass m, the vector sum of the moments of all external forces acting on the system about some arbitrary axis A--A fixed in space is equal to the time rate of change of angular momentum of the system about that axis!

Euler Theory Torque exerted by flow on blade row = shaft output torque = Rate of change of Angular momentum of fluid = t Angular momentum is moment of linear momentum of angular velocity, Vq Define, L as Angular momentum :

Application of RTT to A SSSF-CV For a steady flow through a turbo-machine: Inlet rate of angular momentum : Exit rate of angular momentum at exit: Change in Rate of angular momentum:

Euler Turbine Equation Euler Theory: Power Generated : A change in Whirl Velocity of fluid is critical to establish Power Exchange between fluid and rotor in a turbo-machine !