An Important Engineering Form of Newton's Laws!!!?!?!?!

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Euler Turbine Equation

Simple Turbine : Harvester of Useful Fluid Energy

in

out

T

outin mmCMSSSF ::

CV

out

out

in

inCV WgzV

hmgzV

hmQ

22

22

CVoutoutinin Wmm

out

out

in

in

Vhm

Vhm

22

22

outoutinin mm

Macro Analysis of A Turbine

outoutinin hmhm ,0,0

Cartesian View of Momentum Equation

FVmVmMMin

inout

outinout

This is a vector equation and will have three components in x, y and zDirections.

X – component of momentum equation:

xin

inxout

outx FVmVm ,,

Z – component of momentum equation:

zin

inzout

outz FVmVm ,,

Y – component of momentum equation:

yin

inyout

outy FVmVm ,,

X – component of momentum equation:

xin

inxout

outx FVmVm ,,

Axial Momentum Equations

Axial Momentum Equation: inxoutxx VVmF ,,

A turbomachine cannot tolerate this force doing any work.

The axial force should be completely absorbed by a thrust bearing.

Any remaining traces of this force can cause mechanical or aerodynamic damages.

Concurrent Designs for Minimum Axial Thrust

Normal Momentum Equations

Resultant radial Force:22

zyr FFF

The radial force component has little do with steady state aerodynamic performance of a turbo-machine.

Special Casings are designed to reduce this force.The remaining force is normally absorbed in a journal type bearing.

inyoutyy VVmF ,,

inzoutzz VVmF ,,

Pump Impeller and casing

Radial Force on Impeller

Best Efficiency Point (B.E.P.): The point on a pump's performance curve that corresponds to the highest safe efficiency. At this point, the impeller is subjected to minimum radial force promoting a smooth operation with low vibration and noise.

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.

kVjViVV zrˆˆˆ

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, i.e.

Torque exerted by flow on blade row = shaft output torque = Rate of change of Angular momentum of fluid =

Euler Theory:

Define, L as Angular momentum :

Angular momentum is moment of linear momentum of angular velocity, V

The Boeing 747 Cruising at an Altitude of 10 km

Turbo-jet Engine

Even 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: