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Euler’s Work Equation

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

*kVjViVV rx *ˆˆˆ

**Euler Theory:
**

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*dt
dL
*

*Define, L as *Angular momentum

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

*rmVL *

For a steady flow through a turbomachine:

Inlet rate of angular momentum : 11*rVm *

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Exit rate of angular momentum at exit: 22*rVm *

Change in Rate of angular momentum: 1122 *rVmrVm *

1122 *rVrVm *

Power : 1122 *rVrVmP *

A change Whirl Velocity of fluid is only responsible for Power Exchange between fluid and rotor in a turbo-machine !

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This is known as Euler Turbo-machinery Equation.

1201021122 *zzghhmrVrVmP *

*CVCV Pgz
V
*

*hmgz
V
*

*hmQ *

2

2

1

2

22

First Law for Steady Flow Steady State Turbo Machine:

Avoid heat transfer across surface of a turbo-machine.

*CVPgz
V
*

*hmgz
V
*

*hm *

2

2

1

2

22

Conservation of Energy

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Read Through Euler Turbo-machinery Equation

• A Change in total enthalpy is proportional to change in tangential flow speed or tangential engine speed.

• For engines with little change in mean radius (inlet to exit), the change in total enthalpy is due to change in tangential flow speed of the fluid.

• Creates a small change in enthalpy of fluid. • For engines with large change in radius, the change in enthalpy

is to a large degree due to the change in radius. • The centrifugal/centripetal effect. • Creates Large change in enthalpy of fluid. • How to select a suitable type of action for a resource/demand.

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Vai Vri

Vre Vae

Euler’s Vision of Newton’s Laws of Motion

Jet can lose/gain power both by Impulse and Reaction.

**One important and essential accessory in all these cases is initial flow
velocity i.e., design of a nozzle.
**

**How to select among impulse/reaction?
Or Degree of Reaction?
**

U

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Hero’s Aeolipile Vs Banca’s Wheel

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Pure Reaction Vs Pure Impulse

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In search of A Suitable Machine

• Primary characteristics of a source.
• The effect: Head, *H (m) or **p
*• The Capacity: Discharge *Q (m3/s) *or Power, *P (kg.m2/s3 )*.
• Natural Parameter: Acceleration due to gravity: *g (m/s) (for
*

*Hydraulic machines)
*• *Density of fluid: **(kg/m3).
*

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A Matching Parameter for the Euler’s Machine with Resource

,:, *gHPfGI *
For hydraulic power generating machines

:, *pPfGC * For compressible power Generating fluid machines
:, *pQfCC *

,:, *gHQfCI *
For hydraulic power consuming machines

For compressible power Consuming fluid machines

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Machine-Fluid Matching Parameter:

:, *pQf *
For compressible power consuming fluid machines

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*kji pQ *

*j
i
*

*m
kg
*

*sm
kg
*

*s
m
*

3

23 .

*j
i pQ *

Step by Step Elimination of Fundamental Dimensions

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*ji
*

*s
m
*

*s
m
*

2

23

2 3

2

23

*s
m
*

*s
m
*

*i
*

Take *j = -3/2
*

3

33

*s
m
*

*s
m
*

*i
*

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Take *i = 1/2 *& *j = -3/4
*

4 3

2

22 1

3

*s
m
*

*s
m
*

4

3

2 1

*pQn f
*

4 3

*p
*

*Q
n f
*

2 3

2 3

2 1

2 3

*s
*

*m
*

*s
*

*m
*

*fn
s *1

Speed demand by resource fluid

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Time Scale of a Machine to Resource

Speed: *N (rpm) or n (rps) of a turbo machine:
*

Scale Time Machine Scale Time Resourcescale timeessDimensionl

*n
*

*n f
*1

1 scale timeDimension -Non

This is named as Specific Speed, *Ns
*

4 3

*p
*

*mn
Ns
*

4 3

1

*p
*

*Q
n f
*

4 3

*p
*

*mnNs
*

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