# Euler’s Work Equation-Turbomecines-Lecture Slides, Slides for Turbomecines. B R Ambedkar National Institute of Technology

## Turbomecines

Description: Main topics for the course are how to create electricity, design of Kaplan turbine and runner, hydraulic turbines, flow momentum, Fracnis turbine, Euler work equation, ancient power resources, Pelton turbine plant, HEPP development, redial inflow turbine. This lecture includes: Euler, Work, Equation, Torque, Flow, Angular, Momentum, Shaft, Output, Whirl, Velocity, Conservation, Energy, Steady
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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 : 11rVm 

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Exit rate of angular momentum at exit: 22rVm 

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 

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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• 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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