Similarity Laws for Turbo-machinery

P M V SubbaraoProfessor

Mechanical Engineering Department

From Inception to Utilization….

Buckingham, E. The principle of similitude. Nature 96, 396-397 (1915).

The purpose of Dimensional Analysis

• Want to determine which variables to study.

• Want to determine the parameters that significantly affect the system.

• Reduce the cost/effort of experimental analysis by studying the most important groups of variables.

• The ideas can be used for any physical system.

• This will help in the design of scale test models

Similitude & Dimensional Analysis

•Scale model to prototype design and analysis.•Used to select proper turbo-machine (axial, radial or mixed flow,…)•Used to define performance parameters

Similarity Laws

• GEOMETRIC– Linear dimension ratios are the same everywhere.

– Photographic enlargement

• KINEMATIC (ϕm = ϕp)– Same flow coefficients

– Same fluid velocity ratios (triangles) are the same

• DYNAMIC (ψm = ψp)– Same loading coefficient

– Same force ratios (and force triangles)

• Energetic (m = p)– Same power coefficient

– Same energy ratios.

Euler’s GENERIC TURBOMACHINE (turbine, compressor, pump, ….)

•List the n physical quantities (Qn) with dimensions and the k fundamental dimensions. •There will be (n-k) π-terms.•Select k of these quantities, none dimensionless and no two having the same dimensions. •All fundamental dimensions must be included collectively in the quantities selected.

Fundamental Quantities for Turbo-machines

The First Non-dimensional Parameter

dcba QDN 1

31 DN

Q

Flow Coefficient or Capacity Coefficient ()

3DN

Q

“the dimensionless ‘swallowing’ capacity of the machine”

Flow Velocity Vs Blade Speed

Volumetric flow rate (Q) can be related to the fluid velocity :

A particular value of implies a specific relationship betweenfluid velocity and blade/impeller speed.

Efficiency (η) vs Flow coefficient (ϕ)

Design Innovations for Better Performance

Strategies to Capture More Power from Wind

Pitch-Controlled Variable-Speed Wind Turbine Generation

Grid Acceptable Power

The second Non-dimensional Parameter

dcba pDN 2

222 DN

p

•p corresponds to the energy per unit volume of the fluid.

• N2D2 relates to the rotor or impeller dynamic pressure (K.E. per unit volume).

•Loading Coefficient

Load Coefficient or Head Coefficient

For compressible fluid machines :

2 :t Coefficien Load

U

h

For incompressible fluid machines :

22 :t Coefficien Head

DN

gH

Selection of Load Coefficient for an Axial Flow Compressor

Accepted Technology for Hydro Power generation

Universal Design Chart for Power Consuming Turbo-machines

The Third non-dimensional Parameter

dcba PDN 3

52 DN

P

Role of Power Coefficient : Wind Turbines

Size Vs Capacity of A Wind TUrbine

Design Upgradation

Similarity of Model & Prototype

Design for Best Efficiency for Pumps & Fans

Dimensions for Performance

2/1

4/1

Q

gHDDs