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 DNQ

Flow Coefficient or Capacity Coefficient ()

3DNQ

“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 DNp

•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 LoadU

h

For incompressible fluid machines :

22 :t Coefficien HeadDN

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 DNP

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

QgHDDs