Similarity Laws for Turbo-machinery

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Similarity Laws for Turbo-machinery

• P M V Subbarao
• Professor
• Mechanical Engineering Department

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

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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

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Similarity Laws

• GEOMETRIC
• Linear dimension ratios are the same everywhere.
• Photographic enlargement
• KINEMATIC (?m ?p)
• Same flow coefficients
• Same fluid velocity ratios (triangles) are the
  same
• DYNAMIC (?m ?p)
• Same loading coefficient
• Same force ratios (and force triangles)
• Energetic (Pm Pp)
• Same power coefficient
• Same energy ratios.

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Eulers GENERIC TURBOMACHINE (turbine,
compressor, pump, .)

• List the n physical quantities (Qn) with
  dimensions and the k fundamental dimensions.
• There will be (n-k) p-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.

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Fundamental Quantities for Turbo-machines
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The First Non-dimensional Parameter
Flow Coefficient or Capacity Coefficient (f)
the dimensionless swallowing capacity of the
machine
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Flow Velocity Vs Blade Speed
Volumetric flow rate (Q) can be related to the
fluid velocity
A particular value of f implies a specific
relationship between fluid velocity and
blade/impeller speed.
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Efficiency (?) vs Flow coefficient (?)
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Design Innovations for Better Performance
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Strategies to Capture More Power from Wind
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Pitch-Controlled Variable-Speed Wind Turbine
Generation
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Grid Acceptable Power
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The second Non-dimensional Parameter

• Dp corresponds to the energy per unit volume of
  the fluid.
• rN2D2 relates to the rotor or impeller dynamic
  pressure (K.E. per unit volume).
• Loading Coefficient

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Load Coefficient or Head Coefficient
For compressible fluid machines
For incompressible fluid machines
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Selection of Load Coefficient for an Axial Flow
Compressor
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Accepted Technology for Hydro Power generation
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Universal Design Chart for Power Consuming
Turbo-machines
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The Third non-dimensional Parameter
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Role of Power Coefficient Wind Turbines
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Size Vs Capacity of A Wind TUrbine
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Design Upgradation
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Similarity of Model Prototype
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Design for Best Efficiency for Pumps Fans
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Dimensions for Performance