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Similarity Laws for Turbo-machinery
P M V SubbaraoProfessor
Mechanical Engineering Department
From Inception to Utilization….
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.
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.
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
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