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Similitude and Dimensional Analysis III

Hydromechanics VVR090

Analysis of Turbomachines

• pumps (centrifugal, axial-flow)

• turbines (impulse, reaction)

Dimensional analysis useful to make generalizations about similar turbomachines or distinguish between them.

Relevant variables with reference to power (P):

• impeller diameter (D)

• rotational speed (N)

• flow (Q)

• energy added or subtracted (H) [H] = Nm/kg = m2/s2

• fluid properties such as viscosity (m), density (ρ), elasticity (E)

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Archimedean Screw Pump

Rotodynamic Pumps

Radial flow pump (centrifugal)

Axial flow pump (propeller)

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Turbines

Kaplan

Pelton

Dimensional Analysis for Turbomachines

Assume the following relationship among the variables:

{ }, , , , , , , 0μ ρ =f P D N Q H E

Buckingham’s P-theorem:

3 fundamental dimensions (M, L, T) and 8 variables imply that 8-3=5 P-terms can be formed.

Select ρ, D, and N as variables containing the 3 fundamental dimensions to be combined with the remaining 5 variables (P, Q, H, m, and E).

Possible to use other variable combinations that contain the fundamental dimensions.

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Buckinghams’ P-Theorem

ρ, D, N combined with m yields:

1Π = μ ρa b c dD N

Solving the dimensional equations gives:

2

1 ReρΠ = =

μND

Derive other P-terms in the same manner:

ρ, D, N combined with E Æ2 2 2 2

22 2 Mρ

Π = = =N D N D

E a

ρ, D, N combined with P Æ

ρ, D, N combined with Q Æ

ρ, D, N combined with H Æ

3 3 5Π = =ρ P

P CN D

4 3Π = = QQ C

ND

5 2 2Π = = HH C

N D

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Summarizing the results:

{ }3 5 ' , ,Re,M=ρ Q H

P f C CN D

Or:

{ }

{ }

3

2 2

'' , ,Re,M

''' , ,Re,M

=

=

P H

P Q

Q f C CND

H f C CN D

Previous analysis: ρ∼P QH

Form a new P-term:

{ }3' , ,Re,MΠ = =ρ

IVPQ H

Q H

CP f C CQH C C

Incompressible flow with CQ and CH held constant:

{ }Re= = ηρ

VH

P fQH

hH = hydraulic efficiency

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Alternative Approach

Assume that the relationship between P and ρ, Q, and H is known, and that h includes both Re and mechanical effects.

Assume the following relationship (incompressible flow):

{ }, , , , 0η =f D N Q H

An alternative dimensional analysis gives:

2 2 3' ,⎧ ⎫= η⎨ ⎬⎩ ⎭

H QfN D ND

Typical Plot of Experimental Data

Spread represents variation with h (effects of Re)

2

3

∼∼

H DQ D

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Alternative Dimensionless Terms

3/ 4=sN Q

NH

Specific speed (pumps):

(represents actual speed when machine operates under unit head and unit flow)

• common to relate h to Ns

• characterize classes of pumps etc

Specific speed (turbines):

5/ 4=

ρsN PN

H

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Application of Dimensional Analysis to Pipe Friction

Assume the wall shear stress (τo) depends on:

• mean velocity (V)

• diameter (d)

• mean height of roughness projections (e)

• fluid density (ρ)

• fluid viscosity (m)

The following relationship should hold:

{ }, , , , , 0τ ρ μ =of V d e

Buckinghams’ P-Theorem

V, d, ρ combined with τo yields:

1Π = τ ρa b c doV d

Solving the dimensional equations gives:

1 2

τΠ =

ρo

V

Further analysis gives:

2Π =ed 3

ρΠ =

μVd

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The following relationship may be derived:

2 ' , ' Re,τ ⎧ ⎫ρ ⎧ ⎫= =⎨ ⎬ ⎨ ⎬ρ μ ⎩ ⎭⎩ ⎭o Vd e ef f

V d d

relative roughness=ed

Hydraulically smooth

and rough flow

Darcy-Weisbach Friction Formula

Frictional losses in a pipe:

2

2=L

L Vh fd g

Energy equation:

τ=ρ

oL

h

LhgR

'' Re,⎧ ⎫= ⎨ ⎬⎩ ⎭

ef fd

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Moody Diagram