similitude and dimensional analysis iii - lth and dimensional analysis iii ... derive other p-terms...
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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