Similitude and Dimensional Analysis III Analysis of Turbomachines

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)

1 Archimedean Screw Pump

Rotodynamic Pumps

Radial flow pump (centrifugal)

Axial flow pump (propeller)

2 Turbines

Pelton

Kaplan

Dimensional Analysis for Turbomachines

Assume the following relationship among the variables:

fPDNQH{ ,,,,,,,μρ E} = 0

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.

3 Buckinghams’ P-Theorem

ρ, D, N combined with m yields:

ab c d Π=μρ1 DN

Solving the dimensional equations gives:

ρND2 Π= =Re 1 μ

Derive other P-terms in the same manner:

22 22 ρND ND 2 ρ, D, N combined with E Æ Π= = =M 2 E a2 P ρ, D, N combined with P Æ Π ==C 3 ρND35 P Q ρ, D, N combined with Q Æ Π ==C 4 ND3 Q H ρ, D, N combined with H Æ Π= =C 5 ND22 H

4 Summarizing the results:

P 35= fCC',,Re,M{}QH ρND

Or:

Q = fCC''{} , , Re, M ND3 PH

H 22= fCC'''{}PQ , , Re, M ND

Previous analysis: PQH∼ ρ

Form a new P-term:

P CP IV Π=',,Re,M3 = fCC{}QH ρQH CQH C

Incompressible flow with CQ and CH held constant:

P ==ηf V {}Re ρQH H

hH = hydraulic efficiency

5 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):

fDNQH{ ,,,,η=} 0

An alternative dimensional analysis gives:

HQ⎧⎫ =ηf ',⎨⎬ ND22⎩⎭ ND 3

Typical Plot of Experimental Data

Spread represents variation with h (effects of Re)

HD∼ 2 QD∼ 3

6 Alternative Dimensionless Terms

Specific speed (pumps):

NQ N = s H 3/4

(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):

NP N = s ρH 5/4

7 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:

fVde{τρμ=o ,,,,,} 0

Buckinghams’ P-Theorem

V, d, ρ combined with τo yields:

abcd Π=τ1 oVd ρ

Solving the dimensional equations gives:

τ Π= o 1 ρV 2

Further analysis gives:

e Vdρ Π= Π= 2 d 3 μ

8 The following relationship may be derived:

τ⎧⎫o Vdρ e⎧ e ⎫ 2 ==ff',⎨⎬⎨⎬ 'Re, ρμVdd⎩⎭⎩⎭

e = relative roughness d

Hydraulically smooth

and rough flow

Darcy-Weisbach Friction Formula

Frictional losses in a pipe:

LV2 hf= L dg2

Energy equation:

τo L hL = ρgRh

⎧ e ⎫ ff= ''⎨ Re, ⎬ ⎩⎭d

9 Moody Diagram