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) 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 10.