Dimensional Analysis & Performance Parameters

Units and DimensionsQuantities Units Dimensions

Mass Kg M

Length m L

Time S T

Area m2 L2

Volume M3 L3

Volume flow rate m3 /s L3 /T

Mass flow rate Kg/s M/T

Velocity m/s L/T

Acceleration m/s2 L/T2

Force (Newton, N) Kg m/s2 ML/T2

Pressure (Pascal, Pa) N/m2 , Kg/ms2 M/LT2

Torque Nm, Kg m2/s2 ML2/T2

Heat, work, enthalpy, energy (Joules, J) J =Nm =Kg m2/s3 ML2/T2

Power (Watts) J/s =Nm/s = Kg m2/s3 ML2/T3

Dynamic Viscosity Ns/m2 =Kg/ms M/LT

Density Kg/m3 M/L3

Kinematic viscosity M2/s L2 /T

Rotational speed Rev./s or rad./s 1/T

Buckingham’s ∏-theoremNo. of variables ---- nNo. of non-dimensional groups or dimensionless numbers (also known as π-terms) are given by

k =n – m where m is no. of primary dimensions, m<ne. g. y1, y2 y3 =f(x1, x2,x3,x4,x5)For three primary dimensions, M,L & TNo. of π – terms = n-m =(3+5) -3 =5

π1, π2 =f(π3, π4, π5) More dimensionless groups can be formed by the combination of π-terms already determined

INCOMPRESSIBLE FLOW MACHINES

gH ,P, η = f(N,D, s/l, h/l, Q, ρ, μ)

η is non-dimensional, so not considered for π- theorem

gH/(ND2), P/(ρN3D5), η = f(Q/ND3, ρND2/μ, s/l, h/l,…….)

Head coeff. Power coeff. Volume flow coeff. Form of Reynolds no. Re φ=Cx/u , for axial flow machines

π5 = (Q/ND3)1/2/(gH/N2D2)3/4 =NQ1/2/(gH)3/4 NsP = N√Q/(H3/4)

π6 = (P/ρN3D5)1/2/(gH/N2D2)5/4 = (1/ρ1/2g5/4) . [N√P/H5/4 ] NsT= N√P/H5/4

Dependent variablesHead (gH)Power (P)Efficiency(η)

Independent variablesRotor speed (N)Rotor Diameter (D)Lengths (s, l, h etc.) or length ratios s/l, h/l, etcDischarge (Q)Fluid density (ρ)Fluid viscosity (μ)

COMPRESSIBLE FLOW MACHINESFor compressible fluids, compared to incompressible ones, the following variables are taken1) Mass flow rate m……. for Q (because of changing volume flow rate)2) Pressure ratio pr for gHThe additional variables because of compressibility3) γ ratio of specific heats4) a01 stagnation speed of sound at entry to the machine5) ρ01 gas density

p01/p02 , P, η = f(N, D, s/l, h/l, m. ρ01, μ, γ, a01 )p01/p02 , P/(ρ01 N3 D5), η = f[ρ01ND2/μ , m/ρ01ND3 , s/l, h/l, ND/a01, γ]

Pressure ratio p01/p02 , can be replaced by ratio of static pr. If the dynamic pr. Is negligible at entry and exitFlow coeff. φ = m/(ρ01 ND3) α m/(ρ01.a01.D2 ) =mRT01/{(p01/RT01)√(γRT01).D2}

α m√T01/p01 Speed parameter; ND/a01 =(D/√γR).(N/T01) α N/T01 Power coefficient; Pˆ =P/(ρ01N3 D5) = [m CpΔT0]/[ρ01 ND3].(ND)2

α CpΔT0/(ND)2 Taking φ = m/(ρ01 ND3), which is α ΔT0/T01 already considered & ND α a01 α T01

p01/p02, η, ΔT0/T01 = f{(m√T01)/p01 , N/T01 }