TABLE 7-5 (from Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, McGraw-Hill, 2006) Some common established nondimensional parameters or Πs encountered in fluid mechanics and heat transfer; A is a characteristic area, D is a characteristic diameter, f is a characteristic frequency (Hz), L is a characteristic length, t is a characteristic time, T is a characteristic (absolute) temperature, V is a characteristic velocity, W is a characteristic width, W is a characteristic power, ω is a characteristic radial frequency or angular velocity (radians/s). Other parameters and fluid properties in these Πs include: c = speed of sound, cp, cv = specific heats, dp = particle diameter, DAB = species diffusion coefficient, h = convective heat transfer coefficient, hfg = latent heat of evaporation, k = thermal conductivity and also k = ratio of specific heats, P = pressure, Tsat = saturation temperature, V = volume flow rate, α = thermal diffusivity, β = coefficient of thermal expansion, λ = mean free path length, μ = viscosity, ν = kinematic viscosity, ρ = fluid density,
ρf = liquid density, ρp = particle density, ρs = solid density, ρv = vapor density, σs = surface tension, and τ = shear stress.
Name Definition Ratio of significance
3 Archimedes ρs gL gravitational force Ar = ()ρρ− number μ 2 s viscous force LL length length aspect ratio AR= or or WD width diameter hL convective heat transfer Biot number Bi = k conduction heat transfer
2 g ( ρρfv− ) L gravitational forces Bond number Bo = σ s surface tension forces
PP− v Ca (sometimes σ c ) = 2 ρV pressure - vapor pressure cavitation number ⎛⎞2()PP− inertial pressure sometimes v ⎜⎟2 ⎝⎠ρV 8τ wall friction forces Darcy friction f = w factor ρV 2 inertial forces F drag force drag C = D D 1 2 coefficient 2 ρVA dynamic force V 2 kinetic energy Eckert number Ec = CTP enthalpy
ΔΔPP⎛⎞pressure difference Euler number Eu= ⎜⎟sometimes 221 inertial pressure ρρVV⎝⎠2
Fanning 2τ w wall friction forces fF = friction factor ρV 2 inertial forces Fourier αt physical time Fo (sometimes τ ) = number L2 thermal diffusion time Froude VV⎛⎞2 inertial forces Fr= ⎜⎟ sometimes number gL ⎝⎠gL gravitational forces
Grashof gTLβ Δ 32ρ buoyancy forces Gr = number μ 2 viscous forces
CTT( − ) sensible energy Jakob number Ja = P sat hfg latent energy
Knudsen λ mean free path length Kn = number L characteristic length Name Definition Ratio of significance k α thermal diffusion Lewis number Le == ρCDP AB D AB species diffusion F lift force C = L lift coefficient L 1 2 2 ρVA dynamic force V flow speed Mach number Ma (sometimes M) = c speed of sound Nusselt Lh convection heat transfer Nu = number k conduction heat transfer ρLVC LV bulk heat transfer Peclet number Pe ==p k α conduction heat transfer W power power number N = P ρD53ω rotational inertia Prandtl ν μC viscous diffusion Pr == P number α k thermal diffusion PP− static pressure difference pressure C = ∞ p 1 2 coefficient 2 ρV dynamic pressure
32 Rayleigh gβρΔTL CP buoyancy forces Ra = number kμ viscous forces
ρVL VL inertial forces Reynolds Re == number μ ν viscous forces Richardson Lg5 Δρ buoyancy forces Ri = number ρV 2 inertial forces roughness height roughness ε ratio L characteristic length μ ν viscous diffusion Schmidt Sc == number ρDDAB AB species diffusion VL overall mass diffusion Sherwood Sh = number DAB species diffusion c enthalpy specific heat k (sometimes γ ) = P ratio cv internal energy h heat transfer Stanton St = number ρCVp thermal capacity
ρ dV2 particle relaxation time Stokes number Stk (sometimes St) = pp 18μL characteristic flow time Strouhal fL characteristic flow time St (sometimes S or Sr) = number V period of oscillation ρVL2 inertial forces Weber number We = σ s surface tension forces