TABLE 7-5 (From Engel and Cimbala, Fluid Mechanics: Fundamentals

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 Darcy friction 8τ w wall friction forces f = factor ρV 2 inertial forces drag FD drag force C = 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 FL lift force C = 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 pressure PP− ∞ static pressure difference C = p 1 2 coefficient 2 ρV dynamic pressure 32 Rayleigh gβρΔTL CP buoyancy forces Ra = number kμ viscous forces Reynolds ρVL VL inertial forces Re == number μ ν viscous forces Richardson Lg5 Δρ buoyancy forces Ri = number ρV 2 inertial forces roughness ε roughness height ratio L characteristic length Schmidt μ ν viscous diffusion Sc == number ρDDAB AB species diffusion Sherwood VL overall mass diffusion Sh = number DAB species diffusion specific heat cP enthalpy k (sometimes γ ) = ratio cv internal energy Stanton h heat transfer 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 .

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