Dimensionless Numbers

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Dimensionless Numbers 3.185 November 24, 2003 Note: you are not responsible for knowing the different names of the mass transfer dimensionless numbers, just call them, e.g., “mass transfer Prandtl number”, as many people do. Those names are given here because some people use them, and you’ll probably hear them at some point in your career. Heat Transfer Mass Transfer Biot Number M.T. Biot Number Ratio of conductive to convective H.T. resistance Ratio of diffusive to reactive or conv MT resistance Determines uniformity of temperature in solid Determines uniformity of concentration in solid Bi = hL/ksolid Bi = kL/Dsolid or hD L/Dsolid Fourier Number Ratio of current time to time to reach steadystate Dimensionless time in temperature curves, used in explicit finite difference stability criterion 2 2 Fo = αt/L Fo = Dt/L Knudsen Number: Ratio of gas molecule mean free path to process lengthscale Indicates validity of lineofsight (> 1) or continuum (< 0.01) gas models λ kT Kn = = √ L 2πσ2 P L Reynolds Number: Ratio of convective to viscous momentum transport Determines transition to turbulence, dynamic pressure vs. viscous drag Re = ρU L/µ = U L/ν Prandtl Number M.T. Prandtl (Schmidt) Number Ratio of momentum to thermal diffusivity Ratio of momentum to species diffusivity Determines ratio of fluid/HT BL thickness Determines ratio of fluid/MT BL thickness Pr = ν/α = µcp/k Pr = ν/D = µ/ρD Nusselt Number M.T. Nusselt (Sherwood) Number Ratio of lengthscale to thermal BL thickness Ratio of lengthscale to diffusion BL thickness Used to calculate heat transfer coefficient h Used to calculate mass transfer coefficient hD Nu = hL/kf luid Nu = hD L/Df luid Grashof Number Ratio of natural convection buoyancy force to viscous force Controls the ratio of lengthscale to natural convection boundary layer thickness 3 2 3 2 Gr = gβΔT L /ν Gr = gβC ΔCL /ν Peclet Number Ratio of convective to diffusive heat/mass transport in a fluid Used to determine plug flow/perfect mixing (CSTR) continuous flow model validity Pe = RePr = U L/α Pe = RePr = U L/D Rayleigh Number Ratio of natural convective to diffusive heat/mass transport Determines the transition to turbulence 3 3 Ra = GrPr = gβΔT L /να Ra = GrPr = gβC ΔCL /νD 1 .

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