(Phys) meaningful ratio of parameters in a system arranged so that the dimensions cancel. This technique is useful in identifying factors in complex system and in modeling Cr scaling. A well-known and early example of a dimensionless group is the Mach number, the ratio of the flow velocity to the local velocity of sound, used in describing flow: systems. A large number of dimensionless groups are possible. The most complete list of such parameters has been compiled by Catchpole and Fulford, containing 285 dimensionless groups. Several useful and representative dimensionless numbers are given below. (1) Fanning Friction Factor: The ratio of the wall shear stress to the number of velocity heads. Used in calculating conduit flow. f = (d/⁴L) (2p/r^ov²), where d = characteristic diameter, L = characteristic length, p = pressure drop, r^o = fluid density and v = mean velocity. (2) Froude Number: The ratio of inertial force to gravitational force. Fr = v²/gL where g = acceleration of gravity. (3) Grashof Number: A number used in calculating free or natural convection heat transfer processes, given by Gr = (gr⁰²BTL³)/u² where B = thermal coefficient of volume expansion, T = temperature difference and u = viscosity. Written as Gr/Re², the group expresses the ratio of buoyancy forces to inertial forces; i. e. , Gr = (buoyancy forces) (inertial force)/(viscous forces)². (4) Lewis Number: The ratio of the energy transported by conduction to that transported by diffusion, given by Le = k/r^oDCp, where k = coefficient of thermal conductivity, r⁰ = density, D = diffusion coefficient and Cp = specific heat at constant pressure. The Lewis number is also expressed as the ratio of the Schmidt number to the Prandtl number Le = Sc/Pr. (5) Mach Number: The most important parameter in compressible flow theory, the Mach number is the ratio of the fluid velocity to the velocity of sound M = v/a where M < 1 is subsonic flow, and M > 1 is supersonic flow. (6) Nusselt Number: A number used in calculating forced convective heat transfer, given by Nu = hL/k, where h = heat transfer coefficient k = thermal conductivity. The Nusselt number Also called: Biot number) may be interpreted as a dimensionless temperature gradient averaged over the heat-transfer surface. It is often expressed as a function of Re, Pr, and geometry Nu = Nu (Re, Pr, L/d). (7) Peclet Number: The ratio of heat transfer by convection to heat transfer by conduction Pe = (Cpr^ovL)/k. (8) Prandtl Number: The ratio of molecular diffusivity of momentum to diffusivity of heat Pr = (Cpu) /k. The heat transfer analog of the mass transfer Schmidt number. (9) Reynolds Number: The ratio of inertial forces to viscous forces Re = r^ovd/u. (10) Rossby Number: The ratio of inertial force to Coriolis force Ro = v/2WeL sin e, where We = angular velocity, and e = angle between the axis of the earth's rotation and the direction of fluid motion. (11) Schmidt Number: The ratio of kinematic viscosity to molecular diffusivity Sc = u/r^oD = n/D = u/r^o where n = kinematic viscosity. The Schmidt number is the mass transfer analog of the heat transfer Prandtl number. (12) Stanton Number: The ratio of the Nusselt number to the product of the Reynolds number times the Prandtl number St = Nu/(Re Pr)