Dimensionless Number

Dimensionless numbers are widely applied in the field of mechanical and chemical engineering. According to the Pi-Theorem, the functional dependence between a certain number of variables (e.g.: n) can be reduced by the number of independent dimensions occuring in those variables (e.g. k) to give a set of independent, dimensionless numbers (e.g. p=n-k). The dimensionless numbers can be derived by dimensional analysis.

Example: Stirrer

The power-consumption of a stirrer is a function of the density and the viscosity of the fluid to be stirred, the size and the particular geometry of the stirrer given by the diameter of the stirrer, and the speed of the stirrer. Therefore, we do have n=5 variables representing our example.

Those n=5 variables are build up by k=3 dimensions being:

• Length L [m]
• Time T [s]
• Mass M [kg]

According to the Pi-Theorem, the n=5 variables can be reduced by the k=3 dimensions to form p=n-k=5-3=2 independent dimensionless numbers which are in case of the stirrer

• Reynold-Number (This is the most important dimensionless number)
• Power-Number

There are literally thousands (to be precise: infinite) dimensionless numbers including those being used most often: (in alphabetical order indicating the field of use)

• Archimedes-Number Motion of fluids due to density differences
• Biot-Number Surface vs volume conductivity of solids
• Capillary-Number fluid flow influenced by surface tension
• Deborah-Number Rheology of viscoelastic fluids
• Drag-Coefficient Flow resistance
• Euler-Number Hydrodynamics (pressure forces vs. inertia forces)
• Friction-Factor Fluid Flow
• Froude-Number Wave and surface behaviour
• Grashof-Number Free convection
• Laplace-Number Free convection with inmiscible fluids
• Mach-Number Gasdynamics
• Nusselt-Number Heat transfer with forced convection
• Ohnesorge-Number Atomization of liquids
• Peclet-Number Forced convection
• Power-Number Power consumption by agitators
• Prandtl-Number Forced and Free convection
• Reynolds-Number Characterizing the flow behaviour (laminar or turbulent)
• Sherwood-Number Mass transfer
• Stokes-Number Dynamics of particles
• Strouhal-Number Oscillatory flows
• Weber-Number Characterization of mulitphase flow with strongly bended surfaces
• Weissenberg-Number Viscoelastic flows