# Dimensionless Number

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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

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