Buckingham-Pi theorem

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Buckingham-Pi Theorem

The Buckingham method, sometimes referred to as the Buckingham pi theorem, is a procedure used to obtain a proper set of dimensionless numbers. The theorem states that the difference between the number of variables (n) and the number of dimensions (j), equals the number of dimensionless groups (k). The general process with the Buckingham method is to establish all of the variables involved in the particular process of interest (diameter, viscosity, heat capacity, etc.), and arrange these variables into dimensionless groups or pi groups. To find the pi groups the "core" groups must first be determined. Variables included in the core groups must contain all of the needed dimensions (j) and no two variables in the core group can have the same exact dimensions. It is very important to be thorough when selecting the variables involved in the analysis, for there is no way to actually prove that all variables have been included. Convenient dimensions in SI units include length(L) [m], mass(M) [kg], temperature() [K], and time(t) [s].


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Procedure

Procedure BPT.jpg

Common Variables

Variable Symbol Dimensions Variable Symbol Dimensions Variable Symbol Dimensions
Mass Transfer Coefficient Heat Capacity Thermal Conductivity
Diffusivity Diameter Drag Force F
Density Volume Expansion Coefficient Temperature Difference ΔT
Viscosity Gravity Surface Tension σ
Velocity Angular Velocity Ω Mass Avg. Velocity G
Characteristic Dimension Pressure P Kinematic Viscosity α
Heat Transfer Coefficient Roughness Power W

Summary of Dimensionless Numbers

General Number Symbol Name Variables Description or Usage
NRe Reynolds number Inertial to viscous forces or convective to molecular momentum transfer
NPe Peclet number Convective to molecular conductive heat transfer
NPr Prandtl number Momentum to thermal diffusivity
NPe, mass Peclet, mass Convective to molecular mass transfer
NSc Schmidt number Momentum to mass diffusivity
NLe Lewis number Thermal to mass diffusivity; also, ratio of Schmidt to Prandtl number
NEu Euler number Pressure to inertial forces or momentum generation to convective momentum transfer
NFr Froude number Inertial to gravitational forces or convective momentum transfer to gravitational momentum transfer
- - Convective heat transfer to viscous dissipation heat generation
NBr Brinkman number Viscous dissipation heat generation to molecular conductive heat transfer
NDm1 Damkohler 1 number Chemical reaction generation to convective mass transfer
NDm2 Damkohler 2 number Chemical reaction generation to molecular diffusion mass transfer
NWe Weber number Inertial to surface forces
NNu Nusselt number Total heat transfer to molecular heat transfer
NSh Sherwood number Total mass transfer to molecular mass transfer
Fanning friction factor Shear stress at the wall to the kinetic energy of flow
NSt Stanton number Total heat transferred to total heat capacity:
NSt, mass Stanton number
NAr Archimedes number Fluidization
NBi Biot number Unsteady-state heat conduction
Nb blend number Agitation
JH Colburn heat Colburn factor for heat transfer analogy
JM Colburn mass Colburn factor for mass transfer:
Nco Condensation number Condensation
NDn Dean number Flow in curved tubes
NDe Deborah number Flow of elastic fluid
CD Drag coefficient Flow past immersed bodies
NFo Fourier number Non-dimensional time parameter
NGr Graetz number Heat transfer, laminar forced convection
NGr Grashof number Reynolds number times the ratio of buoyancy force to viscous force (natural convection heat transfer)
NKn Knudsen number Flow of gasses at low pressure
NMa Mach number Flow above the speed of sound
Npo Power number Agitation
Np Pumping number Agitation
NSl Strouhal number Periodic flows
NVK Von Karman number Eliminates Velocity in correlations for Δp