Non-Dimensional Parameters
This material is taken from (Bird, et. al. 2007). The non-dimensional
parameters are all over the field of fluids and thermal hydraulics.
Their importance is, in my opinion, overstated. Any paper worth its salt
will describe the non-dimensional number and its calculation before
using it as a parameter. However, the idea that one can create
parameters comparing effects without some sort of unit change is
important, and several numbers, such as \(Re\) and \(Pr\) are
seen a lot in the literature.
Table 1: Non-Dimensional Parameters and their definitions
| Symbol | Name | Formula | Physical Meaning |
|---|
| $Re$ | Reynold's | $\frac{\rho vD}{\mu}$ | $\frac{\text{inertial force}}{\text{viscous force}}$ |
| $Pr$ | Prandtl | $\frac{\nu}{\alpha}=\frac{c_{p}\mu}{k}$ | -- |
| $Gr$ | Grashof | $\frac{g\beta\Delta TD^{3}}{\nu^{2}}$ | -- |
| $Br$ | Brinkman | $\frac{\mu v^{2}}{k\Delta T}$ | $\frac{\text{heat production by viscous dissipation}}{\text{heat transport by conduction}}$ |
| $Pe$ | Peclet | $RePr$ | $\frac{\text{convection heat transport}}{\text{conduction heat transport}}$ |
| $Ra$ | Rayleigh | $GrPr$ | -- |
| $Ec$ | Eckert | $\frac{Br}{Pr}$ | -- |
| $Fr$ | Froude | $\frac{\rho v}{pgD}$ | $\frac{\text{inertial force}}{\text{gravitational force}}$ |
| -- | -- | $\frac{Gr}{Re^{2}}$ | $\frac{\text{buoyant force}}{\text{inertial force}}$ |
| $Ma$ | Mach | $\frac{v}{c}$ | -- |
| $Nu$ | Nusselt | $\frac{hD}{k}$ | $\frac{\text{convection}}{\text{conduction}}$ |
| $We$ | Weber | $\frac{Dv^{2}\rho}{\sigma}$ | $\frac{\text{inertial force}}{\text{surface tension}}$ |
| $Bi$ | Biot | $\frac{hD}{k}$ | $\frac{\text{convection}}{\text{conduction}}$ |
Bibliography
