# Summary

Dimensionless numbers play an important role in analysing fluid dynamics and heat and mass transfer problems. They provide a method by which complex phenomena can be characterised, often by way of a simple, single number comparison. This article provides a summary of dimensionless numbers and the formulae used to calculate them.

# Definitions

c_{p} | : | Specific heat at constant pressure |

D | : | Diameter |

D_{AB} | : | Binary mass diffusion coefficient |

g | : | Gravitational Acceleration |

h | : | Heat transfer coefficient |

h_{m} | : | Convective mass transfer coefficient |

H | : | Enthalpy |

k | : | Thermal conductivity |

L | : | Length |

L_{C} | : | Characteristic length |

p | : | Pressure |

T | : | Temperature |

t | : | Time |

u | : | Mass average fluid velocity |

V | : | Volume |

V | : | Fluid velocity |

Greek Characters:

: | Thermal diffusivity | |

: | Volumetric thermal expansion coefficient | |

: | Surface tension | |

: | Density | |

: | Shear stress | |

: | Dynamic viscosity | |

: | Kinematic viscosity |

Subscripts:

: | Actual or adiabatic process | |

: | Fluid properties; saturated liquid conditions | |

: | Saturated liquid conditions | |

: | Based on characteristic length | |

: | Mean value over cross section | |

: | At surface conditions | |

: | At saturated conditions | |

: | Saturated vapor conditions | |

: | Difference in conditions for vaporisation | |

: | Free/bulk stream conditions |

# Introduction

Dimensionless numbers are scalar quantities commonly used in fluid mechanics and heat transfer analysis to study the relative strengths of inertial, viscous, thermal and mass transport forces in a system.

Dimensionless numbers are equal for dynamically similar systems; systems with the same geometry, and boundary conditions. This makes also them a powerful tool for scaling operations from model to pilot and beyond.

This article provides a quick reference of dimensionless numbers and how to calculate them. To learn about the derivation and use of dimensionless numbers, see our article on Dimensionless Numbers and Dimensional Analysis.

# Index of Dimensionless Numbers

Name | Category | Definition | Description |
---|---|---|---|

Biot number (Bi) | Heat transfer | Ratio of internal thermal resistance to boundary layer thermal resistance | |

Mass transfer Biot number (Bi_{m}) | Mass transfer | Ratio of the internal mass transfer resistance to the mass transfer resistance at the boundary layer | |

Bond number (Bo) | Fluid Dynamics | Ratio of gravitational and surface tension forces. Also known as the Eötvös number (Eo). | |

Coefficient of friction (C_{f}) | Fluid Dynamics | A dimensionless surface shear stress showing the relationship between the forces of friction between two objects and their normal reaction forces. | |

Eckert number (Ec) | Heat Transfer | The ratio of kinetic energy of a flow to the boundary layer enthalpy difference or more generally the heat dissipation potential of an advective flow. | |

Fourier number (Fo) | Heat Transfer | Characterises transient heat conduction, it is the ratio of the heat conduction rate to the rate of thermal energy storage in a solid. | |

Mass Transfer Fourier number (Fo_{m}) | Mass Transfer | Characterises transient mass diffusion, it is the ratio of species diffusion rate to the rate of species storage. | |

Friction Factor (f) | Fluid Dynamics | Dimensionless pressure drop for internal fluid flow | |

Grashof number (Gr_{L}) | Fluid Dynamics/Heat Transfer | Ratio of buoyancy to viscous forces acting on a fluid | |

Colburn number, heat (j_{H}) | Heat Transfer | Dimensionless heat transfer coefficient | |

Colburn number, mass (j_{H}) | Mass Transfer | Dimensionless mass transfer coefficient | |

Jakob number (Ja) | Heat Transfer | Ratio of sensible heat to latent energy absorbed during liquid-vapour phase change | |

Lewis number (Le) | Mass Transfer | Ratio of thermal diffusivity to mass diffusivity | |

Nusselt number (Nu) | Heat Transfer | Dimensionless temperature gradient at the surface | |

Peclet number, heat (Pe_{L,h}) | Heat Transfer | Dimensionless independent heat transfer (ratio of advective heat transport to convective heat transfer) | |

Peclet number, mass (Pe_{L,m}) | Mass Transfer | Dimensionless independent mass transfer (ratio of advective mass transport to diffusive transfer) | |

Prandtl number (Pr) | Heat Transfer | Dimensionless independent heat transfer | |

Rayleigh number (Ra) | Fluid Dynamics | Ratio of thermal transport via diffusion vs thermal transport via convection | |

Reynolds number (Re) | Fluid Dynamics | Ratio of inertial to viscous forces | |

Schmidt number (Sc) | Mass transfer | Ratio of momentum and mass diffusivities | |

Sherwood number (Sh_{L}) | Mass transfer | Dimensionless concentration gradient at the surface (ratio of convective mass transfer to diffusion rate) | |

Stanton number (St) | Heat transfer | Ratio of heat transferred into a fluid to the thermal capacity of the fluid | |

Stanton number, mass (St_{m}) | Mass transfer | Dimensionless number characterising the species mass transfer in forced convective flows | |

Weber number (We) | Fluid Dynamics | Ratio of inertia to surface tension forces |

# Further Reading

- Perry's Chemical Engineers' Handbook, Eighth Edition
- Chemical Engineering Volume 1, Sixth Edition: Fluid Flow, Heat Transfer and Mass Transfer
- Fundamentals of Heat and Mass Transfer

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