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

cp:Specific heat at constant pressure
D:Diameter
DAB:Binary mass diffusion coefficient
g:Gravitational Acceleration
h:Heat transfer coefficient
hm:Convective mass transfer coefficient
H:Enthalpy
k:Thermal conductivity
L:Length
LC: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

NameCategoryDefinitionDescription
Biot number (Bi)Heat transfer

Ratio of internal thermal resistance to boundary layer thermal resistance
Mass transfer Biot number (Bim)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 (Cf)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 (Fom)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 (GrL)Fluid Dynamics/Heat Transfer

Ratio of buoyancy to viscous forces acting on a fluid
Colburn number, heat (jH)Heat Transfer

Dimensionless heat transfer coefficient
Colburn number, mass (jH)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 (PeL,h)Heat Transfer

Dimensionless independent heat transfer (ratio of advective heat transport to convective heat transfer)
Peclet number, mass (PeL,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 (ShL)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 (Stm)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

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

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