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 $(J/kg.K)$
$D$	:	Diameter $(m)$
$D_{AB}$	:	Binary mass diffusion coefficient $(m^2/s)$
$g$	:	Gravitational Acceleration $(m/s^2)$
$h$	:	Heat transfer coefficient $(W/m^2.K)$
$h_m$	:	Convective mass transfer coefficient $(m/s)$
$H$	:	Enthalpy $(J/kg)$
$k$	:	Thermal conductivity $(W/m.K)$
$L$	:	Length $(m)$
$L_C$	:	Characteristic length $(m)$
$p$	:	Pressure $(Pa)$
$T$	:	Temperature $(K)$
$t$	:	Time $(s)$
$u$	:	Mass average fluid velocity $(m/s)$
$V$	:	Volume $(m^3)$
$V$	:	Fluid velocity $(m/s)$

Greek Characters:


$\alpha$	:	Thermal diffusivity $(m^2/s)$
$\beta$	:	Volumetric thermal expansion coefficient $(K^{-1})$
$\sigma$	:	Surface tension $(N/m)$
$\rho$	:	Density $(kg/m^3)$
$\tau$	:	Shear stress $(N/m^2)$
$\mu$	:	Dynamic viscosity $(N s/m^2)$
$\nu$	:	Kinematic viscosity $(m^2/s)$

Subscripts:


$a$	:	Actual or adiabatic process
$f$	:	Fluid properties; saturated liquid conditions
$l$	:	Saturated liquid conditions
$L$	:	Based on characteristic length
$m$	:	Mean value over cross section
$s$	:	At surface conditions
$sat$	:	At saturated conditions
$v$	:	Saturated vapor conditions
$vap$	:	Difference in conditions for vaporisation
$\infty$	:	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	$\displaystyle \frac{hL_C}{k}$	Ratio of internal thermal resistance to boundary layer thermal resistance
Mass transfer Biot number ( $Bi_m$ )	Mass Transfer	$\displaystyle \frac{h_m L_C}{D_{AB}}$	Ratio of the internal mass transfer resistance to the mass transfer resistance at the boundary layer
Bond number (Bo)	Fluid Dynamics	$\displaystyle \frac{g(\rho_l - \rho_v)L_C^2}{\sigma}$	Ratio of gravitational and surface tension forces. Also known as the Eötvös number (Eo).
Coefficient of friction ( $C_f$ )	Fluid Dynamics	$\displaystyle \frac{2\tau_s}{\rho V^2}$	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	$\displaystyle \frac{V^2}{c_p(T_s - T_{\infty})}$	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	$\displaystyle \frac{\alpha t}{L^2}$	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	$\displaystyle \frac{D_{AB} t}{L^2}$	Characterises transient mass diffusion, it is the ratio of species diffusion rate to the rate of species storage.
Friction Factor (f)	Fluid Dynamics	$\displaystyle \frac{\Delta p}{\left(\frac{L}{D}\right)\left(\frac{\rho u^2_m}{2}\right)}$	Dimensionless pressure drop for internal fluid flow
Grashof number ( $Gr_L$ )	Fluid Dynamics / Heat Transfer	$\displaystyle \frac{g\beta(T_S - T_{\infty})L_C^3}{\nu^2}$	Ratio of buoyancy to viscous forces acting on a fluid
Colburn number, heat ( $j_H$ )	Heat Transfer	$\displaystyle \text{St.}\text{Pr}^{2/3}$	Dimensionless heat transfer coefficient
Colburn number, mass ( $j_H$ )	Mass Transfer	$\displaystyle \text{St}_m\text{Sc}^{2/3}$	Dimensionless mass transfer coefficient
Jakob number (Ja)	Heat Transfer	$\displaystyle \frac{c_p(T_s - T_{sat})}{\Delta H_{vap}}$	Ratio of sensible heat to latent energy absorbed during liquid–vapour phase change
Lewis number (Le)	Mass Transfer	$\displaystyle \frac{\alpha}{D_{AB}}$	Ratio of thermal diffusivity to mass diffusivity
Nusselt number (Nu)	Heat Transfer	$\displaystyle \frac{hL_C}{k_f}$	Dimensionless temperature gradient at the surface
Peclet number, heat ( $Pe_{L,h}$ )	Heat Transfer	$\displaystyle \frac{VL_C}{\alpha} = \text{Re}_L\text{Pr}$	Dimensionless independent heat transfer (ratio of advective to convective heat transfer)
Peclet number, mass ( $Pe_{L,m}$ )	Mass Transfer	$\displaystyle \frac{VL_C}{D_{AB}} = \text{Re}_L\text{Sc}$	Dimensionless independent mass transfer (ratio of advective mass transport to diffusive transfer)
Prandtl number (Pr)	Heat Transfer	$\displaystyle \frac{c_p \mu}{k} = \frac{\nu}{\alpha}$	Dimensionless independent heat transfer
Rayleigh number (Ra)	Fluid Dynamics	$\displaystyle \text{Gr.Pr} = \frac{g\beta\Delta T L^3_C}{\nu\alpha}$	Ratio of thermal transport via diffusion vs convection
Reynolds number (Re)	Fluid Dynamics	$\displaystyle \frac{VL_C}{\nu} = \frac{\rho V L_C}{\mu}$	Ratio of inertial to viscous forces
Schmidt number (Sc)	Mass Transfer	$\displaystyle \frac{\nu}{D_{AB}}$	Ratio of momentum to mass diffusivities
Sherwood number ( $Sh_L$ )	Mass Transfer	$\displaystyle \frac{h_m L_C}{D_{AB}}$	Dimensionless concentration gradient at the surface (ratio of convective mass transfer to diffusion rate)
Stanton number (St)	Heat Transfer	$\displaystyle \frac{h}{\rho V c_p} = \frac{\text{Nu}}{\text{Re.Pr}}$	Ratio of heat transferred into a fluid to the thermal capacity of the fluid
Stanton number, mass ( $St_m$ )	Mass Transfer	$\displaystyle \frac{h_m}{V} = \frac{\text{Sh}}{\text{Re.Sc}}$	Dimensionless number characterising species mass transfer in forced convective flows
Weber number (We)	Fluid Dynamics	$\displaystyle \frac{\rho V^2 L}{\sigma}$	Ratio of inertia to surface tension forces

Index of Dimensionless Numbers

Summary

Definitions

Introduction

Index of Dimensionless Numbers

Further Reading