# Reynolds Number

## Introduction and definition of the dimensionless Reynolds Number - with online calculators

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The Reynolds Number, the non-dimensional velocity, can be defined as the ratio of

- the inertia force
*(ρ u L),*and - the viscous or friction force
*(μ)*

and interpreted as the ratio of

- the dynamic pressure (
*ρ u*), and^{2} - the shearing stress (
*μ u / L*)

and can be expressed as

Re = (ρ u^{2}) / (μ u / L)

= ρ u L / μ

= u L / ν (1)

where

Re = Reynolds Number (non-dimensional)

ρ = density (kg/m^{3}, lb_{m}/ft^{3}^{ })

u = velocity based on the actual cross section area of the duct or pipe(m/s, ft/s)

μ = dynamic viscosity (Ns/m^{2}, lb_{m}/s ft)

L = characteristic length (m, ft)

ν =μ/ρ= kinematic viscosity (m^{2}/s, ft^{2}/s)

### Reynolds Number for a Pipe or Duct

For a pipe or duct the characteristic length is the hydraulic diameter.

*L = d _{h}*

*where*

*d _{h} = hydraulic diameter (m, ft)*

The Reynolds Number for a duct or pipe can be expressed as

Re = ρ u d_{h}/ μ

= u d_{h}/ ν (2)

where

d_{h}= hydraulic diameter (m, ft)

#### Reynolds Number for a Pipe or Duct in Imperial Units

The Reynolds number for a pipe or duct can also be expressed in Imperial units

Re = 7745.8 u d_{h}/ ν (2a)

where

Re = Reynolds Number (non dimensional)

u = velocity (ft/s)

d_{h}= hydraulic diameter (in)

ν = kinematic viscosity (cSt) (1 cSt = 10^{-6}m^{2}/s )

The Reynolds Number can be used to determine if flow is laminar, transient or turbulent. The flow is

**laminar**- when*Re < 2300***transient**- when*2300 < Re < 4000***turbulent**- when*Re > 4000*

In practice laminar flow is only actual for viscous fluids - like crude oil, fuel oil and oils.

### Example - Calculating Reynolds Number

A Newtonian fluid with a dynamic or absolute viscosity of *0.38 Ns/m ^{2}* and a specific gravity of

*0.91*flows through a

*25 mm*diameter pipe with a velocity of

*2.6 m/s*.

The density can be calculated using specific gravity like

ρ = 0.91 (1000 kg/m^{3})

= 910 kg/m^{3}

The Reynolds Number can then be calculated using equation *(1)* like

Re = (910 kg/m^{3}) (2.6 m/s) (25 mm) (10^{-3}m/mm) / (0.38 Ns/m^{2})

= 156 ((kg m / s^{2})/N)

= 156 ~ Laminar flow

1 (N) = 1 (kg m / s^{2})

### Related Mobile Apps from The Engineering ToolBox

- free apps for offline use on mobile devices.

### Online Reynolds Calculator

#### Density and the absolute (dynamic) viscosity is known

The calculator below can be used if the density and the absolute (dynamic) viscosity of a fluid is known. The calculator is valid for incompressible flow - flow with fluids or gases without compression - as typical for an air flow in a HVAC systems or similar.

The default values are for air at *60 ^{o}F*,

*2 atm*pressure and density

*0.146 lb*, flowing

_{m}/ft^{3}*20 ft/s*between two metal sheets with characteristic length

*0.5 ft*. Dynamic (absolute) viscosity is

*1.22 10*.

^{-5}lb_{m}/s ft#### Kinematic viscosity is known

The calculator below can be used when the kinematic viscosity of a fluid is known.

The default values are for water at *20 ^{o}C* with kinematic viscosity

*1.004 10*in a schedule 40 steel pipe. The characteristic length (hydraulic diameter) of the pipe is

^{-6}m^{2}/s*0.102 m*.

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