# Reynolds Number

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

*inertia force (ρ u L)*to*viscous or friction force (μ)*

and interpreted as the ratio

*dynamic pressure (ρ u*to^{2})*shearing stress (μ u / L)*

Reynolds Number can therefore be expressed as

Re = ρ u L / μ

= ρ u^{2}/ (μ u / L)

= 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 Flow in 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 the flow in a duct or pipe can with the hydraulic diameter 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 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 other oils.

### Example - Calculate 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*.

Density can be calculated from the specific gravity of the fluid and the density of the specific gravity reference water * 1000 kg/m ^{3} * - as

ρ = 0.91 (1000 kg/m^{3})

= 910 kg/m^{3}

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 absolute (dynamic) viscosity is Known

This calculator can be used if density and absolute (dynamic) viscosity of the fluid is known. The calculator is valid for incompressible flow - flow with fluids or gases without compression - as typical for air flows in HVAC systems or similar. The calculator is generic and can be used for metric and imperial units as long as the use of units are consistent.

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 kinematic viscosity of the fluid is known. The calculator is generic and can be used for metric and imperial units as long as the use of units are consistent.

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

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

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

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