Reynolds number: Definition, Significance, Formula, Examples [with Pdf]

What is Reynolds number?

The term Reynolds number is the ratio of inertia force to the viscous force present in the fluid.
It is denoted by the symbol Re.

Mathematically, Reynolds number is given by,

`R_{e}=\frac{\rho. u.L}{\mu }`

Where,
ρ = Density of fluid
u = Average velocity of fluid
L = Characteristics length
μ = Dynamic viscosity of the fluid

The formula of Reynolds number can be also written as,

`R_{e}=\frac{uL}{\nu }` – – – – [`\because \nu=\frac{\mu }{\rho }`]

Where `\nu ` is kinematic viscosity.

For the fluid passing through a pipe with a circular cross-section, the Reynolds number can be expressed as,

`R_{e}=\frac{\rho uD}{\mu }` or `\frac{uD}{\nu }`

Where d is the diameter of fluid flow.

Significance of reynolds number in heat transfer:

The Reynolds number has the following importances:-

1) The Reynolds number indicates the relation between inertia forces and viscous forces in the fluid flow.
2) Based on the Reynolds number, the criteria was developed to identify the type of flow.
The criteria for finding the type of flow is given below:

If Re < 2300, then the flow is laminar.
If 2300 < Re < 3500, then the flow is transient.
If Re > 3500, then the flow is turbulent.

3) Reynolds number is used in convective heat transfer for the criteria of kinematics and dynamics similarity.

What is critical reynolds number?

The critical Reynolds number is the Reynolds number at which the fluid starts to change its nature from laminar to turbulent.

The value of the critical Reynolds number is different for different geometries.

A) For flow through the circular pipe, the fluid starts to change its nature from laminar to turbulent at Re = 2300, therefore for flow through the circular pipe, the critical Reynolds number is 2300.
`Re_{cr\ (\text{pipe})}` = 2300

B) For flow over a flat plate, the fluid changes its nature from laminar to turbulent at Re = `5 \times 10^{5}`, and hence the critical reynolds number for flow over a flat plate is `5 \times 10^{5}`.
`Re_{cr\ (\text{plate})}` = `5 \times 10^{5}`.

Reynolds number for laminar and turbulent flow:

One of the advantages of the Reynolds number is that it helps to find the nature of the fluid flow.
The value of Reynolds number for different flow nature was calculated by performing experiments.

1) Reynolds number for flow through a circular pipe:

For flow through the pipe, the limiting value of Reynolds number for different flow natures are,
For laminar flow: Re < 2300
For transition flow: 2300 < Re < 3500
For turbulent flow: Re > 3500

2) For flow over the flat plate:

For the fluid flowing over a flat plate, the limiting value of Reynolds number for different flow natures are,
For laminar flow: Re < 5× `10^{5}`
For turbulent flow: Re > 5× `10^{5}`

Reynolds number examples:

1) Water at 6 m/s flows through the pipe of 25 mm diameter. The water has a dynamic viscosity of 0.001 Pa.s. Check the flow is in a laminar or transition state.

Solution:-

Given:-
`\nu ` = 0.6 m/s
D = 25 mm
`ρ_{water}` = 1000 Kg/m³
μ = 0.001 Pa.s = 0.001 N.s/m²

The reynolds number is given by,

`R_{e}=\frac{\rho vD}{\mu }`=`\frac{1000\times 0.6\times 0.025}{0.001}`

Re = 15000

As Re > 3500, Therefore the flow is turbulent.

2) An oil of the density 800 Kg/m³ and viscosity 0.7 poise flows through the pipe of diameter 180 mm with a velocity of 1.8 m/s. Check the type of flow.

Solution:-

Given:
ρ = 800 Kg/m³
μ = 0.7 poise = 0.07 N.s/m²
D = 180 mm = 0.18 m
V = 1.8 m/s

The reynolds number is given by,

`R_{e}=\frac{\rho uD}{\mu }`=`\frac{800\times 1.8\times 0.18}{0.07}`

Re = 3702

As 2300 > Re > 3500, The flow is in the transition state

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