## What is the Nusselt number?

The Nusselt number is the dimensionless number used in convective heat transfer analysis.

The Nusselt number is the ratio of heat transfer by convection to the heat transfer by conduction within a fluid (When the fluid is considered stationary).

Nussel number is denoted by the symbol Nu and it is given by,

`\therefore Nu =\frac{Q_{\text{Convection}}}{Q_{\text{Conduction}}}`

The Nusselt number is also known as the ratio of conductive resistance to the convective resistance.

`Nu =\frac{R_{\text{Conduction}}}{R_{\text{Convection}}}`

The term nusselt number is considered as a **unitless quantity**.

## Nusselt number equation:

As we have seen before,

`Nu =\frac{Q_{\text{Convection}}}{Q_{\text{Conduction}}}`

Where,

`Q_{\text{Conduction}} = KA\frac{\Delta T}{L}`

`Q_{\text{Convection}} =hA\DeltaT`

Now by putting these values in equation of Nusselt number we get,

Nu = `\frac{hA\Delta T}{KA\frac{\Delta T}{L}}`

Nu = `\frac{hL}{K}`

∴ This is required equation for Nusselt number,

Here,

h = Convective heat transfer coefficient, (w/m².K)

L = Charecteristics length, (m)

K = Thermal conductivity of fluid, (w/m.K)

For the natural convection, the equation for the Nusselt number can be written in the form of Grashoff number and Prandtl number as,

Nu = `C(Gr)^{m}(Pr)^{n}`

Here the values of C, m, and n are determined experimentally for different cases.

For the forced convection, the equation of the Nusselt number is written in the form of Reynolds number and Prandtl number as,

Nu = `C(Re)^{m}(Pr)^{n}`

Where the values of C, m, and n are determined experimentally for different conditions.

## Nusselt number significance:

The significances of Nusselt number are as follows:-

1] The Nusselt number gives the relation between heat transfer by conduction and heat transfer by convection.

2] When Nu = 1, It means that the rate of heat transfer by conduction is equal to the rate of heat transfer by convection or we can say that fluid is stationary.

3] If the Nu > 1, Then the heat transfer by convection is greater than heat transferred by convection.

## Nusselt number examples:

*Following are the parameters for the fluid flowing through the pipe. Find the relation between conduction heat transfer and convective heat transfer that takes place within the fluid.h = 20 W/mKLc = 4.5 mK = 6 W/mK*

**Solution:-**

The nusselt number is given by,

Nu = `\frac{hLc}{K}`

Nu = `\frac{20\times 4.5}{6}`= 15

The Nusselt number is also given by,

Nu = `\frac{Q_{\text{Convection}}}{Q_{\text{Conduction}}}`

`\therefore Q_{\text{Convection}}= 15 Q_{\text{Conduction}}`

This is the relation between the heat transfer by conduction and convection.

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