NUSSELT NUMBER

The 'Nusselt number' is a dimensionless number that measures the enhancement of heat transfer from a surface that occurs in a 'real' situation, compared to the heat transferred if just conduction occurred. It is named after Wilhelm Nusselt, a German engineer, who was born 25 November 1882, in Nurnberg, Germany.
Typically, the Nusselt number (Nu) it is used to measure the enhancement of heat transfer when convection takes place.
: in perpendicular to the flow direction
where

★ ''L'' = characteristic length, which is simply Volume of the body divided by the Surface Area of the body (useful for more complex shapes)

★ ''k_f'' = thermal conductivity of the "fluid"

★ ''h'' = convection heat transfer coefficient
Selection of the significant length scale should be in the direction of growth of the boundary layer. A salient example in introductory engineering study of heat transfer would be that of a horizontal cylinder versus a vertical cylinder in free convection.
Several empirical correlations are available that are expressed in terms of Nusselt number in the elementary analysis of flow over a flat plate etc. Sieder-Tate, Colburn and many others have provided such correlations.
For a local Nusselt number, one may evaluate the significant length scale at the point of interest. To obtain an average Nusselt number analytically one must integrate over the characteristic length. More commonly the average Nusselt number is obtained by the pertinent correlation equation, often of the form Nu = f(Ra, Pr).
The Nusselt number can also be viewed as being a dimensionless temperature gradient at the surface.
The mass transfer analog of the Nusselt number is the Sherwood number.

Contents

Empirical calculations

Free convection at a vertical wall

Free convection from horizontal plates

Forced convection in pipe flow

References

Empirical calculations

Free convection at a vertical wall

Cited as coming from Churchill and Chu Fundamentals of Heat and Mass Transfer, , Frank P., Incropera, Wiley, , p 493

ight]^{4/9} ,} quad Ra_L le 10^9

Free convection from horizontal plates

For the top surface of a hot object in a colder environment or bottom surface of a cold object in a hotter environment
$overline\{Nu\}\_L\; =\; 0.54\; Ra\_L^\{1/4\}\; ,\; quad\; 10^4\; le\; Ra\_L\; le\; 10^7$
$overline\{Nu\}\_L\; =\; 0.15\; Ra\_L^\{1/3\}\; ,\; quad\; 10^7\; le\; Ra\_L\; le\; 10^\{11\}$
For the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotter environment
$overline\{Nu\}\_L\; =\; 0.27\; Ra\_L^\{1/4\}\; ,\; quad\; 10^5\; le\; Ra\_L\; le\; 10^\{10\}$

Forced convection in pipe flow

The ''Dittus-Boelter'' equation (for turbulent flow), with n=0.4 for heating of the fluid, and n=0.3 for cooling of the fluid:
$Nu\_D\; =\; 0.023\; Re\_D^\{4/5\}\; Pr^\{n\}$

References