@wikipedia
Dimensionless quantity characterising the ratio of thermal convection to thermal conduction in fluids across (normal to) the boundary with solids:
{\rm Nu} = \frac{\rm Convective \ heat \ transfer}{\rm Conductive \ heat \
transfer} = \frac{U}{\lambda / L} =\frac{U \cdot L}{\lambda } |
where 
is the convective heat transfer coefficient of the flow, 
is the characteristic length, 
is the thermal conductivity of the fluid.
Stagnant Fluid
For Stagnant Fluid the Nusselt number is a constant number (OEIS sequence A282581):
Natural Convection
In Natural Fluid Convection becomes dependant on Rayleigh number 
and Prandtl number 
: 
:
\mbox{Nu}_D= \left[ 0.825 + \frac{0.387 \, \mbox{Ra}_D^{1/6}}{ \left[ 1+ (0.492/\mbox{Pr})^{9/16} \right]^{8/27}} \right]^2 |
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Churchill and Chu
Churchill, S.W. and Chu, H.H.S. (1975) Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate. International Journal of Heat and Mass Transfer, 18, 1323-1329.
http://dx.doi.org/10.1016/0017-9310(75)90243-4 |
| |
\mbox{Nu}_L= 0.68 + \frac{0.663 \, \mbox{Ra}^{1/4}}{ \left[ 1+ (0.492/\mbox{Pr})^{9/16} \right]^{4/9}} |
|
Churchill and Chu
Churchill, S.W. and Chu, H.H.S. (1975) Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate. International Journal of Heat and Mass Transfer, 18, 1323-1329.
http://dx.doi.org/10.1016/0017-9310(75)90243-4 |
| |
In case of natural convection in the annulus the Nusselt number becomes also dependant on the annulus geometry:
{\rm Nu}_{ann} = \frac{2 \cdot \epsilon({\rm Ra})}{\ln (r_{out}/r_{in})} |
|
where
Forced Convection
In Forced Fluid Convection the Nusselt number becomes dependant on Reynolds number 
and Prandtl number : 
.
{\rm Nu}=3.66 + \frac{ 0.065 \cdot {\rm Re} \cdot {\rm Pr} \cdot {D/L} }{ 1 + 0.04 \cdot ({\rm Re} \cdot {\rm Pr} \cdot {D/L})^{2/3} } |
|
Mills
A.F. Mills, Heat Transfer, Second Edition, Prentice-Hall, New Jersey, 1999. |
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Laminar flow in pipeline with diameter  and length . |
{\rm Nu}=0.023 \cdot \mbox{Re}_D^{3/4} \cdot \mbox{Pr}^{0.4} |
| | |
{\rm Nu}=\frac{ (f/8) \, ({\rm Re} - 1000) {\rm Pr} }{ 1 + 12.7 \, (f/8)^{1/2} \, ({\rm Pr}^{2/3} -1) } |
|
Gnielinski
Gnielinski, Volker (1975). "Neue Gleichungen für den Wärme- und den Stoffübergang in turbulent durchströmten Rohren und Kanälen". Forsch. Ing.-Wes. 41 (1): 8–16. |
| 
 is Darcy friction factor
|
{\rm Nu}=0.3 + \frac{0.62 \, \mbox{Re}^{1/2} \, \mbox{Pr}^{1/3} }
{\left[ 1+ (0.4/\mbox{Pr})^{2/3} \right]^{1/4}}
\left[ 1 + \left( \frac{\mbox{Re}}{282000} \right)^{5/8}\right]^{4/5} |
|
Churchill–Bernstein
Churchill, S. W.; Bernstein, M. (1977), "A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow", Journal of Heat Transfer, 99 (2): 300–306, Bibcode:1977ATJHT..99..300C, doi:10.1115/1.3450685 |
| |
Both numbers naturally arise in modelling the heat exchange between solid body and fluid.
Both numbers have similar definition except that Nusselt number is based on thermal conductivity of the fluid while Biot Number is based on thermal conductivity of the solid body.
Normally Nusselt number indicates whether conductive or convective heat transfer dominates across the interface between solid body and fluid.
While Biot Number indicates whether significant thermal gradient will develop inside a solid body based on the ratio of heat transfer away from the surface of a solid body to heat transfer within the solid body.
See also
Physics / Thermodynamics / Heat Transfer
[ Heat Transfer Coefficient (HTC) ] [ Heat Transfer Coefficient @model ]
[ Dimensionless Heat Transfer Numbers ]
[ Prandtl number ] [ Rayleigh number ] [ Reynolds number ] [ Biot Number ]
References
