@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 

Stagnant Fluid

For  Stagnant Fluid the Nusselt number is a constant number (OEIS sequence A282581):

{\rm Nu}=3.6568

Natural Convection

In Natural Fluid Convection becomes dependant on Rayleigh 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

\mbox{Nu}_L= 0.68 + \frac{0.663 \, \mbox{Ra}^{1/4}}{ \left[ 1+ (0.492/\mbox{Pr})^{9/16} \right]^{4/9}}

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 

{\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} }

{\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) }

{\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}

Relation to Biot Number  

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