Dimensionless quantity characterising the ratio of thermal convection to thermal conduction in fluids across (normal to) the boundary with solids:

(1)	$\begin{array}{l}\displaystyle {\rm Nu} = \frac{\rm Convective \ heat \ transfer}{\rm Conductive \ heat \ transfer} = \frac{U}{\lambda / L} =\frac{U \cdot L}{\lambda }\end{array}$

where $\begin{array}{l}U\end{array}$ is the convective heat transfer coefficient of the flow, $\begin{array}{l}L\end{array}$ is the characteristic length, $\begin{array}{l}\lambda\end{array}$ is the thermal conductivity of the fluid.

Stagnant Fluid

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

(2)	$\begin{array}{l}\displaystyle {\rm Nu}=3.6568\end{array}$

Natural Convection

In Natural Fluid Convection becomes dependant on Rayleigh number $\begin{array}{l}\rm Ra\end{array}$ and Prandtl number $\begin{array}{l}\rm Pr\end{array}$ : $\begin{array}{l}\mbox{Nu} = f (\mbox{Ra}, \mbox{Pr})\end{array}$ :

(3)	$\begin{array}{l}\displaystyle \mbox{Nu}_D= \left[ 0.825 + \frac{0.387 \, \mbox{Ra}_D^{1/6}}{ \left[ 1+ (0.492/\mbox{Pr})^{9/16} \right]^{8/27}} \right]^2\end{array}$

Churchill and Chu

All convection regimes in pipelines

$\begin{array}{l}\mbox{Ra}_D \leq 10^{12}\end{array}$

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

Churchill and Chu

Laminar convection

$\begin{array}{l}\mbox{Ra} \leq 10^9\end{array}$

In case of natural convection in the annulus the Nusselt number becomes also dependant on the annulus geometry:

(5)	$\begin{array}{l}\displaystyle {\rm Nu}_{ann} = \frac{2 \cdot \epsilon({\rm Ra})}{\ln (r_{out}/r_{in})}\end{array}$

where

$\begin{array}{l}\epsilon({\rm Ra})\end{array}$	Natural Convection Heat Transfer Multiplier
$\begin{array}{l}\rm Ra\end{array}$	Rayleigh number
$\begin{array}{l}r_{out}\end{array}$	inner radius of outer pipe
$\begin{array}{l}r_{in}\end{array}$	outer radius of inner pipe

Forced Convection

In Forced Fluid Convection the Nusselt number becomes dependant on Reynolds number $\begin{array}{l}\rm Re\end{array}$ and Prandtl number $\begin{array}{l}\rm Pr\end{array}$ : $\begin{array}{l}\mbox{Nu} = f (\mbox{Re}, \mbox{Pr})\end{array}$ .

(6)	$\begin{array}{l}\displaystyle {\rm Nu}=3.66 + \frac{ 0.065 \cdot {\rm Re} \cdot {\rm Pr} \cdot {D/L} }{ 1 + 0.04 \cdot ({\rm Re} \cdot {\rm Pr} \cdot {D/L})^{2/3} }\end{array}$

Mills

Laminar flow in pipeline with diameter $\begin{array}{l}D\end{array}$ and length $\begin{array}{l}L\end{array}$ .

(7)	$\begin{array}{l}\displaystyle {\rm Nu}=0.023 \cdot \mbox{Re}_D^{3/4} \cdot \mbox{Pr}^{0.4}\end{array}$

Dittus-Boelter

Turbulent flow in pipeline

$\begin{array}{l}\mbox{Re} \geq 10,000\end{array}$

(8)	$\begin{array}{l}\displaystyle {\rm Nu}=\frac{ (f/8) \, ({\rm Re} - 1000) {\rm Pr} }{ 1 + 12.7 \, (f/8)^{1/2} \, ({\rm Pr}^{2/3} -1) }\end{array}$

Gnielinski

Transitional flow and turbulent flow in rough pipeline

$\begin{array}{l}{\displaystyle 3000\leq \mathrm {Re}\leq 5\cdot 10^{6}}\end{array}$

$\begin{array}{l}0.5\leq \mathrm {Pr} \leq 2000\end{array}$

$\begin{array}{l}f\end{array}$ is Darcy friction factor

(9)	$\begin{array}{l}\displaystyle {\rm Nu}=0.3 + \frac{0.62 \, \mbox{Re}^{1/2} \, \mbox{Pr}^{1/3} } {\left[ 1+ (0.4/\mbox{Pr})^{2/3} \right]^{1/4}} \left[ 1 + \left( \frac{\mbox{Re}}{282000} \right)^{5/8}\right]^{4/5}\end{array}$

Churchill–Bernstein

All flow regimes in pipelines

$\begin{array}{l}\mbox {Re} \cdot \mbox {Pr} \geq 0.2\end{array}$

Accuracy $\begin{array}{l}\sim 20 \%\end{array}$

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.

Page tree

Stagnant Fluid

Relation to Biot Number

References

Page tree

Nusselt number = Nu

Stagnant Fluid

Relation to Biot Number

References