Mathematical model of Heat Transfer Coefficient through the annulus gap between concentric pipes filled with fluid:

(1)	$\begin{array}{l}\displaystyle U = \frac{\lambda_{ann}}{d_h} \, {\rm Nu}_{ann}\end{array}$

where

$\begin{array}{l}\lambda_{ann}\end{array}$	thermal conductivity of fluid in the annulus
$\begin{array}{l}d_{ann}\end{array}$	annular hydraulic diameter
$\begin{array}{l}{\rm Nu}_{ann}\end{array}$	dimensionless Nusselt number (Nu)

The Nusselt number (Nu) correlations are:

Stagnant fluid

Natural Convection

Forced Convection

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

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

(4)	$\begin{array}{l}\displaystyle {\rm Nu}= c \cdot \mbox{Re}_D^p \cdot \mbox{Pr}^{0.4}\cdot \left( \frac{\mu}{\mu_w} \right)^{0.14}\end{array}$

where

$\begin{array}{l}\mbox{Re}\end{array}$

$\begin{array}{l}\epsilon({\rm Ra})\end{array}$

$\begin{array}{l}\mbox{Pr} = \nu / a\end{array}$

$\begin{array}{l}{\rm Ra}\end{array}$

$\begin{array}{l}\nu\end{array}$

$\begin{array}{l}a\end{array}$

$\begin{array}{l}r_{out}\end{array}$

inner radius of outer pipe

$\begin{array}{l}r_{in}\end{array}$

outer radius of inner pipe

$\begin{array}{l}\zeta = r_{out}/r_{in}\end{array}$

$\begin{array}{l}p = 1.013 \cdot\exp \left[ -0.067 \cdot \zeta \right]\end{array}$

$\begin{array}{l}\displaystyle c = \frac{0.03 \, \zeta^{1.86}}{0.063 \, \zeta^3 -0.674 \, \zeta^2 +2.225 \, \zeta - 1.157 }\end{array}$

Reference

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