Comparison of Equations for Nusselt Number of Combined Entry Length

Recent editions of Incropera et al's Fundamentals of Heat and Mass Transfer provide different correlations for average Nusselt number in a combined entry problem. That is, neither velocity nor thermal profiles are fully developed. Both assume constant surface temperature with all properties evaluated at average temperature, excluding $$ \mu_s $$.

6th edition
$$\overline{Nu}_{D}=1.86(\frac{Re_{D}Pr}{L/D})^{1/3}(\frac{\mu}{\mu_s})^{1/4}$$

for $$0.60\leq Pr \leq 5 $$

and $$0.0044 \leq (\frac{\mu}{\mu_s}) \leq 9.75 $$

7th edition
$$\overline{Nu}_{D}=\frac{\frac{3.66}{\tanh(2.264Gz_{D}^{-1/3}+1.7Gz_{D}^{-2/3})}+0.0499Gz_{D}\tanh(Gz_{D}^{-1})}{\tanh(2.432Pr^{1/6}Gz_{D}^{-1/6})} $$

for $$ Pr \geq 0.1 $$

with $$ Gz_{D}\equiv(D/L)Re_{D}Pr $$

Usefulness of Graetz number
The Graetz number is a dimensionless number which greatly aids in comparing these equations. By plotting each $$\overline{Nu}_{D}$$ vs. $${Gz_D}^{-1}$$ we can avoid having to vary each parameter Re, D, and L independently. It's also accepted that fully developed velocity conditions exist for $${Gz_D}^{-1} \geq 0.05$$

Correction Factor

 * $$(\frac{\mu}{\mu_s})^{1/4}$$

The 6th edition presents the correlation with a correction factor based on variation in a fluid's viscosity due to temperature. The 7th edition introduces this correction a page later and mentions its importance when dealing with viscous liquids, especially oils. This is simply a scaling factor so we'll analyze the different equations without the correction factor. Similar to the 6th edition's correlation can be written in terms of the Graetz number, similar to the 7th. They respectively become equations A and B below.

Equation B's dependence on additional Pr
When written in terms of $$Gz_{D}^{-1}$$, Eqn. B is dependent on the Pr in addition to the Graetz number's dependence. This means that the equations must be plotted for varying Prandtl numbers, with only Eqn.B varying between plots. This leads me to believe that the newer equation, B, is generally more accurate as it is more responsive to the Prandtl number.

Valid intervals of Pr
The range of valid Prandtl numbers for Eqn. A is more restrictive. Eqn. B is valid for Pr as low as 0.1, compared to A's Pr=0.6. Both equations are valid with Pr as high as 5, where the fully developed velocity assumption becomes valid and a separate correlation could be used. However, Eqn. B is shown to be valid even past Pr=5.

Results
$$\overline{Nu}_{D}$$ vs. $${Gz_D}^{-1}$$ is plotted below for various Prandtl Numbers. A vertical bar appears at $${Gz_D}^{-1}=0.05$$. At this $${Gz_D}^{-1}$$ and larger the fully developed velocity assumption is valid and these equations would no longer be used. Regardless of Pr, equations A and B are most similar near $${Gz_D}^{-1}=0.05$$ with increasing divergence as $${Gz_D}^{-1}$$ decreases.



Pr=0.1 is valid for B while outside of the valid interval for A. The discrepancy between the equations is clear.

While still outside of the valid range for A, it is clear the two equations are converging.



Pr=0.6 begins the valid interval for Eqn. A. The equations are very similar leading up to $${Gz_D}^{-1}=0.05$$.

With Pr=1 the equations are almost identical in the region leading up to the fully developed region.



Pr=3 shows that the equations begin to diverge but are still reasonably accurate near $${Gz_D}^{-1}=0.05$$.

Pr=5 represents the largest Pr that either equation should be used for. They still agree reasonably well leading up to the fully developed region.



Here we see that beyond $${Gz_D}^{-1}=0.05$$ Eqn. B is nearly identical to an additional equation provided for the purely thermal entry length problem.

Here with P=10, the lack of an upper Pr number bound on Eqn. B. is validated.