In the previous lesson we found different relations to estimate the heat transfer due
to convection for internal flows.
However, in many real-world applications, the heat transfer coefficient is not known.
Hence, we need to rely on expressions and correlations that let us estimate the Nusselt number,
so that we can then calculate the heat transfer coefficient
and predict the convective heat transfer in the system.
For certain applications, we may find that the predicted heat transfer may not be sufficient,
hence, we need to use designs that can enhance it. So, let’s go and explore these topics.
Let’s start our analysis with laminar fully developed flows.
Consider a steady, incompressible flow with constant properties through a circular pipe with constant diameter.
If we analyze an infinitesimal small control volume, we can assume that the heat conduction acts only
in the radial direction and the thermal energy advection occurs only in the axial direction.
We can write the thermal equilibrium in this form and then simplify it into this final relation.
Under the assumption of constant surface heat flux the equation can be reduced in this form.
These terms in the expression form the inverse of the heat transfer coefficient.
Hence, we can see that the Nusselt number is constant.
In the case of constant surface temperature,
the equation leads to an expression that can only be solved numerically.
However, the Nusselt number for this case can be derived,
and even in this case it is a constant value. This means that for laminar fully developed flows
the heat transfer coefficient is independent on the Reynolds and Prandtl numbers.
To estimate the Nusselt number in the entrance region for laminar flows we need to use some
correlations since some of the assumptions we made before are not satisfied anymore.
The axial heat transfer is not negligible and the advection acts also in the radial direction.
In this case we can have two different situations:
1) the thermal entrance length problem,
where the thermal profile is developing and the velocity profile is fully developed;
2) the combined entrance length problem, where both the profiles are developing.
Here are the correlations that we could use for these two cases, for the validity ranges that are shown here.
We can see how the Nusselt number is now a function of Reynolds and Prandtl numbers.
For turbulent flows, we need to rely on empirical correlations to estimate the Nusselt number and
obtain the heat transfer coefficient. Depending on the operating conditions
we can find several correlations. Here are some of the commonly used ones.
The first one here, has two expressions depending whether the fluid is heating or cooling,
that are good for small temperature and properties variations.
The different exponents are used to take into account the dependency of viscosity with temperature.
Heating a fluid would reduce its viscosity near the wall enhancing the heat transfer,
while cooling it down increases the viscosity and causes a reduction of the heat transfer.
When temperature and properties variation gets larger, this other correlation is more accurate.
Lastly, this correlation can be used for
a large range of Reynolds numbers, including transitional flows.
In general, for turbulent flow it is common to assume that the Nusselt number in the entrance
region is equal to the one in the fully developed region, since the entrance length is relatively short.
But in the case of short tubes it is better to use appropriate correlations instead.
We analyzed how we can estimate the heat transfer through pipes, but can we enhance it?
The answer is yes. There are multiple ways to do so. One way to accomplish this is by increasing
the surface area where the fluid and the solid are in contact. Using longitudinal fins
or helical ribs we can directly accomplish that.
Another approach is to increase the flow turbulence
by adding a coil-spring wire inside the tube, so that the mixing can help transfer the heat.
Also swirling flows have a better heat transfer rate. Twisted tapes
can orient the flow making it swirl so that a tangential velocity component is created,
increasing the speed of the flow near the wall.
Secondary flows can be very important to
enhancing heat transfer. A simple way to generate them without using any insert is to coil the tube,
so that the curvature would naturally generate secondary motions. A pair of counterrotating
vortices that forms greatly enhances the heat transfer rate, they reduce the entrance length,
and make the fluid temperature more uniform across a cross-section of the tube.
For this reason, this design is commonly used for chemical processes to produce pharmaceuticals or personal care products.
All these heat transfer enhancements come at a cost.
The pressure losses increase, so a more powerful fan or pump would be needed to drive the fluid.
Also, designers must consider that inserts may break, and the pieces could damage
expensive machinery, hence only sturdy design should be considered.
This concludes our lesson on forced convection correlations and enhancements in internal flows.