Every time a hot fluid moves through a cold tube to be cooled down,
or a cold fluid is pushed through hot pipes
to extract the system heat, forced convection is used as the main heat transfer mechanism.
In this lesson we will explore some fundamentals of thermal analysis for internal flows.
When a fluid enters a pipe, a boundary layer starts to form from the inlet due
to viscous effects, and it grows as the fluid flows downstream. The velocity profile of the
fluid changes until the edge of the boundary layer reaches the pipe centerline. This region,
where the velocity profile is not constant, is called the hydrodynamic entrance region.
Past this region the velocity profile is fully developed and does not change anymore.
The shape of the profile and the length of the entrance region depend on the nature of the flow,
whether it is laminar or turbulent. For internal flows we can say that for Reynolds number based
on the pipe diameter smaller than 2300 we have laminar flows. In this case,
the hydrodynamic entrance length is a function of the Reynolds number.
A flow can become fully turbulent for Reynolds numbers above 4000. The entrance
length for turbulent flows is proportional to the 1/6th power of the Reynolds number.
For large Reynolds numbers the entrance length varies between about 10 and 60 pipe diameters.
For this analysis, we will assume that past 10 pipe diameters the entrance effects become negligible
for practical applications. Hence, beyond that length a turbulent flow would be fully developed.
If the fluid has a different temperature than the pipe,
a thermal boundary layer would also form due to convection. Let’s assume in our case that we are
looking at a cold fluid entering a hotter pipe. As the fluid moves downstream the thermal boundary
layer grows and reaches the centerline. Past this point the fluid becomes thermally fully developed,
and the developing region is called thermal entrance region. The shape of the profile
depends on the wall conditions. In this case we can see what the profile would look like
in the case of constant surface temperature or constant heat flux at the surface.
The thermal entrance length for laminar flows has the same expression as the hydrodynamic one
except for the Prandtl number.
This tells us that if the Prandtl number
is larger than 1 the thermal boundary layer would grow slower than the hydrodynamic one.
For Prandtl numbers lower than 1 the opposite would happen. In the case of turbulent flows,
instead, the entrance length is about 10 pipe diameters. The turbulence enhances the transport
of heat through the fluid, so that velocity and thermal layers have about the same thickness.
Since velocity and temperature profiles vary over a pipe cross section, it is useful to
simplify the analysis using mean values instead. The mean velocity is defined as the velocity
that would provide the current mass flow rate in a tube, given its cross-sectional area and
the fluid density. The mean or bulk temperature is the temperature that makes this product equal
to the thermal energy that the fluid is transporting through a cross-section of the tube.
We can now start our thermal analysis for internal flows.
Let’s consider a section of a pipe.
We assume that the fluid flow presents negligible variations of density,
and that the viscous dissipation and axial heat conduction are negligible too.
In this way the total convection heat transfer can be defined as the difference between
the advected thermal energy that enters and exits the tube.
If we look at an infinitesimal control volume in the pipe, the equation can be reduced
to its differential form.
Then expressing the heat transfer in terms of the heat flux
through the wall of the pipe, we can obtain an expression for the mean temperature variation.
As we would expect, the fluid temperature would increases or decreases depending on
whether the wall temperature is higher or lower than the fluid’s.
This equation can be used to find an expression for the mean temperature along the duct
to estimate the heat transfer. We can analyze two cases
that resemble common real-world applications.
Let’s start with the case of constant heat flux.
In this case, for a pipe of constant diameter, dTm/dx
is constant. Hence, the mean temperature varies linearly along the tube.
The temperature difference between the fluid and the wall varies along the entrance region, while
it is constant in the fully developed region since the heat transfer coefficient there is constant.
The convective heat transfer can be estimated by simply integrating the heat flux along the tube.
Let’s now look at a constant wall surface temperature. In this case it is convenient to
consider the temperature difference ΔT
between wall and fluid. With simple math we can obtain this final relation that shows that the
temperature difference decays exponentially along the tube.
We can apply this expression to the entire tube to estimate the heat transfer due to convection.
In this case we obtain an expression of “q” in terms of the log mean temperature difference.
If we know the average convection heat transfer coefficient,
we can then calculate the heat transfer for this case.
This concludes our lesson on the basics of forced convection in internal flows.