As a heating fluid moves through pipes, it warms the pipes and increases their temperature
above the ambient temperature. This causes the pipe to start exchanging heat with the
surrounding air mostly through external natural convection, for large Rayleigh numbers. Hence,
heat is released into the air and the fluid temperature reduces, causing a heat loss.
If the heat loss is too large, a proper insulation of the pipes would be needed. But how can
we estimate that? In this lesson we will learn how to analyze
systems exchanging heat with a still fluid surrounding them. We will focus on external
natural convection for different canonical shapes such as cylinders, plates and spheres
that can be representative of many everyday objects. We will introduce correlations that
can be used to estimate the average Nusselt number and thus the heat transfer for the
system.
Let’s start by defining what is external natural convection. When a system surrounded
by a still fluid is at a different temperature than the fluid, external natural convection
occurs. The fluid near the object heats up or cools down, its density changes and the
buoyant force makes the fluid parcels move along the gravitational field direction.
The heat transfer analysis for external natural convection consists of defining and using
the appropriate correlations to obtain the average Nusselt number, estimate the average
heat transfer coefficient and calculate the overall heat transfer.
Most of the correlations can be generalized in this form where the Rayleigh number is
calculated using an appropriate characteristic length and estimating the fluid properties
at the film temperature. The exponent n is typically ¼ for laminar flows and 1/3 for
turbulent flows. This means that in turbulent flows the heat transfer coefficient is independent
of the body characteristic length.
- Let’s start with our first geometry, the vertical plate. We can use these simple correlations
valid for laminar or turbulent flows. However, we can introduce a more complex relation that
is valid for any value of the Rayleigh number. If more accuracy is needed in the laminar
range, this correlation could be used instead. Looking at these correlations we can see that
they are also functions of the Prandtl number, and that a constant term is present. This
constant takes into account the effects of heat conduction for flows at small Rayleigh
number. These correlations are valid for isothermal
vertical plates, however they can be used also for constant surface heat flux, when
the temperature is not constant along the plate. The trick is to estimate the Nusselt
and Rayleigh numbers using the temperature difference at the half length of the plate.
The value of Delta T_L/2 can be obtained by combining the average Nusselt number relation
with the expression for the average heat transfer coefficient and solving numerically for Delta
T_L/2. The correlations could also be applied for
vertical cylinders as long as the diameter is much larger than the boundary layer thickness.
This condition is satisfied if this relation is true for the case in analysis. L is the
cylinder height and D is the diameter. When the cylinder is too slender, the curvature
affects the boundary layers growth and enhances the heat transfer due to natural convection.
If we rotate the plate by an angle theta, we obtain an inclined plate (like a tablet
on a stand). The gravitational vector now has a component normal to the plate. For the
top side of a cold plate or the bottom side of a hot one, the behavior is similar to the
vertical plate, but the heat transfer is reduced due to the smaller gravitational component
parallel to the plate. For theta between 0 and 60 degrees, we can estimate the heat transfer
on these sides of the plates substituting g with g cosine theta when calculating the
Rayleigh number. On the opposite sides of the plates, instead, the normal component
of g pulls the fluid away from the plate and creates three-dimensional patterns. For the
cold plate the fluid moves down and gets replaced by hotter fluid, while for the hot plate the
fluid moves up and colder fluid replaces it. This motion enhances the heat transfer due
to natural convection. Correlations for these sides of the plates can be found in literature
on a case-by case basis.
An extreme case for the inclined plate is the horizontal plate. Lands or lakes are good
examples for atmospheric convection over horizontal surfaces. If we consider a hot plate, the
air in contact with the plate gets heated, its density reduces, and the buoyant force
pulls it away from the top side of the plate while cooler fluid replaces it. This greatly
enhances the heat transfer. On the opposite side however, the heat transfer is poor. The
hot fluid is blocked by the plate and must move sideways before reaching the end of the
plate and finally move upward. The opposite pattern can be seen for a cold plate.
Adams introduced these correlations for the two sides of the plates. The accuracy can
be further improved if the characteristic length is defined as the ratio of the plate
area by its perimeter.
Let’s move now to study cylinders. If we analyze a section of a hot cylinder, we can
see that the fluid would form a boundary layer around the body, starting from the bottom
side and then moving upward around it. Near the top of the section the fluid separates
into an ascending plume. The opposite can be seen for a cold cylinder.
These correlations provide us the average Nusselt number along the cylinder circumference
either using different combinations of coefficient or as a single formula over a large range
of Nusselt numbers. In this case the characteristic length is the cylinder diameter.
Last, the sphere… A pattern like the one seen in the section of a cylinder can be seen
for the hot or cold spheres. The fluid moves around the sphere and then transforms into
an ascending or descending plume. This correlation can be used for the heat transfer analysis.
The constant 2 corresponds to the heat transfer due to conduction for a sphere in an infinite
medium. This concludes this lesson on external natural
convection.