There are a number of different correlations for free convection depending on the flow
regime and the geometry of the object.
So in this screencast, we'll look at free convection over a horizontal plate.
So when I talk about correlations, what I'm actually looking at is a correlation for the
Nusselt Number, which is equal to h, times L which is some characteristic length, divided
by k which is the thermal conductivity of the fluid.
So this Nusselt number is usually a function of the Rayleigh number, as well as the Prandtl
number.
So our goal is to find the appropriate correlation for the Nusselt number, thereby being able
to solve for h, and from there, we can use Newton's law of cooling that says q, or the
heat transfer rate, is equal to this heat transfer coefficient, times the cross sectional
area, times some delta T. And in free convection, that delta T is going to equal T of the surface
minus the temperature of the ambient air.
Free convection flow over a horizontal plate does not have just one correlation for the
Nusselt number.
We have four different possibilities.
So the first thing we have is a plate whose temperature is higher, its T surface is greater
than T infinity, and we call this a heated plate.
When we're looking at free convection over a heated plate, we have two possibilities
as well.
We have from the upper surface as well as the lower surface.
In addition, we can have what's known as a cooled plate, so this is when T surface is
less than T infinity, and again, it has different correlations for both the upper and lower
surface.
The reason you have this is that the physical phenomena that's occurring is due to the top
surface of a cooled plate or the bottom surface of a heated plate will impede the flow of
the fluid, since the fluid must move horizontally before it can ascend or descend from the edges
of the plates.
So what we're going to do is we're going to use the same conditions that we used for a
previous screencast that looked at a vertical plate with free convection.
So we're going to look at both surfaces of a heated plate, so on the heated plate, the
surface temperature is 90 degrees C, and the ambient temperature is 25 degrees C. Now,
as I mentioned before in screencasts, the dimensionless parameter that guides free convection
is what's known as the Rayleigh number.
So some of the properties that we're going to have to look up include the kinematic viscosity
as well as the thermal diffusivity, and the thermal expansion coefficient.
And in order to look these up, we have to have the right temperature, so what we use
for these correlations is the film temperature, which is just the average, or Ts plus T infinity,
divided by 2, in Kelvin.
So our film temperature for this particular situation is approximately 330 K.
So let's go ahead and look up all of these properties, as well as this property, our
k, which again is the thermal conductivity of the fluid.
If you notice, these are exactly the same properties as we used in our previous screencast
about vertical plates, so you would think that the Rayleigh numbers would be exactly
the same.
However, for horizontal plates, it's suggested that our characteristic length, that L, should
be the surface area of the plate divided by its perimeter, so we'll end up calculating
that, so that L is going to equal 0.3 meters times 0.6 meters, divided by our perimeter,
which is 2 times 0.3 meters, plus 2 times 0.6 meters, and just to refresh your memory,
or just in case you didn't see the vertical plate, what we've done is taken a plate that's
0.6 meters high and 0.3 meters wide and basically flipped it over to make it into a horizontal
plate.
And so when we look at this length, it's equal to 0.1 meter.
For a vertical plate, the characteristic length is the height of the plate, which in that
case was equal to 0.6 meters.
Now, let's go back and put all of these values into our Rayleigh number, and what we do,
we end up with a Rayleigh number that's equal to 3.055 x 10^6.
Now that we have our Rayleigh number, we can decide what correlations to use.
For our upper surface, the correlation for the Nusselt number is equal to 0.54, Rayleigh
to the 1/4, which equals 22.6.
If we look at the lower surface, the appropriate correlation is 0.27 Rayleigh to the 1/4, which
is equal to, as you can probably figure out, 11.3.
So notice the difference that the impediment of the plate makes to the bottom surface,
showing the effect of the horizontal flow.
So then we go ahead and we calculate our h, which is equal to our Nusselt number times
our k, divided by, and again this is our characteristic length, we find that the h of the upper plate
is equal to 6.78, and the h of the lower plate is equal to 3.39.
And again, the bars over there show that it is an average heat transfer coefficient.
So the heat transfer rates are again the same, that the upper plate, 79.3 watts, is twice
that of the lower plate, which is 39.7 watts.
In comparison, our vertical plate had a heat transfer rate of 48.6 watts.
So why is this?
Well, since a vertical plate is the same direction as gravity, fluid motion is caused only by
buoyancy.
For the lower surface of a heated plate, there's a reduction in buoyancy which then decreases
the heat transfer.
However, for the upper surface, heat transfer is enhanced by the three dimensional flow
that occurs due to boundary layer displacement.
Therefore, we would expect to have this order of heat transfer rates, where the upper plate
has the highest heat transfer rate, the lower of the horizontal plate has the lowest, and
that vertical plate is in between.