Free convection, or natural convection, originates when a body force, gravity, acts on a fluid
containing density gradients that are caused by temperature differences.
A typical example is a heated vertical plate, and a quiescent, or still, fluid.
So here's our plate, who's surface temperature is greater than that of the surrounding air,
so T(surface) is greater than T(infinity).
Because density increases with an increase in temperature the fluid is going to be less
dense nearer to the plate, and so what happens then, here's the fluid, the fluid will then
rise due to these buoyancy forces.
We have our determining dimensionless parameter that describes free convection, is known as
the Grashof number.
The Grashof number is similar to the Reynolds number in forced convection.
Our g is gravity, beta is known as the thermal expansion coefficient, which we'll talk about
when we do some example problems, and then as you can see the rest of these, our delta
T here, it's between the surface of the plate and the air, or whatever fluid it is outside
the length of the plate, divided by our kinematic viscosity squared.
What this number is, it's a ratio of the buoyancy forces to the viscous force, so a higher Grashof
number would indicate a larger presence of free convection.
Another important parameter is the Raleigh number, which is actually equal to the Grashof
number times the Prandtl number.
Just as a note that Prandtl number, which is considered a dimensionless parameter, cause
it's the ratio of the momentum in thermal diffusivity, because it's dependent only on
the fluid, i.e. there's no length scale, it's considered to be a property, and we look it
up.
So this number is used to determine the transition, the Raleigh number from laminar to turbulent
flow.
If the Raleigh number is greater we have turbulent flow.
So correlations for free convection are all functions of the Raleigh number.
So let's look at some correlations for simple geometries.
Probably the most general of the correlations for finding the Nusselt number, and again
in any convective heat transfer problem our ultimate goal is to find this so we can use
Newton's law of cooling.
In order to find our h we need to find our Nusselt number, which is a dimensionless parameter,
and this is equal to h times the length scale, whatever that is, which I'll put as L here,
divided by the thermal conductivity of the fluid.
As I said, probably the most general correlation for the Nusselt number is some constant times
the Raleigh number, raised to some n.
So these constants, c and n, can be looked up, but in general n equals one fourth if
the flow is laminar, and it equals one third if the flow is turbulent.
In addition, for laminar flow our c equals 0.59, and for turbulent flow it equals 0.10.
A more suitable correlation that can be used for both turbulent and laminar flow for a
vertical plate is as follows, and what makes this particular correlation useful is that
it can be used over the entire range of the Raleigh number.
However, we can get an even more accurate correlation just for laminar flow by using
this one, the thing to keep in mind however that in order to use this the Raleigh number
has to be less than 10 to the ninth, so of course we have different correlations, other
ones depending on the geometry, and one of these is for a horizontal plate.
So the ones we were looking at before were for vertical plates, now we're looking at
horizontal, but here we have different choices.
First, it could be whether the plate is hotter or colder than the quiescent air, and then
which side of the plate is up, so some typical correlations include those below.
So these first two right here are for horizontal plates, where it's either the upper surface
of a heated plated, in other words the temperature of the plate surface is greater than that
of the outside fluid, or the lower surface of a cooled plate, and these are used for
different values of the Raleigh number.
For the other two situations you only have one correlation, so this is the lower surface
now of the hot plate, or the upper surface of the cold plate.
In addition we have correlations for long horizontal cylinders, as well as spheres.
So here we have that long horizontal cylinder, and here we have the correlation below for
the sphere, and one of the things that you might notice compared to other forms of convection,
except for the vertical plate there's really only one choice that we're given to use to
find our Nusselt number, and therefore our heat transfer coefficient, and then our heat
transfer.
So the secret to solving these free convection problems is first of all to determine whether
it is free convection.
So there's two things you should look for, obviously the surface temperature has to be
different than the surrounding fluid, the other thing you look for are terms such as
still air, or quiescent air.
Then you determine the geometry, calculate the Raleigh number, choose the appropriate
correlation, and then solve for our h.
One thing also to keep in mind is that all the properties are calculated at the film
temperature, which is T(s)+T(infinity) divided by 2, where this temperature is in Kelvin.
So in future screencasts we'll take a look and use some of these correlations in a practical
application.