In this screencast, we choose a correlation for the Nusselt number, where the Nusselt
number is equal to h times the characteristic length divided by the thermal conductivity
of the fluid, and this Nusselt number is chosen depending on the geometry as well as the conditions
of the particular situation. From the Nusselt number, we're able to calculate an h, and
from the h we're able to solve for the Biot number, and from the Biot number we can decide
which method we can use to solve for a transient problem. So here we have a gold sphere with
a 50 millimeter diameter, and it's dropped into a water bath. We have an initial sphere
temperature as well as a center-line temperature which is expected to be 310 kelvin. So how
do we know that this is a transient problem? Well one of the things that can tell us is
this word "initially." Any time you see "initially," or the expression "how long," you can assume
that it is a transient problem. For a sphere, there's only one correlation for the Nusselt
number, which makes our life a little easier. This is what's known as the Whitaker correlation.
It is a function of both the Reynold's number as well as the Prandtl number. Here's our
plan. The first thing that we're going to do is look up the properties, cause in order
to solve for the Reynold's number, we're going to need properties. And for this particular
correlation, the properties are looked up at the temperature of the bath, or what we
call T infinity. Then we'll be able to solve for the Reynold's number. Once we have the
Reynold's number, we're going to use our correlation so that we can solve for the Nusselt number.
Once we have the Nusselt number, we can find our convective heat transfer coefficient.
Then we're going to use h to find the Biot number. Then when we have our Biot number,
this will tell us what method we need to use to solve for our time. Let's start by
finding the properties at the temperature of the water bath, which is - if we take a
look back here - 290 kelvin. So here are the properties that we're going to need to calculate
the Reynold's number. And the Reynold's number is the density times the velocity times the
diameter, divided by the viscosity. So based on the problem statement, as well as these
properties, we come out with a Reynold's number that equals 92,513. Now we can use the Whitaker
correlation. And this correlation says that the Nusselt number is equal to 2, plus 0.4
Reynold's to the 1/2 plus 0.06 Reynold's to the 2/3, this is multiplied by Prandtl number
to the 0.4, and finally this entire thing is multiplied by the viscosity divided by
the viscosity at the surface, raised to the 1/4. So when we put in our Reynold's number
that we calculated, as well as the Prandtl number that we looked up, we end up with a
Nusselt number that equals 822. This allows us to find our h, which is the Nusselt number
times the thermal conductivity of the fluid, divided by the diameter. Our h is going to
equal 9830 watts per meter squared kelvin. Now we have to check the Biot number, which
is equal to our h times the radius divided by 3 (when we check the Biot number, the characteristic
length for a sphere is r divided by 3), and this is divided by the thermal conductivity
of the gold. And we end up with a Biot number of 0.2584. This is larger than 0.1, so we
need to use an analytical solution. Note that when we've done transient problems before,
we've always been given the h, or the convective heat transfer coefficient. Here, we use our
knowledge of Nusselt number correlations, and the geometry and properties of our particular
situation, to find that h for ourselves.