In this screencast, we look at laminar flow over a flat plate, and we try to solve for
the surface temperature of this plate. Depending on the fluid, the velocity, and the length
of the plate, the regime can transition from laminar to turbulent, but here we are going
to assume that we have laminar flow over the entire plate. So this screencast is how to
find a surface temperature using an empirical correlation of the form C times Reynolds to
the m, Prandtl to the n. If you recall from a previous screencast on how to do convective
problems, we should remember that this is an empirical correlation where the C, the
m and the n vary with geometry, as well as our flow regime. In this particular case,
we find that our Nusselt number has a correlation of 0.664 Reynolds to the 1/2, Prandtl to the
1/3 as long as our regime is laminar and our Prandtl number is greater than or equal to
about 0.6. And in looking up our properties, we use the film temperature, which is the
surface temperature of the plate plus the temperature of the fluid flowing over the
plate, divided by two. Let's take a look at how to use this. Let's say we have an isothermal
flat plate, the length of the plate is 1 meter long, our fluid is air, it's at 400K, it has
a velocity of 10 meters per second and an average heat flux of 2500 watts per meter
squared. And what we're interested in is, what is the surface temperature of this plate?
First of all, why do we use the heat flux rather than the heat transfer rate? We don't
know the cross-sectional area, so we have to present this in watts per meter squared.
So our governing equation is that the heat flux equals the average heat transfer coefficient,
times Ts minus T infinity. So we know the heat flux, we know T infinity, we're trying
to find that surface temperature, and therefore what we need is a correlation for h. If we
use this Nusselt number correlation, this equals our h bar times our length divided
by the thermal conductivity of the fluid. Therefore we can solve for our h bar, or our
heat transfer coefficient, using the Nusselt number times the thermal conductivity of the
fluid divided by the length of the plate. The first thing that we're going to need to
do is guess a surface temperature of the plate so that we can look up our properties, because
if you recall as I showed you before, our film temperature where we look up the properties
is the surface temperature plus the flow temperature divided by two. So let's guess a surface temperature
of 600K, since we know that the surface temperature should be greater than that of the fluid temperature.
So our film temperature where we look up things is going to be 500 K, and we look up the kinematic
viscosity which is 38.79 times 10 to the minus 6 meters squared per second, because we're
going to need this in the Reynolds number. We look up the Prandtl number because we're
going to need this in our correlation for our Nusselt number. And finally, we look up
the thermal conductivity of the fluid because we're going to need this to find our h. So
the first thing we're going to do is solve for the Reynolds number. So the Reynolds number,
which is, as I should you before up here, the velocity times the length divided by the
kinematic viscosity. So let's put those numbers in, and we find that the Reynolds number is
1.58 times 10 to the 5. And what this indicates to us is not only can we put it in our correlation,
but that our correlation which is valid for laminar flow, is a correlation that we could
use. Our Nusselt number equals 0.664 Reynolds raised to the 1/2 times the Prandtl number
raised to the 1/3, and if we do that, we find that our Nusselt number equals 297. So now
we use that Nusselt number to find our heat transfer coefficient, using again the definition
of the heat transfer coefficient. So our h bar equals 12.1 watts per meter squared K.
Now we have everything that we need to solve for our surface temperature, which is our
heat flux divided by our average heat transfer coefficient plus the temperature of our fluid,
so that equals 607 K. So we go back to see what we guessed our surface temperature to
be, and we guessed it to be 600K. Our actual surface temperature is very close to that,
or 607 K. The property change between 600 and 607 K or Kelvin is minimal, therefore
we can use this as our answer and congratulate ourselves on a good guess for the surface
temperature.