In this screencast we're going to describe what a thermal boundary layer is and how it's
created.
The idea of a boundary layer, or a hydrodynamic layer, has been introduced in fluid mechanics.
So basically what you have is a flat plate, which is stationary, and then you have fluid
flow that comes in, and it's at a constant velocity that we'll call U(infinity), and
as soon as it comes to the stationary plate what has to happen is that the particles right
here take on the velocity of the plate, they are stationary themselves.
What happens is a layer builds over this plate where the velocity is changing throughout
this, what we call boundary layer.
So we have the same situation in heat transfer, however thermal boundary layers are caused
by a difference in temperature, not velocity, between the plate and the fluid around it,
and so a good example, and we're going to look at something similar, so we have here
an isothermal flat plate, and we have flow that's coming in at some U(infinity), and
more importantly this flow comes in at a uniform bulk temperature that we'll call T(infinity).
However let's say here T(s) is greater than T(infinity), so as the fluid at this bulk
temperature contacts this hotter plate, the fluid particles take on the temperature of
the plate.
What happens is they now exchange energy with the particles above them, and because of this
temperature gradients form in the fluid.
So if we're looking at this in the y-direction what we have are temperature gradients dT/dy.
So if we compare it to fluids we have this hydrodynamic, let's go back to this boundary
layer, and what it has is it has velocity gradients, and what this causes is shear stress.
In our thermal boundary layer what we have here are temperature gradients, and what that
causes is heat transfer, which we'll call q.
Like the hydrodynamic layer there's a point where these gradients are no longer significant,
and that marks the height of the boundary layer.
We call this height del, and for the thermal boundary layer it's del(sub T), and it's defined
as for the hydrodynamic at y equals del, our u of y is equal to 99% of our bulk velocity.
For our thermal boundary layer at y equals del(sub T), and now we write this as T(s)-T
divided by T(s) minus T(infinity) is equal to 0.99.
From our knowledge of conductive heat transfer we can easily find the flux at the surface
as well as define for the first time what's known as h, or the convective heat transfer
coefficient.
So what we're going to do is we're going to apply Fourier's law at the surface, which
if you remember from our picture, that's at y=0.
So why can we do this?
Because at the surface there's no fluid flow, so how does the heat transfer occur?
Solely by conduction.
So we write that q at the surface here is equal, and now we're using Fourier's law,
minus k, note it's k of the fluid, because that's where the conductive heat transfer
is going through, dT/dy, at y=0.
In addition we know from an energy balance that the heat flux at the surface using Newton's
law of cooling is equal to this convective heat transfer coefficient, T(s), minus T(infinity),
and if we equate the two of these what we end up with is that our convective heat transfer
coefficient is equal to the thermal conductivity of the fluid, dT/dy, at y=0 divided by T(s)-T(infinity),
and note that there's no area term here, because we're dealing only with the heat flux.
Taking a look at this how do h and q change as we travel down the plate.
So here's our plate, and we already defined that y is in this direction, and we want to know
what happens as x increases.
So if you take a look there at the expression for h note that the denominator T(s)-T(infinity),
stays constant, since it's an isothermal plate at T(s) and our bulk temperature is T(infinity).
However, that del, or that boundary layer, in this case thermal boundary layer, grows
along the plate as the particles are exchanging energy, and therefore what's going to happen
is that the temperature gradients are getting smaller.
In other words dT/dy at y=0 decreases as we go down the plate, and if dT/dy decreases,
yet our T(s)-T(infinity) stays constant, what you'll see is not only does h decrease as
we go down the plate, the surface flux, which we have here decreases as well.
So in our next screencast we'll look at an example where we can actually quantitatively
show these trends.