Thank you for joining us for this informative American Institute of Chemical Engineers webinar,
"Transport Phenomena: Heat Transfer". Our speaker is Dr. David Foster.
He spent over 30 years as industrial scientist engineer at Eastman Kodak Company while also teaching
fundamental transport phenomena courses at the undergraduate and graduate level at the
University of Rochester for the past 14 years.
The 3 modes of energy transfer that we are going to discuss today are conduction,
convection and radiation. In conduction we have no motion of matter. It's really a molecular
interaction, and pure metals are the best conductors. And they are the best conductors
because they have free electrons available.
Convection we are going to define as the energy exchange between a surface and an adjacent fluid.
I'm going to talk pretty deeply about forced and free convection. Forced convection is a function
of flow past a surface with a fan or a pump. And then we are going to talk very briefly about
radiation, where no medium is required, and it is really a maximum where the surfaces are
and it is really a maximum when the surfaces are separated by a perfect vacuum.
Let's begin with Fourier's Law. Fourier's Law is fundamental to any heat transfer process.
We have fundamentally a heat transfer rate in the x-direction divided by the area normal to the
direction of the heat flow. And that ratio is sometimes termed a heat flux in the x-direction.
This being equal to the negative, meaning that the heat flow is in the direction negative to the temperature gradient;
and then we have the thermal conductivity and then the temperature gradient, and here I've got
temperature in the x-direction. In the equation at the bottom, you'll notice that I've got
a much more general equation.
Most of the time we talk about it as Fourier's Rate Equation or sometimes you'll see it
talked about as Fourier's First Law of Heat Conduction, where the heat flux is proportional
to a temperature gradient. And I've got it here as a del operator, so I can talk about it not only
as an x-, y-, and z-component, but I can also term it as rectangular, cylindrical and spherical coordinates.
We need to think about thermal conductivity, and we need to think about it very carefully, because
it is inherent to what we ought to understand.
Thermal conductivity fundamentally is going to be independent of the direction of the equation,
and Fourier's Law is going to apply to isotropic or continuous mediums only.
So if I've got a medium that is not continuous, I need to break Fourier's Law up into different pieces,
and we will talk about that in a moment. It's primarily a function of temperature, and I like to
equate it to being a term similar to viscosity.
Viscosity, as we all know, we warm up maple syrup and it flows a lot easier than when it's cold.
Thermal conductivity, although not something that we can visualize perhaps as well as viscosity,
has the same kind of properties. The two equations that are most common for gases in
this kind of an area to determine the thermal conductivity are these two equations pictured here.
The first is for a monatomic gas, and the second is Chapman's equation.
Both of them contain parameters that can be looked up very easily, everywhere from
Boltzmann's constant to the collision diameter.
These terms are available in various books like Perry's Handbook, if you are an old guy like me,
or on the Internet in you are someone who is a lot younger and has never even heard of Perry's Handbook.
All right, we need to talk about thermal conductivity in the liquid and the solid phases,
because this is where most of our work is going to be. Some of us deal with gases, but a lot of times
we are going to look at liquids and solids. Unfortunately there is not a simplified equation for
liquid phases like there is for gas phases. In fact there is no universally accurate mathematical model
that currently exists. There are empirical models that are out there. If you search the literature
you will find that people have come up with some nice empirical models that they use.
However,they are very specific to very specific systems. So, necessity of understanding what your
thermal condictivity is in your system is going to be paramount. Solid phases are very similar,
and most solids of interest have been evaluated, which is very fortunate.
The solid phase thermal conductivity is certainly a result of molecular interactions and free electrons.
Let's immediately do an example. If I've got a styrofoam freezer in a refrigerator that has an area
of a meter squared, being used to cool food, and I want to know what thickness of syrofoam insulation
that I need to limit the heat loss to 400 watts, if the inner and outer surface temperatures are
-10 and 33°C respecively. This is a very nice example to illustrate Fourier's Law and we have
dt/dx, that we translate into being ΔT, or the change in temperature, divided by the length
that we want to solve for. We have the conductivity measurement and we have the area.
And q in this case is going to be the heat loss that we are going to be able to tolerate in our system.
Rearranging this equation, solving for L, we find out that our styrofoam insulation has got to be
about 3.23 cm. One other note that I just want to make is that this styrofoam number for insulation,
the k value, is available in the literature. So like I mentioned a minute ago, many values for solids
are available in the literature and easy to look up.
So we need to have about 3.23 cm in order to maintain that 400 W of heat loss.