Thanks to CuriosityStream for sponsoring this video. When any object is heated, like the
rotor in a car disc brake for example, which gets hot because of friction, the
distribution of temperature within the body will change as thermal energy
flows from hotter to colder areas. This re-distribution of thermal energy is
called thermal conduction. It's one of three ways in which heat transfer can occur, along with
convection and radiation. Understanding conduction is really important for so many engineering
applications, so let's learn more about it!
On the molecular scale thermal energy is really
just the random motion of the atoms and molecules that make up an object. As temperature increases
the motion becomes more and more significant, and close to absolute zero
there's barely any motion at all. Because the atoms of a material are connected to
each other by bonds, forming a lattice structure, vibrations of the atoms travel through the
lattice, which results in thermal energy being redistributed from areas of higher energy
to areas of lower energy. This is the mechanism through which heat conduction occurs. The stiffer
the atomic bonds and the more regular the lattice structure, the more easily energy is transferred
between atoms through these vibrations. A second mechanism for conducting
heat occurs in metallic materials, because they contain free electrons
which can travel through the lattice, colliding with other particles and re-distributing
energy in the process. The presence of these free electrons explains why metals are such
good thermal conductors - they benefit from the combined effect of the movement of free
electrons and vibrations of the atomic lattice. In gases and liquids conduction occurs mostly
through collisions of atoms and molecules. In the study of conduction we're usually
interested in determining the heat transfer rate, which is the amount of energy that flows
through a defined area of an object each second. It's designated using the letter q,
and has units of Joules per second, or Watts.
We'll start by looking at the simple
case of heat transfer through a wall that has an area A and a thickness L, and
separates a hot room from the cold outside. We'll assume that the wall surface inside the
room is maintained at a temperature T1 and that the outside surface is at a temperature T2.
Thermal energy always flows from higher to lower temperatures, so heat transfer will occur
from the hot side of the wall to the cold side. If the rate of heat transfer is large, a lot
of energy will be lost through the wall. We can calculate the heat transfer
rate q using Fourier's law. It tells us that q is proportional to
the gradient of the temperature profile DT/dx, the wall surface area A, and to a constant
of proportionality k, called thermal conductivity. Thermal conductivity is a material property that
defines how well a material conducts heat. It has units of Watts per Meter-Kelvin. More energy is
lost through the wall if the thermal conductivity or the wall surface area are increased.
Since thermal energy always flows from higher to lower temperatures, the minus sign is
included in Fourier's law so that the calculated heat transfer rate is a positive quantity
when the temperature gradient is negative. The more negative the temperature gradient,
the larger the heat transfer rate. If we assume that the temperature profile
through the wall is linear, the temperature gradient is constant and is just the difference in
temperature divided by the thickness of the wall, so we can easily calculate the heat transfer
rate. For a hot temperature of 25 degrees, a cold temperature of 5 degrees, and a solid steel wall
that's 5 centimeters thick and has an area of 2 square meters, for example, the heat loss would be
36 kW, which is 36 thousand Joules every second. It's often more convenient to remove
the area term from Fourier's law. Instead of giving us the heat transfer rate
in Watts, Fourier's law in this form gives us the heat flux, in Watts per square meter. Fourier's law isn't a fundamental equation
that can be derived from first principles, but is really just how thermal conductivity is defined. Since k is measured experimentally for different
materials, all of the complexities associated with the underlying mechanism of conduction,
like the vibrations of the lattice structure, are abstracted away, and don't need to be
considered in the engineering analysis. Gases and non-metallic liquids tend to
have very low thermal conductivities, because the large spacing between
molecules means that thermal energy is less easily transported. Non-metallic solids are
next, followed by alloys and pure metals, which have very high thermal conductivities
because they contain free electrons. Diamond is an interesting material. It has
very high thermal conductivity despite not being metallic. This is because its very regular
crystalline lattice and strong atomic bonds allow lattice vibrations to travel very efficiently.
On the other end of the spectrum there are materials like aerogel that are engineered to
have extremely low thermal conductivities. Here's how they compare to a
few other common materials. These are the thermal conductivities at room
temperature, and although conductivity does vary with temperature, in many thermal
analyses constant values can be assumed. So far we've only looked at a one dimensional
case, where heat transfer occurs in a single direction - the temperature field T
is a function of the x coordinate. But we'll also encounter two-dimensional
cases, where temperature is a function of the x and y coordinates, or three-dimensional
cases where it's a function of x, y and z. To understand how we can apply Fourier's law to
these cases we first need to understand how heat flows in two and three dimensions. Conduction
is always driven by a temperature difference, which means that no heat will flow along
isotherms, which are lines of constant temperature in two dimensions, or surfaces
of constant temperature in three dimensions. Since there can't be a component of the heat
flux vector that's aligned with the isotherms, in isotropic materials heat must always
flow perpendicular to the isotherms. If we define n as the normal to the isotherm
at a specific point, we can reformulate Fourier's law into a more general form that
gives us the heat flux in that direction. We can also express this in terms of the x, y and
z coordinates. Here the heat flux is a vector, and i, j and k are unit vectors
in the x, y and z directions. The term in the brackets represents the
gradient of the temperature distribution, and can be expressed more
compactly using the Del operator. This is the general form of Fourier's law.
Fourier's law is really useful, but you might have spotted a problem, which is that it can
only be applied if the temperature field T is known. We were able to apply it for the case
of one-dimensional heat transfer through a wall, but only because we could guess that
the temperature profile would be linear. In the vast majority of cases we'll need
to figure out what the temperature field is before we can apply Fourier's law.
This can be done using the Heat Equation, an equation that describes how heat will flow
throughout an object. By solving this equation for a specific set of boundary and initial conditions
the temperature field T can be determined. Like so many governing equations in engineering,
the heat equation is a partial differential equation. But it's not as complex as it seems
- it's really just a simple energy balance. Consider a single point within a larger
body, that we'll represent as a small volume. The terms on the left represent the net rate
of thermal energy transfer into the volume, and the term on the right represents the rate of
change of the energy that is stored within it. If more energy flows into the volume than
flows out of it, the thermal energy it stores will increase, which results in an
increase in the temperature at this point. By re-arranging the equation to group
all of the material properties together, we can see that the change in
temperature at a single point over time depends on how the temperature
is distributed around that point, and on a constant of proportionality that
depends on the properties of the material. You might be wondering why conduction
depends on second partial derivatives. To illustrate this let's look at a simple function
T that defines the temperature along a distance x. Differentiating T with respect to x gives
us the slope of a line tangent to the curve. Where the temperature increases suddenly, the derivative is large, and where the curve
changes direction the derivative is zero. The second derivative gives us the slope of
a line tangent to the first derivative curve. This means that positive values of the second
derivative correspond to convex areas of the temperature curve, and negative values correspond
to concave areas. The heat equation is saying that heat will flow away from the concave areas of
higher temperature, where the second derivative is a negative value, and towards the convex areas of
lower temperature, where it's a positive value. The same idea applies in three dimensions. An
area where the sum of second partial derivatives is negative is an area of high temperature
relative to its surroundings, so there will be a flow of thermal energy out of that area and
into the surrounding areas of lower temperature. Because the sum of the second partial
derivatives is quite long to write out, the Heat Equation is often written
using the Laplace operator. The three material properties in the heat equation
are the thermal conductivity k, the material density rho and the specific heat capacity, cp.
The specific heat capacity is the energy that needs to be absorbed by a unit mass of a
material to raise its temperature by 1 Kelvin. Multiplying cp by the material density
gives the volumetric heat capacity, which is expressed per unit
volume instead of per unit mass. Dividing thermal conductivity by the volumetric
heat capacity gives us a useful parameter that describes the ability of a material to
conduct heat, versus its ability to store it. This parameter is denoted using the letter
Alpha, and is called thermal diffusivity. If a material has high thermal diffusivity,
because it's highly conductive, or because it has low specific heat capacity,
the heat will diffuse through the body much faster compared to a material with a lower
value of thermal diffusivity - only a small proportion of the energy that moves through
it is needed to raise its temperature. This is what the Heat Equation looks like
when rho, cp and k are replaced with alpha. There's one more term that needs to be added
to obtain a truly generalised form of the heat equation. In some cases the material itself is
a source of heat. This can occur in power cables for example, where the electric current
passing through the cable generates heat. This additional energy input needs to
be accounted for in the energy balance, and we do this by adding a term to the equation. Now that we've got a generalised form of the
heat equation, we can simplify it to meet the requirements of the particular scenario being
analysed. If we're analysing a steady state case for example, the temperature distribution
doesn't change with time, so the transient term is equal to zero. If there's no internal
heat generation we can remove that term too. And if it's a two dimensional case, for
example, the z term can also be removed. To illustrate how we can solve
problems using the Heat Equation, let's look at the example of conduction through
a wall that we saw at the start of the video. It's a one-dimensional case, with no internal
heat generation, and we're only interested in the steady-state temperature distribution, so
the heat equation ends up being quite simple. This is a second order differential equation,
and it can be solved by integrating twice, remembering to add the two constants
of integration along the way. We don't need to consider the initial
conditions because this is a steady-state case. All we need to do to calculate C1
and C2 is apply the boundary conditions. At x=0, T is equal to T1
and at x=L it's equal to T2. We end up with an equation that
defines the temperature distribution, which is linear like we assumed
at the start of the video. That was pretty easy, but solving
the heat equation analytically is challenging for anything much more
complex than a one dimensional case, so specialist software is routinely used by
engineers to perform this type of analysis. These tools use numerical methods to obtain
approximate solutions to the heat equation. One way that quite complex one-dimensional
problems can be solved analytically involves applying the concept of thermal
resistance. It's a really powerful technique that simplifies the analysis of things like
conduction through several different layers, and I've covered it in a separate video that you
can watch right now over on Nebula. The video goes into how the concept of thermal resistance
can be used to solve heat transfer problems, including those involving both conduction and
convection, and how we can use it to understand the critical insulation thickness problem, the
counter-intuitive observation that increasing the thickness of the insulation around a pipe can
increase heat losses instead of reducing them. As many of you will already know, Nebula
is a streaming service built by a group of independent educational creators. You can
find all of my normal videos on Nebula, without any of the ads, but I've been
uploading bonus companion videos there too, including the one on thermal resistance,
but also videos on dimensional analysis, self-buckling and pendulums, and I'll be adding
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CuriosityStream has thousands of high quality documentaries, but one series in particular
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And that's it for this look at conduction
and the heat equation. Thanks for watching!