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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
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And that's it for this look at conduction and the heat equation. Thanks for watching!