An Introduction to the Heat Conduction Equation

In this post, we will begin to look at the heat conduction equation. This is also commonly referred to as the heat diffusion equation or just the heat equation. It’s a very special equation as it mathematically describes the way heat flows within a solid. It can be used to find the temperature distribution; the temperature at every point in the solid and on its surface. Solid objects considered include walls, pipes, heat sinks, engine blocks and much more.

The temperature distribution within a solid can be found with the heat conduction equation.

This is useful for engineers as once the temperature distribution is known, we can use this for a wide variety of things. The heat flux (which is dependent on the temperature gradient) and thus the flow of heat within the solid can be better understood. This can help us to achieve our performance goals. The maximum or minimum temperature within a solid can be found to assess performance and safety. Mechanical stresses imposed by thermal expansion can also be found via the temperature distribution.

It would be great to come up with an equation for the temperature distribution within all kinds of solids with all kinds of boundary conditions and initial conditions. In essence, this can be done by solving the heat equation, but as we will come to realise, this is not a simple task!

The heat equation is a partial differential equation and can be most generally expressed in a Cartesian coordinate system with Eq. 1.

\frac {\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)+\frac {\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right)+\frac {\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) = \rho c \frac {\partial T}{\partial t} \tag*{Eq. 1}

This is a parabolic partial differential equation and is typically used to represent systems that vary with time. In the context of the heat equation, this means the system has transient heat conduction so the temperature at each point within the solid may vary with time. Considering the case where there was no time variation in the system, we would arrive at Eq. 2.

\frac {\partial}{\partial x}\left(k\frac{\partial T}{\partial x}\right)+\frac {\partial}{\partial y}\left(k\frac{\partial T}{\partial y}\right)+\frac {\partial}{\partial z}\left(k\frac{\partial T}{\partial z}\right) = 0 \tag*{Eq. 2}

This is no longer a parabolic partial differential equation, it is known as an elliptic partial differential equation. These types of partial differential equations are typically used to represent systems that are in equilibrium. In the context of the heat equation, this means that the system is at steady state so the temperature at each point no longer varies with time, but they still may vary in space.

These equations can also be expressed using the Laplacian or Laplace operator. A few different symbols denote this special differential operator: $\nabla^2$ and $\Delta$ . Using this operator we can now very compactly represent Eq. 1 through Eq. 3.

\nabla^2T=\Delta T={\frac {\partial ^{2}T}{\partial x^{2}}}+{\frac {\partial ^{2}T}{\partial y^{2}}}+{\frac {\partial ^{2}T}{\partial z^{2}}}

\alpha \nabla^2T=\frac {\partial T}{\partial t} \tag*{Eq. 3}

In Eq. 3, $\alpha = \frac{k}{\rho c}$ and is known as thermal diffusivity. It combines various material properties together – the thermal conductivity ( $k$ ), density ( $\rho$ ) and specific heat capacity ( $c$ ). Note that Eq. 1 and Eq. 3 are the same, they are just represented in different ways. We could similarly use the Laplacian to simplify Eq. 2 into the famous Laplace’s equation as shown in Eq. 4.

\nabla^2T=0 \tag*{Eq. 4}

In any case, whether we have a parabolic or elliptic partial differential equation, they will both be difficult to solve. Many instances of these cannot be explicitly solved. As we continue our discussion into the heat equation, we will realise that only simplified versions can be analytically solved. Even then, some of the analytical solutions are too complicated to practically use. The more complex versions of this equation require numerical solutions which are best handled by computers (we will also discuss some of these as well).

Now that we have a better understanding of what the heat equation is, in the next few posts we will look at deriving the heat equation to see exactly where it comes from.