Thermal analysis is a powerful tool in an FEA analyst’s arsenal. It is useful to analyze problems such as heating of chips in a PCB or the heating effects of a battery. There are two main types of thermal analysis – steady-state and transient. In this blog, we will discuss both of these types and when to use them.

We are only going to talk in the context of uncoupled thermal analysis in this blog (i.e. stress, deformation, and electrical field are not considered) – perhaps coupled thermal analysis could be a future topic!?

## A Quick Background

Heat transfer can be expressed using the governing heat transfer equation, which is as follows for one dimensional heat conduction through a solid:

where:

In words, the heat generation of a system Q is output as conduction and stored internally within the system. Different boundary conditions that can be used to solve this equation include a specified surface temperature, surface heat flux, insulation, symmetry, convective heat transfer, and radiation.

## Steady-State Thermal Analysis

In steady-state thermal analysis, the problem has no meaningful time scale. In other words, the internal energy term is omitted. In Abaqus, you can specify a time increment and a time period, but it does not correlate to real-world time – this is similar to an implicit method in structural FEA. The time period merely linearly changes the scale of the fluxes and boundary conditions applied.

The heat conduction equation can be written in matrix form for a steady-state thermal analysis as follows:

You'll notice the form is similar to that used in an implicit structural FEA; Ku = F. Because the internal energy term is omitted, the only material property needed to run a steady-state thermal analysis is thermal conductivity.

What exactly is thermal conductivity? As can be implied from the name, thermal conductivity is the ability of a material to conduct heat. As an analogy, imagine serving coffee in a Styrofoam cup vs a ceramic mug. When you touch the outside of the Styrofoam cup, your hands will not feel as warm compared to when you touch the outside of the mug. This is because the ceramic mug has a higher thermal conductivity and thus transfers heat better.

Also, because the internal energy term is omitted, steady-state thermal analysis only solves the governing heat transfer equation for when dT/dt = 0. This corresponds to the equilibrium, or steady state of the system.

## Transient Thermal Analysis

Unlike in steady-state thermal analysis, time is significant in transient thermal analysis, more akin to an explicit method in structural FEA. Also like an explicit method, transient thermal analysis requires you to break the analysis down into small increments to properly capture the time-dependent behavior. Pretty much any analysis that can be run as a steady-state analysis can also be run as a transient analysis, as the transient solution will converge to the steady-state solution over a long period of time. However, this is very inefficient, so if a problem can be run easily as a steady-state thermal analysis, it is generally preferred.

The heat conduction equation can be written in matrix form for a transient thermal analysis as follows:

In this equation, C is the product of density and specific heat. In Abaqus, this is solved using the backward Euler formula, which is an implicit method:

In addition to thermal conductivity, specific heat and density of each material must also be defined when running a transient thermal analysis.

What exactly is specific heat? It is the quantity of heat required to raise the temperature of a unit mass of a substance by one degree. The higher a substance’s specific heat, the more energy it takes to increase its temperature. As an example, we use metal pans for cooking food because their relatively low specific heat allows them to get hot quickly. But, when we take those pans out of the oven, we use an oven mitt because its higher specific heat prevents them from heating up as quickly, allowing us to remove the pan from the oven without burning our hands.

Since transient thermal analysis involves loads that are a function of time, initial temperatures are very important. A hot drink left out to cool will take a lot longer to reach a steady state than a lukewarm drink!

## An Example

To demonstrate the differences between steady-state and transient thermal analysis, let’s look at an example of a simple PCB with two chips. The PCB is mounted on a plastic plate with its bottom face held at 293K (room temperature). Each chip is a volumetric heat source with a heat power of 0.1W.

On the top, we have the steady state results. Looking at the upper right corner, the analysis took a time period of '1'. However, in steady-state, time does not mean anything. Each frame is simply a scale factor of the final result; it is not how the temperature actually changes over time. Again, similar to a linear implicit structural analysis.

On the bottom, we have the transient results. The analysis took three hours (in simulation time) to reach the steady-state result. The time in transient thermal analysis corresponds to real-life time. The time animation shows how the temperature of the system actually changes in real life.

## A Note About Nonlinearity

Some nonlinearities, such as latent heat, require the thermal analysis to be transient. Latent heat is the heat absorbed or released by a substance during a phase change without changing its temperature. For example, when ice melts in a glass, all the energy used to melt the ice is absorbed as latent heat and the drink stays at 0°C.

Sometimes, steady-state cases that have severe nonlinearities are more effectively solved as transient cases. This is because the internal energy term acts as a stabilizer.

## Which One Should You Use?

As previously discussed, whether you use steady-state or transient thermal analysis is essentially dependent on whether time-based behavior matters or not. This is summarized below:

Use steady-state thermal analysis when:

Time behavior is irrelevant

You’re only interested in the equilibrium results

Use transient thermal analysis when:

The effects of time are important

You’re interested in the temperature at a specific time

There are severe nonlinearities in the model

## Final Thoughts

Hopefully, this post has provided some useful insight as to the differences between steady-state and transient thermal analysis. As we've discussed, both are valuable tools, but it's important that the benefits and limitations of each are well understood before you start your analysis, or else you might end up with a lukewarm cup of coffee when you wanted it to be hot!

Whether it’s steady-state or transient thermal analysis, our team can help with any of your thermal simulation questions. Reach out to us today!

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