# How long does it take before a cup of coffee gets cold?

Stian Mork &
Eirin Holmstrøm

In an exciting and engaging day at work it is easy to forget the nice warm cup of coffee that you “just” poured. And quite often we’ve had to pour out half full cups, because the coffee has got too cold.

The question is: how long does it take from pouring the coffee into the cup until it becomes “undrinkable” by its temperature?

One method to figure this out would of course be to use a stopwatch, but as Application Engineers at PLM Group with access to all sorts of great software, there are of course more analytic approaches that can be used to find the solution to this question.

### SOLIDWORKS Simulation Study

We start by creating a CAD model of a coffee cup.

The following parameters are known:

• Temperature coffee = 95 grades Celsius
• Temperature surroundings = 20 grades Celsius
• Natural convection to air = 15 W/m^2.K

It is natural to assume that the coffee’s heat loss is a combination of convection to the air, and conduction to the cup.

To reduce the problem size, we’ve taken advantage of the circular symmetry and used 2D thermal analysis for this calculation. By utilizing this capability in SOLIDWORKS Simulation, we can reduce the computation time significantly by only looking at half the cross section of the cup.

We define Ceramic porcelain as the material for the cup and water as the material for the coffee.

Next we define the boundary conditions given the know parameters mentioned above.

We wanted to look at the changes over time (how long it takes for the temperature in the coffee to decrease to 45 degrees Celsius). We therefore specify this to be a Transient study. We set the total solution time to 1 hour (3600 sec.) with a time increment = 60 sec. (meaning that we will plot data every minute.)

After running the study, we are able to animate the temperature distribution. Even if the study is run in 2D, we still have the ability to interpret the results in 3D using 3D plots.

At the total time of 1 hour we can clearly see how the heat from the coffee has been transferred to the cup through conduction and the surroundings by convection. It is only in the centre of the coffee that we still have a temperature above 50 degrees Celsius.

We can find that after 1 hour the coffee is coolest along the edge of the cup, and that the temperature is a little below 32.5 degrees Celsius.

### Results

Since this is a transient analysis, we can easily find at what time this occurs by using a response graph. Looking at the graph we ca see that it crosses 45 degrees at 1225 sec = 20.5 minutes.

The conclusion is therefore that if we want to enjoy our coffee while it has a temperature above 45 degrees, we must drink it within 20 minutes.

But, let’s take this a bit further.

### What about if we have a pre-heated cup?

Pre-heating the cup is an old trick to keep the coffee warm for longer time. So of course, this must be tested as well.

We therefore create a study where we change the initial temperature of the cup to 90 degrees Celsius.

If we look at the graph below, we can see that preheating the cup did not give more than 1.5 degrees difference after 20 minutes.

We can also see that the preheating has the highest effect within the 13 first minutes (768 seconds), after that point the temperatures between the two experiments equalizes. (This behaviour is as expected since the larger the temperature difference between the object and the surroundings the quicker the cooling).

### Bonus study

As a bonus Stian wanted to test how much time it takes before the cup becomes unpleasantly warm to touch. He therefore did a study of 20 seconds after the coffee was poured into the cup. In this study he found that the cup is heated by the coffee and has a temperature of 60 degrees on the outside after about 12 seconds, at this point we’d better use gloves to hold it.

All studies within this experiment were conducted with SOLIDWORKS Thermal studies which are a part of the SOLIDWORKS Simulation Professional package. No coffee cups were hurt while conducting the research.

Enjoy your coffee and work day!