Holly Huellemeier, Engineer, Research

Whether you're an at-home coffee brewmaster or a drive-thru coffee connoisseur, you've probably experienced that awful feeling of burning your tongue on a piping hot cup of coffee. Most coffee drinkers prefer to drink their coffee at temperatures between 68 and 79 °C, but coffee brewing temperatures are generally higher.

When I brew coffee at home, I target a brewing temperature just below the boiling temperature of water (e.g., 91 – 96 °C) using my temperature-controlled electric kettle. Of course, this setting depends on the roast of the coffee. Lighter roasts (my favorite) require a higher brew temperature, whereas darker roasts should be brewed at a slightly lower temperature.

So how long does it take for coffee to cool from optimal brewing temperature to optimal drinking temperature? Sounds like a heat transfer question for some HTRI research engineers!

We can approach this problem using a lumped capacitance model, where heat flow is modeled in a “circuit” consisting of resistors and capacitors. For this analogy, heat flow, temperature, thermal resistance, and thermal capacitance are modeled as electrical current, voltage, resistance, and capacitance, respectively. In the lumped capacitance model, temperature nodes are positioned at points or regions that are assumed to have a uniform temperature. This assumption is valid for cases with a small Biot number.

For example, the cooling of coffee in a mug can be approximated by the circuit described in Figure 1. Temperature nodes are located at the center of the cup wall (Tmug) and in the bulk coffee (Tcoffee), while the ambient temperature (T∞) is assumed to be constant.

Figure 1. Cup of coffee with two-node lumped-capacitance model for transient analysis
Figure 1. Cup of coffee with two-node lumped-capacitance model for transient analysis
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Figure 1. Cup of coffee with two-node lumped-capacitance model for transient analysis
Figure 1. Cup of coffee with two-node lumped-capacitance model for transient analysis

Applying conservation of energy requires tracking multiple modes of heat transfer, including convection, conduction, radiation, and evaporative cooling, which are described by the thermal resistances. The change in temperature as a function of time at each node can be described by an ordinary differential equation (ODE), and the set of ODEs (two in this case) can be solved simultaneously using an iterative method, such as the Runge-Kutta method.

Let's consider a situation where hot coffee at 95 °C is added to a mug that is at room temperature (21 °C). With some assumptions [1], the equivalent circuit model describes a non-linear cooling curve. To test our model with real data, we set up a little experiment by adding hot coffee to a glass mug and then measuring the temperature of the coffee using a handheld digital thermometer for 10 minutes. The results are plotted in Figure 2.

Figure 2. Measured vs. modeled temperature vs. time for cooling coffee in mug
Figure 2. Measured vs. modeled temperature vs. time for cooling coffee in mug
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Figure 2. Measured vs. modeled temperature vs. time for cooling coffee in mug
Figure 2. Measured vs. modeled temperature vs. time for cooling coffee in mug

The data demonstrate an exponential decay in temperature, following the behavior of the model. Closer examination may register more rapid cooling initially (0 to 3 min), then a slower cooling curve (3 – 10 min). The rapid initial cooling may be explained by the initial pouring action of coffee into the mug. Rapid pouring is expected to have a higher convective heat transfer coefficient (Rconv,int) than the convective heat transfer coefficient associated with stagnant fluid (i.e., at longer times after the fluid settles).

If the optimal drinking temperature is 68 – 79 °C, our model suggests the coffee is prime for sipping about 2 – 5 minutes after brewing. This is rather a short window for optimal coffee enjoyment (~3 min)!

And if you are like me, lukewarm or chilled coffee is betrayal in a mug. So, after a long morning meeting, I might be at the microwave giving my mug a quick 30-second zap (bye-bye, coffee aromatics).

Fortunately, the cooling of coffee can be slowed by using a vacuum-insulated thermos. We can thank Scottish scientist Sir James Dewar for inventing a vessel surrounded by a vacuum jacket (also known as the Dewar flask) in 1892 to slow the warming of a chilled liquid.

Can you figure out how to modify our circuit analogy to incorporate a vacuum-insulated thermos? Or how about adding a lid to our cup? The possibilities are endless!

Stay tuned for more articles about your morning cup of joe!

Nomenclature

Ccoffee, Thermal capacitance of coffee, J/K

Cmug, Thermal capacitance of coffee, J/K

Rcond,table, Conductive thermal resistance of table, K/W

Rconv,ext, External convective heat transfer resistance, K/W

Rconv,int, Internal convective heat transfer resistance, K/W

Revap, Evaporative heat transfer resistance, K/W

Rmug, Conductive thermal resistance of mug, K/W

Rrad, Radiative heat transfer resistance, K/W

T∞, Ambient temperature, K, °C

Tcoffee, Coffee temperature, K, °C

Tcoffee,measured, Experimentally measured coffee temperature, K, °C

Tcoffee,model, Modeled coffee temperature, K, °C

Tmug, Mug temperature, K, °C

Tmug,model, Modeled mug temperature, K, °C

References

  1. J.-S. Conderet, Teaching transport phenomena around a cup of coffee, Chem. Eng. Edu. 4(2), 137 – 143 (2007).