The fact that near boiling water cools down quicker than warm water used to be a well-known kitchen knowledge bit. Like my grandma who wasn't a physicist at all knew it. I guess in some places (particularly those where people microwave water) that part of culture is lost cause there's at least a whole generation which hasn't done cooking.
What is meant by "cools down quicker"?<p>Will near-boiling water drop 10 temperature points in a shorter time than the warm water? Yes.<p>Will it reach 10C faster than the warm water? No.
Alternatively, lack of science education, or rather, people forgetting what they learned during chemistry and physics courses.<p>Newton's law of cooling is something I recall being taught in high school in the U.S., and it implies exactly what one observes with boiling warm vs. warm water. Namely, that boiling water will initially drop temperature quicker than warm water.<p>Intuitively, a higher temperature implies a higher average kinetic energy in the molecules, and that leads to faster energy transfer than at lower temperatures.
I remember mentioning this to my high school chemistry teacher and was told I was wrong. I think I even lost points on a test.
To me the neat bit isn't that it got the exponential decay right - that's pretty standard, its that it realised there were two different timescales for the decay and got ball-park numbers for them pretty well.<p>This is the kind of model you would expect from a simple cylindrical model of the coffee cup with some inbuilt heat capacity of its own.<p>However, those decay coefficients are going to be very dependent of the physical parameters of your coffee cup - in particular the geometry and thermal parameters of the porcelain. There's a lot of assumptions and variability to account for that the models will have to deal with.
<p><pre><code> Does that seem hard? I think it’s hard. The relevant physical phenomena include at least..,
</code></pre>
In most engineering problems, the starting point is recognizing that usually one or two key things will dominate and the rest won’t matter.
There's a simple differential equation often taught in intro calc courses, "Newton's Law of Cooling/Heating," which basically says that the rate of heat loss is proportional to the difference in temperature between a substance and its environment. I'm curious what that'd look like here. It's a very simple model, of course, not taking into account all the variables that Dynomight points out, but if a simple model can be nearly as predictive as more complex models...<p>I'm also curious to see the details of the models that Dynomight's LLMs produced!
The appendix lists the equations transcribed from the raw answers.<p><pre><code> LLM T(t) Cost
Kimi K2.5 (reasoning) 20 + 52.9 exp(-t/3600)+ 27.1 exp(-t/80) $0.01
Gemini 3.1 Pro 20 + 53 exp(-t/2500) + 27 exp(-t/149.25) $0.09
GPT 5.4 20 + 54.6 exp(-t/2920) + 25.4 exp(-t/68.1) $0.11
Claude 4.6 Opus (reasoning) 20 + 55 exp(-t/1700) + 25 exp(-t/43) $0.61 (eeek)
Qwen3-235B 20 + 53.17 exp(-t/1414.43) $0.009
GLM-4.7 (reasoning) 20 + 53.2 exp(-t/2500) $0.03</code></pre>
It looks like a lot of them are missing something big. I'd think the two big ones are the evaporative cooling as you pour into the cup, and heating up the cup (by convection) itself. The convective cooling to the air is tertiary, but important (and conduction of the mug to the table probably isn't completely negligible). If there's only one exponential, they're definitely doing something wrong.<p>I'd like to see a sensitivity study to see how much those terms would need to be changed to match within a few %. Exponentials are really tweaky!
That model doesn't explain the relatively sharp drop in the beginning.
It absolutely does. The model that came closest simply used that model twice in the same equation. One for the cup and one for the air.
Are you sure? I believe Newtown's law of cooling says the temperature will drop sharply at the beginning:<p>dT/dt = -k<i>(T_0 - T_room)<p>so T(t) = T_room + (T_0 - T_room) </i> exp(-kt)<p>exp(-x) has a fast drop off then levels off.
<a href="https://www.electronics-tutorials.ws/rc/time-constant.html" rel="nofollow">https://www.electronics-tutorials.ws/rc/time-constant.html</a><p>scroll down, these graphs just don't look similar.
Ha. My university professor used this in a lab to catch people who slack off.<p>There is another factor here: convection. Its speed depends on the viscosity of the fluid and the temperature difference both. And viscosity itself depends on the temperature, so you get this very sharp dropoff.
It does? There is a fast drop followed by a long decay, exponential in fact. The cooling rate is proportional to the temperature difference, so the drop is sharpest at the very beginning when the object is hottest.
probably dominated by the cup as the ambient temperature initially and then as air/the counter top as the ambient temperature on the longer time scale, once the cup and the liquid near equilibrium
On a related note, I have been working on an app that helps determine the correct grinder setting when dialing in espresso. After logging two shots with the same setup (grinder, coffeee machine, basket etc), it then uses machine learning (and some other stuff that I am still improving) to predict the correct setting for your grinder based on the machine temperature, the weight of the shot etc.<p><a href="https://apps.apple.com/ph/app/grind-finer-app/id6760079211">https://apps.apple.com/ph/app/grind-finer-app/id6760079211</a><p>Its far from perfect when it comes to predictions right now but I expect to have massive improvements over the coming weeks. For now it works ok as an espresso log at least.<p>I'm hoping after a few tweaks I can save people a lot of wasted coffee!
Me and the wife (en_GB - draw your own conclusions!) love a decent coffee but can't be arsed with too much wankery over it. We have owned a few kitchen built in units and I've messed with a couple of grinders and espresso pots in the past.<p>Wifey found a kitchen built in unit a few years ago and it is still doing the job, very nicely.<p>Let's face it, what you want is a decent coffee and you have to start from that point, not what sort of bump or grind (that's grindr).<p>I want a cup of coffee with:
- Correct volume - sometimes a shot, mostly an "Americano" - I'm British don't you know
- Correct temperature - it'll go really bitter if too hot. Too cold - ... it'll be cold.
- Crema - A soft top is non negotiable
- Flavour - Ingredients and temperature (mostly)<p>The unit we have now manages bean to cup quite reasonably, without any mensuration facilities. I have made coffee for several Italians and they were quite happy with the results.
Funnily enough I have built essentially the exact same thing in HomeAssistant. Shot collection is completely automated as I have a LM Linea Micra and Acaia Lunar scales (Both have integrations that use Bluetooth). You should consider support for bluetooth scales etc!<p><a href="https://i.imgur.com/a5ztsco.jpeg" rel="nofollow">https://i.imgur.com/a5ztsco.jpeg</a>
Missed chance to call it Grind Finr
That initial drop reminds me of one of the things that stuck to me from my thermodynamic lectures / tests: If you want to drink coffee at a drinkable temperature in t=15min, will it be colder if you add the milk first or wait 15min and then add milk? (=waiting 15 min because the temperature differential is greater and causes a larger drop). Almost useless fact, but it always comes up when making coffee.
The problem is both highly complex, but fairly easy to model. Engineers have been doing this for over a century.<p>Of all the cooling modes identified by the author, one will dominate. And it is almost certainly going to have an exponential relationship with time.<p>Once this mode decays below the next fastest will this new fastest mode will dominate.<p>All the LLM has to do, then, is give a reasonable estimate for the Q for:<p>$T = To exp(-Qt)$<p>This is not too hard to fit if your training set has the internet within itself.<p>I would have been more interested to see the equations than the plots, but I would have been most interested to see the plots in log space. There, each cooling mode is a straight line.<p>The data collected, btw, appears to have at least two exponential modes within it.<p>[The author did not list the temperature dependance of heat capacity, which for pure water is fairly constant]
Irrelevant to your specific cup of coffee its giving you a generic answer.
" Does that seem hard? I think it’s hard. The relevant physical phenomena include at least"<p>Imo no, this seems like something that would be in multiple scientific papers so a LLM would be able to generate the answer based on predictive text.
A full model of a cup of water cooling is, in fact, incredibly difficult.<p>Impossible, since it is chaotic.<p>But a T(t) model should not be too hard for an LLM with a basic heat transfer book in its training set.
Is this for real?<p>This is like someone with no background in physics or engineering wondering "can a LLM predict the trajectory of my golf ball". They then pontificate about how absolutely complex all of the interacting phenomenon must be! What if there was wind? I didn't tell it what elevation I was at! How could it know the air density!? What if the golf ball wasn't a perfect sphere!!? O M G<p>And then being amazed when it gets the generic shape of a ballistic curve subject to air resistance.<p>This speaks far more to the ignorance of the author than something mind boggling about the LLM.
It isn't that surprising that it works well, this problem is fairly well known and some simple heat equations would lead to the result, about which there is a lot of training data online.
nice benchmark! coffee-based Turing test.
The water temperature drops quickly because the room temperature ceramic mug is getting heated to near equilibrium with the water. If you used a vacuum sealed mug(thermos) then the water temp would drop a bit but not much at all initially.
... and so another benchmark is born.