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...
I'm also curious to see the details of the models that Dynomight's LLMs produced!
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.
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!
It's a mix of course, but I think it should be mainly that and evaporative cooling. Evap is _very_ effective but will fall off rapidly as you get away from boiling. The conduction into the mug will depend a lot on the mug material but will slow down a lot as the mug approaches the water temperature.
I'd be very interested in seeing separate graphs for each major component and how they add up to the total. Even asking the LLMs to separate it out might improve some of their results, would be interesting to try that too.
Yes, since they didn't explicitly list the evaporative cooling when the coffee was poured into the cup, I suspect it was not included (as if the coffee started in the cup). That means that the starting temperature is off and screws up all the other calculations.
The evaporative cooling as you pour into the cup is when the coffee is at the highest temperature and has the most surface area even though it only takes a few seconds. One could test this either by including it explicitly in the requested calculation, or by putting the fill spout directly at the bottom of the cup when filling.
That will be the dominating term eventually. But the initial sharp temperature drop is mostly due to the coffee mug being at room temperature and having a ~significant mass.
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.
Apparently the act of pouring has a huge effect on temperature because of the surface area :: volume ratio of the fluid as it streams (and turbulence after striking the bottom). The site above claims a single pour can drop it 20-30 degrees. There may be a similar effect here.
Ha. My university professor used this in a lab to catch people who slack off.
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.
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
* Get less scared about applying to do stuff! I'm leaving my longtime job---I've taught advanced math to super-smart high schoolers; I'm quitting to be a visiting professor at Deep Springs College for a semester and then ???---and in the past, fear of applying to things (jobs, grad schools, writing residencies) has been a major blocker.
* Learn complex analysis!
* Get a better workflow for writing my notes to myself (e.g., Obsidian) and for publishing my blog/website (have a marginally-functional Hugo instance right now). Small thing, but the kind of important-but-not-urgent thing that it's easy to put off!
Mmmm it's something I've done for a long time; I've been at the same school for seven years; it's a ton of fun but also gets repetitive... I want more challenges!
I've used Sage for years to run the backend (calculations/computations/graphics/prototyping) for a multivariable calculus class I teach. It's not perfect, but as a lightweight, Python-style CAS to do all sorts of "standard" calculations, it's very easy to use!
Elite private high school in SFBay (not tech, but all of our families are tech/VC). Everyone is getting raises of max(8%, $6700), working out to an average per-employee raise of 8.6%. Inflation is the main stated reason.
(FWIW this means that my salary, teaching undergrad-level math classes to high schoolers, is going from $74K to $81K.)
reply