hot cup
April 4, 2019 11:32 AM   Subscribe

In a hot mug of liquid, is the liquid at the top hotter or cooler than the liquid at the bottom?
posted by griphus to Grab Bag (17 answers total) 12 users marked this as a favorite
It's got to be cooler at the top - ceramic is an insulator, while the liquid at the top will lose heat directly through the air and also through the loss of hot steam. I'm not sure how much of a difference in temperature there would be, or the depth of the gradient - I imagine that depends on a lot of factors. (This obviously only applies if the room is cooler than the liquid!)
posted by showbiz_liz at 11:55 AM on April 4, 2019 [4 favorites]

My herbal tea habit and basic understanding of thermodynamics means it’s cooler at the top. The heat seeks cool and exchanges temperature with the open air and will work its way down. When I forget about my tea for a little too long I have to slog through the top chilly layer to get to that warm leaf liquid.
posted by Crystalinne at 12:00 PM on April 4, 2019 [1 favorite]

Nthing cooler at the top (my source being the ridiculous smartmug that a relative got me for Christmas - it has a temperature sensor in the bottom and the temperature goes down when I stir it and the cooler liquid at the top gets mixed in).
posted by wuzandfuzz at 12:05 PM on April 4, 2019 [8 favorites]

Yeah but heat rises. Perhaps not enough to overcome the other factors (more rapid heat transfer to the air; insulation deeper within ceramic and liquid) that would lead the top of the cup to cool faster, but perhaps enough to shift the Area of Most Warmth (AoMW) up from the bottom of the cup toward the middle somewhere.

/overthinking a cup of tea
posted by notyou at 12:15 PM on April 4, 2019

As the fluid cools at the edges and top of the mug, it becomes more dense and falls: convection currents form. Comparatively hot fluid at the bottom and center of the cup rises to the top and gets pushed to the edges, where it then cools and falls back down.
posted by cyclopticgaze at 12:20 PM on April 4, 2019 [7 favorites]

The top will generally be cooler if my assumptions about what you mean are correct.

If the cup is a normal hot beverage cup, open on top, filled with something like recently boiled water, and has settled for a minute or so at normal room temp, then yes, the top will be cooler by a bit, for the reasons outlined above (great evidence wuzandfuzz!).

It could in principle be the other way; say if you have an ultrathin container that is highly conductive, and you had a lid on top, and the air was very cold. And maybe blew air over the bottom, etc. But most real life situations, it will be a bit cooler. That's why you blow on stuff to cool it down, right? And stirring also cools pots, etc.

It's true that hot air rises when we're talking about a pocket of hot air in a much cooler air volume, but heat doesn't intrinsically rise, it's stuff that's moving.
Water doesn't change nearly as much in density as temp changes, and the initial differences will be tiny, so hot water doesn't rise the same way a hot air balloon does. In my experience convection action is weak and often not present in a hot cop of coffee, though I'd welcome evidence to the contrary!
posted by SaltySalticid at 12:21 PM on April 4, 2019 [4 favorites]

Best answer: This questions is a lot more challenging to answer than you might think. You're looking at a system that is:

a) transient (conditions change over time)
b) multi-phase (liquid water converting to vapor and vapor condensing back to water)
c) multi-fluid (air and water vapor mixing)

There's convection on both sides of the interface - the liquid in the mug and the air at the surface are both constantly in motion. However, we can make a few assertions and think about their implications. (Let's assume the walls of your mug are perfect insulators to make this easier.) For example:

1) At t=0 (you have just poured hot water in to your mug), the temperature of the liquid in the mug is constant and homogeneous. No heat transfer has taken place, so all of the water in the mug is in equilibrium.
2) At some t_cold>0, the temperature of the liquid has dropped to room temperature and is again constant and homogeneous. Similar situation to point 1.

Starting at t=0, let's increment time by some arbitrarily small amount. There must be some heat transfer between the surface of the liquid and the room - that interface is not a perfect insulator, and there's a temperature gradient (liquid is hotter than room air). Heat will only transfer at the interface, so at this instant the very surface of the liquid in the mug is colder than the rest of the liquid by some (very, very tiny) amount.

You now have set up a temperature gradient in the cup (and also in the air, but let's ignore that for now). The temperature gradient will induce convective activity in the liquid. This will encourage mixing (again, nature doesn't like gradients) and draw heat up to the surface, where it can be transferred with the air.

So yes, between pouring hot water and the mug becoming completely tepid then the surface will always be colder than the bottom of the mug (again, assuming you've got a perfectly insulating mug). The question now is how much colder, and that will depend on how quickly heat can transfer between the water and the air versus how well the natural convection of the water can eliminate that temperature gradient in the mug. The amount of math required to figure that out is more than I am capable of dealing with right now (and we've already made several large assumptions here to simplify things).

I have a Masters in fluid dynamics but have not had to deal with this in quite some time.
posted by backseatpilot at 12:28 PM on April 4, 2019 [52 favorites]

again, assuming you've got a perfectly insulating mug
Yeah but that's a pretty bad assumption, especially for normal ceramic mugs (not vacuum sealed thermal mugs etc). Also the convection is not necessarily fast, and I think you've ignored conductive cooling? I don't have a fancy mug with temp indicator but I'm going to go do some tests :)
posted by SaltySalticid at 12:49 PM on April 4, 2019

Best answer: Ok y'all, preliminary observations have come in.

Case 1: two ceramic cups, from the same source, so about as identical as I can get. Boiled water, added 6 oz to each, set one on tile counter and one on wooden cutting board in the same small kitchen. Waited 2 minutes, took temp at top (1/8" deep) and bottom (1/8" from bottom) of each, using a meat thermometer.

Results 1:
Tile-based cup: 92F at top, 150F at bottom
Wood-based cup:100F top, 140 F bottom

Case 2: One large stein, glass. Poured water and waited one minute.
Results 2: 120 F top, 165 F bottom.

Preliminary Conclusion: conductive transfer matters here, both into the cup, cup to air, and cup to surface. That's why the tile and wood are so different. Also you can see convection in tea and black coffee in clear cups, it's weak.

It certainly can go the other way in difference scenarios, and it would be fun to see if anyone can get one! I tested the one that I thought was most like OP's idea. Also this is not a great data set, and I'd be a lot happier with it if I had a few different thermometers and tested them in different orders, depths etc. I have a PhD in applied math, and I have studied fluid dynamics and thermodynamics, but I am no expert in mathematical physics. I also think of myself as more of a scientist, so when I see that things can go a lot of ways and it's really easy to grab some preliminary observations, I'll do that :)
posted by SaltySalticid at 1:13 PM on April 4, 2019 [36 favorites]

That's why the tile and wood are so different.

We obviously come from different backgrounds because I would say 10% or less is well within the bounds of "basically the same." Engineering, government work, etc.
posted by backseatpilot at 1:19 PM on April 4, 2019 [8 favorites]

"basically the same."
You are totally right, I should not make any inference on the difference! Especially given the poor quality of the data. Also my first comment was wrong where I said convection was "often not present", of course it's technically there, but that doesn't mean a temp gradient cant last for quite a while. In that light, I should mention the stein is now 112F at top, 139F at bottom. I think the gradient just gets shallower and shallower as it approaches equilibrium.
posted by SaltySalticid at 1:27 PM on April 4, 2019 [1 favorite]

When people did those experiments to demonstrate that the vortex which establishes itself when you pull the plug on a big enough tub really does go in opposite directions in the Northern and Southern Hemispheres, they had to wait days for the currents resulting from filling the tank to subside enough to see the signal, so I think we can assume the pattern of circulation produced by filling the cup will persist over the time between filling and taking the first few sips.

And I think that pattern will resemble the pattern of convection going from bottom to top, but will probably go down in the center and up on the sides whereas convection resulting from heating at the bottom might be the opposite.

But all that really doesn't answer the question except to suggest that layers that don't mix will probably not form . . .
posted by jamjam at 2:17 PM on April 4, 2019

Water (like most things) is also thermally conductive, which provides another path for evening out a gradient. Though I would expect convection to have far more of an effect.

Initial heat loss will be dominated by evaporation - every gram that evaporates takes at least 2 kJ out of the bulk liquid. 2 kJ is enough to change the temperature of an entire liter of water by 0.5 C, or of a cup by 2 C. So if you really want to see the extreme, blow lots of air over the surface to encourage evaporation.
posted by Jobst at 2:30 PM on April 4, 2019 [3 favorites]

I wonder if some of these thermal images of coffee mugs might lead to something of interest.
posted by daisyace at 4:36 PM on April 4, 2019

If I was to hypothesize based on the times I have spilled tea and coffee on myself, the uppermost liquid is as hot as liquid can possibly become before turning into gas. I have yet to try spilling liquid from the bottom of the cup on myself, and am not sure how that would even work.

But yes, if you want some further guessing for your study, the liquid at the bottom will be hottest, since it is insulated by the mug, and insulated by the surface the mug is sitting on. The heat will be dumped first through the path of least resistance, i.e. the cool air at the top of the mug.
posted by turbid dahlia at 4:52 PM on April 4, 2019

Seems like an experiment for those folks who have the multi-probe thermometers for their meat smoking. Or it seems like a good science fair project for some young maker. A handful of thermocouples and a Raspberry Pi to collect data and make pretty graphs.

I know heat loss depends on what the mug is sitting on. My coffee stays warmer longer when I put it on a cork-like coaster rather than directly on the tile surface of my table. And it really stays warmer a lot longer when it's in a BODUM® Tumbler (dual wall vacuum).
posted by zengargoyle at 9:25 AM on April 5, 2019

Huh. All the science above suggests I'm wrong, but: in my experience, if a mug of tea is left for a long time, the bottom is colder than the top. My partner makes me a cup of tea in the morning, and it can be up to 30-45 mins before I wake up properly & try to drink it. In those cases, I find the top is still luke-warm and drinkable, but the bottom half of the mug is cold.
posted by yesbut at 3:57 AM on April 6, 2019

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