Tall, Dark, and Mysterious

4/1/2006

Drinking (tea) with mathematicians

File under: I Made It Out Of Clay, Queen of Sciences. Posted by Moebius Stripper at 9:02 pm.

This morning, I made myself some tea in one of the matching mugs from my dinner set, waited an inordinately long time for it to start to cool down, and then downed the rest of it in a few gulps because it was quickly becoming too cold to drink. Not the most satisfying tea-drinking experience, but that’s what I get (I think) for throwing narrow cylinders.

So I started to wonder what shape of mug, made out of clay of constant thickness, would be ideal for drinking hot beverages. Specifically: I would like to pour myself some tea, wait a fixed time t for the surface of the tea to cool down to a drinkable temperature before beginning to sip, and then sip at a fixed (and preferably constant) rate until the tea is gone, with the tea remaining at a constant temperature. It seems like this should be doable; at the very least, I don’t think it’s not a ridiculous thing to ask for: the surface of the tea cools much faster than the tea below, and so there should be a way to sip tea from the appropriate vessel in such a way that each sip exposes another layer of tea that cools down to the temperature of the previous layer, just as I’m drinking it. It also seems clear that the top should be wider than the bottom, as the tea at the bottom will not be completely insulated by the tea above. Also, the heat transfer through the mug cannot be ignored: the specific heat capacity of clay is around a third of that of water, and only a third more than that of air.

…that’s as far as I got: I don’t have the physics background to set up this problem, though I spoke with someone who did, and he came up with some rather grisly equations that I don’t have the calculus background to solve. I also don’t know what simplifications could reasonably be made. But if anyone does, and can come up with a reasonable shape of mug that is conducive to tea-drinking, I’ll make you such a mug.

19 Comments

  1. I wrote down the differential equations that need to be solved with the approximation that there is only heat transfer between the tea and the air, and not between the tea and the mug. (Actually, those are the grisly equation that MS mentions having seen). However, I do not think that this approximation is valid.

    - oxeador — 4/1/2006 @ 9:27 pm

  2. Colder water is denser, so you always have colder water at the
    bottom of the cup. As for better designs for a tea cup, try to research “Japanese-style tea cups (bowls)” aka pialas.

    - MGM — 4/2/2006 @ 12:20 am

  3. Colder water is denser, so you always have colder water at the bottom of the cup.

    Aw, nuts. So, in other words, it’s not me, it’s physics. (But it seems, empirically, that the cool-water-is-denser property is overridden by the water-exposed-to-air-cools-faster property. I’m thinking that we can ignore the former in model, though I never had very good instincts for which sorts of approximations would end up causing problems later on.)

    Thanks for the tip on Japanese-style cups; a quickie Google search revealed nothing of use, but I’ll try again later.

    - Moebius Stripper — 4/2/2006 @ 8:45 am

  4. That “cold water is denser” bit might mean that there are actually convection currents in the cup as it cools. The tea at the surface cools, as does the tea in contact with the clay. The cooled tea sinks towards the bottom of the cup–probably along the sides–and the hotter tea replaces it at the top.

    However, you might be able to get away with ignoring these convection currents, and model the tea in the cup as having approximately-homogenous temperature, with two vectors of heat loss–one through the clay, the other through the air.

    The tea will lose heat energy to the clay, which will gain heat energy (hence, temperature). The tea will also lose heat energy by evaporation, and by heating of the air in the room. Convection in the air in the room will not cause an appreciable rise in temperature–thus the temperature of the air in the room can be considered constant for the model. You can probably also ignore heat loss due to evaporation in your model.

    Since we’re ignoring convection in the cup, I’m assuming we can also ignore differing thickness of the walls of the cup from top to bottom, and the base (which is probably thicker).

    I see that the relative surface area of the cup with respect to the air at the top is the big problem. Heat transfer happens according to an exponential function of the difference in temperatures T1-T2, with constant k in the exponent determined by the specific heat of the two substances.

    If the surface area of the sides of the mug are too large, and the surface area at the top is too small, the tea will lose too much heat to the sides in the process of cooling. Thus, by the time the top is cool enough to drink, the tea lower down in the mug will have lost too much heat energy to remain hot for drinking.

    The relative surface area of the sides and top is defined by the ratio between radius of the cylinder and the height of the cylinder. Thus, a quick-and-dirty approximation could probably be made from those two numbers.

    How long this approximation will remain close (how far linearization will remain within a small distance from the actual function) is anyone’s guess.

    I suspect you’re not asking a math question–you’re asking an engineering questions which can be solved with math. :)

    I myself have noticed that my intake of tea from a professionally-manufactured cup is usually in increasing sizes. That is, the temperature in the tea drops enough to allow me to take larger portions of the tea, while it is usually still hot to the touch.

    What are the dimensions of the cups you are using? (I assume that my purchased cups use a similar material.) I typically use cups of approximately 3 inches (7.62 cm) diameter. The rim typically has 1/4-inch of thickness (6.35 mm), and the cylinder has a outside height of about 3.5 inches (9.525 cm). Cylinder walls thicken slightly towards the bottom, but I don’t have measurements handy for that factor…or for the thickness of the base.

    - karrde — 4/2/2006 @ 9:40 am

  5. Here’s my own take on it — not math, but going back to traditional ways of serving tea (oddly enough, I’m currently reading =A History of the World in 6 Glasses= and I just started the section on tea.)

    First off, I keep the majority of the tea in my teapot, and use one of the small teacups you gave me (I’m guessing this one is the same size.)

    Secondly, I usually heat up the teacup before pouring tea in it. I pour some of the boiling water all around the cup.

    Thirdly, the kind of teapot I use (I can’t find it online right now) has a little platform to set your teacup on while the tea is steeping, to keep the bottom of the teacup warm while you wait.

    I like the Japanese teacups mentioned above (I have a set of porcelain cups from Japan), but I think they’re best for green tea, which is made at a lower temp than black tea, but hotter than herbal tea. If that matters. I think they would be just fine for herbal tea.

    - meep — 4/2/2006 @ 10:46 am

  6. I think post #5 summarizes the optimum way of have a hot cup of tea to drink for an extended period of time. At least, this is how tea is consumed in Russia/Central Asia. Now, living in California has taught me to appreciate cold tea (no ice please) but I can see
    how you might need to keep yourself warm in Arctic Canada.

    If you want to have just one big mug of tea, I do not think there is a way to keep the liquid warmer at the bottom unless you start heating the cup itself. The fact that you can start drinking tea before you can touch the surface of the cup (post #4) only means that the inner surface of your mouth developed higher heat tolerance (beware the damage to your teeth).

    - MGM — 4/2/2006 @ 12:29 pm

  7. PS. As someone who drinks 6-8 cups of tea/day I find this subject very interesting. See also http://www.stillsitting.com/sitting-around/enlarge-small-teaset.html for a picture of pialas/japanese teabowls.

    - MGM — 4/2/2006 @ 12:53 pm

  8. You can’t ignore convection: in a 10-cm tall mug, the tea hits critical Rayleigh number at temperature differences of about 10^-4 degrees. For any measureable temperature difference between the top and bottom of the mug, the system will convect very efficiently. Rayleigh number does go as the cube of the mug height, so if you wanted a very shallow mug you could ignore convection - but that’s silly, no one likes shallow mugs. And then evaporation would cool your tea way too fast, anyway.

    In an efficiently convecting system, what you have is an isothermal core, where convection is working, surrounded by thermal boundary layers where heat transfer takes place by conduction. In a teacup these boundary layers are very thin - less than a tenth of a millimeter for measurable temperature differences, and they get thinner with increasing temperature difference - so when you sip, you’re actually drinking from the isothermal core. You’re therefore concerned with the bulk cooling rate of the system, not with the thermal boundary layer.

    So I think the answer is: use meep’s method, and/or a lidded thermos.

    - yami — 4/2/2006 @ 2:07 pm

  9. It also seems clear that the top should be wider than the bottom. . .

    but now, in attempting to solve one problem you have introduced another, that of a top-heavy cup being more prone to tipping over. I favor small round cups, refilled from the teapot. An alternative would be a larger mug that rests on a warming unit, either electric or a heated disk of soapstone, that would slow the rate of cooling.

    Now, what about leveraging computational fluid dynamics to design a teapot whose spout neither dribbles nor overshoots the target when pouring?

    - Molly — 4/3/2006 @ 11:21 am

  10. I have a teacup I like very much. It is thin-walled, but it keeps tea hot a long time, and maintains temperature while drinking. It is made of ordinary ceramic material. It does not have a lid.

    Imagine a somewhat elongated teardrop. Cut off the bottom, just below the teardrop’s thickest point. That’s the base of your teacup. Cut off the top to make the opening (which will, obviously, be narrower than the base). Flare the opening a bit, so it will fit your lip. Put on a handle. The end result should fit in the same box as a regular coffee mug. Capacity about twelve ounces.

    Why does it work? Here are my ignorant guesses:

    1. The shape of the tea in the vessel approximates a sphere. Okay, it’s a poor approximation, but better than a cylinder. Of course the sphere has optimal relation of volume to surface area.
    2. The bulging, in-curving walls deaden convection currents within the tea. The bulging part is a “dead zone” which will shed disruptive vortices into the currents. Also, because the top area of the tea is smaller than the bottom, currents have to accelerate and decelerate as they cycle. Convection is less efficient than it would be in a cylinder.
    3. Contrary to suggestions above, evaporation is a significant mechanism of heat loss; perhaps the most significant. The smaller opening limits air exposure, and again the internal shape of the vessel reduces convection of air when the tea is partly drunk.

    - dipnut — 4/3/2006 @ 12:08 pm

  11. Designing the perfect cup

    Next part in the geeky handicraft series: under the heading Drinking (tea) with mathematicians, Moebius Stripper asks for a scientific design of the perfect teacup.

    - Qulog 2.0 — 4/4/2006 @ 4:53 am

  12. If you want to reveal the grisly system of DEs to me, I could try to coax a solution out of Maple, or at least a numeric approximation.

    - saforrest — 4/4/2006 @ 7:53 pm

  13. One trick that used to work with coffee was to place a spoon in the cup to enhance the cooling, and then sip the coffee (without the spoon!) from that point on.

    Now that I’ve switched to decaf tea, I’ve noticed that mugs that have a diameter almost as large as the height of the cylindrical portion of the mug cool the tea too quickly. Narrow and tall seems to be a better combination.

    Maybe what we should be looking for is a shape that cools slowly throughout the drinking experience, and if we have to wait too long to begin drinking, well, that’s what ice cubes are for.

    - Rex — 4/5/2006 @ 12:30 pm

  14. I’m with dipnut. I tried about 20 or so mug shapes (all hand-thrown by me) and settled on a narrow top, a fat bottom, and roughly tear-drop shape.

    Hey - where’s the preview button? Am I blind?

    - llewelly — 4/5/2006 @ 4:15 pm

  15. When I brew tea (which is often) I like the water to be as hot as possible, boiling or practically boiling, during the steeping of the leaves, in order to get the best transfer of the phytonutrients, caffeine, flavors, and so forth from the leaves. But then when the brewing is done I have to wait for ages for the tea to cool to a suitable drinking temperature.

    I’ve found a nice solution to this problem. I always keep an extra mug in the freezer, and immediately following the end of the steeping phase, I remove the icy cold mug from the freezer and pour the scalding tea into it, which causes the tea to cool, almost immediately, to just above suitable drinking temperature. The addition of one or two ice cubes at that point lowers the tea to exactly the right termperature to drink. (You can’t just use ice cubes alone without the frozen mug, because it would require too many ice cubes, watering down the tea too much, causing the mug to overflow, or both.)

    Since this trick can be done immediately after the steeping is over, I am in no danger of wandering off (as I am wont to do), letting too much time elapse and coming back only to find my tea too cool to be enjoyed. Instead, I can partake of my tea right away!

    - lobster — 4/6/2006 @ 9:01 pm

  16. This is a fairly ugly mathematical problem.

    1) as others have noted you can’t ignore convection
    2) or evaporation
    3) and even if you tried to just have raidient heat the temp/surface area caluclations are nasty with lots of assumptions (e.g. do you take into account the conductivity of the table or coaster?)

    - Francis — 4/12/2006 @ 12:52 am

  17. Y’know, I’ve been thinking… and the problem I have with my tea is that the bottom is saccharine while the top is unsweetened. Which makes me wonder: is this because the sugar just takes too long to dissolve, or because the teacup is chemically stratified? If the latter, than you *can* ignore convection. I suspect that in reality it’s the former, because I really only notice this in seismo seminars, where they only provide crappy straws to stir with and the sugar is a larger grain than my usual substitute powder. But if you were interested in drinking tea with highly variable concentration of additives, maybe with some xanthan gum to make it more viscous, then you could set it up to ignore convection all you wanted!

    - yami — 4/13/2006 @ 6:46 pm

  18. Hmm, I don’t take sugar in my tea, but now that I think of it, the bottom of the tea tends to be more concentrated than the top. But I don’t know offhand if tea is a suspension or a solution; and if the latter, how much more or less soluble tea is than sugar.

    AND OMG I AM NOT ADDING XANTHAN GUM TO MY TEA EVER KTHXBYE.

    - Moebius Stripper — 4/13/2006 @ 7:22 pm

  19. Here’s the teapot I use on this page:
    http://www.forlifedesign.com/wholeleaf.htm

    I set my teacup on top of the infuser lid, which is metal. And so my teacup is nice and toasty by the time I pour my tea.

    - meep — 4/16/2006 @ 4:49 am

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