Comments on: Drinking (tea) with mathematicians http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/ Just your typical female vegan capitalist mathematician ceramicist cyclist's weblog. Tue, 16 May 2006 04:35:11 +0000 http://wordpress.org/?v=1.5 by: oxeador http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35869 Sat, 01 Apr 2006 21:27:10 -0800 http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35869 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. 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.

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by: MGM http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35872 Sun, 02 Apr 2006 00:20:03 -0800 http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35872 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. 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.

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by: Moebius Stripper http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35873 Sun, 02 Apr 2006 08:45:48 -0700 http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35873 <blockquote>Colder water is denser, so you always have colder water at the bottom of the cup.</blockquote> Aw, nuts. So, in other words, it's not me, it's <i>physics</i>. (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.

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.

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by: karrde http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35874 Sun, 02 Apr 2006 09:40:16 -0700 http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35874 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 <i>T1-T2</i>, with constant <i>k</i> 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. 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.

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by: meep http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35875 Sun, 02 Apr 2006 10:46:54 -0700 http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35875 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 <a href="http://talldarkandmysterious.ca/pottery/v/inprogress/finished/IM000640.JPG.html" rel="nofollow">this one</a> 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. 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.

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by: MGM http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35876 Sun, 02 Apr 2006 12:29:31 -0700 http://talldarkandmysterious.ca/archives/2006/04/01/drinking-tea-with-mathematicians/#comment-35876 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). 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).

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