# A mathematical model for natural cooling of a cup of tea/coffee/ etc.

#### eli28

Finally - did it ?

hey, I am think I am done thanks to all of you.
I will be happy to see if you think it makes sense.
I am attaching a pdf file.

thanks again

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#### oz93666

hey, I am think I am done thanks to all of you.
I will be happy to see if you think it makes sense.
I am attaching a pdf file.

thanks again
I don't see any accounting for the relative humidity of the air ... a BIG factor ...

This must be for a cup of liquid with a thin film of oil to prevent evaporation??

You might just as well pluck a number out of the air ...

Will you be conducting an experiment to see if the model is accurate??

#### benit13

I don't see any accounting for the relative humidity of the air ... a BIG factor ...
The relative humidity only affects evaporation losses. As discussed previously, that term is being approximated by a convection term instead.

Eli:
i) you should add to your assumptions that you're assuming that evaporation thermal losses are approximated using convection losses and that water loss from the cup is negligible.
ii) an e-folding time of about 0.76 hours looks reasonable to me. If I leave my coffee for about an hour at my desk, it's temperature goes down a fair bit

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#### eli28

After finishing developing the theoretical background and optimizing the geometry of the cup I have to verify the results. I will tell you about the results when we conduct the experiments.

#### eli28

benit13, thanks again for your help.
I will add these assumptions as well.

#### Woody

Off at a Slight Tangent

I always find when I initially make a coffee, I want it now!
But I have to wait until it cools a bit,
so I find something to occupy my time until it cools...
When I do get back to my cup it has often gone below optimum drinking temperature.

What we need is a temperature sensor alarm (app) which will ring you when your coffee is ready to drink.

I think I already mentioned (in a post here some years ago) my other idea,
for a cup with a phase change material enclosed in its base
which changes from solid to liquid at the optimum drinking temperature,
the latent heat of melting of this material would then initially help to cool the cup,
but would then gradually re-release the energy back, to keep the coffee warm, as the phase change material solidified again.

Unfortunately I suspect that the volume of phase change material required to significantly buffer the temperature would make the cup unacceptably bulky.

However If any one out there wants a killer app, help yourself...

1 person

#### Woody

Too hot for me!

I had a thermos cup once,
it did keep the drink at drinking temperature longer,
but took much too long to get to drinking temperature.

Hence the idea of "borrowing" some of that excess early heat
and then paying it back gradually to keep the drink warmer toward the end.

#### benit13

I had a thermos cup once,
it did keep the drink at drinking temperature longer,
but took much too long to get to drinking temperature.

Hence the idea of "borrowing" some of that excess early heat
and then paying it back gradually to keep the drink warmer toward the end.
Can't you just add a little bit of cold water to the cup? I do that for my coffee and my mum does it with her tea.

#### Woody

That's just too easy...

1 person