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

#### eli28

1st Experiment results

hey everyone,
Thanks for helping me.
I am attaching a photo with 2 graphs:
1. the calculated temperature according to the developed equations.
2. Experimental results.

in addition, I am attaching a photo of the experiment.

I would be grateful to hear your opinion regarding the results and their adaptation to the theory.

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#### oz93666

That's a pretty good match ! ...

I would expect at high temp the heat loss from evaporation would be higher than the model allowed for , and at low temperatures the reverse ...

That's what we see , I'm just surprised the difference wasn't greater.

Really need to repeat experiment at different humidity levels.

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#### Woody

Tweaks

As Oz says that is not bad,
but there is obviously a (small) temperature dependent systematic error from somewhere.

There are a couple of options if you want to try to improve the model further:

1) Guess what might be causing the deficiency and tweak the coefficients of that term up and down to see if you can obtain a better match.
Sometimes having obtained a better match via trial and error you can back justify the changes.

2) Repeat the experiment many times under variously varying conditions to try to define the required tweaks.

Note that these are not necessarily mutually exclusive.
If mathematical tweaks to the model indicate a particular sensitivity,
you can then try to design a modification to the experiment, to investigate that possibility.

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#### benit13

Nice work!

In addition to Oz's suggestion about latent heat, try upping the convective heat loss term for the side walls of the cup. In building thermal transport models, one of the biggest uncertainties is the convective heat transfer coefficient for air-wall interfaces. It will consequently be unsurprising if you find the same is true for your coffee cup.

#### eli28

1st Experiment results - Continue

I haven't repeated the experiment yet, but tried to fit the curves unsuccessfully.
I am adding the excel file in which there are 4 variables (marked in yellow and red) that can be changed - Water density, Cp, h, k.

Maybe you may have better insights...

Thanks

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#### eli28

an assumption to the consistent deviations

hello everybody,

As I posted before I got some results.
I repeated the experiment, but with a different material (the glass is made of).
I still get an unexplained deviation that disturbs me...
I keep thinking what was modeled wrong...? (Sadsmile)
The only main thing I can think of is the fact that there is evaporation in my problem. what do you think?

I am attaching again:
1. the theoretical equations I developed and the explanations attached.
2. photos of the experiments with the calculated and the experimental data.

thank you.

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#### Woody

Evaporation will certainly carry (heat) energy away from your tea
and would have a greater effect at higher temperatures,
which might explain the steeper initial experimental slope than seen in the calculated curve.

To investigate how significant evaporation is to your model,
You could try placing just a couple of drops of cooking oil on the surface,
just enough to form a very thin film,
which will reduce evaporation without otherwise influencing the results (much).

Or just place some cling film over the top of the cup.
One would not expect a thin piece of cling film to significantly insulate the cup
but the initial evaporation will rapidly increase the humidity in the space between the surface of the tea and the cling film,
and the increased humidity will then inhibit further evaporation.

Perhaps place your cup on a (sensitive) weighing machine
and see if you can measure the mass lost to evaporation.

have a look at:https://en.wikipedia.org/wiki/Evaporation

3 people

#### eli28

I will try it, thanks