# Formulas related to “Cooling a cup of coffee with help of a spoon” [closed]

I was recently going through some of the top voted questions on the thermodynamics tab. And, I came across Cooling a cup of coffee with help of a spoon. I found this question really interesting. In fact, I plan on carrying out my own experiment related to it.

However, to write my experiments and results in a formal research paper for my school, I must have an investigation involving a few independent and one dependent variables. Therefore, I cannot work with comparing methods of cooling coffee, as they are not quantifiable.

My dependent variable would be time, since I want to find the fastest method of cooling coffee.

What are the dependent variables I can use?

It would be great if you guys could provide some relevant thermodynamics formula related to cooling rate which can be applied to this scenario of cooling a cup. I have thought about optimal stirring speed or area of stirrer, but I am yet to find a mathematical relationship between them and the time taken for cooling.

Note: Although this question is tagged as homework. It is not. It is just a request for help in an experiment I may carry out in school.

Note: I am not only restricting methods of cooling to methods using spoons. The method used for cooling a cup of coffee does not need to include a spoon.

• This is more appropriate for the physics chat session, since it does not contain a specific question, but aim for discussing the setup instead. – rmhleo Sep 5 '19 at 13:28
• @rmhleo IF you don't mind, can you please tell me how to join a chatroom? – NoLand'sMan Sep 5 '19 at 16:20
• @NoLad'sMan on the top right corner of the website click on the Stack Exchange icon, in the drop window there are the different communities you belong to. Above the Physics community (current community) there is a chat text which is the link to the chat. I am not completely sure if this looks the same way to you, this is how I find it from my account. – rmhleo Sep 12 '19 at 23:22

It would be great if you guys could provide some relevant thermodynamics formula related to cooling rate which can be applied to this scenario of cooling a cup.

The 'go to' Law for this kind of cooling is Newton's Cooling Law:

$$\boxed{\frac{\text{d}Q}{\text{d}t}=-hA\Delta T}$$

where:

1. $$\frac{\text{d}Q}{\text{d}t}$$ is the heat energy loss per unit of time (aka the heat flux) of the cup. Obviously the higher the heat flux, the faster the cup will cool down (faster temperature loss).
2. $$\Delta T$$ is the temperature difference between the 'hot' object (the cup) and the 'cold' object (the surrounding air).
3. $$A$$: the surface area shared between the 'hot' object (the cup) and the 'cold' object (the surrounding air). The heat flows from hot to cold through that surface.
4. $$h$$ the heat transfer coefficient.

Having understood that in order to maximize cooling we need to maximize the product of the three factors on the RHS of the equation we can now suggest some factors to investigate.

1. $$\Delta T$$ can be maximized by minimizing the temperature of the air. Experiment by cooling the cup in the refrigerator v. normal ambient air.
2. It's is well know that the heat transfer coefficient $$h$$ can be increased by introducing turbulence, both inside the cup and outside of it. Experiment with controlled stirring, perhaps at different speeds. Experiment with air fans blowing on the cup.
3. Greater surface area promotes heat flux and thus cooling rate. Note that a sphere has the lowest surface area to volume ratio of all regular shapes. Elongated cylindrical shapes will have higher surface area to volume ratios. Consider experimenting with cooling fins to boost $$A$$.

With these suggestions carry out some screening experiments to identify the most important factors affecting cooling rate. Finally, by combining those, the fastest cooling rate scenario can be determined.

This kind of investigation is very well suited for the application of Factorial Experimental Designs (FED).

In this approach factors suspected to influence one or more measuring responses ('cooling rate' in our case) are identified. To each factor is assigned two values (symbolically represented by $$-$$ and $$+$$).

We've identified 4 factors and can assign two values to each of them. This allows the running of a so-called $$2^{4-1}$$ FED, which requires a mere $$8$$ experiment runs.

The FED matrix looks like this ($$A$$, $$B$$, $$C$$ etc represent the factors):

Once the $$8$$ runs have been performed, the effects of each factor, as well as first order interactions (e.g. $$AC$$), are easily calculated.

The advantage of this kind of experiment design is that it yields a high degree of information compared to one-variable-at-a-time designs.

• Awesome, thanks! – NoLand'sMan Aug 31 '19 at 2:55