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:
- $\frac{\text{d}Q}{\text{d}t}$ 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).
- $\Delta T$ is the temperature difference between the 'hot' object (the cup) and the 'cold' object (the surrounding air).
- $A$: 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.
- $h$ 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.
- $\Delta T$ can be maximized by minimizing the temperature of the air. Experiment by cooling the cup in the refrigerator v. normal ambient air.
- 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.
- 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.
