How to calculate the time it will take for a specific object to reach a certain temperature? I work at a bakery. We have certain products that have to be put in the refrigerator for x number of hours before we can use them. The cakes we use for cakeballs have to refrigerate for 12 hours, apparently... But no one has calculated this to know whether or not that's true. They're just guessing. I don't think it will take 12 hours for our sheet pans to reach 49 degrees Fahrenheit(the temp of the fridge), but I have been out of school for a while and am pretty rusty on my formulas. I was wondering if someone could help me with the formula. I honestly would like to do the math myself, so a simple formula is all I ask for. I hope someone can help me solve this!
 A: Say we have an object initially ($t=0$) at $T_0$, surrounded by a large quantity of air at a constant temperature $T_{\infty}$.
The temperature evolution $T(t)$ of the object can be found approximately by means of lumped thermal analysis. This method assumes the temperature of the cooling object is uniform (no internal temperature gradients).
In that case we can use Newton's cooling law to calculate the heat loss rate leaving the object as:
$$\frac{\mathrm{d}Q}{\mathrm{d}t}=hA[T(t)-T_{\infty}],\tag{1}$$
where $h$ is the heat transfer coefficient and $A$ the surface area shared by the object and the surrounding air.
If the object cools down by an infinitesimal amount $\mathrm{d}T(t)$, then it loses an amount of heat $\mathrm{d}Q$:
$$\mathrm{d}Q=-mc_p\mathrm{d}T(t),\tag{2}$$
where $m$ is the mass of the object and $c_p$ its specific heat capacity (the minus sign is needed because $\mathrm{d}T(t)<0$).
Inserting $(2)$ in $(1)$ and shortening $T(t)$ to $T$ we get the differential equation (DE):
$$-mc_p\frac{\mathrm{d}T}{\mathrm{d}t}=hA[T-T_{\infty}]\tag{3}$$
This is a first order, linear DE which solves with separation of variables. So integrating between $(T_0,T)$ and $(0,t)$ we get:
$$\ln\frac{T-T_{\infty}}{T_0-T_{\infty}}=-\alpha t\tag{4},$$
where:
$$\alpha=\frac{hA}{mc_p}$$
From $(4)$ we can then extricate $T$:
$$\boxed{T=T_{\infty}+(T_0-T_{\infty})e^{-\alpha t}}\tag{5}$$
Note that the object will only reach $T_{\infty}$ for $t \to +\infty$.

There are two main difficulties with this calculation:

1. Material constants:
The required material constants that populate $\alpha$ may neither be easy to obtain or measure.
2. Internal temperature gradients:
For larger objects the assumption of thermal uniformity may no longer be valid, as the outer layer of the object will reach higher temperatures earlier. In that case we need to take internal transient heat conduction into account, which is mathematically much more demanding.
For those reasons, constructing experimentally determined cooling curves may be far more practical and accurate.
