# Given two identical bottles, how can I get the ratio of times to get them to $D$ degrees given that one is inside 4C and the other is inside -2C

Given two identical bottles, starting both at $T_0$, how can I get the ratio of times to get them to $D$ degrees (for example 4.5C) given that one is in the refrigerator at temperature A (for example 4C) and the other in the freezer at temperature B (for example -2C).

That is I am looking for a function:

$$f(D) = parametric \ equation\ with\ parameters\ A,\ B,\ T_0\ and\ unknown\ D$$

That given on the X axis the different target temperatures, gives me the ratio between the time the slowest (refrigerator) bottle and the fastest (freezer) bottle take to get to D.

I am interested in the ratio of times, I think the target D would have a range of values that is bigger than A.

I think that the decrease in heat should follow a negative exponential growth because the instantenous rate of change depends on the current temperature by a constant factor (heat exchange rapidity factor, based on surface area and materials).

But which kind of function is a ratio of negative exponentials? How do all those parameters modify it? I guess the function as an asymptote for $D = A$ because the first bottle will never get to a temperature equal to that of the environment and the closer it will get the slower it will become in changing temperature, but I am not sure of that.

• Assuming D is greater than A and B, is D less than the new thermal environmental temperature? And extracting time from an exponential growth type of function is going to yield a logarithm. Jun 26 '17 at 12:19
• @BillN "is D less than the new thermal environmental temperature" I was interested in the simplest case where the bottles are hot and the target is between T_0 and the new ambient temperature. The target cannot be lower then A and B in this case. It would be interesting to analyze the case where the bottles start colder than the freezer and must get to a temperature higher then their starting point (I think the target D must be "less than the new thermal environmental temperature" in this case otherwise it cannot be reached) Jun 26 '17 at 12:28

Let's make some simplifying assumptions about the thermal evolution:

• heat transfer coefficient is the same for both cases
• perfect mixing of substance in bottle so there is no "insulating" effect of more rapid cooling (eg if ice formed on the inside wall of the bottle without all the liquid reaching 0 °C, it could act as an insulating blanket)
• Heat transfer proportional to the temperature difference

Call the target temperature $T_1$ and the temperature of the fridge $T_A$ and freezer $T_B$. Then the equation for temperature with time is:

$$T(t) = T_A + (T_0-T_A)e^{-t/\tau}$$

Where $\tau$ is the time constant for the cooling curve, which depends only on factors that are the same for both bottles.

Rearranging, we find the time needed to reach $T_1$ by setting $T(t)=T_1$. Then you get

$$\frac{T_1-T_A}{T_0-T_A}=e^{-t/\tau}\\ t = \tau\left(\log{\frac{T_0-T_A}{T_1-T_A}}\right)$$

As you can see, when $T_1$ gets close to $T_A$, the time grows.

You can now do the same thing for $T_B$ and take the ratio. Interestingly, that result does not contain $\tau$.