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.