Please help- heat conduction and Newton's law of cooling

Question

Criterion: The heat transfer coefficient is the same for the freezer and the refrigerator.

Two bowls of the same size are filled with room-temperature coffee and then placed one in the freezer and the other in the less colder refrigerator. After a minute, the bowls are changed, left in their new places for another minute, and finally, both are taken out. Which bowl cools down better during the experiment?

Thank you for any help.

Answer 1

The rate of cooling is in proportion to the difference in T between the bowls and the environment. This means cooling is faster in the freezer.

Sorry, misread the problem

Newton's Law of Cooling: dT/dT = -U(T-Tf)

General solution for 0 to 1 min.

T=(Tb - TF)e^(-Ut) + TF. (F= freezer)

For 1 minute

T(1) =(Tb - TF)e^(-U) + TF

Substitute f to indicate refrigerator.

For second minute we can build the solution for 1 minute from new initial temperatures

Let's look at the one that went from Freezer to fridge

At 2 min setting t=1 as 0

((Tb-TF)e^(-U)-TF-Tf)e^(-U) + Tf and the same with F and f switched for the other bowl

If you equate the temperatures, I get that they are equal if e^(-2U) =1. If >1 the freezer first dominates as it is clear when U is high enough that T after 1 min can be less than T fridge and T warms up in the second minute but has to end less than Tf.

Interesting problem. Not sure I got through it unscathed.