Modification of Newton's Law of Cooling

Question

Yesterday I randomly started thinking about Newton's Law of Cooling. The problem I realized is that it assumes the ambient temperature stays constant over time, which is obviously not true. So what I tried doing was to modify the differential equation into a system of differential equations, and taking the heat capacity of each into account. $$ \frac{dT_1(t)}{dt} = C_2(T_2(t)-T_1(t))\\ \frac{dT_2(t)}{dt} = C_1(T_1(t)-T_2(t))\\ T_1(0) = T_{10}\\ T_2(0) = T_{20} $$ which had the following solution: $$ T_1(t)={\frac {C_{{2}}T_{{20}}+C_{{1}}T_{{10}}}{C_{{1}}+C_{{2}}}}+{\frac {C_{ {2}} \left( -T_{{20}}+T_{{10}} \right) {{\rm e}^{- \left( C_{{1}}+C_{{ 2}} \right) t}}}{C_{{1}}+C_{{2}}}}\\ T_2(t)={\frac {C_{{2}}T_{{20}} +C_{{1}}T_{{10}}}{C_{{1}}+C_{{2}}}}-{\frac {C_{{1}} \left( -T_{{20}}+T_{{10}} \right) {{\rm e}^{ -\left(C_{{1}} +C_{{2}} \right) t}}}{C_{{1}}+C_{{2}}}} $$

When I plotted the two equations out they seem to be right. They also follow the fact that at any point in time $C_1\Delta T_1=C_2\Delta T_2$. The end behaviour also seem to be correct. However, in cases where $C_1\rightarrow\infty$ you would expect it to behave similarly to Newton's Law of Cooling, but in reality $T_2$ drops its temperature in a very short period of time, which doesn't seem to be right. I tried looking this up but couldn't find much on this topic. If anyone can point out what I did wrong that would be great. I've had very little experience with thermodynamics but I do realize how limited these models are at describing real world scenarios. This was more for the fun of playing with differential equations.

Answer 1

The flaw in your reasoning seems to be that $C$ is not in fact heat capacity. In Newton's Law of Cooling, the proportionality constant would be related inversely to the heat capacity of the two heated liquids/gasses/materials, and directly to the heat conductance of the object separating the two materials. A material with a higher heat capacity would have a smaller temperature change for a given temperature difference, and a thin piece of metal separating the materials would result in a much larger $\frac{dT}{dt}$ than a thick piece of styrofoam would.

When bringing $C_1$ to $\infty$, you are actually decreasing the heat capacity and increasing the conductance, both of which would cause $T_2$ to drop quickly as you observed.