Newton's Law of Cooling Flawed?

[MHrtz]

Newton's Law of Cooling (not the formal definition):

(change in time) = -ln ((Tf - S)/(Ti - S)) / k

Tf = final temperature
Ti = initial temperature
S = temperature of environment
k = heat transfer coefficient

Say that you wanted to cool something (such as a person) to a negative temperature (Tf would be negative) and the temperature of the environment was positive. This would mean that you would have to take the -ln (-#). Obviously, you can't do this. In another situation, say the surrounding temperature was the same as the final temperature. This would mean that you would have to take the -ln (0) which can't be done. How can I apply this formula to these situations?

 

[Mapes]

Hi MHrtz, welcome to PF. Perhaps that's an indication that it's not possible to cool something by warming it?

For the other part: conduction can only bring an object's temperature asymptotically close to the surrounding environment's. In other words, you can get arbitrarily close to the ambient temperature, but in theory it would take infinite time ($-\ln 0$) to reach it. Does this help answer your question?

 

[willem2]

what you actually compute here is the time that an object takes to cool from $T_i$ to
$T_f$. The fact that you get no answer in the first case is to be expected because the object wil never cool to a temperature below the environment.
You get no answer in the second case, because you try to compute the time that your object reaches $T_f$, but your object is at $T_f$ all the time.

 

[MHrtz]

Ok, I see what you mean. I guess I was too focused on the formula itself rather than what it implied. Thank you for the help.

 

[PJC01]

I would like to clarify that Newton's Law of Cooling is not flawed, but rather it has limitations in its application. The law is based on the assumption that the temperature difference between the object and its surroundings is small, and that the rate of cooling is proportional to this temperature difference. In situations where these assumptions are not met, the law may not accurately predict the cooling rate.

In the case of trying to cool something to a negative temperature, it is important to note that this is not physically possible. The law is not intended to be applied to such extreme scenarios and therefore, trying to use the formula in this way would result in an invalid calculation. Similarly, when the surrounding temperature is the same as the final temperature, the temperature difference becomes zero and the logarithmic function in the formula becomes undefined. This shows that the law is not suitable for predicting cooling in this specific scenario.

In order to apply the formula correctly, it is important to ensure that the assumptions of small temperature difference and proportionality hold true. In cases where these assumptions are not met, alternative equations or methods may need to be used to accurately predict the cooling rate. As a scientist, it is crucial to consider the limitations and assumptions of any scientific law or formula before applying it to a specific situation.