Tuesday, April 30, 2019

homework and exercises - What does 1/k represent regarding Newtons Law of Cooling?



What does 1/k represent regarding Newtons Law of Cooling?
I know k represents the cooling constant. I think the inverse of k is the time taken for the liquid to cool from its maximum temperture to surrounding temperature. Any clarification would be most appreciated.



Answer



In Newton's law of cooling, the constant $k$ appears in most solutions schematically as,$^\dagger$


$$T(t)\sim e^{-kt}$$


Clearly, the argument of the exponential must be dimensionless, hence the constant $k$ has dimensions,


$$[k]=\frac{1}{[\mathrm{time}]}$$


The constant $k$ is a measure of the rate of cooling, with the same dimensions as a frequency.




$\dagger$ We have assumed the simplest formulation of Newton's law of cooling, wherein the system of differential equations are linear, with constant coefficients.



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