Application: Cooling and heating

Ordinary differential equations: Separable differential equations

Application: Cooling and heating

We look at an application of solving an ODE by separation of variables and studying the asymptotic behaviour of solutions

Cooling or warming In the context of Newton's model for the temperature (\(T\)) profile of an object in an environment with constant temperature \(T_\text{amb}\) we have \[\frac{\dd T}{\dd t}=-c\cdot \left(T-T_\text{amb}\right)\] where \(c\) is a nonzero constant, and the asymptotic behaviour of the solution is not so strange. Of course, you expect that the object reaches a temperature equal to the ambient temperature in the course of time. If the starting temperature \(T(0)=T_0\) is less than the ambient temperature \(T_\text{amb}\), then the temperature \(T\) raises slowly to \(T_\text{amb}\) according to the formula \[T(t)=T_\text{amb}+\left(T_0-T_\text{amb}\right) e^{-c\, t}\] The same formula means that when the start temperature \(T_0\) is greater than the ambient temperature \(T_\text{amb}\), the temperature of the object decreases exponentially and approaches the ambient temperature.

The formula for the solution of the differential equation can be derived by separating variables. First we rewrite the differential equation in differential form with separate variables \[\frac{\dd T}{T-T_\text{amb}}=-c\,\dd t\] Then we integrate both sides: \[\int_{T_0}^{T(t)}\frac{\dd T}{T-T_\text{amb}}=\int_0^t-c\,\dd t\] We compute the integrals: \[\Bigl[\ln|T(t)-T_\text{amb}|\Bigr]_{T_0}^{T(t)}=\Bigl[-c\cdot t\Bigr]_0^t\] So \[\ln|T(t)-T_\text{amb}|-\ln|T_0-T_\text{amb}|=-c\cdot t\] Thus: \[\ln\left|\frac{T(t)-T_\text{amb}}{T_0-T_\text{amb}}\right|=-c\cdot t\] We rewrite: \[\frac{T(t)-T_\text{amb}}{T_0-T_\text{amb}}=e^{-c\cdot t}\] So: \[T(t)-T_\text{amb}=\left(T_0-T_\text{amb}\right)\cdot e^{-c\cdot t}\] Finally: \[T(t)=T_\text{amb}+\left(T_0-T_\text{amb}\right)\cdot e^{-c\cdot t}\]