Newton's law of cooling ($y^{\prime }=-k(y-T_a)$)

Newton's law of cooling models an object's temperature $y$ using a positive constant rate $k$ and ambient temperature $T_a$. The displayed ordinary differential equation assumes one temperature represents the object, the environment remains at $T_a$, and the proportional rate stays fixed over the modeled range. It also describes warming when $y<T_a$. These are modeling assumptions: ^[ref] explain the proportional law through a tangent line approximation and caution about large temperature differences.

Dimensional analysis requires $k$ to have units of inverse time. The product $k(t-t_0)$ in the exponential function solution is therefore dimensionless. The equilibrium point $y=T_a$ attracts both warmer and colder starting states: the excess temperature undergoes exponential decay. See initial value problem for computed cooling times, direction field for the family of trajectories, and separable differential equation for a derivation.

For equal initial temperatures and the same Newton's law of cooling coefficient $a>0$, compare a unit Dirac delta function cooling impulse at time one with unit-rate cooling on the interval from one to two. The impulse produces a temperature jump of minus one. The finite-duration forcing gives a continuous temperature with change $-(1-e^{-a(t-1)})/a$ while it is active. Relative to the common unforced cooling curve, the impulse contributes $-e^{-a(t-1)}$. Consequently the temperatures cross at $t=1+\log (1+a)/a$, strictly between one and two. After time two the impulse-cooled body is warmer: the later cooling receives less subsequent exponential attenuation.

In an ideal linear temperature-dependent destruction model with equal warming and cooling durations, normalized temperature is $1-e^{-\alpha t}$ during warming and $(1-e^{-\alpha T})e^{-\alpha (t-T)}$ during cooling. Integrating the destruction rate gives $D(T)$, the logarithm of the surviving-count reduction factor. If the endpoint destruction rate and thermal relaxation constant stay fixed, changing the temperature gap alone leaves this integral unchanged. This is a property of the stipulated model.

Under Newton's law of cooling with fixed ambient temperature $T_0$ and $k>0$, an instantaneous jump $\beta$ at $t_{\ast }$ sets $T(t_{\ast }+)=T(t_{\ast }-)+\beta$. The subsequent temperature is $T_0+[T(t_{\ast }-)+\beta -T_0]e^{-k(t-t_{\ast })}$. Equivalently, for an initial temperature $T_i$, the full solution isThis piecewise model has the distributional derivative equation $T^{\prime }+k(T-T_0)=\beta \delta (t-t_{\ast })$, with the Dirac delta distribution representing the added temperature.

A heat transfer coefficient relates heat flux through a boundary to a temperature difference: $F=\lambda \Delta T$. Its dimensions are power per area per temperature. In a quasistatic ice layer, atmospheric boundary resistance $1/{\lambda }_A$ and conductive resistance $h/k$ add in series, giving effective conductance $(1/{\lambda }_A+h/k{)}^{-1}$.