Frequently Asked Questions
What is Newton's law of cooling?
It is an empirical rule stating that the cooling rate of an object is proportional to the temperature difference between the object and its surroundings. It is expressed by the differential equation $dT/dt = -k(T - T_{env})$, with the solution being the exponential function $T(t) = T_{env} + (T_0 - T_{env})e^{-kt}$. This empirical law was observed by Newton in the 17th century, and in modern heat transfer theory, it corresponds to convective heat transfer $Q = hA(T - T_{env})$.
How is the cooling constant k determined?
It is expressed as $k = hA/(mc_p)$. Here, h is the heat transfer coefficient (W/m²K), A is the surface area (m²), m is the mass (kg), and $c_p$ is the specific heat (J/kgK). A container with insulation has a small h and thus a small k. A thin metal cup has a large h and also a small mass m, resulting in a large k. To determine k experimentally, plot temperature vs. time data on a logarithmic scale and find the slope.
Are there conditions where Newton's law of cooling does not hold?
When the Biot number (Bi = hL/λ) exceeds 0.1, the temperature gradient inside the object cannot be ignored, and the lumped capacitance assumption breaks down. Also, if the temperature difference is very large, radiative heat transfer becomes dominant (proportional to $T^4$), making the linear approximation invalid. In such cases, it is necessary to use transient heat conduction analysis with partial differential equations or combine the Stefan-Boltzmann radiation law.
How is 'time of death estimation' done in forensic medicine?
The Henssge and Marshall formula is used. By measuring the rectal temperature T of a corpse, the elapsed time is back-calculated from $T_{body} = T_{env} + (37 - T_{env})e^{-kt}$. The cooling constant k of the human body varies with body weight, clothing, room temperature, and ventilation conditions, but for a standard weight, k is approximately 0.05–0.1/h. However, this is only a guideline, and actual forensic medicine combines multiple indicators.
Why can an insulated tumbler keep drinks hot for hours?
A vacuum-insulated tumbler reduces heat loss due to convection and conduction to nearly zero by creating a vacuum between the inner and outer walls. The mirror finish on the inner wall also reflects radiative heat. As a result, k becomes extremely small (less than 1/10 that of a regular cup), and τ becomes several hours or more. The principle of vacuum insulation is the same as that of the thermos (invented by Dewar in 1892), and in CAE, it is also applied to the thermal design of spacecraft.