Newton's law of cooling revisited - Cartan

1074 M Vollmer
cooling time, i.e. transition temperature difference, where the dominant cooling changes from
radiation to convection. For 10 W m −2 K this happens at �T = 137 K (after ≈840 s in a T(t)
plot)) and for 30 W m −2 K it happens at �T = 415 K (after 112 s). Qualitatively it makes
sense that the higher this transition temperature difference, the larger the range of validity
of Newton’s law of cooling. For the convective heat transfer of 100 W m −2 K and ε = 0.1
(see figure 5), convection would dominate radiation right from the beginning, which easily
explains the linear plot.
From the theoretical analysis, it is clear that radiative cooling should lead to deviations
from Newton’s law of cooling above critical temperature differences, which in some of the
discussed cases were below 100 K. This raises the question of why many experiments reported
the applicability of Newton’s law for a temperature difference range of up to 100 K. The answer
which is proposed here is simple: if one waits long enough, any cooling process can probably
be described by a simple exponential function.
To my knowledge, this statement has not been proven theoretically for the cooling
of objects involving nonradiative heat transfer. The idea behind it may be motivated for
convective cooling of simple shaped objects irrespective of the size or Biot number by the
following argument. The cooling of objects such as spheres, cylinders, or plates of any size
(not just small objects), which start at a given initial temperature and which are in contact
with a fluid (radiative heat transfer is neglected) can be described by a series expansion of
exponential functions [2]. It is found that for sufficiently long times, a single term in the series,
i.e. a single exponential function, describes the temperature distribution within the objects. If a
suitable average temperature is defined, the single exponential function will therefore describe
the cooling process.
With this result from pure convective cooling in mind, theoretical cooling curves involving
nonlinear radiative heat transfer were analysed by using series expansions of exponential
functions as fit functions.
Figure 7 (left) depicts the theoretical temperature plot for αConv = 10 W m −2 K, ε = 0.9
and 40 mm Al cubes of figure 5. These theoretical data were fitted (figure 7 (right)) using a
third-order exponential fit of the form
�T (t) = T0 + A1 · e −t/τ1 + A2 · e −t/tτ2 + A3 · e −t/tτ3 . (11)
The agreement of such a simple fit is extremely good as can be seen from figure 8, which depicts
the difference between the theoretical plot and the third-order exponential fit. Disregarding
the first 10 s, deviations are below 1 K.
The fit includes three functions with different amplitudes and time constants (the constant
term of 0.07 is practically zero and unimportant for the discussion). The first one has the
smallest amplitude (≈162) and a time constant of only 78.7 s. The second contribution has
an appreciable amplitude (≈253) but also a relatively small time constant of 297 s. The
third contribution has an amplitude slightly larger than the second one (281) but decays with
a much longer time constant of about 1010 s. Due to exponential decay an amplitude is
suppressed to less than 1% after five time constants. Hence, the first contribution already
contributes less than about 1 K after 400 s of cooling. Similarly, the second contribution
becomes less important with time. For times longer than 1500 s it only contributes less
than 1.7 K. This happens at a temperature difference of about 65 K, which means that for
longer times, i.e. smaller temperature differences, a single exponential provides a very good
approximation to the cooling curve. Hence, if we assume that it is a general property of the
combined convective and radiative cooling that the resulting T(t) curve can be approximated by
a superposition of exponential functions with different time constants, one does indeed expect
that after the exponentials with small time constants have died away, a simple exponential