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Heat Transfer at Macro- and Nano- Scales
January 21, 2016
I was just nine years old when
Earth's first
artificial satellite,
Sputnik 1, was launched. Sputnik marked the true start of
space exploration, and it was the precursor to the many
space observatories now in
Earth orbit. While
astronomical observations at infrared wavelengths are somewhat possible atop high
mountains and in
dry climates,
infrared observations of faint sources are only possible using space observatories outside the
atmosphere.
Infrared is one way to measure the
temperature of a
planet, but
Pluto is at such a great distance from the Earth that it wasn't until 1987 that its temperature was measured. A measurement using the
Infrared Astronomical Satellite (IRAS) showed an average temperature of about 45
kelvin (K),[1] which is half the temperature of
liquid argon.
Pluto, photographed on July 14, 2015, by NASA's New Horizons spacecraft. This is a composite of images taken by the Ralph/Multispectral Visual Imaging Camera.
(NASA / Johns Hopkins University Applied Physics Laboratory / Southwest Research Institute image, via Wikimedia Commons.)
As a
child I had a few
books on
planetary astronomy, and they listed the temperatures of the planets, including that of Pluto, before Sputnik's launch, and long before any temperature measurement of that planet (Yes, in those days,
Pluto was a planet). Where did the value for Pluto's temperature come from?
Knowledge of a few
physical laws will often lead to some rather accurate
estimates of unknown quantities. As I wrote in an
earlier article (Estimation, December 21, 2011),
Enrico Fermi was a master of such
back-of-the-envelope calculations. There's a subset of such problems called "
Fermi problems," which are unusual estimates of quantities based on whatever information is at hand. One example of a Fermi problem that I've done is estimating how many people live in
our county based on the number of
supermarkets.
Since all the bodies in our
Solar System are heated by the
Sun, we estimate Pluto's temperature by considering the
heat transfered to it from the Sun. To do this, we use the
Stefan–Boltzmann law relating the
energy of an ideal radiator, called a
black body, with its temperature. We assume that all the energy at Pluto comes from the Sun, and we also assume that Pluto is in
thermal equilibrium; that is, it radiates as much energy as it receives.
We could
calculate the solar
radiance received at Pluto, and start plugging numbers into the Stefan–Boltzmann equation, as follows,
where
j is the black-body emissive power (equal to the solar radiance in our model),
T is the
absolute temperature, and
σ, called the
Stefan-Boltzmann constant, is expressed as a combination of some
fundamental constants,
where
k is the
Boltzmann constant,
h is
Planck's constant, and
c is the
speed of light. The best known value for this constant is 5.67037 x 10
−8 watt meter−2 kelvin
−4.[2]
Instead of concerning ourselves with the absolute radiance at Pluto, it's easier to
ratio its values with those of Earth, whose temperature we know quite well. The Sun's radiance diminishes with distance according to the usual
inverse-square, 1/r2, law. We can get the ratio of the radiance at Pluto to that at Earth by taking the ratio of the squares of their distances from the Sun (see figure).
An astronomical unit (au), the average distance from the Earth to the Sun, is a convenient unit of distance in the Solar System.
The Earth is about 93 million miles (150 million kilometers) from the Sun.
(Created with Inkscape.)
Pluto has a highly
elliptical orbit, with an
aphelion of 49.32
astronomical units (au) and a
perihelion of 29.66 au, and it's presently about 33 au from the Sun. When we look at the r
2 ratio of the Earth and Pluto distances, we see that the level of the Sun's radiance at Pluto is just 0.092% of that at the Earth. That's the reason why the
New Horizons spacecraft was
powered by a
radioisotope thermoelectric generator and not
solar panels. I wrote about radioisotope thermoelectric generators in an
earlier article (Radioactive Heat, July 26, 2011).
The fourth-root of the ratio of radiance gives us the ratio of the temperatures of Earth and Pluto. Assuming 300 K for Earth's temperature gives us about 52 K for Pluto's temperature. This is quite close to the 45 K value found in the IRAS measurement mentioned above. The agreement is surprising, since Earth and Pluto are not strictly black bodies, and this reaffirms our faith in
educated guessing.
We need to be careful when we extend this method into the
nanoscale, since the distance between objects becomes comparable to the
wavelength of the radiation. A team of
scientists,
mathematicians, and
engineers from the
University of Michigan (Ann Arbor, Michigan), the
Universidad Autónoma de Madrid (Madrid, Spain), the
Massachusetts Institute of Technology (Cambridge, Massachusetts), and the
Donostia International Physics Center (DIPC, San Sebastiá, Spain) has examined the heat transfer between
nanoparticles, and they've discovered an interesting effect. The heat transfer happens 10,000 times faster than at the
macroscopic scale.[3-4]
Ultra high vacuum scanning thermal microscope used to measure heat transfer at the nanoscale.
(University Michigan image by Joseph Xu.)
While the wavelength of
visible light is about 500 nm, the scale investigated in this study was 10 nm, comparable to the
diameter of
DNA.[3-4] The enhancement of heat transfer at the nanoscale has been known for
decades, but
experiments have been difficult to do, and the exact mechanism was not understood.[4] In the
mid-20th century,
Russian physicist,
Sergei Rytov, proposed a
theory called "fluctuational
electrodynamics" to describe heat transfer at small distances, but his theory had never been tested at the nanoscale.[4]
For this study, samples at 305
°F were probed with the tip of a customized
scanning thermal microscope coated with the same
materials and held at 98 °F lower temperature. The tip was brought from 50 nanometers distance to touching while the temperature of the tip was measured.[3-4] Measurements were made at gaps as small as two nanometers.[3] The materials tested were
silica,
silicon nitride and
gold, and all these showed gap-size-dependent enhancements of the radiative heat transfer.[3]
The cause of the rapid heat transfer appears to be an overlap of the
surface and
evanescent waves of each member of the couple. Both of these types of waves carry heat.[4] Says
Bai Song, a
graduate student in
mechanical engineering at the University of Michigan and an
author of the study,
"These waves reach only a small distance into the gap between materials... and their intensity at the extreme near-field is enormous compared to the electromagnetic waves at larger distances. When these waves from two different devices overlap, that's when they allow tremendous heat flux."[4]
The
data obtained were consistent with fluctuational electrodynamics.
Pramod Reddy, a
professor of mechanical engineering at the University of Michigan and
co-principal investigator of this study, remarks
"These results disprove current dogma in nanoscale heat transfer, which holds that radiative heat transfer in single digit nanometer-sized gaps cannot be explained by existing theory."[4]
There are potential applications for such enhanced nanoscale radiative heat transfer. Since the heating process is rapid, it might be used for heat-assisted
magnetic recording.[4] The process might be utilized to make more efficient
thermoelectric devices, which convert a temperature
gradient to
electricity.[4] This
research was funded by the
U.S. Department of Energy, the
Army Research Office, the
National Science Foundation, the
Spanish Ministry of Economy and Competitiveness, and other organizations.[4]
References:
- H.H. Aumann and R.G. Walker, "IRAS observations of the Pluto-Charon system," Astronomical Journal, vol. 94, no. 4 (October, 1987), pp. 1088-1091.
- Stefan-Boltzmann constant on the NIST CODATA web site.
- Kyeongtae Kim, Bai Song, Víctor Fernández-Hurtado, Woochul Lee, Wonho Jeong, Longji Cui, Dakotah Thompson, Johannes Feist, M. T. Homer Reid, Francisco J. García-Vidal, Juan Carlos Cuevas, Edgar Meyhofer, and Pramod Reddy, "Radiative heat transfer in the extreme near field," Nature, Advance Online Publication (December 7, 2015), doi:10.1038/nature16070.
- Heat radiates 10,000 times faster at the nanoscale, University of Michigan Press Release, December 10, 2015.
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