### Re: Exceptions to Newton's Law of Cooling

Date: Tue Aug 9 20:57:22 2005
Posted By: David Coit, Aerospace Engineer, Naval Air Warfare Center - Weapons Division
Area of science: Physics
ID: 1121605675.Ph
Message:
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Lucia,

Newton's Law of Cooling applies to heat transfer by convection only, and
states that the heat flow rate is proportional to the difference in surface
temperature and free stream temperature (ie temperature of the fluid well
away from the surface). Note I said convection only, and there are two
other ways that heat is transferred in this universe: conduction and
radiation. Also, even some types of convection don't *technically* follow
Newton's Law of Cooling, as I'll try to explain later. First let's run
through the easy ones.

Conduction:
When heat is transferred through a single solid object or multiple solid
objects that are in direct contact with each other, it is called
conduction. Conduction is governed by Fourier's Law, which states that the
rate of heat transfer is proportional to the temperature gradient (dT/dx)
at any point in the solid. This sounds very similar to Newton's Law of
Cooling, and in fact it is. The major difference between conduction and
convection is that the constant of proportionality in Fourier's Law is
based only on one property of the solid, the thermal conductivity, while
the constant of proportionality in Newton's Law of Cooling is based on the
thermal conductivity of the fluid as well as other properties of the flow
field (like pressure, velocity, etc.). We almost always know (or can
measure) the thermal conductivity of a substance, but the proportionality
constant in Newton's Law of Cooling (or "convective heat transfer
coefficient") can be a big, ugly equation involving several variables, or
there might not be an analytical solution for it at all! This is a whole
subject in-and-of itself, and in fact I spent a whole semester in a
graduate-level class on convection deriving the equations for the
convective heat transfer coefficient for various cases.

All things that have a temperature above absolute zero emit thermal energy
in the form of radiation. They radiate in the portion of the
electro-magnetic spectrum that we call "infra-red" which we cannot see, and
the amount of energy they radiate is governed by the Stefan-Boltzmann Law.
This law states that the amount of energy radiated per unit surface area is
proportional to the absolute temperature to the fourth power. The constant
of proportionality in the Stefan-Boltzmann Law is called the
Stefan-Boltzmann Constant (what a coincidence!) and, as the name implies,
its value does not change. However, the Stefan-Boltzmann Law really applies
to so-called "blackbodies" which are imaginary, ideal objects that absorb
all radiation that hits them and reflects none. For real objects the amount
of energy radiated varies depending on the direction relative to the
surface as well as the emittance (or emissivity) of the surface. Note
though that the emitted radiation is proportional to the surface
temperature to the fourth power, and is not dependant on the temperature of
the surroundings. However some heat will also be absorbed from the
surroundings, and that will depend on the temperature of objects in the
line of sight of the surface.

Free Convection:
Thermodynamicists differentiate between "free" or "natural" convection
(where there is no driving pressure difference or gradient) and "forced"
convection (where there *is* a pressure difference that drives the flow),
and they do it because the convective heat transfer coefficients (which I
will simply call "h" in this paragraph) are different for each case. I said
above that the flow field properties help determine h, and this is just one
example of that. Without getting too deep in this very complicated subject,
let me just point out that if you solved the fluid dynamic equations and
figured out the heat transfer for free convection, you would find that it
is proportional to the temperature difference between the surface and the
free stream *to the 5/4 power* if the flow is laminar, and to the 4/3 power
if the flow is turbulent. So strictly speaking, Newton's Law of Cooling
does not apply to free convection. However, it is still useful to use the
form of Newton's Law of Cooling, and thermodynamicists do this by simply
dumping the extra deltaT^1/4 (for laminar flow) or deltaT^1/3 (for
turbulent flow) into h. Admittedly, this can lead to confusion since we
started by saying that Newton's Law of Cooling was valid for convective
heat transfer, then we go on to show that it's not actually valid for *all*
convective heat transfer. It might be easier to think of Newton's Law of
Cooling as just a way of defining h, ie h is defined as the heat transfer
divided by the difference in surface temperature and free stream
temperature for all convective flows. In this definition, h might have a
small dependance on the temperature difference, and that's just fine. The
bottom line is that thermodynamicists can learn a lot about the heat
transfer for a given situation if they know h for that case, and this is
precisely because we have a rigid definition of h (ie Newton's Law of
Cooling).

Reference:
Heat Transfer, 2nd Ed., by A.F. Mills

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