Heat Transfer and Mathematics
Who Cares About Heat Transfer?
Heat transfer is part of our day-to-day lives. It is estimated
that as much as
86% of
all electricity generated uses steam heat to power turbines.
Our homes are made comfortable with proper insulation that keeps us
warm in the winter and cool in the summer. A meal you eat at a
restaurant or cafeteria might include vegetables that were most likely
cooked in a
steam-jacketed kettle. And when you reach for a piece of
paper, it is likely that the paper mill that produced it
used steam to power turbines or dry the wet mushy pulp into a
finished dry piece of paper. All of these processes involve heat
transfer. And heat transfer is predicted and modeled by
chemical engineers that must understand the physics and the
calculus-level mathematics involved.
Three Types of Heat Transfer - Conduction, Convection, and
Radiation
In conductive heat transfer, there is heat passing from
molecule to molecule without any motion of the molecules themselves.
In a home heating situation, this would be analogous to heat being
lost within a home to the outside with no wind blowing outside and no
fans or blowers moving around inside.
In convective heat transfer, there is movement of molecules
to actually move heated molecules and replace them with non-heated
molecules. In the home heating example, this would consist of a
wind striking the home exterior and removing heated molecules from
near its surface. Also, convection ovens use this type of heat
transfer to quickly cook foods by blowing hot air over the food.
In heat transfer by radiation, there is an emission of
electromagnetic waves which carry energy away from the emitting object1.
Although similar to conduction, radiation does not require a medium.
So one could derive heat via radiation from a heated object placed in
a vacuum. When you stand next to a wood burning stove, much of the
heat you obtain is via radiation since air itself is not a good
conductor of heat.
It is very common for all three types of heat transfer described
about to occur in a single situation.
Heat Transfer By Conduction Alone
In order to accurately predict the amount of heat energy delivered,
one must apply laws of heat transfer to the medium being heated as
well as the metal barrier separating the steam from the medium. This
is summarized with the equation shown below.
Note that you could apply the above equation to walls of a home if
there were neither wind outside nor fans blowing air around inside.
If the temperature difference between inside and outside your home is
doubled, the rate at which heat passes through your wall is doubled.
So for example, if it is 26.6oF outside (-3oC),
you will lose heat through your walls twice as fast if your thermostat
is set at 84.2oF (29oC) than if it set at 55.4oF
(13oC). This is because the temperature difference at
29oC is 29 - (-3) = 32oC vs. a difference of 13
- (-3) = 16oC when the temperature is set at 13oC.
Convection Heat Transfer
Heat transfer by convection is modeled by
Note that this equation is derived using a bit of calculus, some of
which is shown at
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/stefan2.html#c1
Convective Heat Transfer
As one might expect, due to the dynamic nature of the heat flow, the
prediction of heat transfer via convection can be fairly complex. The
equations depend on the nature of the flow and the region where the
flow occurs, among many other things. Needless to say, a lot of
mathematics is involved.
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