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# Heat Transfer and Mathematics

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.