Aerodynamics:

Conservation of Energy

The law of conservation of energy is one of the most fundamental in all of physics.

It is also very important to we aviators. The most common theoretical explanation of why airplanes fly is embodied in an equation called Bernoulli's equation, which is just a special case of the "law of conservation of energy."

The simplest form of the law of conservation of energy is:

PE + KE = constant

In this equation PE stands for Potential energy and KE stands for Kinetic Energy.

Potential Energy

There are many types of potential energy. For example a battery contains potential electric energy. But for us the most important type is potential energy relative to the earths gravitational field. As such an object that is at a higher altitude has more potential energy than one at a lower altitude.

All types of energy are in units of force x distance. In the case of potential energy the force in question is the weight of the object and the distance is the altitude of the object. Usually we use the surface of the earth as the "zero point" for potential energy rather than the center of the earth, but you can actually use any altitude as long as you remember that you are measuring potential energy relative to that reference altitude. Objects above the reference altitude have positive PE and those below the altitude have negative PE. The units of PE are:

PE units = lb. x ft.

Weight is a force and like all forces it is calculated using Newton's second law (F = ma.) The acceleration is due to gravity, and is always represented by the small letter "g." Thus:

Weight = mg [m is the mass, and g is acceleration due to gravity.] The value of g is about 32 ft/sec².

The unit of mass is called the slug. A slug is defined so that: mass(in slugs) = weight(in pounds) / g. The units of slugs = lb / ft / sec²

So potential energy is calculated by the simple equation:

PE = mgh [h is the height in feet, m is mass in slugs and g is acceleration due to gravity in ft/sec². ]

From the above equation you can see that the units of PE can be "expanded" as the units of mass times acceleration times height. I.E. slugs x ft/sec² x ft. This is usually simplified by combining the ft units. It is important to realize that the following units are equivalent. One simply uses the more common pound unit, while the other uses the less common slug unit:

So the units of PE are: lb x ft or slugs x (ft/sec)²

Kinetic Energy

Kinetic Energy is the energy an object has by virtue of its motion. If another automobile drives into your automobile you will see kinetic energy converted into bent steal, noise, and sound.

Kinetic energy is defined as:

KE = mV²/2

Notice that the units of KE are exactly the same as the units of PE, i.e. lb x ft or slugs x (ft/sec)²

It is important to realize that all energy, even electric energy is measured in these same units. (Those somewhat familiar with these units will notice that energy has the same units as torque and work.)

Energy Conservation

You know the old saying, what goes up must come down. It refers of course to the observation that if you throw something upwards, no matter how hard you throw it it eventually comes back down.

Energy conservation is more properly called the law of energy transfer. In physics energy transfer is called work. In other words in physics we say that if you change the energy state of an object you have performed work on it. I.E. if you change its altitude you are doing work, or if you change its velocity you are doing work. Unless you do one of these two things you are NOT doing any work.

Bernoulli's Equation

Daniel Bernoulli was a contemporary of Isaac Newton, but they never met. Bernoulli independently discovered the law of conservation of energy, but limited its application to the specific case of fluids flowing in pipes. In part that explains why Newton is more famous.

To understand Bernoulli's equation we just need to realize that the potential energy of a volume of fluid is PE/volume and the kinetic energy of the same volume of fluid is just KE/volume. Thus bernoulli's equation is:

PE/volume + KE/volume = constant/volume

Just as PE of an object depends on its height, the PE of a volume of fluid depends on its height within a plumbing system. In other words the fluid in an interconnected piping system is under more pressure in the basement and less pressure in the attic. The KE energy of a volume of fluid depends on the velocity of the fluid, just like the KE energy of an object. A real world example is using a pressure washer, like the one at the local car wash, to clean something. The fluid (water) leaves the nozzle at high speed and can be used to "blast" things clean. The much slower stream of water leaving a conventional garden hose is much less effective for cleaning because it has less energy, and therefore can do less work.

It is very important to understand the units of Bernoulli's equation. Remember that the unit of energy is lbxft, so the unit of energy per volume is lbxft / ft³ (remember that volume is measured in units of ft³.) One of the ft cancels so the units of energy per volume of a fluid are:

energy per volume units are lb/ft²

You should recognize these units as everyday pressure. You are probably more used to measuring pressure in units of lb/in² (psi) but lb/ft² is just psi/144 (there are 144 square inches in a square foot.)

Density

In fluids or gases density () is simply defined as mass/volume (units of slugs/ft³) Density is represented by the greek letter rho (.)

Therefore mass of a volume is easily calculated:

mass = x volume

KE of a Volume

As already explained above the units of energy per volume is pressure. But, it is more informative when considering the KE of a volume to remember that KE is calculated by the formula:

KE = mV²/2

Substituting the formula for mass of a volume then the KE/Volume is just:

KE/Volume = x volume x V²/(2 x volume) [obviously the volume cancels - see below]

This is called the dynamic pressure i.e. the pressure due to motion. Dynamic pressure is represented by the letter q. You can see that the volume units cancel each other and we have the following equation for dynamic pressure:

q = V²/2

Bernoulli's Equation in Aviation

In the nineteenth century it was discovered that Bernoulli's equation doesn't just apply to liquids. It applies equally well to gasses. Unlike liquids the density of gasses change quite easily, but Bernoulli's equation takes that into account accurately. The most useful form of Bernoulli's equation for us in our study of aerodynamics is:

Ps + q = constant

In this equation Ps stands for static pressure. This is the sort of everyday pressure we are used to measuring with instruments such as tire pressure gauges, or the suction gauge on the instrument panel, or the manifold pressure gauge etc (although these usually aren't in units of lb/ft², so we would have to apply a conversion factor.) q represents dynamic pressure as defined above. So, the long version of Bernoulli's equation is:

Ps + V²/2 = constant

Just like the conservation of energy equation we started with above (PE + KE = constant) Bernoulli's equation tells us that static pressure and dynamic pressure can swap back and forth, but if no work is being done on the system (for example with a pump) then the total pressure will remain constant. In other words if the air flowing around an airplane speeds up, thus increasing the dynamic pressure, then the static pressure would have to decrease, just like the PE of the falling ball above. A real world example would be air escaping from a balloon; as the air escapes it has an increase in velocity, therefore the pressure inside the balloon must decrease. Another example is that of an air compressor; we can see that as air escapes from the nozzle, gaining KE, the pressure in the tank must decrease. The only way to prevent this is to do work on the system. In the case of a compressor a motor runs a pump that keeps adding energy to the system to replace the lost energy.