Rocket Science

Most of the time when you think of a rocket, you think of something blasting off into space. That is what the largest rockets are used for. There are also much smaller rockets that do other things. The military uses thousands of rockets a year and fireworks make up a very large percentage of the world's rocket supply. When you shoot a rocket, you generally want to apply a force to something to cause it to move. The study of moving things is called dynamics. (If you want to study the forces in things that don't move, that is called statics.)

Newton's Laws

An early scientist by the name of Sir Isaac Newton studied the motion of objects and came up with three observations. These observations became known as Newton's Three Laws and started a branch of science known as Newtonian Physics. Water rockets travel slowly (250 miles per hour) compared to the speed of light (186,000 miles per SECOND) so the laws of Newtonian physics work quite well for our simulation purposes. (If we were simulating events that occur near the speed of light we would have to take into account relativistic effects. This branch of dynamics is called Einsteinan physics named after physicist Albert Einstein.) Newton's first law is: Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it. Note here that a uniform state of motion can also be standing still. The magnitude of the velocity vector just happens to be zero. Newton's first law is a simple statement of the laws of momentum and inertia. Newton's second law is: The relationship between an object's mass (M), it's acceleration (A), and the applied force (F) is F=MA. This is the fundamental equation for all dynamics problems. The force F is the total or summation of all forces acting on the mass M. Newton's third law is: for every action there is an equal and opposite reaction. This is illustrated by imagining two ice skaters standing in a line. If the skater in back pushes the skater in front, the front skater will go forward and the back skater will go backward. That is sort of what happens with rockets. As the exhaust is pushed out of the nozzle, the rocket is pushed in the opposite direction with equal force.

Rocket Thrust

To calculate the acceleration of a rocket, we must know the thrust and mass. Since F=MA, to get acceleration, divide both sides of the equation by M and you get A=F/M. The equation for thrust on a rocket is F=mass flow rate times exit velocity. The mass flow rate is how much mass is leaving the rocket each second. The exit velocity is how fast the exhaust is traveling. Often times, the mass flow rate changes from moment to moment. Also, as mass leaves the rocket, the mass of the rocket is decreasing. The acceleration of a rocket is continually changing and a prediction must take all this into account. During a water rocket launch, there are three phases of thrusting; water flow, sonic (choked) air flow, and subsonic (unchoked) air flow. The study of water flow is called incompressible fluid dynamics. (Water isn't really incompressible but it compresses so little that it can be treated as such for most applications.

Water Flow

The thrust produced during the water flow portion of a water rocket flight can be calculated by using the Bernoulli equation. This equation simply accounts for the energy content of a fluid in a steady state process. (There are many types of energy that are constantly trading places. Just a few types of energy are potential, kinetic, electrical, chemical, thermal, nuclear, photonic, acoustic.) The Bernoulli equation has many terms that take into account many types of energy exchange. For our purposes here, we can justifiably neglect most of these terms because that type of energy exchange is not taking place. The relationship that represents the water flow in a water rocket is:

where P=bottle pressure,

=water mass density, V=water velocity, and g=gravitational constant. If we assume that the initial water velocity inside the bottle (V1) is zero and the gauge pressure outside the bottle (P2) is zero, this equation reduces to:

. If we use the units of slugs/ft^2 in our density term, the gravitational constant cancels out to make

. We compute exit velocity then by

. The mass flow rate is the throat area times velocity times density.

But thrust is

but

which yields

We have simplified the equations by neglecting fluid turbulence, fluid viscosity, fluid friction, and other things. These neglected factors combined only account for a very small percentage of the total thrust so this approximation is pretty good. The instantaneous change in weight of the water rocket as previously mentioned is

but

. These relationships hold during the water expulsion portion of the thrust phase.

Air Flow

Once the water is expelled, the remaining compressed air exits producing thrust. Studying compressed gas flow is called compressible fluid dynamics.

Sonic (Choked) Flow

When studying compressible fluid dynamics, there are three types of flow fields; subsonic, sonic, and supersonic. Momentarily, if the launch pressure is high enough, a water rocket can have all three types occurring in different locations in and around the bottle. Sonic flow at the throat (or minimum cross section of the vent) occurs when the ratio of pressure inside and outside the bottle is above the critical pressure ratio. The equation for the critical pressure ratio is:

where

=critical pressure,

=ambient pressure, and

=ratio of specific heats (1.4 for air)
For water rockets using air as the pressurizing agent, the critical pressure is

This means that if, after all the water has been exhausted, the remaining air pressure in the bottle is above 27.8 psia (or 13.1 psig), you will get sonic or choked flow at the throat. The air inside the bottle will still be subsonic, but the air expanding outside the bottle will be supersonic. You can sometimes hear a crack or bang just after launch with a high pressure launch. This is a mini sonic boom. Great way to attract attention at the start of a launch party.
When the air pressure is above the critical pressure, and you have sonic flow at the throat, the mass flow rate can be calculated by the following equation:

where

=mass flow rate

=throat area

=chamber pressure

=gravitational constant

=characteristic gas velocity
The characteristic gas velocity is a measure of the thrust producing abilities of a given gas. The equation for C* is:

where

=ratio of specific heats (1.4 for air)

=gravitational constant

=absolute temperature of gas

=universal gas constant

=molecular weight

One interesting characteristic of sonic flow is that the conditions at the throat are independent of the downstream pressure. Noticed that the mass flow rate equation does not have ambient pressure in it. Once critical pressure is achieved, the mass flow rate will not increase with a decrease of ambient pressure.
The actual exhaust velocity (but not mass flow rate) of supersonic flow depends upon the expansion ratio of the nozzle and surrounding conditions. The ratio of exit plane area to throat area is called the expansion ratio. You have seen rockets that have an exit cone. This increases the gas velocity even further to produce more thrust. If you have choked (sonic) flow at the throat, the conditional relationships along the exit cone are: as the cross sectional area increases, the pressure and temperature drops and the velocity increases. There is, of course, a limit on how far you can expand the gases. If you expand to much, the exit plane pressure will be below ambient pressure and this will have a diminishing influence on the thrust. It can be shown (through a lot of calculus) that the optimum thrust is achieved when the exit plane pressure just equals ambient pressure. If the exit plane pressure is greater than ambient, the flow is said to be underexpanded. Conversely, if the exit plane pressure is below ambient, the flow is said to be overexpanded. For our water rockets, we don't usually have an exit cone because the duration of choked flow is so short. Our water rockets have an expansion ratio of one. The equation for thrust for compressible gases is:

where

=chamber pressure

=throat area

=thrust coefficient
The equation for thust coefficient is:

where

=expansion cone half angle

=ratio of specific heats

=exit plane pressure

=ambient pressure

=chamber pressure

=expansion ratio
To find the exit plane pressure

, we have to solve the following relationship by numerical iteration:

where

=expansion ratio

=ratio of specific heats

=exit plane pressure

=chamber pressure
As you can see, many calculations are required to determine the actual rocket thrust.

Subsonic (Unchoked) Flow

Once the chamber pressure drops below the critical pressure, the flow becomes unchoked. Now the downstream pressure has an influence on the conditions at the throat. The mass flow rate equation for unchoked flow is:

As all the compressed air is expelled, the thrust drops to zero and the water rocket has attained its maximum velocity. Gravity and wind resistance slow the water rocket down to the stopping point and the rocket falls back to earth.

Simulation Results

For a simulation of a 2 liter bottle with initial air pressure of 100 psig and 20% water volume the following are the results:

	Launch	Water Burnout	Supersonic/Subsonic Transition	Deceleration	Pressure Equilibrium
Time (sec)	0.0	0.042	0.074	0.088	0.104
Altitude (ft)	0.0	3.2	12.2	16.9	22.4
Pressure (psig)	100	69.2	13.1	4.1	0.0
Velocity (ft/sec)	0.0	202	325	332	326
Acceleration (ft/sec/sec)	0.0	12,294	1,102	0	-671

Water burnout is when all the water is exhausted and just air remains in the bottle. There is still pressure (69 psi) which continues to provide thrust. Since the pressure is above the critical pressure, choked flow occurs at the throat and you have supersonic exhaust. When the pressure drops to the critical pressure (13.1 psi) the flow changes to unchoked but there is still a little energy left in the air. When the chamber pressure gets down to 4.1 psi, the drag on the bottle equals the thrust produced so you get no net acceleration. This is the point of maximum velocity. By the time the pressure inside the bottle equalizes, the velocity has dropped by 6 ft/sec and gravity and drag will bring the bottle to zero velocity. Notice that 1/3 of the velocity is obtained by the residual air in the bottle after the water is gone.

This is but a brief introduction into the world of rocket science. Alot of fun as you can see!!

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