Mark H. Lienau
11400 N. Cowboy Trl.
Prescott, AZ 86305
Phone 928-708-9123
Water Rockets
How to Design, Build, Launch, and Predict
Chapter 1
The Basic Design
The EngineThe engine needs to do one thing, provide thrust. In the Water Rocket this will be done with a combination of air pressure and water. There are many sizes of engines possible and half the fun is experimenting with this facet of the design. For this paper we will use a small size motor.
Now let’s select the container to use. What to use? Soda bottle, Water Bottle, Glass bottle, Plastic bottle, Can? So what do we need it to do, and how do we decide?
Well the container needs:
1. To direct the water (propellant) out one end smoothly
2. It needs to be light
3. It needs to be able to contain the pressure necessary to push the water out.
There are several choices in the supermarket for instance, and we’ll choose to use a 20 oz water bottle. It is light (made of plastic), it is strong (able to withstand about 80 pounds per square inch of pressure or so before it breaks), and it has a nice smooth transition out the top so water can exit nicely. The illustration below shows the principal.
The Engine will push the rest of the rocket up when the water is let out of the bottle. The launch pad thus becomes an important aspect of the design and construction since it is the apparatus that will let the air into the engine and the water out when a launch is desired. The launch pad will be covered in the last section.
The StructureThe Rocket structure must do several things to make for a successful flight.
It must::
1. Hold the engine so that the thrust is taken from the engine to the rest of the structure without failure.
2. Provide the proper drag areas relative to the center of mass for the vehicle to fly stably.
3. Allow the vehicle to be recovered without undue damage from landing
Well in the rocket we will discuss here we won’t worry too much about recovery except to say that is one of the main functions of the nose cone. The nose cone will crush absorbing the energy of landing so that the main structure may be reused again.
So let’s discuss the other two functions in some detail.
1. As it turns out even a simple water rocket will produce enough initial impulse that we need to pay attention to how we will retain the engine. I have found that a metal coat hanger wire works pretty well and it’s relatively easy to make. The desire here is we want the engine to push on the wire and the wire to push on the rocket structure. Caution: dowels, pencils, or large paperclips are just not sufficient to withstand the impulse. The diagram below illustrates the principle involved here
The arrows indicate that the engine bottle pushing forces on the wire and intern the wire pushing forces on the rocket tube sides. To keep the engine from simply falling out the bottom pre-launch we simply use tape to secure the bottom of the engine to the tube.
2. The last issue with which we must deal if the design of the fins and placement of the rocket CG. This is the key stability issue for all rocket flight. In general terms it is said that the CP (center of pressure) must be rearward of the rocket CG (center of gravity). Why? You might ask, is this important. Well let’s think of the CP as the center of all the drag forces. The diagram below I think will make this concept clear.
As you can see the Resultant drag force is composed of three major parts, the nose cone drag, the body drag, and the fin drag. The fin drag is the largest component when the rocket is skewed relative to it’s direction of flight as in the diagram above. You can see if the rocket skews or rotates for any reason about it’s CG the Resultant drag force tends to correct the skew and point the nose directly into the direction of travel again if the CG if forward of this center of pressure or Resultant Drag force. So for any rocket weight distribution you can see that the area and length of the fins can be designed so that this corrective drag force is far enough back to correct any flight path deviations. For heaven’s sake don’t put fins near the nose cone as that makes things worse.
In a water rocket we do have a unique issue with the weight distribution as you might guess when you consider that when filled, the water sits at the lowest point in the engine as the pressure waits to push it out. Indeed, if this distribution persisted in flight we would have a stability issue. The saving grace is that the water exits so fast that the rocket will have only moved up only a small distance before the weight is gone. Later in the analysis section you will be able to readily see this. In addition, the launch tower adds directional stability during these first few milliseconds of operation due to its design.
Let’s talk some about fin design. If only two fins are used I think you can see that the two fins would form a plane in which the rocket could rotate easily and loose its directional stability. Therefore, always use at least three fins so that a single plane can not be formed. You can experiment with short fins, long fins and strange shaped fins. You can even spin the rocket with twisted or tilted fins.
The Launch TowerThere are several designs for the launch tower from the simple to the complex. The one we shall discuss here is one of the more complex, but it works well, allows you to pressurize the engine to a known pressure before launch, launch exactly when desired, and use a compressor so that quick multiple re-launches are possible. It was designed as far as I know by Wesley Wong of Santa Clara, CA. It is also easily positioned and anchored to the ground so that repeatable launch directions are possible. The launcher consists of an air delivery system with direction alignment tube, a nozzle sealing system, and a trigger. Let’s review these systems one at a time.
1. The air delivery system takes compressor air from the source and delivers it up through the rocket nozzle into the engine bottle. This is basically a series of ½ inch ID PVC pipes and fittings. The diagram below illustrates:
Note that all the pipes are allowed to fill with air and even the hose that goes back to the compressor and shut off valve. The strategy is to fill the engine bottle with air to a specific pressure determined by the compressor regulator and then close the supply valve. This is a very important consideration when we begin to formulate the mathematical model for the rocket performance. The total volume of air available to push the water out at launch must be calculated with these considerations. The pipe that extends up into the engine bottle had several functions.
a. Supply air to the engine bottle.
b. Prevent water from leaking down into the air supply pipes.
c. Guide the rocket up during the first few milliseconds.
In fact this pipe modifies the thrust obtained during lift off somewhat and has the downside effect of adding retarding friction to some degree. This will be discussed later.
Clearly this aspect on the launcher could be improved in the future.
2 The nozzle sealing system consists of one or two garden hose washers and a hold down mechanism. The hose washers are located at the bottom of the engine bottle guide tube and the hold down mechanism is shown below. It is perhaps the most complex part of the system. The function is to grab the lip of the engine bottle neck and pull it down a controlled distance thus compressing the soft rubber washers.
As illustrated, the slider collar rides up and down as the handle is raised and lowered. This intern pulls the lip grabbers down into the fixed containment collar the bottle lip with them and compresses the washers, thus, sealing the engine bottle and preventing water leakage until a launch is desired. The overcenter pivot prevents the seal from coming loose prematurely.
3. The Trigger is composed of the handle in the above illustration and the launch spring added below. As the handle is pulled to the left with a string the overcenter pivot unlocks and allows the launch spring to push the lip grabber up sharply thus allowing the engine bottle pressure to push the rocket upward freely.
Chapter 2
How to build
The EngineTo create the engine you will need a 20 oz water bottle. Many other sizes are possible but for the purposes of this paper let’s start simple. Drain the water and remove the cap and its plastic seal ring. That’s it! You’re done with the engine.
The Rocket StructureThe structure will consist of a body, fins, and a nose cone. We will not get fancy here with recovery systems and parachutes. You will also fashion an engine support to retain the engine.
Making the parts1) The body of the rocket will consist of a cardboard tube 3 inches in diameter and about 19 inches long. You can get these at Papermart.com.
2) The nose cone will be cut from cardboard flat sheet about .010 thick to the pattern below and rolled into shape and taped together on the tab. You might crease the tab first to make rolling the cone easier. The material may be any color or thickness that handles easily, I just happened to use the material listed.
3) The fins are cut from Corrugated cardboard and shaped as shown below. Make 3 or 4 what ever you like. Many variations are possible and experimenting is half the fun. Color these as you wish and smooth the edges that will not attach to the rocket body with sandpaper so the air flows smoothly over them.
4) The engine support is fashioned from a piece of coat hanger. Cut a straight piece about 4 inches long and bend one end about ½ inch from the end down at a right angle.
Assembling the rocket1) Measure up from one end of the cardboard tube about 5.5 inches and punch a small hole thru large enough to insert the Engine support wire. Make another hole on the opposite side of the cardboard tube to accept the engine support rod all the way thru the tube. Now bend the end of the straight end of the wire down so that it lies flush against the side of the cardboard tube. Place a piece of inch wide or wider tape all the way around the tube body securing both ends of the wire to the tube body. This can be a colored piece of tape for decoration or just clear.
2) Attach the nose cone onto the opposite end of the body from the engine support wire as shown in the figure. Again just use tape and smooth it out as mush as possible. Air flow and drag are important.
3) Insert the engine/bottle into the other end and tape as shown taking care to tape as neatly as possible.
4) Attach the fins with some heavy glue like Goop or Shoe-Goo equally around the body next to the engine as shown in the figure. Use tape to help hold the fins as the Goop dries. Tape neatly here as well.
The Rocket Launch TowerAs described in the design section above we shall now describe how to assemble the Launch tower. Get familiar with your local plumping store as most of the parts will come from there. The launch tower consists of an air delivery system with direction alignment tube, a nozzle sealing system, and a trigger. So let’s start in order.
Making the parts1) The air delivery system may be made from either PVC or copper or galvanized. In this case we chose to use PVC for cost as this was a cub scout launcher. Most of the structure is made from ½ inch schedule 40 pipe and fittings. The diagram below shows how this section is put together. The only tricky part here is the insertion of the ½ inch PVC tubing into the Slip-to-threaded coupling (as noted on the drawing) and the fabrication of the launch tube. The ½ inch PVC tubing used for the insert and the piece used for the launch tube must be drilled first internally. A 5/8 in diameter drill must be used to enlarge the internal hole slightly. The launch tube must be drilled to a depth of about 1 inch and the insert piece must be completely drilled out. This is to allow the ½ inch CPVC tube to fit properly into these pieces. Note the extra tube with the hose washers called the launch tube is made separately and simply fit down into the rest of the system just before launch. Frequently this tube is knocked off then the rocket is launched. Care should always be taken to make sure that the tube is smooth and fits easily into the engine bottle throat area. Lubrication here can help assure a clean launch. Another aspect to be careful with is the length of the tube. Not all bottles used for the engine may be the same length and it must be assured that the tube does not hit the end of the inside of the engine bottle. This would prevent the bottle neck from sealing tightly against the hose washers on the launch tube.
2) The Nozzle sealing system works in concert with the air delivery system in that the launch tube contains the hose washers that provide the seal against the engine bottle neck. The rest of the sealing system provides the hold down force necessary to assure the seal. The illustrations below show how to make the release handle and other related parts.
The handle is made from a 2 inch inside diameter PVC pipe about 10 inches long. The handle shape can be more or less free form with the holes being the important features. Just take care to make the lower portion of the handle ( where the pull string is attached ) thick enough to be strong and the portion above it cut back so it clears the containment collar when released.
The slider collar is made from a copper pipe fitting called a Bushing. The size to use is a ¾ to 1 inch bushing. This bushing is necessary because of its thickness. It will allow you to drill and tap the necessary holes right into the material
Note that the bushing will probably have a lip or internal ridge on it. File or cut this off . The inside diameter should be smooth and consistent. It will need to slide freely over the ½ inch PVC pipe riser. The taped holes are centered in the length of the bushing and are all equally spaced around the circumference. The two larger holes in line with each other and two smaller holes at 90 degrees to the larger ones. The placements are not super critical as there are plenty of adjustments available later to make the system work.
The containment collar is made from a 1 ¼ PVC slip-slip coupling. This is drilled through with a ¼ diameter hole as shown and attached to the handle and the top of the air delivery system thru the ½ inch threaded-to-slip coupling. The illustration below shows how the collar is drilled.
The threaded-to-slip coupling at the top of the air delivery system is drilled through with a 3/16 dia hole as shown below thru the non threaded half of the coupling. To assemble the handle to the containment collar to the threaded-to-slip coupling simply use a ¼ inch bolt about 3 inches long. The assembly procedure will follow after all the pieces to the sealing system have been fabricated..
The last piece to make will be the Lip Grabbers. These are made from two pieces of copper strapping cut about 9 inches long. The strapping is typically about 5/8 inch wide and comes punched with alternating holes of two different sizes. It usually is packaged in rolls and needs to be straightened out before use. One end of the strapping on each piece needs to be bent over in a small “L” shape about 1/8 inch long as shown.
The lip grabbers need to be fastened to the sliding collar next with solder and two 10-24 pan head screws about ¼ inch long. Use washers as necessary to keep the screws from going too far thru the sliding collar. Assemble the straps to the sliding collar as shown and when they are aligned square to the collar and to each other solder them to the collar with copper pipe solder.
Now it’s time to assemble the rest. Attach two 3/16 inch turn buckles about 4 inches long to the other two holes in the lip grabber assembly above with 5/16 inch screws ½ inch long and lock washers. Make sure the screws don’t go all the way through so that they would scrape on the riser. Leave the screws loose enough so that the turnbuckles will pivot. Slide the lip grabbers above down over the ½ inch riser making sure that the screws do not hit the PVC riser. Now put the 3 inch long ¼ inch diameter bolt through the handle upper hole and then thru the containment collar. You next need to thread the bolt into the hole in the threaded-to-slip coupling so that the bolt makes an air seal as it passes from one side of the riser to the other. Continue threading the bolt thru until it goes through the other side of the containment collar and the upper hole in the handle. Secure the far end of the bolt with a nylock nut, not too tight. Next attach the free ends of the turnbuckles to the lower holes in the handle with two 5/16 diameter bolts 1 inch long and two nylock washers. Put the bolts in from the inside of the handle. See the illustration below.
The last step here is to add the epoxy putty to the ends of the lip grabbers in order to mold them to the bottle neck and the inside of the containment collar. The illustration below shows the putty applied to the copper tips of the lip grabbers. Simply mold the putty into the bottle neck and up against the inside of the containment collar. Remove the bottle after the putty has set up somewhat so that it can still be scrapped with a sharp knife.
The putty should only grab the lip of the bottle neck from above as shown. Remove any putty from the bottle neck threads below the main neck lip so the bottle can exit smoothly upward during a launch. The putty should be contained tightly by the containment collar when the bottle is pulled down in place by the lip grabbers. The function of the collar is to keep the pressure from spreading the grabbers and launching the rocket prematurely.
3) The trigger is made up of the handle (previously installed) and the launch spring with fixed collar. The spring is about 4.5 inches long with 11.5 turns, 1 inch ID, and 1/8 inch wire diameter. The fixed collar is nothing more than a small piece of water hose secured with a SS hose clamp of about 1 inch diameter. .
The clamp allows the collar to be adjusted up and down so that the spring can be given sufficient force when compressed to pop the lip grabbers upward when the handle is pulled.
This completes the assembly of the launch tower.
Chapter
3 How to LaunchOperation of the Rocket and launcher is pretty simple now. We shall set up the launch tower, fill the rocket with fuel and mount it, pressurize the motor and launch the rocket, and track it to altitude.
Set up the launch tower1. Find a level place on which to set up the launch tower. The tower is then staked down into the ground on that level place. Tent stakes make good implements for this purpose.
2. The Launch tower is plugged into an air hose connected to a compressor with pressure gauge. Plug in your compressor and start it up. You should have an on/off valve between the compressor and your launch tower so that the air flow can be controlled.
3. Run a string from the launch handle out about 10 or 12 feet from the launcher
Fill the rocket with fuel and mount the rocket1. To start, fill the rocket engine about one third full with water.
2. Now make sure the lip grabbers are all the way up and quickly slide the rocket down the launch tube and onto the sealing washers
3. Pull the handle down gently guiding the lip grabbers onto the rocket engine bottle lip.
4. Snap the handle all the way down and over center so that the rocket is secured onto the launcher.
5. Pay the trigger string out away from the launcher and get ready to pull.
Pressurize the motor and launch the rocket1. With the compressor running and connected to the launcher first adjust the output pressure to the desired launch pressure, about 60 psi.
2. Open the supply valve so that the air can now flow to the launcher, making sure that all the connections are tight and no water is spraying from the launcher or the rocket.
3. Once the pressure gauge returns to 60 psi (the set output pressure) shut the supply valve.
4. Now simply pull the trigger string until the launch handle pops out and the rocket lifts off.
Track the rocket to altitude
1. To track the rocket you first must stand at a measured distance from the launcher, say 50 feet.
2. Attach a string with weight to the protractor and let it hang as shown.
3. Using a protractor as shown above sight up the straight edge at the rocket as it rises.
4. When the rocket reaches it’s maximum altitude press your finger on the string so that it can’t move relative to the protractor (note the angle that the string makes with the straight edge of the protractor).
5. Read the angle. The angle should be about 60 degrees if you made the rocket and launcher as suggested. Now you can compute the altitude as follows:
Altitude = 5+(50/ Tan angle)
The 5 is the height you have the protractor off the ground in feet as you sight in the rocket. The 50 is the distance in feet you are standing from the rocket launcher. The angle is as measured in step 5 above in degrees. Tan is short for tangent ( it’s a trigonometric function on a scientific calculator)
The altitude will come out in feet.
Chapter 4
How to Predict
Here we shall explore how to write the equations necessary to model the rocket’s expected maximum altitude. We will also compare the results to those measured in the chapter above. Once a successful model is achieved you will be able to use it to test the effects of fuel level, rocket mass, launch pressure, engine size, and a host of other parameters. The Model will be constructed using an excel spread sheet once we have laid down all the governing equations here. The first thing to realize is that the source of energy is entirely the compressed air. Water is not a propellant, it’s for show and maybe some initial stability.
I realized after some time that the correct detail equations for modeling the thrust are very difficult to write as sonic flow is produced and the mixed nature of the water/air flow really complicates things.
To resolve this, I finally realized that the key was to consider the problem from only an energy perspective.
So let’s begin:After all E=f P dv is all the energy available
To compute this we first have to find a suitable expression for the pressure P. If we use the ideal gas law and assume the initial pressure in the rocket is P1 and the space taken up by the air is V1 and the pressure when it’s totally exhausted, P2 is simply the atmospheric pressure of 14.7 psi. Since the exhaust is done rapidly the best assumption is that the expansion is adiabatic and no heat flows in or out of the system during the expulsion of the air/water mixture.
This then can be used to formulate the following expression for P.
P= P
1V
1g / V
gIf we now integrate this expression over the interval from P1 to P2 we get an expression for the energy contained in the compressed air E.
E= P
1V
1/(
g-1) (1– (P
1/P
2)(1- g )/ g) Note: The pressure must be expressed in absolute terms.
Now that we have an expression for all the energy that is available to propel the rocket upward let’s assume that this is all instantly converted to kinetic energy when the rocket is launched. We, therefore, can compute the launch velocity as:
V
i2=( 2E/M)
Where M is the mass of the rocket structure and any water added.
As the rocket propels itself skyward with this initial velocity, air drag begins to act to slow the rocket down along with gravity of course. Eventually the rocket reaches its maximum altitude.
The drag force is expressed as:
D= ½ ρ
air V
2 C
d S (where V is the velocity of the rocket as it moves up)
Therefore, the deceleration due to this drag is :
A=D/m (where m is the empty mass of the rocket because the water is all expelled very quickly at launch)
Now using the classical projectile equation:
X=-½at
2+V
it
We can solve this for the maximum altitude condition and we get:
X
m=V
i2/2a
We can substitute in for the a:
X
m= V
i2 /2(g+A) (where g is the acceleration due to gravity)
Since this makes the problem of predicting the maximum altitude extremely nonlinear since A is dependant on the velocity of the rocket and much harder to compute let’s assume that the velocity used there can be replaced with 1/3 of the launch velocity Vi. If we do this the equation can be reduced to:
X
m= V
i2 /2(g+½ ρ
air V
i2 C
d S /9m)
You might be able to conclude from this that any water just reduces the launch velocity and, therefore, reduces the maximum altitude. As it turns out some water does help to stabilize the rocket dynamically when it first leaves the rocket launcher due to its mass. The extra mass helps offset any sideways forces from the launcher as the grabbers release it. Further if you then bend the fins to cause the rocket to spin as it moves through the air you will find that it goes much higher as air drag is more uniform and the rockets trajectory is more nearly vertical all the way up.
I hope that this little paper has been instructive to read and perhaps that you made the launcher and had some fun. An excel spreadsheet program is available too or from the equations above you can make your own.
