Comparing the Thrust for DifferentNozzle Shapes in Model Water Rockets

Haris AminDepartment of Physics, Wabash College,

Crawfordsville, IN 47933(Dated: March 15, 2007)

This paper illustrates the use of a numerical model with a finite-element method implementationto qualitatively describe the relationship between the length of a nozzle in a model water rocketand the thrust it exerts. We found that constricting the flow of the exiting water in the directionnormal to the nozzle by increasing the length of the nozzle allows more thrust to be achieved.

Model water rockets are often used to demonstratesome basic principles of physics such as thrust, air drag,and projectile motion. A model water rocket tradition-ally consists of a chamber (bottle), filled with a certainamount of water, whose pressure is increased via an airpump till the thrust of the pressurized air and water ex-iting the nozzle force it to launch. What may seem asquite a simple system is actually fit to provide a wealth ofknowledge about the physics of rockets and thrusts to ele-mentary school and undergraduate physics studies alike.In this Letter, we will be providing a numerical modelthat we developed to observe the effects of the nozzlelength on the thrust of a model water rocket. We alsovalidate our numerical model by comparing its results toexperimental data.

Our numerical model is based on Navier-Stokes fluidflow. Navier-Stokes equations seek to establish relationsamong the rates of change or fluxes. They establish thatchanges in momentum infinitesimal volumes of a fluid aresimple the product of changes in pressure and dissipativeviscous forces acting inside a fluid [? ]. Navier-Stokesequations are written as,

~∇ · ~u = 0, (1)

µs∆~u + Re(~u · ~∇~u) + ∆p− ~∇ · ~τ = ~f, (2)

where ~u describes the velocity profile of the fluid, µs de-scribes the viscosity of the fluid.Re describes the turbu-lence of the system, ~τ describes the stress tensor in thefluid, p describes the pressure in the system, and ~f de-scribes the the forces acting on the system as a whole. Equation 1 mainly insures that mass is conserved inthe system. These differential equations allow you to de-scribe the balance of forces acting at any given regionin the fluid [? ] . In the case of an ideal fluid, zeroviscosity, these equations state that the accelerations isproportional to the derivative of internal pressure. Amodel water rocket is essentially such a system.

Our system is essentially a nonlinear system. Our nu-merical solver will involve a finite element method withleast-squares implementation. This method allows us toapproximate a finite space with piecewise polynomials.

This basically enables us to approximate a nonlinear sys-tem with polynomial approximations. We use quadraticpolynomial approximation for our numerical model.

2 L = 8 units

1.5 L = 6 units

4Units 1 unit

FIG. 1: This model illustrates a fluid flow contraction prob-lem. The fluid is given an initial velocity profile. In this casewe gave our fluid a parabolic velocity profile. The fluid passesfrom the first channel to the thinner second channel where itsrate of flow increases. The numerical model also ensures thatmass is conserved. Note that to simulate a water rocket withdifferent nozzle lengths, the numerical model sampled velocityprofiles at two locations, L and 1.5 L.

The domain of our numerical model consists of a largechamber and a smaller chamber resembling a model wa-ter rocket. The large chamber represents the body ofthe water rocket while the small chamber represents thenozzle of the rocket. Figure 1 illustrates our contractionproblem. The thrust T of a rocket due to the ejection ofmass from nozzle is

T = |vedM

dt|, (3)

where ve is the exhaust velocity of the ejected mass anddMdt is the rate at which mass is ejected from the rocket

from its frame of reference. Our numerical model allowsus to gain a qualitative understanding of how the thrustof the rocket would vary when the nozzle (second cham-ber) is varied in length. The thrust correlates with thehorizontal component of ve. Comparing two nozzles bylooking at the ve profiles at L and at 1.5L, as illustratedin the Figure 1 , the ve profile with a smaller vertical com-ponent and larger horizontal component gives us morethrust. This enables us to gain a qualitative understand-ing of how the length of the nozzle actually correlateswith the thrust of model water rocket.

Figure 2 and Figure 3 illustrate the absolute values ofthe vy profiles at L and 1.5L respectively. As you can seethe vy profile at 1.5L is a lot smaller in magnitude than

the vy profile at L. From our numerical model, we wereable to extract the vx profiles at L and 1.5L. Since allproperties of our fluid such as density and viscosity wereconsistent in the nozzle at both L and 1.5L, we can allowthe thrust at both points to be directly proportional tothe average of the respective vx profiles. Taking this intoaccount we found the thrust at L , TL to be 2.56 ∗ 10−2

and the thrust at T1.5L to be 3.02∗10−2. The normalizedTL, T ′

L, is 8.50∗10−1. These values agree with the afore-mentioned thought that smaller vy values correspond toa larger vx profile which in turn corresponds to a greaterthrust. However, in order strengthen the foundation ofour numerical model for its more qualitative perspectiveon the correlation between nozzle length and thrust of awater rocket, we need an experimental data and proce-dure.

5 10 15 20Point

0.0005

0.001

0.0015

0.002

0.0025

0.003

vy

y_velocities.nb 1

FIG. 2: This is the magnitude of vy sampled 20 times at L.

5 10 15 20Point

5µ 10-8

1µ 10-7

1.5µ 10-7

2µ 10-7

vy

y_velocities.nb 1

FIG. 3: This is the magnitude of vy sampled 20 times at 1.5L

Figure 4 illustrates the setup four model water rocket.It consists of a large PVC pipe chamber with an air nozzleattached to it and a hole for the pressurized water to es-cape at the bottom. This large chamber is attached to aforce plate that will measure the force exerted on it whenpressurized water escapes the chamber. In order to com-pare the results obtained from this setup to our numeri-cal model, we have a smaller PVC pipe that behaves asthe nozzle for our water rocket. We can then measure theforce exerted on the force plate with this smaller chamberattached to simulate our numerical model at 1.5L, andwithout the smaller chamber to simulate our numerical

model at L.

Force Plate

Air Nozzle

Large PVC Chamber

Small PVC Nozzle

5.24 cm

0.84 cm

0.69 cm

35.62 cm

20.20 cm

Water

FIG. 4: Pressurized air is pumped through the air nozzle.As the pressure in the larger PVC chamber increases, thewater exits the hole in the first chamber. As the water exitsthe first chamber it exerts a force against the force plate.Two scenarios were considered her. One where the watersimply exits the first chamber into the atmosphere. Secondwhere the water exits the first chamber and then enters thesecond smaller PVC pipe chamber before being released intothe atmosphere.

After taking several measurements with and withoutthe small chamber with the same volume of water and atthe same pressure, we can fit the force versus time graphto analyze how the water exits the rocket. Knowing thatthe force-time graph represents,

F = gM − T, (4)

where force F , thrust T , and mass M are functions oftime, we can obtain T and an average for the thrust ex-erted by the rocket. Recalling Equation 3 and knowingthat,

ve =1

ρAnozzleM ′, (5)

where ρ is the density of the fluid, water, and Anozzle isthe cross-sectional area of the nozzle through the waterexits we can rewrite Equation 4 as

F = gM − 1ρAnozzle

M ′2, (6)

where

T =1

ρAnozzleM ′2. (7)

Then all that is left to do is to solve the differential equa-tion, Equation 7, with an easy numerical solver to deter-mine the thrust.

Figure 5 and Figure 6 represent the running averageforce with respect to time of our model water rocket with-out the small chamber and with the small chamber re-spectively. The peak of each graph represents when theall of the water had exited the model rocket. The forceapproaching the peak represents the thrust of the rocketin each case as the mass (water) exits the rocket withrespect to time. Solving Equation 7 for both cases, wefound the thrust in the rocket without the small cham-ber. T1 to be 5.42± 0.89 N and the thrust in the rocketwith the small chamber, T2 to be 16.27± 1.10 N. An im-portant fact to note is that the inner radius of the smallPVC chamber is smaller than the radius of the hole inthe large chamber as illustrated in Figure 4. This wastaken into account in solving our differential equation foreach case.

The normalized T1, T ′1 is 0.33. This is not very similar

to T ′L at all. This can be due to the fact that the data

obtained form our numerical model does not deal withdifferent size radii for the water to exit through. Thoughour numerical model did not take different sized exit holesinto account, our experimental results still reinforce thekey qualitative analysis provided by our numerical model.Our numerical model was successful at illustrating howthe length of the nozzle is able to restrict the flow of thewater to the direction normal to he nozzle.

The success of our numerical model to provide a qual-itative perspective on water rockets and nozzles is not atits end. This model could very well be modified to ex-amine the effect of different nozzle shapes on the thrustof a water rocket. Such shapes could include a conic,parabolic, or even an hourglass-sahped nozzle. Similarly,one can check the validity of the numerical model qualita-tively by comparing it to experimental data. This modelallows us to further examine simple model water rocketsin a more depth. Such analysis fosters a better under-standing of the physics of rockets.

[] White, Frank M., Fluid Mechanics Third Edition,McGraww-Hill Inc, 1994.

[] Finney, G. A., Analysis of a water-propelled rocket: Aproblem in hors physics.,American Association of PhysicsTeachers, June 1999.

[] Prusa, Joseph M., Hydrodynamics of a water rocket,SIAM Review Vol.42, No.4, 2000

[] Yawn, James, Model Rocket Mo-tors,http://www.jamesyawn.com/modelrocket/drill/index.html

0

2

4

6

8

10

12

14

0 0.5 1 1.5 2

Running Average Force (N)

Run

ning

Ave

rage

For

ce (

N)

Time (s)

FIG. 5: Water exits the first chamber into the atmosphere.Notice that the force-time relationship is linear as the waterexits the first chamber. A running average of 50 runs wastaken to reduce the simple harmonic oscillating effect of thewater rocket system when the pressurized water was exitingthe system.

0

2

4

6

8

10

0 0.5 1 1.5 2

Running Average Force (N)

Run

ning

Ave

rage

For

ce (

N)

Time (s)

FIG. 6: Water exits the first chamber and into the secondsmaller chamber before entering the atmosphere. Notice thatthe force-time relationship is quadratic as the water exits therocket. Again a running average of 50 runs was taken to re-duce the simple harmonic oscillating effect of the water rocketsystem when the pressurized water was exiting the system.