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\begin{titlepage}
\vspace*{1in}
\begin{center}
\LARGE Performance Optimization of a Pressure-Driven Rocket
\end{center}
\vspace*{1in}
\begin{center}
Michael D. Mendyke\vspace*{2em}\\Submitted in partial fulfillment of
the requirements of 16.621/.622\vspace*{1em}\\The Massachusetts
Institute of Technology\vspace*{1em}\\  
December 15,1988\vspace*{1in}\\
Partner: Eric L. Shukan\hfill Advisor: Prof. Andreas Von Flotow
\end{center}
\end{titlepage}

\begin{center}
{\Large Abstract}
\end{center}
This paper describes the derivation of a performance model for a water
fueled, pressurized rocket and an experiment used to test the validity
of the model. the experiment consisted of launching the rocket
multiple times with different initial conditions, measuring its speed
during flight at a known height. The results of the experiment
indicate that the performance model is accurate for the range of
pressures tested. At the end of the paper are recommendations for
further experimentation and study. 

\newpage

\tableofcontents
\newpage

\section{Introduction}

\subsection{Purpose}
Analytical models can be useful in predicting the performance of real
systems, and in providing a basis for related models. This paper will
discuss the derivation of a model for the performance of a water
fueled, pressurized rocket and the experimental procedure used to test
the model.

\subsection{Background}
The rocket used in this project is commonly found in toy stores.
Constructed from hard, translucent red plastic, it is an ellipsoid
with a short cylindrical tube on one end which acts as a convergent
nozzle. It is about $15$ cm long and $4$ cm wide, and weighs about
$20$ grams.

When the rocket is partially filled with water, pressurized and
released, the air inside expands, forcing the water out of the nozzle
and generating upward thrust. It is clear that the two parameters
which drive the thrust are the initial volume of water (if normalized
to the total internal chamber volume can be called the {\em volume
fraction}) and the initial internal pressure.

For a given initial pressure the volume fraction can be varied from
$0$--$1$, and it can be demonstrated that there is an optimum value.

If the volume fraction is too small, the water runs out before the
internal pressure fully equilibrates with the external pressure. This
is inefficient as air expanding through the nozzle produces much less
thrust than water, due to its lower density.

If the volume fraction is too large, the internal pressure reaches $1$
atm before all the water is expelled, and thus the remaining water has
acted only as additional weight. The optimum value occurs somewhere
between these two limits. 

It was originally predicted that the optimum volume fraction for a
given pressure would occur at the point where the water ran out {\em
at the same time} the internal pressure reached $1$ atm. However, when
the theoretical data was determined, maximum performance occurred when
the water ran out {\em just before} the pressure equilibrated.
Apparently at the predicted optimum condition the weight of the
additional water outweighed the additional thrust gained, resulting in
a slightly reduced performance.

The performance characteristic of interest is, of course, the final
height reached by the rocket for a given set of initial conditions.
However, it is a difficult quantity to accurately measure (a typical
height for this rocket is $30$ meters).

It is easier to measure the velocity of the rocket at some height
after it has stopped producing its own thrust and is being accelerated
only by gravity. By simple energy balance this is equivalent to
measuring the final height. The faster the rocket is going, the higher
it will eventually go.

Therefore, the theoretical model derived in the next section will
determine the total velocity imparted on the rocket for a given set of
initial conditions.

\newpage
\section{Theoretical Model}

\subsection{List of Variables}

Mass of rocket structure                            \hfill $M_{0}$\\
Initial mass of water in rocket chamber             \hfill $M_{0w}$\\
Initial mass of air in rocket chamber               \hfill $M_{0a}$\\ 
Total mass of rocket                                \hfill $m(t)$\\      
Mass flow rate                                      \hfill $\dot{m}(t)$\\ 
Total internal volume of rocket                     \hfill $V_{0}$\\
Initial volume of water in rocket                   \hfill $V_{0w}$\\
Initial volume of air in rocket                     \hfill $V_{0a}$\\
Volume of air in rocket                             \hfill $V(t)$\\
Water volume fraction                               \hfill $\theta$\\
Density of water                                    \hfill $\rho_{w}$\\
Density of air in rocket                            \hfill $\rho_{c}(t)$\\
Pressure of air in rocket                           \hfill $P_{c}(t)$\\
Atmospheric pressure                                \hfill $P_{a}$\\
Temperature of air in rocket                        \hfill $T_{c}(t)$\\
Height of water in rocket (referenced from throat)  \hfill $h(t)$\\
Cross--sectional area of water at $h(t)$            \hfill $A(t)$\\
Exit area                                           \hfill $A$\\
Exit speed of water relative to rocket              \hfill $u(t)$\\
Exit acceleration of water                          \hfill $\dot{u}(t)$\\
Inertial speed of rocket                            \hfill $v(t)$\\
Thrust                                              \hfill $F(t)$\\
Rocket acceleration                                 \hfill $\frac{dv}{dt}$\\
Ratio of specific heats                             \hfill $\gamma$\\
Normalized gas constant                             \hfill $R_{g}$\\
Acceleration of gravity                             \hfill $g$\\ 
Mach number at throat                             \hfill $M(t)$\vspace*{1em}\\

At $t=0$, $\rho_{c}=\rho_{0}$, $P_{c}=P_{0}$, and $T_{c}=T_{0}$.

\newpage

\subsection{Introduction}

The goal of this analysis is to determine the total velocity gained by
the rocket, given $P_{0}$ and $\theta$. This is done in two sections.

The Air--Water Case determines the $\Delta v$ generated by the water
being forced out of the nozzle. If the internal pressure is still
greater than $1$ atm when all the water has been ejected, we must then
add the $\Delta v$ produced by the expanding air (the Air--Only Case).

To simplify the calculations, we make the following basic assumptions:

\begin{enumerate}

\item The gas flow within the nozzle is isentropic. This is a
reasonable assumption since the thrust is nearly adiabatic (it occurs
within several thousandths of a second).

\item Viscous effects are ignored. This is an admittedly large
assumption, but for this simple model is considered acceptable.

\item The gas inside the rocket obeys the ideal gas law. At no point in
the experiment does the gas experience extremely high pressure or low
temperature, hence, it is reasonable to assume ideality.

\item Drag on the rocket is small compared to thrust. The speed of the
rocket does not generally exceed $30 m/s$. If we assume that the drag
coefficient of the rocket is close to 1 (the rocket is well
streamlined), then the drag is only about $0.1$ \% of the thrust.

\item The mass of the gas inside the rocket is very small compared to
the mass of the water and the rocket itself. In general, the air mass
accounts for less then $0.01$ \% of the total mass. If there is no
water (i.e. the Air--Only Case) the mass of the rocket is assumed to
be $M_{0}=$ constant.

\end{enumerate}

\subsection{Air--Water Case}
Given the initial conditions $P_{0}$, $\theta$:

We know that 

\begin{equation}
F_{t}=\rho_{w}u^{2}(t)A=m(t)\frac{dv}{dt}
\end{equation}

Now $A$ and $\rho_{w}$ are constant, so to determine the force we need
only determine $u(t)$. So using Bernoulli's equation we have

\begin{equation}
\frac{1}{2} \rho_{w} u(t)^{2} = (P_{c}(t) - P_{a}) + \rho_{w}gh(t)+\rho_{w}\frac{dv}{dt}-\rho_{w}\int_{0}^{h(t)}\dot{u}(t)\frac{A(t)}{A}dt
\end{equation}
 
where

$P_{c}(t)-P_{a}$ is the stagnation pressure gradient

$\rho_{w}gh(t)$ is the gravity pressure gradient

$\rho_{w}\frac{dv}{dt}$ is the rocket acceleration pressure gradient

$\rho_{w}\int_{0}^{h(t)}\dot{u}(t)\frac{A(t)}{A}dt$ is the pressure
gradient due to the water accelerating relative to the rocket

By substituting Eq.~$2$ into Eq.~$1$ and solving for $\frac{dv}{dt}$ we
obtain: 

\begin{equation}
\frac{dv}{dt}=\frac{2A(P_{c}(t) - P_{a} + \rho_{w}gh(t)-\rho_{w}\int_{0}^{h(t)}\dot{u}(t)\frac{A(t)}{A}dt)}{m(t)-\rho_{w}h(t)}
\end{equation}

We find $P_{c}(t)$ from the isentropic relation

\begin{equation}
P_{c}(t)=P_{0}\left[\frac{V_{0a}}{V(t)}\right]^{\gamma}
\end{equation}

But $V(t)$ is the initial volume of air plus the volume of water that
has left, or

\begin{equation}
V(t)=V_{0a}+\int_{0}^{t}u(t)Adt=V_{0}-V_{0w}+\int_{0}^{t}u(t)Adt
\end{equation}

Thus,

\begin{equation}
P_{c}(t)=P_{0}\left[\frac{V_{0}-V_{0w}}{V_{0}-V_{0w}+\int_{0}^{t}u(t)Adt}\right]^{\gamma}
\end{equation}

or in terms of the volume fraction $\theta$

\begin{equation}
P_{c}(t)=P_{0}\left[\frac{1-\theta}{1-\theta+\frac{(\int_{0}^{t}u(t)Adt)}{V_{0}}}\right]^{\gamma}
\end{equation}

And $m(t)$ is just the initial mass of structure and water minus the
amount of water expelled:

\begin{equation}
m(t)=M_{0}+\rho_{w}A_{0}\theta -\int_{0}^{t}\rho_{w}u(t)Adt
\end{equation}

The acceleration of the rocket is therefore fully described by
equations $(3)$, $(7)$ and $(8)$. The speed can be determined by
integrating $\frac{dv}{dt}$ numerically. Appendix A contains the code
used in this project.

If there is still pressure in the rocket greater than $1$ atm, an
additional $\Delta v$ is gained by the expanding gas. This possibility
is discussed next. 


\subsection{Air--Only Case}

The velocity gained by expanding air through the nozzle can be
added by superposition to the velocity gained from the Air--Water
Case. Hence to integrate for $\frac{dv}{dt}$ the time can be
initialized to zero. 

Given the initial chamber conditions---$\rho_{0}, P_{0}, T_{0}$:

Find the nozzle Mach number from the isentropic pressure ratio

\begin{equation}
\frac{P_{0}}{P_{a}}=\left(1+\frac{\gamma-1}{2}M^{2}(t)\right)^{\frac{\gamma}{\gamma-1}}
\end{equation}

From $\frac{P_{0}}{P_{a}}$ determine the nozzle conditions $\rho$ and $T$

\begin{equation}
\frac{P_{0}}{P_{a}}=\left(\frac{\rho_{0}}{\rho(t)}\right)^\gamma=\left(\frac{T_{0}}{T(t)}\right)^\frac{\gamma}{\gamma-1}
\end{equation}

From $M$ and $T$ determine $u(t)$
\begin{equation}
u(t)=M(t)\sqrt{\gamma R_{g} T(t)}
\end{equation}

Since

\begin{equation}
\dot{m}(t)=\rho(t) u(t) A
\end{equation}

we obtain an expression for the acceleration of the rocket from the force:

\begin{equation}
F(t)=\dot{m}(t)u(t)=\rho(t) u^{2}(t) A \rightarrow \frac{dv}{dt}=\frac{F(t)}{M_{0}}
\end{equation}

Integrate over time to determine the new chamber density:

\begin{equation}
\rho_{c}(t) = \rho_{0} - \frac{\int_{0}^{t}\rho(t) u(t) A dt}{V_{0}}
\end{equation}

Now that $\rho_{c}(t)$ is known, solve for $P_{c}(t)$ and
$T_{c}(t)$ using isentropic flow and the ideal gas law:

\begin{equation}
\frac{P_{c}(t)}{P_{0}}=\left[\frac{T_{c}(t)}{T_{0}}\right]^{\frac{\gamma}{\gamma-1}}
\end{equation}

\begin{equation}
P_{c}(t)=\rho_{c}(t)R_{g}T_{c}(t)
\end{equation}

and with these new chamber conditions
$(\rho_{c}(t),P_{c}(t),T_{c}(t))$ the process repeats from step $1$,
summing up $\frac{dv}{dt}$ until the chamber pressure reaches $1$ atm.
Note that since the nozzle is convergent, the flow may become choked
at Mach $1$. The algorithm to solve this case numerically is located
in appendix A.

\subsection{Theoretical Results}

The results predicted by the performance model are presented on the
following three pages. Figure $1$ gives a family of curves
representing the speed of the rocket at a height of $86.7$ cm as a
function of initial volume fraction, for several different initial
pressures. The value of $86.7$ cm was chosen for two reasons. At this
height, the rocket has finished thrusting and is moving only under the
influence of gravity. Thus it is a trivial matter to determine the
final height the rocket would attain for a given point on the curves.
Also, this was the most convenient height to use for purposes of
constructing the apparatus used to test the theory. 

Note that as discussed in the introduction to this paper, for very
small and very high values of $\theta$, the speed is also small. The
line connecting the peak of each curve is the theoretical line of
maximum performance.

Figure $2$ is an expanded view of the line of maximum performance.
This curve basically indicates how much water should be put in the
rocket to make it go the highest for a given pressure. Note that the
curve is roughly exponential in nature. As pressure increases, the
amount of thrust increases, allowing more water to be accelerated, and
so the volume fraction approaches $1$. As pressure decreases, the
weight of the water begins to outweigh the thrust it gives, and so
the volume fraction approaches zero. 

Figure $3$ shows the maximum speed at $86.7$ cm that the rocket can
achieve as a function of pressure. 

\section{Test Procedure}

\subsection{Introduction}

As noted in the previous section, it is essential that the measurement
of the rocket's speed is taken at a point {\em after} the rocket has
stopped thrusting. In this way we can be sure that the rocket is
experiencing identical conditions (i.~e. internal pressure is $1$ atm,
gravity is the only force present) each time we take the measurement.
The model predicted that for initial pressures of $3$ and $4$ atm, the
rocket would finish thrusting at no higher than $50$ cm.

Because of the difficulty in placing instrumentation of the rocket, it
was desirable to measure the speed in a manner which would not have an
adverse effect on the performance of the rocket. It was decided that
rather than try to measure the instantaneous speed at a precise
height, it would be sufficient to measure the average speed over a
small height interval. By noting the time of flight between heights,
the average velocity is simply

\begin{equation}
\bar{v}=\frac{\Delta h}{\Delta t}
\end{equation}

The average speed was measured in this experiment using a laser-net
timing apparatus, as described in the next section. 

\subsection{Apparatus}

\subsubsection{Laser--Net Timer}

The framework for the timer was made out of Dexion. It was mounted on
a lab table at a height of $86.7$ cm. Four Dexion arms extended over
the edge of the table, one pair above the other, each pair about two
feet apart and parallel to the ground and each other. Four long, thin
rectangular mirrors were mounted, one to each arm, each mirror facing
its counterpart. A four milliwatt laser (passed through a beam
splitter to produce two beams) was reflected off of each set of
mirrors at a very small angle, so as to produce two parallel
``planes'' of laser light. A schematic of this setup is shown in
figure $4$ following.

At the ends of each pair of mirrors was placed a photodiode whose
output was connected to a digital timer. Each time the rocket traveled
between each set of Dexion arms, it impinged upon the laser and
dropped the intensity of light reaching the photodiodes. This had the
desired effect of starting, then stopping the timer as the rocket passed
through first the lower, then the upper set of arms. This elaborate
method of bouncing the laser beams back and forth between several
mirrors was necessary to insure that the rocket would actually hit
both beams and trigger the timer. Basically, this method gave the
person launching the rocket a target window to launch through, rather
than the unlikely target of two single beams. 

The single largest difficulty with this setup lay in the alignment of
the mirrors. Because of imperfections in the mirrors, the laser beams
would not behave ideally, sometimes producing parabolic surfaces
rather than flat planes of light. In one case the light even began to
bounce back upon itself, in the direction opposite the location of the
photodiode. these problems were alleviated by clamping additional
Dexion beams perpendicular to the arms and twisting them about the
axis of each arm, producing torque and altering the shape of the
mirrors to control the generated patterns of laser light.

\subsubsection{Launch System}

Included with the rocket at purchase is a plastic hand pump, at the
end of which is a well-designed clamp to hold the pressurized rocket
in place until the operator is ready to launch. While the clamp was
suitable for our purposes, the pump was not, having no way to measure
the initial pressure generated inside the rocket. Therefore, the clamp
was removed from the pump, attached to a thick rubber tube and affixed
firmly to the ground beneath the Dexion apparatus. The other end of the
tube was attached to a tank of gaseous nitrogen with a pressure
regulator. While ambient air is not pure nitrogen, the properties of
the two gases are very close, and nitrogen was considered appropriate
for use as the pressurant within the rocket. 

\subsection{Test Procedure} 

Since the rocket was launched over $150$ times, the launch procedure
quickly became routine. The first step was to check that the mirrors
were aligned so that the laser beams hit the photodiodes, and measure
the distance between the laser nets. The second step was to weigh in
the appropriate amount of water for the particular case. It was
easiest to simply weigh in the water rather than measure its volume. 

Step three was to place the rocket on the launch clamp and pressurize
it. And step four was launch the rocket and record the time interval.
The entire process could be done in about three minutes. 

\subsection{Testing Difficulties}

Even though the launch clamp was fixed to the ground, there was a
small amount of rotation possible, and this could account for the
rocket traveling at angles up to $5$ degrees from the normal. This
occasionally created problems in successfully launching the rocket
between both laser nets, particularly at higher pressures. 

While the setup was designed with the firm conviction that all of the
water would be gone from the rocket by the time it reached the
mirrors, at higher pressures and high volume fractions the mirror
apparatus still managed to become soaked (as did the person launching
the rocket). It was finally realized that while the water was indeed
leaving the rocket below $50$ cm, the water still had a net {\em
upward} velocity, and was therefore rising several meters before
falling, unfortunately, all over the apparatus. 

\section{Data Precision}
There are three basic sources of error in the experiment. They are:

\begin{enumerate}

\item $V_{0w} \rightarrow \pm 1 cm^{3}$

\item $\Delta t \rightarrow \pm 0.1 ms$

The timer had to be very accurate, as $\Delta t$ was typically less
than $20 ms$.

\item $\Delta h \rightarrow \pm 0.02 m$

\end{enumerate}

The distance between the laser beams was the single largest source of
error, due to both the difficulty in aligning the beams in parallel
planes and the difficulty in launching the rocket normal to the
ground. 

The initial internal pressure has an associated error of $\pm 0.1$
atm. However, this does not cause uncertainty in the measured speed of
the rocket. Instead, it creates uncertainty in the rocket's operating
point. 

The raw data has been placed in Appendix B. 

\section{Results}

The results of the experiment are presented on the next two pages.
Data was collected for initial pressures of three and four
atmospheres. It had been hoped to also test the five atmosphere case,
but this prospect was abandoned due to time constraints. As noted in
the previous section, there was uncertainty as to the operating point
of the rocket within $\pm 0.1$ atm, and so each graph shows two lines
generated by the theoretical model for one-tenth atmosphere above and
below the ``actual'' tested pressure. 

\newpage

\section{Conclusions}

As is evident by Figures $5$ and $6$, the experiment seems to validate
the performance model, at least for the four and five atmosphere
cases. The most obvious recommendation for further research is to try
many more pressures, particularly high pressures. It would be
interesting to press the limits of this theory and the associated
assumptions. At very high pressures the gas may no longer be assumed
to behave ideally. At very low pressures and high volume fractions
viscous effects may be more prevalent. At a high enough pressure, the
water may even choke the nozzle, at which point this performance model
breaks down utterly. 

While I am dubious as to the direct potential of this model to serve
as a propulsion system in space, I do believe that this model provides
a useful basis from which other more complicated and rigorous models
can begin. 

\newpage

{\noindent \LARGE Appendix A\vspace*{1em}\\Computer code for Numerical
Analysis}

\newpage

{\noindent \LARGE Appendix B\vspace*{1em}\\Raw Data}




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