Why do we use water?
Newton's third law of motion tells us that
throwing a mass in one direction causes us to move in the
opposite direction. Water being heavy serves as a good mass
to throw. This is also known as a 'reactive mass'. Water is
inexpensive, available everywhere and is human
friendly. Being a liquid it allows us to easily control the
rate at which we throw it.
Why can't we fill up the whole rocket with
water?
Water being essentially incompressible
cannot store much energy when it is pressurised. Air on the
other hand can store lots of energy when pressurised but is
not very heavy and therefore doesn't work well as a reactive
mass. In a water rocket the air provides the stored energy
and the water provides the reactive mass. In order to
achieve maximum performance a balance needs to be found
between the amount of stored energy and the amount of
reactive mass the rocket carries. Too much water and the rocket is
too heavy, and not enough water
means there isn't a lot of reactive mass to throw. Because
the water and air share the same volume we have to trade one
for the other.
Working Out the Optimum Amount
The optimum amount of water depends on a
number of interdependent factors such as weight of the
rocket, its drag coefficient, launch pressure, nozzle size
etc. There is no simple rule to calculate the optimum
amount of water for any particular rocket. Additionally you also need to take into consideration
your desired goal for the rocket flight. You may be after
the highest altitude, or carrying a heavy payload, or
achieving the highest acceleration. These decisions will
influence the optimum amount of water you use in your
rocket.
So how can we work out the optimum?
Using Simulators
By far the easiest way to work out the
optimal amount of water is to use one of the many
water rocket simulators available online.
Procedure
The following procedure can be used with
most simulators:

Enter your
rocket and launcher parameters into the simulator and choose about 1/3 the
capacity for the water fill. Most simulators have
various extra parameters such as nozzle efficiency,
launcher volume, air humidity, drag coefficient etc.
that are needed to calculate the rocket performance. If you don't know what
these are for your rocket then generally
accepting the default values is close enough.

Run the simulation and the simulator will give
you the predicted performance. Take a note of the
performance result you are interested in.

Increase the water
amount and rerun the simulation. The simulator will give you
either higher or lower
performance results.

If the simulator gives you lower
performance then try to decrease the water amount and
try again. If the simulator gives better performance
then try increasing the water amount again.

By repeating these steps a number of times you
can find the peak performance for a particular water fill
amount.
NOTES:

Some simulators [1,4] allow you to enter a range of
values and the simulator will calculate the optimum amount
for you.

Most simulators, however, have limitations and
will give less accurate results if you use extreme
parameter values. Don't expect useful results if you are
trying to simulate a 2000psi rocket or huge nozzles for
example.

Some of the better simulators are quite
accurate at predicting flight performance, but if your
rocket does not fit the typical profile, you can still
work out the optimal water fill by flying your rocket number of
times, varying the water amount and measuring your
desired performance parameter.
Optimum Water Amounts
Now let's have a look at some of these
rocket parameters and how they affect the optimum. Because
there is a large variety of possible water rocket
configurations we have chosen a pair of rockets that
represent ones in common use:
Rocket A 
Capacity 
2 Liters 
Nozzle 
9 or 22 mm 
Weight 
150 or 300
grams 
Diameter 
110 mm 
Drag
Coefficient 
0.3 

Rocket B 
Capacity 
8 Liters 
Nozzle 
9 or 22 mm 
Weight 
700 or 1000
grams 
Diameter 
90 mm 
Drag
Coefficient 
0.3 

The following series of graphs describe the
relationships between different rocket parameters and how
they affect the optimum amount of water. Since we could
optimize for different performance characteristics, we have chosen to optimize the amount of
water for achieving the greatest altitude. For each point on
the graph a simulation was run with the given parameters and
we let the simulator calculate the optimum water amount.
Rocket Weight
The following graphs show how the optimum changes
with respect to the rocket's dry weight.
Graph 1  Optimum water amount vs. weight of rocket 
Rocket A
Graph 2  Optimum water amount vs. weight of rocket 
Rocket B
You can see from Graph 1 that as expected
the maximum altitude gained decreases as the dry weight of
the rocket increases. However the optimum amount of water
increases as well. At 2Kg the proportion of water is almost
58%. In graph 2 you can see that there is a big difference
between the 22mm and 9mm nozzles in terms of the optimum
water percentage. The graph also shows the optimum weight for rocket A
is around 150 grams and about 175 grams for rocket B to
achieve the greatest altitude.
Nozzle Size
The following graphs show how the optimum changes
with respect to the nozzle size.
Graph 3  Optimum water amount vs. nozzle size 
Rocket A
Graph 4  Optimum water amount vs. nozzle size 
Rocket B
Graph 3 shows that for very small nozzles
the optimum amount of water is quite low in proportion to
the total volume of the rocket. This is expected because the thrust generated by a small nozzle isn't sufficient to
lift a lot of weight. Notice also that for rocket A as you
increase the nozzle size, the optimum amount of water
slightly decreases for very little gain in altitude. Large nozzles
mean higher acceleration but also higher drag. For rocket B
though, you get higher performance as you increase the
nozzle size and the optimum water amount also increases.
Pressure
The following graphs show how the optimum changes
with respect to the launch pressure.
Graph 5  Optimum water amount of 150gram vs. pressure  Rocket A
Graph 6  Optimum water amount of 300gram vs. pressure  Rocket A
Graph 7  Optimum water amount of 700gram vs. pressure
 Rocket B
Graph 8  Optimum water amount of 1000gram vs.
pressure  Rocket B
In the diagrams above, both common nozzles
of 9mm and 22mm are shown. You can see that initially as
pressure increases so does the optimum water amount, but
then decreases again as the pressure increases.
There is a large difference between the 22mm nozzle and 9mm
nozzle. As the pressure increases the optimum amount of
water peaks and the difference between 9mm and 22mm
decreases.
Altitude
The next two graphs show the predicted
altitude over a range of different water amounts at constant
pressure.
Graph 9  Predicted altitude vs. water amount  Rocket
A
Graph 10  Predicted altitude vs. water amount  Rocket
B
As you can see in graphs 9 and 10 the
optimal amount of water peaks at different amounts depending
on the the weight and size of nozzle. In graph 10  while a
heavy rocket with large nozzle peaked at 36% a lighter
rocket with a smaller nozzle peaks at 23%. You can also see
that the water amount is not critical near the peak
altitude. In graph 9 for the 150 gram rocket you can see
that water fill from 20%40% results in only a small change
in altitude of less than 10 feet or 4% of the overall
altitude.
Practical Considerations
The following practical tips suggest why you
may not necessarily want to use the optimum amount of water.
Stability
When you run a simulation to calculate the
optimal amount of water you'll notice that there is quite a
broad range of volumes of water around the optimum that give
almost the same peak altitude (See Graphs 9 and 10). You may
want to consider using less water than the optimum in your
rocket that will result in higher acceleration and allow the
rocket to reach a stable speed sooner.
This is particularly important if you are
using reduced nozzles. Reduced nozzles have two drawbacks
when it comes to rocket stability. They produce a lower take
off thrust at the same pressure meaning that the rocket
accelerates slower and hence reaches a stable speed later
before the fins can actually start correcting the rocket's flight
path. The other is the weight of the water. The heavy water
near the tail can significantly move the center of gravity
back on the rocket and cause the rocket to be unstable. With
a reduced nozzle the water remains in the rocket longer and
hence the center of gravity is further back for more of the
flight. This means your rocket needs to use bigger fins and
a longer guide rail compared to a higher accelerating
rocket.
Pressure chamber restrictions
The construction of your rocket may also
influence how much water you should use. If you build a
large volume rocket out of bottles joined for example with
Robinson couplings or tornado tubes, the optimal amount of
water may exceed the capacity of the lowest bottle. Filling
the rocket with water above the coupling will result in an
effective rocket nozzle the size of the Robinson coupling
while there is water still above the coupling. This may
result in a loss of thrust on takeoff if the nozzle is larger than the
lowest coupling.
In these situations only fill the bottom
bottle with water allowing only air to pass through the
coupling in order to restore some of the expected thrust on takeoff.
Can I add something to the water to make it
heavier?
Yes you can, but it will typically result in
lower expected altitude. Although the rocket will have
higher thrust because it is expelling a heavier mass, it is
also having to accelerate the heavier mass still in the
rocket.
A water rocket actually benefits from a
lower density liquid compared to water. The graphs below
cover a range of liquid densities from alcohol (0.785g/cm3)
to brine (1.23g/cm3) which are readily available. There are
both lower and higher density liquids available, but these
also tend to have other less desirable properties.
Graph 9  Predicted altitude vs liquid density
 Rocket A
Graph  Predicted altitude vs liquid density
 Rocket B
For each liquid density a simulator was used
to work out the optimum amount of liquid.
Conclusion
As can be seen from the graphs above the
optimum amount of water can greatly vary from 10%60%
depending on your rocket and launcher configuration,
however, most rockets will not fall into these extremes. For
larger nozzles the optimum is closer to 33% while for
smaller nozzles the optimum is closer to 25%.
References
Simulators
25th May 2011 