The nozzle constant is a measure of energy loss when the
fluid flow is constricted by the nozzle. The more that the
fluid has to go around tight corners, the more energy is
lost and the less thrust you get.

The best way to determine this is by experiment but as most
people haven't got access to a fluid dynamics lab, an educated
guess is going to be your best bet.

Sudden contractions in the flow can have a detrimental effect
on the flow as well as the fluid is 'reshaped' in the pipe as it
flows. If the larger pipe diameter is 'D' and the smaller pipe
diameter is 'd':

The acceleration of the incompressible fluid into the nozzle is
what generates thrust. If you consider the body of the pressure
vessel as a pipe full of fluid that is travelling along it, a certain
volume will be flowing past any arbitrary point during a given
period.

If you attach a smaller diameter pipe to it, that same volume
will still flow along it but by being a smaller diameter, it will
now be travelling faster. In other words, it will have been
accelerated.

We know how much is travelling through the pipe and we know
the diameters of the two pieces of pipe, along with the density
of the fluid so now, we also know the mass which means that
we can use the formula...

...where the force F is equal to the mass multiplied by the
acceleration.

However, it is not as simple as that. If you look along the axis
of the pipe constriction, and imagine the flow of the water into
the constriction, the water flows in from the outsides towards
the middle and as it does so, you can imagine it getting
more 'crowded'.

Now, imagine viewing the constriction from the side so that
the water flows from left to right. If the constriction occurs
over a long distance, the water has a chance to speed up
gradually and any imaginary 'chunk' of water only has its
shape distorted over a long distance and the difference of
speed within it is only going to be small. The amount of
distance the 'chunk' has to move sideways is only going to
be small.

However, if the change is abrupt, some of it will be moving a
lot faster than other bits of it, distorting the 'chunk' and the
more it is distorted, the more energy is turned to heat. The
water still has to flow into the constriction is it just that with
less of its energy ending up as kinetic energy, for a given
pressure, less liquid flows through it.

So, you can see that both the amount of constriction and the
type of entry into the smaller pipe affect the efficiency of the
constriction and that is what this value represents.

You need to measure the flow rate from a pressure vessel as
the water flows through it and measure the thrust. You can
calculate what the thrust would be and compare that with the
results to work out the constant 'K'.

You can hold a pop bottle with the bottom cut off, upside
down, held by a retort stand that is mounted on a set of
scales. Plug the nozzle and almost fill it with water. Then,
pull the plug and make note of the flow rate - marks put on
the side of the pop bottle - and at the same time, the weight
on the scales.

• The time for the water level to go past the marks
  will give you the rate of flow for a particular level;
• The height of water above the end of the nozzle
  will give you the pressure; and,
• The weight on the scales at a given moment in
  time will tell you the combined mass of the whole
  thing plus the mass of water supported by
  it minus the thrust.

One way of taking these measurements is to video it and
you can analyse it afterwards and go back over things if you
think you have missed something. If you have to use two
cameras, you can synchronise them by clapping your hands
once if you can do that in their field of view, using a flash of
light if you can't move them and they aren't looking at the
same point in space. You don't need the sound as it is all
visual.

The formula for the pressure you would expect in there were
no losses is...

The losses in that are your losses for your nozzle.

The nozzle itself is basically a constriction so you have fluid
flowing towards a central line and then changing direction -
the more efficiently it can do that, the less loss. The
acceleration in a vertical axis is what provides the thrust for
the nozzle so the centre of acceleration is where this
constriction is, not the end of the nozzle. Water is in effect
incompressible so the nozzle on a water rocket simply aims
the jet of water, it does not gain extra thrust from it in the way
that the cone on a rocket engine does - that is a compressible
fluid that is expanding and is optimised for external air
pressure hence the nozzles on vacuum engines being larger
than those on sea-level engines.

One other thing to consider is the behaviour of the fluid
itself. In fluid mechanics, there is a dimensionless value called
Reynolds number (Re) which is a ratio of inertial force to
viscous force ...

Re =	ρ V D
	μ

... where:

ρ is the density of the fluid;
V is the velocity of the fluid;
D is the diameter of the pipe; and,
μ is the dynamic viscosity of the fluid.

With a value below 2,000, the flow in the pipe is laminar
because the viscous forces dominate but above around 4,000,
the inertial forces dominate. between 2,000 and 4,000, there is
a situation where there is some turbulent and some laminar
flow.

The importance of this is that when you have long pipes full of
fluid, the pressure drop along them is different between
laminar and turbulent flow.

When you work out Re for the nozzle of a water rocket, it is well
above 4,000 and some people might be tempted to argue that
the formulae for turbulent flow should be used. However, the
key point against that argument is that when entering a pipe
that has an Re above 4,000, it takes a pipe length equal to
around 10 pipe diameters for turbulent flow to establish itself
and as a result, the flow inside a water rocket nozzle is in fact
laminar.