People have always pulled their hair out when fueling hybrid launchers, because inaccuracies multiply themselves, and misunderstandings can kill the whole thing in the water.

A lot of people attempt to use a standard fuel meter at first. For a hybrid, I will go out on a limb here and state that it is nearly impossible to do so with a hybrid and higher mixes.

So what do we do? Read on.

All examples in this article work with propane.

This document uses \KaTeX for rendering mathematics. Subsequently, this page may load a little slowly, and javaScript is obviously required to view it correctly.

Since Google Chrome has decided to gimp MathML, this is where we're at. Sorry.

All values returned by these equations are in absolute pressure, not gauge pressure. I am also using the value of 14.7 for atmospheric pressure, which may not be the same as where you live.

To find gauge pressure (the pressure you would read on a pressure gauge), simply subtract 14.7 from returned values.

The simplest way for smaller launchers is to use a syringe. Ideally, you would rig up a check valve setup so that fuel can be injected into the chamber without displacing air. This can be done by epoxying the tip of a luer-lock or taper tip syringe into a schrader cap, and perhaps cutting the spring in the valve core so it cracks at a lower pressure.

When fueled this way, the amount of fuel required would simply
be *knV _{c}*, where

Since the fuel injected into the chamber raised the pressure by a certain amount, you can't simply add air until your gauge reads n atmospheres. You have to add the pressure of fuel to your final gauge pressure. In PSIA (actual pressure, subtract 14.7 to get gauge pressure), this works out to be:

\begin{aligned} 14.7V_{c} + 14.7(0.042nV_{c}) &= P_{i}V_ {c}\\ 14.7V_{c}(1 + 0.042n) &= P_{i}V_{c}\\ P_{i} &= 14.7(1 + 0.042n) \end{aligned}

In the above formula, n would be fuel/air "compression", although this is different from the actual gauge compression. For a 2x mix, use 2. For a 4x mix, use 4, etc. So a better name would be mix number...

Some people can't be bothered to rig up a check valve like the above method, and simply inject fuel into an open plug, screw hole, etc. The problem with this is that it displaces air upon fuel injection. This has to be accounted for to prevent headaches.

It turns out that it is rather simple. If you recall from the combustion launcher section's page on metered fueling, this is accounted for by simply using another fuel ratio about equivalent to 0.0403. It turns out this holds true for fueling hybrid launchers with a syringe as well, but let's prove it.

We start out with roughly the same system of equations:

V_{c} - V_{f} = V_{a} \quad,\qquad V_{f} = 0.042V_{a} + 0.042(n - 1)V_{c}

We start off with a simple one: The volume of air left in the
chamber is equal to the chamber volume minus the volume of
fuel. The second equation is not so intuitive.
0.042*V _{a}* means that, for your first
"

Now, we simply solve the system to eliminate
*V _{a}*:

\begin{aligned} V_{f} &= 0.042(V_{c} - V_{f}) + 0.042(n - 1)V_{c}\\ 1.042V_{f} &= 0.042V_{c}[1 + (n - 1)]\\ V_{f} &= \frac{0.042nV_{c}}{1.042}\\ V_{f} &≈ 0.0403nV_{c} \end{aligned}

By simplifying the above expression, you'll realize that it's the same as syringe fueling without displacement, but using a fuel ratio of about 0.0403 instead if 0.042.

This is a method of volumetric metering first used by SpudBlaster15 on the Spudfiles forums. It works almost like a traditional fuel meter, except air is injected through the meter pipe and into the chamber after fuel is, which mixes air completely with propane.

This requires that the fuel meter be "burped" with propane between shots, to purge air from the meter, but it makes for a fairly accurate meter... especially if you know how to operate it.

Time to whip out Boyle's law and PV constants again...

P_{m}V_{m} = (14.7)0.042V_{c} + 14.7(0.042)( n - 1)(V_{c} + V_{m})

Whoa, let's take a step back here. What this is saying is that
the total PV constant of the meter pipe is equal to the PV
constant of fuel required to react with the air in the
combustion chamber (for the first "*n*"), plus the PV
constant of the fuel required to react with subsequent
atmospheres of air injected (which fill the entire volume of
the system, not just the combustion chamber). Solving this for
*P _{m}* yields...

P_{m} = \dfrac{0.6174[V_{c} + (n - 1)( V_{c} + V_{m})]}{V_{m}}

As earlier, this is only one half of accurate fueling. You also
need to know how much air to add. Since the volume increases
after fuel injection as well, the standard 0.6174*n* for
gauge pressure increase won't work. The amount of pressure
increase depends on the size of your meter pipe and chamber
this time. Let's figure that out...

The sum of PV constants of air can be represented as
14.7(*n* - 1)(*V _{c}* +

This sum can be proven by dividing the PV constant of fuel by the PV constant of air. Since The PV constant is directly related to number of moles provided temperature is the same, this will yield the f/a ratio for propane, giving us the green light to move forward...

\begin{aligned} \frac{0.6174[(n - 1)(V_{c} + V_{m}) + V_{c}]}{ 14.7(n - 1)(V_{c} + V_{m}) + 14.7V_{c}} &= 0.042\\ \frac{0.6174}{14.7} × \frac{(n - 1)(V_{c} + V_{m}) + V_{c}}{(n - 1)(V_{c} + V_{m}) + V_{c}} &= 0.042\\ 0.042 &= 0.042 \end{aligned}

Now that we know the PV constant for air, we simply divide by the combined volume of chamber and meter to obtain the partial pressure of air required.

P_{a} = \dfrac{14.7[(n - 1)(V_{c} + V_ {m}) + V_{c}]}{V_{c} + V_{m}}

Now that we know the partial pressure of air, we need to know the partial pressure of fuel after equalization. This is as simple as dividing the PV constant of fuel by the total volume of chamber and meter, the same as with air.

P_{f} = \dfrac{0.6174[(n - 1)(V_{c} + V_{m}) + V_{c}]}{V_{c} + V_{m}}

Now that you have the partial pressures of fuel and air, simply add them together to find the final pressure (in PSIA) you need to fill your hybrid's chamber to when fueling.

Quite hands-down the simplest way (if not a little expensive) to fuel a hybrid launcher, and chamber volume doesn't even need to be taken into account.

Take a look at the first problem we tackled in this article...
pressure rise during fuel injection. You'll notice the change
in pressure scales linearly with each higher *n* value
(mix number). By subtracting 14.7 from the value when
*n* = 1, we can determine this pressure rise is 0.6174
PSI per mix number. Notice PSI has no absolute or gauge
restraint on it, it's simply the change in PSI.

So, simply multiply your mix number by 0.6174, inject propane
until this pressure is achieved (using an accurate gauge or
transducer), and then add air until the final pressure is
*n* atmospheres plus 0.6174*n* PSI.

It's as simple as that. You might have troubles finding an accurate enough gauge or transducer, and you'll probably wind up spending a good deal of money...

These are the most commonly used methods for fueling a hybrid launcher which will allow enough accuracy to create an ignitable mixture in the combustion chamber. There are other ways people have tinkered around with, like flow-based metering (effectively like a carburetor in an older engine), etc., as well. All of your volumes (chamber volume, meter volume) should be measured accurately by filling with water and decanting into a graduated cylinder... do not rely on calculating the volume of a cylinder for your meter pipe or chamber... your hair will thank you.