# Hybrid Fueling

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.

## Note on Display

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.

## A Note On Absolute Pressure

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.

## Syringe Fueling

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 knVc, where k is the fuel/air ratio for the fuel you're using (recall 0.042 for propane), n is the compression of the final mixture in atmospheres, and Vc is the volume of the combustion chamber. After fuel is injected, things get a little more complicated, but not too much so.

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...

### The Hard Way

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. 0.042Va means that, for your first "n", the fuel volume injected would be 0.042 of the volume of air, not the chamber volume. 0.042(n - 1)Vc simply states that for "all other n's", you would use 0.042 of the chamber volume, since each atmosphere of air added is equivalent to one chamber volume at STP.

Now, we simply solve the system to eliminate Va:

\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.

## "Air-Through-Meter" Fuel Metering

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 Pm 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 0.6174n 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)(Vc + Vm) + 14.7Vc. That is to say, you start off with the chamber volume of air at STP, and then add an atmosphere of air to both the chamber and meter volume combined for each subsequent mix number.

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.

## Manometric 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 0.6174n 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...

## Conclusion

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.