But what if the aft grain is small? It’d pack more propellant in the motor, but bad things could also happen. Common industry wisdom says keep a port to throat throat to port (thanks James!) area ratio of 0.5; McCreary is a bit sportier in “Experimental Composite Propellant“, going for a diameter ratio of 0.75 (and thus an area ratio of ~0.56). Of course, many rocketeers have been known to push this limit, even so far as to have a port the same size or smaller than the throat. And it works, sometimes. So what happens as the port size is increased?

There are some simple gas effects that we can model fairly easily using ideal rocket assumptions. The major item is a pressure differential that develops between the head end and the aft end of the grain. If you’ve ever bought an Aerotech 38/1080 motor and wondered why it needs a seal disk, while a 38/240 does not, it’s because of this pressure differential issue. (The seal disk keeps the gas leak from focusing over the edge of the liner, eroding it quickly and leading to forward end heat problems.) This pressure differential develops because mass is being added down the length of the rocket motor, and this mass needs to be accelerated by the gas flow to sonic velocity at the nozzle throat. Since temperature is constant throughout the motor, the energy lost to accelerating the gas shows up as a pressure loss.

How do we calculate this? The first thing to do is to figure out the gas velocity over the aft end of the grain. We can turn to our old isentropic flow friends to help us out — we know the port to throat area ratio Ap/At, so all we have to do is apply the area-Mach relation and we’ll have what we’re looking for there:

With the Mach number in hand, now we move on to using a simplification of the compressible flow god equation (topic for another post…) neglecting all terms except mass addition, to give the pressure ratio as a function of Mach number. (Think of this as the mdot version of Fanno flow or Rayleigh flow.) Considering the head end of the motor as location 1 and the aft end as location 2, we get:

So why does this matter, and how do you figure stuff out with it? Well, first of all, it affects the thrust of your motor — since there is energy lost in the system, neglecting erosive effects, you are losing thrust when compared to a simple lumped-parameter ballistic calculation. Calculating phi allows you to figure out exactly what your thrust loss is, since you want to be using the static pressure at the nozzle end to calculate thrust; in the words of my favorite propulsion professor, “no matter your vices and sins upstream, your thrust is determined by the pressure feeding the nozzle.” This also contributes to additional stress on the head end of the motor; if your have a burn that is already “on edge” in terms of your hardware strength, you could be in trouble here, too. (This is particularly important if you have a handle on the erosive behavior of the motor; though you might be designing in an erosive spike at startup, the combination of the spike and the increased head end pressure might be enough to blow the bulkhead.)

You can calculate the pressures at each end by simply using a conservation of mass flowrate — we’re still in steady-state ballistics land, so (mdot in) = (mdot out). Assuming a constant (conservative) burning rate for the propellant down the length of the motor:

Of course, this analysis does not take into account various propellant combustion effects. Having a port smaller than the throat will cause the choke point to be in the grain at startup, rather than in the nozzle throat. This can have a nasty effect on motor performance, what with shock waves forming and all, and can also lead to serious erosive burning effects as the gas velocity increases and the boundary layer thins. That’s the somewhat unpredictable part; it depends on propellant quality, composition, and rheology. But at least it gives us some insight into what exactly is going on when we tighten up the port in search of better performance.

I’ve been working on a new blue formula as of late. Consider this the first “official release”. It was conceived in January, first fired in April, and first fired with data last weekend. It’s a smokeless variant of the typical blue I’ve flown in the past — 1% CuO, 2% Al — with the goal of getting something “Blue Thunder“-like. The story of the first flight is up in a blog post on OurPlanet.

First reactions from the flight were: (1) Not quite as fast as Blue Thunder, but certainly nice nonetheless, and (2) Oh my, look at that flame separation. Data from static testing showed that this burn was only at ~570 PSI, but the flame stands off nicely, even at low chamber pressures. It turns out to be about as fast as Kosdon fast, at least per the data. Here’s a typical curve from a Loki 38/480 firing:

Delivered Isp is somewhere around 200 sec, so nothing too special — just another “knob” propellant to add to the fray. Formula and rate data after the jump.

“SA Blue” (named such as I envisioned doing this in a C-slot for a Standard Arm I’m building) is 2% Al 400, 1% CuO, 80% solids with all 200 mic AP. Pretty much as simple as you can make it. But I like how it performs; not much to no slag on the nozzle, and a lot tamer than high CuO formulations (Amarillo Blue, I’m lookin at you…). Initial data says r = 0.0175P_c^0.42 (P_c in PSI) with a theoretical C* of 4632 ft/sec, safe to run between Kn = 200 and 300 in most situations. Maybe one day I’ll try it in a 2550 — right now it looks like just over 900 PSI and a 1.8 sec burn. L1500 anyone? Maybe after it’s better characterized…