The goal of any rocket motor designer is to pack as much propellant into the motor casing as possible. After all, you’ve only got so much room for motor — might as well make the most of it. One popular way to increase volumetric loading is to step the cores of the grains as they go down the length of the motor, putting a large port diameter near the nozzle throat and a smaller port diameter up near the head end of the motor, where the mass flux and port velocity is low. As the gas accelerates down the length of the grain, opening up the port lowers the mass flux to hopefully mitigate the effects of erosive burning.

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 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:

Of course, this is implicit for Ma in terms of Ap and At, so you need to break out Excel with a goal seek or your other favorite numerical solution method to get an answer.

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:

or, for stagnation conditions 0 (as exist at the head end of the motor):

Both of these equations confirm what our intuition from the previous paragraph said: increasing Mach number leads to decreasing pressure down the grain. We can simplify this knowing that the flow velocity at the head end of the motor must be zero, and thus static conditions are equal to stagnation conditions. Define the motor pressure ratio, phi, as

and we should be all set.

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:

(more accuracy would be cool in the form of a ballistic element model, but that’s also for another post), we can now solve for Ph and Pa. If we take McCreary’s recommendation (Ap/At = 0.563) for a test drive here with a typical propellant, we get a phi of 0.935. This translates, in a motor with a steady-state Pc of ~950 psi, to an actual head end stagnation pressure of ~1050 psi and aft end static pressure of ~900 psi — that’s fairly significant.

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.

Posted April 30, 2010 in Propulsion Theory. Add Comment / Trackback