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:

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.

The linear model for erosive burning (as posited in Zucrow & Hoffman, Green, and other sources) posits a direct linear correlation between local Mach number (or, assuming a homogeneous chamber temperature, local gas velocity) and increased burn rate, due primarily to compressibility. Several nonlinear models exist, too (Renie & Osborn, Lenoir & Robillard, King, Radzan & Kuo…), which are also related to gas velocity by way of mass flux. The

In small-scale metal-laden high solids loading propellant systems (as we in hobby rocketry build and fly), heat transfer effects seem to play a larger role in accelerating the burn than compressibility does, as simulations of pure grain length-induced erosivity achieve the greatest success using nonlinear methods. My gut feeling says the reasons for this are twofold. First, the physical properties of typical EX propellant allow it to withstand minor surface disturbance. Secondly, most EXers tend to follow very conservative design rules, ensuring the Mach number throughout the core of the motor is far below 1.

John DeMar’s simulations (and test data) shown above are based on the Lenoir-Robillard model. From Humble, Henry, and Larson:

Probably the best-known correlation for this effect is that of Lenoir and Robillard:

   

where

   

and

   

are new constants which must be determined experimentally, and

   

is the length of the grain. In addition, the parameter

   

is the bore mass flux (kg/m^2-s), and the 0.8 exponent comes from the effects of convective heat transfer. We consider the Lenoir-Robillard model as an erosion based on mass flux.

So that works. However, one of my strange long-term obsessions is the double-taper grain geometry. Ideally, the geometry is based such that the flow chokes in the core on its way to the nozzle to generate a large pressure (read: thrust) spike at startup. My concern is choking the flow for too long and ultimately blowing up the motor. Any simulations require a good erosive burning model, but I hesitate to jump straight to the Lenoir-Robillard equation, because it seems that compressibility is suddenly very important to the motor’s regression profile. (This is the part where I do more static testing so I don’t force my foot too far into my mouth.) Perhaps an appropriate course of action would be to modify Lenoir-Robillard with the aforementioned linear model proposed by Green, into something like

which means we now have THREE constants to characterize. Nifty.

I might just go out, cast up a double taper motor, and fire it instead. If the motor works, it’ll help me figure out whether this is a reasonable train of thought. If the motor blows up, well at least I’ll have a little bit of pressure data to see how things were beginning to shape up!

Let’s start with mass flux. Mass flux (

) is simply a measurement of flow over an area at a certain velocity, and has units of mass per time-area. In SI form, this turns out to be

Conveniently, this has the same units as, and is in fact equal to

where

is the density of the stuff, and

is the stuff’s velocity.

In a rocket motor, there is equal pressure throughout the motor, and as the combustion products can be considered relatively homogeneous, it is a reasonable approximation to consider

to be constant. Thus it seems that

is essentially directly proportional to flow velocity

, assuming non-choked conditions.

Perusing a gas dynamics book one day (like all good children of science?), I happened upon our good old friend, Vielle’s Law. Except it was weird. There was something else involved. Zucrow and Hoffman had written it this way:

At first glance, it’s much different from the one I learned from McCreary and Wickman, but really it’s the same thing with an extra constant on the front —

— a constant relating to velocity.

Light bulb goes on. That’s an erosive burning modifier.

Using simple 1-D fluid models, it makes terrific sense. Thus it seems that the addition of Oxamide (or any other burn rate modifier) isn’t just tampering with just the values for

and

— it’s messing with the erosive constant

as well. Man, this is worse than Robert Goulet in that Corn Nuts commercial. Looks like you’ll need a few more test burns, Scott!

After this delightful epiphany, I then continued to ponder. Why is mass flux such a big deal, rather than flow velocity? I mean, it’s not like we can directly measure either one in an operating motor — why pick mass flux to be the anointed child?

Well, for now, I’ve settled on the conclusion that it’s easier to simulate mass flux than it is to simulate flow velocity. Total mass flux can easily be calculated using a simple surface area model, while local flow velocity (note the word local — the equation requires the flow velocity at the propellant element under consideration to solve the new erosive rate) requires many assumptions about path, temperature, and upstream behavior of the gas. Numerical flow code like FLUENT can probably do it. I can’t (easily).

So mass flux is a decent method by which to figure out what the overall picture of the operating motor is, erosive-ly speaking. It also allows the configuration of unique grain designs (double-taper, anyone?) by solving the mass flux at the internal choke point(s), allowing the designer to figure out how fast the odd points will wear away, and perhaps give some insight as to what’s going to happen when the motor comes on. And mass flux seems to have a nice, solid basis in the world of fluid dynamics. But in the end, it’s still just a model. Testing of each unique design is still required.

Darn. Another excuse to fire more rocket motors.