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.