After more research, discussion, and testing, the linear erosive model (see this post for more info) doesn’t seem to hold 100% true. Sometimes. This post is more of me thinking out loud about the subject in general, and attempts to tie up most of the theories into a neat, easily accessible post. So, here goes.

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!

Tags: Double-Taper, erosive burning, Mass Flux, theory

This entry was posted on Monday, December 22nd, 2008 at 1:52 am and is filed under Propulsion, Theory. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.