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Mass Flux and Erosive Burning
There has been a lot of discussion as of late regarding erosive burning and its relation to mass flux. Many have suggested that propellants with certain additives (notably, Oxamide) have a better response to high mass flux conditions, and have proven with ample results that it in fact does. While the effect these additives have is noticeable, it does not seem to scale linearly with pressure, as I might have expected thanks to the basic interpretation of Vielle’s law. After doing some thinking and some research, I happened upon some useful information on how to characterize the effect mass flux has on erosivity.
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:
— 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.
The density is not constant at the surface where the combustion transition (solid, melt, gas, combustion) is taking place. This is why the classic choked flow m-dot is not the same as mass flux.
There are many models, from simple to wildly complex, to describe erosive burning and related phenomenon such as acceleration affects and resonant burning. The “1+kv” linear approximation isn’t all that good. There are two that I use in my simulations: Lenoir-Robillard(1957)/King(1975) and Kriedler(1964). L-R/K says it is nonlinear and is affected by flow(+), burn rate(-), solid density(-), and length(-). Such as: 1 + Ka V^0.8 exp(-Kb rb rho / V) / L^0.2
Kriedler says there’s a threshold mass flux that depends on pressure, and there’s less erosive affect at higher pressures and higher br exponents. Such as: 1 + (K1 + K2 p)(V-Vth)/p^n.
A good reference is: Razdan and Kuo, 1983, “Erosive Burning of Solid Porpellants”, AIAA.
Here are a couple examples of the non-linear simulation. This one is the L-R/K model which matches the very erosive burn simulation: http://thrustgear.com/MotorSim/bg6gr_38.GIF
This sim is the gradual affect of the Kreidler model which is more useful for mildly erosive propellants: http://thrustgear.com/MotorSim/motorsim_example1.JPG
The only way to find the threshold and constants is to burn several motors with increasing mass flux while trying to hold everything else the same.
-John DeMar
At what point does the gas velocity begin to increase appreciably as you leave the burning surface front? Is it still within the transition region, or beyond that?
I did forget to mention Radzan and Kuo, Renie and Osborn, King, or any of the other myriad of erosive burning papers that have been published in the AIAA journal. Many of the more modern studies rely extensively on numerical simulation techniques (PEM combustion) and assume that the AP/binder gas is premixed before erosivity becomes a problem — so I think the density model still applies.
Granted, I’m not sure of the accuracy of the
model — though the Zucrow and Hoffman text is held in high regard, I’m not aware of anybody firing motors to try and corroborate the equation. Looks like a good static testing exercise 
One other thing I did think about last night as I was falling asleep is the fact (duh) that in a core-burning or BATES model, the mass flux is always highest across the face of the grain closest to the nozzle. I’ve been struggling with modeling mass flux in two dimensions (
as a function of grain length, and 
as a function of burn depth), but this would simplify things — at least for characterization motors. Don’t know why I didn’t bring that up before.
Maybe a fun experiment would be to try and predict the time at which the aft-most grain gets completely consumed and spit during erosive motor operation. Seems simple enough — make a motor with short aft grains, figure out the rate difference between head end and nozzle end flow, and go from there. Hmm.