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BATES Grain Length Derivations

Using Richard Nakka’s model for surface regression of a BATES grain:

A_{b,i} = N[\frac{1}{2}\pi(D^2-d_o^2)+\pi L_od_o]

A_{b,f} = N\pi D(L_o-2t) where t= \frac{1}{2}(D-d_o) or the initial web thickness.

Then to achieve a neutral burn we want the initial K_n to equal the final K_n so:

N[\frac{1}{2}\pi(D^2-d_o^2)+\pi L_od_o]= N\pi D(L_o-2t)

Substituting in for t and simplifying:

\implies \frac{3D^2}{2}-\frac{d_o^2}{2}+L_od_o=DL_o+d_oD

As Mr. Nakka points out, D is constrained based on motor size and liner thickness and d_o is constrained by erosive burning considerations and the port-throat-ratio. Thus we treat these as constants and solve for L_o, the length of a singular BATES grain which, given fixed core and outer diameters, will be a parameter in determining K_n.

\implies L_o = \frac{\frac{d_o^2}{2}-\frac{3D^2}{2}+d_oD}{d_o-D}

Using some tricky algebra, we rewrite this as

L_o = \frac{(d_o-D)(3D+d_o)}{2}\cdot\frac{1}{d_o-D}

And arrive at the result: L_o = \frac{1}{2}(3D+d_o), giving us the proper length of a single BATES grain in order to achieve a symmetric K_n profile.

Surface Regression Model Used

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