Redrocketman's Blog

This article from Maurizio may be of some help...

One of the things that has always puzzled me about black powder rocket makers, was their trial and error approach to designing a spindle for a core-burning rocket. A very well known and respected (by myself also) maker of black powder rocket spindles once confided in me about an instance in which he and another pyrotechnics master were trying different spindle length and nozzle throat combinations for a specific fuel they had developed. The trial and error they went through might have been avoided, or at least reduced, if they’d had a concrete way of quantifying such parameters. Well, I bring glad tidings; such a way exists.

High powered rocket motor manufacturers use a very important variable dubbed “Kn” in the manufacture of their rocket motors. Kn is a very simple, but critical value, and it represents the ratio of two very important surface areas; Kn = (surface area of burning fuel, divided by the rocket nozzle throat area). Kn becomes a very important value in determining a rocket motor’s internal working pressure, and subsequently, propellant burn rate, thrust, and even exhaust temperature. In order to determine Kn, we must obviously be able to calculate the surface area of the burning fuel, to a reasonable approximation (you usually won’t have to be accurate to sub-millimeter dimensions). As this area changes in step with the burning of fuel, it may be convenient to calculate a minimum, and a maximum value for Kn. To calculate the minimum Kn, at the start of burning, we can use the average diameter of the spindle, assuming that it has a reasonably linear taper to it, and to calculate maximum Kn, we will use the inside diameter of the rocket motor tube. The equation to calculate a reasonable approximation of the surface area of the burning propellant assumes that this area is in the form of a burning cylinder with only one side capped; Surface area at start = (Pi * the average spindle diameter * Core length) + 2 * Pi * (one half the spindle’s tip diameter, squared). This value is then divided by the nozzle throat area, which is; Nozzle Throat Area = Pi * (one half the throat diameter, squared). Surface area at the end of the burn, yielding the maximum Kn, will have the exact same formulas, but slightly different variables; Surface area at end = (Pi * inside motor tube diameter * Core Length) + 2 * Pi * (one half the motor tube’s inside diameter, squared).

Let’s do the following example:

Let’s say we have a motor with an internal tube diameter of 1.5 inches, a nozzle throat diameter of .625 inches, and a spindle length of 14 inches. The spindle’s average diameter is .469 inches (the average of its maximum and minimum diameters), and the tip of the spindle is machined into a round tip, rather than a flat one. First, we calculate the throat area, as this does not change appreciably – unless the nozzle is made of marginal materials, or is specifically designed to erode; Throat area = 3.141 * (.3125)2 = .307 square inches. Next, starting (minimum) burning area: Amin = (3.141 * .469 * 14) + (2 * 3.141 * (.3125)2), = 20.63 square inches, plus .61 square inches, for a total of 21.24 square inches. We can now calculate the starting, or minimum, Kn;
Min Kn = 21.24/.307 = 69.2

We now calculate the maximum burning surface area, and thus the maximum Kn ;
Amax = (3.141 * 1.5 * 14) + (2 * 3.141 * (.75)2) = 66 square inches, plus 3.5 square inches, for a total of 69.5 square inches. We can now calculate the final, or maximum Kn:
Max Kn = 69.2/.307 = 225.4.

So, here we are; we’ve come up with these two values, but what good are they for us? Is there any way they can help us in the real world? Fortunately, the answer is a resounding “YES”. A specific fuel can be used in many different sized rockets, as long as the Kn value is respected (with the exception of very TINY motors). A rocket maker can adjust a very wide range of diameters, spindle length needs, and thrust needs for any fuel he wishes, so long as he knows the Kn range in which that fuel operates. If one knows that a particular formula worked well in, say, a 1” diameter rocket with a spindle length and diameter that gave a Kn of 125, he can design his spindle, throat, and tube to adhere to this criteria, for any size rocket motor he wishes to make. In the case of a nozzleless rocket, recently covered in an article by Skylighter, one uses the inside diameter of the fuel grain for both the nozzle diameter, and the spindle diameter. For an end burner, one uses the following values: Amin = Pi * (one half the motor tube’s inside diameter, squared), and Amax = 2 * Pi * (one half the motor tube’s inside diameter, squared). Maximum and minimum Kn will then be solved accordingly. As a side note, BP works very poorly in end burners with a Kn of less than 35 – 40, even in a very fine grain size. At minimum Kn = 45, things start picking up, and by min Kn = 55, performance becomes acceptable. I have seen end burner kits marketed by several companies with Kns in the low to middle 20s, and the people who try to make end burner rockets with them, tend to be very disappointed, unless they resort to adding exotic, or very high energy fuels to the BP. Another note about Kn involves the use of very small nozzle throat sizes. While using a very small nozzle will, in fact, raise the Kn and the motor’s internal pressure to high levels, such designs are best avoided because smaller nozzles are far less efficient than their larger brothers, mainly owing to the turbulence created when the hot exhaust gases go supersonic through the throat. Further, in the internal pressure versus thrust balance, a nozzle throat area is reduced by the inverse square of the ratio of a smaller vs. a larger nozzle, reducing thrust accordingly, while increasing internal pressure, and efficiency, only in a linear fashion. Hence the preference for larger nozzle throats. Last but not least, if one is making a spindle from scratch, by all means, don’t omit the expansion bell at the base of the spindle! Popularly known as the “15-degree taper”, this controlled expansion of the motor’s exhaust gasses improves thrust significantly, especially in motors with high internal pressures, where the proper shaping of an expansion bell can increase motor impulse by as much as 20%!!!!

Thanks heaps. That's is one of the best explanations of a topic I have read for a long time