**THE PROPELLANT:NOZZLE AREA RATIO**

**
A Practical Guide to K _{n} **

John S. DeMar

john_demar@hotmail.com

Feb. 2, 2007

You’ve probably come across the term “Kn” all the time in amateur/experimental rocket motor design.

*What does Kn mean? *

*How
is it calculated? *

*What
does it affect? *

*How
do I control Kn when designing motors? *

*What other factors can
affect Kn? *

The following is my attempt to answer these questions. Please follow up with any additions, corrections, or further questions.

**What does Kn mean?**

* The definition of Kn is the ratio of the
burn area of the propellant to the area of the nozzle throat. *

The abbreviation comes from a German phrase. It doesn't really stand for anything easy to remember in English. So, don't get hung up on that! Just pronounce it as “Kay En”, and everyone will know what you’re talking about. (Which reminds me, whatever happened to the old stereotype that all Rocket Scientists had to have a German accent?)

**How is Kn calculated?**

Add up all the
exposed (uninhibited) surfaces of the propellant to get the burn area (the
inside core area, plus the top & bottom faces of all the grains, for
example). Call this area **A**_{B} for burn area. Then, find the area of the nozzle's
throat: **A**_{T} = ¼ pi * D_{T}^{2} where D_{T}
is the throat diameter. If it has
multiple throat openings, add up all the throat areas.

**Kn = A**_{B}** / A**_{T}
Make sure you use the same units for both area calculations.

Keep in mind that Kn
is an instantaneous, time-varying value ... it is continually changing as the
propellant burns. Depending on the grain
geometry, the Kn may increase or decrease (or both) during the total burn time
of the motor. The ** intial Kn** is important
because it affects how easily the motor will ignite. The

Here’s a simple Bates grain Kn Calulator program. If you prefer a spreadsheet that plots the Kn curve, try this one by Jimmy Yawn.

**What does Kn affect?
**

The pressure in
the combustion chamber at any point in time is directly related to the Kn. As the Kn increases, the chamber pressure
increases. This relationship is not
linear – doubling the Kn does not double the pressure. The burn rate exponent is the fundamental
reason for this nonlinearity, typically producing *three times* the chamber pressure when the Kn is *doubled*.

During the start-up
phase of the motor burn, just after ignition, a sufficient initial Kn provides
a transition to the equilibrium phase of combustion. If the initial Kn is too low, the motor may
not reach steady-state burning. As the
motor begins to come up to pressure, the combustion gases begin to flow. When there is gas flow there is reduced
pressure. If the pressure drop is high
enough, the motor will cease to ignite (a *“chuff”*). If there is sufficient residual heat and
pressure, the motor will ignite again and continue *chuffing* until the Kn increases enough to transition to steady
combustion. Here’s a video of classic
chuffing (and a good lesson in motor retention!).

As a practical guideline, an initial Kn of 220 is sufficient to produce reliable ignition. If the propellant is more energetic (finer AP or catalyzed), a lower initial Kn (180 to 200) could be enough. If the propellant has low-energy additives (fuel rich or low-energy effects chemicals), a higher Kn (240 or higher) may be required.

**How do I control Kn when designing motors? **

Choosing the *grain geometry* is the primary way to
control the overall Kn curve for a motor design. The way the surface area of the propellant
changes as the burn progresses is how the Kn curve will be shaped. Also, the nozzle throat diameter will scale
the overall Kn curve inversely proportional to the *square* of the throat diameter (~1/ D_{T}^{2}). For example, doubling the nozzle throat
diameter will scale the Kn curve to ¼ its magnitude. Again, programs such as Burnsim are great tools to experiment with
to learn the affects of varying the grain lengths, core diameters, and nozzle
throat size. There is no need to guess
these parameters, then fire the motor, and destroy an expensive casing!

Although the topic
of grain geometry is another long discussion, let’s look at what happens when
varying the dimensions of Bates grains (multiple cylinders with central core). As a nominal starting point, choose the
length of the grain for an equal initial Kn and final Kn (symmetrical burn,
assuming non-erosive nozzle and no erosive burning). Here’s the equation: L = (3D_{O} + D_{i })/2,
where L is the grain length, D_{O} is the outside diameter of the
grain, and D_{i} is the inside core diameter. Decreasing the length of the grain will
produce a regressive curve; increasing the length of the grain will cause a
progressive curve. In both cases, the
difference from the initial Kn to the max Kn will be greater. It is easy to see how there may be advantages
and disadvantages in taking either approach.
A regressive curve could have good start-up characteristics, high thrust
to get the rocket off the pad, followed by a longer ‘tail’ with lower
thrust. A progressive curve could have a
low initial mass flux to prevent erosive burning, then accelerate the rocket as
it gets further into the burn. However,
the higher peak pressure could stress the case, or the lower average pressure
could reduce the efficiency of the motor (reduced delivered I_{SP}). The same affects may be analyzed by looking
at variations in the core diameter of the grains. A larger core diameter will flatten the Kn curve
(and therefore the pressure and thrust curves); a smaller core will begin with
a lower Kn and have a more pronounced “hump” to the curve. The disadvantage of the larger core is
reduced propellant mass (low volume loading), reduced burn time, and lower
total impulse. The smaller cores may not
provide high enough initial Kn for reliable ignition, and may also produce a
pressure peak beyond the safe operating conditions for the motor casing. Try some “what if” simulations to see these
affects.

More advanced grain
geometries (or at least non-conventional ones) may be used to tailor thrust
curves. The Kn calculations for some of
these can get quite complicated. A *star grain*, for example, has a high
initial Kn from all the peaks and valleys, but then the Kn decreases as the
grain core becomes more cylindrical. If
a star grain is quite long, the curve will again increase in Kn toward the end
of the burn. The combinations of grain
configurations are seemingly endless, as long as all the practical limitations
are considered (especially erosive burning and physical support of the grain
under acceleration).

Once you are satisfied with a general shape of the curve, the Kn curve is easily shifted by adjusting the nozzle throat diameter. When testing a motor design in the test stand, it’s a good idea to keep all other factors constant while testing at two or three different Kn’s. This requires having various sized nozzles available. An increase of 10% in the throat diameter will decrease the Kn by about 17%. A decrease of 10% in the throat diameter will increase the Kn by about 23%. As you can see, it doesn’t take much to affect the Kn, which in turn will change the pressure by a higher percentage.

**What other factors can affect Kn?**

** Erosive Nozzle:**

If the nozzle throat is made from erosive material (plastic, clay, etc.) the throat area will increase during the burn, effectively decreasing the Kn as compared to a non-erosive nozzle material. Aerotech motors have grain dimensions that allow for the erosive one-time use nozzles. A flat burn with an erosive nozzle will become a progressive burn with a non-erosive (graphite) nozzle. This will give a very high pressure (and possible failure) if the plastic nozzle is replaced with a graphite nozzle without adjusting the grain geometry.

** Inhibited Grain Ends:**

If some of the grain surfaces are kept from burning early on as planned, the Kn profile may change drastically. This is a common problem when the ends of Bates grains are packed tightly against each other at the beginning of the burn. The result is not intuitive! Give it a try in Burnsim by inhibited one end of each grain. You should see a lower initial Kn and a much higher max Kn, with an associated high peak chamber pressure. This is a common failure mode in solid rocket motors. One solution is to put grain spacers between the grains. This may be done using thin o-rings of the same diameter as the grains (the o-rings only need to be effective during the first few tenths of a second of the burn). Another solution is to shape the ends of the grains with concave surfaces instead of flat ends. Either way, it allows complete ignition of all surfaces. Otherwise, the ends will burn slower than the core, causing a larger internal surface area inside the core as the motor burns. This increases the max Kn and peak pressure later in the burn, possibly over-pressurizing the case.

**Kn isn’t everything:**

Using the same grain dimensions and nozzle throat size (same initial and peak Kn) with different propellants is a fairly common practice. It’s very convenient to have a range of effects and burn rates without changing the hardware design. Keep in mind that the faster propellants at the same Kn will have a higher chamber pressure. Combine this with a hotter-burning propellant and insufficient liner/closure seals, it’s possible to bulge the casing near the closure. Or, if extreme, the casing may shear at the hot spot. If in doubt, reduce the Kn with untested formulas that could run ‘hot’ or may have voids.