Solid Rocket Stability

SOLID ROCKET COMBUSTION STABILITY1

Section 12.5.3 of Theory of Aerospace Propulsion, by Pasquale Sforza, discusses the burning stability of solid rocket motors, and demonstrates in a qualitative fashion that the pressure exponent, n, should be less than unity for a stable motor. This document arrives at this criterion by a more mathematically rigorous means. Begin with the surface regression rate:

n 1 r˙  apc

Again remember that this is an empirical rule. There is no straightforward way to derive it from first principles. The physical processes involved are extremely complex, and include heat transfer to the grain by conduction, convection and radiation, decomposition of the solid, mixing and finally combustion in the gas phase. First assume that the rocket is operating in steady-state:

p A* 2 m˙  r˙A   c g b p c*

Substituting in the regression rate from equation 1, we have:

p A* 3 apn A   c c b p c*

In this expression, Ab is the burning area, which is defined as the surface area of the grain that is regressing. Solving for the chamber pressure, pc, gives:

A 4 p1n  ac* b c p A*

This equation says that given the propellant properties we can compute the chamber pressure, and the mass flow of the propellant. Combined with a nozzle model, this enables us to compute the thrust of the rocket, as well as its specific impulse. Equation 4 applies if the operating point is stable, that is if the pressure and mass flow are steady in time. If the parameters, a, n, c*, and p, are we defined, we can get pc from it. Note however, that pc depends on Ab, and that it can be sensitive to unintended variations in grain configuration. Cracking of the grain, or its de-bonding from the case can lead to increases in burning area, very high pc and an explosion. This is probably the most frequent cause of solid rocket failures.

Even supposing Ab is as intended, there is still the question of whether the operating point is stable. We have assumed that there is steady pc. To determine whether this is a valid assumption, we may first construct a model that allows unsteady behavior of pc. In first

1 approximation, unsteadiness implies a variation in the mass stored in the chamber, which modifies the mass balance above, which was used to determine pc.

d p A* 5.1  V   r˙A   c dt c c b p c* d p A* 5.2  V   apn A   c dt c c c b p c*

d dVc 1 dpc 5.3 cVc   c Vc dt dt RTc dt

Here, we also make the assumption of constant chamber temperature.

d 1 dpc 5.4 cVc   cr˙Ab Vc dt RTc dt

d n 1 dpc 5.5 cVc   capc Ab Vc dt RTc dt

Note that in the above expressions, all quantities with the subscript ‘c’ refer to chamber conditions, and V is the volume of the chamber. We can compare the relative size of these two terms. Their ratio is

 apn A apn A 6.1 c c b  c b Vc dpc 1 dpc Vc RTc dt pc dt

This is a ratio of two time scales. If we take:

V 6.2 L  c Ab

We take this as the characteristic length of the chamber, we have:

n 6.3 apc 1 dp L c pc dt

In physical terms, this is a pressure change time divided by a burning time. This ratio is much, much less than unity. We can drop the first term and write the approximation:

* 6.4 Vc dpc n pc A  apc Ab  p  * RTc dt c

2 Now suppose the chamber pressure consists of a steady part plus a part that varies in time:

pc  pco pc t 7.1

We assume make the assumption:

pc  pco 7.2

Now we can write:

* 8.1 Vc dpc n pco pc A  apco pc  Ab  p  * RTc dt c n V dp  p  p A*  p  8.2 c c  apn 1 c  A   co 1 c  co   b p *   RTc dt  pco  c  pco  V dp  p  p A*  p  8.3 c c  apn 1 n c A   co 1 c  co   b p *   RTc dt  pco  c  pco  V dp  p  p A*  p  8.4 c c   apn A n c   co  c  p co b   *   RTc dt  pco  c  pco  * 8.5 Vc dpc  n1 A   apco Abn p  * pc RTc dt  c  * 8.6 1 dpc RTc  n1 A   apco Abn p  *  pc dt Vc  c 

We have divergence is the quantity in brackets is greater than zero, and stability if the quantity is less than zero. We can simplify the expression further to write:

A* aA n A* 9.1 apn1 A n    b  co b p * p A * c ac* b c p A* A* A* 9.2 apn1 A n   n 1 co b p c* c* * 9.3 1 dpc RTc A  * n 1 pc dt Vc c

Now we can see, we have stability if n is less than unity, and instability if n is greater than unity. The time scale for growth is approximately:

3 * 10.1 Vcc * RTc A

We can also write this in terms of the characteristic length, L, as:

Lc* 10.2

RTc

For typical values, this is about 1 millisecond, which is much shorter as compared to rocket burning times, so we can conclude that if n is less than unity the chamber pressure will relax from perturbation in a time on the order of 1 millisecond. Conversely is n is greater than unity, the rocket will explode in about the same time.