A.8.1.1.4 Non-Catastrophic Failure Propulsion
pg. 1
The D&C group uses the Gaussian Probability Method to account for the uncertainties in our
launch vehicle. To execute this task, each group of the team provides standard deviations of
various parameters that have some effect on the size of the launch vehicle. The propulsion group
provides standard deviations of propellant mass and mass flow rates in liquid engines, hybrid and
solid motors. The D&C group uses these standard deviations to vary the mass flow rate and
propellant mass at the start of each simulation to obtain the correct size of our launch vehicle.
In solid motors, there exist many factors which create variations in several parameters such as
propellant weight, burning rate, density, characteristic velocity and throat area. According to
Professor Heister, in missile applications, limiting variations in ballistic parameters can result in
improved motor case and insulation designs which minimize inert mass.1 Our team wants to
minimize the inert mass because it lowers our GLOW.
Industry standard scales and load cells can determine propellant mass so that the industry can
verify the specification weight ranges of ±0.3% or 3σ. Professor Heister’s paper predicts that
propellant mass values can range from 0.08% to 0.12%. However, because of the small variation
of the parameter, it does not have a huge effect on the burning time dispersions. 2 The Monte
Carlo simulation uses the recommended standard deviation of 0.12%.
To determine the burning rate, manufacturers burn small strands of propellant in small ballistic
test motors with different throat sizes. The chamber pressure varies in these small ballistic test
motors. The plot of ln(rb) versus ln(pc) is an empirical representation of the burning rate behavior
of a propellant shown as curve (a) in Fig. 8.1. The related equation is of the burning rate (rb), St.
Robert’s Law, shown in Eq. (A.8.1.1.4.1).3
rb  ap cn
(A.8.1.1.4.1)
where a represents the burn rate coefficient and n represents the burn rate exponent. However,
the equation above does not address the complex thermochemical and combustion processes that
occur when a propellant actually burns.4 Unlike propellant mass, the variation of the burning rate
is difficult to determine because there exists little research that assesses the variations present
within a batch of propellant tested under constant pressure conditions. According to Humble,
most composite propellants behave as in curves (a) or (d) in Fig. 8.1.
Author: Dana Lattibeaudiere
A.8.1.1.4 Non-Catastrophic Failure Propulsion
pg. 2
Fig. 8.1 Sample of observed burning rate behavior of solid propellants.
(R. W. Humble, G. N. Henry, W. J. Larson)5
Curves (b) and (c) do not apply because we do not use a double-base propellant in our launch
vehicle. Additionally, temperature sensitivity and throat erosion make it difficult to obtain the
burning rate from full-scale firings.
The parameters  p [%/K], measures temperature sensitivity of burn rate as shown in Eq.
(A.8.1.1.4.2) below.6
p 
 ln( rb )
T
(A.8.1.1.4.2)
pcconst
where T represents the temperature of the propellant grain precombustion.
According to Humble, at higher propellant temperatures, the increased internal energy within the
propellant leads to small increases in burning rate as compared to normal temperature
conditions.7 Note that in most situations, the small ranges in temperature which make this effect
small, but not negligible. Eq. (A.8.1.1.4.3) accounts for this small effect.8
rb  e
( p T )
apcn
Author: Dana Lattibeaudiere
(A.8.1.1.4.3)
A.8.1.1.4 Non-Catastrophic Failure Propulsion
pg. 3
where ∆T represents the difference in temperature from the assumed “standard” condition of 15˚C.
Typical values range from 0.001 to 0.009 per degree Kelvin.9
Erosion can occur in either of two ways. Erosive can occur because of mass flux shown below in
Eq. (A.8.1.1.4.4), the Lenoir-Robillard model.10
rb  ap 
n
c
G 0.8
0.2
L
  p rb
e
G
(A.8.1.1.4.4)
where α and β represent experimentally determined constants, L represents the length of the
grain and G represents the bore mass flux (kg/m2s). Compressibility can also cause erosion to
occur where the Mach number (M) influences the burning rate as shown in Eq. (A.8.1.1.4.5).11
rb  apcn (1  kM )
(A.8.1.1.4.5)
where k represents the empirical constant that addresses the erosive effects. Fig. 8.2 shows that
erosive burning enhances the burning rate.
Fig. 8.2 Pressure-time curve with and without erosive burning.
(George P. Sutton, Oscar Biblarz)12
Despite the above factors which affect the burning rate, manufacturers use cured strands of
propellant fired at constant pressure to standardize the burning rate of production batches
Author: Dana Lattibeaudiere
A.8.1.1.4 Non-Catastrophic Failure Propulsion
pg. 4
although ambiguities arise such as bore centerline offset, mandrel misalignment, etc. Using this
technique, manufacturers suggest a burning rate standard deviation of 1% (1σ) which the Monte
Carlo simulation uses.13
References
1-2. Heister, S., D., Davis, R., J., “Predicting Burning Time Variations in Solid Rocket Motors,”
Journal of Propulsion and Power, Vol. 8, No. 3, 1992, pp. 564-565.
3-8., 10-11. Humble, R. W., Henry, G. N., Larson, W. J., “Solid Rocket Motors,” Space
Propulsion Analysis and Design, 1st ed., Vol. 1, McGraw-Hill, New York, NY, 1995, pp. 327331.
12. Sutton, G., P., Biblarz, O., “Solid Propellant Rocket Fundamentals,” Rocket Propulsion
Elements, 7th ed., Vol. 1, Wiley, New York, NY, 2001, pp. 434.
13. Heister, S., D., Davis, R., J., “Predicting Burning Time Variations in Solid Rocket Motors,”
Journal of Propulsion and Power, Vol. 8, No. 3, 1992, pp. 568.
Author: Dana Lattibeaudiere