A.8.1.1.3 Propulsion
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A.8.1.1.3 Propulsion
We use the Gaussian Probability Method to account for the uncertainties in our launch vehicle.
To execute this task, each category of design provides standard deviations of various parameters
that have some effect on the size of the launch vehicle. We provide standard deviations of
propellant mass and mass flow rates in liquid engines, hybrid and solid motors. Then we use
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 We want to minimize
the inert mass because it lowers our GLOM.
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.1 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. 8.1.1.3.1. The related equation is of the burning rate
(rb), St. Robert’s Law, shown in Eq. (A.8.1.1.3.1).2
rb  ap cn
(A.8.1.1.3.1)
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.2 Unlike propellant mass, the variation of the burning rate
is difficult to determine because there exists little research that assesses the variations present
Authors: Dana Lattibeaudiere, Stephanie Morris
A.8.1.1.3 Propulsion
2
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. 8.1.1.3.1.
Fig. 8.1.1.3.1 Sample of observed burning rate behavior of solid propellants.
(R. W. Humble, G. N. Henry, W. J. Larson)2
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.8.1.1.3.2) below.2
p 
 ln( rb )
T
(A.8.1.1.3.2)
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.
Note that in most situations, the small ranges in temperature make this effect small, but not
negligible. Eq. (A.8.1.1.3.3) accounts for this small effect.2
Authors: Dana Lattibeaudiere, Stephanie Morris
A.8.1.1.3 Propulsion
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rb  e
( p T )
apcn
(A.8.1.1.3.3)
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.2
Erosion can occur in either of two ways. Erosion can occur because of mass flux shown below in
Eq. (A.8.1.1.3.4), the Lenoir-Robillard model.2
rb  apcn 
G 0.8
0.2
L
  p rb
e
G
(A.8.1.1.3.4)
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.8.1.1.3.5).2
rb  apcn (1  kM )
(A.8.1.1.3.5)
where k represents the empirical constant that addresses the erosive effects. Fig. 8.1.1.3.2 shows
that erosive burning enhances the burning rate.
Fig. 8.1.1.3.2 Pressure-time curve with and without erosive burning.
(George P. Sutton, Oscar Biblarz)3
Authors: Dana Lattibeaudiere, Stephanie Morris
A.8.1.1.3 Propulsion
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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
although ambiguities arise such as bore centerline offset and mandrel misalignment. Using this
technique, manufacturers suggest a burning rate standard deviation of 1% (1σ) which the Monte
Carlo simulation uses.3
In liquid propellant engines, there are many variables in each launch and usage of the engine.
The main variables that can be calculated and are useful for the Monte Carlo simulation are the
mass of the propellant and the mass flow rate of the propellant. Mass flow rate is directly related
to the thrust of the engine is a standard deviation of thrust. The Vanguard satellite launching
vehicle is a very similar vehicle to the launch system that we have created4. The Vanguard report
publishes their scientifically found standard deviations and average values for each of their
stages. Using these numbers, we calculate our percent standard deviation for propellant mass and
mass flow rate with the Eq.( A.8.1.1.3.6).
% Deviation = (Standard Deviation / Average Value) 100%
(A.8.1.1.3.6)
The standard deviation for our vehicle is the product of the nominal value and the percent
deviation. The results for calculating the percent deviations can be seen in Table A.8.1.1.3.1.
Table A.8.1.1.4.1 Liquid Propellant Standard Deviations
Average Value4
Propellant Mass
16,351 kg
Mass Flow Rate
110.7 kg/s
Standard Deviation4 Percent Deviation
120
0.7340 %
0.545
0.4923 %
For hybrid motors, we cannot find any historical standard deviations. In order to still be
conservative with the hybrid propellant standard deviations the two standard deviations for solid
and liquid propellants are combined together to obtain a conservative hybrid standard deviation.
Table A.8.1.1.3.2 shows the data for each propellant and the resulting hybrid percent deviation.
Authors: Dana Lattibeaudiere, Stephanie Morris
A.8.1.1.3 Propulsion
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Table A.8.1.1.3.2 Propellant Percent Deviations
Propellant Mass
Mass Flow Rate
Solid Propellant1
0.12 %
1.0 %
Liquid Propellant4
0.734 %
0.4923 %
Hybrid Propellant
0.854 %
1.4923 %
The standard deviation values are included in the Monte Carlo simulation and used to calculate
the deviation of thrust for the launch vehicle.
References
1. 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.
2. 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. 327-331.
3. Sutton, G., P., Biblarz, O., “Solid Propellant Rocket Fundamentals,” Rocket Propulsion
Elements, 7th ed., Vol. 1, Wiley, New York, NY, 2001, pp. 434.
4. Martin Company, The Vanguard Satellite Launching Vehicle, Engineering Report No.11022.
April 1960, pp. 26, 208.
Authors: Dana Lattibeaudiere, Stephanie Morris