1. Study, Development and Application of Solid
Rocket Balistic Models
Graduate
Filippo Facciani
Supervisor
Prof. Ing. Fabrizio Ponti
Co-supervisor
Ing. Roberto Bertacin

2. 1. Introduction: Solid Rocket Motors (SRM)
Solid Rocket Motor:
Propulsion system based on the generation of
thrust from the conversion of Enthalpic Energy to
Kinetic Energy
Igniter
Grain
Components:
• Igniter
• Propellant Grane
• Case
• Thermic Protection
• Nozzle
Case and
Thermal
Protections
𝓕 = 𝑷 𝟎 𝑨⋆ 𝑪 𝓕
𝐶ℱ is specifically related to the Nozzle
and gives reason of its performance
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Nozzle
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3. 1. Introduction: Internal Balistic
Internal Balistic: Subject act to study the development of the ducted flow internal to the SRM
Combustion Chamber (CC)
CC Gas Mixture:
• Inert filling gases
Mass Addition:
• Combustion hot gases
• Ablation gases
• Igniter gases
Mass Subtraction:
• Gases leaving the nozzle
Geometric Parameters:
• CC Volume
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4. 1. Introduction: Internal Balistic Phases
The operative life of an SRM can be devided in:
• Ignition transient
• Quasi steady state
• Tail off transient
Quasi Steady State:
• Igniter is off
• Ablation of Thermic Protections is negligible
Influencing Parameters:
• Combustion gases hot flow
• Nozzle flow
𝒎𝒈 = 𝝆𝒑 𝑺𝒃 𝒓𝒃
Courtesy of “Modeling and Numerical Simulation of Solid Rocket
Motors Internal Ballistics”, Enrico Cavallini
𝒏
Combustion Ratio: 𝒓 𝒃 = 𝒂𝑷 𝟎 + 𝒓 𝒃𝒆
𝑷 𝟎 𝑨⋆
𝒓 𝒃𝒆 = 𝒇 𝒖
𝒎𝒏=
𝑪⋆
The Combustion Surface development in time determines the Combustion Gases Mass Flow
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5. 2. Scope: Deisgn and Realization of a Combustion Simulator
Scopo:
Realize an SRM Combustion Simulator able to break through the current limits
Key Parameters:
 CC Pressure
 Axial Velocity
 Combustion Surface
Fluid dynamic
Geometric
Stato dell’arte:
Balistic Models
• 0-D: parameters are averaged in space and
function of time
• 1-D Stationary: parameters are function of
the axial position only
• 1-D non-Stationary: parameters are
function of both the axial position and time
Combustion Surface Regression Models
• Analytic
• Based on Simmetry or Periodicity
Isotropic
Current Limits: Isotropy forbids the use of
anisotropic inputs from sofisticaded Balistic
Models
Solution: develop of two cross-linked models, and Internal Balistic Model and a Regression
one, interdependent and able to work with Anisotropic geometries
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6. 3. Simulator Map
Input/Output:
• Burn Rate
• Mesh Superficiale
Grain Configuration
from CAD modelling
Surface Mesh
Generation
Amplification
Factors
Combustion Chamber
Fluid Dynamicss
Time
Step
Burn Rate
Distribution
Grain Surface
Regression
Ballistic Models
0-D + 1-D
Surface Remeshing
Procedures
Igniter
Nozzle Dynamics
Stability Control
Graphical
Visualization
Thermal Protections
Updated Surface
Mesh
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7. 3. 0-D Unsteady Balistic Model
Use:
• L𝐨𝐰 𝑳 𝑫 𝒉 ratios
• 𝑨 𝒑 𝑨⋆ > 𝟒
geometries where 𝒓 𝒃𝒆 can be neglected
Ipothesis:
•
•
•
•
•
•
Fluid dynamic parameters are function of time only
Ideal gasses
Heat flux through the propellant grain is negligible
No chemical reactions within the control volume
Inviscid Fluid
Subsonic Flux
𝑑𝑝
𝑅
=
𝑄
+
𝑑𝑡
𝑐 𝑣 𝑉 𝑙𝑜𝑠𝑠
Continuity Equation
Energy Equation
𝑗
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𝑚𝑗ℎ𝑗 −
𝑗
𝑐 𝑣 𝑝 𝑑𝑉
𝑅 𝑑𝑡
𝑑𝑇
1
𝑑𝑝
𝑑𝑉
𝑝
=
𝑉
+ 𝑝
−
𝑚 𝑎𝑐𝑐 + 𝑚 𝑔 + 𝑚 𝑃𝑇 − 𝑚 𝑛
𝑑𝑡
𝜌𝑉𝑅
𝑑𝑡
𝑑𝑡
𝜌
𝑢2
𝑎𝑐𝑐
𝑚 𝑗 ℎ 𝑗 = 𝑚 𝑎𝑐𝑐
+ 𝑐 𝑝 𝑎𝑐𝑐 𝑇 𝑎𝑐𝑐 + 𝑚 𝑔 𝑐 𝑝 𝑔 𝑇 𝑔 + 𝑚 𝑃𝑇 𝑐 𝑝 𝑃𝑇 𝑇 𝑃𝑇 − 𝑚 𝑛 𝑐 𝑝 𝑛 𝑇 𝑛
2
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8. 3. 0-D Balistic Model: Inputs
From the Regression Model:
𝒎𝒈 = 𝝆𝒑
𝒅𝑽
,
𝒅𝒕
𝑺 𝑷𝑻
𝒅𝑽
𝒅𝒕
𝒎 𝑷𝑻 = 𝝆 𝑷𝑻 𝒓 𝒂𝒃𝒍 𝑺 𝑷𝑻
𝑸 𝒍𝒐𝒔𝒔 = 𝒉 𝒆𝒒 𝑺 𝑷𝑻 𝑻 𝒈𝒂𝒔 − 𝑻 𝒔𝒖𝒑 𝑷𝑻
𝑺 𝑷𝑻 is calculated through analysis of the intersection
between the radius of the Combustion Surface and
the Case profile
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9. 3. 0-D Balistic Model: Application to BARIAs
initial Geometry:
Analytical Regression:
Balistic Prediction:
• Phase of Interest: Quasi-Steady-State
Good Match
• Tail Off: discrepancy due to the Nozzle
Physical Model. Such a model just
describes sinic conditions.
• Qualitative Trend: optimal match with
the expected trend .
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10. 3. 0-D Balistic Model: Zefiro 9
Data Provided by AVIO (Sponsor):
• Igniter properties
• 𝐷⋆ (𝑡) trend (experimental)
• p(𝑡) trend (experimental)
• HUMP e Scale Factor corrective factors
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Dati Found in Literature
• Therma Protection Characteristics
Data Calculated from the Mesh
• 𝑟 every section, in order to calculate
intersections with the Case.
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11. 3. 0-D Balistic Model: Zefiro 9
Geometry:
• Overall results satisfactory
• Considerations: the simulation was carried on using an
isotropic approach. Therefore, anisotropies in the cobustion
velocity direction have been considered using an HUMP
factor
Errore ~ 4%
• Two deviations from the reference curve:
Regression:
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12. 3. 0-D Balistic Model: Zefiro 9
Geometry:
Error due to remeshing
Regression:
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13. 3. 0-D Balistic Model: Zefiro 9
Geometry:
Deviation in the final part of the
Steaty State phase due to the lack of
knowledge about the Thermal
Protections
Regression:
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14. 4. 1-D Non-Stationary Balistic Model
Use:
geometries where 𝒓 𝒃𝒆 can not be neglected
• H𝐢𝐠𝐡 𝑳 𝑫 𝒉 ratios
• 𝑨 𝒑 𝑨⋆ < 𝟒
Ip:
•
•
•
•
•
•
•
Properties of the gas mixture are uniform in a given motor section
Velocity components normals to the motor axis are neglectable
Inviscid ideal fluids
The only thermic flux is through exposed PT surfaces
No chemical reactions inside the Control Volume
Subsonic flux
No abrupt discontinuities in combustion chamber geometry
Continuity Equation
Momentum Equation
𝜕(𝜌𝐴 𝑝 ) 𝜕(𝜌𝑢𝐴 𝑝 )
𝑚 𝑎𝑐𝑐 𝐴 𝑝
𝑚 𝑃𝑇 𝐴 𝑝
+
= 𝑟𝑏 𝑃𝑏 𝜌 𝑝 +
+
𝜕𝑡
𝜕𝑥
𝑉
𝑉
3
𝜕(𝜌𝑢𝐴 𝑝 ) 𝜕[(𝜌𝑢2 + 𝑝)𝐴 𝑝 ]
𝜕𝐴 𝑝
𝑚 𝑎𝑐𝑐 𝑢 𝑎𝑐𝑐 𝐴 𝑝
+
= 𝑝
+
−
𝜏 𝑤 𝑖 𝑃𝑖
𝜕𝑡
𝜕𝑥
𝜕𝑥
𝑉
𝑖=1
Energy Equation
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𝜕(𝜌𝐸𝐴 𝑝 ) 𝜕[ 𝜌𝐸 + 𝑝 𝑢𝐴 𝑝 ]
𝑚 𝑎𝑐𝑐 ℎ 𝑎𝑐𝑐 𝐴 𝑝
𝑚 𝑃𝑇 ℎ 𝑃𝑇 𝐴 𝑝
+
=
+
𝜕𝑡
𝜕𝑥
𝑉
𝑉
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15. 4. 1-D Balistic Model: Inputs
From the Triangular Mesh:
• 𝒓 𝒎𝒆𝒂𝒏 𝑨 𝒑 𝑷 𝒃 𝑽
• 𝑺 𝒃 𝑺 𝑷𝑻
• To calculate the mean radius the sart shape
is approximated to the circonference of
equivalent area. When the Case surface is
exposed, 𝑟 𝑚𝑒𝑎𝑛 will be equivalent to the
Case radius
•
𝑺 𝑷𝑻 will be calculated the same way as per
the 0-D case
•
𝑺 𝒃 is found by redistributing on the
calculation nodes of 1/3 of the triangular
elements adjacent to every vertex assigned
to every specific node
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16. 4. 1-D Balistic Model: Cilindrical Geometry
Geometry:
• Reference curve: results from the 0-D
model
• Analytic regression
Thermic Protection
• Small size motors
Ablation is
• Short combustion time
neglegible
C𝐨𝐦𝐩𝐚𝐫𝐢𝐬𝐨𝐧 𝐨𝐟 𝟒 𝐭𝐞𝐬𝐭𝐬 𝐰𝐢𝐭𝐡 𝐯𝐚𝐫𝐢𝐚𝐛𝐥𝐞 𝜟𝒙
Qualitative trend in agreement with
Expectations
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Quantitative values converging to the 0-D
model ones for 𝜟𝒙 = 0.01L
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17. 4. 1-D Balistic Model: Star-Aft Geometry
Reference values: results from the 0-D model
Inputs: geometry, nozzle throat diameter
evolution in time.
The match between the 1-D and the 0-D
model is not good.
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18. 4. 1-D Balistic Model: Star-Aft Geometry
Complexive trend: the blue line trend agrees
with the one of a cilindric geometry.
Cause:
the
star-shaped
section
are
reconducted to geometric shapes with
equivalent area
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19. 4. 1-D Balistic Model: Star-Aft Geometry
Complexive trend: the blue line trend agrees
with the one of a cilindric geometry.
Cause:
the
star-shaped
section
are
reconducted to geometric shapes with
equivalent area
Use of circular sections determines an
underestimation of 𝑷 𝒃 in the first part of the
Quasi Steady State, an overextimation in the
second one.
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20. 4. 1-D Balistic Model: Star-Aft Geometry
Effect of the remeshing:
• 1-D response is delayed
compared to the 0-D
• Late and incomplete damping
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21. 4. 1-D Balistic Model: Star-Aft Geometry
Effect of the remeshing:
• 1-D response is delayed
compared to the 0-D
• Late and incomplete damping
Cause: the damping factor is artificial and
embedded withing the MacCormack integration
method.
The artificial viscosity is triggered by the
pressure gradient
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22. 4. 1-D Balistic Model: Star-Aft Geometry
Effect of the remeshing:
• 1-D response is delayed
compared to the 0-D
• Late and incomplete damping
Cause: the damping factor is artificial and
embedded withing the MacCormack integration
method.
The artificial viscosity is triggered by the
pressure gradient
The damping factor is not reacting to the geometric
perturbations, but only to the pressure gradients induced
by it.
When these gradiants become low again, the damping
ends indipendently from the permanence of geometric
stimuli.
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23. 5. Conclusions and Future Developments
0-D model:
• Results are generally satisfactory
• Influence of remeshing is localised
Solutions:
• Improve Remeshing techniques
• Higher Triangular Mesh density
• Filtering of the numeric noise introduced by the
geometric parameters.
1-D non-Stationary Model:
• Good results with analytic geometries
• Results are not good with complex geometries due to the interface and the dynamic
behaviour.
Solutions:
1. Develop of algorithms to reorder the section’s point cloud: this will allow to avoid errors
introduced by evaluating geometric parameters using equivalent circular shapes.
2. Modify the Damping term in order to have it triggered directly from the geometric
perturbations introduced by the remeshing
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24. 5. Conclusions and Future Developments
Observations:
The 1-D non-stationary model highlighted unexpected consequences of using the Anisotropic
Regression Model: the effect of the geometric noise on the model were not expected, nor met
in literature.
They can be bypassed developing an hybrid model, mixin the 0-D and a 1-D Stationary model.
This model will get advantage form the 0-D fast response and the capability of the 1-D model to
calculate axial distribution along the motor axis for the relevant parameters.
At every iteration, the results from the 0-D model will initialised the 1-D stationary model.
0-D Model
1-D Stationary Model
𝒓 𝒃 distributions
This will allow to limit the remeshin effects while still being able to achieve distributions for
the fluid dynamic quantities of interest and, therefore, of the combustion ratio
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