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### P1

1. 1. Study, Development and Application of Solid Rocket Balistic Models Graduate Filippo Facciani Supervisor Prof. Ing. Fabrizio Ponti Co-supervisor Ing. Roberto Bertacin
2. 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 10/10/2012 Nozzle Study, Development and Application of Solid Rocket Balistic Models 2
3. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 3
4. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 4
5. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 5
6. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 6
7. 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 10/10/2012 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 Study, Development and Application of Solid Rocket Balistic Models 7
8. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 8
9. 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 . 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 9
10. 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 10/10/2012 Dati Found in Literature • Therma Protection Characteristics Data Calculated from the Mesh • 𝑟 every section, in order to calculate intersections with the Case. Study, Development and Application of Solid Rocket Balistic Models 10
11. 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: 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 11
12. 12. 3. 0-D Balistic Model: Zefiro 9 Geometry: Error due to remeshing Regression: 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 12
13. 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: 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 13
14. 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 10/10/2012 𝜕(𝜌𝐸𝐴 𝑝 ) 𝜕[ 𝜌𝐸 + 𝑝 𝑢𝐴 𝑝 ] 𝑚 𝑎𝑐𝑐 ℎ 𝑎𝑐𝑐 𝐴 𝑝 𝑚 𝑃𝑇 ℎ 𝑃𝑇 𝐴 𝑝 + = + 𝜕𝑡 𝜕𝑥 𝑉 𝑉 Study, Development and Application of Solid Rocket Balistic Models 14
15. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 15
16. 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 10/10/2012 Quantitative values converging to the 0-D model ones for 𝜟𝒙 = 0.01L Study, Development and Application of Solid Rocket Balistic Models 16
17. 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. 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 17
18. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 18
19. 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. 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 19
20. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 20
21. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 21
22. 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. 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 22
23. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 23
24. 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 10/10/2012 Study, Development and Application of Solid Rocket Balistic Models 24