Monte Carlo Method for Uncertainties Quantification of SRM Internal ...

Monte Carlo Method for Uncertainties Quantification of SRM Internal Ballistics Model Davide Viganò1, Filippo Maggi2 Politecnico di Milano, Milan, I-20156, Italy and Adriano Annovazzi3 AVIO S.p.A., 00034 Colleferro, Italy

Solid rocket motors are systems commonly adopted for launcher missions. Compactness, reliability, readiness, and construction simplicity make the technology very appealing for commercial service. The lack of mission flexibility makes reliability of predictions and reproducibility of performances a primary goal. An analysis of SRM performances uncertainties is offered throughout the implementation of a numerical model of motor internal ballistics based on Shapiro’s equations. The model is then used with the Monte Carlo method to evaluate performance statistics. A comparison with experimental data is presented.

Nomenclature A a ct c* Isp Isv M n p

- area - pre-exponential coefficient of Vieille law - time coefficient - characteristic velocity - specific impulse - volumetric specific impulse - Mach number - exponential coefficient of Vieille law - pressure

rb T t V w x η µ σ

- burning rate - temperature - time - speed - mass flow rate - Cartesian coordinate - efficiencies - mean value - standard deviation

I. Introduction A new era of space exploration is now beginning. Private space companies, aiming for manned flights, are taking place in a sector that was driven by governments for decades. The commercialisation of space is creating easier and cheaper ways to space. The consequent increase in the number of launches makes reliability of launch vehicles a critical factor. In this context solid propulsion still has an active role for launcher applications and embarked systems. It is a proven technology having decades of launches and a Technology Readiness Level of 9. In addition, it can achieve very high thrusts and high volumetric specific impulses1. All propulsion systems are subjected to some level of performance variability from nominal values which can become critical for solid rocket motors. A solid rocket motor is not throttleable, once ignited it cannot be shut down (in a non-destructive way) or re-ignited1,2. Their thrust profile is decided a priori and cannot be changed during flight. Propellants used for space applications are a complex mixture of oxidizer salts, metallic powders and a binder2. Their performance can be sensitive to casting, raw material lot variability or environmental factors. This kind of behaviour imposes strict requirements on reliability. In the aforementioned context a tool was developed to analyse and predict the performance uncertainties for a solid rocket motor. This tool is able to treat statistically all the major sources of uncertainties for a solid rocket motor, and on this basis, to calculate motor performances, deriving their statistical distribution. Due to the complexity of the problem an analytical solution would rely on very simple models of internal

1

MSc Graduate, Aerospace Sciences and Technologies Department, 34 Via La Masa. Assistant Professor, Aerospace Sciences and Technologies Department, 34 Via La Masa. 3 Solid Propulsion Design Department, Via Ariana km 5.2, Associate Member. 2

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ballistics. A numerical modelling of the motor is required to use more sophisticated models. The Monte Carlo method allows to treat statistically numerical problems3.

II. Statistics of Rocket Motor When we usually speak of rocket engine we refer to their nominal parameters, such as thrust, specific impulse, MEOP (Maximum Expected Operative Pressure) and so on. In most of our applications a rocket engine never works at nominal parameters. It works, hopefully, close to them, and statistics helps in defining the bounds of confidence for our predictions. The practical interest for such analysis is very wide, spanning from mission analysis for a single rocket to thrust imbalance analysis for the design of passive and active control systems in case of multiple strap-on boosters (e.g. Ariane V). Before introducing the Monte Carlo method, the distributions that represent input data have to be assumed. For the construction parameters here used (ballistic coefficients, characteristic velocity efficiency, nozzle efficiency, propellant hump, propellant axis offset, and propellant mass) their distribution is known apriori, and these data can be represented with Gaussian distributions3,4. For what concerns output performances, a data analysis has been made using probability papers. An example is reported in Figure 1. The results showed that also performances can be represented with Gaussian distributions.

Figure 1. Example of probability paper, in this case data fits properly a Gaussian distribution. The Monte Carlo method allows to treat numerically a problem that can be impossible to be analysed with analytical statistics3. The method reduces the problem to a “black box” which, in this case, takes as inputs the construction parameters of the Solid Rocket Motor and gives as outputs its performances. The Monte Carlo method samples the parameter distribution with N points which are used as inputs for the model of the Solid Rocket Motor, then, the output data are analysed statistically. In the previous case, only one parameter is treated statistically, while all other construction parameters are kept nominal. To be able to analyse eventual cross effects, multiple input distributions should be take into account. A possible solution is the use of Latin Hypersquares3 (see Figure 2). With this technique m input distributions are sampled with N points, but instead of analysing all Nm possible combinations, each value is taken only once from each distribution reducing to only N calculations.

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Figure 2. A heuristic rapresentation of Latin Hypersquares.

III. The Motor Model The inside of a rocket motor is an extremely complicated environment2. Flow speed ranges from zero, at motor head, to thousands of meters per second, at nozzle exit with consequent strong Mach number variations. Temperatures are in the order of 3000 K with flow that changes its thermochemical properties. Propellant ignition, mass injection, variable geometry, erosive combustion, nozzle erosion and possible acoustic instabilities represent only some of the processes that should be considered. Some assumptions and simplifications are required. Modelling all of the previous phenomena in detail would be extremely complex from a numerical viewpoint and prohibitive for computing power requirements. The model implemented in this work consists of a simple, robust and expandable module which can execute fast parametric evaluations. Quasi-one dimensional flow is assumed since the longitudinal flow is predominant inside the motor2, though some transverse effects are considered in integral form (e.g. cross section variation). Local secondary behaviours, like vortexes, boundary layers, local propellant geometry, are lost but these are beyond the goal of the model. The code is quasi-steady because solid rocket motor behaviour is quasistationary1, apart from short ignition and tail-off transients. Limited computational power is required for time integration. Quasi steady Shapiro’s equations5 are suitable for implementation in the aforementioned model. A detailed description of these equations is left to next paragraph. However, the reader should remember that smooth solution of Shapiro’s equations are obtained for subsonic or supersonic flow, but not for sonic. A problem arises at the nozzle section where, in general, rocket motors are sonic. For this reason, the motor was split in to two different entities. The combustion chamber, with its complete model, is a subsonic flow region till the beginning of the nozzle. On the other side a zero dimensional model is used to simulate the nozzle. Asher H. Shapiro, in his work5, presents an elegant, yet simple method to solve complex problems for quasi-one dimensional flows. A series of analytical solutions for simple conditions could be derived and are already available in the open literature. For example Rayleigh flow considers only heating, while Fanno lines represent a set of solutions where the source terms of the equation set derives from friction only. But in real problems (e.g. our motor) all these phenomena occur all together so that a numerical solution is required. Shapiro’s equations can then take into account different source terms, such as: - Area change. - Wall friction. - Heat exchange. - Drag of internal bodies. - Chemical reaction. - Change of phase. - Mixing of gases. - Changes in molecular weight and specific heat.

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On the other hand, very few assumptions are required: - One dimensional, steady flow. - Changes in stream properties are continuous. - Gas is semi-perfect, i.e., specific heat is dependent only by temperature and composition. Shapiro considered a flow inside a duct and a simple control volume between two section at an infinitesimal distance dx. He derived for this control volume a series of basic physical equations in differential logarithmic form. The resulting ordinary differential equations describe the evolution of relevant gas properties in the frame of reference. A scheme is reported in Figure 3.

Figure 3. Shapiro’s control surfaces defined for different methods of gas injection and liquid evaporation5. Despite Shapiro’s equations can withstand various phenomena, only few of those (Eq. 1 to 4) were implemented in this work. The resulting set of ODEs is as follows:  k 1 2   k 1 2  21  M  2 1  kM 2 1  M  dM 2 dA 2 2     dw   M2 1 M 2 A 1 M 2 w

(1)

dV 1 dA 1  kM 2 dw   V 1 M 2 A 1 M 2 w

(2)









dT k  1M 2 dA k  1M 2 1  kM 2 dw   T 1 M 2 A 1 M 2 w

(3)

 k 1 2  2 kM 2 1  M  2   dw 2 1 M w

(4)

dp kM 2 dA   p 1 M 2 A

Where cross section area, injected mass flow, and their differential are independent variables. But, while area is easily known, the injected mass flow is proportional to the local regression rate of propellant which is expressed through Vieille’s law2 (Eq. 5): rb  ap n

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(5)

This expression is a function of pressure, which is unknown at the beginning of calculations. The implemented solution uses a 0-dimensional model of combustion chamber to guess pressure (which is assumed constant along x axis). This guess pressure is then used to evaluate mass flow so that Shapiro’s equations can be solved. Finally, a check is performed on mass balance. If difference is above desired tolerance, pressure calculated with Shapiro’s equations is used as guess, and evaluations iterated until convergence. Simulations showed that motors with low L/D ratio does not require any iteration. The obtained Shapiro’s ODEs are solved using Matlab solver, ode45 which is a 4th order Runge-Kutta solver with variable integration step6. The computational domain is 1D, resulting in a simplified grid. Motor length, from head-end to the beginning of the nozzle, is divided into equally spaced cells, which number can be varied. Each cell contains a series of information about its geometry (position, port area and perimeter), propellant (thickness burned both in radial and lateral directions), flow information (speed, pressure, and density) and finally a marker to define which kind of cell it is (fixed duct, burning propellant or, for special cases, lateral burning propellant). Cell information represents the core of the code. All these data are rearranged in matrixes so that history of each single cell can be reproduced and analysed. Time discretisation is not linear, due to variation of combustion velocity it was established to calculate Δt at each step with Eq. 6. t  ct

x max rb 

(6)

In this way, the time step is related to combustion velocity and to coefficient ct defined by the user. This choice was made to assure a proper number of time-steps for every calculation. In order to properly model the whole motor, the code required some features that are able to cope with the most important phenomena. Here follows a list of implemented features: - Lateral combustion, which is widely used in solid rocket motors, from large Solid Rocket Boosters to small test motors1. - Erosive combustion, which is a phenomena related to high flow field speed inside combustion chamber, leading to an anomalous increase in local propellant combustion rate7-9. This issue significantly affects rockets with high L/D ratio, for example SRBs. For small L/D ratios, erosive combustion can be neglected. In this case, erosive combustion was modelled using the Lenoir & Robillard semi empirical equation. - Variable thermochemistry, due to pressure variation during firing. Changes in gas flow properties occur and so, in the end, performances. All thermochemistry calculations are performed using NASA software CEA (Chemical Equilibrium Analysis). A thermochemical tabulation is implemented for the scope. - Hump, which represents a local variation of propellant ballistics (Figure 4). The vast majority of high performance civil propellants are a mixture of oxidizer salt, metallic fuel, and inert or energetic binder [2]. Macroscopically, they look homogeneous but microscopically they are far from that. Viscous effects during production of propellant can lead to local variation of propellant microstructure. This effects is summarized in a variation of propellant regression rate along propellant thickness.

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Figure 4. Example of Hump distribution. - Propellant geometry, which is a key factor in a solid rocket motor. Burning surface and its evolution regulate pressure, mass flow rate, and so thrust. Since the advent of modern solid rocket motors, all sorts of port geometries have been tried to withstand mission requirements. The model can handle general types of grain shapes, tabulating their behaviour upstream, by a dedicated code10.

Figure 5. Example of central perforation evolution in time. - Propellant axis displacement, which can be generated by error in propellant casting. This error can be higher in test motors where the mandrel is placed manually. This variation can be take into account into the code. - Characteristic velocity efficiency, C* , which is a figure of merit of propellant and combustion chamber1. Its efficiency can be take into account by the code. - Nozzle efficiency, which can take into account different sources of losses. Main contributions are two dimensional flow (velocity in radial direction) and two phase flow. In our case, nozzle efficiency is an imposed parameter.

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- Motor tail-off which can be important in some specific applications (for strap-on boosters). It can be interesting how combustion chamber behaves after all propellant is burned, so it is required to treat the so called “burn-out”. When there is no more propellant inside the chamber, the code moves from the quasi-onedimensional quasi-steady, model to a simple zero-dimensional, unsteady, model to perform these calculations. Every numerical model needs to be verified. The chosen test case was the Baria motor. This test rocket motor is part of the bigger family of BATES motors11 which are small scale motors used for the experimental evaluation of propellants. The Baria motor has been used by Avio S.p.A. for decades to test the Ariane 5 propellant. With this wide number of firings, it represents a reliable and trustworthy source of experimental data. The Baria motor is conceived for propellant testing. It is designed to withstand several firings, changing only the graphite nozzle and the internal propellant-loaded liner. To correctly test propellants it is designed to have a quasi-neutral burning surface and, consequently, quasi-constant pressure trace. A simple convergent nozzle is used. Different throat diameters can be used to vary the internal pressure. At the motor head the igniter and the pressure probe are placed. The propellant cartridge is simple. It is a 290 mm long single cylindrical perforation with 30 mm of propellant thickness. To ensure quasi-constant burning surface, propellant sides are not inhibited and participates to combustion. TRASDUTTORE PRESSURE TRANSDUCER DI PRESSIONE

290 IGNITER ACCENDITORE

LINER LINER

NOZZLE UGELLO

100

160

GRAPHITE INSERT GRAFITE

PROPELLANT PROPELLENTE STRUCTURAL CASE INVOLUCRO Figure 6. Baria motor schematics (not in scale).

Validation tests were performed by comparing model results with experimental data obtained from BARIA rocket motor. During real Baria motor firing, motor head pressure is recorded while thrust is disregarded since the motor is divergent-less. Internal pressure trace was used for the code validation. After calculations, the model pressure trace was compared with the one from experimentation, the results showed good agreement between two curves at different pressure levels (see Figure 7).

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Figure 7. Comparison between experimental and simulated pressure traces.

IV. Simulations and Results A list of performances considered in this work is reported hereafter: - Burning time - Specific impulse - Volumetric specific impulse - Total impulse - Mean thrust - MEOP (Maximum Expected Operative Pressure) - Mean pressure Some of them (mean thrust, specific impulse, volumetric specific impulse, total impulse) are important performance parameters. Other data gain particular interest in solid rocket motors because they cannot be modified during operation by any means (burning time, mean pressure, and Maximum Expected Operative Pressure). The alteration of such parameters can be mission-critical, if beyond certain limits. For each simulation 200 points were used for the Monte Carlo method. This number define the size of the sample that is used to estimate mean values and standard deviations of performances distributions. It is heuristically clear that a bigger sample means more “accurate” evaluations, on the other hand it does require more time to do every test. Two hundred points was selected as a good compromise between precision and required time for each simulation, of about 8/9 hours on a desktop PC. We can use confidence intervals to verify the fidelity of the choice3. A confidence interval tells us the band inside which the real value falls with a probability of γ. For example, if we use γ=95%, we get that every 95 samples over 100 our estimate is inside that interval. When the standard deviation of the population is not a-priori known, an estimate s must be used. In this case, the confidence interval of the mean value is: X  t 1    / 2; n  1  s    X  t 1    / 2; n  1  s

(7)

Where t(p,ν) is the t-student distribution percentile and s is the standard deviation estimate. For full explanation refer to3. The reader should keep in mind the difference between the estimates of mean value, standard deviation, and their real value. Taking as an example the burning time from the complete simulation with all parameters treated statistically, and a confidence level of 95% we get an interval of: [4.0144; 4.1941] s When just 100 points for the Monte Carlo method were used, the interval becomes: [4.0084; 4.1931] s The difference is very small so that a further increase in Monte Carlo points would not bring appreciable results.

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The numerical model of the Baria motor was made using real data of the internal geometry (see Figure 6 ), discretised with 500 grid cells along the X axis. The real motor nozzle is only convergent because only the internal pressure is requested for this kind of burning rate tests. Instead, a convergent – divergent nozzle is used in the model, with an area ratio of 4. This change does not interfere with combustion chamber behaviour but increases thrust and specific impulse values, without altering their distributions. For all the simulations of this chapter, a nozzle diameter of 25.25 mm was used, which correspond to the “medium pressure Baria” used in Avio S.p.a., with an expected mean operating pressure of about 45-50 bar. Here follows a list with the investigated parameters: - Variation of ballistic parameter a - Variation of ballistic parameter n - Variation of c* efficiency - Variation of nozzle efficiency - Variation of Hump - Variation of propellant length - Variation of propellant axis offset - Variation of all parameters All simulations were post-processed to extract statistical parameters from data. All test showed to be well approximated by Gaussian distributions. In table 1 data from all simulations are reported in the form of relative standard deviations, mean values are omitted since we are investigating uncertainties. Table 1. Results from all simulations in the form of performances relative standard deviations (in percentage), first row represents the relative standard deviations used for input parameters. (*) Axis offset mean value is 0, its relative standard deviation is defined with respect to combustion chamber maximum radius. 100*σ/µ Input std Burn time Mean Isp Mean Isv Total impulse Mean thrust MEOP Mean pressure

a 0,4660 0,7597 0,0000 0,0000 0,0000 0,7597 0,7598 0,7562

n 0,7207 0,6878 0,0000 0,0000 0,0000 0,6878 0,6977 0,6846

η-c* 0,5 0,3290 0,5146 0,5146 0,5146 0,8441 0,8441 0,8453

η-nozzle 0,5 0,0000 0,4887 0,4887 0,4887 0,4887 0,0000 0,0000

Hump 0,6667 0,0347 0,0091 0,0091 0,0325 0,0117 1,0026 0,0099

offset 0,0167* 0,1510 0,0002 0,0002 0,1448 0,0062 0,0024 0,0061

mass 0,2 0,1214 0,0019 0,0019 0,1894 0,3108 0,3076 0,3099

Total n.a. 1,1098 0,6759 0,6759 0,7186 1,4400 1,4021 1,3975

Data from simulations allow to understand how each parameter influence performances distributions, for example, specific impulse is appreciably affected only by efficiencies, or, the only relevant effect of Hump is on the MEOP. Another important result is that, from actual data, appears that each parameter influence independently performances distributions. This can be seen using random variables algebra3, in the hypothesis of complete independences, overall standard deviation con be written as:  tot 



2 j

(8)

j

Table 2. Comparison between complete simulation relative standard deviations (in percentage) and those from random variables algebra. 100*σ/μ Burn time Mean Isp Mean Isv Total impulse Mean thrust MEOP Mean pressure

r.v. algebra 1,0941 0,7097 0,7097 0,7493 1,4486 1,6960 1,3606

complete simulation 1,1098 0,6759 0,6759 0,7186 1,4400 1,4021 1,3975

difference % -1,4117 4,9998 4,9998 4,2774 0,5938 20,9636 -2,6389

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From Table 2 it is possible to appreciate the small difference between random variables algebra data and those from complete simulation. The small correlation factors (except for MEOP) suggests that the effects of single parameters are independent from each other. It is also noticeable that burn time and mean pressure have an opposite trend with respect to other performances, further investigations are left for future works. All results presented up to now are from numerical simulations. A comparison with experimental data12 is required to evaluate the goodness of the method implemented. As said before, the Baria motor has been in use for years and a vast quantity of experimental data have been collected. To ensure maximum fidelity the chosen experimental data are those used to extrapolate the ballistic coefficients used for the numerical model. From each batch of propellant 27 Baria motor firings at 3 different pressure levels are made, so that 9 sets of data are available for the medium pressure model. Due to limitations in data availability only comparisons on burning time, MEOP, and mean pressure could be performed. The comparison is here reported: Table 3. Relative standard deviations from Monte Carlo method and from experimentation. 100*σ/μ Burn time MEOP Mean pressure

complete simulation 1,1098 1,4021 1,3975

experimental 0,8736 1,1344 1,0327

difference % 27,0342 23,5964 35,3228

From Table 3 it is possible to appreciate that the Monte Carlo method overestimates relative standard deviations, with respect to experimental data, of about 30%. Comparisons with other propellant batches showed similar results. The reader should keep that ballistic coefficients used in the model are affected by experimental errors during their extrapolation from real firing data.

V. Conclusion The numerical model made during this work proved out to be a good approximation of the Baria motor. It is a robust and simple tool capable of dealing with a wide variety of phenomena that occur inside a solid rocket motor. Its capability to execute fast and reliable simulations is perfect for its implementation in a Monte Carlo method where a large number of simulations is required. The pressure traces obtained from the model were compared to those from real firings giving a good reproducibility. Only single cylindrical perforations were used in the current work. However, the model is also capable to deal with more complicated geometries in two dimension (e.g. star-grained rockets). Other properties were implemented in this work but the type of simulated motor did not take advantage of those features (e.g. erosive burning). The analysis of data from Monte Carlo simulations showed the different influence of parameters over performances. It is now possible to view how all the geometrical and chemical uncertainties propagate and how much each parameter is relevant to mission reliability. The simulations showed the importance of ballistic parameters and efficiencies. While phenomena that were of great interest, like Hump, axis offset and propellant length, exhibited minor influences over final performances. The comparison between statistical data from simulations and real Baria motor firings showed good agreement, despite the Monte Carlo method shows a certain overestimate of performance uncertainties. Further investigations should look for the input parameters distributions that gives the same experimental distributions, or at least, minimise the error.

References 1

G.P. Sutton, O. Biblarz, Rocket Propulsion Elements, eight edition, Wiley, Hoboken, New Jersey USA, 2010. L.T. DeLuca, Problemi Energetici in Propulsione Aerospaziale, Notes for Students, Politecnico di Milano, Milano, Italy, 2012. 3 S. Beretta, Affidabilità delle Costruzioni Meccaniche, Strumenti e metodi per l’affidabilità di un progetto (in Italian), Springer, Milano, Italy, 2008. 4 A.J. Wheeler, A.R. Ganji, Introduction to Engineering Experimentation, third edition, Pearson, New Jersey, USA, 2010. 5 A.H. Shapiro, The Dynamic and Thermodynamic of Compressible Fluid Flow, Ronald Press, New York, USA, 1953. 6 A. Quarteroni, R. Sacco, F. Saleri, Matematica Numerica, third edition, (in Italian), Springer, Milano, Italy, 2008. 7 NASA technical report SP-8039, Solid Rocket Motor Performance Analysis and Prediction, National Aeronautics and Space Administration, Houston, Texas, USA, 1971. 2

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B.A. McDonald, The Development of an Erosive Burning Model for Solid Rocket Motors Using Direct Numerical Simulation, Doctor of Philosophy thesis in Aerospace Engineering, Georgia Institute of Technology, Georgia, USA, 2004. 9 R. Badrnezhad, F. Rashidi, Solid Propellant Erosive Burning, Paper 1111, 8th World Congress of Chemical Engineering, Montreal, Canada, 2009 10 NASA technical report SP-8076, Solid Propellant Grain Design and Internal Ballistics, National Aeronautics and Space Administration, Houston, Texas, USA, 1972. 11 R.S. Fry, Evaluation of Methods for Solid Propellant Burning Rate Measurements, NATO/RTO Advisory Report, AVT Working Group 16, 2002. 12 F. Maggi, Riduzione dati e analisi balistica del motore Baria. MSc Thesis (in italian), Politecnico di Milano, Milan, Italy, April 2002.

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