Faculteit der Exacte Wetenschappen

TNO Defensie en veiligheid

Bachelorproject verslag Natuur- en Sterrenkunde, omvang 12 EC,uitgevoerd als stage bij TNO defensie en veiligheid in de periode

26-04-2010 tot 01-07-2010

Solid propellant grain geometrydesign, a model for the evolution

of star shaped interfaces

Auteur:Arnon Lesage,5795656

Begeleiders:Ir. Francois Bouquet

Dr. Rudolf Sprik

August 9, 2010

Contents

1 Introduction 5

2 Solid rocket motors 62.1 Passive regulation . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Thrust and mass flow . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Burn rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Pressure and burn rate . . . . . . . . . . . . . . . . . . . . 102.3.2 Temperature and burn rate . . . . . . . . . . . . . . . . . 102.3.3 Erosive burning . . . . . . . . . . . . . . . . . . . . . . . . 112.3.4 Other effects . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Mass flow and pressure . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Grain geometry 133.1 Propagating interface . . . . . . . . . . . . . . . . . . . . . . . . . 143.2 Interface propagation methods . . . . . . . . . . . . . . . . . . . 153.3 Cellular automaton . . . . . . . . . . . . . . . . . . . . . . . . . . 153.4 Fast marching algorithm . . . . . . . . . . . . . . . . . . . . . . . 183.5 Geometric solution of a parametric star . . . . . . . . . . . . . . 183.6 The chosen method . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Geometric evolution of a parametric star - Building the model 214.1 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Initial shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Evolution of the shape . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Intersection points . . . . . . . . . . . . . . . . . . . . . . . . . . 294.5 Burn area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.6 Achievements and Limitations of the algorithm . . . . . . . . . . 314.7 Results and comparison with the current software, GDP[1] . . . . 324.8 Improvements and beyond . . . . . . . . . . . . . . . . . . . . . . 32

5 Conclusion 36

1

Nomenclature

m Mass flow

r Burn rate

K Temperature sensitivity of pressure

c Chamber gas density

e Exhaust gas density

p Propellant mass density

() The angle from the origin to a point on the off-center circle

The angle from the origin of the off-center circle to a point on the off-center circle

i Angles representing the limits of the angular range over which the piece-wise functions of r() work

i(bd) Angles representing the limits of the angular range over which the piece-wise functions of r(, bd) work

a angle used in geometry,= N

At Nozzle throat area

a0 Propellant constant that depends on initial grain temperature

Ab Burn Area

Ae Nozzle exit Area

At Throat Area

b angle used in geometry,= k1N

bd Burn Depth

c Angle used in geometry,= k2N

F Thrust

Fr1 The initial Radius of the first fillet

Fr2 The initial Radius of the second fillet

IR Inner radius

k Specific heat ratio

k1 Fraction of angles, =ba

2

k2 Fraction of angles, =ca

N Number of star points

n Combustion index

OR Outer radius

Pa Outer pressure

Pc Combustion chamber pressure

Pe Exit pressure

Pa Point angle

R Gas constant, or later on circle radius

r() Radial function describing the initial interface

Rc The radius function of an off-center circle

ri() Radial function describing a single piece of the geometry

ri(, bd) Radial function describing a single piece of the geometry as it evolvesas a function of burn depth

rfi() Radial function describing the fillet piece of the geometry

rfi(, bd) Radial function describing the fillet piece of the geometry as it evolvesas a function of burn depth

Tc Combustion temperature

Vc Grain cavity volume

Ve Exit gas velocity

3

Samenvatting

Vaste stuwstof motoren bieden geen mogelijkheid om zich actief te reguleren.Actief reguleren houdt in dat het mogelijk is om tijdens het vliegen de stuwkrachtte veranderen naar wensen. Na het ontsteken, verbrandt de stuwstof en leverthet een bepaalde stuwkracht. De stuwkracht is afhankelijk van de hoeveelheidverbrandingsoppervlak van de stuwstof. Door de stuwstof een bepaalde ge-ometrische vorm te geven, is het mogelijk de hoeveelheid stuwkracht toch passiefte kunnen reguleren. In tegenstelling tot actief reguleren, houdt passief regulerenin dat de stuwkracht veranderingen vantevoren, dus voor het ontsteken wordenbepaald. Tijdens het verbranden van de stuwstof, zet de vaste stuwstof zich omin gas, en verandert de geometrische vorm van de stuwstof massa. De stuwstofgeometrie moet dan ook na het veranderen nog steeds aan de stuwkrachteisenvoldoen. Hiervoor is een model nodig dat de manier waarop de geometrie veran-dert kan simuleren. Hiermee kan het model berekenen wat de stuwkracht is vaneen bepaald geometrie ontwerp tijdens zijn gehele verbranding. Het model dathier wordt bestudeerd zal de eerste stap zetten naar een optimalisatie model.Het optimalisatiemodel, het einddoel, is een model dat een stuwstof geometrieberekent die aan bepaalde prestatie-eisen voldoet. De stap die in dit onder-zoek wordt bestudeerd is niet de optimalisatie, maar het berekenen van deprestaties van cilindrische geometrie, gebaseerd op twee dimensionale sterge-ometrie. Het model is uitgewerkt in MatLab, in opdracht van TNO Defensieen Veiligheid. Het model berekent de prestaties op een manier die optimal-isatie mogelijk maakt. Dit was nog niet mogelijk met de bekende methodes, endaarom werd het huidige model ontwikkeld.

4

1 Introduction

TNO Defence Security and safety provides many solutions to the overall safetyand security as a strategic partner of the Ministry of Defence. TNO has a longhistory in researching propellants. The research began in the middle of the1980s, with the design of igniters and starter engines for the Ariane 5 mainengine (Vulcain). The Ariane 5 is a civilian rocket used as a carrier of satellitesinto orbit.

Over the years, development provided detailed methods by which the per-formance of a given propellant and propellant-geometry can be determined.The currently used software, GDP[1], is able to calculate the performance andburn behavior in detail. Yet, it does not provide fast analysis or a basis foroptimizations of the propellant geometry, also known as the grain geometry.Optimization is an essential ingredient for the development of grain geometry.The grain, which is the propellant bulk, is developed when the requirements ofthe rocket are known. It is therefore needed to have a method that calculatesa grain-geometry based on the given performance requirements. Further cal-culations and fine tuning of the geometry can then be done using the currentsoftware models.

The project is split into two separate projects. The first is the developmentand implementation of a model that provides fast performance calculations ofa given propellant and propellant-geometry. Different approaches are availablewhich will be discussed. The chosen approach is implemented in MATLAB.Fast analysis of the grain is the key ingredient of the chosen approach. Speed isimportant because in the next step, optimization, simulation of many differentgrains takes place. It is therefore desired to have an approach that minimizesthe time of a single analysis to under a second. The second part of the projectis the optimization phase. A grain geometry will be calculated that providesthe requirements for the mission at hand. The subject discussed is the first partof the project, the analysis of a grain geometry. In (4.8) a discussion followsthat provides some points of thought for the second part of the project, theoptimization.

The model produced will concentrate on burn area evolution, as a functionof burn depth. This means it will not take into account all factors affectingthe burn rate, and with that the production of thrust. These factors will bediscussed though in order to understand, why these can be examined separatelyand how future implementation is possible.

5

2 Solid rocket motors

A solid rocket motor (SRM) is a machine that provides thrust. Every SRMprovides a different amount of thrust depending on payload, destination andother factors. But thrust is not the only performance variable of a SRM. Forexample the mass of the propellant is also a major factor. As the propellantburns, its mass decreases, allowing higher acceleration. Therefore performanceassessment models also look at specific impulse, mass flow and other variables.This model will only examine a few aspects of the performance of a SRM,although many others can also be examined.

The solid rocket motor is one of two major classes of motors, where liquidpropellant is the other kind. What makes solid unique versus the liquid propel-lant, is the simplicity of the motor. The solid propellant contains the fuel andoxidizer. It is therefore enough to ignite the propellant and no other chemicalhas to be added for combustion to take place. There are different propellantcompositions, usually double base or composite, that all have different proper-ties. Burn rate, ignition temperature, flame temperature and other propertiesthat can all be found using experimentation and modeling.

The simplicity of the solid propellant is also its drawback. Requiring nothingexternal to burn, the SRMs burning cannot be regulated in-flight by providingmore or less fuel, as in engines. A form of active, real ti