5. Frequency response: >100 kHz. 1. Make and model: NI PXI-6123 DAQ. 2. Platform: NI PXIe-1062Q. 3. Controller: NI PXIe-8370. 4. Measuring frequency: 500 ...
I
Military Technical College Chair of Rockets
Design of Solid Motor for Predefined Performance Criteria By Capt. Eng. Anwer Elsayed Anwer Hashish Military Technical College
Under Supervision of Maj. Gen. (Ret.) Assoc. Prof. Mohamed Allam S. Al-Sanabawy Military Technical College Maj. Gen. (Ret.) Assoc. Prof. Hamed Mahmoud Mahmoud Abdalla Military Technical College Col. Dr. Mahmoud Yehia Mohamed Ahmed Military Technical College This thesis is submitted as a partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering
Cairo 2018
I
Abstract In solid propellant rocket propulsion, the design of the propellant grain is a decisive aspect. The grain design governs the entire motor performance and, hence, the whole rocket mission. The ability to decide, during design phase, the proper grain design that satisfies the predefined rocket mission with minimum losses is the ultimate goal of solid propulsion experts. This study aims to accomplish the two following objectives. On one hand, to develop an optimized tool for predicting the performance of rocket motor with specified grain configuration. On the other hand, to estimate the grain geometry for predefined performance through optimization. The objective of optimization is to minimize the deviation between the required and predicted motor performance. A hybrid optimization technique is used as an optimization method which consists of Genetic Algorithm (GA) to find the global optimum (minimum root mean square error between the predefined and calculated performances) and Simulated Annealing (SA) optimization method to find the accurate local optimum. An internal ballistic prediction model (IBPM) for the pressure time curve of the rocket motor is created using MATLAB. Then, it is linked to GA and SA optimizers. The coupled (optimized) model is used either to maximize the prediction accuracy or to find the design that best satisfies the predefined needs. Static firing tests are designed and conducted to validate the proposed tool. It is found that the developed program is capable of predicting rocket motor performance or grain design with acceptable accuracy for the preliminary design phase.
II
Acknowledgment After thanking God for His help in finishing this study, I would like to express my sincere gratitude to my advisors Assoc. Prof. Mohamed Allam S. Al-Sanabawy, Assoc. Prof. Hamed Mahmoud Abdalla and Dr. Mahmoud Yehia Mohamed Ahmed for their continuous support during my master study and related research, for their patience, motivation, and immense knowledge. Their guidance helped me during all the course time of research and reduction of this thesis. I could not have imagined having a better supervisors and mentors for my master study. Besides my advisors, I would like to thank the rest of my instructors: Dr. Osama Kamal, Dr. Mostafa Sameer, Dr. Hatem Belal and Dr. Ahmed Ayad for their insightful comments and encouragement that incented me to widen my research from various perspectives. Cordial gratitude to the staff of Abu Zaabal Company for Specialty Chemicals (Factory-18) which helped me a lot in my experimental work that would not be done without their facilities and help. I would like to thank Eng. Metwally Barakat, Mr. Daniel Nathan, Mrs. Hanaa Abd Elglel, Eng. Saeed Mohamed and Ahmed Yehia for their support in experimental work. I thank my department colleagues for the stimulating discussions, for the sleepless nights during which we were working together before deadlines, and for all the fun we have had in the last four years. Also, I thank my friends in the Military Technical College for their support and encouragement. Finally, I would like to thank my family: my parents and my wife for supporting me spiritually throughout the preparation of this thesis and my life in general.
Anwer Elsayed Hashish Cairo 2018
III
Table of Contents Abstract
I
Acknowledgment ..................................................................................................... II Table of Contents
III
Nomenclature .......................................................................................................... VI List of Figures ......................................................................................................... XI List of Tables ......................................................................................................... XV Chapter 1 Introduction............................................................................................. 1 Chapter 2 Background and Literature Survey ..................................................... 5 2.1
Main Parts of Solid Propellant Rocket Motor ......................... 6
2.2
Definition of SPRM Key Performance Parameters ............... 10 2.2.1 Thrust and thrust coefficient ......................................................................11 2.2.2 Total impulse ..............................................................................................12 2.2.3 Characteristic velocity.................................................................................12 2.2.4 Specific impulse ..........................................................................................12 2.2.5 Nozzle expansion ratio................................................................................13 2.2.6 Pressure time curve: ...................................................................................14 2.2.7 Chamber pressure and MEOP ....................................................................14 2.2.8 Burning rate ................................................................................................15
2.3
Grain Burn-Back Analysis .................................................... 16
2.4
Internal Ballistic Performance prediction in SPRM .............. 19
2.5
Preliminary Design Methods of SPRMs ............................... 26
2.6
Optimization Techniques....................................................... 31
2.7
Applications of Optimization in Solid Rocket Motor Design 35 2.7.1 Enhancing performance of a specific grain design .....................................35
2.7.2 Predicting the grain design for a given performance criteria.....................40 2.7.3 Enhancing the accuracy of IB prediction ....................................................43
2.8
Ranges of Key Design and Performance Parameters of SPRM 44
IV Chapter 3 Methodology ......................................................................................... 49 3.1
Over-View of the Research Methodology and Work Plan .... 49
3.2
Internal Ballistics Prediction Model (IBPM) ........................ 52 3.2.1 Inputs to the internal ballistics model ........................................................53 3.2.2 IBPM calculation procedure .......................................................................54 3.2.3 IBPM flowchart ...........................................................................................60 3.2.4 Validation of IBPM ......................................................................................62
3.3
Optimized Internal Ballistic Prediction Model (OPTIBPM) 64
Chapter 4 Experimental Work ............................................................................. 67 4.1
Introduction ........................................................................... 67
4.2
Manufacturing Processes of Composite Propellant............... 67 4.2.1 Mixing operations .......................................................................................68 4.2.2 Casting of the grains ...................................................................................70 4.2.3 Curing and finishing ....................................................................................71
4.3
Determination of Propellant Properties ................................. 73 4.3.1 Assembly of standard two-inch test motor ................................................73 4.3.2 Test Procedure............................................................................................75 4.3.3 Experimental measurements of propellant ballistic properties .................76 4.3.4 Thermochemical calculation of the used propellant ..................................78
Chapter 5 Optimization Results and Experimental Validation
81
5.1
Introduction ........................................................................... 81
5.2
Optimization of Internal Ballistic Prediction Model ............. 82
5.3
Design of a Star Grain for Predefined Performance Criteria Using OPTIBPM ................................................................... 87
5.4
Design of Dual Thrust Tubular Grain for Predefined Performance Criteria Using OPTIBPM
............................ 90
5.4.1 Experimental validation of OPTIBPM grain ................................................94
Chapter 6 Conclusions and Future Work........................... ................................ 101
V 6.1
Conclusions ......................................................................... 101
6.2
Future Work ......................................................................... 103
References.............................................................................................................. 104 Appendix A
IBPM for Star Grain Geometry ....................................... 108
Appendix B
IBPM for Dual Thrust Motors ......................................... 116
Appendix C
Isp and CEA Results ......................................................... 123
Appendix D
Burn-Back Analysis ........................................................... 134
VI
Nomenclature Symbols Burning rate coefficient Burning area of star grain Critical cross-section area of the nozzle Exit cross-section area of the nozzle Port area of solid grain at each burning step Characteristic exhaust velocity Thrust coefficient d
Inner diameter of the two-inch motor grain Internal diameter of the first grain Internal diameter of the second grain Initial critical diameter of nozzle
D
Outer diameter of solid propellant grain Erosion rate of nozzle critical section Fillet radius
F
Thrust force Gravitational acceleration at sea level Specific impulse Total impulse Burning step
L
Length of the grain Length of star grain Length of the first grain Length of the second grain Mass of the propellant
VII Gas Mach number at the nozzle end of the grain ̇
Mass flow rate
̇
Rate of discharge of gases ̇
Rate of generation of gases ̇
Propellant mass flow rate
n
Burning rate exponent Number of star points Atmospheric pressure Average pressure Combustion chamber pressure Discharge pressure Exit pressure Gas pressure at head end of the grain Maximum pressure Pressure at nozzle end Stagnation pressure of flowing gases Average burning rate of the grain Burning rate at the head end of the grain Total burning rate at the nozzle end considering erosive burning
R
Universal gas constant Grain inner radius
t
Burning time Delay time until beginning of nozzle erosion
T
Initial temperature Combustion chamber temperature Final chamber volume Initial volume of combustion chamber
VIII Exit exhaust velocity Gases velocity inside combustion chamber Flow velocity of gases at the nozzle end Threshold velocity of gases Web thickness
Greeks Erosive burning coefficient Erosive burning pressure coefficient Specific heat ratio of the combustion gases Angle fraction Nozzle expansion ratio Star point angle Density of the solid propellant Density of gases Time increment Distance burnt Tensile stress Ultimate stress Yield stress Subscripts b
Burning
cr
Critical
f
Final
h
Head
i
Initial
n
Nozzle
IX Abbreviations: 1-D
One dimensional
3-D
Three dimensional
ACO
Ant colony optimization
AL
Aluminum
AP
Ammonium perchlorate
BB3D
Ballistic burn-back three dimensional
CAD
Computer aided design
CEA
Chemical equilibrium analysis
CFD
Computational fluid dynamics
DBP
Di butyl phthalate
DOE
Design of Experiment
DOP
Di octyl phthalate
DOS
Di octyl sebaciate
DTRMs
Dual thrust rocket motors
F3DBT
Fast Three Dimensional Burn back Tool
GA
Genetic algorithm
GDB
Grain design ballistics
GEOM
Dimensioning of grain and regression of burning surface
HMDI
Hexamethylene diisocyanate
HTPB
Hydroxy terminated poly butadiene
IB
Internal Ballistics
IBPM
Internal ballistic prediction model
IPDI
Isophorone diisocyanate
IPT
Area under the curve between the pressure and the time
Isp
Specific impulse program
LSM
Level set method
X MAPO
Trismethyl aziridinyl phosphine oxide
MEOP
Maximum Expected Operating Pressure
ODE
One Dimensional Equilibrium
PIBAL
Propulsion and Internal Ballistic Software
RMSE
Root mean square error
SA
Simulated annealing
SAPS
Simulated annealing with pattern search optimization
SPP
Solid performance program
SPRM
Solid propellant rocket motor
SQP
Sequential Quadratic Programming
STL
Standard Template Library
TB
Actual time of burning the propellant
TCPSP
Calculation of thermochemical properties
TD
Delay time
TEMP
Ambient temperature
UBRDO
Uncertainty based robust design optimization
XI
List of Figures Fig. No.
Title
Page No.
Fig. 1.1 Solid propellant rocket motor construction ............................................ 1 Fig. 2.1 Roadmap of Chapter 2 ............................................................................ 5 Fig. 2.2 SPRM main parts.................................................................................... 6 Fig. 2.3 Solid propellant grains categories ........................................................ 10 Fig. 2.4 Pressure time curve .............................................................................. 14 Fig. 2.5 SPPMEF flowchart [17] ....................................................................... 21 Fig. 2.6 Software architecture according to Laimek [20] .................................. 24 Fig. 2.7 Performance estimation procedure by Schumacher[21] ..................... 25 Fig. 2.8 Preliminary design procedure by Schumacher[21] ............................. 27 Fig. 2.9 Selection of grain type according to [25] ............................................ 29 Fig. 2.10 Preliminary design process of a SPRM [25] ...................................... 29 Fig. 2.11 Grain geometry and performance module [30] .................................. 36 Fig. 2.12 Hybrid optimization algorithm [30] ................................................... 37 Fig. 2.13 Grain design process by Kamran [10] ................................................ 37 Fig. 2.14 Framework of hybrid optimization [34] ............................................. 39 Fig. 2.15 Framework of uncertanity-based optimization [34] ........................... 39
XII Fig. 2.16 Design optimization tool architecture[38].......................................... 42 Fig. 3.1 Work methodology flowchart .............................................................. 51 Fig. 3.2 Pressure time profile phases ................................................................. 52 Fig. 3.3 Internal ballistic prediction model flow chart ..................................... 61 Fig. 3.4 Star grain of tested motor [47] ............................................................. 62 Fig. 3.5 Nozzle structure with graphite insert [47] ............................................ 62 Fig. 3.6 Literature vs. theoretical pressure time curve of star grain [47] ......... 64 Fig. 3.7 OPTIBPM tool flowchart ..................................................................... 66 Fig. 4.1 The ingredients before mixing inside the mixing container ................. 69 Fig. 4.2 Mixer double wall vessel ...................................................................... 69 Fig. 4.3 The vertical mixer with two blades ...................................................... 69 Fig. 4.4 The slurry after premixing .................................................................... 69 Fig. 4.5 The slurry after adding the curing agent .............................................. 70 Fig. 4.6 Assembly of the vessel to the vacuum chamber and the shaker before casting ................................................................................................... 71 Fig. 4.7 Curing furnace ...................................................................................... 72 Fig. 4.8 Cast grain before finishing ................................................................... 72 Fig. 4.9 Finished grains ..................................................................................... 72 Fig. 4.10 Assembly drawing of two-inch test motor ......................................... 74
XIII Fig. 4.11 Two inch motor .................................................................................. 74 Fig. 4.12 Results of static firing tests and curve fitting using MATLAB ......... 77 Fig. 4.13 Comparison of results ......................................................................... 79 Fig. 4.14 Variation of chamber and exit temperatures with chamber pressure . 79 Fig. 4.15 Variation of specific impulse and thrust coefficient with chamber ..................................................................................................................... 80 pressure Fig. 5.1 Road map of chapter 5.......................................................................... 82 Fig. 5.2 Convergence history of GA optimization for the first application ...... 84 Fig. 5.3 Convergence history of SA optimization for the first application ....... 84 Fig. 5.4 Validation of OPTIBPM for the first application................................. 85 Fig. 5.5 Convergence history of GA optimization ............................................ 88 Fig. 5.6 Validation of OPTIBPM grain geometry prediction for the second application .......................................................................................... 89 Fig. 5.7 Predefined performance of SPRM for third application ...................... 90 Fig. 5.8 Predefined pressure-time profile of SPRM for the third application ... 91 Fig. 5.9 Proposed geometry of the candidate grain design ................................ 92 Fig. 5.10 Convergence history of GA optimization .......................................... 93 Fig. 5.11 Predefined and predicted performance for the third application ........ 93 Fig. 5.12 Manufactured cores for the third application ..................................... 95
XIV Fig. 5.13 Solid propellant grains after casting for the third application ............ 95 Fig. 5.14 Test motor components before assembly for the third application .... 96 Fig. 5.15 Test motor after final assembly for the third application ................... 96 Fig. 5.16 Schematic of test motor assembly for the third application ............... 98 Fig. 5.17 Static firing test of SPRM for the third application ........................... 98 Fig. 5.18 A comparison between the static firing test and optimized pressure ........................................................................ 99 time history for the third application s Fig. D.1 Geometric parameters and burn-back phase.......... ........................... 134 Fig. D.2 Port area of star grain during second phase........................................ 135 Fig. D.3 Port area of star grain during third phase.............................. ............. 137 Fig. D.4 Port area of star grain at sliver phase ................................................. 139 Fig. D.5 Tubualr grain configuration ............................................................... 140
XV
List of Tables Table. No.
Title
Page No.
Table 2.1 Burn-back methods [8] ...................................................................... 17 Table 2.2 Ranges of rocket motor diameter [44] .............................................. 45 Table 2.3 Slenderness ratio for rocket motor [4] .............................................. 46 Table 2.4 Neutrality for different propellant grain configurations [43] ............ 47 Table 2.5 Main characteristics of common grain configuration [4, 43] ........... 48 Table 3.1 Case study parameters [47] ............................................................... 63 Table 4.1 Composition of the propellant ........................................................... 68 Table 4.2 Experimental results .......................................................................... 76 Table 4.3 Composition details for Isp program ................................................ 78 Table 5.1 Optimized model parameters and lower and upper bounds. ............. 86 Table 5.2 Star grain design parameters, lower and upper bounds .................... 88 Table 5.3 Design parameters, lower and upper bounds .................................... 92 Table 5.4 Optimized and experimental grain parameters of the third application .................................................................................................................................. 94 Table 5.5 Static firing test results for the third application ............................... 99
1
Chapter 1 Introduction A solid propellant rocket motor (SPRM) is simply a devise which generates the required thrust for rocket motion. The chemical energy stored in the propellant (in solid state) is converted to pressure and thermal energy by combustion. The propellant grain contains both fuel and oxidizer so the SPRM is air independent. An electrical signal initiates the igniter that produces hot high pressure gases that ignite the main propellant grain. The hot high pressure gases are accelerated through the nozzle which converts the thermal energy and pressure in the combustion chamber into kinetic energy. Fig. 1.1 shows the main components of SPRM.
Fig. 1.1 Solid propellant rocket motor construction The thrust produced by the solid propellant rocket motor grain is solely dependent on its chemical composition and geometry. In fact, the thrust profile (variation of thrust with time) can be predicted if the time variation of the area of
2 burning surfaces of the grain is determined. On the other hand, the grain design can be tailored to yield the required thrust profile. This tailoring includes specifying the grain configuration, dimensions the surfaces of the grain to be allowed to burn, and the inhibited surfaces. Predicting the performance parameters of a defined rocket motor is a mature topic. Many researchers have discussed this aspect and numerous reliable tools have been developed. Currently, it has become routine task to predict the overall mission parameters of a rocket equipped with a definite solid propellant grain. However, in the context of designing a motor to serve as the power plant of a rocket, what is definite a priori is the rocket mission parameters rather than the motor design parameters. Defining the solid propellant grain design that satisfies a predefined mission is not a straightforward one-shot task. It also relies on designer experience, up or down scaling from a baseline motor, or trial and error. The need to develop a reliable procedure (tool) for defining the grain design for predefined mission requirement is clearly evident. However, as will be shown in the survey, no much effort has been devoted in this respect. Such tool is thought to be beneficial for SPRM designers in many aspects. On the one hand, the designers need not to be highly experienced. On the second-hand, in cases involving novel design concepts and unconventional mission, a suitable baseline may not be available. A systematic deign procedure is indeed more efficient than the trial-anderror approach. Many researchers attempted to automate the design process of solid rocket motors and to find the optimal grain design. A lot of studies has been made in this topic since 1960 till now. These studies were seeking forward to develop efficient design tools that minimize efforts in the preliminary design of a solid propellant rocket motor. Nowadays, with the help of new CAD programs and optimizing
3 techniques, many solid propellant performance prediction tools have been developed. However, such tools are not available in the open literature and are treated as classified materials related to national security of countries. The main objective of the present study is to develop and test a design procedure for a solid propellant rocket motor grain for predefined performance criteria. The one adopted in this study is the pressure-(or thrust-) time profile. The core idea is to couple a reliable internal ballistics prediction tool with an efficient optimizer. The resulting algorithm searches for the grain design parameters that ensure minimum deviation from the predefined performance merits. The proposed algorithm is implemented and assessed in three applications. In the first application, the prediction accuracy of the developed internal ballistic tool is maximized. In the other two cases, the grain design with predefined performance is optimized for two grain configurations. A computer code is developed to be used as internal ballistic prediction model. In addition, a set of static firing tests of standard test motors is carried out to assess and validate the proposed approach. The thesis is organized in six chapters while supplementary data is presented in three appendices. A comprehensive review on the literature is presented in Chapter Two. The review focuses on the aspects of SPRM design, analysis and optimization. Chapter Three discusses the methodology adopted in developing the proposed approach. This includes both the internal ballistics model and the used optimization algorithm. The experimental work is explained in detail in Chapter Four. These details include the type and characteristics of the used propellant, the manufacturing process, and all aspects of static firing tests. Chapter Five includes the main results and contributions of the study. The outcomes of the developed optimization-internal ballistics model are explained in
4 the three cases considered. The main conclusions of the study are summarized in Chapter Six along with recommendations for future studies. The supplementary data include: In Appendix A, computer code in MATLAB for SPRM targeting maximum prediction accuracy concerning star grain geometry is listed. In Appendix B, computer code in MATLAB for SPRM with predefined performance concerning dual thrust motors with tubular grain geometry is presented. Finally, Appendix C includes results of thermochemical calculations using Isp and CEA codes for the propellant used in performing the experiments.
5
Chapter 2 Background and Literature Survey The presented chapter aims to introduce the basics and fundamentals of solid propellant rocket propulsion and related performance parameters to the readers. The chapter also surveys the previous efforts devoted to understand and predict the SPRM performance. Methods for SPRM design proposed by researchers are surveyed. Since it is the scope of the presented work, optimization applications in the field of SPRM are discussed. The roadmap of this chapter is illustrated in Fig. 2.1 below. Chapter 2
Main parts of solid propellant rocket motor
Definition of SPRM key performance parameters
Grain burn-back analysis
Internal ballistic performance in SPRM
Design methods of solid propellant rocket motors
Optimization techniques
Application of optimization in solid rocket motor design
Ranges of key design and performance parameters of SPRM
Fig. 2.1 Roadmap of Chapter 2
6
2.1 Main Parts of Solid Propellant Rocket Motor As schematically illustrated in Fig. 2.2, the design of SPRMs is mainly composed of igniter (1), front grid (2), combustion chamber (3), solid propellant grain (4), rear grid (5), and converging-diverging nozzle (6).
Fig. 2.2 SPRM main parts Generally, the motor case is a cylindrical vessel containing the solid propellant, igniter and insulator. The combustion takes place in the motor case; therefore, sometimes it is referred to as combustion chamber. The area of passage of gases inside the combustion chamber is akeyparameterandisreferredtoas“portarea”. This area depends on the grain configuration and increases during the grain combustion. The motor case must be capable of withstanding the internal pressure resulting from the grain combustion, that range approximately from 30 to 300 bar [1], with a sufficient safety factor.
Therefore, motor case is usually made from high
specifications metals / nonmetals (e.g., composite materials). In addition to the
7 stresses due to the pressure in the chamber, thermal stresses may sometimes be critical. If the motor case also serves as flight vehicle airframe, axial, bending loads and inertial forces should also be taken into consideration, especially after burn out of the motor. In a rocket motor, the thrust is obtained by discharging high energy gases (produced upon propellant combustion) through the nozzle. The performance of the rocket motor depends mostly on how much of the total energy of the product gases is converted into useful work through the nozzle. So, the nozzle is a very important subcomponent of a rocket motor. In a converging-diverging nozzle, Fig. 2.2, the high pressure product gas in the combustion chamber accelerates through the converging part of the nozzle. While accelerating, the pressure and temperature of the gas drop and the velocity of the gas reaches the speed of sound at the throat of the nozzle. Downstream of the throat, the gas further accelerates through the diverging part, reducing the pressure. Finally, the gas is discharged to atmosphere through the exit of the nozzle. The discharged gas has a velocity greater than speed of sound. Thus, this type of nozzles is also called“supersonicnozzles”.Thehighvelocitydischargedgasgivesmomentumto the rocket motor. The front and rear grids are used with free standing grains to keep them fixed inside the chamber with their longitudinal axis coincident with that of the rocket motor. In order to reduce the risk of misfiring of rocket motors, the propellant is designed to be usually insensitive to environmental conditions and external disturbances. Thus, a more sensitive igniter is used to initiate rocket motors on demand. The igniter usually consists of an electric squib, an electrically ignited primary charge, and a more energetic secondary charge, which ignites the solid propellant.
8 Solid propellant grains Solid propellant rocket motors use a fuel-oxidizer mixture, the propellant, in solid state as the energy source. Solid propellant grains can be categorized in many ways namely; according to production method, chemical composition and shape. According to production methods, the propellants can be subcategorized into two groups: cast and free standing. Cast propellants are prepared in special mixing chambers in the form of highly viscous liquid and are then poured inside the motor case.Themotoristhen“baked”,whichhelpsthepropellantto plasticize. A mandrel is placed inside the motor case to give the propellant the desired shape. The mandrel is eventually removed, leaving the cavity required for motor operation. Free standing propellants are prepared by the same method as cast propellants. However, instead of pouring the propellant directly inside the motor case, the propellants are poured inside some molds having the desired geometry. Then the propellants are “baked” like the cast propellants. The plasticized propellants are placed inside the motor case and fixed using mechanical or chemical methods [2]. Usually this kind of production method is used to obtain large burning areas, therefore larger thrust for shorter time. The second way of categorizing solid propellants is by chemical composition; double base or composite propellants. Composite propellants are the most commonly used type of propellants in solid rocket motors. They have high burning stability, high energy and relatively low response to temperature changes [2]. Composite propellants are less hazardous to manufacture and handle than doublebase propellant. Theterm“composite”indicatesthat,thepropellantismadeoftwo or more substances in a heterogeneous mixture, without any chemical binding between them.
9 Composite solid propellants are manufactured from a two-phase mixture. The first is a synthetic polymeric matrix (fuel and binder) and the second is the oxidant which is normally inorganic salt. The recent trends in manufacturing composite propellant indicate that they are commonly composed of Ammonium Perchlorate as an oxidizer, Aluminum as a fuel, organic polymer and plasticizer as a fuel binder. The typical percentages of these ingredients are 60-72%, 5-22%, 8-16%, respectively. Some additives are used to improve some features. For instance, metallic fuels (e.g., boron) increases the specific impulse and the combustion stability. Plasticizers: (e.g., dibutyl phthalate, DBP, dioctyl phthalate, DOP, and, dioctyl sebaciate, DOS) are added to improve the elongation of propellant, processing, chemical and mechanical properties and decrease viscosity. Curing (cross linking) agent helps to form interlocks chain between long chains. The percent of curing agent is small amount (0.2-3.5%). Typical examples are HMDI and IPDI. Bonding agent (e.g., MAPO) is a typical example and causes adhesion between solid particles (oxidizers) and the fuel binder in the required matrix. Its percent is more than 0.1%. Wetting agent (e.g., Waxes and Silicon oil) cause wetting for particles so it can be covered by solution. Burning rate modifiers help in accelerate or decelerate the combustion at the burning surface and increase or decrease the value of propellant burning rate such as Lead salts, lead sterate, ferrous oxide and Lithium fluoride. Its percent is about (0.2-3) %. Double base propellants are the other type of solid propellant. The term “double base” indicates that two kinds of base propellants are mixed to obtain this propellant. In double base propellants, both the oxidizer and the fuel are of the same molecule chain. Therefore, a chemical binding is present between the oxidizer and fuel. The two kinds of base propellants usually used are Nitrocellulose and
10 Nitroglycerine. Also, some stabilizing and plasticizing agents are added to the mixture at production stages. The other way of categorizing SPRM grains is according to their geometry. Typical sub-categories are end burning, tubular, multi-tubular, star, dog bone, and slotted. The different ways of categorizing solid propellant grains are summarized in Fig. 2.3.
SP grains
Production
Free Standing
End burning
Composition
Cast
Double base
tubular
multi-tubular
Geometry
Composite
star
dog bone
slotted
Fig. 2.3 Solid propellant grains categories
2.2 Definition of SPRM Key Performance Parameters The SPRM design involves numerous parameters. These parameters can be classified into distinct categories as follows [3]: 1. Properties of solid propellant, this category include the following parameters: Enthalpy, total impulse, specific heat ratio, propellant material, burning rate, characteristic velocity, propellant density, production technology and ignition.
11 2. Mission requirements which include both thrust and thrust coefficient. 3. Grain geometry: that includes web fraction, propellant geometric configuration, volumetric loading coefficient, slenderness ratio, and sliver. 4. Nozzle geometry, this category includes: exit area, throat area, nozzle shape, convergence and divergence angles and expansion ratio. 5. Other ballistic parameters includes: combustion chamber material, combustion pressure, exit pressure, combustion temperature, burning time and motor diameter. Definitions of performance parameters that are of special concern in the present work are presented below. 2.2.1 Thrust and thrust coefficient
The thrust force of a rocket motor, , is expressed as: ̇ where ̇
is the mass flow rate of gases through the nozzle exit,
(2-1) is the exit
velocity, Ae is the exit cross-section area of the nozzle, Pe is the exit pressure, and Pa is the atmospheric pressure. The rocket nozzle is usually designed so that the exhaust pressure is equal to the ambient pressure to maximize the thrust. Such nozzle is referred to as a nozzle with optimum expansion or adapted nozzle. The thrust coefficient,
, is defined as
(2-2)
12 where
the critical cross-section area of the nozzle throat and
is the
combustion chamber pressure. 2.2.2 Total impulse
Total impulse, , is defined as the integral of the thrust over the motor operating time,
. It is expressed as: ∫
(2-3)
The total impulse can be accurately determined experimentally by calculating the area under the thrust-time curve. 2.2.3 Characteristic velocity
This term is related to the efficiency of combustion and is used to compare the relative performance of different rocket motor designs and propellants. The characteristic velocity,
, has the following expression: √
where
√
, R and
the gas products, respectively.
(2-4)
are the gas constant and specific heat ratio of
is the combustion temperature. The value of
usually ranges between 800 and 1800 m/s [2, 4]. 2.2.4 Specific impulse
Specific impulse
is a measure of the impulse or momentum change that can
be produced per unit weight of the propellant consumed. It is an important ballistic parameter of the propellant that controls the overall performance of a rocket
13 propulsion system. It is analogous to the kilometers per liter parameter used with automobiles [5]. The specific impulse is defined as: ∫ ∫ ̇ where rate,
̇
is the thrust force, t is the burning time,
is the gravitational acceleration at sea level,
velocity and
(2-5) ̇ is the propellant burning is the characteristic exhaust
is the thrust coefficient.
Alternatively, the specific impulse is defined as the total impulse per unit weight of the propellant. (2-6)
The typical values of
range from 200 to 230 sec for double base propellants
and from 220 to 270 sec for composite propellants [2]. 2.2.5 Nozzle expansion ratio
Nozzle expansion ratio
is defined as the ratio of nozzle exit area to the nozzle
critical cross-section area as given below. (2-7) It is an important nozzle design parameter used in the thrust calculations since it determines flow properties at the nozzle exit.
14 2.2.6 Pressure time curve:
For the evaluation of motor performance, the static tests allow to measure the pressure and thrust histories. The measurement records are then analyzed to get detailed performance data. It is important to introduce the basic definitions on a typical pressure-time or thrust-time curve. These definitions include: ignition delay time, ignition rise time or thrust build up time, burning time, action time and tail-off time. All these definitions are illustrated in Fig. 2.4.
Fig. 2.4 Pressure time curve 2.2.7 Chamber pressure and MEOP
Chamber pressure is the static pressure measured at the head (forward) end of the chamber [2]. It is the main parameter affecting the thrust of the rocket motor. By increasing the chamber pressure, it is possible to obtain high thrust levels. However, due to the structural limitations of the motor case and other components, maximum chamber pressure is limited for the design activities. In addition, high chamber pressure environments leads to undesirable combustion instability [6]. The upper limit on chamber pressure is usually termed as Maximum Expected Operating
15 Pressure (MEOP). To avoid another form of combustion instability, coughing, a lower limit is imposed on combustion pressure. Eventually, the chamber pressure is recommended to range from 30 to 300 bar. 2.2.8 Burning rate
During the rocket motor operation, the burning surface of the propellant grain marches in a direction normally perpendicular to the surface. Burning rate is the linear regression rate of the flame edge, measured at a specific time and a specific distance on the propellant burning surface. The steady-state burning rate of a propellant (excluding the ignition phase and thrust tail-off) is defined by the ratio of minimum web to be burned over steady-state burning time [5]. The web is the minimum distance traveled by the flame edge from the start of combustion to the time when the flame reaches the outside contour of the grain. The burning rate of the propellant depends on propellant composition, combustion chamber pressure and initial temperature of the solid propellant grain [4]. For a grain of a given composition, the relation between burning rate and its governing parameters is referred to as the “burning law” of the propellant. It has different expressions, however; the most commonly used (and adopted here) is the following empirical equation: (2-8) where
is the burning rate coefficient, n is the burning rate exponent ,
camber pressure and
is the temperature sensitivity coefficient.
and
is the are the
initial and normal grain temperatures, respectively. In some cases, the burning rate is modified by an additional burning term referred to as the“erosiveburning”rate.Thisrateisdependenton velocityofgas flow parallel to the burning surface and kinematics of motor motion (longitudinal
16 and lateral accelerations and spin). There are many forms to estimate the erosive burning. The following form is one of the most well-known [7]. ( ̇ where ,
are constants and
flow rate through the nozzle, grain, and
(
)
̇ )
is the grain length,
(2-9) ̇ is the discharge mass
is the propellant density,
is the port area of the
is the grain length.
2.3 Grain Burn-Back Analysis In SPRM, the value of chamber pressure, and hence thrust, for a given grain composition is dependent on the area of burning surface. Hence, it is crucial to understand the variation of this area during motor operation. The grain burn back analysis aims to calculate the evolution of the grain burning surface during the combustion process of the rocket. From the burn-back analysis, some very useful data can be obtained. On one hand, the mass of the propellant at any time can be estimated. On the second hand, the sliver fraction necessary for calculating the tail off thrust can be derived. The location and level of expected thermal loading can be deduced. The burning and port areas at any time along the whole grain can be calculated. They are used in understanding the erosive burning characteristic. According to Ata [8] The grain burn-back analysis can be undertaken by four different methods. These methods can be classified based on their main relative features as shown in Table 2.1
17 Table 2.1 Burn-back methods [8] Method
Capability
Computation time
Accuracy
Software
Human labor
requirement
requirement
None
Low
Simple and Analytical some complex
Very low
Accurate
geometries
Numerical All geometries
Drafting All geometries
3-D burn-back
High
Very high
Based on mesh number
Accurate Based on grid
All geometries
Low
and triangle number
CAD and mesh
High
programs CAD programs CAD programs
Very high
Low
The analytical method is the simplest method and the least human labor demanding. So, it can be easily used in design optimization processes and preliminary design phase. In the detailed design phase, the grain geometries become more complex so the numerical method is used for burn-back analysis but it requires a CAD program and high computational time. At the end of the detailed design, the drafting method can be used to validate the grain burn-back analysis which is given by the other methods. Puskulcu [3] compared solid modeling codes for the numerical burn-back
analysis in his study on finocyl and tapered grains. Two solid models were used: AutoCAD Mechanical Desktop software and Unigraphics NX software.
18 With the help of the LISP language, Puskulcu managed to control AutoCAD commands externally. The commands used to draw the initial geometry are inserted into the code for one time. Then, by changing the parameters inside the code, the geometry can be updated easily. Using LISP, the process of obtaining geometries can be done automatically if the parameters are changed in order. In contrast, Unigraphics does not require an external code to perform burn-back analysis. The grain geometry is modeled parametrically for one time. Then by changing the parameters, burn-back of the initial geometry is obtained. Reddy and Pandey [9] investigated numerical grain burn-back analysis of 3D star grain geometries for solid rocket motor. The design process involved parametric modeling of the geometry in CAD software through dynamic variables that defined the complex configuration. Initial geometry was defined in the form of a surface which defined the grain configuration. Grain burn back was handeled by making new surfaces at each web increment and calculating geometrical properties at each step. They used CAD, Pro-E and ANSYS design Modular for the development of initial grain geometry with initial parameters. Then, the parameters that changed during the burn-back were adapted for every burn step. The change of the volume of the grain geometry gave the amount of propellant burned for that interval. Dividing this volume by the thickness, the burn area was acquired. Using this data and internal ballistic parameters, the pressure inside the rocket motor was obtained. Grain burn-back of a 3D radial slot grain was conducted by Kamran [10]. This was achieved by creating new surfaces at each web increment and calculating geometrical properties at each step.
19 During geometric and regression modeling, CAD software was linked to MATLAB via Visual Basic. MATLAB sent variable array to CAD software enabling automatic creation of the grain geometry. In the set of studies by Raza and Wang [11-13], CAD was used to automatically create, calculate, and update the grain geometrical properties through surface offset and boolean functions. MATLAB was used by Visual Basic to pick and update the geometry obtained by CAD software. Ata [8] developed a three-dimensional grain burn-back simulation with minimum distance method using STL (Standard Template Library) geometry output. The developed burn-back simulation tool named F3DBT (Fast three Dimensional Burn-back Tool) was claimed to yield accurate and efficient grain burn-back analysis. By comparing it with analytical and numerical ones, the minimum distance burn-back method has proven to be more practical for both detailed design and preliminary design phases. Shekhar [14] developed closed form burn-back equations for canonically trimmed tubular propellant grain which gives a neutral burning profile instead of the normal tubular propellant grain which gives a progressive burning. Wichard [15] employed level set method (LSM) to perform grain burn-back analysis in solid rocket motor simulations.
2.4 Internal Ballistic Performance prediction in SPRM The performance of a solid propellant motor can be directly measured through experiments. However, experiments have the drawbacks of high cost, high risk and low flexibility of measurements. Hence, it is vital to rely on other approaches to predict (with reasonably high accuracy) the SPRM performance parameters. This
20 motivated the researchers to develop SPRM performance prediction models. Nonetheless, such models must be validated by comparing their output with experimental measurements. According to Coats et al.[16], there are three broad categories of models namely, simple, engineering, and full-up. In simple models, the entire combustion process is treated as a single entity.in engineering models, a portion of the combustion process is idealized and other parts are treated in detail. In full-up (or research models), all important combustion processes are modeled. The simple model should contain the following sub-modules: performance using equilibrium thermochemistry, loss models, and motor model. It should include also libraries of propellants, grain shapes, material properties, and sample cases of known non-classified systems. In contrast, the engineering model should comprise modules to perform two main tasks. The first task deals with solid rocket motor including grain design, internal ballistics, nozzle throat erosion, propellant combustion, as well as combustion instability option. The other task treats nozzle flow field including nozzle inlet, throat, exhaust nozzle, two phase flow, and finite rate chemistry. Solid performance program (SPP) code by NASA is considered by many researchers as the reference internal ballistic prediction tool. Following SPP, many efforts were devoted to enhance its outcome. Terzic et al. [17] developed a computer program SPPMEF based on SPP for prediction of internal ballistic performance of solid propellant rocket motor. This program consists of four modules. The first module (TCPSP) calculates thermochemical properties of the propellant. The second module (NOZZLE) estimates nozzle dimensions and predicts the specific impulse. The third module
21 (GEOM) estimates the dimensions of grain and regression of burning surface. The last module (ROCKET) gives prediction of average delivered performance. Fig. 2.5 shows the structure outline of the SPPMEF. The program was verified with experimental results obtained from standard ballistic rocket test motors and experimental ones including rocket motor (57 mm), rocket motor (128 mm) with various grain configurations, and rocket motor (204.7 mm).
Fig. 2.5 SPPMEF flowchart [17]
22 Coats et al. [18] described the improvements to the nozzle performance modules within SPP. Both full and parabolized Navier-Stokes nozzle flow solvers were added to the code and these results were compared to the solutions adopted by SPP. The developed nozzle module has many capabilities regarding chemical kinetics, equilibrium, accuracy and stability of calculations, and robustness. The results of the modified code were compared with the motor firing data. Combustion modeling of solid propellant rocket motor and the future improvements in the modeling tools were outlined by Coats et al.[16]. Dunn and Coats[19] used the Solid Performance Program (SPP) for predicting the delivered performance of different types of 3-D grains. They developed Grain Design and Ballistics (GDB) module that calculated the ideal pressure-thrust history, and subsequently modified these ideal values based on the nozzle performance efficiencies. The Ballistic Module was automatically provided with burning surface area, cross sectional area, and perimeter from the Grain Design Module. Equilibrium gas properties were automatically transferred from the One Dimensional Equilibrium Chemistry (ODE) Module. The Ballistic Module solved the one dimensional flow Problem with energy and mass addition. The average absolute difference between predicted and measured performance loss was ~0.45%, with a standard deviation of ~0.52%. The estimated confidence level in the measured data yields a deviation in specific impulse within ± 0.5%. Shekhar [14] derived a mathematical form to estimate pressure time profile of canonically trimmed tubular grain. The performance prediction was validated by a
23 static firing test of a small propellant grain. The neutrality was obtained for a slant angle of 10 degrees and for aspect ratio of 4.5. Wichard [15] developed a one dimensional internal ballistics code to simulate the complete operational phase of motors with arbitrary complex grain designs. Monte-Carlo integration techniques were applied to calculate the burning surface and port area parameters of the burning surface. The internal ballistics solver (SPP) was the chosen as a basis for the code. A full motor simulation of a novel design was conducted using the coupled solver. The results were compared to actual test bed data. Laimek and Pawgasame [20] used object-oriented programming framework and software architecture for solving complex calculation of thrust profile and visualization of burning solid propellant. They used an external CAD program to design the geometry of the grain and divided propellant mass into several segments during thrust profile calculation. The program used to calculate thrust profile of solid propellant is illustrated in Fig. 2.6.
24
Fig. 2.6 Software architecture according to Laimek [20] Similarly, Schumacher[21] proposed a model for SP performance prediction. Fig. 2.7 shows the flowchart of that model. The performance estimation technique was validated by comparing the results with three real firing tests of solid rocket motors (ORBUS 6, ORBUS 21and the STAR 48 B).
25
Fig. 2.7 Performance estimation procedure by Schumacher[21]
26
It can be concluded that a functional SP prediction tool should include the following modules: thermochemical calculation of the propellant, geometry definition, burn-back analysis, and pressure and thrust profile estimation. Inputs to such tool should include: propellant composition, burning law, grain geometry, and motor design parameters. Additional modules can be considered for erosive burning and nozzle erosion.
2.5 Preliminary Design Methods of SPRMs According to Gandia [22], the design of a solid propellant rocket motor evolves through three major phases; preliminary design, trade studies, and final design. The overall goal of the preliminary design is to develop a motor to serve as a baseline for later trade study sensitivity analyses. The necessary input parameters for the motor included propellant properties, motor diameter, various lengths, various thicknesses, nozzle diameter, and material. Trade studies are concerned with examining the dependence of motor performance on changes of the input parameters. These studies provide the sensitivity analysis of performance with respect to input parameters. The final design utilizes trades and preliminary design to develop a compliant solution as far as mission requirements are concerned. The preliminary design phase, which is the key phase of design, is iterative by nature. Schumacher[21] proposed the scheme shown in Fig. 2.8 that illustrates the flowchart of design processes in this phase.
27
Fig. 2.8 Preliminary design procedure by Schumacher[21]
28 Sforzini [23] developed an automated technique which minimizes the sum of the squares of the differences, at various times, between a desired thrust-time trace and that calculated with a mathematical performance model of a solid-propellant rocket motor. Up to 14 design parameters were varied during the search. He used a simple model of the SRM's performance to evaluate the best match design for the desired performance requirements using three different types of star grains; standard star, slotted tubes and wagon wheel configurations. He used an optimization computer program with pattern-search technique which could treat up to 15 geometric variables. Pires and Alonso[24] developed computational system for solid propellant rocket motor design. The system determined the propellant type, burning characteristics, grain geometry, and material for combustion chamber for a certain mission. Their work focused on achieving certain mission (range, speed, time, and so on). Dauch and Rib [25] presented in details the different modules of the Propulsion and Internal Ballistics software (PIBAL) and showed through an application how a complex three dimensional grain could be designed and evaluated. This software has two distinct modules: a Grain Design Module and a Ballistics Module. They concluded that the design of a SRM is mainly based on the choice of a general grain shape associated with a type of propellant as indicated in Fig. 2.9. The entire process of a SRM preliminary design is an iterative process as shown in Fig. 2.10.
29
Fig. 2.9 Selection of grain type according to [25]
Fig. 2.10 Preliminary design process of a SPRM [25]
30 Konecny [26] presented a different method of preliminary estimation of main rocket and rocket motor parameters based on demands of unguided rocket trajectory. The first step in the design, ballistic design, proposed the rocket having the minimum possible weight for transportation of given effective payload for the required tactical-technical requirements. The next step was the determination of the principal parameters of rocker motor. Main characteristics, to be determined are total impulse and propellant charge weight. Steyn et al. [27] developed a tool for preliminary design to be used as an input for the more detailed design. First step was to define required motor performance based on user requirements. Then, thrust time curve, mass flow rate, and pressure time curve were calculated. The grain geometry was chosen according to the booster or sustainer. Star grain geometry was chosen for the booster and tubular grain for the sustainer. The proposed tool was applied on a 200 mm caliber missile. Gandia [22] designed a solid propellant rocket motor for an air-to-air missile with a total mass less than 100 kg, a payload mass of 30 kg, a total length less than 4 m, and a flight altitude around 3000 m. He developed an analytical tool for the preliminary design of the rocket motor. The results of his analytical preliminary design were compared to the simulation results from the BurnSim Software. The motor diameter was sized based on dimensions similar to those of the AIM-9 family of Sidewinder missiles. It was observed that the preliminary design did not meet the requirements of the project. So, the trade studies were utilized to determine which aspects of the motor and missile design could be improved to meet the requirements. Several trades have been made for the development of material, propellant and dimensions. Recommendation for motor casing material and propellant burn rate were attained. Stress check was done to validate the usage of Titanium as the
31 selected material. With these changes, the final design was compliant to the specified mission parameters.
2.6 Optimization Techniques Optimization is the methodology of finding the set of design operating parameters that yields the best outcome in some sense. Over the years, a wide variety of mathematical techniques of optimization has been developed and has evolved in many aspects. However, optimization techniques can be broadly classified into three groups. They are, direct (random) search, gradient-based, and gradient-free (evolutionary) techniques. It is well-established that, compared with the first two techniques, the evolutionary techniques require evaluation of objective functions only. Continuity and differentiability of functions are not required. Thus the method can be useful for non-differentiable problems, and problems for which gradients cannot be calculated or are too expensive to calculate. It is also possible to implement the algorithm on parallel computers to speed up the calculations. The deficiencies of these techniques are the unknown rate for reduction of the target level for the global minimum, and the uncertainty in the total number of trials and the point at which the target level needs to be reduced. Focus is made here to explain the underlying principles of two evolutionary techniques implemented in this research, namely genetic algorithm (GA) and simulated annealing (SA). Genetic algorithm Genetic algorithms are based on Darwin’s theory of natural selection. The specific mechanics of the algorithm use the language of biology, and its
32 implementation mimics genetic operations. The basic idea of the approach is to start with a set of designs, randomly generated using the allowable values for each design variable. Each design is also assigned a fitness value that represents the performance merit value of this design. Penalty function is used to express for constrained optimization problems. From the current set of designs, a subset is selected randomly with a bias allocated to more fit members of the set. Random processes are used to generate new designs using the selected subset of designs[28]. The size of the design set is kept fixed. Since more fit members of the set are used to create new designs, the successive sets of designs have a higher probability of having designs with better fitness values. The process is continued until a stopping criterion is met. The various terms associated with the algorithm will be explained in the following paragraph. Population is the set of design points at the current iteration. It represents a group of designs as potential solution points. The number of designs in a population is called the population size. Generation is an iteration of the genetic algorithm. A generation has a population size that is maintained. Chromosome is used to represent a design point. Thus, a chromosome represents a design of the system; whether feasible or infeasible. It has the form of a vector that contains values for all the design variables of the system. Gene is used for a scalar component of the design vector which represents the value of a particular design variable. Once the design variables are encoded in a chromosomal manner and a fitness measure for discriminating good solutions from bad ones has been chosen, genetic algorithms start to evolve solutions by using the following basic procedures [29]: 1. Initialization: The initial population of candidate solutions is usually
generated randomly across the search space.
33 2. Evaluation: Once the population is initialized or a population is created, the fitness values of the candidate solutions are evaluated. 3. Selection: Selection allocates more copies of those solutions with higher fitness values (better designs) into mating pool and thus imposes the survival of the fittest mechanism on the candidate solutions. The main idea of selection is to prefer better solutions to worse ones. 4. Recombination (Crossover): Recombination combines parts of two or more parental chromosomes to create new, possibly better design alternatives (i.e. offspring). There are many ways of accomplishing this, and
competent
performance
depends
on
properly
designed
recombination mechanism. 5. Mutation: While recombination operates on two or more parental chromosomes, mutation locally but randomly modifies chromosome by changing genes. Mutation performs a random walk in the vicinity of a candidate solution. 6. Replacement:
The
offspring
population
created
by
selection,
recombination, and mutation replaces the original parental population. Many replacement techniques such as elitist replacement, generationwise replacement and steady state replacement methods can be used. 7. Steps 2 to 6 are repeated until a terminating condition is met. Simulated annealing Simulated annealing (SA) is a stochastic approach for locating a good approximation to the global minimum of a function. The approach is named after the annealing process in metallurgy. This process involves heating and controlled cooling of a material to increase the size of its crystals and reduce their defects. At high temperatures, the atoms become loose from their initial configuration and move randomly, perhaps through states of higher internal energy, to reach a
34 configuration having absolute minimum energy. The cooling process should be slow, and enough time needs to be spent at each temperature, giving more chance for the atoms to find configurations of lower internal energy. If the temperature is not lowered slowly and enough time is not spent at each temperature, the process can get trapped in a state of local minimum for the internal energy. The resulting crystal may have many defects or the material may even become glass with no crystalline order [28]. The simulated annealing method for optimization of systems emulates this process. Given a long enough time to run, an algorithm based on this concept finds global minima for continuous-discrete-integer variable nonlinear programming problems. The basic procedure for implementation of this analogy to the annealing process is to generate random points in the neighborhood of the current best point and evaluate the problem functions there. If the cost function (penalty function for constrained problems) value is smaller than its current best value, the point is accepted and the best function value is updated. If the function value is higher than the best value known thus far, the point is sometimes accepted and sometimes rejected. The point’s acceptance is based on the value of the probability density function of the Bolzman-Gibbs distribution. If this probability density function has a value greater than a random number, then the trial point is accepted as the best solution even if its function value is higher than the known best value. In computing the probability density function, a parameter called the temperature is used. For the optimization problem, this temperature can be a target value for the optimum value of the cost function. Initially, a larger target value is selected. As the trials progress, the target value (the temperature) is reduced (this is called the cooling schedule), and the process is terminated after a large number of trials. The acceptance probability steadily decreases to zero as the temperature is
35 reduced. Thus, in the initial stages, the method sometimes accepts worse designs, while in the final stages the worse designs are almost always rejected. This strategy avoids getting trapped at a local minimum point.
2.7 Applications of Optimization in Solid Rocket Motor Design In the literature, optimization techniques were utilized by the researchers in the field of SPRM design. The applications where optimization was implemented can be categorized into three applications, namely enhance performance of a grain design, predict a design for a given performance, and enhance accuracy of IB prediction. In what follows, the efforts related to the three applications are discussed. 2.7.1 Enhancing performance of a specific grain design
Many optimization objectives have been acquired through numerous optimization techniques. One objective was to minimize the propellant mass. Nisar [30] used a hybrid optimization technique (genetic algorithm and sequential quadratic programming) on 3D finocyl grain involving 18 parameters. Similarly, Fredy [31] used GA on different grain geometries (end burning, tubular, star, etc.) which had up to 8 parameters. In contrast, Kamran[32] investigated different optimization objectives such as maximum volumetric loading fraction, minimum sliver fraction and maximum total impulse using GA on convex star grain with 6 parameters. In another study, Kamran [10] also used GA to find the maximum average thrust of 3D grain configuration with radial slots having 24 different parameters. Brooks[33] studied the optimal design of star grain comprising six design parameters. The optimization objectives were volumetric loading, sliver and burning neutrality. He concluded that the volumetric loading coefficient would be
36 maximum for five or six star points. For a given number of star points, neutrality of the thrust time history increased and sliver fraction decreased as the web fraction increased. Neutrality increased as volumetric loading fraction increased. Khurram [30] developed a method to design and optimize 3D finocyl grain configuration. A mathematical modeling of the geometry was developed for the design process of grain configuration to satisfy the requirements of mass of propellant, thrust, burning time, volumetric loading fraction and web fraction. A hybrid optimization technique composed of genetic algorithm and sequential quadratic programming (SQP) was implemented for achieving the final optimal design. The objective function of the optimization was to minimize the mass of the propellant for the required performance criteria. Fig. 2.11shows the flowchart of the module used for the performance prediction of SRMs. The interference between this module and optimization algorithm is illustrated in Fig. 2.12.
Fig. 2.11 Grain geometry and performance module [30]
37
Fig. 2.12 Hybrid optimization algorithm [30] Kamran [10] optimized the design of 3D radial slot grain configuration for upper stage solid rocket motors to minimize the inert mass by adopting a high volumetric loading with minimum possible sliver. Computer aided design was used for parametric modeling of the geometry during the design process. Equilibrium pressure method was used to calculate the internal ballistics. Genetic algorithm (GA) was used as the optimizer. The flowchart for the grain design process is illustrated in Fig. 2.13.
Fig. 2.13 Grain design process by Kamran [10]
38 Kamran [32] optimized neutral and non-neutral star grain configuration for geometric and ballistic objective functions with respect to volumetric loading fraction, sliver fraction and web-time total impulse. He used almost the same technique in his previous paper [10]. CAD module performed parametric modeling of the star grain. Equilibrium pressure method calculated the internal ballistics and genetic algorithm was used as an optimization technique. For the cases of neutral and non-neutral burning, star web burn-out time and average pressure proved to be driving variables, respectively. In addition, the nonneutral configuration provided higher web time total impulse compared to neutral star configuration. Ballistic objective function proved superior over geometrical objective function. Applications of optimization techniques in the design of dual-thrust motors were discussed by Raza and Wang in a number of studies [11-13, 34, 35]. In [34, 35], they used robust design to optimize the mean performance and reduce the performance variation in terms of thrust ratio and total impulse. That approach consisted of 3D geometric design coupled with complex internal ballistics, hybrid optimization, worst-case deviation, and efficient statistical approach. The performance of such propulsion system was determined by internal ballistics through either motor grain burn-back analysis with CAD modeling or employing analytical expressions. They optimized their design by using two different methods. In the first method, hybrid optimization (Simulated Annealing with Pattern Search) was implemented. In the other method, uncertainty-based robust design optimization was utilized. Fig. 2.14 and Fig. 2.15 illustrate the flowcharts of both methods.
39
Fig. 2.14 Framework of hybrid optimization [34]
Fig. 2.15 Framework of uncertanity-based optimization [34]
40 Raza and Wang concluded that the hybrid approach of Genetic Algorithm and Simulated Annealing had shown excellent performance about the efficiency from the optimization view point. The values of performance parameters achieved by robust design were less than those achieved by optimal one but insensitive to variations. Raza and Wang in [11-13] discussed a methodology for the design of a 3-D wagon wheel. A hybrid genetic algorithm and simulated annealing were used for optimization. The objective function of optimization was to maximize the average thrust ratio and the total impulse. They argued that the design optimization methodology showed promising results in optimizing the grain geometry design variables and motor performance. The hybrid approach of Genetic Algorithm and Simulated Annealing had shown excellent performance in enhancing the efficiency as well as in the overall optimization of the process. Peretz and Berger [36] studied range extension of a 150 mm air-to-air missile by thrust profile optimization with a maximum weight constraint. The optimization code used a quasi-Newton algorithm (successive quadratic-programming method) to maximize range or minimize motor mass with operational and technological constraints. 2.7.2 Predicting the grain design for a given performance criteria
As discussed in sec 2.5, the proper design of solid propellant rocket motors (SPRMs) involves multi-disciplinary algorithms to develop efficiently and accurately the designs related to the required performance parameters. Over the years, researchers developed tools for the preliminary design of SPRMs. Generally,
41 these tools comprise three steps: geometric modeling, burn-back analysis and optimization. Steyn et al. [27] developed a tool for preliminary design of a SPRM based on required motor performance. They optimized the grain design by using level set method to create a grain burn-back module. Yücel et al.[37] developed a method to obtain the optimum design of threedimensional grains of solid rocket motors for satisfying an objective thrust versus time profile and maximizing the total impulse under constraints of chamber pressure and propellant mass. They developed 3-D analytical burn-back code (BB3D) for the geometric model and burn-back analysis. For the performance prediction of the rocket motor, the internal ballistics solver (0DSOLVER) was developed. Genetic algorithm (GA) and complex methods were used for optimization as they are derivative free methods. A 3-D finocyl grain with 8 axial slots at the fore end and a radial slot at the aft end was modeled as a case study. The objective function was to minimize the sum of differences between the objective and computed thrust values at specified times during motor operation divided by average objective thrust and total number of time data. Acik [38] developed a tool for the design of solid rocket motors. It involved geometric modeling of the propellant grain, burn-back analysis, ballistic performance prediction analysis of rocket motor and the mathematical optimization algorithm. A direct search method with complex algorithm was used in this study. The optimization algorithm changed the grain geometric parameters and nozzle throat diameter within the specified bounds, finally achieving the optimum results. The
42 objective function was to minimize the error between the desired and computed thrust values. The sum of the squares of the error values at specified time instance during motor operation divided by the average desired thrust and total number of data was adopted as the measure of error.
Fig. 2.16 Design optimization tool architecture[38] Static firing test results of a test motor having tubular propellant grain geometry with two burning ends were used for validation of ballistic solver and implemented with the tool. Optimization tool validation was done by comparing the results with known solutions and predesigned rocket motor results. Fredy [31] aimed to find the optimal SRM design to maximize the mission performance in terms of thrust. A genetic algorithm (GA) optimization method was used. Sensitivity analysis of the optimized solution has been conducted using Monte Carlo method to evaluate the effect of uncertainties in design parameters.
43 2.7.3 Enhancing the accuracy of IB prediction
In all cases, researchers rely on theoretical techniques to predict the performance of SPRMs. The accuracy of such tools is a crucial aspect as far as credibility of these tools is concerned. This motivated researchers, in many cases, to improve the accuracy of the tool they use via optimization. In this respect, the optimization technique is used to minimize the root mean square error (RMSE) between the desired and computed performance merit. In [37-40], different optimization methods were used such as complex method, pattern search and genetic algorithm. Sforzini [39] used pattern search for optimizing the computed thrust-time profile of a 3D finocyl grain with 10 parameters. Both Acik [38] and Yücel [37] used complex method to find the minimum RMSE but in different cases. Acik [38] optimized different grain geometries (end burning, internal burning tube, slot, slot-tube, star and star-tube) with geometric parameters up to 9. Yücel [37] optimized a 3-D finocyl grain with 8 axial slots at the fore end and a radial slot at the aft end with 11 parameters. He also used genetic algorithm on his case study. Recently, Gawad [40] used genetic algorithm to find the minimum RMSE but on DTRM with tubular grain with two different diameters and sloped grain near its head end with 10 parameters. In summary for the previously detailed literature, optimization objectives were varied according to required performance merits. They can be summarized in the following: Minimize the mass of the propellant for the required performance criteria [10, 30]. Minimize the inert mass by adopting a high volumetric loading with minimum possible sliver [33].
44 Maximize the average thrust ratio and the total impulse in dual thrust rocket motors [11-13, 37]. Maximize range or minimize motor mass with operational and technological constraints [31, 36]. Minimize root mean square error between the desired and predicted thrust profiles [34, 35, 38]. It is clear that many studies implemented GA as the optimization method. This may be justified by its ability to define the global optimum inside the domain of study. For more accurate results, researchers refine optimization results via a hybrid optimization technique with a method for global search followed by a method using local search superiority. Khurram [30] used a hybrid optimization technique which consists of a genetic algorithm for optimizing global minimum and sequential quadratic programming algorithm for fine search of optimum solution. Raza and Wang Liang [34] also used hybrid optimization technique that consists of Simulated Annealing with Pattern Search. Kiyak [41, 42] used ant colony optimization (ACO) with local search capability to optimize star grain parameters.
2.8 Ranges of Key Design and Performance Parameters of SPRM Acceleration The initial acceleration for existing powerful solid rockets ranges from 250 to 350 m/s2 [26]. Higher acceleration is achieved by small propellant mass whereas lower acceleration is associated with large propellant mass.
45 Combustion chamber pressure profile Normally, the pressure inside the combustion chamber ranges from 30 to 250 bar. For short duration of rocket operation time, high pressure can be selected while low pressure is associated with long duration. The normal chamber pressure can be taken to be 60 bar for burning time more than 2 sec and 100-200 bar for burning time from 0.5 to 2 sec. [1] The higher the operating pressure, the higher is the specific impulse of the motor, but also the higher is the thickness and mass of the case. There must be a tradeoff and optimization of the motor design [43]. Slenderness ratio for rocket motor In case of salvo rockets, more slender constructions with smaller calibers are chosen. In case of tactical guided rockets, shorter constructions but with greater calibers are selected [26]. The nominal values of motor diameter are found to be dependent on rocket range and explained in Table 2.2. Table 2.3 shows the nominal values of motor slenderness (length-to-diameter) ratio categorized according to rocket application. Table 2.2 Ranges of rocket motor diameter [44] Range (Km)
1000
2000
10000
Diameter (m)
1.4 : 1.8
2.0 : 2.4
3.0 : 5.0
46 Table 2.3 Slenderness ratio for rocket motor [4] Category
Application
Large boosters and
Space launch vehicles and lower
second stage motors
stage of ballistic missiles
High altitude
Upper stages of ballistic missiles
motors
and space launch vehicles High acceleration: antitank and unguided artillery
Tactical missiles
Motor slenderness ratio 2-7
1-2
4-13
Modest acceleration guided or unguided: air-to-surface, surface-to-
5-10
air, guided artillery, air-to-air. The motor slenderness ratio controls the maximum vehicle performance and maximum cost effectiveness. It’s determined by tradeoff optimization among minimum inert weight, minimum vehicle drag loss, and case buckling stability and stiffness requirements[45]. For a complete missile, the fineness ratio is found to vary from 8 to 14. If the fineness ratio is too small, the drag increases reducing the range. In contrast, a missile with very high fineness ratio will have a low stiffness structure [44] Propellant grain configuration alternatives Neutrality is solely dependent of grain configuration and dimensions. It is illustrated in Table 2.4.
47 Table 2.4 Neutrality for different propellant grain configurations [43] Burning area evolution
Grain configuration
Wagon wheel high neutrality
Dual level
comments Short burning time, low volumetric loading fraction
Star
Significant sliver
Tube
Very low volumetric loading fraction
End burning
Long burning time, low thrust
Slotted tube
Ratio of the two levels can be adjusted
Star
by varying the geometry
End burning with
Volumetric loading fraction may reach
annular slots
0.88 and l/d may reach 10
The recommended values of grain design parameters: web fraction, volumetric loading fraction, and sliver fraction are listed in Table 2.5. The table also summarizes the main performance parameters in a qualitative comparison.
48 Table 2.5 Main characteristics of common grain configuration [4, 43] Volumetric Configuration
loading
Burning area
fraction
Burning
Sliver
Web
area
fraction
fraction
neutrality
(%)
(d/w)
Star
0.75 : 0.84
Intermediate
Good
5 : 10
3.5 : 5.5
Slotted tube
0.75 : 0.85
Large
Good
0
3
Wagon wheel
0.5 : 0.7
Very large
Excellent
5 : 10
6 : 12
Dual propellant
0.9
Intermediate
Excellent
End burning
0.98 : 1
small
Excellent
0
1
Internal burning
0.8-0.95
Intermediate
Good
0
1.2:2
Dendrite
0.55-0.7
Very large
Good
5 : 10
5:10
0.75-0.85
Large
Excellent
0
2:3.5
Dog bone
0.7-0.8
Very large
Excellent
5 : 10
3.5:5
Rod and tube
0.6-0.85
Large
Good
0
2:3.5
Lower than 5
2.5 : 3
Internalexternal burning
Motor operating time The motor operating time varies significantly according to application. It varies from 0.25:1 sec for highly accelerated anti-tank and unguided artillery to 60:120 sec for space launch vehicles [4]. Others Values of the mass of propellant to the total motor mass range from 0.4 to 0.95 and 0.86 for modern designs [1]. Burning rates ranges from 1 to 50 mm/s [1].
49
Chapter 3 Methodology As stated in chapter 1, the objective of the present work is to develop an approach capable of predicting the design of solid propellant grain for a predefined performance. To fulfill this objective, the methodology of work comprises seven main steps.
3.1 Over-View of the Research Methodology and Work Plan 1. A burn-back model is developed to handle star and tubular grains as illustrated in Appendix D. 2. An internal ballistics prediction model (IBPM) is developed based on the governing physical principles and relations. The model is designed to handle any grain configuration through a sub-model of grain burn-back analysis model. However, only star and tubular grain configurations are considered. The internal ballistic model can handle both single and dual thrust profiles. The IBPM code for star and tubular grains is supplemented in Appendix A and Appendix B respectively. 3. The developed internal ballistic model is validated to assess its accuracy of prediction. This is done by comparing its prediction with experimental measurements of a star grain from the literature. 4. The successfully validated internal ballistic model is coupled with the optimization algorithm. The resulting coupled model (OPTIBPM) is implemented in two applications. 5. The first application is to maximize the accuracy of prediction of internal ballistic model by tuning its parameters. The quality of the optimized model is assessed by comparing with the experimental measurements of the star grain from the literature.
50 6. The second application is to predict the grain geometry for a predefined thrust-time profile. Two cases are considered: A single-level neutral pressure-time profile using a star grain. This case is conducted to confirm the validity of the proposed model by reproducing the experimental pressure-time profile of the star grain from the literature. Upon success of this case, the other case follows. A dual level pressure-time profile. This is the main case of the work. A thrust time profile is assumed and a suitable grain configuration is assumed and its dimensions are predicted by the model. 7. The predicted grain configuration is manufactured and tested experimentally. The resulting measured pressure-time profile is compared with both of the required and that predicted by the optimized model. Fig. 3.1 illustrates the different activities according to the work methodology. The present chapter is devoted to discuss in detail the aspects of the first four of the steps listed above. For the sake of coherence, the results of application on OPTIBPM (step 5 to 7) are discussed separately in chapter 5. All details of the setup and preparation of own experiments are presented in chapter 4.
51
Burn-back model of Star grain and tubular grain
Development of IBPM
Experiment from literature
Validation of IBPM success Development of OPTIBPM
Applications of OPTIBPM
Optimizing prediction accuracy of IBPM
Validation of OPTIBPM
Assessment
Prediction of grain configurations and Geometry for predefined performance
Star grain
Tubular grain
success
Developing own experiments
Fig. 3.1 Work methodology flowchart
Assessment
52
3.2 Internal Ballistics Prediction Model (IBPM) A mathematical model for the internal ballistics of the solid propellant grain is developed based on the mass balance of the gas products [1, 4]. The developed model adopts the following assumptions: The flow along the combustion chamber and nozzle is isentropic. The gas products are ideal gases. Regression of surface over the grain length is linear. The computations are performed at two stations; at the head (forward) and nozzle ends (back ward) of the grain. The fillet radius of the star point angle is equal to zero and the grain erosive burning is accounted for. The typical pressure time profile can be divided into three phases; the initial (ignition) pressure rise, the quasi-steady state phase, and the exhaust phase. These phases are illustrated schematically in Fig. 3.2 .
Fig. 3.2 Pressure time profile phases
53 In the initial pressure rise (ignition) phase, the igniter is activated to bring the chamber pressure and propellant surface temperature to a level sufficient to ignite the propellant grain surface. The initial pressure rise is dependent on the igniter charge rather than the main propellant grain. It is thus overlooked in the model. The quasi steady state operation phase generally occupies the longest time in the motor operation. In the present analysis, this phase starts immediately after ignition. The phase ends at the moment when the burning gases reach the inner wall of the combustion chamber. For the case of a star grain, this corresponds to the end of phase three. Finally, during tail off, the burning surface decreases sharply in two distinct regimes. In the first regime, the mass of gases produced by combustion still represents a fraction of flow discharge through the nozzle. This phase is characterized by high port area in the nozzle end section together with a reduced mass flow rate in consequence of reduced burning surface. The following conditions are thus assumed: (1) reduction in gas velocity, (2) absence of erosive burning and (3) absence of pressure drop along the combustion chamber. In the second regime, after the burning is completed, the remainder of combustion gases simply exhausts out of the nozzle. This phase is characterized by: (1) zero burning surface and (2) the rate at which the chamber pressure decreases with time is relatively high. 3.2.1 Inputs to the internal ballistics model
The following parameters should be available as inputs to the model: i.
Geometric parameters: Detailed dimensions of the grain
54 Nozzle critical diameter Motor case inner diameter ii.
Thermochemical parameters: Specific heat ratio of the gas products, Gas constant of the gas products, R Adiabatic flame temperature of the propellant, Characteristic velocity of the propellant, The values of these parameters are derived using experimental measurements,
chemical analysis, or theoretical thermochemical calculations. iii.
Ballistic parameter; they include the burning low parameters: Pressure exponent, n Burning law coefficient, Temperature sensitivity coefficient, The values of these parameters are derived using experimental measurements of
the propellant burning. 3.2.2 IBPM calculation procedure
Quasi steady state phase The grain burn-back analysis explained earlier is performed to calculate the burning and port areas of grain configuration. Analysis of internal ballistics is initiated by a guess. The gas Mach number at the nozzle end of the grain, assumed to be
, is
. The final Mach number value is evaluated iteratively by
the following equation until the difference between the old and new Mach number values at any iteration is as small as 0.000001.
55
[
where ,
,
(
)]
(3-1)
are specific heat ratio of the combustion gases, port area of the
grain at each burning step and critical section area of the nozzle, respectively. The flow velocity of gases at the nozzle end,
(
√
where and
is then estimated as follows:
(3-2)
)
is the characteristic velocity of the propellant defined as
√
√ The isentropic stagnation pressure of flowing gases,
is estimated as an
initial value from the relation:
[
Where
,
,
]
(3-3)
are propellant density, burning area of the grain at each step
and the pressure exponent of the propellant, respectively. Hence, the pressure at nozzle end,
is:
56
(
(3-4)
)
Now, the rate of discharge of gases (through the nozzle),
̇ is calculated as:
̇
(3-5)
Hence the gas pressure at head end of the grain,
is:
̇
The burning rate at the head end of the grain,
(3-6)
can be obtained by:
(3-7)
where is the burning rate coefficient of the propellant. As stated earlier, the model also accounts for augmentation of burning rate of propellant by the erosive burning. This is expected to take place (if any) at the nozzle end of the grain port. The total burning rate at the nozzle end due to applying the erosive burning rate is evaluated iteratively via the following relation until the difference between the new and old burning rates could be considered negligible:
57
( ̇
where , gases,
are constants and
̇ )
(
)
(3-8)
is the grain length. The rate of generation of
̇ is estimated using the following equation: ̇
(3-9)
The discharge mass flow rate is obtained more accurately iteratively until the difference between the new and old value is negligibly small using the following equations [4]:
̇
where
̅ ̇
∑
̅
,
(̅ ̅
and
)
(3-10)
are the initial free volume of combustion
chamber and j is the burning step. The rate of change of chamber pressure ( ) is computed as follows: ̅ ̅
̅ ̅
(3-11)
The bars on the symbols in the equations above indicate the respective average values between two successive iteration steps.
58 During the quasi-steady state phase, the chamber pressure varies due to the change in the grain burning surface area. The computation of the pressure-time curve requires iteration because the burning surface is a function of the distance burnt
during a time increment
distances
. The grain web is divided into equal
. Hence the time increment for the calculations is:
(3-12)
As stated earlier, the first phase of the exhaust phase is characterized by no pressure drop along the chamber. This is formulated as:
Hence: The rate of change of chamber pressure is obtained from:
(3-13)
̅
In the second phase of the exhaust, the pressure is computed using the relation: [
where
,
]
(3-14)
are the final free volume of the combustion chamber and the
discharge pressure, respectively.
59 The discharge mass flow rate in this phase can be obtained from:
̇
(3-15)
Threshold velocity model In handling erosive burning, IBPM adopts the concept of threshold velocity below which the erosive burning is negligible. The dependence of the threshold mass velocities on pressure was shown by Kreidler [46]. It can be expressed in the form:
(3-16)
where
(Kg m-2 s-1, bar)
The above relationship is used for a preliminary check for a given rocket motor flow and pressure conditions for using the erosive burning model. If the gas velocity is greater than the threshold velocity the erosive burning model will be used. Otherwise, the IBPM will skip the erosive burning model. Nozzle erosion model IBPM also accounts for nozzle erosion. The nozzle critical diameter,
, is
assumed to increase with time, t, according to the relation:
(3-17)
60 Where
is the initial critical diameter of the nozzle,
of nozzle critical section and
is the erosion rate
is the delay time for the onset of erosion.
3.2.3 IBPM flowchart
The following flow chart in Fig. 3.3 illustrates the procedure of calculation of the pressure time curve for the solid propellant rocket motor.
61
start
Input: propellant characteristics and grain geometry
Begin calculation at initial surface of the propellant y=0
Y 2 bar
true false
Start of sliver and tail off pressure calculation loop
Pressure time-profile of pressure and tail off phase
Draw the overall pressure- time profile
Fig. 3.3 Internal ballistic prediction model flow chart
62 3.2.4 Validation of IBPM
A grain with star perforation geometry used by Maklad [47] is adopted in this study to validate the performance prediction program. The star grain of tested motor is shown in Fig. 3.4 while the structure of the motor nozzle with graphite insert is shown in Fig. 3.5. The parameters of the propellant of the star grain are listed in Table 3.1.
Fig. 3.4 Star grain of tested motor [47]
Fig. 3.5 Nozzle structure with graphite insert [47]
63 Table 3.1 Case study parameters [47] Propellant characteristics:
Propellant characteristic velocity
1560
m/s
Pressure exponent
0.42
---
0.0000113
---
Density of the burning propellant
1680
kg/m3
Specific heat ratio of the combustion gases
1.24
---
36
mm
Initial free volume of the combustion chamber
0.004286
m3
Final chamber volume
0.017356
m3
No of star points
7
---
Star point angle
74
degree
Angle fraction
0.5058
---
Grain inner radius
23.5
mm
Fillet radius
1.6
mm
Web thickness
33.5
mm
1.60411
m
Burning rate coefficient
Motor characteristics: Initial critical diameter of nozzle
Star grain parameters:
Length of star grain
Fig. 3.6 presents a comparison between the experimental and predicted pressure time profiles of the star perforated grain in concern. The IBPM manages to capture the general trend of the pressure-time profile. The deviation from the experimental results may be attributed to accuracy in experimental measurements, uncertainties in models of erosive burning, critical section erosion, and burning law parameters. The underlying assumptions of the IBPM may also contribute to its accuracy.
64 It is found that the root mean square error (RMSE) between the theoretical and experimental pressure-time profiles during the entire operation time is ~6.5 %. Neglecting the tail-off pressure phase, the RMSE is about 3.2 % in the steady state phase. It can be concluded that the IBPM can be reliably used to predict the pressure-time profile of a solid propellant grain of a complicated configuration. 180 160 140 120
Pressure (bar)
100 80 60 IBPM Experimental [47]
40 20
0 0
0.5
1
1.5
2
2.5
3
3.5
4
time (s)
Fig. 3.6 Literature vs. theoretical pressure time curve of star grain [47]
3.3 Optimized Internal Ballistic Prediction Model (OPTIBPM) The validated IBPM is coupled with the optimization algorithm to form the optimization model OPTIBPM. Here, the genetic algorithm is used as the optimizer. The model works as follows: 1. The parameters to be optimized and the desired performance criteria are identified. Here, the pressure time-profile is adopted as the criteria.
65 2. The upper and lower bounds of each parameter are input to the model. In addition, the pressure-time profile to be satisfied via optimization is input to the model. 3. The optimization algorithm manipulates the design parameters, to forward them to the IBPM to generate the predicted pressure time profile. 4. The IBPM calculates the error between predicted and target pressuretime profile. The root mean square error (RMSE) is taken as the merit of error evaluation. 5. The optimization algorithm modifies the design parameters. 6. The steps 3,4 and 5 are repeated until the error of pressure-time profile prediction is minimized. 7. The global optimum attained by GA is enhanced by applying simulated annealing SA that locates the optimum design parameters more accurately. Genetic algorithm settings can be summarized in the following: Population size is 200. Fitness scaling is rank. Selection function is stochastic uniform. Reproduction elite count is 10 and crossover fraction is 0.8. Mutation function and crossover are constraint dependent. Stopping criterion used to terminate the optimization iterations is one of two conditions. The first condition is when the number of iterations reaches 600. The other condition is when the average change in the fitness function value over stall generations is less than function tolerance. Function tolerance is chosen to be 1e-6. Fig. 3.7 shows the main processes undertaken by OPTIBPM.
66
Initial guess of parameters Parameter bounds
New generation of parameters by optimizer
Matlab Optimizer (genetic algorithm)
Grain geometry
Burn back analysis
Ballistic parameters Propellant properties Nozzle characteristics
Internal ballistic prediction model
no Optimum solution yes end
Fig. 3.7 OPTIBPM tool flowchart
67
Chapter 4 Experimental Work 4.1 Introduction The present chapter is devoted to presenting the setup of experiments that have been conducted in the context of the present research. These experiments were used to validate the predicted grain dimensions that satisfy the predefined thrust-time profile via OPTIBPM. The chapter is also intended to serve as a concise reference for the researchers in the field of experimenting solid motor propulsion especially using composite propellants. Manufacturing of composite propellant involves complicated processes which will be illustrated in detail in this chapter. These processes are performed in the research mixer at Military Factory 18. Experimental tests are also conducted to determine the ballistic properties of the propellant. These properties are key inputs to the OPTIBPM.
4.2 Manufacturing Processes of Composite Propellant In the present study, the composite solid propellant used in the design of the solid rocket motor has the chemical composition as indicated in Table 4.1. Ammonium Perchlorate (AP) with three different particle sizes is selected. This trimodal AP composition is selected to increase the packing of the propellant and to increase the specific impulse of the manufactured propellant [2].
68 Table 4.1 Composition of the propellant Ingredient
Chemical designation / symbol percentage
Binder
HTPB
10.401
Bonding agent
MAPO
0.2
Plasticizer
DOS
2.6
Fuel
AL(2 micron)
17
AP(400 micron)
21
AP(200 micron)
22
AP(7-11 micron)
26
IPDI
0.799
Oxidizer
Curing agent 4.2.1 Mixing operations
The mixing operation consists of kneading of the solid phase (oxidizer and fuel) and the liquid one (binder) to get a homogenous moldable mixture. The premixing is done in a container equipped with a mixer. After they are weighed, Aluminum is poured into the container with the fuel binder Hydroxy Terminated Poly Butadiene (HTPB), plasticizer and bonding agent. The mixture is continuously agitated to ensure a good mixing. Fig. 4.1 shows the mixing containers with ingredients prior to mixing.
69
Fig. 4.1 The ingredients before mixing
Fig. 4.2 Mixer double wall vessel
inside the mixing container Following the mixing process, kneading process is performed to produce homogeneous slurry. The kneading process requires the slurry to be heated. This is achieved using a mixer with double wall system as shown in Fig. 4.2 and a vertical mixer with two blades, Fig. 4.3. The final slurry is shown in Fig. 4.4.
Fig. 4.3 The vertical mixer with two blades
Fig. 4.4 The slurry after premixing
70 The oxidizer, Ammonium Perchlorate (AP) is prepared by grinding to obtain different particle sizes. After that, AP is weighted and separated into four quarters to be added one by one to the mixture. Finally, the curing agent is added to finish the mixing operation and prepare the mixture for the casting process. Fig. 4.5 shows the final product.
Fig. 4.5 The slurry after adding the curing agent 4.2.2 Casting of the grains
The molding cases are assembled and fitted in closed vessels before beginning the casting process. Then, the feeding pipelines are assembled and checked by the operator. The casting process is done in the closed vessel under vacuum as shown in Fig. 4.6. The slurry is casted in the molding cases under gravity and with a very slow flow rate using a valve. A shaker is used to vibrate the molding cases to eliminate air bubbles and cavities.
71
Fig. 4.6 Assembly of the vessel to the vacuum chamber and the shaker before casting 4.2.3 Curing and finishing
Curing is intended to accelerate the hardening of the propellant. This is done by raising the temperature of the propellant to a moderate degree in furnaces as shown in Fig. 4.7. Microscopic changes in the propellant taking place in the curing process improve the properties of the propellant grains. Typically, the curing time varies from 7 to 10 days under 55 degrees Celsius. After curing the grain is removed from the mold as shown in Fig. 4.8. Final grains are shown in Fig. 4.9.
72
Fig. 4.7 Curing furnace
Fig. 4.8 Cast grain before finishing
Fig. 4.9 Finished grains
73
4.3 Determination of Propellant Properties The density of the propellant is measured to be 1730 kg/m3. The propellant ballistic properties are measured experimentally in standard small-scale two-inch test motors whereas the compositions and thermochemical properties of the combustion gas products are theoretically calculated. The maximum and minimum expected pressures, the corresponding burning rates, and the propellant characteristic velocity are obtained experimentally. This is done using a standard two inch test motor. The main task of this motor is to determine the burning rate of the propellant at different combustion pressures. A tubular composite propellant grain is cast inside a special casing so that the burning is allowed only from inside and from the two ends. 4.3.1 Assembly of standard two-inch test motor
The following steps summarize the assembly of the test motor: Propellant is cast into the casing of length 108 mm, outer diameter 64 mm and inner diameter 60 mm. A mandrel of diameter 34 mm is used to create the inner grain diameter. The casing and mandrel are fixed to a special fixture. Propellant curing is secured before dismounting from the fixture. Propellant is X-ray inspected to ensure that it is crack-free and bubblefree. The test motor is assembled with the propellant charge, the igniter, nozzle, closure, sealing and pressure transducer adaptor as shown in Fig. 4.10. The pressure at the head end is measured while the pressure at nozzle end is calculated theoretically. The motor is designed in such a way that further
74 modifications are possible regarding the nozzle critical section to allow investigation of different combustion pressures.
Fig. 4.10 Assembly drawing of two-inch test motor
Fig. 4.11 Two inch motor
75 4.3.2
Test Procedure
An electric signal (about 2-3 DC) volt is initiated from the control room to the igniter. As the burning proceeds from inside inner and the two end surfaces of the grain, it is expected to realize a nearly neutral burning. For reliability reasons, two pressure transducers are connected to the point of measurement. The output pressure-time curve is displayed and recorded for further data processing with the help of LABVIEW software, the duration of the test ranges from 2 to 5 s. Input data of test: The following are the inputs for the post-processor of data: length of the grain (L) mass of the propellant (mp) initial temperature (T) the estimated operation time (t) the max expected pressure (P) the web thickness (w) inner diameter (d) critical diameter (dcr) Output data of test: The following are the outputs of the test the ambient temperature (Temp) the throat diameter (Throat) the mass of the propellant (WT) the delay time (TD) the actual time of burning the propellant (TB)
76 T50% the maximum pressure (Pmax) the area under the pressure-time curve (IPT) the average pressure (Pav) P 50% R 50% the characteristic velocity (C*) 4.3.3 Experimental measurements of propellant ballistic properties
Seven static firing tests were performed at normal temperature (20oc) using different nozzle throats to determine the burning law of the propellant. By varying the nozzle throat, the combustion chamber pressures (P 50%) as well as the burning rates (R 50%) are varied. Plotting the associated values of P 50% and R 50% yields the burning law of the propellant at the given temperature. The table shown in Table 4.2 lists the results of the seven tests. Table 4.2 Experimental results ABU ZAABAL COMPANY FOR SPECIAL CHEMICALS, LABORATORY SECTOR, STATIC TEST STAND DATE11/04/2017 GR.no
Temp
Throat
WT.
TD
units
C
mm
kg
msec
14/B1
20
9.50
0.360
11/B1
20
9.00
12/B1
20
9/B1
TB
T50%
Pmax
sec
sec
bar
39
1.890
1.789
0.358
17
1.774
9.00
0.360
9
20
8.50
0.358
10/B1
20
8.50
2/B2
20
3/B2
20
IPT
Pav.
P50%
R50%
C*
bar.s
bar
bar
mm/s
m/sec
42.4
81.0
42.84
45.26
7.07
1592
1.719
52.1
89.6
50.51
52.12
7.50
1590
1.890
1.715
50.2
88.9
47.02
51.82
7.52
1571
18
1.720
1.659
59.9
100.6
58.51
60.66
7.78
1594
0.358
19
1.697
1.648
62.5
102.1
60.14
61.93
7.83
1615
8.00
0.379
22
1.549
1.489
75.1
111.2
71.78
74.68
8.50
1476
7.00
0.334
37
1.398
1.352
119.2
137.7
98.49
101.8
8.99
1587
In all tests, two pressure transducers were used to measure the internal pressure of the combustion chamber. All pressure transducers were calibrated prior to test.
77 Pressure transducers have the following specifications: 1. Make and model: HBM P6AP absolute pressure transducer. 2. Type: full bridge strain gauge. 3. Upper pressure limit range: 200 bar. 4. Output at nominal pressure: 2 mv/v. 5. Frequency response: >100 kHz. Data acquisition system: 1. Make and model: NI PXI-6123 DAQ. 2. Platform: NI PXIe-1062Q. 3. Controller: NI PXIe-8370. 4. Measuring frequency: 500 kHz. The values of R 50% are plotted against those of P 50% in Fig. 4.12. A power-law curve fit is introduced to generate the trend line representing the burning law of the propellant. MATLAB [48] is used to generate the curve fit. Eventually, the burning law of the used propellant had the form:
burning rate (mm)
9.5 9
8.5 8 7.5 7 40
50
60
70
80
90
100
110
pressure (bar)
Fig. 4.12 Results of static firing tests and curve fitting using MATLAB
78 4.3.4 Thermochemical calculation of the used propellant
Thermodynamic and thermochemical properties of the propellant were derived theoretically using conventional packages. Here, ISP [49] and CEA [50] codes were implemented and the outputs of the two codes were compared at six different combustion pressures namely 30, 42, 50, 60, 75, and 100 bar. These results were compared with the experimental data. The inputs to Isp code are the ingredients of propellant, their heats of formation, and percentages in weight in the whole mixture. Table 4.3 lists the inputs. Table 4.3 Composition details for Isp program Propellant Heat of Formation
Weight
AP
-70.6900
69.0000
AL
0.0000
17.0000
HTPB
-14.6411
10.4010
MAPO
-223.6000
0.2000
DOS
-344.2100
2.6000
IPDI
-88.8000
0.7990
The CEA tool assumes equilibrium composition during expansion from a “theoretical”finiteareacombustor.InputstotheprogramarethesameinputsofIsp tool. All results of thermochemical calculations based on Isp and CEA are listed in Appendix C. The main results of the thermochemical calculations are illustrated in Fig. 4.13 to Fig. 4.15 as estimated by Isp and CEA. Fig. 4.13 below compares the propellant density and characteristic velocity values measured experimentally and calculated theoretically using Isp and CEA. Fig. 4.14 shows the variation of chamber temperature and exit temperature with chamber pressure. The variation of specific
79 impulse and thrust coefficient with chamber pressure is shown in Fig. 4.15. The typical relations are evident in these figures. The comparison proved a close agreement in general between measured and calculated results. 1800 1750 1700 1650
Exp Isp
1600
CEA 1550 1500 1450 C*
Denisty
Fig. 4.13 Comparison of results 3500 3300 3100 Tc(Isp)
Temp (K)
2900
Tc(CEA)
2700
Texit(Isp)
2500
Texit(CEA)
2300 2100 1900 1700 1500 0
20
40
60
80
100
120
Pressure (bar)
Fig. 4.14 Variation of chamber and exit temperatures with chamber pressure
80
290
1.7 1.65
280 270
1.55 1.5
260
1.45 250
1.4 specific impulse(Isp)
240
1.35
specific impulse(CEA)
1.3
Cf(Isp)
230
thrust coefficient
specific impulse (sec)
1.6
1.25
Cf(CEA)
220
1.2 0
20
40
60
80
100
120
Pressure (bar)
Fig. 4.15 Variation of specific impulse and thrust coefficient with chamber pressure
81
Chapter 5 Optimization Results and Experimental Validation 5.1 Introduction The present chapter is devoted to presenting the main results of the study. These results are the output of the developed Optimized Internal Ballistic Prediction Model (OPTIBPM). The results are presented in the following order: 1. Optimized prediction accuracy of OPTIBPM is assessed based on experimental measurements of Maklad [47]. 2. The ability of OPTIBPM to predict the dimensions of a star perforated grain for a predefined pressure-time profile is addressed. Validation is made by comparing with the case of Maklad [47]. 3. The validated OPTIBPM is utilized to predict the dimensions of a dualthrust tubular grain for a predefined thrust-time profile. The grain is manufactured and experimentally tested. The measured pressure-time profile is compared with the predefined profile. The road map of the present chapter is illustrated below.
82
Experiment from literature
Applications of OPTIBPM
Optimizing prediction accuracy of IBPM
Prediction of grain configurations and Geometry for predefined performance
Validation of OPTIBPM
Star grain
Assessment
Tubular grain
success
Developing own experiments
Assessment
Fig. 5.1 Road map of chapter 5 5.2 Optimization of Internal Ballistic Prediction Model As shown in sec. 3.2.4, the developed prediction model (IBPM) involves uncertainties in the input ballistic parameters of the propellant. Six parameters are considered here namely; burning rate coefficient, , pressure exponent, , erosive burning coefficient,
, erosive burning pressure coefficient,
nozzle critical section,
, erosion rate of
, and delay time for the onset of erosion,
. The
model parameters are implemented through the following relations. The burning rate of the propellant ( ̇
)
(
̇ )
(5-1)
Nozzle erosion model (5-2)
83 The prediction accuracy of the model is thus dependent on the values of these parameters. The set of values of these parameters that maximize the model accuracy is attained by optimization. The six parameters in concern are allowed to vary within their respectable ranges. The values for
and
are arbitrarily chosen to
involve the baseline values provided by the experimental work [47], Table 3.1. Values of other parameters are specified based on previous experience of the supervisors [40]. A hybrid optimization technique is used to get the minimum Root Mean Square Error (RMSE) between the theoretical and experimental pressure-time profile in the quasi steady-state phase using Genetic Algorithm globally and Simulated Annealing locally. The optimization is conducted using MATLAB optimization toolbox. Fig. 5.2 shows the convergence history of genetic algorithm optimizer. The solution is found to converge after 200 iterations. Fig. 5.3 shows the function value convergence during the simulated annealing optimization. The solution is found to converge after 3245 iterations. The pressure-time profiles of the star grain as predicted by IBPM and OPTIBPM are plotted in Fig. 5.4. The experimentally measured profile is shown for comparison. The RMSE of prediction during the steady-state phase is improved to 2.38 % upon using GA. Over the entire operation time the RMSE is 6.5%. The RMSE is improved during the steady-state phase to 2.36 % upon using SA. The slight improvement in the SA optimization phase indicates that GA has reached the global optimum solution with high accuracy.
84
Fig. 5.2 Convergence history of GA optimization for the first application
Fig. 5.3 Convergence history of SA optimization for the first application
85 It can be seen from Fig. 5.4 that the difference between IBPM and OPTIBPM is negligibly small. Having optimized the model uncertain ballistic parameters, the remaining discrepancies between OPTIBPM prediction and the measured data may be attributed to other factors. These factors may be the inaccurate values of geometric parameters of the grain and the possible error in measurement reported by Maklad [47]. They may also include the underlying assumptions of the IB model. 180 160
pressure (bar)
140 120
100 80 60
OPTIBPM Experimental [47] IBPM
40 20 0 0
0.5
1
1.5
2
2.5
3
3.5
4
time (s) Fig. 5.4 Validation of OPTIBPM for the first application Table 5.1 lists the optimized values of model parameters in concern. For the sake of comparison, the corresponding baseline values, lower and upper bounds of variation are also listed.
86 Table 5.1 Optimized model parameters and lower and upper bounds. Model parameters
Symbols
Burning rate coefficient Pressure exponent erosive burning coefficient
Optimized solution
Lower
Upper
Base line
Bound
Bound
values[47] Global (GA)
100 e-07
120 e-07
0.41
0.43
295e-07
315 e-07
113e-07
Local (SA)
110.13 e-07
110.12 e-07
0.422
0.421
308e-07
304.91 e-07
304.81 e-07
0.42
erosive burning pressure
140
160
150
154.909
154.859
1e-04
3 e-04
2e-04
1.053 e-04
1.049 e-04
0.01
0.9
0.7
0.0256
0.0254
6.508%
2.38 %
2.36 %
coefficient Erosion rate of nozzle critical section
[m/s]
Delay time until begin of erosion
[
]
Root mean square error (steady state
RMSE
-
-
phase) In the preliminary design phase of SPRM, the configuration of the grain is unknown and is to be tailored towards satisfying the predefined performance requirements. These requirements are set by the user needs and are mainly the thrust-time history that is expected to fulfill the rocket mission.
87 Having maximized its accuracy of prediction, the optimized model OPTIBPM is implemented in a different application. The model is used to predict the geometric parameters of a grain for a predefined pressure-time profile. The following two sections are devoted to illustrating this application.
5.3 Design of a Star Grain for Predefined Performance Criteria Using OPTIBPM In the first case, the experimental pressure profile by Maklad [47] is used as the performance criterion (user’s needs) to be fulfilled by a star-perforated grain, Fig. 3.6 (experiments only). All six geometric parameters of a typical star perforated grain are involved in the optimization; they are all illustrated in Fig. 6.1. Theuser’sneedsalsoimplythatthe grain outer radius,
, is taken as
having a fixed value of 57 mm. The genetic algorithm is used as the optimizer to get the required star grain geometry. The optimizer runs within certain bounds for each design parameter as illustrated in Table 5.2. The source of these bounds is a result of a survey on similar systems. It takes 259 iterations to get the optimized star grain geometry. The convergence history of the GA optimizer is shown in Fig. 5.5. Upon optimization, the root mean square error is found to be 2.49 % in the quasi-steady state phase.
88
Fig. 5.5 Convergence history of GA optimization Table 5.2 Star grain design parameters, lower and upper bounds Design
Lower
Upper
Optimized
Experimental
Bound
Bound
solution GA
case
5
8
7
7
65
80
73.08
74
Angle fraction (-)
0.4
0.6
0.5206
0.5058
Fillet radius (mm)
1
2
1.71
1.6
30
38
31.43
33.5
1.5
1.8
1.5917
1.60411
parameters No. of star points (-)
Symbols
N
Star point angle (deg)
Web thickness (mm) Length of star grain (m) Root mean square error (steady state phase)
RMSE
2.49%
89 It is clear that the OPTIBPM manages to predict the grain geometry with a remarkable accuracy. The error in the prediction is less than 2.5%. This proves the validity of the IBPM as a design tool of grains for predefined performance criteria. Fig. 5.6 shows the pressure time curves for both the predicted star grain geometry and the experimental one. It is clear that the two curves are closely matched. The differences between the two profiles at the end of regressive and progressive burning phases are 5 bar and 8 bar, respectively. By comparing the predicted pressure profiles in Fig. 5.4 and Fig. 5.6, a significant relative accuracy is proven to be achieved by optimizing the grain geometric parameters. This confirms the hypotheses that the discrepancies in Fig. 5.4 may be partially attributed to inaccuracy in grain geometry definition by Maklad [47], model assumptions and measurements errors. 180 160
Pressure (bar)
140 120 100 80 60 OPTIBPM Experimental [47]
40 20 0 0
0.5
1
1.5
2
2.5
3
3.5
time (s)
Fig. 5.6 Validation of OPTIBPM grain geometry prediction for the second application
90
5.4 Design of Dual Thrust Tubular Grain for Predefined Performance Criteria Using OPTIBPM In this other case the pressure-time profile shown in Fig. 5.8 is adopted to meet the user needs. The user requirements also impose a motor inner cavity diameter of 60 mm and length of 220 mm. This profile assumes a dual-thrust profile. A high thrust (boost) is needed at the beginning of motor operation to accelerate the rocket to a nominal speed in 1 sec the thrust is reduced (sustain) to maintain the normal flight speed for another 1.7 seconds before burn-out. Such dual-thrust profile can be achieved using two grains of different configurations and/or different compositions. In the present study, two grains of the same chemical composition with different geometric parameters are adopted. The chemical composition of the two grains is taken as the one developed in the framework of this research and detailed in chapter 4. 6000 5000
thrust (N)
4000 3000 2000 1000 0 0
0.5
1
1.5
2
2.5
time (sec)
Fig. 5.7 Predefined performance of SPRM for third application
3
91 Since OPTIBPM uses pressure-time profile as input, it would be needed to deduce the pressure-time profile from the predefined thrust-time profile. To do so, the pressure is estimated using the relation: (5-3) A nozzle having a critical cross section diameter of 10 mm is assumed. The deduced pressure time history is shown in Fig. 5.8. 120
pressure (bar)
100 80 60 40 20 0 0
0.5
1
1.5
2
2.5
3
time (sec)
Fig. 5.8 Predefined pressure-time profile of SPRM for the third application From the previous experiences, the grain geometry is assumed to comprise two separate tubular grains with the same outer diameter (60mm), different internal diameters, and different lengths as shown in Fig. 5.9. Hence, four design parameters of solid propellant grain are involved in this study which are:
………..internaldiameterofthefirstgrain.
………..internaldiameterofthesecondgrain.
………..lengthofthefirstgrain.
………..lengthofthesecond grain.
92
Fig. 5.9 Proposed geometry of the candidate grain design Two arrangements of the grains are possible. Based on experience and survey of similar systems, the grain with smaller inner diameter is placed at the head end of the motor. OPTIBPM is implemented with ballistic and thermochemical properties of the developed propellant as inputs. The bounds of the design parameters are shown in Table 5.3. The RMSE between the predefined performance and estimated performance of the optimized solution is ~5.9%. The convergence history of the GA optimizer is shown in Fig. 5.10. A comparison between the predefined performance and the estimated performance is illustrated in Fig. 5.11. Table 5.3 Design parameters, lower and upper bounds Design parameters Lower Bound Upper Bound Optimized solution GA (mm)
50
150
105
(mm)
10
60
19
(mm)
50
150
110
(mm)
10
60
41
RMSE
~5.9%
93
Fig. 5.10 Convergence history of GA optimization 140 120 Predefined
pressure (bar)
100
OPTIBPM
80
60 40 20
0 0
0.5
1
1.5
2
2.5
3
time (s)
Fig. 5.11 Predefined and predicted performance for the third application The grain developed by OPTIBPM manages to fulfill the nominal values of boost and sustain pressures. The durations of boost and sustain phases are also satisfied. The difference between the two profiles can be explained as follows. The
94 OPTIBPM grain generates a progressive rather than a neutral pressure in the boost phase. In the sustain phase, the grain produces a slightly regressive rather than a neutral pressure. Nonetheless, the nominal values in both phases are satisfied with acceptable accuracy. The total pressure impulse (integral of pressure with time) based on user’s needs and OPTIBPM is 162.6 and 160.2, respectively. The difference in the total pressure impulse is 1.5%. 5.4.1 Experimental validation of OPTIBPM grain
Design of Test Motor A modified two inch test motor was manufactured and tested to validate the performance of the grain developed by OPTIBPM. During the course of manufacturing, some modifications were made on the design parameters to scope with optimized solution. These modifications were done in order to simplify and economize the manufacturing processes by using the available cores and molds for casting the solid propellant grains. Table 5.4 below presents a comparison between the dimensions of the grains predicted by OPTIBPM and those of the manufactured grains. Table 5.4 Optimized and experimental grain parameters of the third application Design parameters Optimized solution GA Experimental case (mm)
105
108
(mm)
19
20
(mm)
110
108
(mm)
41
40
The following parts of the test motor were designed and manufactured: -
Steel casing of length 267 mm.
-
Teflon mandrels of diameters 20 and 40 mm as shown in Fig. 5.12.
95 -
Steel rings of width 5 mm to be used as spacers in the modified twoinch motor.
-
Propellant grains with two inner diameters as illustrated in Fig. 5.13.
Fig. 5.12 Manufactured cores for the third application
Fig. 5.13 Solid propellant grains after casting for the third application The components of the manufactured test motor are shown in Fig. 5.14. The final view of the assembled test motor is shown in Fig. 5.15.
96
Fig. 5.14 Test motor components before assembly for the third application
Fig. 5.15 Test motor after final assembly for the third application
Motor casing stress check The calculation of the safety factor of the steel motor casing is necessary to make sure that no failure of the rocket motor will take place during experiment. Steel 37 is used for the motor casing material. The maximum tensile stress,
, is given by
(5-4)
97 where P is the maximum expected pressure in combustion chamber, d is the inner diameter of the two-inch motor, and t is the thickness of the motor case. The safety factor is expressed as: (5-5) where
is the yield stress.
The properties of the material were acquired by using KEY TO STEEL program which showed that the yield strength of the material is
= 235 MPa.
The maximum expected pressure for the motor was estimated to be P = 200 bar, diameter of the motor casing was d = 65 mm, and its thickness was t = 7.5 mm. Hence,
These calculations for the safety factor of the motor casing show that it is safe for the experiments in concern. Static firing test results During the assembly of the test motor, the two solid propellant grains were assembled to the motor such that the grain with smaller inner diameter is placed at the head end in order to avoid the erosive burning. The motor parts are assembled as shown in Fig. 5.16.
98
Fig. 5.16 Schematic of test motor assembly for the third application The assembly of the motor to the test stand and the fixation of the pressure transducers are shown in Fig. 5.17. The results of the static firing test are illustrated in Table 5.5. A comparison between the static firing test, optimized pressure time history and predefined performance is illustrated in Fig. 5.18.
Fig. 5.17 Static firing test of SPRM for the third application
99 Table 5.5 Static firing test results for the third application ABU ZAABAL COMPANY FOR SPECIAL CHEMICALS, LABORATORY SECTOR, STATIC TEST STAND, DATE 24/04/2017 GR.no
Temp
Throat
WT.
TD
units
C
mm
kg
msec
2*7/B1
20
10.00
0.792
51
TB
T50%
Pmax
sec
sec
bar
2.624
1.322
117.1
IPT
Pav.
P50%
R50%
C*
bar.s
bar
bar
mm/s
m/sec
156.8
59.77
118.6
14.63
1554
140 120 Experimental
100
pressure (bar)
Predefined 80
OPTIBPM
60 40 20 0 0 -20
0.5
1
1.5
2
2.5
3
time (s)
Fig. 5.18 A comparison between the static firing test and optimized pressure time history for the third application The pressure profile of the manufactured grains is in a close similarity with that of the grains developed by OPTIBPM. An over pressure in the boosting phase is shown in the experimental pressure time curve shown in Fig. 5.18. This over pressure may be attributed to the hump effect which occurs in many internal burning grain configurations and may reach 8% of the planned value. The hump effect has the role of changing the burning rate as a function of the propellant thickness. As an explanation of this effect, it may a result from the distribution of
100 the solid fuel particles during manufacturing processes [2]. The boost-sustain transition, sustain profile, and intervals of both phases are remarkably matched in experimental and OPTIBPM approaches.
101
Chapter 6 Conclusions and Future Work 6.1 Conclusions Providing the designers of solid propellant rocket propulsion with a reliable design tool yields many advantages. Such tool automates the design process and increases the reliability of the results even with less-experienced designers. The objective of the present work was to develop and assess a reliable design tool. The tool is comprised of an internal ballistics prediction model (IBPM) coupled with an optimization algorithm to form an integrated optimized model (OPTIBPM). IBPM is implemented to predict the pressure time performance of star grain configuration and the validation of the model is done by using a static firing test. A hybrid GA-SA optimization is performed to find the minimum RMSE between the IBPM and the experimental pressure time curve. After performing the hybrid optimization technique the tool becomes more accurate. Also, GA was applied alone to estimate the geometry of star grain for predefined experimental results. The developed IBPM for dual thrust rocket motor is used to predict the dimensions of a solid propellant grain to satisfy predefined performance criteria. GA is used as the optimizer in this case. The optimized parameter values of RMSE were modified to match the available experimental and manufactured resources. Composite solid propellant grains are manufactured. Then static firing tests were performed and the experimental results were compared with the theoretical and optimized results. The comparison shows a good estimation of the grain design parameters.
102 The main findings and conclusions of the work are: For the problems examined, the global optimization method managed to locate the global optimum with high accuracy. The following local search yielded unnoticeable change in the optimum solution. The developed internal ballistic model provides a fair simulation of the physics taking place in the motors investigated. The uncertainties in defining the propellant geometric and ballistic features impact the accuracy of prediction. Accuracy can be restored using the developed optimized prediction approach. Production availability can pose constrains on realizing the optimum grain design for a predefined performance merit. In such case, the producible design is adopted. Defining the grain geometric and ballistic parameters with maximum accuracy is crucial. This can be achieved by increasing the quality of experiments involved in specifying the ballistic properties of the propellant. The developed model managed to propose the design and dimensions of a solid propellant grain for predefined criterion. The capability of defining the instantaneous burning surface area of a grain is a crucial. Due to its importance, the step has drawn the attention of the researchers.
103
6.2 Future Work By the end of this work, future work can be extended to the following points; Different grain geometries can be added to the IBPM so it can easily predict performance of different grain geometries. Predicting the grain configuration with different shapes for predefined performance criteria. Use different algorithms for optimization to enhance the methodology of selection of motor parameters and architecture. Use CAD modeling to investigate the burn-back analysis of the grains. Add more erosive burning models to the IBPM. Add the ignition time rise phase to the IBPM. Divide the solid propellant grain into segments and calculate the pressure at the start and end of each segment.
104
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123
Appendix C Isp and CEA Results Isp results AFAL SPECIFIC IMPULSE PROGRAM THE COUNTER TO LIMIT THE NUMBER OF ITERATIONS IS NOW SET AT 50. ANY DATA POINT WHICH EXCEEDS THIS COUNTER SHOULD BE VIEWED CRITICALLY BEFORE ACCEPTANCE. PROPELLANT AP AL HTPB MAPO DOS IPDI
HF DENSITY WEIGHT -70.6900 1.9500 69.0000 .0000 2.7000 17.0000 -14.6411 .9380 10.4010 -223.6000 1.0800 .2000 -344.2100 .9100 2.6000 -88.8000 1.0600 .7990
GRAM ATOMS/100 GRAMS H 3.8761 O 2.3877 N .6301 P .0009 ENTHALPY = -45.66069 CSTAR = 5140.69 PRESSURE (PSIA) EPSILON ISP ISP (VACUUM) TEMPERATURE(K) MOLECULAR WEIGHT MOLES GAS/100G CF PEAE/M (SECONDS) GAMMA HEAT CAP (CAL) ENTROPY (CAL) ENTHALPY (KCAL) DENSITY (G/CC) ITERATIONS
.5973 CL
MOLES .5873 .6301 .1039 .0009 .0061 .0036
.5873 C
VOLUME 35.3846 6.2963 11.0885 .1852 2.8571 .7538 .9721 AL
DENSITY =1.768
CHAMBER 435.113 .000 .000 .000 3301.694 27.625 3.620 .000 .000 1.177 47.834 233.973 -45.661 3.01893E-03 12
THR(SHIFT) 251.286 1.000 104.818 197.095 3114.484 27.839 3.592 .656 92.277 1.176 47.732 233.972 -58.282 1.86262E-03 3
EXH(SHIFT) 14.505 5.543 239.797 269.323 2326.997 28.415 3.519 1.501 29.526 1.193 43.304 233.973 -111.719 1.46878E-04 9
MOLES/100 GRAMS PROPELLANT AP
HF DENSITY WEIGHT -70.6900 1.9500 69.0000
MOLES .5873
VOLUME 35.3846
124 AL HTPB MAPO DOS IPDI
.0000 -14.6411 -223.6000 -344.2100 -88.8000
2.7000 .9380 1.0800 .9100 1.0600
GRAM ATOMS/100 GRAMS H 3.8761 O 2.3877 N .6301 P .0009 ENTHALPY = -45.66069 CSTAR = 5151.30 PRESSURE (PSIA) EPSILON ISP ISP (VACUUM) TEMPERATURE(K) MOLECULAR WEIGHT MOLES GAS/100G CF PEAE/M (SECONDS) GAMMA HEAT CAP (CAL) ENTROPY (CAL) ENTHALPY (KCAL) DENSITY (G/CC) ITERATIONS
17.0000 10.4010 .2000 2.6000 .7990
.5973 CL
.6301 .1039 .0009 .0061 .0036
.5873 C
6.2963 11.0885 .1852 2.8571 .7538 .9721 AL
DENSITY =1.768
CHAMBER 609.158 .000 .000 .000 3327.805 27.708 3.609 .000 .000 1.176 47.873 231.556 -45.661 4.20599E-03 11
THR(SHIFT) 351.337 1.000 105.147 197.493 3134.514 27.910 3.583 .657 92.346 1.175 47.760 231.556 -58.362 2.59415E-03 3
EXH(SHIFT) 14.505 7.157 249.658 276.943 2242.564 28.463 3.513 1.559 27.284 1.197 42.486 231.556 -117.263 1.52666E-04 13
MOLES/100 GRAMS PROPELLANT AP AL HTPB MAPO DOS IPDI
HF DENSITY WEIGHT -70.6900 1.9500 69.0000 .0000 2.7000 17.0000 -14.6411 .9380 10.4010 -223.6000 1.0800 .2000 -344.2100 .9100 2.6000 -88.8000 1.0600 .7990
GRAM ATOMS/100 GRAMS H 3.8761 O 2.3877 N .6301 P .0009 ENTHALPY = -45.66069 CSTAR = 5153.85 PRESSURE (PSIA) EPSILON ISP ISP (VACUUM)
.5973 CL
MOLES .5873 .6301 .1039 .0009 .0061 .0036
.5873 C
VOLUME 35.3846 6.2963 11.0885 .1852 2.8571 .7538 .9721 AL
DENSITY =1.768 CHAMBER 725.189 .000 .000 .000
THR(SHIFT) 417.863 1.000 105.387 197.691
EXH(SHIFT) 14.505 8.135 254.436 280.502
125 TEMPERATURE(K) MOLECULAR WEIGHT MOLES GAS/100G CF PEAE/M (SECONDS) GAMMA HEAT CAP (CAL) ENTROPY (CAL) ENTHALPY (KCAL) DENSITY (G/CC) ITERATIONS
3340.830 27.750 3.604 .000 .000 1.176 47.892 230.306 -45.661 4.99516E-03 11
3144.211 27.945 3.578 .658 92.304 1.175 47.774 230.305 -58.420 3.07973E-03 4
2185.221 28.487 3.510 1.588 26.066 1.197 42.332 230.307 -120.030 1.56801E-04 11
MOLES/100 GRAMS PROPELLANT AP AL HTPB MAPO DOS IPDI
HF DENSITY WEIGHT -70.6900 1.9500 69.0000 .0000 2.7000 17.0000 -14.6411 .9380 10.4010 -223.6000 1.0800 .2000 -344.2100 .9100 2.6000 -88.8000 1.0600 .7990
GRAM ATOMS/100 GRAMS H 3.8761 O 2.3877 N .6301 P .0009 ENTHALPY = -45.66069 CSTAR = 5159.13 PRESSURE (PSIA) EPSILON ISP ISP (VACUUM) TEMPERATURE(K) MOLECULAR WEIGHT MOLES GAS/100G CF PEAE/M (SECONDS) GAMMA HEAT CAP (CAL) ENTROPY (CAL) ENTHALPY (KCAL) DENSITY (G/CC) ITERATIONS
.5973 CL
MOLES .5873 .6301 .1039 .0009 .0061 .0036
.5873 C
VOLUME 35.3846 6.2963 11.0885 .1852 2.8571 .7538 .9721 AL
DENSITY =1.768
CHAMBER 870.266 .000 .000 .000 3354.068 27.793 3.598 .000 .000 1.175 47.911 229.001 -45.661 5.97999E-03 11
THR(SHIFT) 500.812 1.000 105.607 197.887 3153.959 27.981 3.574 .659 92.279 1.175 47.789 229.001 -58.473 3.68436E-03 3
EXH(SHIFT) 14.505 9.303 259.204 284.069 2124.941 28.505 3.508 1.616 24.865 1.198 42.164 229.001 -122.843 1.61354E-04 10
MOLES/100 GRAMS PROPELLANT AP AL
HF DENSITY WEIGHT -70.6900 1.9500 69.0000 .0000 2.7000 17.0000
MOLES .5873 .6301
VOLUME 35.3846 6.2963
126 HTPB MAPO DOS IPDI
-14.6411 -223.6000 -344.2100 -88.8000
.9380 1.0800 .9100 1.0600
GRAM ATOMS/100 GRAMS H 3.8761 O 2.3877 N .6301 P .0009 ENTHALPY = -45.66069 CSTAR = 5166.54 PRESSURE (PSIA) EPSILON ISP ISP (VACUUM) TEMPERATURE(K) MOLECULAR WEIGHT MOLES GAS/100G CF PEAE/M (SECONDS) GAMMA HEAT CAP (CAL) ENTROPY (CAL) ENTHALPY (KCAL) DENSITY (G/CC) ITERATIONS
10.4010 .2000 2.6000 .7990
.5973 CL
.1039 .0009 .0061 .0036
.5873 C
11.0885 .1852 2.8571 .7538 .9721 AL
DENSITY =1.768
CHAMBER 1087.780 .000 .000 .000 3369.711 27.844 3.591 .000 .000 1.175 47.934 227.409 -45.661 7.45354E-03 10
THR(SHIFT) 627.179 1.000 105.525 198.113 3166.644 28.021 3.569 .657 92.588 1.174 47.807 227.410 -58.453 4.60220E-03 3
EXH(SHIFT) 14.505 10.970 264.728 288.218 2051.402 28.521 3.506 1.649 23.490 1.199 41.950 227.409 -126.168 1.67231E-04 10
MOLES/100 GRAMS PROPELLANT AP AL HTPB MAPO DOS IPDI
HF DENSITY WEIGHT -70.6900 1.9500 69.0000 .0000 2.7000 17.0000 -14.6411 .9380 10.4010 -223.6000 1.0800 .2000 -344.2100 .9100 2.6000 -88.8000 1.0600 .7990
GRAM ATOMS/100 GRAMS H 3.8761 O 2.3877 N .6301 P .0009 ENTHALPY = -45.66069 CSTAR = 5170.59 PRESSURE (PSIA) EPSILON ISP ISP (VACUUM) TEMPERATURE(K)
.5973 CL
MOLES .5873 .6301 .1039 .0009 .0061 .0036
.5873 C
VOLUME 35.3846 6.2963 11.0885 .1852 2.8571 .7538 .9721 AL
DENSITY =1.768 CHAMBER 1450.380 .000 .000 .000 3388.947
THR(SHIFT) 833.594 1.000 106.024 198.391 3179.917
EXH(SHIFT) 14.505 13.598 271.408 293.264 1957.709
127 MOLECULAR WEIGHT MOLES GAS/100G CF PEAE/M (SECONDS) GAMMA HEAT CAP (CAL) ENTROPY (CAL) ENTHALPY (KCAL) DENSITY (G/CC) ITERATIONS MOLES/100 GRAMS
27.906 3.583 .000 .000 1.174 47.964 225.357 -45.661 9.90392E-03 11
28.073 3.562 .660 92.367 1.174 47.827 225.357 -58.574 6.10264E-03 3
28.534 3.505 1.689 21.856 1.201 41.662 225.356 -130.283 1.75312E-04 10
128 CEA results ******************************************************************** NASA-GLENN CHEMICAL EQUILIBRIUM PROGRAM CEA2, MAY 21, 2004 BY BONNIE MCBRIDE AND SANFORD GORDON REFS: NASA RP-1311, PART I, 1994 AND NASA RP-1311, PART II, 1996 ******************************************************************** THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION FROM FINITE AREA COMBUSTOR Pin =
435.1 PSIA
Ac/At = 40.9600
REACTANT
WT FRACTION (SEE NOTE) 0.6900000 0.1700000 0.1040100 0.0020000 0.0260000 0.0079900
AP AL HTPB MAPO DOS IPDI
O/F= 0.00000 %FUEL= 0.000000 PHI,EQ.RATIO= 0.000000
Pinj/Pinf =
1.000059
ENERGY KJ/KG-MOL -295766.960 0.000 -61253.760 -935542.400 -1440174.640 -371539.200
TEMP K 298.000 298.000 298.000 298.000 289.000 289.000
R,EQ.RATIO= 1.771045
Pinj/P P, BAR T, K RHO, KG/CU M H, KJ/KG U, KJ/KG G, KJ/KG S, KJ/(KG)(K)
INJECTOR 1.0000 30.000 3329.24 3.0054 0 -1910.69 -2908.90 -34481.3 9.7832
COMB END THROAT 1.0001 1.7325 29.996 17.316 3329.22 3146.53 3.0051 0 1.8508 0 -1910.75 -2441.90 -2908.95 -3377.47 -34481.2 -33225.0 9.7832 9.7832
EXIT 30.000 1.0000 2327.00 1.4798-1 -4686.47 -5362.23 -27452.0 9.7832
M, (1/n) MW, MOL WT (dLV/dLP)t (dLV/dLT)p Cp, KJ/(KG)(K) GAMMAs SON VEL,M/SEC MACH NUMBER
27.731 27.731 27.964 28.631 25.663 25.663 25.797 26.265 -1.01876 -1.01876 -1.01441 -1.00235 1.3445 1.3445 1.2748 0.0000 4.0040 4.0040 3.6113 0.0000 1.1320 1.1320 1.1356 0.9977 1063.0 1063.0 1030.7 821.1 0.000 0.010 1.000 2.870
PERFORMANCE PARAMETERS Ae/At CSTAR, M/SEC CF Ivac, M/SEC Isp, M/SEC
58.546 1572.5 0.0069 92067.2 10.8
1.0000 1572.5 0.6555 1938.4 1030.7
5.4714 1572.5 1.4984 2643.0 2356.2
129 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION FROM FINITE AREA COMBUSTOR Pin = 609.2 PSIA Ac/At = 43.5500 CASE =
Pinj/Pinf =
O/F= 0.00000 %FUEL= PHI,EQ.RATIO= 0.000000
0.000000
1.000053 R,EQ.RATIO= 1.771045
Pinj/P P, BAR T, K RHO, KG/CU M H, KJ/KG U, KJ/KG G, KJ/KG S, KJ/(KG)(K)
INJECTOR 1.0000 42.000 3357.59 4.1857 0 -1910.69 -2914.10 -34420.5 9.6825
COMB END THROAT 1.0001 1.7338 41.996 24.225 3357.57 3168.93 4.1853 0 2.5781 0 -1910.75 -2445.14 -2914.15 -3384.76 -34420.4 -33128.2 9.6825 9.6825
EXIT 42.000 1.0000 2269.73 1.5190-1 -4919.31 -5577.63 -26895.9 9.6825
M, (1/n) MW, MOL WT (dLV/dLP)t (dLV/dLT)p Cp, KJ/(KG)(K) GAMMAs SON VEL,M/SEC MACH NUMBER
27.822 27.822 28.041 28.666 25.739 25.739 25.861 26.293 -1.01731 -1.01731 -1.01315 -1.00174 1.3137 1.3137 1.2480 1.0423 3.8130 3.8130 3.4439 2.1176 1.1337 1.1337 1.1376 1.1724 1066.6 1066.6 1033.9 878.5 0.000 0.010 1.000 2.792
PERFORMANCE PARAMETERS Ae/At CSTAR, M/SEC CF Ivac, M/SEC Isp, M/SEC
61.950 1575.6 0.0065 97614.2 10.3
1.0000 1575.6 0.6562 1942.7 1033.9
7.1535 1575.6 1.5569 2721.4 2453.0
130 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION FROM FINITE AREA COMBUSTOR Pin = 725.2 PSIA Ac/At = 50.5700 CASE =
Pinj/Pinf =
O/F= 0.00000 %FUEL= PHI,EQ.RATIO= 0.000000
0.000000
1.000040 R,EQ.RATIO= 1.771045
Pinj/P P, BAR T, K RHO, KG/CU M H, KJ/KG U, KJ/KG G, KJ/KG S, KJ/(KG)(K)
INJECTOR 1.0000 50.000 3371.83 4.9702 0 -1910.69 -2916.68 -34382.8 9.6304
COMB END THROAT 1.0001 1.7344 49.996 28.829 3371.81 3180.06 4.9699 0 3.0616 0 -1910.73 -2446.76 -2916.72 -3388.37 -34382.7 -33072.1 9.6304 9.6304
EXIT 50.000 1.0000 2213.49 1.5590-1 -5036.02 -5677.44 -26352.9 9.6304
M, (1/n) MW, MOL WT (dLV/dLP)t (dLV/dLT)p Cp, KJ/(KG)(K) GAMMAs SON VEL,M/SEC MACH NUMBER
27.868 27.868 28.080 28.693 25.777 25.777 25.893 26.315 -1.01659 -1.01659 -1.01253 -1.00128 1.2983 1.2983 1.2347 1.0316 3.7190 3.7190 3.3619 2.0363 1.1346 1.1346 1.1386 1.1767 1068.4 1068.4 1035.4 868.8 0.000 0.008 1.000 2.878
PERFORMANCE PARAMETERS Ae/At CSTAR, M/SEC CF Ivac, M/SEC Isp, M/SEC
71.197 1577.2 0.0057 112292.9 9.0
1.0000 1577.2 0.6565 1944.8 1035.4
8.1331 1577.2 1.5852 2756.7 2500.1
131 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION FROM FINITE AREA COMBUSTOR Pin = 870.2 PSIA Ac/At = 56.6900 CASE =
Pinj/Pinf =
O/F= 0.00000 %FUEL= PHI,EQ.RATIO= 0.000000
0.000000
1.000032 R,EQ.RATIO= 1.771045
Pinj/P P, BAR T, K RHO, KG/CU M H, KJ/KG U, KJ/KG G, KJ/KG S, KJ/(KG)(K)
INJECTOR 1.0000 60.000 3386.36 5.9488 0 -1910.69 -2919.29 -34338.6 9.5761
COMB END THROAT 1.0001 1.7350 59.996 34.582 3386.34 3191.34 5.9485 0 3.6648 0 -1910.73 -2448.40 -2919.32 -3392.02 -34338.6 -33008.9 9.5761 9.5761
EXIT 60.000 1.0000 2154.16 1.6032-1 -5154.72 -5778.49 -25783.1 9.5761
M, (1/n) MW, MOL WT (dLV/dLP)t (dLV/dLT)p Cp, KJ/(KG)(K) GAMMAs SON VEL,M/SEC MACH NUMBER
27.916 27.916 28.120 28.714 25.817 25.817 25.926 26.333 -1.01586 -1.01586 -1.01191 -1.00091 1.2827 1.2827 1.2213 1.0229 3.6243 3.6242 3.2797 1.9673 1.1355 1.1355 1.1397 1.1808 1070.2 1070.2 1037.0 858.2 0.000 0.008 1.000 2.968
PERFORMANCE PARAMETERS Ae/At CSTAR, M/SEC CF Ivac, M/SEC Isp, M/SEC
79.094 1578.7 0.0051 124870.1 8.1
1.0000 1578.7 0.6569 1947.0 1037.0
9.3069 1578.7 1.6135 2792.1 2547.2
132 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION FROM FINITE AREA COMBUSTOR Pin = 1087.8 PSIA Ac/At = 64.0000 CASE = O/F= 0.00000 0.000000
Pinj/Pinf = 1.000026
%FUEL=
0.000000
R,EQ.RATIO= 1.771045
Pinj/P P, BAR T, K RHO, KG/CU M H, KJ/KG U, KJ/KG G, KJ/KG S, KJ/(KG)(K)
INJECTOR 1.0000 75.000 3403.62 7.4135 0 -1910.69 -2922.36 -34278.0 9.5097
COMB END THROAT 1.0001 1.7358 74.996 43.208 3403.61 3204.63 7.4132 0 4.5676 0 -1910.72 -2450.34 -2922.38 -3396.29 -34277.9 -32925.3 9.5097 9.5097
EXIT 75.000 1.0000 2081.41 1.6603-1 -5295.33 -5897.64 -25088.9 9.5097
M, (1/n) MW, MOL WT (dLV/dLP)t (dLV/dLT)p Cp, KJ/(KG)(K) GAMMAs SON VEL,M/SEC MACH NUMBER
27.973 27.973 28.167 28.732 25.865 25.865 25.965 26.348 -1.01500 -1.01500 -1.01118 -1.00058 1.2641 1.2641 1.2055 1.0151 3.5132 3.5132 3.1839 1.9014 1.1366 1.1366 1.1410 1.1852 1072.3 1072.3 1038.9 844.9 0.000 0.007 1.000 3.079
PERFORMANCE PARAMETERS Ae/At CSTAR, M/SEC CF Ivac, M/SEC Isp, M/SEC
88.388 1580.5 0.0046 139699.8 7.2
1.0000 1580.5 0.6573 1949.4 1038.9
10.985 1580.5 1.6462 2833.3 2601.8
PHI,EQ.RATIO=
133 THEORETICAL ROCKET PERFORMANCE ASSUMING EQUILIBRIUM COMPOSITION DURING EXPANSION FROM FINITE AREA COMBUSTOR Pin = 1450.4 PSIA Ac/At = 83.5900 CASE =
Pinj/Pinf =
O/F= 0.00000 %FUEL= PHI,EQ.RATIO= 0.000000
0.000000
1.000016 R,EQ.RATIO= 1.771045
Pinj/P P, BAR T, K RHO, KG/CU M H, KJ/KG U, KJ/KG G, KJ/KG S, KJ/(KG)(K)
INJECTOR 1.0000 100.00 3424.99 9.8484 0 -1910.69 -2926.08 -34188.7 9.4243
COMB END THROAT 1.0000 1.7368 99.997 57.577 3424.98 3220.90 9.8482 0 6.0686 0 -1910.71 -2452.71 -2926.10 -3401.48 -34188.7 -32807.4 9.4243 9.4243
EXIT 100.00 1.0000 1988.43 1.7388-1 -5469.10 -6044.20 -24208.6 9.4243
M, (1/n) MW, MOL WT (dLV/dLP)t (dLV/dLT)p Cp, KJ/(KG)(K) GAMMAs SON VEL,M/SEC MACH NUMBER
28.046 28.046 28.226 28.748 25.925 25.925 26.014 26.360 -1.01395 -1.01395 -1.01029 -1.00032 1.2411 1.2411 1.1861 1.0085 3.3781 3.3781 3.0680 1.8400 1.1380 1.1380 1.1426 1.1899 1074.9 1074.9 1041.2 827.2 0.000 0.005 1.000 3.225
PERFORMANCE PARAMETERS Ae/At CSTAR, M/SEC CF Ivac, M/SEC Isp, M/SEC
112.68 1582.6 0.0036 178329.5 5.7
1.0000 1582.6 0.6579 1952.4 1041.2
13.621 1582.6 1.6856 2883.3 2667.7
134
Appendix D Burn-Back Analysis Burn-back analysis of star grain A typical configuration of star perforated grain is shown in Fig. D.1.
Fig. D.1 Geometric parameters and burn-back phases The star grain geometry is defined by seven parameters which are number of star points, , star point angle, , angle fraction, , grain inner radius, Web thickness,
and star point fillet radius,
, fillet radius, ,
.
The burn-back pattern of the star grain can be distinguished in four phases.
135 First Phase It starts by the initial burning surface and ends as the radius
vanishes. This
phase is very short and is thus neglected in the present work. Second Phase This phase ends by vanishing of star leg. The burning area decreases with burnback yielding regressive burning.
Fig. D.2 Port area of star grain during second phase The area of burning surface in this phase as the surface progresses a distance y is:
where
is the length of star grain, , ,
136
and
The corresponding port area of gases is:
where;
[
]
Third Phase In this phase, the star grain burn-back is similar to an internal tubular burn-back as shown in Fig. D.3.
137
Fig. D.3 Port area of star grain during third phase Hence, for the beginning of third phase:
(
)
The third phase vanishes by end of tube grain burning. The burning area in this phase as a function of y is:
138
where ) [
]
The port area is: ) where (
)
[
]
[
]
Sliver Phase This phase takes place as the outer end of the star grain is reached by the burning surface as shown in Fig. D.4.
139
Fig. D.4 Port area of star grain at sliver phase The burning area in this phase is:
where;
[
]
[
]
140 The port area is: ) where
(
)
Burn-back analysis of tubular grain A typical configuration of tubular grain is shown in Fig. D.5. Here, the tubular grain is inhibited from outside and burns only from inside and the two ends. The grain geometry is defined by three geometric parameters; the grain length, L, the grain inner diameter, d, and the grain outer diameter, D.
Fig. D.5 Tubualr grain configuration
141 The instantaneous burning area of the tubular grain at any burnt distance, y, is calculated from the following relation:
The corresponding port area is:
