To Evaluate the Burn-Perimeter of Star-Shaped Propellant Grain of A Solid Rocket Motor using Dimensionless Equations

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To Evaluate the Burn-Perimeter of Star-Shaped Propellant Grain of A Solid Rocket Motor using Dimensionless Equations

1Aravind Rajan Ayagara, 2Navya Shree Veerapaneni, 3D. Govardhan

1Associate professor, 2U.G Student, 3Head of the department

Aeronautical Engineering, Institute of Aeronautical Engineering, Hyderabad, Telangana, India-500055

Abstract:- The main idea of my work is to discuss and evaluate the geometrical parameter like burn perimeter of the star- shaped propellant grain in detail. Whenever, solid propellant comes into the discussion, the grain design is one of the most important topics to be studied and evaluated. Because the thrust output as per requirement can be obtained by varying the shape of grain. Many scholars did lot of research on different sizes and shapes of grains that can provide the burn rate as per wanted. The star-shaped propellant grain design parameter like burn perimeter by using non-dimensional equations is evaluated. A MATLAB code is written in order to plot the non-dimensional burn-perimeter (S/l) versus burnt distance (y + f)/l for star-shaped propellant grain having star point, n = 6 for = 65-75. The obtained graphs for star points with different opening of star point angle in MATLAB are compared with burn perimeter values measured in CATIA V5 designing tool for corresponding set of values. Later, by taking the geometrical inputs like inner radius(l), fillet radius(f), web thickness(w) for star-shaped grain having star point, n = 6, the grain area regressive is modelled by using CATIA V5 design tool.

Keywords:- Solid Rocket Motor Propellant Grain Burn perimeter Star-shape grain design

  1. INTRODUCTION

    The act of supplying a propelled force to a body that is initially at rest, causing it to change its velocity by overcoming the retarding forces present in the medium of that body is called Propulsion.

    Jet propulsion is divided into two categories: rocket propulsion and duct propulsion. The distinction between rocket and duct propulsion is that rocket propulsion generates thrust by ejecting stored matter (propeller). Non- air-breathing engines are another name for rocket propulsion engines. Air-breathing engines are generally referred to as duct propulsion engines. The majority of the working fluid in a duct propulsion engine is its surroundings, along with some stored fuel.

    Chemical combustion is the primary source of energy for rocket propulsion. Solar radiation and nuclear reactions are two other sources of energy that can be used to propel rocket engines.

    The propellants are the materials that make up the chemical reactions that take place within a rocket motor.

    Depending on whether the propellant is solid or liquid, chemical rocket motors are divided into two groups.

    The propellant in a solid-propellant rocket is entirely contained inside the combustion chamber as one or more formed blocks known as grains, which are supported by the walls or special grids, traps, or retainer pads. A solid- propellant rockets key feature is its simplicity; it doesnt even need a feed system. However, the burning time is limited to a few seconds or fractions of a second and seldom exceeds one or two minutes, though low-performance motors have achieved durations as long as ten or twenty minutes.

    In the case of a liquid propellant, the propellant is said to be hypergolic if the two liquids provided ignite spontaneously as a result of simple mixing. Whenever the two constituents responsible for the reaction are contained in a single liquid (as in catalytic decompositions), this liquid is alluded to as a monopropellant.

    As the applications of rocket motors depend on their special characteristics, knowing those characteristics in order to develop one is important. The main characteristics are indicated below:

    1. The essential characteristic of rocket motors, the one which distinguishes them intrinsically from other types of jet power-plants, is that they do not make use of the surrounding air.

    2. The rocket motor contains no moving parts (except perhaps certain auxiliaries in the feed system); its geometry is fixed and its operations is independent of the speed of flight, in contrast to the other jet systems in which the speed is limited by internal aerodynamic considerations. Only rockets are capable of operating at extremely high Mach numbers.

    3. The high pressure in the combustion chamber (10 to 500 kg/cm2) makes it possible to develop enormous quantities of energy for small volumes. For the same weight and size of powerplant, the rocket motor offers the highest thrust that can be obtained.

    4. The necessity of carrying both the fuel and the oxidizer means that a rocket motor has a high specific consumption. The consumption is between 3 and 6 per ton (metric) of the thrust and per second, as compared with a fuel consumption of 0.25 to 0.3kg per ton and per second in the case of a turbojet at sea level.

      1. Basic Solid Rocket Motor

        A solid propellant rocket is formed by four main components as shown in fig 1.1.

        • A case containing the solid propellant and withstanding internal pressure when the rocket is functioning.

        • The solid propellant charge (or grain), which is typically bonded to the inner wall of the case, and occupies before ignition the greater a part of its volume. When burning, the solid propellant is transformed into hot combustion products. The volume occupied by the combustion products is called a combustion chamber.

        • The nozzle channels the discharge of the combustion products and because of its shape accelerates them to supersonic velocity.

        • The igniter, which can be a pyrotechnic device or a small rocket, starts the rocket operating when an electrical signal is received

          Figure 1.1: Basic solid rocket motor

      2. Thrust law

        It is possible with suitable grain designs, to obtain a thrust which increases (progressive burning), remains constant (neutral), decreases (regressive) or follows a wide variety of laws along the burning time.

        With constant-geometry nozzles, the thrust is always approximately proportional to the chamber pressure (the thrust coefficient does not vary significantly). To show that the previous sentence is correct, a test was recorded of the thrust and pressure versus time. Fig 1.2 represents such a typical recording corresponding to a near-constant thrust, together with some current definitions of durations and thrust (or pressure) values. The integral of the thrust gives the impulse I.

        Figure 1.2: Thrust law.

      3. Propellant burning rate

        The combustion by parallel layers of a solid propellant is a progressive phenomenon entirely localized on the surface of the grain and the burning rate r is defined as the distance travelled per second by the flame front perpendicularly to the free surface of the grain (the ignition is assumed to be homogeneous).

        Therefore, the combustion surface must vary in a foreseeable manner during the burning time and fissures or cracks in the grain must be strictly avoided because they may unexpectedly increase the surface and even induce local pressure build-ups and the associated stresses that tend to spread the fissure.

      4. Erosive burning

    The burning rate r of the solid propellant or propellant increases simultaneously with the increase in gas velocity of crossflow. The increased velocity which is dependent contribution to the burning rate r is said to be Erosive Burning.

  2. LITERATURE REVIEW

    When solid popellant comes into the discussion, the grain design is one of the most important topics to be studied and evaluated. Because the thrust output as per requirement can be obtained by varying the shape of grain.

    Many scholars did lot of research on different sizes and shapes of grains that can provide the burn rate as per wanted.

    Inside the rocket motor, the grain is a developed mass of processed solid propellant. The motor performance characteristics are determined by the propellant product and grain geometrical configuration. The propellant grain has a cast, moulded, or extruded body that reflects hard rubber or plastic in appearance and feel. It will burn on all of its exposed surfaces once ignited, producing hot gases that are then expelled through a nozzle. Just a few rocket motors have multiple grains in a single case or chamber, and even fewer grains have segments of separate propellant compositions. Most rockets, on the other hand, only have one grain.

    Lets look at the definitions and terminologies that are important to grain and were discussed by George P Sutton:[1]

    Fig. 2.1. Classification of grain according to their pressure (or thrust)- time characteristics.

    Configuration: The shape or geometry of a grains initial burning surfaces as they are supposed to function in a motor.

    Cylindrical grain: The internal cross section of a grain remains constant along the axis despite of perforation form. Neutral burning: Period during which the thrust, pressure,

    and burning surface area of the motor remain relatively

    constant, usually within ±15 percent. Many grains burn in a neutral manner.

    Progressive burning: Thrust, pressure, and burning surface area rises during the burn time.

    Regressive burning: Thrust, heat, and burning surface area declines during the burn period.

    Silver: Slivers are any unburned propellant that are left-out after combustion.

    End-burning grains are typically used only in low- thrust, low-performance motors with long burn times. Their chamber diameter and dead weight are indeed high (intensive wall heating). Furthermore, during combustion, their centre of gravity changes. Nevertheless, with very fast- burning propellants, high-thrust, high-performance cigarette-burning motors may be considered. The combustion of a side-burning grain takes place on a straight cylinder shaped by the grains lateral surfaces.

    Since combustion occurs on almost all lateral surfaces, it is also known as unrestricted combustion (Table.2.1: A, B, C, D, E and F). Due to final grain fragmentation, the use of unregulated grains causes extreme wall heating and relatively high losses. The propellant itself is usually used to shield the walls in modern designs (Table.2.1: G, H, I, J, K, L, M, N and O). With this setup, wall heating can be kept to a minimum, if not entirely removed, resulting in very light chambers.

    Side-burning grains are commonly referred to as cylindrical grains; although, such charges do not have to be cylindrical: for example, the axial grain load may be

    supported by a conical portion of the chamber, eliminating the need for the grid.

    The grains that do not fit into any of the previous two categories fall into this group. Its distinguished by the fact that the burning areas generatrixes are no longer parallel straight lines. These types of grains are notoriously difficult to compute because they cannot be reduced to a two- dimensional problem.

    Here we consider star-shape grain for our further discussion. The star shape optimization is usually a trade-off between sliver propellant mass, first-phase neutrality, and volumetric processing. The port area grows as the number of star points grows, with a subsequent reduction in the burning perimeter and the quantity of final sliver. For 3-to-5-star points, volumetric loading fractions in the range of 0.7 to 0.9 can be achieved. With increased volumetric loading, the sliver fraction increases dramatically [3].

    The preliminary design of a solid propellant grain for a solid rocket motor must meet specifications expressed as a thrust versus time curve that takes into account available volume, temperature cycles, and other factors.[4]

    By properly designing the propellant grain, the mechanical and thermal stresses in the grain can be reduced. Also, the physical properties of the propellant grain must thus be carefully evaluated in the choice of grain geometry.[5]

    Many people have explained different theories and generated different equations to make the geometrical calculations for star-shape grain easy so that the parameters like thrust and chamber pressure versus time can be evaluated easily. The design calculations which were done on the star-shape grain are discussed in detail in following sections.

    A

    B

    C

    D

    E

    F

    G

    H

    I

    J

    K

    L

    M

    N

    O

    Table 2.1: Wide variety of grain cross-section.

  3. METHODOLOGY

      1. Internal-burning stars [2-6]

        As mentioned in the previous chapter many people have explained different theories and generated different equations to make the geometrical calculations for star-

        1. First phase

          +

          Table 3.1

          Internal-burning star

          sin

          shape grain easy so that the parameters like thrust and

          for

          cos

          2

          chamber pressure versus time can be evaluated easily. The

          Burning perimeter:

          design calculations which were done are studied in detail in

          sin

          +

          this chapter. The design parameters of internal-burning star = 2 [ + ( +

          ) + (1

          are shown in the form of figure 3.1.

          sin

          2

          2

          ) ]

          2 2

          neutral burning for + = +

          Initial port area:

          2 2 2

          = sin (cos sin ) + (1 )

          2

          2

          sin

          + 2

          [ + (1 )

          2

        2. Second phase

        + 0.5

        ( +

        2

        ) ] 2 2

        Fig. 3.1. Definition of the design parameters of an internal-burning star.

        for

        +

        sin

        cos

        2

        Number of star points n

        Burning perimeter:

        +

        = 2 { [ + sin1 (

        • sin )]

          Web thickness w Maximum inner radius l + f Opening of star points –

          Angular fraction –

          Final losses:

          +

          +

          + (1 ) }

          2

          Fillet radius f

          2 = (1 + )

          From the above fig. 3.1, several conclusions were illustrated.

          1. In view of the progressive nature of combustion, the star with a fillet is identical to the same basic star (same n, l, and ) without fillet which would have

            {sin

            + 2

            [( )

            2

            burnt over a distance f.

          2. With the same initial perforation, the web can be increased without altering the beginning of the curve giving S as a function of y. It follows that the characteristic length to be selected to obtain dimensionless relationships is l.

            + cos ]

            + 2

            + ( ) [

          3. The combustion is divided into two successive phases:

            • During first phase, the straight star points have not yet disappeared and, S can be progressive, neutral or regressive, depending upon the value of .

            • During second phase, the star points have

          =

          + sin1 (

          +

          and

          /2

          +

        • sin

        2

        )] }

        disappeared and, shortly after transition, S becomes strongly progressive. Afterwards, when the burning front has reached the wall, relatively important tail-off takes place.

        (1 + ) /2

        Table 3.1 gives, the dimensionless form of burning perimeter S/l during the first and second phases, the port area Ap/l2 during the first phase and the silver area Af/l2 during the second phase as functions of the reduced burnt distance (y + f)/ l and of the various design parameters.

        Let us call the value of for which neutrality is achieved during the first phase. The methodology used here is to generate a MATLAB code in order to find out burning perimeter, using burn perimeter to inner radius ratio S/l for star point n = 6 as per the conditions given. The MATLAB code is scripted in such a way that, for different combinations of n, , f, l, the graphs are plotted for burn perimeter (S/l) versus burnt distance (y+f)/l.

      2. Modelling and analysis

        As mentioned in the previous section. 3.1, the different combinations of n, , f, l, values are used to design the grain shape using computer aided designing software tool, CATIA- V5.

        CATIA is the dominant current production solution across all engineering firms, like OEMs to independent small manufacturers, through their supply chains. CATIA's capabilities allow it to be used in a diverse range of companies, including aerospace, shipbuilding, plant design, and consumer goods, including design for items as diverse as jewellery and clothing.

        Surface and form design, mechanical engineering, and equipment and systems engineering all benefit from CATIA's collaborative engineering capabilities. From subdivision, styling, and Class A surfaces to mechanical functional surfaces, CATIA offers a suite of surfacing, reverse engineering, and visualisation solutions to develop, alter, and validate complex innovative shapes.

        From 3D sketches, sheet metal, composites, moulded, forged, or tooling components, to the concept of

        mechanical assemblies, CATIA enables the production of 3D parts. It includes methods for completing product specification, such as functional tolerances and kinematics.[7]

        The star shape grain for n = 6, f = 0.5mm, = 0.74314, l = 15mm, w = 25mm, is designed using CATIA V5 software and the initial burning perimeter is measured using measuring tool for different values ranging between 65-75.[8]

        Table 3.2

        Star-shaped Propellant grain parameters

        n = 6, = 0.74314, l = 15mm, f = 0.5mm.

        in degree

        Burning perimeter(S) in mm

        65

        151.852

        66

        150.267

        67

        148.733

        68

        147.247

        69

        145.807

        70

        144.41

        71

        143.056

        72

        141.742

        73

        140.466

        74

        139.228

        75

        138.025

        The star-shape propellant grain for n = 6, = 0.74314, l = 15mm, f = 0.5mm, w = 25mm is designed in CATIA V5 designing tool. The values present in Table 3.2 are the values obtained from CATIA V5 designing tool by measuring the burn perimeter at each star point angle from 65-75.

        Fig.3.2.1 CATIA Model Star-grain =65

        Fig.3.2.2 CATIA Model Star-grain =66

        Fig.3.2.3 CATIA Model Star-grain =67

        Fig.3.2.4 CATIA Model Star-grain =68

        Fig.3.2.5 CATIA Model Star-grain =69

        Fig.3.2.6 CATIA Model Star-grain =70

        Fig.3.2.7 CATIA Model Star-grain =71

        Fig.3.2.8 CATIA Model Star-grain =72

        Fig.3.2.9 CATIA Model Star-grain =73

        Fig.3.2.10 CATIA Model Star-grain =74

        Fig.3.2.11 CATIA Model Star-grain =75

        3.2.1 AREA REGRESSION

        Area regression for star-shaped propellant grain is analysed too.

        For example, let n = 6, l = 15mm, = 67, = 0.74314, w = 25mm, regression of propellant grain area due to combustion is visualized using CATIA V5 tool

        Figure 3.2.1(d) y = 3mm

        Figure 3.2.1(e) y = 4mm

        Figure 3.2.1(f) y = 5mm

        Figure 3.2.1(g) y = 6mm

        Figure 3.2.1(h) y = 7mm

        Figure 3.2.1(i) y = 8mm

        Figure 3.2.1(d) y = 3mm

        Figure 3.2.1(e) y = 4mm

        Figure 3.2.1(f) y = 5mm

        Figure 3.2.1(g) y = 6mm

        Figure 3.2.1(h) y = 7mm

        Figure 3.2.1(i) y = 8mm

        .

        Figure 3.2.1(a) y = 0mm

        Figure 3.2.1(b) y = 1mm

        Figure 3.2.1(c) y = 2mm

        Figure 3.2.1(j) y = 9mm

        Figure 3.2.1(k) y = 10mm

        Figure 3.2.1(l) y = 11mm

        Figure 3.2.1(m) y = 12mm

        Figure 3.2.1(n) y = 13mm

        Figure 3.2.1(o) y = 14mm

        Figure 3.2.1(p) y = 15mm

        Figure 3.2.1(q) y = 16mm

        Figure 3.2.1(r) y = 17mm

        Figure 3.2.1(s) y = 18mm

        Figure 3.2.1(t) y = 19mm

        Figure 3.2.1(u) y = 20mm

        Figure 3.2.1(v) y = 21mm

        Figure 3.2.1(w) y = 22mm

        Figure 3.2.1(x) y = 23mm

        Figure 3.2.1(y) y = 24mm

        Figure 3.2.1(z) y = 25mm

        MATLAB (Matrix Laboratory)

        MATLAB is a fourth-generation scripting language and multi-paradigm numerical computing environment. Matrix transformations, plotting of various functionalities, implementation of algorithms, design of user interfaces, and syncing with programmes written in other languages, such as C, C++, Java, and Fortran, are all achievable with MATLAB, which has been developed by Math Works. While MATLAB is primarily designed for numerical computations, an optional toolbox that uses the MuPAD symbolic engine provides symbolic computing capabilities.

        Simulink, a separate kit for dynamic and embedded systems, adds graphical multi-domain simulation and Model-Based Design.

        Around one million of people used MATLAB across academia and industry in 2004. Users of MATLAB cover a wide range of engineering, science, and economics backgrounds. In both academic and industrial settings, MATLAB is commonly used.

        In this project work, MATLAB code is programmed for a star shaped propellant grain such that the burn perimeter, port area, silver area is obtained for given set of parameters. From y =0 to 25 at 1mm increment, burn perimeter at each interval and corresponding value is evaluated and plots for each value are plotted. The plots obtained for different values are discussed in the next following section.

  4. RESULTS AND DISCUSSION

    The plots for non-dimensional burn perimeter (S/l) versus burnt distance (y + f)/l are obtained from executing the above MATLAB code for n = 6, = 0.74314, l = 15mm, f = 0.5mm and ranging from 65-75.

    IJERTV10IS060016

    www.ijert.org 20

    For n = 6 and 0 = 73

    20

    18

    16

    "'./

    14

    12

    10

    8

    0 0.5 1 1.5 2

    ('y + .f) /f

    For n = 6 and 0 = 74

    20

    18

    16

    "'./

    14

    12

    10

    8

    0 0.5 1 1.5 2

    (y + f)/f

    For n = 6 and 0 = 75

    20

    18

    16

    "'./

    14

    12

    10

    8

    0 0.5 1 1.5 2

    (y + f) /f

    IJERTVIOIS0600I6 www.ijert.org 21

    BURN PERIMETER VALUES OBTAINED FOR n = 6 AND = 65-75

    =65°

    =66°

    =67°

    =68°

    =69°

    =70°

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    /td>

    S in mm

    10.0695

    151.0431

    9.9638

    149.457

    9.8614

    147.921

    9.7622

    146.433

    9.6661

    144.9908

    9.5728

    143.5925

    10.0355

    150.5328

    9.9467

    149.1998

    9.8605

    147.9075

    9.7769

    146.654

    9.6958

    145.4377

    9.6171

    144.2571

    10.0015

    150.0225

    9.9295

    148.9427

    9.8596

    147.894

    9.7917

    146.8751

    9.7257

    145.8847

    9.6614

    144.9216

    9.967483

    149.5122

    9.9124

    148.6856

    9.8587

    147.8805

    9.8064

    147.0962

    9.756

    146.3317

    9.7058

    145.5862

    9.933465

    149.002

    9.8952

    148.4284

    9.8578

    147.867

    9.8212

    147.3172

    9.7852

    146.7786

    9.7501

    146.2508

    9.899447

    148.4917

    9.8781

    148.1713

    9.8569

    147.8535

    9.8359

    147.5383

    9.8150

    147.2256

    9.7944

    146.9151

    9.865429

    147.9814

    9.8609

    147.9141

    9.8559

    147.84

    9.8506

    147.7594

    9.8448

    147.6726

    9.8387

    147.58

    9.923613

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    10.16208

    152.4311

    10.1621

    152.4311

    10.1620

    152.4311

    10.1621

    152.4311

    10.16207

    152.4311

    10.1621

    152.4311

    10.47430

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.82324

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    11.19319

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.57625

    173.6437

    11.5762

    173.6437

    11.5762

    173.6437

    11.5763

    173.6437

    11.5763

    173.6437

    11.5763

    173.6437

    11.968

    179.52

    11.968

    179.52

    11.968

    179.52

    11.968

    179.52

    11.97

    179.52

    11.968

    179.52

    12.36578

    185.4866

    12.3658

    185.4866

    12.3658

    185.4866

    12.3658

    185.4866

    12.366

    185.4866

    12.3658

    185.4866

    12.76789

    191.5183

    12.7679

    191.5183

    12.7679

    191.5183

    12.7679

    191.5183

    12.7679

    191.5183

    12.768

    191.5183

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.58093

    203.714

    13.5809

    203.714

    13.581

    203.714

    13.5809

    203.714

    13.5809

    203.714

    13.5809

    203.714

    13.99053

    209.858

    13.9905

    209.858

    13.9905

    209.858

    13.9905

    209.858

    13.9905

    209.858

    13.9905

    209.858

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    15.2270

    228.4057

    15.2270

    228.4057

    15.2271

    228.4057

    15.2270

    228.4057

    15.2270

    228.4057

    15.2271

    228.4057

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

    =71°

    =72°

    =73°

    =74°

    =75°

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    S in mm

    (S/l)

    S in mm

    9.4824

    142.2363

    9.3947

    140.9205

    9.31

    139.6434

    9.2269

    138.4035

    9.1466

    137.1993

    9.5407

    143.1105

    9.4665

    141.9968

    9.3943

    140.9145

    9.3242

    139.8624

    9.256

    138.8392

    9.599

    143.9848

    9.5382

    143.0731

    9.4790

    142.1856

    9.4214

    141.3214

    9.3653

    140.4792

    9.6573

    144.8591

    9.61

    144.1495

    9.5638

    143.4568

    9.5187

    142.7803

    9.4746

    142.1195

    9.7156

    145.7333

    9.6817

    145.2258

    9.6485

    144.7279

    9.616

    144.2392

    9.584

    143.7595

    9.7738

    146.6076

    9.7535

    146.3021

    9.7333

    145.999

    9.7132

    145.6982

    9.6933

    145.3996

    9.8321

    147.4818

    9.8252

    147.3785

    9.8180

    147.2701

    9.8105

    147.1571

    9.8026

    147.0396

    9.9236

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    9.9236

    148.8542

    10.1621

    152.4311

    10.1621

    152.4311

    10.1621

    152.4311

    10.1621

    152.4311

    10.1621

    152.4311

    10.4743

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.4743

    157.1145

    10.8232

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    10.8232

    162.3486

    11.1932

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.1932

    167.8978

    11.5763

    173.6437

    11.5762

    173.6437

    11.5762

    173.6437

    11.5763

    173.6437

    11.5763

    173.6437

    11.968

    179.52

    11.968

    179.52

    11.968

    179.52

    11.968

    179.52

    11.968

    179.52

    12.3658

    185.4866

    12.3658

    185.4866

    12.3658

    185.4866

    12.36578

    185.4866

    12.36578

    185.4866

    12.7679

    191.5183

    12.7679

    191.5183

    12.7679

    191.5183

    12.7679

    191.5183

    12.7679

    191.5183

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.1732

    197.598

    13.5809

    203.714

    13.5809

    203.714

    13.5809

    203.714

    13.5809

    203.714

    13.5809

    203.714

    13.991

    209.858

    13.9905

    209.858

    13.9905

    209.858

    13.9905

    209.858

    13.9905

    209.858

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.4016

    216.0242

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    14.8139

    222.2078

    15.2271

    228.4057

    15.2271

    228.4057

    15.2270

    228.4057

    15.2271

    228.4057

    15.2271

    228.4057

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    15.6410

    234.6151

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.0556

    240.8342

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.4707

    247.0611

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

    16.8863

    253.2948

  5. CONCLUSION

    The burn perimeter obtained from the MATLAB code are compared with the burn perimeter measured in CATIA V5 software using measuring tool. It is observed that:

        • The values for burn perimeter obtained through MATLAB is exactly same when measured in CATIA designing tool.

        • The results acquired concludes that the particular MATLAB code is correct and can be used to calculate the burn perimeter, port area, silver area, not only for n=6 but also for different star point numbers 7, 8, 9, 10, 11, 12 and with different combinations of , , f, l, w as inputs. But from the observations made, for n = 6, f = 0, = 1, the star point angle is limited to 30.

        • In addition to this, one more observation that is made by using MATLAB code is that, n and values play role on the burn perimeter but the epsilon () and fillet radius f plays a major role and are responsible for transition between phase I and II irrespective of variation in n and values.

  6. FURTHER WORK [9]

    Enrico Cavallini has developed an analysis/simulation capability of SRM internal ballistic for the entire combustion time, with simplified physical models. The burn perimeter obtained through MATLAB code for the design parameters as per propulsion requirements are used to calculate the pressure and thrust versus time using incremental analysis.

    =

    Similarly, it is impossible for n = 5, f =0 and = to

    have 0.875.

        • From the plots obtained, for example n = 6:

    o It is seen that for = 65 and 66, during first phase the burn perimeter is regressive and during the second phase burn perimeter is strongly progressive.

    = ( )

    In order to solve for thrust and chamber pressure, the throat area, burn surface area, mass flow rate discharged through nozzle, are calculated by solving for quasi 1D unsteady flow field conseration, momentum and energy equations at each x increment.

    ()

    +

    ()

    = +

    +

    o For = 67, during first phase the burn perimeter is neutral whereas during second phase i.e., from Figure 3.2.1(h) y = 7mm the burn perimeter is

    ()

    +

    [(2 + )]

    1

    = + 2

    strongly progressive.

    o For = 68, 69, 70, 71, 72, 73, 74 and 75, during first phase the burn perimeter is progressive and after disappearing of star points i.e., during

    ()

    +

    [( + )]

    2

    second phase the burn perimeter is strongly progressive.

    = +

    +

  7. APPENDIX

MATLAB PROGRAMME

clc clear close all

disp('Internal Burning Star with thby2 in Degrees'); disp('—————————————————');

disp('Inut Parameters are ');

disp('—————————————————');

%%

l = 15;

w = 25;

f = 0.5;

eps = 0.74314;

n = 6;

thby2 = deg2rad(68/2); yn = 26;

y = transpose(linspace(0,w,yn)); disp(['Length = ',num2str(l),' mm']);

disp(['Width = ',num2str(w),' mm']);

disp(['Fillet = ',num2str(f),' mm']);

disp(['Epsilon = ',num2str(eps)]); disp(['Number of stars = ',num2str(n)]);

disp(['Theta = ',num2str(rad2deg(thby2*2)),' °']); disp('—————————————————');

%%

disp('—————————————————');

disp('Calculating the combustions parameters for y '); disp('—————————————————');

for i = 1:numel(y) LHS(i,:) = (y(i)+f)./(l);

for j = 1:numel(thby2) const(j,1) = eps*(pi/n);

RHS(j) = sin(eps*(pi/n))/cos(thby2(j)); if LHS(i,:) <= RHS(j)

disp('Combustion in Phase I')

BP(i,:) = ((2*n)*((sin(eps*(pi/n))/sin(thby2(j)))+…

(LHS(i,:)*((pi/2)+(pi/n)-(thby2(j))-…

cot(thby2(j)))+((1-eps)*(pi/n)))));

disp(['BP = ',num2str(BP(i,:)*l),' mm at y =

',num2str(y(i)), ' mm']);

IPA(:,j) = (abs(n*sin(const(j)))*(cos(const(j))-…

(sin(const(j))*cot(thby2(j)))))+((1-eps).*pi)+…

((2*n*(f/l))*((sin(eps*(pi/n))/sin(thby2(j)))+…

((1-eps)*(pi/n))+((0.5*(f/l))*((pi/2)+(pi/n)- (thby2(j))-…

cot(thby2(j)))))); elseif LHS(i,:) >= RHS(j)

disp('Combustion in Phase II') BP(i,:) =

(2*n)*((LHS(i,:)*((pi/n)+asin((1/LHS(i,:))*(sin(const(j)))))

)+…

((1-eps)*(pi/n)));

disp(['BP = ',num2str(BP(i,:)*l),' mm at y =

',num2str(y(i)), ' mm']);

FL(:,j) = (eps*pi*((1+LHS(i,:))^2))-…

n*((sin(const(j))*((sqrt(LHS(i,:)^2-

(sin(const(j))^2)))+…

cos(const(j))))+ ((LHS(i,:)^2)*(const(j)+asin((1/RHS(j))*(sin(const(j)))))));

sigf(i,:) = FL(1,j)/((pi*((1+LHS(i,:))^2))- ((IPA(1,j))/(l^2)));

end end

end

disp('—————————————————');

disp(' End of calculation! '); disp('—————————————————');

% return

%% Plotting results

disp('—————————————————');

disp(' Plotting the Burn Perimeter! '); disp('—————————————————');

ft = fittype( 'smoothingspline' );

opts = fitoptions( 'Method', 'SmoothingSpline' ); opts.SmoothingParam = 0.0233503103188522; [ftr,gof] = fit(LHS,BP,ft);

% return

ind_trans = find(LHS(:,1)>RHS); figure('color','w');

set(gca,'fontsize',14); hold on

plot(LHS, BP,'sk');

plot(ftr,'-k'); plot(LHS(1),

BP(1),'pr','MarkerFaceColor','r','MarkerSize',10); plot(LHS(6), BP(6),'pr','MarkerFaceColor','r','MarkerSize',10); plot(LHS(11), BP(11),'pr','MarkerFaceColor','r','MarkerSize',10); plot(LHS(16), BP(16),'pr','MarkerFaceColor','r','MarkerSize',10); plot(LHS(21), BP(21),'pr','MarkerFaceColor','r','MarkerSize',10); plot(LHS(26), BP(26),'pr','MarkerFaceColor','r','MarkerSize',10); xlabel('$(y+f)/\ell$','Interpreter','Latex') ylabel('$S/\ell$','Interpreter','Latex');

title(['For $n\ =\ $',num2str(n),' and $\theta\ =\ $',… num2str(rad2deg(thby2*2))],'Interpreter','Latex')

lgn = legend('Data','Fit'); set(lgn,'Interpreter','Latex','Location','BestOutside','EdgeCo lor','w');

grid on grid minor box on

REFERENCES

  1. GEORGE P. SUTTON., Rocket Propulsion Elements, California: John Wiley & Sons, 2001, 7th edition, Ch 1, pp. 1-6.

  2. M. Barrere., Jaumotte. A., B. Fraeijs De Veubeke., Vandenkerckhove. J., Rocket Propulsion, Paris: Elsevier, 1960, Ch 1: pp 1-6, Ch 4: pp 190-205, Ch 6: 291-308.

  3. Makled. A., El -Senbawi. M. A., Abdalla. H. M., Evaluation of Design Parameters of The Solid Propellant Star Grain Configuration, ASAT Conference, 10th volume, pp. 243-245, 13- 15 May 2003.

  4. G. Uhrig., B. Ducourneau and P. Liesa., Computer Aided Preliminary Design of Propellant Grains for Solid Rocket Motors,

    AIAA/SAE/ASME/ASEE 23rd Joint Propulsion Conference, AIAA-87-1734, pp. 1-3, June 29- July 2 1987.

  5. JEAN A. VANDENKERCKHOVE., Recent Advances in Solid Propellant Grain Design, ARS Journal, DOI: 10.2514/8.4810, pp. 1-4, 1959.

  6. MAX W. STONE., A Practical Mathematical Approach to Grain Design, ARS Semi-Annual Meeting, DOI: 10.2514/8.7285, pp. 236-239, June 10-13 1957.

  7. Patan Stalin., Y.N.V.Santosh Kumar and SK.Nazumuddin., Design and Geometrical Analysis of Propellant Grain Configurations of a Solid Rocket Motor, International Journal of Engineering Development and Research, ISSN: 2321-9939, Volume 2, Issue 4, pp. 3418-3419, 2014.

  8. Anwer Hashish., Mahmoud Yehia Mohamed., Hamed M. Abdalla and Mohamed Allam Al-Sanabawy., Design of Solid Propellant Grain for Predefined Performance Criteria, AIAA SciTech Forum, m 7-11 January 2019, DOI: 10.2514/6.2019-2014, pp. 6- 10, 7-11 January 2019.

  9. Enrico Cavallini., Maurizio DI Giacinto., Modeling And Numerical Simulation Of Solid Rocket Motors Internal Ballistics, [Ph.D. thesis], Sapienza Universita DI Rome, Rome, Italy, 2010.

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