Ballistics Modeling of Combustion Heat Loss Through Chambers and Nozzles of Solid Rocket Motors

8/22/2019 Ballistics Modeling of Combustion Heat Loss Through Chambers and Nozzles of Solid Rocket Motors

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BALLISTICS MODELING OF COMBUSTION HEAT LOSS THROUGH

CHAMBERS AND NOZZLES OF SOLID ROCKET MOTORS

Stephen Scot Moore

B.S., California State University, Sacramento, 2004

THESIS

Submitted in partial satisfaction ofthe requirements for the degree of

MASTER OF SCIENCE

in

MECHANICAL ENGINEERING

at

CALIFORNIA STATE UNIVERSITY, SACRAMENTO

SUMMER2010


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BALLISTICS MODELING OF COMBUSTION HEAT LOSS THROUGHCHAMBERS AND NOZZLES OF SOLID ROCKET MOTORS

A Thesis

by

Stephen Scot Moore

Approved by:

__________________________________, Committee Chair

Dongmei Zhou, Ph. D.

__________________________________, Second Reader

James Bergquam, Ph. D.

____________________________

Date


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Student: Stephen Scot Moore

I certify that this student has met the requirements for format contained in the University

format manual, and that this thesis is suitable for shelving in the Library and credit is to

be awarded for the thesis.

__________________________, Graduate Coordinator ___________________Kenneth Sprott, Ph. D. Date

Department of Mechanical Engineering


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Abstract

of

BALLISTICS MODELING OF COMBUSTION HEAT LOSS THROUGH

CHAMBERS AND NOZZLES OF SOLID ROCKET MOTORS

by

Stephen Scot Moore

There are several assumptions made when the ballistics of a solid rocket motor

(SRM) is being modeled. Among them is the assumption that the case wall of themotor is adiabatic, i.e., no heat from combustion is lost through the case and nozzle

walls as a solid rocket motor burns. However, this adiabatic assumption is usually

not numerically validated. This work is intended to prove or disprove such anassumptions through computational studies. First, CAD models are built using ProE,

that represent successive layers of a solid fuel as it burns back. Each individual

model is then meshed in the computational fluid dynamics (CFD) preprocessor,

GAMBIT. The individual mesh files are then imported into the CFD program

FLUENT and the simulations are finally run in FLUENT. Heat loss models arecompared to adiabatic models. The results show radiative heat loss is most

significant inside the motor case whereas convective heat loss is greater in thenozzle. Convective losses in the nozzle dominate the overall heat loss. The heat loss

in general does not significantly affect ballistic performance, validating the adiabatic

assumption, however the CAD-CFD method is useful for other ballistics analysis.

_______________________, Committee ChairDongmei Zhou, Ph. D.

_______________________

Date


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ACKNOWLEDGMENTS

The author would like to acknowledge and thank: Aerojets Ballistics group for the

introduction to the Pro Engineering CAD method of SRM modeling, the support of

school of Engineering and Computer Sciences at California State University, Sacramento

for providing the computational resources, and finally Dr. Dongmei Zhou and Dr James

Bergquam for their technical expertise and guidance.


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TABLE OF CONTENTS

PageAcknowledgments................................................................................................................v List of Tables ................................................................................................................... viiiList of Figures .................................................................................................................... ix

Chapter

1 INTRODUCTION ............................................................................................................12 INTRODUCTION TO SRMS AND BALLISTICS ENGINEERING .............................5

2.1 Solid Rocket Motor Basics ......................................................................................52.2 Ballistics Engineering ..............................................................................................82.3 Performance Parameters ........................................................................................13

3 THE SOLID ROCKET MOTOR....................................................................................163.1 Physical and Material Properties ...........................................................................163.2 The Case and Nozzle .............................................................................................173.3 The Grain ...............................................................................................................193.4 Propellant ...............................................................................................................20

4 MODELING METHOD .................................................................................................234.1 CAD Modeling Method .........................................................................................234.2 Mesh Construction .................................................................................................264.3 Computational Fluid Dynamics Model ..................................................................28

5 ADIABATIC AND HEAT LOSS MODELS .................................................................395.1 Adiabatic Model.....................................................................................................395.2 Heat Loss Zones .....................................................................................................415.3 Flow Characteristics and Free-stream Reference ..................................................415.4 Heat Loss Model ....................................................................................................42

6 RESULTS .......................................................................................................................546.1 Heat Loss ...............................................................................................................546.2 Motor Performance ................................................................................................59


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7 CONCLUSION ...............................................................................................................60Appendix A ........................................................................................................................63

SRM Transient Algorithm ...........................................................................................63Appendix B ........................................................................................................................64

ProE CAD Method .......................................................................................................64References ..........................................................................................................................70


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LIST OF TABLES

Page

Table 1 Variables and Symbols .......................................................................................... 3Table 2 Case and Nozzle Thermal Properties ................................................................... 19Table 3 Solid Propellant Properties .................................................................................. 21Table 4 Gas Properties ...................................................................................................... 21Table 5 Al/AP/HTPB Propellant ...................................................................................... 22Table 6 Boundary Layer Mesh Data ................................................................................. 28Table 7 Model Assumptions ............................................................................................. 29Table 8 Boundary Condition Pressure and Time .............................................................. 33

Table 9 Convergence Criteria ........................................................................................... 35Table 10 Run Times per Webstep ..................................................................................... 37Table 11 Percent difference between Free-stream and Wall Temperatures ..................... 48Table 12 Total and Specific Impulse ................................................................................ 59


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LIST OF FIGURES

Page

Figure 1 Basic Solid Rocket Motor .................................................................................... 6Figure 2 Typical SRM ........................................................................................................ 7Figure 3 End-burning Grain ................................................................................................ 9Figure 4 End-Burning Grain with Cylindrical Bore Added.............................................. 10Figure 5 Burn Area and Pressure Comparison ................................................................. 11Figure 6 Action Time ........................................................................................................ 15Figure 7 Solid Rocket Motor under study ........................................................................ 17Figure 8 Nozzle Geometry ................................................................................................ 18

Figure 9 Grain Design ....................................................................................................... 20Figure 10 SRM Burning Back .......................................................................................... 25Figure 11 Burn Area versus Webstep Data....................................................................... 25Figure 12 GAMBIT Generated Mesh ............................................................................... 27Figure 13 Mesh Sensitivity Study Results ........................................................................ 36Figure 14 Percent Error and Runtime ............................................................................... 38Figure 15 Heat Loss Areas ................................................................................................ 41Figure 16 Forward Wall Heat Transfer Coefficient, Wall and Freestream Temperatures 49Figure 17 Throat Heat Transfer Coefficient, Wall and Freestream Temperatures ........... 51Figure 18 Heat Loss from Forward and Converging Walls .............................................. 55Figure 19 Heat Loss from Throat and Diverging Walls ................................................... 56Figure 20 Total Heat Loss................................................................................................. 57Figure 21 Heat Transfer Coefficients- Forward and Converging Walls ........................... 58Figure 22 Heat Transfer Coefficients- Throat and Diverging Walls ................................ 58Figure 23 Modeling Algorithm ......................................................................................... 62Figure 24 Example SRM Grain ........................................................................................ 64Figure 25 Driving Dimension ........................................................................................... 65Figure 26 Related Dimensions .......................................................................................... 65Figure 27 Dimensions changing as the Driving Dimension Changes .............................. 66
http://gaia.ecs.csus.edu/moores/phonymotor/Thesis/Heat%20loss%20through%20SRM%20chamber%20walls%20-%20A.docx%23_Toc267846700http://gaia.ecs.csus.edu/moores/phonymotor/Thesis/Heat%20loss%20through%20SRM%20chamber%20walls%20-%20A.docx%23_Toc267846700http://gaia.ecs.csus.edu/moores/phonymotor/Thesis/Heat%20loss%20through%20SRM%20chamber%20walls%20-%20A.docx%23_Toc267846700


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Figure 28 Measuring the Areas ......................................................................................... 67Figure 29 Creating a Family of Data ................................................................................ 68Figure 30 Family of Data Table ........................................................................................ 69


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Chapter1

INTRODUCTION

There are several assumptions made when the ballistics of a solid rocket motor

(SRM) is modeled. Among them is the assumption that the chamber wall of the motor is

adiabatic, i.e., no heat from combustion is lost through the chamber wall. While this

assumption is useful and reasonably accurate for most motor applications it can produce

errors in performance characteristic predictions especially of motors that have significant

exposed internal chamber area at start-up or during motor operation.

The chamber wall loses heat when exposed directly to hot combustion gases even

with the use of thermal insulation. Some of the internal pressure, and therefore thrust, is

lost when heat escapes through the case walls. Other measures of performance, such as

impulse and burn time are also affected. The use of insulation minimizes the effect,

although its primary purpose is to protect the SRM case from detrimental heat effects.

The effect heat loss has on performance is usually not well known until after detailed

thermal analysis or after static firing tests, both occurring after the initial design is

complete. Any re-design work to compensate for heat-loss can be costly and time

consuming. Accounting for it early in the design provides more accurate performance

predictions and may result in fewer design iterations. Since ballistics design is among the

first of SRM design effort, it makes sense to include accounting for the effect of heat-loss

here.

Ballistic design includes designing and modeling the internal geometry of the solid

rocket motor propellant, or grain. The geometry directly affects pressure and thrust


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profiles and defines the amount of internal case surface exposed to hot combustion gases

during motor operation. Ballistics engineering includes predicting SRM performance by

running computer models. Accurate models of the solid rocket motors are therefore

important. Current ballistic modeling techniques typically do not account for the loss of

heat through chamber walls. It is here that a method of estimating heat loss can be

useful.

This work uses a CAD-CFD approach to determine heat loss from a representative

SRM. The physical SRM description and ballistic performance characteristics are

discussed in Chapter 2 followed by the SRM description in Chapter 3. In Chapter 4 the

CAD and CFD modeling approach are discussed, which includes modeling assumptions,

boundary conditions. Chapter 5 discusses the adiabatic and the heat-loss models. Results

from the heat-loss model are compared to the adiabatic model in Chapter 6. Chapter 7

concludes the thesis.

Table 1provides the symbols and variables used throughout the thesis.


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Table 1 Variables and Symbols

Variable or

symbol

Definition Unit

A area m2

a burn rate coefficient (burn rate at 1.0 Pa) m/s

c specific heat at constant pressure J/kgK

cr curvature n/a

F force or thrust N

Fw view factor (from free-stream gas to wall) n/a

go standard acceleration of gravity, 9.80665 m/s2

h heat transfer coefficient W/m2K

Isp specific impulse s

It total impulse N-sk conductivity W/mK

l distance along a surface m

MW molecular weight kg/kmol

n burnrate exponent n/a

Nul local Nusselt number, hl/k n/a

qconductive conductive heat flux, *( / )k dT dz W/m2

qconvective convective heat flux, )( wTTh W/m2

qradiative radiative heat flux, ww FTT )( 44 W/m

2

Pr Prandtl number, c/k, / n/a

P pressure PaR gas constant, R`/MW kJ/kg-K

R` Universal gas constant (8.314 kJ/kmol-K) kJ/kmol-K

r radius m

rb burning rate m/s

Rel local Reynolds number based distance l, l u*/ n/a

St Stanton number, h/ uc n/a

T temperature K

t time s

tw thickness of the wall m

u free-stream velocity m/s

x webstep distance mz wall thickness m

thermal diffusivity, k/ c m2/s

change n/a

emissivity n/a

1 displacement thickness, 1= 1.72* (*l/u) m

2 momentum thickness, 2= 0.664*(*l/u) m


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Stefan-Boltzmann constant, 5.67e-8 W/m2-K4 W/m2-K4

dynamic viscosity N-s/m2

kinematic viscosity, / m2/s

ratio of specific heats, cp/cv n/a

density kg/m3

Subscripts

a action time

amb ambient conditions external to the motor

c case condition

cr curvature

e nozzle exit

s internal surface

t nozzle throat

w wall condition

free-stream condition


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Chapter 2

INTRODUCTION TO SRMS AND BALLISTICS ENGINEERING

This chapter introduces solid rocket motor basics and a brief introduction to ballistics

engineering. This is to gain some understanding of the language used to describe SRM

systems and to understand the CFD modeling method. The SRM performance

parameters used for comparing the adiabatic model with the heat loss model are defined.

2.1 Solid Rocket Motor Basics

An SRM is one of two basic classes of chemical rockets. Typically, rockets are

propelled with either liquid or solid fuels although there are other types of rockets (to

include, but not limited to, ion and nuclear propulsion). Just as the term liquid in liquid

rocket engines refers to the phase of the fuel, the term solid in solid rocket motor also

refers to the phase of the fuel. The rocket in this study is a solid rocket motor.

SRMs are assembled with several typical components, as shown inFigure 1 [1] and

Figure 2 [2]. The casing provides the basic structure, contains the mass and pressure

produced by the burning solid propellant, and transfers thrust to the payload. Typically,

the case is internally insulated more to protect the motor structure from adverse heat

effects from combustion gases than to prevent heat loss. The converging-diverging

nozzle converts the heat, pressure, and mass flow into thrust. An igniter, which produces

high mass and heat flux, is required to start the solid propellant burning. Finally, the

solid propellant, or grain, is the fuel that produces heat, pressure, and mass flow.


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Figure 1 Basic Solid Rocket Motor

Unlike bi-propellant liquid rocket engines, which must maintain the fuel and oxidizer

separately or spontaneous combustion will occur (i.e. hypergolic combustion), solid fuels

combine the fuel and oxidizer together with a binder material in a single mixture. Solid

propellants tend to be quite stable at ambient temperatures and pressures and it is only

after the application of an adequate ignition source to the grain surface that the fuel

begins to combust sustainably. The fuel/oxidizer/binder mixture casts directly in the case

and is left to cure, or can be extruded, cured, and later installed in the case. The cured

solid propellant is called the propellant grain. The grains internal surface can be

machined but is usually formed by allowing the mixture to cure around a forming core.

The internal surface of the grain is designed to create a specified pressure and thrust

versus time profiles depending on the purpose of the rocket system.


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Figure 2 Typical SRM

There are other components used in SRMs, some of which are shown in the above

figures. Nose cones provided volume to contain payloads and reduce the drag

experienced on the rocket system. Stage motors contain skirts used to attach to other

stages and the nosecone in a motor stack. Fins, or strakes, provide aerodynamic stability

during flight. Thrust termination devices can be employed to open the pressure vessel

ending the production of thrust. The system may also use thrust vector control for

stability and maneuverability. A wide variety of avionics equipment may be employed

on an SRM for guidance and control.

A more comprehensive treatise of rockets in general and solid rockets specifically is

found in Sutton [2].


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2.2 Ballistics Engineering

An introduction to ballistics engineering is required here as this type of design work

is atypical and directly relates to the heat loss modeling method. Only basic concepts are

introduced.

The grain burns on any surface that is exposed to combustion gases. Conversely, any

surface that is covered (by the case wall for instance) does not burn. As the exposed

surfaces burn, they recede normal to the burning surface. For example in the case of a

simple end-burning grain the exposed surface (not covered by the case) burns axially

along the centerline of the motor, as shown inFigure 3. As the grain burns back, the

exposed area does not change until the burning surface reaches the forward dome. A

relationship exists between the burn area and burn distance, as shown in the graph in the

same figure inFigure 3. In this case, it shows the area remains constant throughout the

burn until the dome is reached. If a cylindrical bore is added to the end-burning grain,

the end still burns, however this time there is a change in burn area as the bore grows as

shown inFigure 4. The relationship between the burn area and the burn distance is

clearly modified. If other geometries are cut from the grain then almost any burn area

versus burn distance profile can be obtained.


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Figure 3 End-burning Grain

The burn distances are typically referred to as websteps. The web of a grain is the

largest distance that a burning surface will travel. For example, inFigure 3,the largest

distance the burning surface travels is the axial distance from the aft end to the forward

end so this is its web. InFigure 4,the web is the radial distance from the bore surface to

the case. Although the grain surface is continuously receding as it burns it is represented

by a series of burn distances, or websteps.


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Figure 4 End-Burning Grain with Cylindrical Bore Added

The profile of the burn area, Ab, versus webstep plot relates to the profile the internal

pressure versus time plot. In fact, the shapes are similar except during motor startup and

motor tail-off (end of motor operation) where transient effects become significant. This

is clear when a burn area profile is compared to a pressure profile, provided inFigure 5.

Additionally, the thrust-time profile follows the pressure-time profile. This allows the

ballistics designer to custom fit a specified pressure or thrust profile by modifying the

grain internal geometry while knowing nothing of the fuel.


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Figure 5 Burn Area and Pressure Comparison

The parameters of burn area, Ab, and webstep are easily obtained from any preferred

CAD program (ProE was used here). The burn distance intervals are modeled by

offsetting burning surfaces and can be arbitrarily chosen. Usually, the web is determined

and is divided into approximately fifteen or more websteps. This is heuristic and depends

upon the detail required of the grain being modeled. Only a few websteps are shown in

Figure 3 andFigure 4. The burn area, Ab, is then found for each webstep by using the

analysis tools in the CAD program.

The relationship between the Abprofile and the pressure and thrust profiles becomes

apparent upon examination of the relationship between the burn rate of the propellant and

mass flux. The steady-state burn rate (m/s) simply is

t

xr

b

(1)

If the webstep, x, is known (and it is because it is chosen) all that is needed is the burn

rate to find time step, t. The propellant burn rate follows the relationship


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n

cb aPr (2)

where aand nare empirical constants, which are ideally constant over wide pressure

range. This reveals that the burn rate of the propellant is directly related to the internal

pressure of the motor.

Some manipulation is required to determine pressure at each webstep. The mass flux

off the grain is found using

bbbb Arm

(3)

wherebis density andAbis the burn surface area [2]. The mass flow through the throat

is defined by

c

APm tct

(4)

wherePcis the chamber pressure andAtis the throat area [2]. The factor, c*, is also a

property of the propellant, is considered constant, and will be defined later. If it is

assumed that steady-state conditions exist then

tb mm

. (5)

This assumption is valid except during motor startup and tail-off where transient effects

become significant, however in most cases this causes only negligible error in the results.

It can be shown after some manipulation that


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n

t

bbc

A

caAP

1

1

(6)

Taking all the factors and exponent as constants exceptAb, it is clear that pressure is a

function of the burn area,Ab.

The burn rate, rb, is determined from the pressure using equation (2). Time is then

found by dividing the webstep (arbitrarily defined) by the burn rate. This method

correlatesAb-webstep profile to the pressure-time profile and illuminates why the profiles

are similar. This method determines the pressure profile in the representative SRM.

2.3 Performance Parameters

There are several parameters used to measure motor performance. The ones used

here are total impulse, specific impulse, internal motor pressure, thrust, and burn time.

Total impulse (N-s) is defined as

dtFIt * (7)

whereFis thrust (N) [2]. Specific impulse (s) is defined by

dtmg

I

dtmg

dtFI

o

t

o

sp

****

*

(8)

wheregois standard acceleration of gravity [2]. Its units are simplified to seconds though

specific impulse does not refer to time. The units are accurately defined as thrust per unit

weight flow rate, or Newtons-seconds per go-(kilogram/second)-seconds. The standard


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gravity term is equal to 9.81 m/s2. This simplifies to the units of seconds. Specific

impulse is to rockets as miles-per-gallon is to automobiles.

Internal motor pressure is calculated as previously discussed. Thrust is determined

from

ctF PACF (9)

where

t

e

c

ambe

c

eF

A

A

P

PP

P

PC

1

11

2

11

2

1

2. (10)

CFis the thrust coefficient and is a function of the ratio of specific heats, , exit, chamber,

and ambient pressures, and nozzle area ratio [2].


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The final performance parameter is action time. Action time is based on the internal

motor pressure. At motor start-up the motor pressurizes rather quickly. However, as the

motor begins to burn out the pressure drops asymptotically toward zero making burn

duration rather difficult to determine. Because of this, action time, ta, is defined as the

time the motor pressure first reaches 10% of max pressure to the time when the pressure

drops to 10% of max pressure during the end of motor operation [2]. This is illustrated in

Figure 6.

Figure 6 Action Time


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Chapter 3

THE SOLID ROCKET MOTOR

This chapter defines the solid rocket motor, or SRM, used in this work. It starts by

discussing the physical and material properties of the SRM case and nozzle. The grain

physical properties are defined. The chapter finishes with a discussion of the solid

propellant and its physical and gas properties.

3.1 Physical and Material Properties

The SRM considered in this work (seeFigure 7)is a simplified version of a typical

SRM. It is an example of one that might be used as a stage motor in a multi-stage rocket

system although there is no specific purpose defined here. The enveloping length and

diameter is 1.69 m and 0.48 m, respectively. It employs an insulated case, a converging-

diverging nozzle, and a simple axi-symmetric grain. To add to the simplification, the

igniter has been removed. The igniter usually burns to completion just before the SRM

grain is fully lit so it does not greatly affect heat loss during motor operation. Finally, no

other components, such as nosecones, skirts, thrust vector control, etc. are included as

these components also do not significantly affect internal heat loss. Additionally,

mechanical interfaces are removed. For example, the interface between the nozzle and

the case is usually a mechanical system (circular pattern of bolts, threaded joint, or snap-

ring construction) and uses high temperature o-rings to contain combustion gases within

the case. Since the work here is to model heat loss the design is kept as simple as

possible and includes only those components that have a direct effect on heat loss.


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3.2 The Case and Nozzle

The insulated case is comprised of typical materials used in SRM motors today. The

case is a carbon fiber filament wound case with an internal insulation made of silica filled

EPDM, ethylenepropylene diene terpolymer. It is assumed that the case and insulation

system takes on the insulating properties of the insulation only. The case thickness is

everywhere 12.7 mm.

The converging-diverging nozzle is made using conventional geometry and is also

made of materials typically found in SRM systems. The nozzle and its geometric

characteristics are given inFigure 8. The nozzle is a typical converging-diverging

nozzle. The converging section provides a smooth transition from the spherical aft dome

of the case to the nozzle entrance. The diverging section is conical and has a standard

Figure 7 Solid Rocket Motor under study


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15 half angle. The throat diameter was sized to target a specific maximum pressure.

The average thickness along the nozzle wall is approximately 12.7 mm although the

thickness does increase near the throat for added thermal-structural capability. This

nozzle is made of graphite material, which is also typical in SRM construction. The

material properties of the nozzle are provided inTable 2.

Figure 8 Nozzle Geometry

61.3

15.0

Ae/At = 7.06

Rthroat = 70 mm

Raxial curvature

= 127 mm


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Table 2 Case and Nozzle Thermal Properties

Component Material Specific heat,C (J/kg-K)

Density, (kg/m

3)

Thermal

conductivity, k

(W/m-K)Case Carbon Fiber and

silica phenolic EPDM

lumped

1674.7 977.13 0.24234

Nozzle Graphite 1425 1540.4 89.7

3.3 The Grain

As was previously mentioned, the design of the grain is a simple axi-symmetric

geometry, shown inFigure 9. This type of geometry allows some simplification of CFD

modeling approach. The bore is a truncated cone and the aft end employs a spherical

cutout. The aft end also has a section of end burning grain. The same philosophy of case

and nozzle simplification applies to the design of the grain: since this work focuses on

heat loss of the SRM a simple model is desired.


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Figure 9 Grain Design

3.4 Propellant

The propellant used in this work is a representation of typically used propellant. No

attempt is made to design a propellant as this is a task more suitable for chemists or

chemical engineers than for ballistics engineers. Only the solid propellant properties and

gas properties are required to complete the SRM definition. The solid propellant and gas

properties of the representative propellant are provided inTable 3 andTable 4,

respectively.


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Table 3 Solid Propellant Properties

Symbol Nomenclature Value Unit Notesa Burn rate coefficient 1.39e-05 m/s Calculated from a=rb,ref/Prefn,

where Pref= 6.89MPa

n Burn rate exponent 0.5 n/a

b Density 1799.2 kg/m

c Characteristic velocity 1356 m/s See definition below

Table 4 Gas Properties

Symbol Nomenclature Value Unit Notesk Thermal conductivity 0.231 W/m-K

cp Specific heat at constant

pressure

1286.6 J/kg-K

Dynamic

Viscosity

9.5e-5 kg/m-s

Tf Flame temperature 2756.4 K Stagnation temperature

MW Molecular weight 28 kg/kgmol

The parameter c*(pronounced cee-star), is the characteristic velocity (m/s), as

shown inTable 3. Rearranging equation (4), it is defined by

t

tc

m

APc

(11)

By definition, it relates to the internal pressure, Pc, the throat area, At, and the mass flow

through the throat. However, it relates more to propellant combustion efficiency, which

is independent of the nozzle geometry. It can be shown

1

1

1

2

RTc

. (12)

Since the factors gamma, ratio of specific heats, , gas constant,R, and the absolute

temperature, T, are properties of combustions gases, c*must also be a property of the


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combustion gas [2]. The parameter c*is a thermodynamic property of the propellant

characteristic of the thrust coefficient. It refers to the average velocity of the gas at the

nozzle exit plane for a thrust coefficient, Cf, of 1.0. The exit velocity of the gases from

the motor at any time during operation can be calculated by multiplying Cf by c*. Since

gamma and temperature remain constant over a wide range of pressures therefore c*is

assumed constant as previously discussed.

The propellant properties are approximated from one of the most common

formulations used today-Al/AP/HTPB propellant [2]. Properties are inTable 5.

Table 5 Al/AP/HTPB Propellant

Chemical formula Nomenclature Function Approximate

Mass fraction

Notes

Al Aluminum Fuel 0.17 In powder form

NH4ClO4 Ammonium

perchlorate(AP)

Oxidizer 0.70 Crystal sizes

10250m

HTPB Hydroxyl-

terminatedpolybutadiene

Binder 0.13


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Chapter 4

MODELING METHOD

Now that the SRM definition and the performance parameters have been determined,

this chapter defines the modeling methods. The discussion includes the computer aided

design (CAD) method, the mesh construction, and the computational fluid dynamics

(CFD) method to model the motor.

4.1 CAD Modeling Method

FLUENT, a computation fluid dynamics (CFD) program, is a numerical solver [3]. It

requires the creation and meshing of the model flow volume under study outside of the

program. FLUENT takes advantage of a preprocessor GAMBIT that can be used to

produce a computer model and then used to produces the mesh [4].

Another method, the one used in this study, is to model the flow volume outside the

GAMBIT preprocessor using a CAD program. Once the CAD models are created they

are imported into GAMBIT for meshing. The CAD program, Pro Engineer (ProE)[5], is

used in this study to create the CAD models. They are then exported to GAMBIT to be

meshed.

The case, nozzle, and grain geometries are created in ProE. The geometries of the

case and nozzle are constant throughout the motor operation so the geometries are not

changed within the CAD model. The grain, however, does change in time during motor

operation. The initial grain model is created then, one by one, each successive webstep

of the receding grain is created and saved within ProE. Figure 10provides an example.


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The initial grain is shown in the top graphic and the series of successive websteps

follows. This continues until the grain has completely receded.

Pro Engineer provides several tools used to simplify the modeling method [5]. A tool

allows parametrically relating model dimensions. This tool greatly simplifies the creation

of each webstep model by relating the changing grain dimensions to a control dimension.

Changing the control dimension by any arbitrarily chosen webstep changes the receding

grain surfaces by the same amount. For example if the control dimension is changed 0.02

m the spherical radius, and the bore radii are all changed by the same length. Figure 10

shows the control dimension value and its effect on the grain geometry. ProE can

measure surface areas of the CAD model. As the grain recedes the area changes. The

area of each burning surface is measured for each webstep. A final tool used is ProEs

capability to create a table of data, which can be constructed to automatically correlate

the websteps to each burn surface area. The table can be exported to a spreadsheet where

a complete table of burn area,Ab, versus websteps can be made. The table data sample is

shown inFigure 11. More comprehensive discussion of this CAD modeling method is

provided in Appendix B.


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Figure 10 SRM Burning Back

Figure 11 Burn Area versus Webstep Data


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Each ProE CAD model is exported for meshing to GAMBIT. Each CAD model was

converted to the IGES format due to its ability to communicate to a wide variety of

modeling programs including GAMBIT. Any number of models can be created, however

converting all of them to IGES format is difficult and time consuming. ProE has a

journal feature, which is intended to help the ProE user to recover previously created

work on a model in the event data was lost, during a power outage for example [5].

Journal files can be modified and used to run repeated steps of saving and exporting

models in the IGES format.

4.2 Mesh Construction

GAMBIT generates the meshes to be used in the CFD program FLUENT [4]. The

IGES files imported to GAMBIT are modified to ease meshing. The three-dimensional

models created in ProE are simplified to two-dimensional, axi-symmetric models;

simplifying the geometry by taking advantage of symmetry reduces the computation

time. Boundary conditions are defined. A mesh interval is chosen and a boundary layer

mesh is added. The chosen meshing scheme contains quadrilaterals and triangles.

GAMBIT automatically generates the mesh using the defined parameters. Figure 12

shows the generated mesh for the initial webstep and the boundary conditions applied to

the surfaces. An insert of the CAD model is provided for reference. The mesh differs

from the CAD model in that only the flow volume needs to be meshed so only the

boundaries that contain it are shown. Additionally, to reduce computation time, only half

of the model is needed since the flow conditions on both halves are the same.


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Figure 12 GAMBIT Generated Mesh

As previously mentioned, a boundary layer mesh was added to the flow volume

mesh. The thickness is based on the maximum calculated momentum boundary layer

thickness of less than 6.00 mm. Table 6provides the boundary layer mesh data.

Creating the boundary layer mesh in this manner captures the flow condition and heat

loss near the case wall surface while still interfacing with the rest of the mesh. The

meshing method also allows for a convenient place for measuring the free flow

conditions. Since the momentum boundary layer at any point along any surface is less


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than 6 mm thick then any flow properties measured beyond are free-stream properties

(i.e. free-stream velocity, density, viscosity, etc.). Free-stream properties are then

measured at the outer edge of the boundary layer mesh.

Table 6 Boundary Layer Mesh Data

Row # Row thickness(mm)

Total BL thickness(mm)

1 0.102 0.102

2 0.159 0.261

3 0.249 0.510

4 0.391 0.9015 0.612 1.513

6 0.959 2.472

7 1.503 3.975

8 2.354 6.329

4.3 Computational Fluid Dynamics Model

Computational fluid dynamics models are discussed now that the CAD models and

the meshes are established. Assumptions are introduced, boundary conditions are

defined, turbulence, convergences criteria, mesh sensitivity, and model validation are

discussed.

4.3.1 Assumptions

There are several assumptions made in the CFD model and are summarized inTable

7. Numbers 1-6 require further explanation while 7-12 are self-explanatory.


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Table 7 Model Assumptions

1 No throat growth 7 The external temperature case is

constant at 300K

2 No deformation of the grain due to

operating pressure and temperature

8 The combustion gas follows the

Ideal Gas law

3 The motor grain ignites

instantaneously

9 Steady-state pressure predictions are

calculated for each webstep

4 Chemical reactions go to completion

immediately upon combustion

10 The gas is calorically perfect

(constant specific heats)

5 Heat transfer due to charring and

sloughing off is ignored

11 The combustion gases have constant

properties

6 Emissivity of charred EPDM and

Graphite is 0.95

12 Flow where Reynolds numbers

based on length is below 60,000 islaminar [10]

The throat erosion occurs during the operation of an SRM [2]. Some erosion is due to

the mass flow across the throat and particle impingement abrades the material away or

the throat material may react chemically to the combustion species that contact the throat

material and accelerate erosion. The effect of this enlarges the throat area which affects

the pressure and thrust profiles. Since this work focuses on heat loss in the chamber, the

throat growth is ignored.

During motor operation the pressures involved compresses the grain causing change

in the grain shape [2]. Temperature soaking can have additional effects changing the

shape of the grain due to thermal expansion and contraction [2]. Since deformation does

not relate to the heat loss during motor operation it is ignored.

The motor grain takes time to fully ignite during startup [2]. The igniter operates and

expels hot gas and material onto the motor grain surface igniting the motor. A flame

front speeds across the grain until all exposed burn area is lit. The pressure begins to rise


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as the flame spreads and propellant begins to burn. These events take a measurable

amount of time. This time duration is the ignition transient. However, the ignition

transient is usually small relative to the time the motor is operating, as is the case for this

motor. The transient event is ignored since it does not greatly affect heat loss.

Upon grain ignition, chemical reactions between fuels and oxidizers occur near the

surface of the grain producing combustion gases [2]. Reactions can occur within the gas

flow in the case, the throat, and the exit cone. These secondary reactions can affect gas

flow but the effect on heat loss is not significant.

Major modes of heat transfer from combustion gases to the chamber wall and nozzle

is convective and radiative [6] [7]. Some heat is transferred to the walls conductively by

particles within the gas flow impinging on the walls. Heat is transferred from the walls

by inert insulating material charring and sloughing off during motor operation. It is

assumed that these other modes of heat transfer are not significant.

The emissivity of the internal insulation and the nozzle material at high temperature is

difficult to determine. The internal insulation and the nozzle material char as the motor

operates [2]. Char, carbon, is assumed to have the same emissivity of lampblack at 1000

C, or 0.96 [8]. However, unburned EPDM is a hard rubber material, which has an

emissivity of 0.94 [8]. Averaging these two values gives an emissivity of 0.95. This is

the value used for both the charring internal insulation and the nozzle.


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4.3.2 Model Set-up

The models use the same basic FLUENT solver settings. The models use the

pressure based, implicit solver. The CFD model is set to 2-D axi-symmetric reflecting

the motor geometry. The solver is set to steady-state based on the above assumptions.

The working fluid is viscous, and it follows the ideal gas law so the energy equation is

turned on.

Turbulence is included in the viscosity model since Reynolds numbers reach

23,000,000. The k- turbulence with RNG (renormalization group theory) model was

chosen. This model accounts for a wide range of Reynolds number flow and more

accurately accounts for rapidly strained flows, both of which occur in these models [3].

Default values were used to set up the model as much of the turbulent flow is unknown.

Work by Thakre and Yang used similar values in modeling turbulence in an SRM nozzle

erosion investigation supporting the selection made here [6].

The materials used in the CFD model are defined. These include the solid materials

used in the nozzle, the insulated case, and the working fluid. The nozzle and case

material properties are found inTable 2. The gas properties are foundTable 4.

The motor operating conditions are defined. The external temperature and pressure

are ambient, at 300K and 101.325 kPa, respectively.

4.3.3 Boundary Conditions

The insulated case and the nozzle are modeled as stationary walls. They use a no-slip

shear condition and the default value wall roughness. The wall thermal conditions are


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defined using heat flux values. These values are set to zero in the adiabatic models. In

the heat loss models, they are calculated using heat transfer coefficients and radiative heat

flux (discussed later).

The nozzle exit plane is a pressure outlet. The values in the momentum tab are all set

to default and the backflow total temperature is set to 300K, the same as the external

ambient temperature.

Two boundary conditions include the centerline and the working fluid. The motor

centerline is the x-axis about which the case, nozzle, and grain geometries are rotated.

The grain surface is a mass flow inlet boundary. The mass flow direction is specified

as normal to the surface. Turbulence kinetic energy and dissipation rate are both set to

zero as required in laminar, transpired flow [9]. The total temperature is equal to the

flame temperature, 2756.4K.

The grain boundary mass flow for the individual model is found using theAbdata

obtained by the CAD models in ProE. Using the relationship between chamber pressure,

Pc, and the grain burn surface area,Ab, in equation (6), pressure is found for each

webstep. This pressure is used to determine the mass flow off the grain only. The

pressure reported in the final solution comes from FLUENT (although the difference is

negligible). Mass flow off the grain is found using equation (3). This value is calculated

and used as the boundary condition for each webstep. Table 8 shows some of the results.


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Table 8 Boundary Condition Pressure and Time

For example, this is how the results are obtained for #10 webstep. The burn area,

0.6174 m2is determined from the CAD model as previously discussed. Pressure,Pc, is

found using equation (6) and the parameters a,b, c*, and n. Pressure is then

1 11 1 0.5

2

1.39 5*0.6174*1799.2*13561.853

*0.07

nb b

c

t

aA c eP

A

MPa.

The burn rate is found using equation (2),

0.51.39 5*(1.853 6) 0.01886nb cr aP e e m/s.

The mass flow used in the boundary condition is found using equation (3),

0.01886*1799.2*0.6174 20.95b b b bm r A

kg/s.


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Although it is not specifically related to the boundary definition this is an appropriate

place to discuss the pressure and mass flow correlation to time. Since the webstep and

the burn rate for the #10 webstep are known the change in time from the previous

webstep can be determined using a modification of equation (1),

b

xr

t

,

b

xt

r

0.01016 0.007620.13468

0.01886b

xt

r

s.

The delta-time is added to the time correlated to the #9 webstep to get

10 9 10 0.4216 0.13468 0.5562t t t s.

Pressure and mass flow are now correlated to time.

4.3.4 Convergence Criteria

The CFD solutions had to meet or exceed convergence criteria. These criteria are

defined for the residual monitor parameters and in the mass flow balance between the

grain, mass flow inlet, and the exit plane, mass flow outlet. All webstep models are

converged when the residuals are less than or equal to the values listed inTable 9 and the

mass imbalance between the mass flow off the grain and the mass flow out of the nozzle

exit plane was less than the value inTable 9.


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Table 9 Convergence Criteria

Residual or condition CriteriaContinuity 1e-5

Velocity in X-direction (axial) 1e-5

Velocity in Y-direction (radial) 1e-5

Energy 1e-8

k (turbulence) 1e-5

(turbulence) 1e-5

Mass imbalance 2e-5

4.3.5 Model Validation

It is necessary to compare the CFD model results to other model results in order to

validate the model. Ideally, the model should be validated by live-fire test results,

however, there is no actual pressure or thrust data for this motor since it is a simplified

representation of actual SRMs. The ballistics of this SRM was modeled using an Aerojet

proprietary transient ballistics program called here SRM. The program uses an input of

ballistics parameters including throat area, burn area versus webstep, and gas properties

and uses the algorithm as provided in Appendix A. The pressure results from this

program are inFigure 13,labeled SRM Pressure. This SRM model is validated to this

standard.


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Figure 13 Mesh Sensitivity Study Results

4.3.6 Mesh Sensitivity Study

The SRM free volume for each webstep was meshed individually. A mesh sensitivity

study was performed to ensure acceptable solution accuracy while using computer time

efficiently. An acceptance criterion was defined and several mesh sizes were tested until

the largest interval size was found that produced results that were reasonably insensitive

to mesh density and used computer time efficiently. The efficient use of computer time is

necessary as there is a CFD model run for each of several individual websteps.


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Table 10 Run Times per Webstep

The criteria for choosing the interval size depend upon percent error between the

CFD models and the standard, SRM Pressure. Upon inspection ofFigure 13,it is clear

that the Coarse and Very Coarse solutions do not provide accurate results when compared

to the Fine and Medium meshes so they were rejected immediately. The criteria also

depend on the amount of runtime for each model. The Fine mesh produces accurate

results, but the model runtime is unacceptably long (greater than 30 minutes per webstep)

when compared to the Medium mesh, according toTable 10. When the accuracy and the

run times are compared to the Medium mesh, shown inFigure 14,the Medium mesh

provides the best accuracy for the run times. The choice is the Medium mesh interval as

the mesh for all CFD model websteps.

Interval Size (mm) Time to run (s)

3.81 1800-2700

6.35 200-300

12.7 10-15

19.1


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Figure 14 Percent Error and Runtime


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Chapter 5

ADIABATIC AND HEAT LOSS MODELS

The adiabatic and heat loss models are introduced in this chapter. The adiabatic data

is discussed and a thrust and an impulse calculation example is provided. The heat loss

zones and their associated flow characteristics are defined. The convective heat transfer

models and radiative heat loss models are discussed. The method to determine the

internal surface temperature for each zone is discussed. Examples are given to illustrate

heat loss model calculations.

5.1 Adiabatic Model

Adiabatic solutions of each webstep are calculated. They use the assumptions, model

set-up, and boundary conditions discussed in Chapter 4. Freestream conditions, including

velocity, density, and temperature are found from FLUENT. FLUENT solutions of

pressure, mass flow, and velocity are also collected and used to calculate thrust and

impulse. These values are compared against the heat loss models to determine the effect

heat loss has on the SRM.

Typical freestream values are included in the heat loss model discussion but an

example is useful to illustrate pressure, thrust, and impulse calculations. This example

occurs in the adiabatic model approximately 3.5 seconds into the motor operation. Case

pressure data from FLUENT is collected, averaged, and found to be 3.18 MPa. The

thrust coefficient is computed using equation (10), and assuming the nozzle exit pressure

equals the ambient pressure, the thrust coefficient is


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t

e

c

ambe

c

eF

A

A

P

PP

P

PC

1

1

12

1

1

2

1

2

45.1000,180,3

1013251

13.1

2

13.1

)3.1(2 3.113.1

13.1

13.12

.

Thrust is found using equation (9) and is

71000,180,3*070.0**45.1 2 ctF PACF kN.

Pressure and thrust are calculated in the same manner for each webstep and for both the

adiabatic and heat loss models.

Impulse is found for the entire motor operation (although an impulse can be found up

to this point if desired). Total impulse and specific impulse are calculated using

equations (7) and (8), respectively. By numerical integration, the adiabatic total impulse

and specific impulse for the complete motor operation are

000,612* dtFIt N-s

and

2.189

****

*

dtmg

I

dtmg

dtFI

o

t

o

sp s.

Total and specific impulse are found for the heat loss model using this same method.


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5.2 Heat Loss Zones

There are four regions where heat is lost through the motor case and nozzle, as shown

inFigure 15. The forward case wall and the converging wall grow in area as the grain

burns. The converging wall includes the wall surface from the aft face of the grain to

pointA. The throat area includes the region from pointAto the throat. This area is

markedly different from the rest of the aft case region as the gas begins to experience

high acceleration here. The diverging section includes the region aft of the throat. The

gas experiences additional acceleration here.

Figure 15 Heat Loss Areas

5.3 Flow Characteristics and Free-stream Reference

The flow in the heat loss zones are modeled as external flow along a flat plate, and

the free-stream conditions are taken near the wall surface instead of the motor axis. The

flow is modeled as external because it is defined as a wall bounding one side of a

boundary layer and free-stream conditions bounding the other [10]. Even at the smallest

radius, at the throat, the boundary layers do not converge at any time during the motor


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operation. This condition requires the flow to be treated as external flow. The boundary

layers (momentum and displacement) along any of the heat loss surfaces do not exceed

more than about 6 mm. For example, the largest displacement boundary layer occurs

near the throat at 3.95 seconds into motor operation. The displacement thickness here is

1

( / )* (9.5 5 / 4.329)*0.1911.72 1.72 5.954

0.350

l e

u

mm.

The momentum thickness is

2

( / )*0.664 2.299

l

u

mm

and will always be less than the displacement thickness. Therefore, the free-stream

conditions are taken at points along a line parallel to and 6 mm from each wall. This is a

better representation of local free-stream conditions than the motor centerline flow as this

flow is so far removed that it does not adequately represent the local freestream

conditions. This is especially true when considering the flow at the forward wall where

the flow is normal to the gas flow along the motor centerline.

5.4 Heat Loss Model

The heat loss model definition requires heat flux to be defined for each heat loss zone

at each webstep. Heat flux boundary condition requires knowledge of heat transfer

parameters since the models use a combination of convective and radiative heat loss.

Convective heat loss is found using the relationship

)("

wconvective TThq (13)


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Radiative heat transfer for the heat loss zones is found by

wwradiative FTTq )(44"

. (14)

The total heat loss is then

radiativeconvectivetotal qqq"""

, (15)

or

wwwtotal FTTTThq )()(44"

. (16)

Based on the above equations, and knowing Tfrom the adiabatic models, it is necessary

to find the convective heat transfer coefficients, effective emissivity, view factor, and the

internal wall temperatures for each heat loss zone.

5.4.1 Convective Heat Transfer Coefficients

The forward wall heat transfer coefficient is found assuming laminar flow across a

flat plate. The assumption of flow along a flat plate is not technically correct, as the gas

actually flows radially inward, however the error introduced does not significantly affect

the total motor heat loss. The flow within the zone has low Reynolds numbers as the

motor begins to operate. For example, at 10% into motor operation the largest Reynolds

number is

411.255.9

053.0*62.2*35.14**Re e

e

lul

.


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As the motor approaches 50% into the burn, some area becomes turbulent, Re=300,000,

but laminar flow dominates most of this zone. The flow returns to Reynolds numbers (

Documents

Ballistics Modeling of Combustion Heat Loss Through Chambers and Nozzles of Solid Rocket Motors