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 of
the requirements for the degree of
MASTER OF SCIENCE
in
MECHANICAL ENGINEERING
at
CALIFORNIA STATE UNIVERSITY, SACRAMENTO
SUMMER
2010
BALLISTICS MODELING OF COMBUSTION HEAT LOSS THROUGH
CHAMBERS 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
ii
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.
Department of Mechanical Engineering
iii
___________________
Date
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 the
motor 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 an
assumptions 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 are
compared to adiabatic models. The results show radiative heat loss is most
significant inside the motor case whereas convective heat loss is greater in the
nozzle. 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 Chair
Dongmei Zhou, Ph. D.
_______________________
Date
iv
ACKNOWLEDGMENTS
The author would like to acknowledge and thank: Aerojet’s 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.
v
TABLE OF CONTENTS
Page
Acknowledgments................................................................................................................v
List of Tables ................................................................................................................... viii
List of Figures .................................................................................................................... ix
Chapter
1 INTRODUCTION ............................................................................................................1
2 INTRODUCTION TO SRMS AND BALLISTICS ENGINEERING .............................5
2.1 Solid Rocket Motor Basics ......................................................................................5
2.2 Ballistics Engineering ..............................................................................................8
2.3 Performance Parameters ........................................................................................13
3 THE SOLID ROCKET MOTOR....................................................................................16
3.1 Physical and Material Properties ...........................................................................16
3.2 The Case and Nozzle .............................................................................................17
3.3 The Grain ...............................................................................................................19
3.4 Propellant ...............................................................................................................20
4 MODELING METHOD .................................................................................................23
4.1 CAD Modeling Method .........................................................................................23
4.2 Mesh Construction .................................................................................................26
4.3 Computational Fluid Dynamics Model ..................................................................28
5 ADIABATIC AND HEAT LOSS MODELS .................................................................39
5.1 Adiabatic Model.....................................................................................................39
5.2 Heat Loss Zones .....................................................................................................41
5.3 Flow Characteristics and Free-stream Reference ..................................................41
5.4 Heat Loss Model ....................................................................................................42
6 RESULTS .......................................................................................................................54
6.1 Heat Loss ...............................................................................................................54
6.2 Motor Performance ................................................................................................59
vi
7 CONCLUSION ...............................................................................................................60
Appendix A ........................................................................................................................63
SRM Transient Algorithm ...........................................................................................63
Appendix B ........................................................................................................................64
ProE CAD Method .......................................................................................................64
References ..........................................................................................................................70
vii
LIST OF TABLES
Page
Table 1 Variables and Symbols .......................................................................................... 3
Table 2 Case and Nozzle Thermal Properties ................................................................... 19
Table 3 Solid Propellant Properties .................................................................................. 21
Table 4 Gas Properties ...................................................................................................... 21
Table 5 Al/AP/HTPB Propellant ...................................................................................... 22
Table 6 Boundary Layer Mesh Data ................................................................................. 28
Table 7 Model Assumptions ............................................................................................. 29
Table 8 Boundary Condition Pressure and Time .............................................................. 33
Table 9 Convergence Criteria ........................................................................................... 35
Table 10 Run Times per Webstep ..................................................................................... 37
Table 11 Percent difference between Free-stream and Wall Temperatures ..................... 48
Table 12 Total and Specific Impulse ................................................................................ 59
viii
LIST OF FIGURES
Page
Figure 1 Basic Solid Rocket Motor .................................................................................... 6
Figure 2 Typical SRM ........................................................................................................ 7
Figure 3 End-burning Grain ................................................................................................ 9
Figure 4 End-Burning Grain with Cylindrical Bore Added.............................................. 10
Figure 5 Burn Area and Pressure Comparison ................................................................. 11
Figure 6 Action Time ........................................................................................................ 15
Figure 7 Solid Rocket Motor under study ........................................................................ 17
Figure 8 Nozzle Geometry ................................................................................................ 18
Figure 9 Grain Design ....................................................................................................... 20
Figure 10 SRM Burning Back .......................................................................................... 25
Figure 11 Burn Area versus Webstep Data....................................................................... 25
Figure 12 GAMBIT Generated Mesh ............................................................................... 27
Figure 13 Mesh Sensitivity Study Results ........................................................................ 36
Figure 14 Percent Error and Runtime ............................................................................... 38
Figure 15 Heat Loss Areas ................................................................................................ 41
Figure 16 Forward Wall Heat Transfer Coefficient, Wall and Freestream Temperatures 49
Figure 17 Throat Heat Transfer Coefficient, Wall and Freestream Temperatures ........... 51
Figure 18 Heat Loss from Forward and Converging Walls .............................................. 55
Figure 19 Heat Loss from Throat and Diverging Walls ................................................... 56
Figure 20 Total Heat Loss................................................................................................. 57
Figure 21 Heat Transfer Coefficients- Forward and Converging Walls........................... 58
Figure 22 Heat Transfer Coefficients- Throat and Diverging Walls ................................ 58
Figure 23 Modeling Algorithm ......................................................................................... 62
Figure 24 Example SRM Grain ........................................................................................ 64
Figure 25 Driving Dimension ........................................................................................... 65
Figure 26 Related Dimensions .......................................................................................... 65
Figure 27 Dimensions changing as the Driving Dimension Changes .............................. 66
ix
Figure 28 Measuring the Areas ......................................................................................... 67
Figure 29 Creating a Family of Data ................................................................................ 68
Figure 30 Family of Data Table ........................................................................................ 69
x
1
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
2
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 1 provides the symbols and variables used throughout the thesis.
3
Table 1 Variables and Symbols
Variable or
symbol
Definition
Unit
A
a
c
cr
F
F∞→w
go
h
Isp
It
k
l
MW
n
Nul
q”conductive
q”convective
m2
m/s
J/kgK
n/a
N
n/a
m/s2
W/m2K
s
N-s
W/mK
m
kg/kmol
n/a
n/a
W/m2
W/m2
q”radiative
area
burn rate coefficient (burn rate at 1.0 Pa)
specific heat at constant pressure
curvature
force or thrust
view factor (from free-stream gas to wall)
standard acceleration of gravity, 9.80665
heat transfer coefficient
specific impulse
total impulse
conductivity
distance along a surface
molecular weight
burnrate exponent
local Nusselt number, hl/k
conductive heat flux, k *(dT / dz )
convective heat flux, h(T  Tw )
radiative heat flux,  (T4  Tw4 ) Fw
Pr
P
R
R`
r
rb
Re l
St
T
t
tw
u∞
x
z
α
Δ
ε
δ1
δ2
Prandtl number, μc/k, ν/α
pressure
gas constant, R`/MW
Universal gas constant (8.314 kJ/kmol-K)
radius
burning rate
local Reynolds number based distance l, l u∞* ρ/ μ
Stanton number, h/ u∞ρc
temperature
time
thickness of the wall
free-stream velocity
webstep distance
wall thickness
thermal diffusivity, k/ ρc
change
emissivity
displacement thickness, δ1 = 1.72*√(ν*l/u∞)
momentum thickness, δ2 = 0.664*√(ν*l/u∞)
n/a
Pa
kJ/kg-K
kJ/kmol-K
m
m/s
n/a
n/a
K
s
m
m/s
m
m
m2/s
n/a
n/a
m
m
W/m2
4
σ
μ
ν
γ
ρ
Subscripts
a
amb
c
cr
e
s
t
w
∞
Stefan-Boltzmann constant, 5.67e-8 W/m2-K4
dynamic viscosity
kinematic viscosity, μ/ρ
ratio of specific heats, cp/cv
density
action time
ambient conditions external to the motor
case condition
curvature
nozzle exit
internal surface
nozzle throat
wall condition
free-stream condition
W/m2-K4
N-s/m2
m2/s
n/a
kg/m3
5
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 in Figure 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.
6
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 grain’s 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.
7
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].
8
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 in Figure 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 in Figure 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 in Figure 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.
9
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, in Figure 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. In Figure 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.
10
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 in Figure 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.
11
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 and Figure 4. The burn area, Ab, is then found for each webstep by using the
analysis tools in the CAD program.
The relationship between the Ab profile 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
rb 
x
t
(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
12
rb  aPc
n
(2)
where a and n are 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

mb  rb  b Ab
(3)
where ρb is density and Ab is the burn surface area [2]. The mass flow through the throat
is defined by

mt 
Pc At
c
(4)
where Pc is the chamber pressure and At is 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


mb  m t .
(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
13
 1 


 aAb  b c   1n 

Pc  
 At 
(6)
Taking all the factors and exponent as constants except Ab, 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
correlates Ab-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
I t   F * dt
(7)
where F is thrust (N) [2]. Specific impulse (s) is defined by
I sp 
I
 F * dt 
g *  m * dt g *  m * dt
t

o

(8)
o
where go is 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
14
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
F  C F At Pc
(9)
where
  1 
  1  

 


2 2  2   1    Pe      Pe  Pamb  Ae
 


.


CF 
1   P 
 
  1    1 
Pc
At
c 





(10)
CF is the thrust coefficient and is a function of the ratio of specific heats, γ, exit, chamber,
and ambient pressures, and nozzle area ratio [2].
15
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
16
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 (see Figure 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 convergingdiverging 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 snapring 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.
17
Figure 7 Solid Rocket Motor under study
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 in Figure 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
18
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 in Table 2.
61.3
15.0
Ae/At = 7.06
Rthroat = 70 mm
Raxial curvature
= 127 mm
Figure 8 Nozzle Geometry
19
Table 2 Case and Nozzle Thermal Properties
Component
Material
Case
Carbon Fiber and
silica phenolic EPDM
lumped
Graphite
Nozzle
Specific heat,
C (J/kg-K)
Density, ρ
(kg/m3)
Thermal
conductivity, k
(W/m-K)
1674.7
977.13
0.24234
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 in Figure 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.
20
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 in Table 3 and Table 4,
respectively.
21
Table 3 Solid Propellant Properties
Symbol Nomenclature
Value
Unit
Notes
a
Burn rate coefficient
1.39e-05
m/s
Calculated from a=rb,ref/Prefn,
where Pref = 6.89MPa
n
ρb
c*
Burn rate exponent
Density
Characteristic velocity
0.5
1799.2
1356
n/a
kg/m3
m/s
See definition below
Table 4 Gas Properties
Symbol Nomenclature
Value
Unit
k
cp
0.231
1286.6
W/m-K
J/kg-K
9.5e-5
kg/m-s
2756.4
28
K
kg/kgmol
µ
Tf
MW
Thermal conductivity
Specific heat at constant
pressure
Dynamic
Viscosity
Flame temperature
Molecular weight
Notes
Stagnation temperature
The parameter c* (pronounced “cee-star”), is the characteristic velocity (m/s), as
shown in Table 3. Rearranging equation (4), it is defined by
c 
Pc At
(11)

mt
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
c 
RT
 1
 2   1
 

   1
.
Since the factors gamma, ratio of specific heats, γ, gas constant, R, and the absolute
temperature, T, are properties of combustions gases, c* must also be a property of the
(12)
22
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 in Table 5.
Table 5 Al/AP/HTPB Propellant
Chemical formula
Nomenclature
Al
NH4ClO4
Aluminum
Ammonium
perchlorate
(AP)
Hydroxylterminated
polybutadiene
HTPB
Function Approximate
Mass fraction
Fuel
0.17
Oxidizer 0.70
Binder
0.13
Notes
In powder form
Crystal sizes
10–250µm
23
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 10 provides an example.
24
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 ProE’s
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 in Figure 11. More comprehensive discussion of this CAD modeling method is
provided in Appendix B.
25
Figure 10 SRM Burning Back
Figure 11 Burn Area versus Webstep Data
26
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.
27
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 6 provides 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
28
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 #
1
2
3
4
5
6
7
8
Row thickness
(mm)
0.102
0.159
0.249
0.391
0.612
0.959
1.503
2.354
Total BL thickness
(mm)
0.102
0.261
0.510
0.901
1.513
2.472
3.975
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 in Table
7. Numbers 1-6 require further explanation while 7-12 are self-explanatory.
29
Table 7 Model Assumptions
1
No throat growth
2
No deformation of the grain due to
operating pressure and temperature
The motor grain ignites
instantaneously
Chemical reactions go to completion
immediately upon combustion
Heat transfer due to charring and
sloughing off is ignored
Emissivity of charred EPDM and
Graphite is 0.95
3
4
5
6
7
The external temperature case is
constant at 300K
8 The combustion gas follows the
Ideal Gas law
9 Steady-state pressure predictions are
calculated for each webstep
10 The gas is calorically perfect
(constant specific heats)
11 The combustion gases have constant
properties
12 Flow where Reynolds numbers
based on length is below 60,000 is
laminar [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
30
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.
31
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-ε 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 in Table 2. The gas properties are found Table 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
32
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 the Ab data
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.
33
Table 8 Boundary Condition Pressure and Time
For example, this is how the results are obtained for #10 webstep. The burn area,
0.6174 m2 is determined from the CAD model as previously discussed. Pressure, Pc, is
found using equation (6) and the parameters a, ρb, c*, and n. Pressure is then
 aA  c 
Pc   b b 
 At 

 1 


 1 n 


1


 1.39e  5*0.6174*1799.2*1356  10.5 

 1.853 MPa.

 *0.07 2


The burn rate is found using equation (2),
rb  aPc n  1.39e  5*(1.853e6) 0.5  0.01886 m/s.
The mass flow used in the boundary condition is found using equation (3),

mb  rb b Ab  0.01886*1799.2*0.6174  20.95 kg/s.
34
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),
rb 
t 
x
,
t
t 
x
rb
x 0.01016  0.00762

 0.13468 s.
rb
0.01886
The delta-time is added to the time correlated to the #9 webstep to get
t10  t9  t10  0.4216  0.13468  0.5562 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 in Table 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 in Table 9.
35
Table 9 Convergence Criteria
Residual or condition
Criteria
Continuity
Velocity in X-direction (axial)
Velocity in Y-direction (radial)
Energy
k (turbulence)
ε (turbulence)
Mass imbalance
1e-5
1e-5
1e-5
1e-8
1e-5
1e-5
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 in Figure 13, labeled “SRM Pressure.” This SRM model is validated to this
standard.
36
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.
37
Table 10 Run Times per Webstep
Interval Size (mm)
Time to run (s)
3.81
1800-2700
6.35
200-300
12.7
10-15
19.1
<3
The criteria for choosing the interval size depend upon percent error between the
CFD models and the standard, “SRM Pressure.” Upon inspection of Figure 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 to Table 10. When the accuracy and the
run times are compared to the Medium mesh, shown in Figure 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.
38
Figure 14 Percent Error and Runtime
39
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
40
  1 
  1  



 
2 2  2   1    Pe      Pe  Pamb  Ae
 




CF 
1   P 
 
  1    1 
Pc
c 


 At


 1.31 
 1.31 




2(1.3) 2  2  1.31    101325   1.3  



1 
 1.45 .

  3,180,000 

1.3  1  1.3  1 


Thrust is found using equation (9) and is


F  C F At Pc  1.45 *  * 0.070 2 * 3,180,000  71 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
I t   F * dt  612,000 N-s
and
I sp 
I
 F * dt 
 189.2 s.
g *  m * dt g *  m * dt
t

o

o
Total and specific impulse are found for the heat loss model using this same method.
41
5.2 Heat Loss Zones
There are four regions where heat is lost through the motor case and nozzle, as shown
in Figure 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
point A. The throat area includes the region from point A to 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
42
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  1.72
(  /  )* l
(9.5e  5 / 4.329)*0.191
 1.72
 5.954 mm.
u
0.350
The momentum thickness is
 2  0.664
(  /  )* l
 2.299 mm
u
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
q "convective  h(T  Tw )
(13)
43
Radiative heat transfer for the heat loss zones is found by
q " radiative   (T4  Tw4 ) Fw
.
(14)
The total heat loss is then
"
qtotal
 q "convective  q " radiative
,
(15)
or
q "total  h(T  Tw )   (T4  Tw4 ) Fw
.
(16)
Based on the above equations, and knowing T∞ from 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
Re l 
u *  * l


14.35 * 2.62 * 0.053
 2.11e4 .
9.5e  5
44
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 (<
60,000) as the motor approaches the end of operation.
The flow in the converging wall is similar. The flow for each webstep starts as
laminar and trips to turbulent about 66-75% down the length of the wall. Nevertheless,
laminar flow also dominates this zone. Like the forward wall, the heat transfer coefficient
is found assuming fully laminar flow.
Based on the above discussion and calculating the Prandtl number for this gas as
0.529, the convective heat transfer coefficient for these two regions is found by the local
Nusselt number relationship
Nul  0.332 Pr 0.333 Rel 0.5 .
(17)
This relationship provides reasonable heat transfer coefficients for laminar flow at the
range of Prandtl numbers from 0.5- 15 [10]. Local heat transfer coefficients are found at
each point along the surfaces.
The heat transfer coefficient is different at the throat and diverging walls. The flow
here is turbulent from the throat entrance to the nozzle exit plane at every webstep.
Additionally, the shape of the wall changes with axial position dramatically accelerating
the gas. The heat transfer coefficient for this flow regime can be found using
St 
0.0287 Pr 0.4 R 0.25 (Ts  T )0.25  0.2
 l R1.25 (T  T )1.25  u dl 
s

 
 0

0.2
,
(18)
45
where R is the radius of the converging and diverging walls along the motor axis [10].
5.4.2 Emissivity and View Factor
Radiative heat loss within SRMs is not well understood as emissivity and absorptivity
of combustion products are unreliable [7]. However, a rough estimate of radiative heat
loss can be made by using an effective emissivity, ε. Effective emissivity is found from
both the estimated emissivity of the gas, εgas and the wall, εw [7]. It is determined using
1


1
 gas

1
w
 1.
(19)
Gas emissivity, εgas, is taken as unity [7]. The internal wall insulation and the nozzle
material emissivity is 0.95. The effective emissivity then becomes 0.95. Additionally,
the view factor is taken as equal to unity since the combustion gas radiation impinges on
all surfaces nearly equally [7].
5.4.3 Internal Wall Temperatures
Two methods are used to estimate an internal case wall temperature. First, the total
heat loss equation (15) is used to determine the wall temperature in low Reynolds number
regions in the case. In high Reynolds number regions, a comparison is made to data
found in other works and an estimate is made for the temperature along the throat and the
diverging wall.
The wall temperature at the forward and converging walls is found by starting with
the total heat equation (15). This is rewritten as
46
"
q "convective  q " radiative  qconductive
(20)
since the total heat flux is equal to the amount of heat conducted out of the motor. If the
heat conducted out of the motor is
q "conductive   k
dT
,
dz
(21)
then total heat loss can be expressed as
h(T  Tw )   (T4  Tw4 )  k
dT
.
dz
(22)
Rearranging, it becomes
k 4
k

4
4
  h Tw  Tw  Tamb  hT  T ,
(23)
z
z


where z is equal to the case wall thickness. The right side can be solved as all of the
variables are known. The left side contains the factor, Tw, the value to be found, along
with other known variables. The wall temperature, Tw, exists in both the power of unity
and four making a closed form solution difficult. However, the wall temperature can be
solved numerically. Once the right side is solved the value of Tw is varied until the left
side matches the right side. This technique results in a wall temperature that is 99.6% of
the free-stream temperature for every webstep. Additionally, this value is considered
constant since it is insensitive to the range of heat transfer coefficients calculated
throughout all web steps.
For example, the right side of the equation becomes
0.2423
*3004  500* 2756.4  5.67e  8*0.95* 2756.44  4.49e6 ,
0.0127
47
using an assumed value of 500 W/m2-K for the heat transfer coefficient. The left side is
set-up as
 0.2423

 500  Tw  5.67e  8*0.95* Tw4 .

0.0127


This side is solved by adjusting the value of Tw until its solution matches the right side
solution. Varying the assumed heat transfer coefficient from 0 to 5000 W/m2-K affects
the left and right side solution but has only has only slight effect on the wall temperature
value.
The temperature at the throat and the diverging wall is determined from previous
studies of heat loss in nozzles. Two studies were undertaken to determine heat loss in
SRM nozzles. Data published, including heat flux [9] and heat transfer coefficients [11],
for the nozzles of two different sized SRMs provide information to determine the percent
difference between the freestream gas and the throat and diverging walls. These data,
along with the percent differences in the forward and converging walls, are provided in
Table 11.
48
Table 11 Percent difference between Free-stream and Wall Temperatures
Heat Loss Zone
% Difference
(Tw and T∞)
Forward wall
Converging wall
Throat wall
Diverging wall
99.6
99.6
95.0
87.0
5.4.4 Heat Flux Boundary Condition
The heat loss models use a heat flux definition as a boundary condition on each of the
four heat loss zones. Total heat loss is defined by equation (15), which uses both
convective and radiative heat loss mechanisms. Since the freestream conditions, surface
temperatures, heat transfer coefficients are now known, convective and radiative heat
flux, and total heat flux for each wall can be determined. The following are two
examples as they are useful to illustrate the method used to calculate the total flux
boundary definition.
5.4.4.1 Forward and Converging Wall Heat Flux
This example is calculated at a point approximately centered at the forward wall half
way between the motor centerline and the forward grain surface as shown in Figure 16.
This webstep occurs approximately 3.5 seconds into the motor operation.
49
Figure 16 Forward Wall Heat Transfer Coefficient, Wall and Freestream Temperatures
Freestream data, including velocity, density, and temperature, were collected from the
adiabatic model. With a length along the wall of 0.057 mm, the local freestream velocity
and density of 3.86 m/s and 4.07 kg/m3, respectively, the Reynolds number is calculated
as
Re l 
l * u * 


0.057 * 3.86 * 4.07
 9900 .
9.5e  5
The Nusselt number is
Nul  0.332 * Pr 0.333 Re l
0.5
 0.332 * 0.5290.333 * 99000.5  27 .
The Nusselt number is defined as
Nul 
h*l
.
k
Rearranging to solve for the heat transfer coefficient gives
50
h
Nul * k 27 * 0.231

 109 W/m2-K.
l
0.057
This value is averaged with the others along the surface. The average heat transfer
coefficient is reported in the results.
The local heat transfer coefficients are used to determine the convective heat flux.
Continuing from the heat transfer calculations and adding the freestream and surface
temperatures, the convective heat flux is determined using equation (13)
q "convective  h(T  Tw )  109 * (2756  2745)  1200 W/m2.
This convective heat flux is averaged with the others along the surface.
The radiative heat flux is found using equation (14) and is
q " radiative   (T4  Tw4 ) Fw  5.67e  8 * 0.95 * (2756 4  27454 ) * 1  49,300 W/m2.
Like the convective heat flux, the radiative heat flux is averaged with the others along the
surface.
The average convective heat flux and the average radiative heat flux are summed.
This is the value used in the heat loss boundary condition for the forward wall. The
converging wall boundary condition is calculated using the same method.
5.4.4.2 Throat and Diverging Wall Heat Flux
This example is located at the throat at the same 3.5 seconds into motor operation as
shown in Figure 17.
51
Figure 17 Throat Heat Transfer Coefficient, Wall and Freestream Temperatures
Like discussed previously, freestream data was collected at this point in the adiabatic
model. Freestream velocity, density, and temperature are 570 m/s, 3.42 kg/m3, and 2630
K, respectively.
The Prandtl number is 0.529, radius is 70 mm, and surface temperature is 95% of
freestream temperature or 2499 K. Calculating the integral in equation (18) numerically
across both the throat and the diverging wall (as they are continuous) gives
l
R
0
1.25
(Ts  T )1.25 u dl  28,771.
The local Stanton number is then calculated as
52
St 
0.0287 Pr 0.4 R 0.25 (Ts  T ) 0.25  0.2
 lR1.25 (T  T )1.25  u dl 
s

 
 0

0.2

0.0287 * 0.59 0.4 0.070 0.25 (2630  2499) 0.25 9.5e  50.2
 1300 .
28,7710.2
The definition of the Stanton number is
St 
h
.
u *  * c
Solving for heat transfer coefficient
h  St * u *  * c  1300 * 570 * 3.42 *1286.6  3.26 kW/m2-K.
An average of the local heat transfer coefficients are reported in the results.
Just as in the forward and converging walls, the local heat transfer coefficients are
used to determine the convective heat flux. From the heat transfer calculations and using
the freestream and surface temperatures, the convective heat flux is determined using
equation (13) or
q "convective  h(T  Tw )  3.26e3 * (2630  2499)  753 kW/m2.
Again, this convective heat flux is averaged with the others along the surface.
The radiative heat flux is again found using equation (14)
q " radiative   (T4  Tw4 ) Fw  5.67e  8 * 0.95 * (2630 4  2499 4 ) * 1  4.76 kW/m2.
Like the convective heat flux, the radiative heat flux is averaged with the others along
the surface.
53
The average convective heat flux and the average radiative heat flux are summed and
is used in the heat loss boundary condition for the throat. The converging wall boundary
condition is calculated similarly.
54
Chapter 6
RESULTS
6.1 Heat Loss
Heat loss from the SRM calculated by FLUENT is shown in Figure 18, Figure 19,
and Figure 20. The heat loss is compared against the pressure (also calculated by
FLUENT) developed in the motor with each webstep. This provides a reference of motor
operation as the heat loss progresses.
The heat loss through the forward and converging walls is shown in Figure 18. The
loss increases with time as expected because these two areas are continually increasing as
the grain burns away. The converging wall exhibits the most heat loss of these two zones
and ranges from 15,000 to 56,000 W near burn out.
55
Forward and Converging Walls
60,000
5.0E+06
4.5E+06
Forward wall
Converging Wall
Pressure (Pa)
50,000
4.0E+06
3.5E+06
3.0E+06
30,000
2.5E+06
2.0E+06
Pressure (Pa)
Heat Loss (W)
40,000
20,000
1.5E+06
1.0E+06
10,000
5.0E+05
0
0.0E+00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
Time (s)
Figure 18 Heat Loss from Forward and Converging Walls
Heat loss in the throat and diverging walls are dramatically greater. The loss along
the throat wall remains near level at 65 kW whereas the diverging wall exhibits most of
the heat loss, which ranges from 150 to 320 kW. The heat loss here shows an increasing
trend as well. The trend is more neutral than the forward and converging walls as it is an
effect of increased mass flow, and therefore increased convective effects, at each
webstep.
56
Throat and Diverging Walls
350,000
5.0E+06
4.5E+06
300,000
4.0E+06
3.5E+06
3.0E+06
200,000
Throat Wall
2.5E+06
Diverging Wall
150,000
2.0E+06
Pressure (Pa)
Pressure (Pa)
Heat Loss (W)
250,000
1.5E+06
100,000
1.0E+06
50,000
5.0E+05
0
0.0E+00
0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
Time (s)
Figure 19 Heat Loss from Throat and Diverging Walls
The total heat loss is a superposition of the convective and radiative heat losses from
the four heat loss zones and is shown in Figure 20. The maximum heat loss occurs at
81% into the motor operation at the location of peak pressure. Heat loss here reaches 436
kW. It is clear heat loss in the diverging wall is the chief contributor to overall heat loss
in this SRM.
Convective effects reduce and radiation becomes more significant in the forward and
converging walls. The heat transfer coefficients reduce as the motor operates as seen in
Figure 21. The reduction in convective heat transfer coefficient is likely due to the
increase in exposed internal insulation while at the same time the gas flow across these
57
walls remaining generally steady. Conversely, the heat loss increases in these two walls,
seen in Figure 18, suggesting radiation is a significant source of heat loss within the case.
The significant contribution heat loss by radiation in the case agrees with NASA design
monograph [7].
Convection increases in the throat and diverging walls. The heat transfer coefficient
trends can be seen in Figure 22. The trend is due to the increase of gas flow across these
walls as the motor burns. Convection in the nozzle dominates heat loss in the SRM. The
dominate convection heat loss in the nozzle agrees with a published study by Thakre [6].
5.0E+06
450,000
4.5E+06
400,000
4.0E+06
350,000
3.5E+06
300,000
3.0E+06
250,000
2.5E+06
Total Wall Loss
200,000
2.0E+06
Pressure (Pa)
150,000
1.5E+06
100,000
1.0E+06
50,000
5.0E+05
0
0.0E+00
0.00
1.00
2.00
3.00
4.00
5.00
Time (s)
Figure 20 Total Heat Loss
6.00
7.00
8.00
Pressure (Pa)
Heat Loss (W)
Total Wall Loss
500,000
58
Average Heat Transfer Coefficient
(W/m^2-K)
400
350
300
250
200
Forward wall
150
Converging Wall
100
50
0
0.0
1.3
2.4
3.5
4.4
5.3
6.2
7.2
Time (s)
Figure 21 Heat Transfer Coefficients- Forward and Converging Walls
Average Heat Transfer Coefficient
(W/m^2-K)
3,500
3,000
2,500
2,000
1,500
Throat Wall
1,000
Diverging Wall
500
0
0.0
1.3
2.4
3.5
4.4
5.3
6.2
7.2
Time (s)
Figure 22 Heat Transfer Coefficients- Throat and Diverging Walls
59
6.2 Motor Performance
Ultimately, from a ballistics point of view, it is important to determine the effect heat
loss has on internal pressure, thrust, impulse and burn-time. The difference is quite
small. In fact, the average percent difference between the pressures is only 0.047% with
a maximum of 0.11% at the end of motor operation. The thrust reflects similar levels of
average and maximum percent differences of 0.054% and 0.14%, respectively.
Consequently, the heat loss affects the pressure and the thrust negligibly.
The effect on total impulse and specific impulse is similarly negligible. Table 12
shows the results of these calculations for the adiabatic and the heat loss models. The
heat loss models show total impulse and specific impulse of 0.03% less than the adiabatic
models.
Table 12 Total and Specific Impulse
It (N-s)
Isp (s)
Adiabatic models
612107.6
189.2
Heat loss models
611921.8
189.1
Percent Diff (%)
0.03
0.03
Action time, ta, will not be significantly affected if the impulse shows little change.
The adiabatic action time is 7.701 s, while the heat loss action time runs 0.032% longer at
7.704 s.
The above result justifies the assumption that SRMs are accurately modeled as
adiabatic systems.
60
Chapter 7
CONCLUSION
The heat loss in an SRM was determined and compared to an adiabatic model. The
heat loss source was from convective and radiative effects. Radiation is more significant
than convection at the internal case walls. The opposite is true in the nozzle where the
heat loss is primarily due to convection rather than radiation. The heat loss in the nozzle
accounts for most of the heat loss in this SRM.
Other heat effects can be included in future work to obtain a complete heat loss
model. Effects like cooling effect of inert insulation ablating from the case as it chars
and sloughs off and heat loss due to conduction of condensed species impinging on the
case wall. In this work, neither of these effects was considered as they generally do not
contribute as significantly as convection and radiation, however to obtain a complete heat
loss model they must be included.
The heat loss does not significantly affect ballistics performance. Comparing the
adiabatic performance data to heat loss performance data demonstrate negligible effect
heat loss has on the internal pressure, thrust, impulse, and motor action time.
It is important to point out that the modeling method has value beyond the study of
SRM heat loss. Other ballistic phenomena can be modeled and studied using this
method. The SRM example used here is an axi-symmetric SRM. Typically, grain
designs are much more complicated, much like the example provided in Figure 2. The
use of axial slots (fins) and other possible shapes makes it necessary to model the grain in
61
3-dimensional space. CAD modeling, meshing, and CFD can accommodate this with no
change in the overall algorithm. Transient effects can be included. Heat and mass flow
from an igniter, the heat soak into the motor grain, flame spread along the grain surface,
and motor pressurization and tail-off, all of which are transient events, can be included in
the CFD model as FLUENT can account for transient events [3]. Chemical reactions that
take place generally occur near the surface of the grain during motor operation. This is
not always the case as some reactions continue within the gas flow and through the
nozzle. In addition, reacting metalized propellant can produce condensed species within
the gas flow resulting in gas flow in two phases. Two-phase flow can have considerable
affect on motor performance. The FLUENT can account for reacting gas flow and a
multi-phase working fluid [3].
The modeling method used in this study simplifies ballistics analysis of SRMs. The
overall method, summarized in Figure 23, uses a CAD modeling program, a CFD
preprocessor for meshing fluid volumes, and a CFD program. The CAD program takes
advantage of several tools including relationships between dimensions, analysis tools to
measure areas and webstep distances, and a family of data tool that allows the correlation
of these data. Then any number of individual CAD models representing individual
websteps can be created. The CAD models are then individually exported to the meshing
preprocessor. The meshing preprocessor creates meshes from the imported CAD files.
Mesh files are used in the CFD program where flow analysis can be performed. The use
of user defined programs and journal files can simplify the creation of individual CFD
case files. Post processing can be made easier by addition of user defined programs.
62
Using
relationships,
analysis, and
family of data
tools
Grain design using CAD software
File export (IGES) to meshing
program
Again, using
journal files to
accommodate
repeated steps
Mesh creation, to include
boundary condition definition
CFD model creation
Using Cprograms to aid
post-processing
analysis
Using journal
files to
accommodate
repeated steps
Post processing analysis
Figure 23 Modeling Algorithm
Using Cprogram to aid
the building of
CFD models
63
APPENDIX A
SRM Transient Algorithm

dP RT  
P d MW 
 P dV P dT



 m in  m out  
dt
V 
 V dt T dt MW  dt






RT m in C p ,inTin  m out C p ,outTout  T  Cv ,in m in  Cv ,out m out 
dT




dt
PVCv


m in  m out  V
d
dV

dt
dt
PV  mRT
64
APPENDIX B
ProE CAD Method
The initial grain geometry is created within the CAD program, ProE Wildfire. This
example is a simple grain cast propellant with a burning center bore, spherical cut, and
end and is shown in Figure 24.
Figure 24 Example SRM Grain
A dimension is chosen as the driving dimension. The driving dimension can be any
dimension. In this example the dimension chosen is not part of the model, rather it is part
of a cosmetic sketch, shown in Figure 25. The relationship tool (Tools, Relations) relates
the bore, sphere, and end dimensions to the driving dimension, Figure 26. Now as the
driving dimension changes the bore and sphere radii, and end length changes by the same
amount. The driving dimension is easily changed within the graphics window by doubleclicking on the cosmetic sketch, double clicking on the driving dimension, and changing
65
the value. Regenerating the model is required to update the model with every new value,
as seen in Figure 27.
Figure 25 Driving Dimension
Figure 26 Related Dimensions
66
Figure 27 Dimensions changing as the Driving Dimension Changes
Now burn areas can be measured. Use the analysis feature in ProE to measure the
areas and save each measured area as a feature in the feature tree, Figure 28. This
example has the bore, the sphere and the end areas. The family of data table uses this
data.
67
Figure 28 Measuring the Areas
It is necessary to consider the model symmetry. The model example here is only half
of the full grain. The areas measured in the analysis tool only give the area of the
highlighted surface. In this example, it is necessary to multiply this number by a
symmetry factor of two. Had this example modeled only 1/15th of the full grain (as could
be the case since it is axi-symmetric) then the symmetry factor is 15. Apply this factor to
the measured area to find the full burn area.
Create a family table (Tools, Family Table). Insert the first column as a dimension
and choose the driving dimension. Next, insert the area analysis features, Figure 29. The
family table is now complete. All that is necessary is to add values in the driving
dimension column and click the check button, Figure 30. The driving dimension values
can be changed as necessary.
68
Figure 29 Creating a Family of Data
69
Figure 30 Family of Data Table
The data table can be exported to Excel, or other spreadsheet to complete any further
analysis or for graphing.
70
REFERENCES
1. “Solid Rocket Motor.” http://en.wikipedia.org/wiki/Solid_rocket_motor,
en:User:Pbroks13, 19 May 2008.
2. Sutton, G. P. and Biblarz, O., Rocket Propulsion Elements, 7th ed., John Wiley and
Sons, New York, 2001.
3. FLUENT 6.3 User’s Guide, Fluent Inc., Lebanon, NH, 2006.
4. GAMBIT 2.3 User’s Manual, Fluent Inc., Lebanon, NH, 2006.
5. Pro Engineering, Wildfire 4.0, Parametric Technology Corp., Needham, MA, 2010.
6. Thakre, P., and Yang, V., “Graphite Nozzle Material Erosion in Solid Propellant
Rocket Motors,” AIAA-2007-778 (2007).
7. Twitchell, S. E., Solid Rocket Motor Internal Insulation, SP-8093, NASA, December
1976.
8. Modest, M. F., Radiative Heat Transfer, 2nd ed., Academic Press, San Diego, CA,
2003.
9. Yumusak, M., Vuillot, F., and Tinaztepe, T., “Viscous Internal Flow Applications for
Solid Propellant Rocket Motors,” AIAA-2006-5116 (2006).
10. Kays, W., Crawford, M. E., and Weigand, B., Convective Heat and Mass Transfer,
4th ed., McGraw-Hill, New York, NY, 2005.
11. Ahmad, R.A., “Convective Heat Transfer in the Reusable Solid Rocket Motor of the
Space Transportation System,” Heat Transfer Engineering, 26(10) 2005:30-45.