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Scientific predictability of solid rocket performance:Analyses of the processing parameters.
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Authors Perez, Daniel Lizarraga.
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Scientific predictability of solid rocket performance: Analyses of the processing parameters
Perez, Daniel Lizarraga, Ph.D.
The University of Arizona, 1992
Copyright ©1992 by Perez, Daniel Lizarraga. All l'ights reserved.
V·M·I 300 N. Zeeb Rd. Ann Arbor, MI48106
SCIENTIFIC PREDICTABILITY OF SOLID ROCKET PERFORMANCE:
ANALYSES OF THE PROCESSING PARAMETERS
by
Daniel Perez
Copyright© Daniel Perez 1992
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF AEROSPACE AND MECHANICAL ENGINEERING
In Partial Fulfillment of the Requirements For the Degree of
DOCTOR OF PHILOSOPHY With a Major in Aerospace Engineering
In the Graduate College
THE UNIVERSITY OF ARIZONA
1992
THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE
As members of the Final Examination Committee, we certify that we have
read the dissertation prepared by Daniel l.izarraga Perez.
2
entitled ___ S_C_I_E_N_T_I_F_I_C_P_R_E_D_I_C_T_A_B_IL_I_T_Y __ O_F __ S_O_L_ID __ R_O_C_K_E_T __ P_ERF __ O_RMA ___ N_C_E_: ____ __
ANALYSES OF THE PROCESSING PARAMETERS.
and recommend that it be accepted as fulfilling the dissertation
requirement for Doctor of Philosophy
2/26/92 Date
2/26/92 Date
2/26/92 Date
2/26/92 Date
Date
Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College.
I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. ~ {~
~ ,~1r-~ 2/26/92
Dissertation Director Kumar N. Ramohalli Date
3
STATEMENT BY AUTHOR
This dissertation has been submitted in partial fulfIllment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for pelmission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder.
SIGNED: .~, efQ~.-----:'
4
ACKNOWLEDGMENT
I wish to express my gratitude and appreciation to Dr. Kumar Ramohalli, for his guidance and support throughout this research program. I especially wish to thank him for the many opportunities he made available for me to interact with industry and government in pursuing the work.
I also wish to thank Dr. Russell Petersen, Dr. Terry Triffet and Dr. Sridhar for serving on my committee, and for sharing their knowledge and experience in aerospace engineering. A special thanks to Dr. Petersen for encouraging me to pursue a PhD. I am also grateful to Ken Nichol for reviewing the manuscript and making helpful suggestions.
This research was started by NASA Code M (Marion Kitchens) via grant #8-757 from NASA/MSFC (Richard Brown). Technical direction was provided by Jack Frerick at MSFC. The research was subsequently continued by Code Q (Nonn Shulze) via MSFC (Ted Kublin). The author received a NASA Traineeship from Code XEU for which he thanks Elaine Schwartz and John Lynch.
DEDICATION
I dedicate this dissertation to my parents,· Adolfo and Emma, for giving me
encouragement and support throughout my studies.
5
Gracias por ayudarme a que mis sueiios se hagan realidad. Los quiero mucho, tu
hijo Daniel.
6
TABLE OF CONTENTS
Page
LIST OF ILLUS1RA TIONS ................................................................................... 8
LIST OF TABLES .......................................................................................... ........... 11
ABSTRACf ............................................................................................................. 12
CHAPTER
1. INTRODUCTION ................................................................................................ 14
2. PROCESSING OF COMPOSITE SOLID PROPELLANT ............................... 18 2.1 Background .... ...................... .................................................................. 19 2.2 Processing Studies .............................................. ........ ........................... 32
3. RHEOLOGY OF COMPOSITE SOLID PROPELLANT SUSPENSION ........ 44 3.1 Background ............................................................................................ 45 3.2 Experiments ............................................................................................ 50 3.3 Results .................................................................................................... 57 3.4 Other Studies ............................................................. .............. .......... ..... 66
4. MUL TIPHASE MIXTURE THEORy............................................................... 67 4.1 Background .................................................................................. .......... 68 4.2 Multiphase Mixture Theory ................................................................... 69 4.3 Computational Work .............................................................................. 72 4.3 Results from Program ............................................................................ 79
5. DISCUSSION ...................................................................................................... 88
APPENDIX A: PROPELLANT DATA FOR VARIATION STUDy............... 91 A.I Data Collection .................................... ................................................. 91 A,2 Average Values for 150-Gallon Batch Runs ....................................... 91 A.3 Data ....................................................................................................... 96
7
T ABLE OF CONTENTS-continued
CHAPTER Page
APPENDIX B: RHEOLOGICAL DATA ............................................................... 133 B.! Orifice Viscometer ................................................................................ 133 B.2 Orifice Visometer Data ......................................................................... 136
B.2.! Calibration .............................................................................. 136 B.2.2 Binder ..................................................................................... 145 B.2.3 Monomodal Mixtures ............................................................. 149 B.2.4 Bimodal Mixtures ................................................................... 161
APPENDIX C: COMPUTER PROGRAM FOR THE FLOW SIMULATION .... 173
REFERENCES .................................................................................................. ....... 221
8
LIST OF ILLUSTRATIONS
~pre ~~
2.1 ASRM Cut-Away View..................................................................................... 19
2.2 Oxidizer Particle Size Distribution ................................................................... 22
2.3 150 Gallon Batch Mixer .................................................................................... 24
2.4 150 Gallon Batch Schedule ............................................................................... 26
2.5 Vacuum Casting Technique ................................................................ ...... ........ 27
2.6 Continuous Mixing and Casting Schematic ..................................................... 28
2.7 SRM Performance Data ..................................................................................... 30
2.8 Particle Size Distributions Used for 150 and 1 Gallon Batches ..................... 33
2.9 Temperature Sensitivity of End of Mix Viscosity Samples ............................ 39
2.10 Temperature Sensitivity of End of Cast Viscosity Samples .......................... 40·
3.1 One-pint Baker-Perkins Mixer .......................................................................... 50
3.2 Particle Size Distribution ................................................................................... 51
3.3 SEM Photograph of Coarse ............................................................................... 51
3.4 SEM Photograph of Intermediate ...................................................................... 52
3.5 SEM Photograph of Fine ................................................................................... 52
3.6 Orifice Viscometer ............................................................................................. 53
3.7 Calibration Curve using HT600 ........................................................................ 54
3.8 Calibration Curve using HTI000 ...................................................................... 55
3.9 Changes in Mixing Schedule for Monomodal ................................................. 56
9
LIST OF ILLUSTRATIONS-continued
Figure Page
3.10 Changes in Mixing Schedule for Bimodal....................................... .............. 57
3.11 H1'PB Flow Behavior .............................................................................. ........ 58
3.12 Monomodal Relative Viscosity....................................................................... 58
3.13 Bimodal Relative Viscosity............................................................................. 60
3.14 Flow Behavior for Monomodal Mixture of Coarse ....................................... 60
3.15 Flow Behavior for Monomodal Mixture of Intermediate .............................. 61
3.16 Flow Behavior for Monomodal Mixture of Fine ........................................... 61
3.17 Flow Behavior for Bimodal Mixture of 65/35 Coarse/Fine Ratio ................ 62
3.18 Flow Behavior for Bimodal Mixture of 70/30 Coarse/Fine Ratio ................ 63
3.19 Flow Behavior for Bimodal Mixture of 75/25 Coarse/Fine Ratio ................ 63
3.20 Viscosity dependence on Temperature ........................................................... 65
3.21 Viscosity dependence on Additives ................................................................ 65
4.1 Boundary Conditions .......................................................................... ............ ... 80
4.2 Comparison of Multimodal Flow to a Newtonian Fluid ................................. 81
4.3 Convergence of Monomodal (--) and Newtonian Flow ( ) Properties ............ 83
4.4 Sum of the Error for Monomodal (--) and Newtonian Flow ( ) ..................... 83
4.5 Computational Monomodal Flow...................................................................... 84
4.6 Changes in the Binder Viscosity....................................................................... 85
4.7 Changes in Maximum Concentration ................................................. ............... 86
10
LIST OF ILLUSTRATIONS-continued
~pre ~~
4.8 Changes in Drag Coefficient ............................................................................. 86
4.9 Computational Bimodal Flow........................................................................... 87
B.1. Viscometer Cut-Away View............................................................................ 133
11
LIST OF TABLES
Table Page
2.1 List of Propellant Ingredients . ...... .......... .................. ........ ........ ......................... 21
2.2 150 Gallon Batch Formulation for SRM Propellant ........................................ 22
2.3 Variations in 150 Gallon Batch ........................................................................ 35
2.4 Comparison of Pre-Stored and Un stored Propellants Bum Rate Properties ... 36
2.5 Comparison of Pre-stored and Unstored Propellant Material Properties .. ".... 37
2.6 Viscosity History for 0.01 % Ferric Oxide 150 and 1 Gallon Batches ............ 37
2.7 Viscosity History for 0.04% Ferric Oxide 150 and 1 Gallon Batches ............ 38
2.8 Viscosity History for All 150 Gallon Batches ................................................. 38
B.1 Orifice Plate Configurations .................................................................... "....... 134
12
ABSTRACT
The objective of this dissertation is to present a computational model of the
suspensions composing uncured composite solid propellant. The work examined highly
concentrated suspensions of more than 50% solid volume, with attention to bimodal
mixtures. Investigation of propellant processing was conducted to determine how this
model can be applied to processing. Experimental work was conducted to supply data for
comparison to the computational results. This involved data gathered from an Olifice
viscometer on viscosity and flow behavior. This model is a tool to investigate goodness
of mixing throughout the processing stages of the propellant. The investigation into
processing focused both on mixing and casting of the suspension. By studying this model
for concentration, velocity and thermal behaviors, a better understanding of how well the
propellant composition progresses in processing was obtained.
A multiphase mixture approach was taken. This involved a continuum description
for the mixture and each constituent A Fortran program was written to conSU'uct this
routine. It was run on both a V AXstation 3100, Model 40 using VMS Digital operating
system, and a SUN IPX, using SUN UNIX operating system.
The code examined two-dimensional monomodal and bimodal mixture flows
through a pipe. It examined concentrations between 65% and 75%. Due to the high
concentration, it was necessary to apply all inertial and viscous terms within each
constituent and the entire mixture. Proper boundary conditions and initial conditions to
produce stable runs were found.
13
Both monomodal and bimodal computational results showed good correlations with
the experimental data, although a slight dilatation was produced by the program. No
dilatation appeared in the experimental work. No concentration drop was detected in
either the computational results or experimental work.
Chapter 1
INTRODUCTION
14
The objective of this dissertation is to present a computational model of the
suspensions composing uncured composite solid propellant. The work examined highly
concentrated suspensions of more than 50% solid volume, with attention to bimodal
mixtures. Investigation of propellant processing was conducted to determine how this
model can be applied to processing. Experimental work ~as conducted to supply data for
comparison to the computational results. This involved data gathered from an orifice
viscometer on viscosity and flow behavior. This model is a tool to investigate goodness
of mixing throughout the processing stages of the propellant. The investigation into
processing focused both on mixing and casting of the suspension. By studying this model
for concentration, velocity and thermal behaviors, a better understanding of how well the
propellant composition progresses in processing was obtained.
The technique used to develop the model is based on theories in multiphase
mixtures. This technique was introduced in the '60s by Wallis [1] and Truesdell [2].
Initial interest was in two-phase mixtures of liquid-gas systems, particularly for nuclear
reactor cooling systems. Since then the technique has been applied to liquid-liquid, liquid
solid and many other materials. Computational work in this area grew immensely in the
mid- '70s, especially the work by Spalding [3]. The two most commonly used two-phase
codes, GENMIX2P and PHEONICS, were developed by Spalding in the early '80s. A
very recent summary of multi phase mixture theory and computational developments was
15
done by Soo [4].
Due to the complex nature of these suspensions, the study of highly concentrated
suspensions has acquired a special area in multi phase mixtures theory. Interaction between
the constituents is considered highly important and is expressed in the governing
equations. The basic foundations of multiphase mixture theory are used. Each constituent
of the suspension is evaluated individually with respect to the balance equations of mass,
momentum and energy. Interaction force terms are used in these equations to take into
account the other constituents. The properties of the suspension are then determined as
a function of the constituent properties. This is done by requiri!1g that suspension
properties also obey the same balance equations. For simplicity within this work, chemical
reaction was neglected and the temperature between all the adjacent constituents was
assumed to be equal.
The computational model developed in this dissertation is the first attempt for such
highly concentrated suspensions. It is an extension of the GENMIX2P and PHEONICS
codes. The scheme employed the IPSA (interPhase Slip Algorithm) established in these
codes to handle two-phase aspects of flows. Changes were applied for multiphase flows.
Other changes involved applying inertial and viscous terms to all the constituents. The
scheme is two-dimensional, and although it was applied to a steady state condition, it is
transient in structure.
Work is presented in four parts. The first is a study of propellant processing
techniques. Second is the introduction to rheology of highly concentrated suspensions.
Third is the introduction to multiphase mixture theory and the computational scheme
16
which stems from it. Fourth is the discussion of the results and the future potential of the
technique for a more detailed investigation of propellant processing.
Chapter II presents the background information on the processing of composite
solid propellant. It introduces the terminology and techniques used in batch processing.
A statistical analysis on small-scale and full-scale batch processing is presented. Studies
in propellant oxidizer particle breakage and propellant curing are presented to properly
define the suspension. Variations in propellant performance were detected, caused by
changes in mixing procedure.
Chapter III presents the rheology of highly concentrated suspensions. Background
on monomodal and bimodal mixtures is presented. Experimental work is presented on an
orifice viscometer. The data is used to empirically define the viscosity of the suspension
and to provide the flow behavior for comparison to the computational work. Findings
indicated that complex flow behavior exists for monomodal mixtures, while bimodal
mixtures possess a simple nature. Empirical formulas for the mixtures by Chong [5] show
a good fit with the data.
Chapter IV presents the theory of multiphase mixtures and the computational
scheme developed to model the suspension. Computational results to flow runs and an
analysis of their applicability to processing is also presented. The shear rate data acquired
from the computational runs show good fits with the experimental data. A dilatant
behavior was noted in the monomodal and bimodal mixture in the computational runs.
Convergence is found to occur very rapidly for the multiphase mixtures. Stability is
highly dependent on the interaction and viscous terms within the momentum equation.
17
The computational work is done on a Digital V AXstation 3100, model 40 and a Sun IPX.
Chapter VI presents a summary of results. The computational model proved to
portray the suspension flow well. Confidence in the technique is high. The critical aspect
now lies in applying it to more specific and detailed equipment in order to evaluate the
best possible processing technique.
18
CHAPTER 2
PROCESSING OF COMPOSITE SOLID PROPELLANT
Solid Rocket Motors (SRMs) play an important role in the U.S. Space Program.
They have proven to be low-cost and reliable means of delivering payloads into space.
Their history of success in space and defense launch systems has placed them in high
regard within the growing commercial market. There is no doubt that they will remain an
integral part of future space endeavors.
In light of this fact, NASA has established the Solid Propellant Integrity Program
(SPIP). The main objective of this program is to further improve the success rate and
advance the technology of U.S.-built SRMs. NASA's present attention focuses on
composite solid propellants, primarily due to their use on the space shuttle. This vehicle
is relied upon highly for large payload deliveries. Two solid-booster strap-ons are used
in each mission, containing more than 500,000 kg (1,100,000 lbs) of propellant apiece.
The work within this dissertation stems from efforts to improve processing for this
system.
This chapter describes in general the processing of composite solid propellant. It
supplies the terminology and technology of this field. It also presents studies issued to
identify critical areas of concern in processing to improve propellant performance through
prediction and repeatability.
19
2.1 Background
Figure 2.1 shows the highlights for a space shuttle Advanced Solid Rocket Motor
(ASRM). The motor is to be manufactured in three segments as shown. In each segment,
propellant is casted into the casing and then allowed to cure. The inner wall of the casing
has a polymeric liner which bonds to the propellant. The segments are then shipped to the
launch facility where they are then assembled into the boosters,
c.nrldgo LNded HTPB Propeltant IgnUar • 29 lewer leak paths
than RSRM • T aiIorod 6011 Ign410n
Kavlar/EPDM lnaulatlon • Added margins at
CI1IlcaI joints • Imploved J'!MIals • Optimized far ,trip waund • 2,000 Ib lighter than
ASRM mater1al
Propellant Grain • Proven In continuous mix • Proven propeny ropeatabilcy • Fewer expo!MId persannel
9N~.3C Staol CaN • I~ved fraclure toughness • Higher resistance to
stress corrosion • Assured multiple reu!MI • Weld ability
Noale • 1 less jalnllhan RSRM • 1 less InllllJlhroat ring
than ASAM • Improved process
ablative materials • Eliminates ll11x!MIal cowl
and bool assembly • Nearly 5,000 Ibs lighter
than ASAM
Figure 2.1 ASRM Cut-Away View.
20
The ingredients which can be used to compose the propellant are numerous. A list
of some of the ingredients which are presently used is shown in table 2.1. The propellant
is basically made of high concentrations of solid oxidizer particles, held together by a
polymeric binder. In the present shuttle boosters, the oxidizer is ammonium perchlorate
(AP) and the binder is polybutadiene-acrylic acid-acrylonitrile (PBAN). The ASRM will
use hydroxyl-tenninated polybutadiene as the binder. Curing agents are used to harden
the binder and solidify the propellant. To improve specific impulse, metal powders are
often added. Other additives may also be placed into the propellant to improve burning
characteristics, material strength properties and even processing workability. These last
additives together usually compose less than 5% by volume of the propellant.
The key ingredient in the propellant is the oxidizer. Since it is the source of
energy, high concentrations are desired. Concentrations as high as 65% by volume of
propellant are used. With the addition of the metal powders, solid concentrations of 750/0
are sometimes reached. It is important that this be done without hampering the structural
integrity of the propellant, which is governed by the binder. Care must be taken to
determine the best packing of the oxidizer and still allow for the proper curing. Zero
permeability and porosity are also sought. To satisfy aU these requirements, bimodal
mixtures of oxidizer particle sizes are used.
Table 2.2 shows the formulation of the present shuttle booster propellant.
Figure 2.2 shows the size distribution corresponding to the oxidizer. The solid
concentration is 75%. The additives DOA and ferric oxide are the plasticizer and burning
rate catalyst, respectively. DER-331 is the epoxy curing agent. The coarse and fine are
Table 2.1 List of Propellant Ingredients.
oxidizer
binder
AP: ammonium perchlorate AN: ammonium nitrate NP: nitronium perchlorate KP: potassium perchlorate RDX: cyclotrimethylene trinitramine HMX: cyclotetramethylene tetranitramine
PBAN: polybutadiene-acrlic acid-acrylonitrile PS: polysulfide PVC: polyvinyl chloride PU: polyurethane CTPB: carboxyl terminated polybutadiene HTPB: hydroxyl terminated polybutadiene
curing and/or crosslinking agents DER-331: diepoxide of bisphenol A TDI: toluene-2,4-diisocyanate MAPO: tris{ 1-(2-methyl) aziridinyl} phosphine oxide IPDI: isophorone diisocyanate
bonding agent MAPO: tris{1-(2-methyl) aziridinyl} phosphine oxide TEA: triethanolamine
plasticizer DOA: dioctyl adipate IDP: isodecyl pelargonete DOP: dioctyl phthalate
burning rate catalyst Fe20 3: ferric oxide FeO(OH): hydrated-ferric oxide LiF: lithium flouride
metal fuel AI: aluminum Mg: magnesium Be: beryllium B: boron
combustion instability suppressant AI: aluminum Zr: zirconium zrC: zirconium carbide
21
22
Table 2.2 150 Gallon Batch Formulation for SRM Propellant.
Ingredients wt. % wt. kg wt. lbs
AP, 200~m 48.99 422.21 930.81
AP, 9~m 21.00 180.99 399.00
AI, 20~m 16.00 137.89 304.00
Fe20 3 0.01 0.09 0.19
PBAN 11.49 99.02 218.31
DOA 0.70 6.03 13.30
DER-331 1.81 15.60 34.39
Total 100.00 861.83 1900.00
100~--------------------------~
80 ~
ri E ::I 0
60 > CII
>- ~ .c 0
40 0
.... () CD c
G:
20
O+-~~~~.-~~~~--~~~m
1 10 100 1000 Particle Diameter, f..Lm
Figure 2.2 Oxidizer Particle Size Distribution.
23
mixed at a ratio of 70/30 to acquire the bimodal mixtures desired. In most cases, the
oxidizer of the coarse size is acquired from the source and then ground to the proper fine
size. This permits the adjustment of the fine sizes and guarantees that the quality remains
the same for all the oxidizer.
Now that the propellant composition has been discussed, it is equally important
to discuss the means by which it is processed. The discussion that follows describes the
conventional means by which propellant is produced in the U.S., particularly the method
currently used to produce the shuttle boosters. A major change proposed in this method
for the ASRM is discussed at the end of the section.
The batch mixer is the principle device used in manufacturing these propellants.
Figure 2.3 shows the propellant, mixing bowl and blades. In this process, the ingredients
are placed in a large mixing bowl in a particular sequence. Time is allowed for mixing
between additions to acquire the proper dispersion of each ingredient. The standard mixer
used is 600 gallons and processes about 3,000 kg (7,000 lbs) of propellant. The figure
shows the scrap-down stage of a 150-gallon batch operation. The bowl is dropped to
remove the blades from the mixture; these blades are then cleaned so that better mixing
can be obtained. Monitoring the state of the mixture is usually done before the addition
of the curing agent, at the end of mix and at the end of casting stages. Samples are
extracted and delivered to the laboratory for examination. Viscosity and density is
relied upon to supply the information on the progress of the mixture.
Remote controls are used to guide the propellant through the process. On-site
personnel are held to a small number and are present only at certain stages of the
24
Figure 2.3 150 Gallon Batch Mixer.
25
mixing. The processing itself is usually performed in an isolated portion of the facility.
Figure 2.3 also shows the most common configuration of the batch mixer. Baker
Perkins Planetary Vertical Blade Mixers have been developed for small- and large-scale
production of these propellants and are the standard equipment used. The blades are
shaped so that the propellant is forced up to the top of the bowl. The rotation for both
blades is planetary about the center. The speed of rotation relative to the center and about
the blade shaft differ from one facility to another and are set by Baker-Perkins.
Figure 2.4 shows a general processing schedule for the formulation shown in
table 2.2. The mixer is identical to the one shown in figure 2.3. All the ingredients are
preheated before they are placed in the bowl. Note that most of the liquid ingredients are
added to the bowl at the beginning of the schedule, and that the particles are added
gradually over a long stretch of time.
Standard procedures also include storage of the propellant for more than 24 hours
before the addition of the curing agent. This is done to improve wetting of the binder to
the oxidizer and results in a noticeable drop in viscosity. In addition, the particles are
introduced in a quasi-mixed manner. For 15 minutes, the coarse particles are introduced,
then the fine, then the coarse, and so on. The process continues until all the oxidizer has
been placed in the bowl.
All these techniques have been used previously in other motors and are
considered to be beneficial to a good mix. (No true verification of benefits r~sulting from
these techniques have been done except from the manufacture of the motors.)
In the casting stage, the motor casing is placed in a vacuum bell and the
r- PBAN I- DOA r- F~~ rAt
Mixing at Atm Mixing at Vacuum Mixing at Atm 5 min r---- 10 min r-- ,min 160·P 1600 P 1600 P
T;tkc
r AP, Coarse 15 r:r:iliJ/ Pine 15 mfu/ Coarse ..•
Mixing at Vacuum Samples Mixing at Atm Mixing at Atm 45 min f-- Mix as Required I+- 15 min 1400 P 140·P 1600 P
r DER-331
Mixing at Vacuum Mixing at Vacuum Mixing at Vacuum 60 min 1400 P
r- IO min 140"F
Take Samples
r- 15 min 140·P
Figure 2.4 150 Gallon Batch Schedule.
r--- Vacuum Casting
Take: Samples
26
propellant is drawn from the bottom of a casting can (usually the mixer bowl). Figure 2.5
shows the technique. This technique is the best method of casting the propellant since no
air is entrapped. Note that a slot former and dispersion cone are incorporated in the
passage.
Rheological considerations are extremely important in this technique since the sole
driving force enabling the propellant to flow is the differential pressure and gravity. Care
must be taken to fix the rate of casting so that void-free propellant is produced. As
propellants are made of higher solid concentrations, the difficulties in casting multiply.
Curing is the last stage in the process and involves placing the casing into an
environmentally controlled oven. Since the temperature of curing is 140°F, the pit is
usually modified to act as the oven. This stage lasts for at least several days, usually
Propellant Casting Can
foOiI;+-I~ Rocket Motor Casing
1ooI~\'---Vacuum Bell
...... -t-l-l<l--- Core
- Pit Wall
L:5~i~J---- Casting Stand Assembly
Figure 2.5 Vacuum Casting Technique.
27
several weeks. In addition, pressure may be applied to the propellant through the core
sometime during curing to generate a pre-stress on the propellant. This is done to
diminish any deformations which might take place at fIring, where the chamber pressure
spikes.
Samples taken at the end of cast are also cured with the motors and tested at
prescribed curing times. This supplies the best time at which the motor can be removed
and stored for the mission.
In the light of new developments associated with tire ASRM, it is proper to discuss
the method of continuous mixing. This is the method proposed for the development of
this booster in the mid-'90s and it should be mentioned. Note that any further discussion
beyond this section and the conclusion will focus on the batch mixer only. The conclusion
discusses the ea~e with which the work presented in this dissertation can be applied
toward continuous mixing.
28
Continuous mixing is the assembly-line approach to making propellant. Figure 2.6
shows the schematics of the technique. Continuous mixers are usually composed of a
twin-screw mixer/extruder, with some modifications to improve mixing. The ingredients
are constantly fed at the front of the mixer at critical sections. The material is then
extruded through the end and cast. The production capacity is more than 2,000 kg/hr
(4,400 lbs/hr). Figure 2.6 corresponds to the schematics for the continuous processing
design proposed for the ASRM. Note that on-line quality control monitors are
Ground .... p{g Oxidizer Supply Bin
Ground .... P Loss-In-We 19M Feeder
Premix From Fuel Prep.
Burning Ral. Control From Futl Prep. (Optional)
Final Fuel Into Tank
{g
ungrOUnd Oxfdlar Supply Bin
Damage Control System to 150 lat. Incident and Put Out Fires; Facility Designed 10 Direct Forc;;e __ -. or Incident .... way From Olner Areas or Facility '//////
Figure 2.6 Continuous Mixing and Casting Schematic.
incorporated and that the process overall is more adaptable to automation than the batch
method.
Both batch and continuous mixing techniques serve to be productive in the
manufacturing of large motors. Both supplied a production efficiency of 95% or better
29
when considering waste. The main advantage in continuous mixing is in the amount of
propellant handled at one particular time. The hazard of ignition is apparent in both
techniques, continuous mixing handles less of the propellant and therefore is less likely
to produce a catastrophic flre.
Unfortunately, the demand for large motors diminished in the late '60s.
Continuous mixing became less attractive due to the start-up and shutdown costs. The last
time it had been applied to production of large systems was in 1965, when production of
the Polaris missile concluded. Batch processing proved more cost efflcient to handle the
demand and therefore became the conventional means of processing. It is only now that
continuous mixing is being considered once more.
Better predictions are the major focus in advancing processing. Processing of
propellant is not very well understood. Each motor is tailored to the specifications. By
determining the factors which cause variations, and by controlling these factors, better
predictions can be acquired. Since more stringent standards and cost-efficient management
are being applied throughout the aerospace field, better prediction is stressed even further.
Additional fuel is supplied by NASA's attempt to introduce these motors for commercial
enterprise.
Predicting the performance of the propellant relies on the theoretical chemistry of
the propellant and is supported by tests on small motors. Tests on the boosters themselves
are rare due to economical reasons. Statistical methods are heavily relied on to determine
the range of deviation which may occur. As mission data are acquired, modifications to
this range are done.
30
Unfortunately, the range allowed often proves to be too large, and in several
incidents has proven to be of major concern to mission specialists.
The best example of the problem of performance variations comes from the data
from the fourth shuttle mission launch. Figure 2.7 shows data from several static and
mission firings of the solid boosters. Using the data acquired from static and smaller
1.04 ....--0:: CD
"'C Q) L-a.. ........ 0:: 1.02 CD
L-0
0+-0
::::l: '-'"
s.: ~ 1.00 0 C
l.J...
Q)
C 0
(/)
0.98 Static Tests
0 3 5
STS 4
STS Data
8 10 Motor
13
Figure 2.7 SRM Performance Data.
15
scale tests, shuttle performance was predicted. All was well until the fourth mission,
where an abrupt drop in the bum rate was apparent. Although this fell well within the
specifications, it caused major concerns. As a means of compensating for the loss in
thrust, the shuttle main engines were used. This proved equal to a loss of more than
5,400 kg (12,000 lbs) of payload capability which could have been otherwise delivered.
The cost in payload was large, but attention is also attracted by the fact that the booster
31
separation altitude targets were missed. Both show that a problem does exist and needs
to be addressed.
The actual cause of the incident was never known. All that could be stated was
that the boosters experienced a drop in bum rate from a pre-lift value of 0.366 to a value
of 0.359 in./sec. No noticeable change in processing the propellant was noted nor any
drop forecast. To correct for the prediction, solid performance was revised downward and
only flight data in conjunction with small-scale tests were to be used. The revised value
for the next mission was 0.365 in./sec.
Problems of this nature are abundant and result from a sensitivity of the propellant
to processing variables. Given what is considered identical processing procedures as
known now, the likelihood of varying performance is stilI possible. The most apparent is
the shuttle mishap in which the mission was severely hampered.
Another example is the "mid-web anomaly". During the firing of a motor, the
mean chamber pressure tends to rise above predicted values and then drop below. The
term comes from the assumption that the time necessary to rise to the maximum pressure
is close to the time needed for the propellant surface to regress to approximately halfway
through the propellant. Time measurements have been quite accurate and the pressure rise
determined to be slightly larger than 8% of the average chamber pressure.
Overall, there are three distinct cases of variation: those associated with propellant
performance variations within a motor; those associated with propellant performance
variations between identical size motors; and those between a small-scale motor and full
scale motor. The first will be defined as within-motor variations, the second as motor-to-
motor variations, and the third as scale-up variations.
2.2 Processing Studies
32
To acquire a direct prospective of the variation problem, on-site studies at the Jet
Propulsion LaboratorylEdwards Air Force (JPLIEAF) facility in California were
completed. The studies were conducted on 60 propellant processing runs of 150-gallon
batch mixtures, and 11 runs of 1 gallon. The propellant formulation examined is shown
in table 2.2.
These runs were conducted by JPLIEAF to manufacture 48 in. dia. motors. An
invitation was made to examine the processing and study the data recorded for each batch
run. Tests on the motors were not available at that time.
The mixing was done using Baker-Perkins Mixers. The mixing schedule is shown
in figure 2.4 for the 150 gallon runs. The I-gallon runs were mixed in a similar fashion,
but with a lower mixing time due to the size of the batch.
Three values for the ferric oxide concentration were used for the 150-gallon runs;
they were 0.01 %, 0.04% and 0.24% by weight. In addition, 24 runs of 0.01 % were used
to study pre-storage. In this particular case, 8 runs were pre-stored for 24 hours, another
8 for 48 hours and the rest for 72 hours.
The I-gallon runs used 0.01 %, 0.03% and 0.04% ferric oxide concentrations.
Three runs were also made with none at all. No pre-storage was done.
The ingredient lots were identified in each run and assumed to be identical,
although concern was directed toward the two lots of AP coarse particles used. Their
particle size distribution is shown in figure 2.8. One was procured by JPL from Thiokol.
33
Both lots varied in their average from 200J.lm. Thiokol had an average of 31OJ.lm while
JPL had 250J.lm.
100~--------------------------,
80 IR Q)
E CII
:J 60 CII f! f! 0
0 0 > 0 u
0
>- U .0 -.J
"- 40 Q)
£l; t:
G:
20
O+-L,-.~~.--.-rrr~r--'~on.m
1 10 100 1000 Particle Diameter, f..tm
Figure 2.8 Particle Size Distributions Used for 150 and 1 Gallon Batches.
Most of the data acquired came from samples taken from the batch after casting.
The data corresponding to slurry viscosity measurements were taken befQre the addition
of the curing agent and from the top of the bowl. Similar samples were taken for end of
mix viscosity. All samples were then allowed to cure in separate molds. The samples
were then machined to the proper shape for material and burn rate tests.
The material strength tests involved measuring Maximum and Breakage Tensile
Stresses, corresponding Elongation values, and a Shore A Hardening test. The tensile tests
were done on a Instron Universal Tester, at a crosshead separation rate of 2 in/min. A
JANNAF uniaxial tensile specimen was milled from the samples with a gage length of
34
2.0 in., a thickness of 0.495 in. and a width of 0.380 in. A plastic strain gage was used
to measure the extension directly. Hardness tests were conducted using a Scleroscope
Model A.
The burn rate tests were completed in a Crawford Bomb Chamber with nitrogen
gas as the surrounding atmosphere. Burn rates were measured by placing two wires
through the sample. They were placed 5 in. apart across the length of the sample. A 0.5
in. dia. specimen was used in these tests.
Viscosities were measured using a Brookfield Rotory Viscometer with a coaxial
cylinder measuring head. In addition, other measurements such as density of the samples,
the temperature of the mixture within the bowl, and the room humidity were measured.
A total of 33 processing parameters were recorded for each batch. Appendix A contains
the data acquired for all runs.
Table 2.3 shows the variations of the 24 pre-stored runs corresponding to a ferric
oxide concentration of 0.01 %. A deviation of less than 1 % of the average value for all
runs existed between samples taken within the same batch. This indicated that the testing
procedures used did not introduce a substantial amount of uncertainty. The table also
shows that the deviations in batch-to-batch and scale-up are quite large and, at least for
the material strength properties, cause some concern. Note that the scale-up properties
exceed the batch-to-batch in all but three cases. Similar findings were encountered for all
concentrations of ferric oxide and both batch sizes. Density values remained unchanged
throughout all the runs. Pressure exponents of the burn rate between the small and large
runs are also found to deviate by as much as 25%. A comparison of the ferric oxide
Table 2.3 Variations in 150 Gallon Batch.
Batch to Batch Variations (Standard Deviation)
Viscosity: 7% Slurry
(before addition of curing agent)
7% End of Mix 11 % End of Cast
Burn rate:
2% 2% 2% 1% 1%
Pressure 350 psia 500 psia 650 psia 750 psia*
1000 psia
Material properties: 6% Max stress 8% Max elong 6% Stress @ Failure 9% Elong @ Failure 1 % Density* 3% Shore A Hardness
*less than 1 %
Scale-up Variations (Difference between 150 and 1 Gallon Batches)
Viscosity: 49% Slurry
(before addition of curing agent)
34% End of Mix 4% End of Cast
Burn rate: Pressure
14% 350 psia 10% 500 psia 8% 650 psia 7% 750 psia 5% 1000 psia
Material properties: 39% Max stress 14% Max elong 43% Stress @ Failure 18% Elong @ Failure 1% Density* 3% Shore A Hardness
35
sensitivity for small and large runs showed similar large differences. To place all this in
the proper perspective, the fourth shuttle misf,'\on experienced only a 3% drop in burn
rate,
Samples were taken at only a few different locations within the mixing bowl so
no conclusion can be made whether within-batch variations existed before casting.
36
Table 2.4 and 2.5 show the results of the pre-storage runs. Burn rate values rose
dramatically with pre-storage yet the material strength properties showed no effect at all.
The viscosity at the end of mix for the pre-storage runs showed only a small drop, a 1
kp drop. (Surprisingly, the change in viscosity between the pre-stored and other batches
showed no significant difference at the end of cast.) The property variations did not differ
in the pre-storage runs between any of the times allowed to store.
One critical finding was the history of the viscosity. Table 2.6, 2.7 and 2.8 show
the viscosities taken throughout the processing schedule. In all runs a dramatic rise in the
end of cast viscosity indicated that either adverse mixing or curing was occurring at
the casting stage of the process. Average values for pre-stored end of mix corresponding
to 7.21 kp, at end of cast this value rose to 11.42 kp. The slurry viscosity before the
0.4 .------------------------,
0.3
0.1
o 350 500 650 750
Pressure, psia
11150 Gallons/0.01 % Ferric O:tide/Pre-slored
E21150 Gallons/0.01% Ferric O:tide/Unslored
1000
Table 2.4 Comparison of Pre-stored and Unstored Propellant Bum Rate Properties.
150 r-------------------------------------------------~
100
SO
o Max S, psi Failure S, psi Ma.'t c, % Failure e, %
• 150 Oalions/O.Ol% Ferric Oxide/Pre-stored
E2l150 Oallons/o.ol% Ferric Oxide!Unstored
SboreA
Table 2.5 Comparison of Pre-stored and Un stored Propellant Material Properties.
30 ~------------------------------------------~
25
20
10
5
o Slurry End of Mix End of Cast
Processing Stages
.150 Oalions/O.Ol% Ferric OxidelUnstored ~ 1 Oalion/O.Ol% Ferric Oxide!Unstored
Table 2.6 Viscosity History for 0.01 % Ferric Oxide 150 and 1 Gallon Batches.
37
30 r-----------------------------------------------~
25
20
10
5
o Slurry End of Mi:t End of Cast
Processing Stages
• 150 Oallon5/0.04% rerric OxidelUnstored ~ 1 Oalion/O.04% Ferric OxidelUnstored
Table 2.7 Viscosity History for 0.04% Ferric Oxide 150 and 1 Gallon Batches.
30 r---------------------------------------------~
25
20
10
5
Slurry End of Mi:t End of Cast Processing Stage
.150 Oalions/0.01% Ferric OxIPre-storcd E21150 Oalions/0.01% Ferric Oxide/Unstored
gggj 150 Oallons/0.04% Ferric O:lidc/Unstored ~ 150 Oalions/0.27% Ferric Oxide/Unstored
Table 2.8 Viscosity History for All 150 Gallon Batches.
38
39
curing agent was added was 7.31 kp. The trend is seen in all formulations. (The rise in
viscosity within the 0.04% and 0.24% ferric oxide formulation throughout the process is
a result of the use of Thiokol coarse lot The 0.01 % formulation is the JPL oxidizer).
In evaluating the sensitivity of the propellant viscosity to the sample temperature,
the phenomenon stated above is seen again. Figures 2.9 and 2.10 show the sensitivity for
the sample viscosities. For the end of mix sample, the temperature sensitivity matched
that for the mixture before the addition of the curing agent. This indicated little curing
at this stage since the slope of the line is proportional to the molecular weight of the
binder. The drop in viscosity after the addition of the curing. agent was simply due to
better wetting. The lack of similar correlation in the end of cast indicated a dramatic
100.-------------------------~
Slurry Before Addition of Curing Agent
~ 10 o CJ (I)
:;: End of Mix
1+Orn~~~noTrno~no~~TM~rn~
0.00290 0.00297 0.00303 0.00310 Inverse Sample Temperature, l/K
Figure 2.9 Temperature Sensitivity of End of Mix Viscosity Samples.
100~-------------------------'
End of Cast 0
o~8 g >-~ 10 o U III
:> End of Mix
l+r~~~Tn~~~~~~~~~~
0.00290 0.00297 0.00303 0.00310 Inverse Sample Temperature, l/K
Figure 2.10 Temperature Sensitivity of End of Cast Viscosity Samples.
40
change in the mixture either due to adverse mixing from casting or curing of the binder.
Finally, a parametric analysis was also completed to determine any other factor
which may playa role in the variations of propellant perfonnance. No other parameter
showed strong and conclusive impact on the propellant than those mentioned above.
In summary, batch-to-batch and scale-up variations were apparent, and to such a
degree as to cause concern for the motor performance. Any factors involved in curing
were not apparent in the mixing stage, although this question is stilI open for debate for
the casting.
Supporting work in this area of processing is small and involves mostly
manufacturing mishaps than direct experimental tests. The following are the cases which
41
should be noted.
In conjunction with the work presented above, Mckay [6] in JPLIEAF extended
the study by taking sa.mples at the end of mix stage. The purpose was to determine if
sufficient mixing was occurring to disperse the curing agent properly. The runs
corresponded to 1, 30 and 150 gallon batch runs of the identical formulations studied
above. The samples were extracted from the top of the mixer so no impact due to casting
was involved. The extraction was at locations very near the wall of the bowl. The samples
were then allowed to cure under normal procedures.
In the results, these samples showed remarkable differences in the progress of
curing. For the 30 and 150 gallon runs, the samples did not cure under the normal
procedures. The uncured region was determined to be between 0.25-1.17 in. away from
the wall for 150 gallons, 0.20-0.62 in. for 30 gallons. The 1 gallon run showed no
problem in this area. The uncured region diminished as the wall temperature was raised
at mixing. The conclusion was that the curing agent had not properly dispersed for the
larger mixtures; in order to improve the situation, the wall temperature was raised so that
the mixture viscosity would drop.
It was assumed that with such small .concentrations of curing agents, a good mix
is not reached in all cases and variations are possible in propellant performance.
In another study [7], particle shattering was assumed to contribute to the changes
in performance. Fortunately studies in the area of grinding AP indicates that the amount
of energy necessary to transfer from the blades to the particles was insufficient for these
batch mixers to generate the force needed to break the particles.
42
Throughout this discussion, much has been said about the mixing of the propellant
but little on the casting. Air Force Rocket Propulsion Laboratory (AFRPL) has been
studying the phenomenon of "mid web anomaly" for some time [8]. As stated previously,
the time at which the pressure rises corresponds to the time it would take the propellant
surface to regress to approximately the center of the web. The times are extremely
repeatable and appear for a diverse number of propellants. The anomaly was found to be
insensitive to motor size. It was also discovered that high energetic fonnulations showed
less of a pressure rise, but these propellants had dramatically lower solid concentrations.
Overall, it was assumed that casting flow patterns established orientation patterns in the
propellant.
They supported this conclusion by conducting tests on the casting technique. A set
of 15 lb motors were manufactured with three variations in vacuum casting: the fIrst was
to place a rod through the annular cavity to disrupt the flow; the second was to rotate the
mold to eliminate propellant flow relative to the wall; and the third was simply to cast
under normal conditions and then remove 50, 60 and 70% of the web. The results were
very interesting. All showed a large reduction in the anomaly. Both the stir and spin cast
modification showed a drop in the pressure rise by one half, with spin cast being the best.
The machined cast showed very little anomaly.
This work was verified by Kallmeyer and Sayer [9]. Similar motors tests were
conducted and showed that the manner in which the propellant is introduced in the casing
and how the flow pattern develops between the core and the casing is crucial in this
phenomenon.
43
Lastly, a most enlightening experiment was conducted by Thiokol by Neilson and
Miles [10]. In their work, slices of propellant were taken from a motor at three particular
orientations. The circumferential and vertical slices showed dramatic rises in bum rate
compared to the radial in all the cases. It was assumed that the orientation of the AP was
critical to the phenomenon and that this phenomenon was sensitive to the flow direction
of the casting technique.
In the studies cited above, two points stand out as critical issues to be investigated.
The first is the mixing stage, where the question is whether the curing agent is properly
mixed in the suspension. The second is the casting stage, where concern is focused on
whether the flow behavior of the suspension adversely changes the homogeneity of the
solids. It is these two areas on which this dissertation will focus.
44
CHAPTER 3
RHEOLOGY OF COMPOSITE SOLID PROPELLANT SUSPENSION
This chapter presents the study which was conducted on the rheology of highly
concentrated suspensions. The suspensions consisted of mono modal and bimodal mixtures.
This was intended primarily to supply the empirical relationships for the viscosities of
both mixture compositions. These relationships are to be used within the next chapter in
the multiphase mixture theory and computational work.
The results introduce a better understanding of composite solid propellant
suspension. The study was directed toward concentrations and particle size distributions
similar to those stated for the propellant in chapter 2. More specific, the emphasis was
on bimodal mixtures of solid concentrations between 65% and 75% by volume. This
covers most of the propellant motors presently in use.
Other areas of interest were also examined on both monomodal and bimodal
mixtures. Investigation of the flow behavior of the material, particularly shear stress
versus shear rate behavior, was conducted. Investigations of the effects of temperature and
additives on the mixture viscosity were also conducted. Investigation of concentration
gradients of the curing ag~nt within the mixer were conducted, but proved unsuccessful
due to the complex nature of the material; the techniques used to determine the
concentration gradients are presented.
45
3.1 Background
Little experimental work has been done in the area of highly concentrated
suspensions. Most work in suspensions deals with monomodal mixtures below 50% solid
concentrations by volume. Most focus on problems of very dilute suspensions. Propellant
concentrations usually lie above 65%. These studies have still proven to be useful,
however, since an extensive amount of information has been collected in identifying those
parameters which influence the rheology of the material, work which can be extended to
high concentrations. This includes concentration, particle size distribution and binder
properties. A summary of all the work done in this area is given by Utracki [11].
One specific study stands alone in high concentrations, the work done in the early
'60s by Chong [5]. His work dealt directly with the problem of propellant rheology and
involved developing a new means of measuring viscosity for propellants. Conventional
rotational viscometers suffer from "wall effects", named due to the small clearanc·e
between the rotor and the cup. This limits the maximum concentration which can be
measured. Chong examined an orifice viscometer and determined that it was best suited
to handle high concentrations. Other instruments have since been developed by industry
to overcome the "wall effect" phenomenon [12], yet none have been so thoroughly studied
for high concentrations as the orifice viscometer. Concentrations as ·high as 75% and of
bimodal composition were examined without encountering the phenomenon of "wall
effects". The flow behavior work presented in this chapter follows from his setup and
focuses on the particle size distributions mentioned in chapter 2. A basic description of
rheology as it pertains to propellants follows.
46
In the rheology of propellants, four factors playa major role in the flow behavior
and viscosity of the material. The first is the concentration of the solid particles; the
second is the particle size distribution; the third is the binder flow behavior and viscosity;
and the fourth is the effects of the propellant additives to the viscosity of the binder.
Particle shape may be considered as a fifth, but under the strict quality control and
procurement specifications now used in industry, it plays less of a role. The particles
representing the AP are highly spherical. and with more advancements in their
manufacture, the factor of shape will continue to lose ground to the first four factors. It
is for thi..; reason that particle shape will not be discussed further.
Concentration of the suspension plays the most important role of the four factors.
The concentration governs whether constituent interaction plays a major role in the
rheology. At low concentrations, below 40%, the rheology is governed entirely by the
binder. Little is due to the particles. For higher concentrations. particle packing determines
the rheology. Experiments on shearing motions indicate that above a certain concentration
in monomodal mixtures, shearing of the material will cause swelling and dilatation. This
is known as the critical concentration and is approximately 52%. The particles are packed
in a cubic lattice but are free to move about relative to adjacent particles. In higher
concentrations. their motion becomes more restrictive. A maximum concentration of 74%
can be reached for monomodal mixtures if the particles form a rhombohedral lattrice. At
this concentration, the particles are locked together very tightly. The swelling arises when
the particles roll over adjacent ones as a result of an applied shear stress.
Particle size distribution is as important as concentration. Values above the
47
maximum concentrations of monomodal mixtures can easily be reached with the use of
bimodal and multimodal mixtures. The advantage is also seen in a lower viscosity and as
a result, a more workable mixture. Theoretical work in packing has indicated a maximum
of 86% for bimodal mixtures. When considering the propellant suspension, the average
size plays little or no role in the rheology. In addition, dilatantancy is less likely due to
the packing arrangement between the fine and coarse particles. Propellant compositions
usually permit only random packing due to the relative size between the coarse and fine.
The impact of particle size distribution is seen in the propellant shown in
chapter 2. Figure 2.8 shows the two lots used. The Thiokol lot has slightly more fine
particles which induce a higher propellant viscosity seen in table 2.8. The JPL lot
represents the standard size distributions for oxidizer coarse and fine particles.
Monomodallots usually have less than a 20l-1.m standard deviation in size distributions by
percent volume. If the distributions were more similar, the higher average particle size
(50% value) for the Thiokol lot would have made little difference.
As for the aluminum, since it composes a much smaller percentage of the
propellant and possess a particle size distribution very close to the fine oxidizer, it most
likely has little influence over the rheology. The propellant can then be portrayed as a
bimodal mixture.
The flow behavior of the binder plays a much lesser role in the flow behavior of
the mixture. Pseudoplastic or bingham flow characteristics in the binder are directly
inherited by the mixture. Viscosity is also effected, playing a role in how the binder
interacts with the particles.
48
Finally, the additives, curing agent and metal powder that are incorporated in the
propellant formulation will effect the viscosity of the binder and indirectly effect the flow
characteristics of the mixture. Chapter 2 showed that the curing agent lowered the
viscosity of the binder as it was dispersed through the suspension at mixing.
The key to the work in chapter 4 is the manner in which these four factors are
related to the viscosity of the mixture. Fortunately there are many empirical relationships
available for both monomodal and bimodal mixtures. A list is available in Utracki [11].
Within this work, two equations were found to best fit the data. Equations 3.1 and 3.2
shows the relationships for monomodal and bimodal, respectively.
I-Lmonomodal suspension - I-L bindeI fn (C solids) Eq. (3.1)
Eq. (3.2)
where
fn (x) - I-LbindeI [1+ 2 (1- ;. 5x ) ]2 X Cmax solids
Eq. (3.3)
Eq. (3.4)
C fine Eq. (3.5) C bindeI + C fine
and where c is the concentration, j.l is the viscosity and cmax solids is the maximum
concentration. The bimodal equation is particularly suitable for the work to be done in
chapter 4. It states that the coarse particles see the binder and fine particles as a single
49
fluid (with viscosity equal to the fine and binder monomodal suspension:
Jljine.bintkrsuspension = JlbintkJn(c;J). Note that this is only true when the coarse to fine average
diameter ratio is greater then 20. Fortunately, this is very appropraiate for propellants.
As for the orifice viscometer, equation 3.6 shows the relationship derived by
Chong [5] to determine the viscosity of the mixture. The equation relates the pressure
applied to the flow rate observed. The shear stress and shear rate terms have been
generalized so that comparison to other devices can be made.
T = F ~ slurryB Eq. (3.6)
where T is the shear stress and S the shear rate, and are calculated as follows:
T = IlP 2
Eq. (3.7)
Eq. (3.8)
!::J> is the pressure drop across the orifice, Q is the volumetric flow rate and Ro is the
orifice radius. F is the orifice coefficient and is determined experimentally. Chong [5]
acquired values of 0.75. Support for this value is also shown in the next section.
One important note, equation 3.6 applies to newtonian behavior. In its application
to the suspension, only the low shear rate region can be investigated. This is where the
suspension is most likely to behave newtonian. Fortunately, this is the region 'which best
portrays the mixing and casting flow of propellants.
50
3.2 Experiments
The mixtures used for this study were mixed in a one-pint Baker-Perking Mixer
shown in figure 3.1.
Figure 3.1 One-pint Baker-Perkins Mixer.
The mixture consisted of a binder and several particle size distributions. It was
composed of HTPB binder and silica-alumina alloy particles. The propellant industry is
in a transition from the use of PBAN to HTPB. It was thought best to implement the new
binder in this study. Except for a lower viscosity, HTPB has the same flow behavior as
PBAN. The specific particle size distributions and the requirement that no reaction occur
between the binder and solids determined silica-alumina alloy to be the best material.
Figure 3.2 shows all particle size distributions chosen. The distributions were picked for
the fine and coarse to correspond to those shown in figure 2.2. Figures 3.3, 3.4 and 3.5
show the shapes of the particles with a scanning electron microscope. All show a large
51
100~--------------------.----.
80 ~
a Q)
E ::J 0
60 > ~
.0
L. 40 QI Q)
~ c: G: 0
8 20
O+-~-r~~r-~~"rrrr--.-rr~m
1 10 100 1000 Particle Diameter, J.Lm
Figure 3.2 Particle Size Distribution.
Figure 3.3 SEM Photograph of Coarse.
52
Figure 3.4 SEM Photograph of Intennediate.
Figure 3.5 SEM Photograph of Fine.
53
percentage of spherical particles corresponding close to those seen in AP. The
intermediate particles show a few more random shapes.
An orifice viscometer was used to examine the flow behavior. A picture of the
apparatus is shown in figure 3.6. In the orifice viscometer, pressure was used to push the
mixture through the orifice. The pressure was raised and then dropped to determine if any
hysteresis existed in the flow behavior. Several orifice sizes were made to measure the
flow and care was taken to use the appropriate one so that no concentration gradients
would be generated. A water jacket surrounded the device to control temperatures and the
suspension temperature was recorded with a thermocouple. Temperature in the bowl and
in the viscometer were measured and found to be quite uniform. Appendix B presents the
data acquired in all the rheological experiments.
Figure 3.6 Orifice Viscometer.
54
The mixtures were extracted from the top of the mixer bowl. No vacuum casting
was used. Some samples were placed in a vacuum to determine if any air was entrapped
due to the method in which the mixture was loaded within the viscometer. Little or no
air was found entrapped in the mixture.
The values and curves for calibrated fluids are shown in figures 3.7 and 3.8. The
orifice viscometer was calibrated with two samples from Brookfield. They were the
highest viscosity samples which could be obtained. In the calculations, the pressure
difference and the volumetric flowrate are proportional in the manner shown in
equation 3.6. The calibration value in equation 3.6 was found to be very close to the
values acquired by Chong [5] with very little error. Therefore a value of 0.75 was used
for the rest of the calculations.
30.0
c a.. HT600 ViSCOSIt~ @25 °c: 578.6 Poise
N Orifice Coefflc ent: 0.7755 0 Best Line Average Error: 13.2 Pa .... x
20.0 Orifice Diameter: (em) III III * 0.276 Q) I... o 0.318 -III c 0.426 I...
* C Q)
.c 1"0.0 V1
Shear rate, 1/s
Figure 3.7 Calibration Curve using HT600.
55
30.0
HT1000 Viscosity @25°c: 974.4 Poise e Orifice Coefficient: 0.7922 a.. Best Line Average Error: 38.5 Pa
N 0 Orifice Diameter: (cm) .... )(
20.0 * 0.276 VI
o 0.318 VI [J 0.426 Q) + 0.790 I-.... VI
l-e w
.r:: 10.0 (f)
Shear rate, 1/s
Figure 3.8 Calibration Curve using HT1000.
These figures also show that the viscosity of the mixture is independent of the
orifice size. The next step was to determine if any concentration drop was present when
extracting the high concentration suspensions through the orifice.
The technique used to determine concentration changes involved using a solvent .
to separate the particles from the binder. Several baths were done for 10 ml of
suspension. In each bath, 50 ml of toluene was used to dissolve the binder.
A separation of the fine and coarse was also conducted, but with no success. A
wet screening technique was the only possible method. Other instruments could not be
used due to the large difference between the coarse and fine diameter size. This particular
method failed due to the small quantity of total solid in the sample, allowing for too much
56
error as a result of residual fine particles on the screen.
Measurement of the solid concentration in mixtures were conducted after going
through the orifice viscometer. No difference in concentration were found between the
extracts and the initial formulation used. The results therefore showed no drop at all in
the solid concentration through the viscometer or as a result of mixing. Both monomodal
and bimodal mixtures were investigated. Other studies [13] have supported these results,
indicating no drop in concentration for mixtures beyond 40%. Subsequently, the 0.246 cm.
dia. orifice was used throughout the rest of the study since it proved best in controlling
the flow and allowing for more data to be gathered.
Several mixing procedures were conducted. The changes are shown in figures 3.9
and 3.10 for monomodal and bimodal mixtures. This was done to determine if the
200.0
0 0..
N 0 150.0 Intermediate Particles ..... x 555@; Solid VI * Stored VI 0 and 0 Unstored Cl) I.. ..... 100.0 VI
I.. 0 Cl) .c (f)
50.0
Shear rate, 1/s
Figure 3.9 Changes in Mixing Schedule for Monomodal.
50.0
c 0..
N 40.0 0 ..... x III 30.0 III Q) ....
+-III
.... c 20.0 Q)
.s::. (/)
10.0
Processing Schedules: * 4 Hours Mixed/Unstored (Std) o 2 Hours Mixed/Unstored [J 6 Hours Mixed/Unstored + 4 Hours Mixed/Stored
70/30 Coarse/Fine Ratio 65% Solid
0
Best Line for Standard Procedure
10.0 20.0 30.0 40.0 50.0 Shear rate, l/s
Figure 3.10 Changes in Mixing Schedule for Bimodal.
57
suspension viscosity is sensitive to the mixing procedure. No distinct differences were
found in the mixtures. Standard procedure for the rest of the mixtures was then prescribed
as four hours of mixing under vacuum at 70°c. Viscosity measurements were done at
60°c, with the exception of the HTPB which was done at 25°c for manageability. The
viscosity of the binder at 60°c is 17 poise.
3.3 Results
Figure 3.11 shows the data acquired for HTPB. The fluid behaves newtonian and
serves to show that no non-newtonian flow behavior in the suspension results from its
use.
Figure 3.12 shows the values of t11e relative viscosity for monomodal mixtures.
10.0
B.O Line for HTPB Viscosity 0 @25°c = 60 Poise
0-
N 0
6.0 ..--x
VI VI Q) L. 4.0 0+-VI
L. 0 Q)
..c: 2.0 U1
o . 0 -I4--'-'--'-'--'-'-"T"'"T""-r-r--r-r--r-r-..-.--,.....,....,.....,......-r-..-r-..-r-.-ri
0.0 40.0 80.0 120.0 Shear rate, 1/s
Figure 3.11 HTPB Flow Behavior.
10000~--------------~8~------~ Particle Size
>.. := VI o u VI
100
A Coarse 0
o Intermediate o Fine
Maximum Concentration 5~ . ••
o o
o
A
;> 10 Maximum Concentrotlon
70% Q)
.~ 0+-o Q)
a:= 1+O~~~~~~~~~~~~~~~
30 40 50 60 70 Solid Concentration by Volume, ~
Figure 3.12 Monomodal Relative Viscosity .
58
59
The empirical relationships from equation 3.1 are shown for two maximum
concentrations. The data fit quite well above concentrations of 50%. The coarse and fine
particles tended toward a maximum concentration of 70%. They seem to generate similar
viscosity curves. The intermediate particles showed a much lower maximum concentration
and tended toward 50%. This may be due to the larger standard deviation in intermediate
particle size distributions. Note that the coarse particles proved to be more scattered than
the rest, this scatter will be discussed later in the section when the flow behavior data is
presented.
Bimodal mixtures proved to be the less complicated to measure, as is seen in
figure 3.13. Little scatter was found in the data. The empirical relationships from
equation 3.2 are shown for two maximum concentrations. Note that a maximum
concentration value of 70% approximates the viscosity of the mixtures. No significant
change in the relative viscosity was discovered for changes in the coarse/fine volume ratio
either. The small differences at the highest total solid concentration seem to indicate that
a lower value of coarse is preferable, since the 65/35 coarse/fine ratio had the lowest
viscosity. The infinite-modal mixture line is simply for reference and indicates that not
much will be gained in lowering the viscosity after bimodal mixtures.
The flow behavior is shown in figures 3.14, 3.15 and 3.16 for the monomodal
mixture. This shows the response of the suspension to pressure forces in the orifice
instrument. The coarse particles showed a large degree of hysteresis, resulting in the
scatter of the viscosity data above. Note that as the concentration rises in the fine
particles, the newtonian behavior is lost. At this particular point, the behavior of the
o a..
N
"i:' CD '0
10000~--------------------------~
Bimodal Mixtures:
c: 1000: CD
75/25 Coarse/Fine Volume Ratio Maximum Volume Concentration for
Uniform Size Particles 5051
'? I::J
(/) ~
>.. -'iii o (J III
C
100:
Bimodal Mixtures: 75/25
:> 10 Coarse/Fine Ratio: Q)
> :;:: o Q)
0::
c 75/25 L1 70/30 o 65/35
1+n~~~~~~~~~~~~~~~
50 60 70 80 90 100 Solid Concentration by Volume, %
Figure 3.13 Bimodal Relative Viscosity.
200.0 ...,-------------------------.
o 150.0 .... x
III III
e Vi 100.0 Io Q)
..c: (/)
50.0
Concentration o 62% Solid c and * 6756 Solid
Direction In which p,ressure was applied and withdrawn
10.0 20.0 30.0 Shear rate, 1/s
Figure 3.14 Flow Behavior for Monomodal Mixture of Coarse.
60
200.0
0 0..
N 0 150.0 */ ..... x
VI VI. 55~ SOLID Q) L. .... 100.0 VI
L. 0 ~ Q)
.J: en I 50.0
" 5~ SOLID 45~ SOLID
-~ ..... 0.0
0.0 10.0 20.0 30.0
Shear rate, l/s
Figure 3.15 Flow Behavior for Monomodal Mixture of Intermediate.
200.0 ,..----------------.
o 0..
N
o 150.0 .-x
VI VI
e Vi 100.0
10-o Q)
.J: en
50.0
55~ SOLID
50~ SOLID
20.0 30.0
Shear rate, l/s
Figure 3.16 Flow Behavior for Monomodal Mixture of Fine.
61
62
material best fits a bingham fluid where a yield shear stress must be reached before any
motion occurs. When the pressure was raised and dropped to determine any hysteresis,
none was detected. The intermediate also showed no hysteresis.
The bimodal behavior is shown in figure 3.17, 3.18 and 3.19. No hysteresis and
no large sway from a newtonian flow is seen. The behavior was very simple.
It is interesting that the viscosities in the monomodal mixture show particle size
as being very important. In the bimodal mixture, the average particle size of the mixture
had little impact on the viscosity. One reason may be that the surface tension of the
binder around the coarse particles is quite low, due to the large radius. The particles are
free to move about even at high concentrations. On the other end, the fine particles
generate a much higher surface tension due to their smaller size and result in a more
20.0
a a..
* ('oj
7Ch15 Solid 0 15.0 ..... x
III III (J) Solid L-..... 10.0 III
L-a (J)
..c: (/)
5.0
10.0 20.0 30.0 40.0 Shear rate, 1/s
Figure 3.17 Flow Behavior for Bimodal Mixture of 65/35 Coarse/Fine Ratio.
20.0
e 7556 Solid 0..
N 16.0 0 ..... x 7~ Solid
CIJ 12.0 CIJ Q) I-..-CIJ
l-e 8.0 Q)
.c Vl
4.0
10.0 20.0 30.0 40.0
Shear rate, 1/s
Figure 3.18 Flow Behavior for Bimodal Mixture of 70/30 Coarse/Fine Ratio.
e a..
N o ..... X
50.0 :r---------------. 7556 Solid
40.0
::l 30.0 e ..-CIJ
I- 7~ Solid g 20.0
..c: Vl
10.0
o . 0 -j-n-rT'1n-r-I"'l'TrrT'1"'TT"TO"TTT'T"""-'-'''''''''-rT'1n-r-n-r-r-r-rl 0.0 10.0 20.0 30.0 40.0
Shear rate, 1/s
Figure 3.19 Flow Behavior for Bimodal Mixture of 75/25 Coarse/Fine Ratio.
63
64
rigid motion of the suspension. A certain amount of shear stress is necessary to overcome
this resistance and allow the material to flow. The intermediate and bimodal mixtures
show no hint of either of these behaviors.
In conjunction with this work, Marine [14] conducted a series of investigations
into the effects of temperature and additives on viscosity. The work involved a falling ball
method to determine the viscosity of the mixture. The ingredients used for the mixture
are the same as those used above.
Figures 3.20 and 3.21 show the viscosity for several temperatures and with the
placement of several propellant additives into monomodal and bimodal mixtures. The
temperature dependence shown in the figures is identical to that found in the PBAN data
in chapter 2 (figure 2.9). The formulation in table 2.2 was followed. Note that the
temperature relationship remains unchanged.
The relationship between viscosity and temperature have been made for polymers
of this type. It is an empirical relationship valid only for small temperature variations of
approximately +/-20°c. The relationship is known as Andrade's equation and is shown in
equation 3.9. The exponent (B) for polybutadiene polymers is 3200°c, the coefficient (A)
is 1.15xlO-3 poise. <I> is the temperature.
~ bindQI = AeBlt Eq. (3.9)
By combining equations 3.1 or 3.2 with equation 3.9, the viscosity for the
suspension can easily be determined.
a..
>:!:: III o U III
:>
10000~--------------------------,
1000
100
Size Distributions of Slurry IJ Fine Particles '" Coarse Particles A (80/20 Coarse to Fine
Volume Ratio)
Slurry Is Composed of: 51~ Solid Particles 49~ HTPB By Volume
d.~030 0.0031 0.0032 0.0033 0.0034 Inverse Temperature, 1/K
Figure 3.20 Viscosity dependence on Temperature.
a..
>--·iii 0 u III
:>
10000~------------------------~
1000
100
LIne for Slurry Without Any AddlllvBs
o
0
* *
o
* *
o o
o
* * *
Additives to Slurry o All Ingredients IJ A02246 A Alrosperse 'If Iron Oxide
'" lOP
Slurry Is Composed of: 63511 Solid Parllcias/37511 HTPS Sy Volume 80/20 Coarse to tine Partlclo Volume Ratio
10+rno~MTrnTr~on~no~MT~Tr~~
0.0030 0.0031 0.0032 0.0033 0.0034 Inverse Temperature, 1/K
Figure 3.21 Viscosity dependence on Additives.
65
66
3.4 Other studies
Before completing this chapter, one other investigation should be stated. This work
focused on the mixer and was intended to determine whether the propellant additives and
curing agent were properly mixed. The ingredients used are the same as those mentioned
above, with the exception of a few mixtures using AP particles.
The use of AP particles was unsuccessful when the separation process was applied.
Microscopic examination showed agglomeration of the particles and a large amount of
residual binder in the suspension examined.
Quantitative analysis of the propellant additives and curing agent also proved
unsuccessful. The process of using solvents to separate the particles from the binder
diluted any curing agent within the mixture and hampered any analysis to determine the
concentration. Work even included using a Fourier Transform Infrared Analysis (FTIR)
to determine the concentration of the curing agent and other additives. The opaque nature
of the mixture hampered any analysis using a die with the mixer.
It is unfortunate that these techniques did not succeed for there are only a very
few available tests which can be conducted on these mixtures. In this particular case, the
porosity of the AP and possibly some chemical reaction between the AP and the HTPB
prevented the separation technique from working.
CHAPTER 4
MULTIPHASE MIXTURE THEORY
67
Results from chapter 3 show that bimodal mixtures possess a simple rheology. No
indication of plastic, dilatant or thixotropic behavior was seen. In fact, they lend
themselves easily to analytical and numerical representation. This chapter presents the
approach taken to develop a computer code which models this material. The bases of the
theory and computational work drew from the study of multiphase mixtures, and extended
it to higher concentrations than have conventionally been examined. Previous work dealt
with material below 50% solid by volume. The theory, computational scheme and results
are presented in this chapter.
The code examined two-dimensional monomodal and bimodal mixture flows
through a pipe. It examined concentrations between 65% and 75%. Due to the high·
concentration, it was necessary to apply all inertial and viscous terms within each
constituent and the entire mixture. The fact that bimodal mixtures were examined made
the work more difficult. In addition, it is unknown if proper boundary conditions can be
found to solve the governing equations.
This chapter will examine the application of the technique to bimodal mixture
specifically, but will focus on the applicability of the technique rather than exploring the
causes of the variations problems mentioned in chapter 2. It is hoped that sufficient
confidence is obtained in the technique so that such exploration can be encouraged.
68
4.1 Background
The application of multiphase mixture theory in the field of engineering is
wide spread and a great degree of confidence has been placed in both the analytical and
numerical studies resulting from its use. Most work in this area emphasizes liquid-liquid
and liquid-gas mixtures.
The bases of this theory were derived from thermodynamics of diffusion, which
focuses on describing mixtures of gases. It has since grown to describe many more
complex mixtures. Much has been written on this subject. The development of the theory
can be found in Wallis [1], Truesdell [2] and most recently in Soo [4]. The bases for
much of the work presented in this chapter come from Truesdell's approach, with special
emphasis on Passman [15], who extended it to suspensions.
Computational work is just as abundant, including the work of Spalding [3].
GENMIX2P and PHEONICS are the result of his two-phase work and have been regarded
as the most accurate codes. Ishii [16] examines the most common methods used in
computational models and serves as a good summary.
In the work presented in this chapter, the Diffusion Model method will be used.
This involves a continuum description for the mixture and each constituent. This method
has proven the best success in the numerical field and permits the easiest means of
obtaining the proper constitutive relation for each constituent. A basic description of the
model will be given in the next section. For further detail, Passman presents the best
theoretical explanation.
It should be noted that the application of inertial and viscous terms to all the
69
constituents adds a factor of difficulty to the computational work described previously.
This is further complicated by the lack of infonuation on the boundary conditions and
several quantitative values within the constitutive relations. No attempts have been
targeted for concentrations above 50% and certainly not considering all force tenus. The
closest work was that of Passman, where a I-dimensional solution to a two-phase
suspension was proposed for the shear generated between two parallel plates. The work
leads the way to multidimensional codes with higher concentrations, in addition to
multiphase.
Before addressing the theory, it is important to note that the best technique used
in the field of propellants arises from assuming that the mixture behaves as a
homogeneous material with bingham fluid behavior [17]. The studies now being done in
continuous mixers correlate with this work. Although this method has proven to supply
a good understanding of mixing and casting, the use of multiphase mixture theory should
generate a clear picture on how the constituents behave individually.
4.2 Multiphase Mixture Theory
The following is a description of the model used within the computational work.
In this model, a continuum description is used on the mixture and each constituent of the
mixture. Each constituent occupies each point of the mixture at the same time. The
constraints on the constituents lie in the following three assumption imposed on the
mixture:
1. All properties of the mixture must be mathematical functions of the properties
of the constituents.
70
2. The motion of each constituent is defmed solely by isolating it from the rest of
the mixture. Allowances are then made for the proper interactions of the other
constituents.
3. The motion of the mixture is governed by the same equations as the
constituents.
The fIrst and the last describe the properties of the mixture. The second describes
the manner in which the constituent properties are to be determined.
The motion of the constituents and the mixture are confined to the conservation
of mass and momentum. The problem under investigation is assumed to have no chemical
reaction. The equations are as follows: (Note that the subscript i refers to a constituent
of the mixture, no subscript refers to the mixture properties. Also note that the notation
for a vector is an overline, double overline for a tensor):
Eq. (4.1)
Eq. (4.2)
where Tj is the partial stress tensor, Bj is the specific body force, M j is the specific
interaction force, pj is the partial density and Vj is the velocity. The constitutive relations
for the constituents are of the form:
Tj = Cj[-1t-Pj+CjAjV'Vj+2V'(<<jVCj)]I
+Cj~ j (VVj+VVj T) -2« jVCjVCj
Eq. (4.3)
where Cj is the concentration, 1t is the interface pressure, ~j is the intergranular contact
71
pressure, Aj is the bulk viscosity, a j is the concentration distribution modulus, lis the
identity tensor and Jlj is the viscosity.
The specific interaction force term is also constrained by:
LMi = 0 Eq. (4.4)
where the value for the particles is:
Eq. (4.5)
where D is the Stokes' drag of the binder on the particle. The specific interaction force
for the binder can be determined using equations 4.4 and 4.5.
The temperature between constituents at each point is assumed to be the same,
therefore the energy equation for the mixture can be used to solve for it. The energy
equation for the mixture is given as:
Eq. (4.6)
where, is the specific heat, <l> is the temperature and k is the thermal conductivity.
In addition to these conservation equations, the constraint of saturation is also
applied.
Eq. (4.7)
The properties of the mixture must be accordingly defined with respect to the
conservation equations of mass and momentum. The detail is left to reference 15. They
are:
72
P 5!! E P j 5!! E C j4> j Eq. (4.8)
Eq. (4.9)
Eq. (4.10)
T 5!! E (Tj-pjVjVj ) +pW Eq. (4 .11)
5!! L [Tj +Cj4> j (W-Vj Vj )]
where <l>i is the local density.
4.3 Computational Work
The technique used to solve for the properties of all the constituents and the
mixture was a finite difference scheme. The equations to be solved included the
conservation equations for all the constituents plus the energy equation of the mixture.
The constraint of saturation was also included. The finite difference expressions of the
governing equations were formulated by using a forward time and centered space (FTCS)
finite differencing scheme.
Before any details of the technique can be presented, several points about the flow
problem investigated must be noted:
1. The flow in a pipe was studied. It was assumed to be axisymmetrical, thus all
the derivatives with respect to the angular direction in the equations were set
to zero. All velocities in the angular direction were also set to zero. The flow
was two-dimensional with axial and radial space. Since symmetry existed at the
center line, only half of the profile was examined.
73
2. The governing equations constituted a closed system. Each constituent has 3
independent unknowns. The mixture has 2 unknowns. The number of governing
equations plus a saturation constraint equals the number of unknowns. If there
are proper boundary conditions, the system of the equations can, theoretically,
be solved.
3. The constitutive relations for the binder and solids corresponded to linear
theory. Microrotation and other high shear effects were neglected. This
condition was well suited to fit the low shear flows the propellant is exposed
to in processing.
4. Within monomodal mixtures, equation 3.1 was used to represent the viscosity
of the solids. Within bimodal mixtures, equation 3.2 was used to represent the
viscosity of the coarse solids, while equation 3.1 was used to represent the fine
solids. This is a very important point. In order to apply this modeling technique
to bimodal mixtures, both coarse and fine viscous terms within momentum
equation must be known. Equation 3.9 was used to allow for the factor of
temperature on viscosity.
5. All constituents were assumed to be incompressible. The particles were assumed
to be rigid.
6. The boundary conditions for the velocities were no slip at the walls. Since no
boundary conditions for the concentrations, pressures and temperature have
been conventionally established, several possibilities are established. Further
discussion is left for the results.
74
A non-dimensionalization of these equations is now presented. The following is
a list of the transformations used (the superscript * refers to a dimensionless variable):
x* = x/L Eq. (4.12)
r* = r/R Eq. (4.13)
Eq. (4.14)
Eq. (4.15)
Eq. (4 .16)
Eq. (4.17)
where L is the length of the pipe, R is the radius of the pipe, Ujnlel is the average velocity
of the mixture entering the pipe, q, b is the binder local density and J,lb is the binder
viscosity.
The axial momentum equation for a constituent is used as an example to show
both the non-dimensional and finite difference scheme. Applying the transformations
above, the dimensionless axial momentum equation becomes:
where T· i = c i [-1t·-P* i+Ci"'* iV'V* i+2V·{a· iVci)]1
Eq. (4.19)
+cilJ.· i (VV· i+VV· j T) -2a· jVCjVCi
75
-. - / 2 B i = BR Uinlot Eg. (4.20)
Eg. (4. 21)
and where Re = PbUjnlt,R/~b and is the binder Reynolds number. For the sake of
convenience, the "*,, superscript and the subscript i in the dimensionless equations will
be omitted from here on. Equation 4.18 is then rewritten into:
Eg. (4.22)
The FfCS finite-difference expression of equation 4.22 for each constituent is as follows:
'" n [ x ~ , J - x ~ , J n-l (. .) x ~ + , J - Vx ~ - , J (
V n (. .) V n-l (. .») ( Vn-1 (. 1 .) n-l (. 1 .») ..,c At +Vx ~,J Ax
n-l( .. )( V~-l(i,j+l)-V~-l(i,j-l»)] _ +Vr ~,J AI -
( T~l (i+l,j) _T~l (i-l,j) )+( T~/ (i,j+l) _T~;l (i,j-l) )
R~x R~I
+"'c n B +Mn-1 .., x x
Eg. (4.23)
where subscript i and j are space indices (representing respectively the location of the grid
in x- and r-direction), superscripts n and n-l are the time indices (n denotes the nIb time
step, meaning new value to be determined, and n-l denotes the (n-l)lb time step, meaning
old value). Solving for the new velocity in equation 4.23, the result is shown in
equation 4.24.
76
V n-1 (' .) Ilt [_Vn-1 ( • • )( ~-l(i+l,j) _V~-l(i-l,j)) =" J.,] +-- " J.,] A $C n uX
( . . )n-l( ~-l(i,j+l) -~-l(i,j-l») -Vr J.,] Ilr +
( T~l(i+l,j) -~l(i-l,j) )+( T~l(i,j+l) -T~l(i,j-l))
Rellx Rellr
Eq. (4.24)
It is in this manner that the axial momentum equation is used. The radial momentum
equation is used to determine the new radial velocity and the mass equation is used to
determine the new concentration.
Finally, it was necessary to impose a convergence criteria to determine the proper
interface pressure. The technique used to acquire the value of 1t was a modification of the
InterPhase Slip Algorithm (IPSA) used by Spalding [3]. Without any knowledge of the
proper interphase pressure, an initial guess must be made to determine the velocities by
the momentum equation. This guess generates an error in the mass equation and results
in mass generation or destruction. A method of correcting the interface pressure guess was
needed.
The modified IPSA developed uses the sum of the mass equations (equation 4.1)
of each constituent, weighted relative to the constituents' local density respectively. Since
the error is due to an incorrect value of the interface pressure, the velocities in
equation 4.1 are modified to account for this error. The transformation is done by using
equations 4.25 and 4.26.
7t exact = 7t iteration+d1t
v/vcact(i,j) = Viic6raCion(i,j}
+ E all nodes (aVi (i, j) /01t) d1t
77
Eq. (4.25)
Eq.(4.26)
where '](xacI is the correct value of the interface pressure, ,(kration is the guessed value, d1t
is the correction, the vract is the correct value of the velocity, V/terati6n is the present
iterated value (resulting from the guessed value of the interface pressure), and the partial
derivative within the summation is derived from equation 4.24 for the axial velocities,
respective to each constituent. The partial derivative for the radial velocities are derived
from a similar equation using the radial momentum equation.
The exact velocity value is then inserted into the sum of the mass equations in the
FITS form and the result is a set of linear equations for each node for the correction to
the interface pressure. The error in the mass equations results in a source term within the
linear equations and guides the direction of the correction.
This method proved successful in liquid-liquid studies where a defined interphase
plane existed between the constituents. This is the first time it would be considered to
determine a mixture with no distinguishing interphase plane.
The computational procedure is outlined as follows:
Step 1: Set initial conditions (at t = 0) and boundary conditions. The solution starts with
the establishment of initial values for the concentrations and velocities for all the
constituents, and the interface pressure and temperature for the mixture.
Step 2: The viscosities of the constituents are determined.
78
Step 3: All new constituent concentrations are calculated using the mass equation. Error
in saturation is eliminated by normalizing all the concentrations to saturation.
Step 4: All new constituent velocities are calculated using the momentum equation and
. the new values of concentration.
Step 5: The errors at each node is determined for each constituent by applying the new
velocities and concentrations to the mass equation, weighted by the local density
of each constituent respectively.
Step 6: A correction to the interphase pressure is determined by using the sum of the
errors for all the constituents. The set of linear equations is solved using a Gauss
Elimination with backward substitution technique.
Step 7: The interphase pressure is corrected in addition to the velocities.
Step 8: The temperature of the mixture is calculated using the new values.
Step 9: Steps 2 through 7 are repeated until the error lies below a set value. Then the next
time step is taken and the program returns to step 1, with the new values
replacing the initial values.
Step 10: When the error has gone below the set value, and when the percentage change
in the concentrations, velocities, interface pressure and temperature are
sufficiently small, the procedure is stopped.
A Fortran program was written to construct this routine. It was run on both a
V AXstation 3100, Model 40 using VMS Digital operating system, and a SUN IPX, using
SUN UNIX operating system. A uniform 102x22 grid system was used. This served to
establish 200 free nodes to evaluate the problem. Care was taken to define the distance
79
between nodes so that it would be large in comparison to the average diameter of the
coarse particles. Non-uniform grid systems were tried but proved to become more
unstable and thus were discontinued.
Note that the calculation of the concentrations and velocities were done in an
explicit manner. The time step serves as the relaxation factor since the flow conditions
examined were steady state. An Alternating Directional Implicit Method (ADI) was tried
in order to speed the convergence but proved to make the procedure unstable. Attempts
to solve the pressure correction equations by similar means proved similar results.
4.4 Results from Program
Figure 4.1 shows the configuration set up for investigating pipe flow. The mesh
size is simplified for illustration purposes.
In the process of generating a stable and convergent code, the boundary conditions
for the concentration and interface pressure were extremely important. Several boundary .
conditions were tried. This included a combination of Dirichlet and Neumann forms at
the wall, inlet and exit. Symmetry was imposed on the center line. The only condition
which produced a working code is shown in figure 4.1. Note that a reference interface
pressure was applied at the inlet. The results which follow arise from imposing these
boundary conditions.
The temperature boundary condition had no dramatic effect on the computational
results, except for the impact that temperature has on the viscosity (equation 3.9). A
constant temperature wall was used. In the worst case when the material entered the pipe
at a temperature 35°c higher than the wall, the viscosity difference which was seen
VXi= Ulnlet
VIi= 0 1t=O Cs= Cl lnlet
r L-__ -,) Gravity
aVxdar = 0 Vrl= 0
a1t/Br = 0 aCl/ar = 0 acI>/ar = 0
Figure 4.1 Boundary Conditions
No Slip a1t/ar = 0 aCi/ar = 0
cI>=cl> wall
aVxdax. = 0 aVrilax. = 0
a1tlax. = 0 aCl/ax = 0 acI>lax. = 0
x
80
between the wall and the center of the material were insignificant As a result, the energy
equation was dropped and isothermal conditions were assumed in all further work.
The initial conditions were uniform concentration and axial velocities; and zero
radial velocities and interface pressure throughout the pipe. Changes to the initial axial
velocity were investigated and is discussed later in this section.
The program for bimodal mixtures is shown in appendix C. This includes the input
values for the properties. The specific gravity of the solids was 2.3, the binder was 1. The
viscosity of the binder was 17 poise at 60°c. When the energy equation was applied, the
specific heat of the mixtures were assumed to be governed by the particles and was 0.3
cal/g_Oc. Similar assumptions were made on the thermal conductivity and was 9xlO-4
81
cal/cm-s-Oc. Parametric studies were done on the constitutive coefficients since present
knowledge of their true values are unknown. Except for these studies, they were set to
zero.
Several programs were made to fully understand the capabilities of the multiphase
mixture and computational technique. Figure 4.2 shows the results for a fluid, a
monomodal and a bimodal code using the algorithm mention in the previous section. The
fluid code was done to compare convergence to a known code. Fluent, a fluid dynamics
simulation code developed by Creare, Incorporated was used to determine the accuracy
of the multiphase codes. As seen in the figure, the fluid code simulated the proper
velocity profile. The monomodal and bimodal behavior are very similar and show a slug
flow behavior.
E u ~
c 0 .-
:!:: en 0
a...
0 .--0 0
e:::
5.0
4.0
3.0
2.0
1.0
0.0 0.0 1.0
Bimodal
2.0 3.0
-7 Gravity
4.0 Axial Velocity, m/s
5.0
Figure 4.2 Comparison of Multimodal Flow to a Newtonian Fluid.
In evaluating the rate of convergence between the fluid and the monomodal code,
it was found that the monomodal code converges much more rapidly than expected.
82
Figures 4.3 and 4.4 show the percent change in the interface pressure, axial velocities and
sum of the error as the codes march forward. Note that the rate of convergence of the
fluid system was slow compared to that of highly concentrated suspensions.
Even after 500 time steps, the fluid code showed the slowest rate of convergence
with the percent changes larger than the multiphase codes at the end. The computational
processing time for 500 time iterations exceeds 9 hours on the SUN system and proved
unnecessary. In any subsequent work, iterations of 100 were used to evaluate the solution
for multiphase codes. The time step used was 0.00001 seconds, the space interval was
0.00228 cm. The stability of the code proved to follow a stability where
11 t _ 11x2 2Re
where the Reynolds number is for the suspension.
Eq. (4.27)
Note that the space interval was set so that it was large in comparison to the
coarse particles. This is very important since a continuum requirement has been imposed
on all the constituents. Assuming the propellant formulation shown in table 2.2, the space
interval was 10 times greater than the coarse particles. The restriction on the space
interval may later prove to be a problem if investigating small sections of the mixing or
casting equipment.
To determine the impact of the initial conditions on the results, several axial
velocities were tried. The basic initial condition used for all codes was a uniform axial
velocity. Parabolic and triangular profiles with the same average velocity as the uniforn1
condition were also used, and although the convergence was much slower, the solution
0.1
I:R 0.01
Q)
01 C 0
.J:: u 0.001 I ..... c I Fluid Axial Velocity Q) u I "'-Q)
0.. 0.0001
I L_~~~~Y~~~ _________ _ I I
Fluid and Parllcle Axial Velocity -------------------
0.00001 o 100 200 300 400 500 Number of Iterations
Figure 4.3 Convergence of Monomodal (--) and Newtonian Flow (-) Properties.
Q) CJl C o
.J:: U ..... c Q) u "'Q)
0..
0.0001
0.00001
0.000001
0.0000001
0.00000001
0.000000001 -------------------
o 100 200 300 400 500 Number of Iterations
Figure 4.4 Sum of the Error for Monomodal (--) and Newtonian Flow (-).
83
84
tended toward the same result as shown in figure 4.2.
Figure 4.5 shows the fit of experimental data of chapter 3 to the computational
monomodal solution. A maximum concentration of 65% was found to fit the data best.
This is. done by using the equations for a capillary viscometer [12]. In this case,
equation 3.8 is still valid if Ro is replaced by the pipe radius Rp and equation 3.7 is
replaced with the following:
R,/l.P T=--
2aL Eq. (4.28)
where & is the length in the axial distance of the capillary between the measured
pressure differences. These terms can now be compared to those of the orifice viscometer
data in chapter 3.
160.0 ~--------------------------~
c a..
N
~ 120.0 x ~
VI VI Q) L-
Vi 80.0 LC Q)
..c: Vl
40.0
>I<
•
55l1S
50l1S
10.0 20.0 30.0 Shear rate, 1/s
Figure 4.5 Computational Monomodal Flow.
40.0
85
A very slight dilatation was noted in all the concentrations. Note that the 62% fine
particle data were similar in slope to the 60% curve. Fits of other data can be easily done
by adjusting the values of drag and maximum concentration. But the dilatant behavior
shown was not expected and appears to increase as the concentration rises.
Figures 4.6, 4.7 and 4.8 show variations in the binder viscosity, maximum
concentration and Stokes' drag for the mono modal code. The dash line in figure 4.8
indicates a strong instability in the code (the units are glcm3-s). No change was detected
in changes of the contact force, concentration modulus and bulk viscosity. It was assumed
that this was not due to computational scheme, but due to the flow conditions which were
being modeled. The velocity profiles of each constituent vary only slightly from each
other and were dependent on the magnitude of the drag term. As the drag coefficient rose,
c a..
N 0 .,.... x . III III Q) L.. ....
V1
L.. C Q)
..c: V1
200.0
100 poise
150.0
100.0
50.0 50 poise
-=-_------- 10 poise 0.0 io""'n-rrr-rTTj''''"'TT"1n-rrrr1OT'''"'TT"1n-rrr-r1OT-r-l
0.0 1 0.0 20.0 30.0 40.0 Shear rate, l/s
Figure 4.6 Changes in Binder Viscosity.
60.0 -r----------------.
e 65~
a.. 50.0 N o .,... x
1/1 1/1 Q) .... -(I) .... e Q)
..c Vl
e
40.0
30.0 75~
20.0
70%
10.0
o. 0 -h-~...,..,....,,........,...,....,....,_,_,_ .......... ..,...,.., ......... ~........_ ......... ..,...,.., ......... ~........I 0.0 10.0 20.0 30.0 40.0
Shear rate, 1/s
Figure 4.7 Changes in Maximum Concentration.
120.0 -r---------------....., 10000
a.. 100.0 N o .-x
1/1 1/1 Q) .... -Vl .... e Q)
.J:: Vl
80.0 100
60.0
40.0
20.0
0.0 -h-T"T""T..,..,...,,........,...,...,..,-,-,-,..,....,...,...,..., ......... .....-........._ ......................................... ........I 0.0 10.0 20.0 30.0 40.0
Shear rate, 1/s Figure 4.8 Changes in Stokes' Drag Coefficient.
86
87
the constituent velocity profiles merged. The largest difference constituted less than 1 %
of the mean value. This is supported by other studies for low concentration [18]. In this
study, concentrations above 40% showed little or no differences in constituent velocities.
Figure 4.9 shows the fit of the bimodal data to the computer code. The figure
shows the flow behavior for several total solid concentrations of a 70/30 coarse/fine ratio
bimodal mixture. A maximum concentration of 90% was found best to fit the data. This
was extremely high. The correlation with the bimodal data is good and matches the results
from the monomodal work above.
C 0..
.... 0 -x
III III Q) "--(f) "-c Q)
.c: (I)
20.0 '"
* * 15.0
* * * * 1 0.0
5.0
*
* *
10.0 20.0 30.0 Shear rate, 1/s
Figure 4.9 Computational Bimodal Flow.
40.0
CHAPTERS
DISCUSSION
88
The computational model developed in this dissertation proved to be successful
in portraying the propellant suspension. The multiphase mi?tture approach taken is new
in that it has applied inertial and viscous terms for all the constituents. Up to now, a
primary constituent was chosen to possess these factors. The motions of the others would
be governed by the drag induced by this fluid (in the interaction force term). This is
inappropiate for highly concentrated suspensions. With a better understanding of the
rheology for both monomodal and especially bimodal mixtures, the fully stated goveming
equations for all the constituents can be used. The advantage which arises from using
equation 3.2 allows this to be done.
The code was long in lines due to the number of constituents considered. Several
methods were tried to minimize storage and computational processing time. In the end,
the simplest of methods was found to be the best and only method. An iteration of the
governing equations served to work best, taking advantage of the time steps in the FfCS
technique. As for correction to the interface pressure, a Gauss Elimination with backward
substitution proved to solve the problem faster than any other method.
The greatest question concerned the boundary conditions. Set values of pressure
and concentration at the wall proved to generate instability at all initial conditions chosen.
The use of zero gradients for both properties was chosen solely due to the stability of the
solution. The argument can be stated that this condition is valid for the fluid constituent,
89
yet it is not clear whether this condition is appropriate for the solids. Should future work
investigate the boundary conditions, the use of IPSA would need to be further modified.
The technique used here does not have the flexibility to have individual constituent
boundary conditions.
A lesser question concerned the dilatant behavior seen in the computational runs.
Although this is known to occur in the flow of solids, such a phenomenon was not
expected due to the newtonian type momentum equations within each constituent. The
slug flow behavior follows in the same light. It would be interesting to apply these
equations to an abrupt opening or a free surface, where the suspension would be free to
swell. This would no doubt result in some concentration changes in the solids, possibly
explaining the casting experiments in the processing studies of chapter 2.
Unfortunately, the application of this investigation would require a tracking of the
free surface and be considered a multiphase mixture problem nested within a two-phase
(gas-suspension) problem. Generation of this model would extend far from present
knowledge and techniques. Any attempt might depend on the development of a technique
analogous to the Volume of Fluid Algorithms (VOFA) which has been used for two-phase
problems [19].
The slug flow shown in figure 4.2 also brings up an interesting question. It may
explain why the curing agent is not reaching the wall of the mixing bowl. Much of the
suspension is undisturb by the presence of the wall in the figure and results in ,little of no
propagation of momentum toward it. A curing agent injected at the center may never
reach the wall but simply lie about the center.
90
It would therefore be important to apply this model to flow conditions similar to
those in the mixer, particularly near the wall at the time the blade sweeps by, to get a
good picture of how the curing agent is dispersed.
In the area of the variation problem, it was clear that the values are large and must
cause concern for mission planning. The work within this dissertation is not intended to
resolve the problem but to indicate a new approach which might help in investigating
processing. What has been gained from this work is a better understanding of the problem
overall, and a means of attacking it with a more scientific approach than has previously
been used. Just from the statements above, more specific and detailed questions are being
asked.
The statistical analysis produced the best picture of the problem. Batch-to-batch
variations and scale-up variations existed in all properties. The lack of any parameter
correlation which would take the blame for the variations was extremely surprising.
Conventional thought targeted mixing time as the number one variable. Yet no
dependence on this variable was seen. A detailed analysis of the processing of 150 gallon
batch runs eliminated the conventional explanations of why variations were occurring.
Particle breakage and propellant curing within the mixer proved to be false. Other factors
such as temperature of the mixture and humidity of the room showed no correlation at
all. More surprising was the dramatic rise in the pre-stored batch burn rate
corresponding to a drop in the end of mix. If only end of cast viscosity was recorded, the
reason for the rise in burn rate would not be as clear. It is now evident that where the
viscosity is taken is important.
91
APPENDIX A
PROPELLANT DATA FOR VARIATION STUDY
A.1 Data Collection
The data collected compose sixty ISO-gallon and eleven I-gallon batch runs of
propellant processed at the JPL/EAF facility. Processing was done by JPL personnel.
Inspection of the processing and testing techniques on several runs was done before
studying the data. Lotus 123 software by Lotus Development Corporation was used to
record the data. This data is presented at the end of this section.
A.2 Average Values for 1S0-Gallon Batch Runs
The following are the average values and standard deviations for main properties
of the propellant. Due to the small amount of data available for the I-gallon batch runs,
average values were not taken. Scale-up comparisons presented in chapter 2 are with runs
1, 2, 4 and 9 for the 0.01% ferric oxide propellant, and 11 for the 0.04% ferric oxide
propellant of I-gallon size. This allowed for a close match in the ingredient lots used
between batch sizes.
0.01 % Ferric Oxide Pre-stored Propellant
Viscosity:
End of Mix: 7.68 kp +/-0.55 kp
Temperature: 143.9 of +/- 3.1 of
End of Cast: 13.28 kp +/- 1.52 kp
Temperature: 141.4 of +/- 2.5 of Casting Time: 1.8 hrs +/- 0.4 hrs
Burn rate:
@ 300 psia: 0.2428 in/s +/- 0.0048 in/s
@ 500 psi a: 0.2791 in/s +/- 0.0059 in/s
@ 650 psia: 0.3045 in/s +/- 0.0046 in/s
@ 750 psia: 0.3197 in/s +/- 0.0028 in/s
@ 1000 psia: 0.3485 in/s +/- 0.0038 in/s
Material properties:
Maximum Stress: 136.2 kpsi +/- 8.3 kpsi
Maximum Elongation: 30.3% +/- 2.4%
Stress at Failure: 130.9 kpsi +/- 8.5 kpsi
Elongation at Failure: 35.6% +/- 3.2%
Density: 0.06420 lbs/sq in. +/- 0.00000 Ibs/sq in.
Shore A Hardness: 69.88 +/- 2.4
92
0.01 % Ferric Oxide Unstored Propellant
Viscosity:
Suspension (Before the Addition of the Curing Agent): 13.72 kp +/- 1.00 kp
Temperature: 147.5 OF +/- 6.8 OF
End of Mix: 8.71 kp +/-0.31 kp
Temperature: 141.6 OF +/- 4.1 OF
End of Cast: 13.57 kp +/- 2.73 kp
93
Temperature: 141.2 OF +/- 2.5 OF Casting Time: 1.8 hrs +/- 0.5 hrs
Burn rate:
@ 300 psia: 0.2294 in/s +/- 0.0064 in/s
@ 500 psia: 0.2606 in/s +/- 0.0050 in/s
@ 650 psia: 0.2868 in/s +/- 0.0054 in/s
@ 750 psia: 0.3032 in/s +/- 0.0046 in/s
@ 1000 psia: 0.3309 in/s +/- 0.0091 in/s
Material properties:
Maximum Stress: 137.7 kpsi +/- 4.8 kpsi
Maximum Elongation: 28.7% +/- 1.5%
Stress at Failure: 131.9 kpsi +/- 5.1 kpsi
Elongation at Failure: 34.7% +/- 2.3%
Density: 0.06419 lbs/sq in. +/- 0.00006 lbs/sq in.
Shore A Hardness: 70.3 +/- 2.5
0.04% Ferric Oxide Unstored Propellant
Viscosity:
Suspension (Before the Addition of the Curing Agent): 24.72 kp +/- 3.08 kp
Temperature: 144.0 OF +/- 3.7 OF
End of Mix: 11.59 kp +/-0.93 kp
Temperature: 137.8 OF +/- 3.2 OF
End of Cast: 17.84 kp +/- 1.43 kp
94
Temperature: 138.1 OF +/- 2.4 OF Casting Time: 2.0 hrs +/- 0.29 hrs
Bum rate:
@ 300 psia: 0.2405 in/s +/- 0.0085 in/s
@ 500 psia: 0.2767 in/s +/- 0.0060 in/s
@ 650 psia: 0.3008 in/s +/- 0.0049 in/s
@ 750 psia: 0.3141 in/s +/- 0.0068 in/s
@ 1000 psia: 0.3450 in/s +/- 0.0069 in/s
Material properties:
Maximum Stress: 145.5 kpsi +/- 6.3 kpsi
Maximum Elongation: 28.7% +/- 1.1 %
Stress at Failure: 141.0 kpsi +/- 6.4 kpsi
Elongation at Failure: 33.4% +/- 1.7%
Density: 0.06420 lbs/sq in. +/- 0.00000 lbs/sq in.
Shore A Hardness: 71.2 +/- 1.5
0.27% Ferric Oxide Unstored Propellant
Viscosity:
Suspension (Before the Addition of the Curing Agent): 28.10 kp +/- 6.52 kp
Temperature: 136.2 of +/- 13.03 of
End of Mix: 11.70 kp +/-1.19 kp
Temperature: 137.3 of +/- 3.61 of
End of Cast: 17.02 kp +/- 1.56 kp
95
Temperature: 140.7 of +/- 2.07 of Casting Time: 1.9 hrs +/- 0.39 hrs
Bum rate:
@ 300 psia: 0.2888 in/s +/- 0.0056 in/s
@ 500 psia: 0.3322 in/s +/- 0.0136 in/s
@ 650 psia: 0.3535 in/s +/- 0.0057 in/s
@ 750 psia: 0.3715 in/s +/- 0.0048 in/s
@ 1000 psia: 0.4052 in/s +/- 0.0038 in/s
Material properties:
Maximum Stress: 148.4 kpsi +/- 3.1 kpsi
Maximum Elongation: 28.4% +/- 0.6%
Stress at Failure: 144.5 kpsi +/- 3.7 kpsi
Elongation at Failure: 33.7% +/- 1.6%
Density: 0.06398 lbs/sq in. +/- 0.00041 Ibs/sq in.
Shore A Hardness: 72.3 +/- 2.8
96
A.3 Data
150 GALLON RUNS
BATCH NO.: SB-178A SB-178B SB-178C SB-179A ~B-179B BATCH WT.: 1800 1800 1800 1900 1900
PROP DESIG: MOD 8 MOD 8 MOD 8 MOD 8 MOD 8 BATCH 10.: 1 2 3 4 5
==:aa=aaaaaadamnn",n.a-= ~-rc-.====.·"=.-=-=-"m·"==-·--"-"=."== p
COMPOSITION PARAMETERS AP(UNGD)70%
SOURCE: JPL JPL JPL JPL JPL LOT NO: 5049 5049 5049 5049 5049
NOM SZ MC: 200 200 200 200 200 WT %: 48.99 48.99 48.99 48.99 48.99
W'l' (LB): 881.82 881.82 881.82 930.81 930.81
AP(GD) 30% SOURCE: JPL JPL JPL JPL JPL
LOT NO.: 5049 5049 5049 5049 5049 NOM SZ MC: 10 10 10 10 10
KEAS SZ MC: 12.7 12.7 12.7 11.2 11.2 WT %: 21 21 21 21 21
WT (LB): 378 378 378 399 . 399
ALUMINUM SOURCE: JPL JPL JPL JPL JPL
LOT NO.: 7676 7676 7676 7676 7676 WT %: 16 16 16 16 16
WT (LB): 288 288 288 304 304
Fe203 SOURCE: JPL JPL JPL JPL JPL
LOT NO.: lB612599 lB612599 lB612599 1B612599 lB612599 WT %: 0.01 0.01 0.01 0.01 0.01
WT (LB): 0.18 0.18 0.18 0.19 0.19
PBAN (1.0 eqs) SOURCE: JPL JPL JPL JPL JPL
LOT NO.: 867 867 867 867 867 WT l: 11.49 11.49 11.49 11.49 11.49
WT (LB): 206.82 206.82 206.82 218.31 218.31
DCA (5% binder) LOT NO.: 48-664 48-664 48-664 48-664 48-664
WT t: 0.7 0.7 0.7 0.7 0.7 WT (LB): 12.6 12.6 12.6 13.3 13.3
DER-331 (1.3eqs) SOURCE: JPL JPL JPL JPL JPL
LOT NO.: WT032293 WT032293 WT032293 WT032293 WT032293 WT t: 1.81 1.81 1.81 1.81 1.81
WT (LB): 32.58 32.58 32.58 34.39 34.39
97
MECHANICAL AND BURNRATE PROPERTIES BATCH HO.: SB-178A SB-178B SB-178C SB-179A SB-179B BATCH ID.: 1 2 3 4 5 _.- •• •• ••• • • .. .. •
VISCOSITY (KPS,F,hrs) SLURRY (VIS, 'l'p)
EOM (VIS,'l'p) 7.8 7.5 7.3 8.5 8 142 142 144 140 140
EOC (VIS,'l'p,Tm) 13.5 13.5 15 13 14 .3
138 140 142 140 140 1.75 1 1.5 1.75
BURN RATE IH/S DAYS CURE:
7 7 7 7 7 PSIA
350 0.248 0.239 0.247 0.247 0.246 500 0.276 0.279 0.28 0.276 0.282 650 0.303 0.305 0.309 0.304 0.301 750 0.318 0.318 0.318 0.322 0.318
1000 0.342 0.345 0.348 0.35 0.355 MECHANICAL PROPERTIES
DAYS CURE: 14 14 14 14 14
Sm PSI: 121.8 126.1 126.3 129.8 131.7
Em %: 31.7 31.6 31.6 33.9 32.4
Sb PSI: 116.2 121. 8 122.2 126.6 119.5
Eb %: 38.5 37.6 38.8 38.5 37.6
DNS Le/IN3: 0.0642 0.0642 0.0642 0.0642 0.0642
SHR A lIARD: 66 65 66 70 66
GROUND AP (GRIND RUN NO •• F • S • S • ,M. T. ) 193 193 193 196 196
ACTUAL SLURRY W'I'. (LB)
98
PROCESSING PARAMETERS BATCH NO.: SB-178A SB-178B SB-l78C SB-179A SB-179B BATCH 10.: 1 2 3 4 5
===-- -==~a:==;=====:=====_n=-= vr. g... . ac;a- •••• •••• a ;::1
MIXING TIMES (min) AP FEED:
43 45 29 32 49 AP VAC:
108 105 105 105 92 AFT STO:
30 48 57 30 40
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN:
132.3 145.3 149.5 141.7 152.7 ALUMINUM:
145 160 158.3 156.7 160.3 AP:
161 159 155 160 160 160 159 158 147 157.5
139.5 142 142 140 150 AFT STO:
135 139.5 136 139 137 CURATIVE:
136.5 133.8 133.8 138.5 135.5
PROCESS TEMP (F) THERMOMETER PBAN:
ALUMINUM:
AP:
CURATIVE:
MIXING ROOM TEMP (F) 75 82 76 72 75
MIXING ROOM REL HUKIOIT~ (\) 39 26 38 21 24
99
SB-179C SB-180A SB-180B SB-180e SB-18lA SB-181B 1900 1850 1850 1850 1850 1850
MOD 8 MOD 8 MOD 8 MOD 8 MOD 8 MOD 8 6 7 8 9 10 11
=o"m=='-__.= __ "======rc-=ocm=."nmnnpm.'.""""".--=--==-=---.-""'"n"
COMPOSITION PARAMETERS AP(UNGD) 70\
JPL JPL JPL JPL JPL JPL 5049 5049 5049 5049 5049 5049
200 200 200 200 200 200 48.99 48.99 -48.99 48.99 48.99 48.99
930.81 906.32 906.32 906.32 906.32 906.32
AP(GD) 30\ JPL JPL JPL JPL JPL JPL
5049 5049 5049 5049 5049 5049 10 10 10 10 10 10
11.2 12.5 12.5 12 11.2 11.2 21 21 21 21 21 21
399 388.5 388.5 388.5 388.5 388.5
ALUMINUM JPL JPL JPL JPL JPL JPL
7676 7676 7676 7676 7676 7676 16 16 16 16 16 16
304 296 296 296 296 296
Fe203 JPL JPL JPL JPL JPL JPL
lB612599 lB612599 lB612599 lB612599 lB612599 lB612599 0.01 0.01 0.01 0.01 0.01 0.01 0.19 0.19 0.19 0.19 0.19 0.19
PBAN (1.0 eqs) JPL JPL JPL JPL JPL JPL 867 867 867 867 867 867
11.49 11.49 11.49 11.49 11.49 11.49 218.31 212.56 212.56 212.56 212.56 212.56
DOA (5\ binder) 48-664 48-664 48-664 48-664 48-664 48-664
0.7 0.7 0.7 0.7 0.7 0.7 13.3 12.95 12.95 12.95 12.95 12.95
DER-331 (1.3eqs) JPL JPL JPL JPL JPL JPL
WT032293 WT032293 WT032293 WT032293 WT032293 WT061303 1.81 1.81 1. 81 1.81 1.81 1.81
34.39 33.48 33.48 33.48 33.48 33.48
MECHANICAL AND BURNRATE PROPERTIES SB-179C SB-180A SB-180B SB-180C
6 7 8 9
VISCOSITY (KPS, F, hrs) SLURRY (VIS,Tp)
EOM (VIS,Tp) 8.3 7.5 7.8 8 142 144 143 145
EOC (VIS,Tp,Tm) 12.5 14.8 13.5 18.5
140 146 145 138 1. 75 2.25 1. 67 3.25
BURN RATE IN/S DAYS CURE:
7 7 7 7
0.243 0.246 0.257 0.246 0.283 0.282 0.294 0.278 0.313 0.304 0.314 0.295 0.326 0.318 0.324 0.318 0.357 0.35 0.352 0.348
MECHANICAL PROPERTIES DAYS CURE:
14 14 14 14 Sm PSI:
133.2 131. 7 139.9 ll6.9 Em \:
32.1 32.7 31.3 32.4 Sb PSI:
127.2 127.9 135.6 111.2 Eb \:
37.3 36.7 34.9 39.6 DNS LB/IN3:
0.0642 0.0642 0.0642 0.0642 SHR A HARD:
73 68 72 68 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
196 201 201 201
ACTUAL SLURRY WT. (LB)
SB-181A 10
8
15
2.33
7
0.234 0.283 0.305 0.322 0.344
14
126.3
33.7
121.7
38.5
0.0642
69
203
SB-181B 11
6.6
12.8
1.75
7
0.243 0.29
0.298 0.317
0.35
14
143.8
31.9
137.6
37.6
0.0642
71
203
100
101
PROCESSING PARAMETERS SB-179C S8-180A SB-180B SB-laOC S8-18lA 58-1818
6 7 8 9 10 11 ==~aa~ __ n=""n% ______ =~=a==~=_=_~_s_n=-'=.==--==· •• -n=r'
MIXING TIMES (min) AP FEED:
29 47 46 39 45 AP VAC:
105 91 90 90 91 90 AFT STO:
30 30 65 30 30 75
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN:
144 141 142.3 105.3 140 143 ALUMINUM:
152 154 157.3 145.3 150 151.5 AP: 161 157.8 157 159.5 155.7 157.5 148 161 158 159 158 161 140 145 139 140.5 141 141.5
AFT STO: 140 143 142 137 135 136.5
CURATIVE: 134 .3 136.8 136.8 138 136.5 137.5
PROCESS TEMP (F) THERMOMETER PBAN:
ALUMINUM:
AP:
CURATIVE:
MIXING ROOM TEMP (F) 74 73 74 73 76 74
MIXING ROOM REL HUMIDITY (% ) 33 33 39 39 40 33
102
SB-181C SB-183A SB-183B SB-183C SB-184A SB-184B 1850 1900 1900 1900 1850 1850
MOD 8 MOD 8 MOD 8 MOD 8 MOD 8 MOD8 12 13 14 15 16 17
===r=m="="-a:a_~~D===;;;a_,-··mS2-=.="'=·=AAn·=·==-='-·--"rM""
COMPOSITION PARAMETERS AP(UNGD) 70%
JPL JPL JPL JPL JPL JPL 5049 5049 5049 5049 5049 5049
200 200 200 200 200 200 48.99 48.99 48.99 48.99 48.99 48.99
906.32 930.81 930.81 930.81 906.32 906.32
AP(GD)30% JPL JPL JPL JPL JPL JPL
5049 5049 5049 5049 5049 5049 10 10 10 10 10 10
11. 2 10.4 10.4 10.4 10.3 10.3 21 21 21 21 21 21
388.5 399 399 399 388.5 388.5
ALUMINUM JPL JPL JPL JPL JPL JPL
7676 7676 7676 7676 7676 7676 16 16 16 16 16 16
296 304 304 304 296 296
Fe203 JPL JPL JPL JPL JPL JPL
1B612599 lB612599 lB612599 lB612599 lB612599 lB612599 0.01 0.01 0.01 0.01 0.01 0.01 0.19 0.19 0.19 0.19 0.19 0.19
PBAN (1.0 eqs) JPL JPL JPL JPL JPL JPL 867 867 867 867 867 867
11.49 11. 49 11. 49 11.49 11.49 11.49 212.56 218. :n 218.31 218.31 212.56 212.56
DOA (5% binder) 48-664 48-664 48-664 48-664 48-664 48-664
0.7 0.7 0.7 0.7 0.7 0.7 12.95 13.3 13.3 13.3 12.95 12.95
DER-3Jl (1. 3eqs) JPL JPL JPL JPL JPL JPL
WT061303 WT061303 WT061303 WT061303 WT061303 WT061303 1.81 1. 81 1. 81 1.81 1.81 1. 81
33.48 34.39 34.39 34.39 33.48 33.48
MECHANICAL AND BURNRATE PROPERTIES SB-181C SB-183A SB-183B SB-183C
12 13 14 15 SB-184A
16 SB-184B
17 _ --0"'"2"===-07== _=_===~=-=c-"==.n".·'.--nr- -.-.e"Z--.,--r-? VISCOSITY (KPS,F,hrs) SLURRY (VIS,Tp)
EOM (VIS,Tp) 8 6.8 6.8 8.8 7.3 7.5
148 152 140 EOC (VIS,Tp,Tm)
13 12.5 13.8 13 12.5 12.5 142 145 143
1. 75 2 2.5 2.167 1.67 2 BURN RATE IN/S DAYS CURE:
7 7 7 7 7 7
0.242 0.24 0.245 0.245 0.24 0.24 0.281 0.275 0.275 0.266 0.278 0.273 0.299 0.302 0.303 0.304 0.309 0.305 0.324 0.317 0.322 0.316 0.322 0.32 0.351 0.349 0.347 0.347 0.349 0.34:;!
MECHANICAL PROPERTIES DAYS CURE:
14 14 14 14 14 14 Sm PSI:
135.2 142 144.1 148.6 138.7 133.7 Em %:
31.2 30.3 29.6 26.9 27.9 30 Sb PSI:
130.4 135.8 138.3 142.3 130.4 128.8 Eb %:
37.3 36.7 34.6 33.3 36.8 35.5 DNS LB/IN3:
0.0642 0.0642 0.0642 0.0642 0.0642 0.0642 SHR A HARD:
70 69 70 69 70 70 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
191 205 205 205 207 207
ACTUAL SLURRY WT. (LB)
103
104
PROCESSING PARAMETERS SB-181C SB-183A SB-183B SB-183C SB-184A SB-184B
12 13 14 15 16 17 ;::;.;;:;..;:::z' - __ ._=aDC_aagg====aca_ =-n r. r" "" MIXING TIMES (min)
AP FEED: 30 48 44 54 48 37
AP VAC: 105 92 90 105 120 105
AFT STO: 30 30 30 30 30 30
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN:
155 148.5 143.3 138 147 147.5 ALUMINUM:
157 151 153 152.3 156 l58 AP: 160 158 157.5 159.5 158.7 156
157.5 146 143 150 l52 l62 140 144 139.5 140.5 140.5 l43.5
AFT STO: 138 138.5 135 l39.5 l36.5 l40
CURATIVE: 135.5 139.5 140.5 139 138.8 136.8
PROCESS TEMP (F) THERMOMETER PBAN:
ALUMINUM:
AP:
CURATIVE:
MIXING ROOM TEMP ( F) 71 74 70 74 73 72
MIXING ROOM REL HUMIDITY (') 33 29 29 28 23 28
105
5B-184C 5B-185A 5B-185B 5B-185C SB-186A SB-186B 1850 1900 1900 1900 1900 1900
MOD 8 MOD 8 MOD 8 MOD 8 MOD 8 MOD 8 18 19 20 21 22 23 ==c===_caa __ ==:~ __ ===== -Mzaon=n=r·r-mr.=.arrrZMn.===---=-=z= =
COMPOSITION PARAMETERS AP(UNGD)70%
JPL JPL JPL JPL JPL JPL 5049 5049 5049 5049 5049 5049
200 200 200 200 200 200 48.99 48.99 48.99 48.99 48.99 48.99
906.32 930.81 930.81 930.81 930.81 930.81
AP(GD) 30% JPL JPL JPL JPL JPL JPL
5049 5049 5049 5049 5049 5049 10 10 10 10 10 10
10.3 11.5 11.5 11.5 10.3 10.3 21 21 21 21 21 21
388.5 399 399 399 399 399
ALUMINUM JPL JPL JPL JPL JPL JPL
7676 7676 7676 7676 7676 7676 16 16 16 16 16 16
296 304 304 304 304 304
Fe203 JPL JPL JPL JPL JPL JPL
lB612599 lB612599 lB612599 lB612599 lB612599 lB612599 0.01 0.01 0.01 0.01 0.01 0.01 0.19 0.19 0.19 0.19 0.19 0.19
PBAN (1.0 eqs) JPL JPL JPL JPL JPL JPL 867 876 876 876 876 876
11.49 11.49 11.49 11.49 11.49 11.49 212.56 218. Jl 218. Jl 218.31 218.31 218.31
DCA (5% binder) 48-664 48-664 48-664 48-664 48-664 48-664
0.7 0.7 0.7 0.7 0.7 0.7 12.95 13.3 13.3 13.3 13.3 13.3
DER-3Jl (1. 3eqs) JPL JPL JPL JPL JPL JPL
WT061303 WT061J03 WT061303 . WT061303 WT061303 WT061303 1. 81 1.81 1.81 1.81 1. 81 1.81
33.48 34.39 34.39 34.39 34.39 34.39
106
MECHANICAL AND BURNRATE PROPERTIES SB-184C SB-185A SB-185B SB-185C SB-186A 5B-186B
18 19 20 21 22 23 ==;=-=-= a_n-'~=_~ __ :;= __ ,==.zn-_a='n'-·.a".r'"""=Aaaaaa"-n==_= VISCOSITY (KPS,F,hrs) SLURRY (VIS,Tp)
EOM (VI5,Tp) 7 7.9 7.5 7.3 8 8.5
142 145 148 143 145 EOC (VIS,Tp,Tm)
13 11. 3 12 13.5 12.3 12.3 140 141 141 137 143
1.5 2 1.75 2 1.2 1.75 BURN RATE IN/S DAYS CURE:
7 7 7 7 7 7
0.238 0.239 0.241 0.246 0.242 0.239 0.279 0.285 0.283 0.278 0.277 0.276 0.296 0.307 0.309 0.303 0.307 0.304 0.317 0.317 0.323 0.317 0.321 0.32 0.345 0.348 0.351 0.345 0.35 0.354
MECHANICAL PROPERTIES DAYS CURE:
14 14 14 14 14 14 Sm P5I:
138.5 148.8 144.4 141 141.3 142 Em \: 29.2 27.6 28.6 23.1 29.6 29
Sb PSI: 133.7 144 139.3 140 137.6 136.5 Eb \: 34.6 33.3 33.9 23.7 32.7 34.5
DNS La/IN3: 0.0642 0.0642 0.0642 0.0642 0.0642 0.0642
SHR A HARD: 72 72 70 72 73 72
GROUND AP (GRIND RUN NO., F . S .5 . ,M. T. ) 207 210 210 210 216 216
ACTUAL SLURRY WT. (La)
107
PROCESSING PARAMETERS SB-184C SB-185A SB-185B SB-185C SB-186A SB-186B
18 19 20 21 22 23 ====~=== __ ==-=-n-=========-~=c, "2m"-==m·==~E.-·----n"=M==== MIXING TIMES (min)
AP FEED: 41 73 63 42 39
AP VAC: 105 105 105 105 105 105
AFT STO: 30 30 30 30 30 30
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN:
144 150.5 146 145 131.5 153 ALUMINUM:
158 158.5 156 157.5 146 158.5 AP:
160.5 161.5 157.5 158 154 161.5 150.5 155.5 153.5 159 155 156
140 140 141 140.5 142.5 140 AFT STO:
135.5 139 143 140 140.5 137.5 CURATIVE:
135.3 135.7 136.7 140.3 136.5 134.8
PROCESS TEMP (F) THERMOMETER PBAN:
ALUMINUM:
AP:
CURATIVE:
MIXING ROOM TEMP (F) 73 78 74 74 75 77
MIXING ROOM REL HUMIDITY (% ) 29 33 34 29 J4 20
108
SB-186C SB-190A 5B-190B 5B-190C 5B-19lA 5B-191B 1900 1900 1900 1900 1900 1900
MOD 8 MOD 8 MOD 8 MOD 8 "MOD 8 MOD 8 24 25 26 27 28 29
=-=?===·-=="m~ ____ a---aa== --nan"nc-----"-'-.-,.--.'_·='---"--" COMPOSITION PARAMETERS AP(UNGD) 70\
JPL JPL JPL JPL JPL JPL 5049 5049 5049 5049 5049 5049
200 200 200 200 200 200 48.99 48.99 48.99 48.99 48.99 48.99
930.81 930.81 930.81 930.81 930.81 930.81
AP(GD)30% JPL JPL JPL JPL JPL JPL
5049 5049 5049 5049 5049 5049 10 10 10 10 10 10
10.3 10.5 10.5 10.5 10.5 10.5 21 21 21 21 21 21
399 399 399 399 399 399
ALUMINUM JPL JPL JPL JPL JPL JPL
7676 7676 7676 7676 7676 7676 16 16 16 16 16 16
304 304 304 304 304 304
Fe203 JPL JPL JPL JPL JPL JPL
lB612599 18612599 18612599 lB612599 lB612599 lB612599 0.01 0.01 0.01 0.01 0.01 0.01 0.19 0.19 0.19 0.19 0.19 0.19
PBAN (1. 0 eqs) JPL JPL JPL JPL JPL JPL 876 876 876 876 876 876
11. 49 11.49 11. 49 11. 49 ll.49 ll.49 218.31 218.31 218.31 218.31 218.31 218.31
DOA (5% binder) 48-664 48-664 48-664 48-664 48-664 48-664
0.7 0.7 0.7 0.7 0.7 0.7 13.3 13.3 13.3 13.3 13.3 13.3
DER-331 (1.3eqs) JPL JPL JPL JPL JPL JPL
WT061303 WT8109252 WT8109252 WTB109252 WT8109252 WT8109252 1. 81 1. 81 1.81 1.Bl 1.81 1.Bl
34.39 34.39 34.39 34.39 34.39 34.39
109
MECHANICAL AND BURNRATE PROPERTIES SB-186C SB-190A SB-190B SB-190C SB-19lA SB-191B
24 25 26 27 28 29 ====aag _____ ac= __ aaaa~ __ ====~_== ----" ......... = "= =m
VISCOSITY (KPS,F,hrs) SLURRY (VIS,Tp)
14 14 .5 14 13.5 12.4 151 132 150 137 157
EOH (VIS,Tp) 7.5 8.8 8.8 8.5 8.7 8.8 146 131 140 142 147 144
EOC (VIS,Tp,Tm) 10.5 15 12.3 14.5 1-2.8 12.3
144 137 140 145 143 142 1.5 3 1.2 2.5 2 1.67
BURN RATE IN/S DAYS CURE:
7 9 9 9 9 9
0.235 0.228 0.223 0.23 0.224 0.226 0.269 0.258 0.257 0.266 0.258 0.256 0.309 0.284 0.283 0.293 0.281 0.284 0.317 0.309 0.3 0.304 0.3 0.305 0.345 0.339 0.306 0.342 0.329 0.33
MECHANICAL PROPERTIES DAYS CURE:
14 9 9 9 9 9 Sm PSI:
141. 8 136.4 137.3 131.6 136 134.7 Em %:
27.7 29.6 28.9 28.9 28.6 30 Sb PSI:
137.2 132.8 130.3 126.4 130.7 127 Eb %:
32.9 33.6 34.4 34.2 35 37.5 DNS LB/IN3:
0.0642 0.0642 0.0642 0.0642 0.064 0.0642 SHR A HARD:
74 70 69 70 70 69 GROUND AP (GRIND RUN NO.,F.S.S.,H.T.)
216 222 222 222 222 222 8.7 8.7
ACTUAL SLURRY WT. (LB) 1870 1860
110
PROCESSING PARAMETERS SB-186C SB-190A SB-190B SB-190C SB-19lA SB-191B
24 25 26 27 28 29 ;=-=====;CQ~=====:========== - = 7='--'-====== .mnT'T'S1==r- =:
MIXING TIMES (min) AP FEED:
49 88 62 49 52 47 AP VAC:
105 90 90 90 90 91 AFT STO:
30
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN:
151 141 134.5 146.5 149 145 ALUMINUM:
146 156.5 149.5 155 149 AP: 164 156 153 145.3 157.5 159.3 138 152 144 133 154.5 152 140 144.5 141.5 140 144 147.5
AFT STO: 141
CURATIVE: 137.3 129 127.5 128.8 130.8 129
PROCESS TEMP (F) THERMOMETER PBAN:
128 132 ALUMINUM:
156 148 150 150 AP:
149 142 160 149 151 141 150 152
CURATIVE: 150 131 140 141 134.5
MIXING ROOM TEMP (F) 73 78 79 79 74 78
MIXING ROOM REL HUH I DIT'i (\) 38 29 29 29 29 39
111
5B-191C 5B-192A 5B-192B 5B-192C 5B-193A SB-193B 1900 1900 1900 1900 1900 1900
HOD 8 HOD 8 HOD 8 HOD 8 HOD 8 MOD 8 30 31 32 33 34 35
-==n== _.,nm " __ en =::.;::====' -no === = = w===z--
COHP05ITION PARAMETERS AP(UNGD) 70'
JPL JPL JPL JPL JPL JPL 5049 5049 5049 5049 5049 5049
200 200 200 200 200 200 48.99 48.99 48.99 48.99 48.99 48.99
930.81 930.81 930.81 930.81 930.81 930.81
AP(GD)30' JPL JPL JPL JPL JPL JPL
5049 5049 5049 5049 5049 5049 10 10 10 10 10 10
10.5 8.9 8.9 8.9 8.9 8.9 21 21 21 21 21 21
399 399 399 399 399 399
ALUMINUM JPL JPL JPL JPL JPL JPL
7676 7676 7676 7676 7676 7676 16 16 16 16 16 16
304 304 304 304 304 304
Fe203 JPL JPL JPL JPL JPL JPL
1B612599 1B612599 1B612599 1B612599 lB612599 lB612599 0.01 0.01 0.01 0.01 0.01 0.01 0.19 0.19 0.19 0.19 0.19 0.19
PBAN (1. a eqs) JPL JPL JPL JPL JPL JPL 876 876 a76 876 876 876
11.49 11.49 11. 49 11.49 11.49 11.49 218.31 218.31 218.31 218.31 218.31 218.31
DOA (5' binder) 48-664 48=664 48-664 48-664 48-664 HA-030-1103
0.7 0.7 0.7 0.7 0.7 0.7 13.3 13.3 13.3 13.3 13.3 13.3
DER-3Jl (1.3eqs) JPL JPL JPL JPL JPL JPL
WT8109252 WT8109252 WT8109252 WT8109252 WT8109252 WT8109252 1.81 1.81 1. 81 1. 81 1.81 1.81
34.39 34.39 34.39 34.39 34.39 34.39
112
MECHANICAL AND BURNRATE PROPERTIES SB-191C SB-192A SB-192B SB-192C SB-193A 5B-193B
30 31 32 33 34 35 ============~==-==============---~:;a~=~~==~n" VISCOSIT'i (KPS,F,hrs) SLURR'i (VIS,Tp)
14 .3 12.5 13.3 12.5 13 15.8 149 154 150 153 148 145
EOM (VIS,Tp) 9.5 8.5 8.5 8.3 8.5 9.1 140 147 141 145 140 143
EOC (VI5,Tp,Tm) 11.5 13.3 12.3 11. 7 22 12.3
).40 144 140 143 143 140 1.5 2 1.5 1.2 1.75 1.5
BURN RATE IN/S DA'iS CURE:
9 12 12 12 14 14
0.228 0.228 0.224 0.229 0.231 0.234 0.26 0.258 0.259 0.253 0.271 0.264
0.289 0.287 0.278 0.282 0.295 0.293 0.31 0.302 0.297 0.295 0.308 0.301
0.337 0.333 0.325 0.33 0.339 0.326 MECHANICAL PROPERTIES DA'iS CURE:
9 14 14 14 14 14 5111 PSI:
128.4 144.7 139.9 142.4 145.8 137.8 EIII \:
31.8 25.9 26.5 27.9 30 27.8 Sb PSI:
121 138.2 137.2 138.1 137.6 131.9 Eb \:
38.8 32.9 30.5 32.2 37.4 34.1 DNS LB/IN3 :
0.0642 0.0642 0.0642 0.0642 0.0642 0.0642 SHR A HARD:
65 75 74 73 70 69 GROUND AP (GRIND RUN NO.,F.S.S.,K.T.)
222 222A 222A 222A 222A 222A 8.7 10.3 10.3 10.3 9.3 10.3
ACTUAL 5LURR'i WT. (LB) 1870 1863 1866 1866 1866 1869
113
PROCESSING PARAMETERS S8-191C S8-192A S8-1928 S8-192C S8-193A SB-193B
30 31 32 33 34 35 ",- - = '=a~z========-" MIXING TIMES (min)
AP FEED: 50 50 50 49 85 48
AP VAC: 91 92 90 90 91 91
AFT STO:
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN: 147.5 149 138.5 139.5 152 150
ALUMINUM: 155 153.5 150 150 157 157 AP:
159.7 150 148 162.8 151.3 162.5 149.5 144 141 166 163 156
138 146 140 136 137 132 AFT STO:
CURATIVE: 120.3 134.3 125.5 127.8 126.7 127.3
PROCESS TEMP (F) THERMOMETER PBAN:
132 132 156 162 ALUMINUM:
157 155 143 147 155 AP: 154 150 149 150 148 160
148.5 152 148 152 150 140 CURATIVE:
140 142 140 140 142 143
MIXING ROOM TEMP (F) 72 79 79 81 75 73
MIXING ROOM REL HUMIDIT~ (%) 32 32 30 29 36 36
114
SB-193C SB-211A SB-211B SB-211C SB-212A SB-212B 1900 1900 1900 1900 1900 1900
MOD 8 MOD 9-T MOD 9-T MOD 9-T MOD 9-T MOD 9-T 36 37 38 39 40 41
====_==~=~;:===z=============~ -'=rr=-=~,-=mrr=nv=cn-a=="=
COMPOSITION PARAMETERS AP (UNGD)70\
J'PL MT MT MT MT MT 5049 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041
200 200 200 200 200 200 48.99 48.96 48.96 48.96 48.96 48.96
930.81 930.24 930.24 930.24 930.24 930.24
AP(GO)30\ JPL UT MT MT MT MT
5049 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 10 10 10 10 10 10
8.9 10.4 10.4 10.4 8.9 8.9 21 21 21 21 21 21
399 399 399 399 399 399
ALUMINUM JPL MT MT MT MT MT
7676 7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 16 16 16 16 16 16
304 304 304 304 304 304
Fe203 J'PL J'PL J'PL J'PL J'PL J'PL
16612599 1B612599 16612599 16612599 16612599 1B612599 0.01 0.04 0.04 0.04 0.04 0.04 0.19 0.76 0.76 0.76 0.76 0.76
PBAN (1.0 eqs) J'PL MT MT MT MT MT 876 7227-0036 7227-0036 7227-0036 7227-0036 7227-0036
11. 49 11.49 11.49 11.49 11.49 11.49 218.31 218.31 218.31 218.31 218.31 218.31
DOA (5\ binder) HA-030-1103HA-030-1103HA-030-1103HA-030-1103HA-030-1103HA-030-1103
0.7 0.7 0.7 0.7 0.7 0.7 13.3 13.3 13.3 13.3 13.3 13.3
DER-331 (1.3eqs) MT begin ECA (1.3eqs) J'PL MT MT MT MT MT
WT8109252 7225-0043 7225-0043 7225-0043 7225-0043 7225-0043 1. 81 1.81 1. 81 1.81 1.81 1.81
34.39 34.39 34.39 34.39 34.39 34.39
MECHANICAL AND BURNRATE PROPERTIES SB-193C SB-211A SB-211B SB-211C
36 37 38 39 SB-212A
40 5B-212B
41 ===========-,-=-==-=-======= - - -- - 7 m =ac-- - rna C-$-=----VISCOSITY (KPS,F,hrs) SLURRY (VIS,Tp)
14.8 19.75 21 22.25 22.3 23.3 144 148 150 144 147 143
EOM (VIS,Tp) 8.5 10.5 10.75 11.5 10.3 10.5 139 140 141 137 142 130
EOC (VIS,Tp,Tm) 12.8 18.25 17.75 17 .5 17.5 17.8
137 143 136 137 139 140 1.5 2.25 2 2 2.3 1.9
BURN RATE IN/S DAYS CURE:
14 14 14 7 14 14
0.248 0.231 0.241 0.218 0.246 0.245 0.267 0.271 0.273 0.269 0.284 0.291 0.293 0.301 0.309 0.298 0.3 0.311 0.307 0.319 0.325 0.308 0.313 0.322 0.335 0.339 0.354 0.336 0.343 0.359
MECHANICAL PROPERTIES DAYS CURE:
14 14 14 14 14 14 Sm PSI:
137.7 140.7 149.7 146.5 148.6 152.7 Em ~:
28.5 26.6 28 27.9 30.8 30.5 Sb PSI:
131.1 135.5 146.7 141. 4 141.6 149.7 Eb \:
35.7 31.5 31 34 35.3 34.4 DNS La/IN3:
0.0643 0.0642 0.0642 0.0642 0.0642 0.0642 SHR A HARD:
70 70 72 71 70 73 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
222A E-36-229 E-36-229 E-36-229 E36.230 E36.230 10.3 7.4 7.4 7.4 7.7 7.7
ACTUAL SLURR'x' WT. (La) 1867 1850 1867 1892 1853 1873
115
116
PROCESSING PARAMETERS SB-193C SB-211A SB-21lB SB-211C SB-212A SB-212B
36 37 38 39 40 41 =~.~.=-m _.=~_m==========~=~~~g~~ ••••• ~ga.~.=3_.am •••••••••••• 3 MIXING TIMES (min)
AP FEED: 47 49 47 49 49 48
AP VAC: 91 91 90 9l 9l l05
AFT STO:
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN: 151.5 139.5 151.5 152 155.5 150.5
ALUMINUM: 156 154 160 160.5 162 l56 AP:
164.3 l58.7 l55 141. 7 154.7 154 162 155 149 122 155 155 128 142.5 l28 l23 132 l2l.5
AFT STO:
CURATIVE: 125.8 124 121.8 115.3 124.7 117.8
PROCESS TEMP (F) THERMOMETER PBAN:
140 l35 l49 l45 l46 l39 ALUMINUM:
160 l58 156 l59 147 AP: 160 l57 l55.5 l41 l49 154 l44 l55 145 142 147 l42
CURATIVE: 140.5 l4l 145 l37 143 139.5
MIXING ROOM TEMP (F) 74 77 77 75 86 89
MIXING ROOM REL HUMIDIT~ (%) 34 34 29 3J 26 26
117
SB-2l2C SB-2l3A SB-2l3B SB-2l3C SB-2l4A SB-2l4B 1900 1900 1900 1900 1900 1900
MOD 9-T MOD 9-T MOD 9-T MOD 9-T MOD 9-T MOD 9-T 42 43 44 45 46 47
====;=======================_=-====Qac=--a~_a»aaa __ --=aaa ___ ma=
COMPOSITION PARAMETERS AP CUNGD) 70%
MT MT MT MT MT MT 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041
200 200 200 200 200 200 48.96 48.96 48.96 48.96 48.96 48.96
930.24 930.24 930.24 930.24 930.24 930.24
AP(GD)30\ MT HT MT MT MT MT
7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 10 10 10 10 10 10
8.9 9.2 9.2 9.2 12 12 21 21 21 21 21 21
399 399 399 399 399 399
ALUMINUM HT MT MT MT MT MT
7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 l6 16 l6 l6 16 l6
304 304 304 304 304 304
Fe203 JPL JPL JPL JPL JPL JPL
lB612599 16612599 16612599 lB612599 lB612599 lB6l2599 0.04 0.04 0.04 0.04 0.04 0.04 0.76 0.76 0.76 0.76 0.76 0.76
PBAN (1.0 eqs) MT MT MT MT MT MT
7227-0036 7227-0036 7227-0036 7227-0036 7227-0036 7227-0036 11.49 l1.49 l1.49 l1.49 11.49 11.49
2l8.31 2l8.31 2l8.31 2l8.31 2l8.31 2l8.31
DCA (5% binder) HA-030-l103HA-030-ll03HA-030-1103HA-030-1l03HA-030-1103HA-030-1103
0.7 0.7 0.7 0.7 0.7 0.7 13.3 13.3 13.3 13.3 13.3 13.3
ECA (1.3eqs) MT MT MT MT MT MT
7225-0043 7225-0043 7225-0043 7225-0043 7225-0043 7225-0043 1.81 1.81 1.81 1.81 1.81 1.81
34.39 34.39 J4.39 34.39 J4.39 34.39
118
MECHANICAL AND BURNRATE PROPERTIES SB-212C SB-213A SB-213B SB-213C SB-214A SB-214B
42 43 44 45 46 47 ======~===-==-=-==========-"n="_ama===== n-a~~CC_X?"="cm T
VISCOSITY (KPS,F,hrs) SLURRY (VIS,Tp)
22.8 19.5 26.3 24.1 27.3 27.8 147 147 144 142 150 143
EOM (VIS,Tp) 13.8 11 11.5 11.3 11. 3 11.5
135 138 138 143 142 137 EOC (VIS,Tp,Tm)
19 18.5 19.3 18.3 21 16.8 140 138 132 140 140 136 1.6 2.8 1.8 2 2.4 1.85
BURN RATE IN/S DAYS CURE:
14 14 14 14 14 14
0.24 0.237 0.24 0.243 0.245 0.234 0.28 0.272 0.268 0.284 0.274 0.273
0.304 0.297 0.301 0.297 0.298 0.295 0.321 0.305 0.312 0.305 0.318 0.312 0.345 0.351 0.349 0.339 0.345 0.335
MECHANICAL PROPERTIES DAYS CURE:
14 14 14 14 14 14 Sm PSI:
151.2 153.2 147.4 141.3 147.3 140 Em %:
27.3 27.9 28.9 27.9 28.9 29.1 Sb PSI:
148.4 151. 3 142.7 138.4 142.7 134.4 Eb %:
30.3 30.5 34 31.8 34.2 35.2 DNS LB/IN3 :
0.0642 0.0642 0.0642 0.0642 0.0642 0.0642 SHR A HARD:
71 73 74 73 70.5 72 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
E36.230 E36/234 E36/234 E36/234 E84.8 E84.8 7.7 10.5 10.5 13.5 13.5
13.04 13.04 ACTUAL SLURACTUAL SUJRRY WT. (La)
1865 1876 1859 1867 1867 1862
119
PROCESSING PARAMETERS SB-212C SB-213A SB-213B SB-213C SB-214A SB-214B
42 43 44 45 46 47 =:;=======--====~~=====- ::;; ::::a;;:;:1==-- ..... ;;;::1: .. -=~m •• gaA3~=-~-=--·-MIXING TIMES (min)
AP FEED: 49 49 51 55 56
AP VAC: 90 92 92 92 91
AFT STO:
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN: 149.5 168 141.5 151 142 150.5
ALUMINUM: 159.5 167 151 153 150 145.5
AP: 161 157.5 139 134.7 139 137.3 155 160 135 136 140 124
127.5 141.5 136 134.5 139.5 124.5 AFT S'1'O:
CURATIVE: 116 113.3 129.8 130 128.3 123.8
PROCESS TEMP ( F) THERMOMETER PBAN:
141 162 152 157 158 156 ALUMINUM:
156 161 142 155 138 142 AP:
157.5 154 142 139 139 139 146 140.1 143 150.4 138
CURATIVE: 141 139 141 140.5 144 140
MIXING ROOM TEMP (F) 98 102 96 86 83 78
MIXING ROOM REL HUMIDITY (\) 23 26 27 27 34 24
120
SB-214C SB-215A SB-215B SB-215C SB-216A SB-216B 1900 1850 1850 1850 1900 1900
MOD 9-T MOD 9-T MOD 9-T MOD 9-T MOD 9-T MOD 9-T 48 49 50 51 52 53
=========-=---m==-=========~==·="·=--'ma •• =""m"=M._=·=== :iiII=== COMPOSITION PARAMETERS AP(UNGO) 70\
MT MT MT MT MT MT 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041
200 200 200 200 200 200 48.96 48.96 48.96 48.96 4'8.96 48.96
930.24 905.76 905.76 905.76 930.24 930.24
AP(GO)30\ MT MT MT MT MT }iT
7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 10 10 10 10 10 10 12 8.9 8.9 8.9 8.8 8.8 21 21 21 21 21 21
399 388.5 388.5 388.5 399 399
ALUMINUM MT MT MT MT MT MT
7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 16 16 16 16 16 16
304 296 296 296 304 304
Fe203 JPL JPL JPL JPL JPL JPL
lB612599 lB612599 1B612599 lB612599 lB612599 lB612599 0.04 0.04 0.04 0.04 0.04 0.04 0.76 0.74 0.74 0.74 0.76 0.76
PBAN (1.0 eqs) MT MT MT MT MT MT
7227-0036 7227-0036 7227-0036 7227-0036 7227-0036 7227-0036 11.49 11. 49 11.49 ll.49 ll.49 11.49
218.31 212.57 212.57 212.57 218.31 218.31
DCA (5\ binder) HA-030-1103HA-030-1103HA-030-1103HA-030-1103HA-030-1103HA-030-1103
0.7 0.7 0.7 0.7 0.7 0.7 13.3 12.95 12.95 12.95 13.3 13.3
ECA (1.3eqs) MT MT MT MT MT MT
7225-0043 7225-0043 7225-0043 7225-0043 7225-0043 7225-0043 1.81 1. 81 1. 81 1.81 1.81 1.81
34.39 33.48 33.48 33.48 34.39 34.39
121
MECHANICAL AND BURN RATE PROPERTIES SB-214C SB-215A SB-215B SB-2l5C SB-216A SB-216B
48 49 50 51 52 53 - - az=m=n2 •• -=a,== =========z:~"-=aw·'mv __ ==="",-= •• m·=-.=."='
VISCOSITY (KPS, F ,hrs) SLURRY (VIS,Tp)
27.5 30.8 27 27.5 25 25.5 140 146 138 140 142 143
EOM (VIS,Tp) 12 13 .3 12 12 12.3 12
138 139 133 137 138 136 EOC (VIS,Tp,Tm)
18.3 19.8 15.8 16.8 17.5 16 136 139 140 139 138 136 2.3 2 1. 75 1.8 2 1.9
BURN RATE IN/S DAYS CURE:
14 14 14 14 14 14
0.239 0.259 0.251 0.242 0.235 0.241 0.275 0.276 0.274 0.278 0.284 0.278
0.29 0.305 0.303 0.3 0.303 0.303 0.3 0.316 0.32 0.32 0.31 0.311
0.336 0.355 0.346 0.347 0.345 0.339 MECHANICAL PROPERTIES DAYS CURE:
14 14 14 14 14 14 Sm PSI:
136 134 138.2 139.3 151.1 150 Em %:
28.1 29.8 30 29.4 28.1 28.4 Sb PSI:
132.4 130.4 132.6 134.9 144.3 145.9 Eb %:
32.9 34.8 35.5 33.9 34.8 33.2 DNS LB/IN3 :
0.0642 0.0642 0.0642 0.0642 0.0642 0.0642 SHR A HARD:
72 70 66 72 70.5 70 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
E64.8 E 36/235 E 36/235 E 36/235 E36-237 E36-237 13 .5 8.3 8.3
13.04 8.9 8.9 6.9 ACTUAL SLURRY WT. (LB)
1873 1819 1809 1812 1666 1668
122
PROCESSING PARAMETERS SB-214C SB-215A SB-215B SB-215C SB-216A SB-216B
48 49 50 51 52 53 ===~-========;;===-======;;;===~ =
an c n = " --MIXING TIMES (min)
AP FEED: 41 40
AP VAC: 91 91 92 105 105
AFT STO:
AVE PROCESS MIX TEMP ( F) THERMOCOUPLE PBAN:
154 147 150.5 148.5 144 146 ALUMINUM:
150 153 154.5 149.5 152.5 AP:
138.7 137.3 133.5 127.3 137 141 128 138.5 124 124 137 139 128 129 123.3 122.5 126.8 125.8
AFT STO:
CURATIVE: 124.7 128.8 119.3 125.7 125 121.3
PROCESS TEMP (F) THERMOMETER PBAN:
153 156 148 150 155 157 ALUMINUM:
139 138 141 154 136 142 AP: 141 137 137 136.5 138 142
140.5 145 138.5 135 141.5 142 CURATIVE:
137.4 140 135 139 139 138.7
MIXING ROOM TEMP (F) 77.5 78 78 77 78 79
MIXING ROOM REL HUMIDITY (~) 27.6 22 22 22 31 31
123
SB-216C SB-219A SB-219B S6-219C S6-220A S6-2206 1900 1950 1950 1950 1900 1900
MOD 9-T MOD 13T MOD 13T MOD 13T MOD 13T HOD 13T 54 55 56 57 58 59
====_~aaaAaaa.~==-====--=--a--R.='--=··M=.aaaaa.--·-'T·-"===
COMPOSITION PARAMETERS AP(UNGD) 70\
HT MT HT HT HT HT 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041
200 200 200 200 200 200 48.96 48.73 48.73 48.73 48.73 48.73
930.24 950.23 950.23 950.23 925.9 925.9
AP(GD)30\ HT MT HT HT MT HT
7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 7229-0041 10 10 10 10 10 10
8.8 8 8 8 10.5 11 21 21 21 21 21 21
399 409.5 409.5 409.5 399 399
ALUMINUM MT MT MT MT MT HT
7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 7228-0033 16 16 16 16 16 16
304 312 312 312 304 304
Fe203 JPL JPL JPL JPL JPL JPL
1B612599 1B612599 1B612599 1B612599 lB612599 1B612599 0.04 0.27 0.27 0.27 0.27 0.27 0.76 5.27 5.27 5.27 5.1 5.1
PBAN (1.0 eqs) MT MT HT HT HT MT
7227-0036 7227-0036 7227-0036 7227-0036 7227-0036 7227-0036 11.49 11. 49 11.49 11.49 11.49 11.49
218.31 224.06 224.06 224.06 218.3 218.3
DCA (5% binder) HA-030-1103HA-030-1103HA-030-1103HA-030-1103HA-030-1103HA-030-1103
0.7 0.7 0.7 0.7 0.7 0.7 13.3 13.65 13.65 13.65 13.3 13.3
ECA (1.3eqs) MT HT MT HT MT HT
7225-0043 7225-0048 7225-0048 7225-0048 7225-0048 7225-0048 1.81 1.81 1. 81 1.81 1.81 1.81
34.39 35.29 35.29 35.29 34.4 34.4
124
MECHANICAL AND BURNRATE PROPERTIES SB-216C SB-219A SB-219B SB-219C SB-220A 56-2206
54 55 56 57 58 59 ============_=====~=========:=a=== -=_1:1 ~ • -==:;;.a:~~
VISCOSIT,{ (KPS,F,hrs) SLURR,{ (VIS,Tp)
25.3 26.5 27 26.3 41 22.5 138 140 lU 140 110 140
EOM (VIS,Tp) 11 10.8 11.8 10.8 14 11.3
137 137 136 139 131 141 EOC (VIS,Tp,Tm)
15.3 16 16 18.8 18.5 17.8 137 143 139 141 143 140 1.8 2 1.6 2.2 2.4 1.6
BURN RATE IN/S DA,{S CURE:
14 14 14 14 14 14
0.242 0.295 0.285 0.288 0.282 0.276 0.329 0.329 0.322 0.325
0.3 0.355 0.354 0.356 0.343 0.353 0.316 0.373 0.374 0.369 0.364 0.378 0.347 0.408 0.399 0.408 0.404
MECHANICAL PROPERTIES DA,{S CURE:
14 14 14 14 14 14 Sm PSI:
152.5 152.5 145.1 146.7 145.5 149.4 Em \:
28.6 28.3 28.3 29.1 27.5 28.9 Sb PSI:
144 148 142.5 145.2 139.2 142.8 Eb \:
34 33.3 32.8 32.5 35.9 35.5 DNS LB/IN3:
0.0642 0.0632 0.0641 0.0639 0.0643 0.0642 SHR A HARD:
70 76.5 74.5 73 70 70 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
E36-237 E36-240 E36-240 E36-240 E36-241 E36-241 8.3 8.6 8.6 8.6 9.1
11.6 11. 6 ll.6 II ACTUAL SLURR'{ WT. (LB)
1861 1932 1931 1927 1865.6 1860
125
PROCESSING PARAMETERS SB-216C SB-219A SB-219B SB-219C SB-220A SB-220B
54 55 56 57 58 59 =====aaaa&_ ••• a •• ga_a==;====Q==_g. ___ a~.a ••• g._._._ ••••• _ ••• __ ._~. MIXING TIMES (lIIin)
AP FEED: 41 40 52 37
AP VAC: 105 105 105 105 105 105
AFT STO:
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN:
144 153 146 154 148.5 148.5 ALUMINUM:
150.5 154.5 151 152 158 155.5 AP: 138 128.5 137 136 140 151
121. 5 120.5 122 122 135.7 135 ll8 117.5 123 122.5 126 125.5
AFT STO:
CURATIVE: 123 126 126.3 126 120.5 123.3
PROCESS TEMP (F) THERMOMETER PBAN:
157 153 142 139 151 147 ALUMINUM:
138 146 143 149 153 155 AP: 140 142 136 137 140 138
136.5 138.5 138 132 145.5 144.5 CURATIVE:
137 139.5 141 140 137 141
MIXING ROOM TEMP (F) 76 83 78 67 75 74
MIXING ROOM REL HUMIDITY (%) 34 36 47 39 24 29
SB-220C 1900
HOD 13T 60
COMPOSITION PARAMETERS AP(UNGD)70%
MT 7229-0041
200 48.73 925.9
AP(GD)30% HT
7229-004l 10 11 21
399
ALUMINUM MT
7228-0033 16
304
Fe203 JPL
1B612599 0.27 5.1
PBAN (1.0 eqs) MT
7227-0036 11.49 218.3
DOA (5% binder) HA-030-1103
0.7 13.3
ECA (1.3eqs) MT
7225-0048 1.81 34.4
126
MECHANICAL AND BURNRATE PROPERTIES SB-220C
60 ~= __ --"-·--·-m".Z·m ======_'====.===============-__ "n==--n-____.
VISCOSITY (KPS,F,hrs) SLURRY (VIS,Tp)
25.3 146
EOM (VIS,Tp) 11.5
140 EOC (VIS,Tp,Tm)
15 138 1.4
BURN RATE IN/S DAYS CURE:
14
0.294 0.356
0.36 0.371 0.407
MECHANICAL PROPERTIES DAYS CURE:
14 Sm PSI:
151.2 Em \:
28.1 Sb PSI:
149.1 Eb %:
32.2 DNS LB/IN3:
0.0642 SHR A HARD:
70 GROUND AP (GRIND RUN NO.,F.S.S.,M.T.)
EJ6-241 9.1
11 ACTUAL SLURRY WT. (LB)
1858
127
PROCESSING PARAMETERS SB-220C
60
MIXING TIMES (min) AP FEED:
AP VAC: 105
AFT STO:
AVE PROCESS MIX TEMP (F) THERMOCOUPLE PBAN: 144.5
ALUMINUM: 154.5
AP: 146 137
121.6 AFT STO:
CURATIVE: 125
PROCESS TEMP (F) THERMOMETER PBAN:
142 ALUMINUM:
155 AP: 138
143.3 CURATIVE:
139.5
MIXING ROOM TEMP (F) 74
MIXING ROOM REL HUMIDITY (\) 32
128
129
ONE GALLON RUNS
BATCH NO. SB-188 SB-189 5B-194 5B-195 5B-196 BATCH WT. 13.9 13.9 10 10 10
PROP DESIG MOD 8 MOD 8 MOD 1 MOD 8 MOD 1 BATCH ID: 1 2 3 4 5
- -~g=-~.a=-~m.=.~D=m - =~CAgD_.a~.D~ •• _~= ••• _._. __ Ga __ a.a=.~Q
COMPOSITION AP(UNGD)70%
50URCE: JPL JPL JPL JPL MT LOT NO.: 5049 5049 5049 5049 7229-0041
NOM SZ MC: 200 200 200 200 200 WT %: 48.99 48.99 49 48.99 49
AP(GD)30% SOURCE: JPL JPL JPL JPL HT
LOT NO.: 5049 5049 5049 5049 7229-0041 NOM SZ MC: 10 10 10 10 10
MEAS SZ MC: 10.3 10.5 10.5 8.3 WT\: 21 21 21 21 21
COMMENTS: GND JPL GND JPL GNO JPL GNO JPL GND JPL
ALUMINUM SOURCE: JPL JPL JPL JPL MT
LOT NO.: 7676 7676 7676 7676 7228-0033 TYPE: 5-392 S-392 S-392 S-392
WT%: 16 16 16 16 16 COMMENTS: SPH SPH SPH SPH GRAN
Fe203 SOURCE: JPL JPL no iron JPL no iron
LOT NO.: lB612599 lB612599 lB612599 WT%: 0.01 0.01 0 0.01 0
PBAN SOURCE: JPL JPL JPL JPL MT
LOT NO.: 876 876 876 876 7227-0036 WT%: 11.49 11.49 11.49 11.49 11.49
DOA SOURCE: JPL JPL JPL JPL JPL
LOT NO.: 48-664 48-664 48-664 48-664 48-664 WT%: 0.7 0.7 0.7 0.7 0.7
DER-3Jl SOURCE: JPL JPL JPL JPL MT
LOT NO.: WTO 61303 WTO 61303 WTO 61303 WTO 61303 7225-0043 WT%: 1.81 1. 81 1.81 1.81 1. 81
COMMENTS: old lot old lot new lot new lot MT lot
130
MECHANICAL AND BURNRATE PROPERTIES BATCH NO. SB-1BB SB-lB9 SB-194 SB-195 SB-196
BATCH ID: 1 2 3 4 5 VISC (KPS) TEMP (F) AFTER MIX CYCLE(slurry)
1ST VISC: 17 17.5 19 17.3 27 2ND VISC: 15.3 23.3 18 24
1TEMP: 140 145 135 137 127 2TEMP: 144 137 140
AFTER CURATIVE ADDITION VISC: 10 10.3 9.B 7.5 II TEMP: 140 14B 136 146 132
EOC VISC: 13.1 12.B lB 15.B 27 TEMP: 130 136 ll8 112
BRNRT IN/S PSIA
350 0.273 0.278 0.236 0.253 0.23 500 0.306 0.308 0.27 0.2B8 0.241 650 0.323 0.328 0.284 0.312 0.263 750 0.333 0.342 0.295 0.321 0.269
1000 0.359 0.365 0.317 0.352 0.29
DNS LB/IN3 : 0.0642 0.0642 0.0642 0.0635 0.0642
SHR A HARD: 72 74 70 72
Sm PSI: 189 169 155 178
Em %: 26 29 29 27
Sb PSI: 187 165 153 174
Eb %: 29 33 32 30
==--r-===--=========;:=======---,=_ ZlA ==::.-::: ;;:;:1 =~;:: ---PROCESS BEFORE CURATIVE ADDITION TEMP (F)
JACKET: 140 134 126 125 121 THER CPL: 144 134 138 124
THERH: 140 136 139 135 AFTER CURATIVE ADDITION TEMP (F)
JACKET: 150 135 140 136 137 THER CPL: 125 131 123
THERH: 140 128 136 140 139 ========~=_=:;=Q===_====================-r CO=========:Q==:=;;=====
131
5B-197 5B-198 5B-199 5B-200 5B-201 5B-202 10 10 10 10 10 10
MOD 1B MOD 8 MOD 88 MOO 8 MOD 8e MOO 80 6 7 8 9 10 11
===================~===========-====-- ===~ ---======;~------ ~== COMPOSITION AP(UNGD)70%
MT MT MT JPL MT MT 7229-0041 7229-0041 7229-00·11 5049 7229-0041 7229-0041
200 200 200 200 200 200 49 48.99 48.99 48.99 48.97 48.96
AP(GD)30% MT MT MT JPL MT MT
7229-0041 7229-0041 7229-0041 5049 7229-0041 7229-0041 20 10 20 10 10 10
6.9 4.8 21 21 21 21 21 21
GND MT GND JPL GND MT GND JPL GND JPL GND JPL
ALUMINUM MT MT MT JPL MT MT
7228-0033 7228-0033 7228-0033 7676 7228-0033 7228-0033
16 16 16 16 16 16 GRAN GRAN GRAN 5PH GRAN GRAN
Fe203 no iron JPL JPL JPL JPL JPL
lB612599 18612599 18612599 1B612599 1B612599 0 0.01 0.01 0.01 0.03 0.04
PBAN MT MT MT JPL MT MT
7227-0036 7227-0036 7227-0036 876 7227-0036 7227-0036 11.49 11.49 11.49 11.49 11.49 11.49
DCA JPL JPL JPL JPL JPL JPL
48-664 48-664 48-664 48-664 48-664 48-664 0.7 0.7 0.7 0.7 0.7 0.7
DER-331 liT liT MT JPL MT MT
7225-0043 7225-0043 7225-0043 WTO 61303 7225-0043 7225-0043 1.81 1. 81 1.81 1.81 1.81 1.81
liT lot MT lot MT lot old lot liT lot MT lot
132
MECHANICAL AND BURNRATE PROPERTIES SB-l97 SB-l98 SB-l99 SB-200 SB-20l SB-202
6 7 8 9 10 11 VISC (KPS) TEMP ( F) AFTER MIX CYCLE(slurry)
24.8 20 26.8 20 22.3 25.3 l34 l38 l34 l29 l40 l33
18.5 l50
AFTER CURATIVE ADDITION 8 9.3 lO.8 9.8 11.8 l1. 3
l48 143 l42 l36 l40 136 EOC
16.8 22.8 25 l5.8 20.8 23.3 ll8 l08 III 117 114 l1l
BRNRT IN/S
0.227 0.25l 0.25 0.252 0.25l 0.27 0.25l 0.279 0.283 0.279 0.28l 0.293 0.272 0.295 0.3 0.311 0.312 0.324 0.28l 0.302 0.315 0.319 0.325 0.327 0.305 0.336 0.342 0.346 0.35 0.351
DNS La/IN3: 0.0642 0.0642
SHR A HARD: 71 71
Sm PSI: l58 l68
Em i: 30 28
Sb PSI: l54 l6l
Eb %: 35 34
-==-----~--=-- - ----========-;;;=:;;----=-- -c- -
PROCESS BEFORE CURATIVE ADDITION TEMP (F)
l45 l50 llO l26 l24 l25 l35 l25 l32 l30 133 l31 l48 140 138 l36 139 135
AFTER CURATIVE ADDITION TEMP (F) l55 117 l28 110 133 l20 130 l30 l32 l30 l29 132 148 l4l 14l l38 l36 l40
=======_==============================~=_=~==~===~D========~D===
B.1 Orifice Viscometer
APPENDIX B
RHEOLOGICAL DATA
133
Flow data were acquired using an orifice viscometer similar to the one used by
Chong [5]. The device was machined in two parts, a cylinder and an orifice plate. The
cylinder was 5 cm (2 in.) in inner diameter and 19 cm (7.5 in.) long. Figure B.1 shows
a schematic of the orifice viscometer and detail of the orifice plate. A water jacket
surrounded the cylinder to control the suspension temperature and a thermocouple was
place at the base, 1 cm above the plate. The thermocouple extended in from the inner
cylinder wall by only 0.5 cm and did not disturb the main flow. Several orifice plates
were machined. Table B.1 lists the configurations.
Pressure
Insulation
Thermocouple
Gravity Orifice diameter
Figure B.l Viscometer Cut-Away View.
134
Table B.l Orifice Plate Configurations.
Orifice Orifice Orifice ID diameter, angle,
cm deg
A 0.2760 35.19 B 0.4220 31.80 C 0.3180 33.09 D 0.5360 35.77 E 0.4260 28.83
Mercury and water manometers were set up to measure the pressure differential
between the orifice chamber and the laboratory. An Ohaus, Galaxy Model4000D weight
scale was used to weigh the sample weight vessels collecting material extruding from the
plate.
The suspension was placed in the viscometer and the distance from the suspension
to the top of the viscometer recorded. When pressure was applied to the device, the
suspension extruded from the orifice and was collected in a sample vessels. Material was
collected continuously as the proper pressure setting was adjusted and steady state
conditions were reached. Then another sample vessel was used to collect material while
the time was recorded to determine flow rates. Both vessels were weighed. At each
pressure setting, the pressure was recorded before and after the time measurement. The
data acquired are given in the following sections. Except for the data calculations shown
below, where shear stress and shear rate data are determined, all other data were
condensed using a software known as Grapher from Golden Software, Incorporated. This
135
included best line fits.
The calibration fluids HT600 and HT1000 were acquired from Brookfield,
Incorporated and corresponded to viscosities of 578.60 and 974.40 poise at 25°c. The
HTPB was acquired from Sartomer Company and corresponded to viscosities of 60 and
17 poise for 25 and 60°c, respectively. This binder had a specific gravity of 0.90. The
Intermediate and Fine particles were acquired from Zeelan Industries, Incorporated and
corresponded to specific gravities of2.1 and 2.3. The information above was supplied by
the sources. This includes the microtrac particle size distributions shown in chapter 3. The
coarse particles were obtained from Grainger, a local supplier of industrial and
commercial supplies. Density measurements gave a specific gravity of 2.5. Due to their
large diameter, the coarse particle size distribution was acquired by sieving.
The following is a sample calculation for the first pressure setting of suspension 1.
Density of suspension = 1.5540 g/cc
Drop in height due to loss of material into sample wt vessel A =
«20.32-19.80)+(29.02-28.64)/2)/(1.5540* 19.64) = 0.0233 cm
Hydrostatic pressure = 1.5540*(19.00-8.64-0.0233)*980.67 = 15752.73 g/cm S2
Shear stress = «2.10*33863.90)+15752.73)/2 = 43351.70 g/cm S2
43.3517 (x102) Pa
Volumetric flowrate = (29.02-28.64)/(1.5540*173.29) = 0.0014 cc/s
Shear rate = 4*0.0014/(3.1416*(0.2760/2)3) = 0.6836
136
B.2 Orifice Viscometer Data
B.2.1 Calibration
Test Fluid 1
Fluid: HT600 Sample wt vessel A: 6.82 g
Orifice diameter: 0.4260 cm Sample wt vessel B: 6.82 g
Initial height: 13.34 cm Orifice Coefficient: 0.7488
Temperature: 25.0oc
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (xI02
) Pa 1/s
2.10 15.95 1.90 1:26.32 14.23 4.90 11.31 4.35 19.79 4.20 0:52.42 21.32 7.47 17.82 2.90 25.00 2.65 0:58.60 27.00 5.32 12.77 1.00 28.35 1.00 1:00.20 29.87 2.92 6.28
137
Test Fluid 2
Fluid: HT600 Sample wt vessel A: 6.81 g
Orifice diameter: 0.4260 cm Sample wt vessel B: 10.70 g
Initial height: 16.21 cm Orifice Coefficient: 0.6475
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa l/s
1.20 12.17 1.20 0:59.98 14.08 2.64 7.42 2.30 14.78 2.30 0:39.03 16.99 3.87 9.82
138
Test Fluid 3
Fluid: HT600 Sample wt vessel A: 7.23 g
Orifice diameter: 0.4260 cm Sample wt vessel B: 7.25 g
Initial height: 14.61 cm Orifice Coefficient: 0.6513
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa l/s
0.95 8.92 0.90 1:10.72 11.51 3.16 7.94 1.95 11.41 1.95 1:00.98 16.56 4.26 10.91 1.00 14.39 0.70 1:01.86 20.37 2.71 8.11 0.85 16.47 0.75 1:03.71 23.76 2.51 7.01 1.70 18.83 1.75 1:01.53 28.18 3.51 9.46 0.80 12.28 0.70 1: 11.24 31.31 2.37 5.79
139
Test Fluid 4
Fluid: HT600 Sample wt vessel A: 22.81 g
Orifice diameter: 0.3180 cm Sample wt vessel B: 31.34 g
Initial height: 14.29 cm Orifice Coefficient: 0.6550
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure rnin:s wt stress rate in. H2O g (x102
) Pa l/s
0.75 23.67 0.10 1:01.47 32.80 2.75 7.52 1.30 26.92 1.60 1:01.75 34.80 3.90 10.26 2.35 28.31 2.85 1:00.78 37.44 5.25 13.76 2.00 30.24 2.00 1:01.55 39.60 4.39 11.12 0.80 31.11 0.95 1:00.86 41.11 2.93 7.86
140
Test Fluid 5
Fluid: HT600 Sample wt vessel A: 33.03 g
Orifice diameter: 0.2760 cm Sample wt vessel B: 41.09 g
Initial height: 13.65 cm Orifice Coefficient: 0.8763
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
1.25 33.68 0.80 1:23.08 42.33 3.82 7.23 1.60 34.30 1.75 1:00.77 43.53 4.58 9.57 2.40 35.06 2.50 1:11.19 44.99 5.50 9.94 3.30 35.84 3.35 1:08.58 46.88 6.53 13.35 4.40 36.84 4.45 1:02.09 48.78 7.83 14.83 5.35 37.79 5.45 1:05.11 51.26 8.96 18.45 6.75 39.32 6.85 1:00.79 53.89 10.61 20.96 8.40 40.48 8.45 1:01.52 57.19 12.53 25.99 5.40 42.00 5.10 1:01.43 59.24 8.48 16.17 2.90 43.06 2.70 1:02.92 60.69 5.36 11.16 0.85 44.36 0.70 1:02.52 61.38 2.78 5.35
141
Test Fluid 6
Fluid: HT1000 Sample wt vessel A: 7.22 g
Orifice diameter: 0.2760 cm Sample wt vessel B: 7.18 g
Initial height: 13.97 cm Orifice Coefficient: 0.9100
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (xI02
) Pa l/s
1.60 7.76 1.60 1:01.47 7.80 4.38 4.89 3.35 8.18 3.35 1:23.88 9.16 6.53 7.86 5.65 9.12 5.70 1:01.71 10.49 9.37 10.44 9.70 10.38 9.80 1:08.00 12.81 14.37 16.53 5.90 11.83 5.70 1:04.88 14.18 9.37 10.23 2.90 12.76 2.65 1:02.36 14.97 5.56 6.14 5.35 13.31 5.55 1:01.22 16.23 8.85 9.97 9.45 14.36 9.80 1:27.61 19.08 13.97 15.76
142
Test Fluid 7
Fluid: HT1000 Sample wt vessel A: 16.40 g
Orifice diameter: 0.3180 cm Sample wt vessel B: 19.10 g
Initial height: 13.97 cm Orifice Coefficient: 0.7425
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
1.00 17.70 0.75 1:05.28 20.00 3.46 4.37 1.20 18.30 1.20 1:02.88 21.10 3.83 5.54 2.70 18.80 2.70 1:02.07 22.70 5.65 8.16 3.30 19.80 3.30 1:01.15 24.40 6.33 8.81 4.40 20.90 4.45 1:01.72 26.50 7.66 10.78 6.35 22.30 6.35 1:00.34 29.10 9.97 13.65 8.40 23.80 8.50 1:01.40 32.30 12.48 16.51 5.30 25.40 4.85 1:00.53 34.50 8.17 11.51 2.20 26.60 2.05 1:00.84 35.70 4.43 6.25
143
Test Fluid 8
Fluid: HT1000 Sample wt vessel A: 27.10 g
Orifice diameter: 0.4260 cm Sample wt vessel B: 35.70 g
Initial height: 14.29 cm Orifice Coefficient: 0.6725
Temperature: 25.0°c
Applied Time Sample Shear Shear pressure rnin:s wt stress rate in. H2O g (x102
) Pa lis
1.10 29.10 0.80 1:40.18 39.60 3.34 5.13 1.60 30.70 1.65 1:23.47 43.50 4.05 6.16
144
Test Fluid 9
Fluid: HT1000 Sample wt vessel A: 32.80 g
Orifice diameter: 0.4260 cm Sample wt vessel B: 49.70 g
Initial height: 18.36 cm Orifice Coefficient: 0.6900
Temperature: 25.0oc
Applied Time Sample Shear Shear pressure rnin:s wt stress rate in. H2O g (xI02
) Pa lis
2.50 34.80 2.50 1:00.84 53.00 4.81 7.15 2.00 36.20 1.85 1:00.42 55.60 4.01 5.67 1.10 37.40 1.00 1:01.98 57.50 2.81 4.04
B.2.2 Binder
Binder 1.
Fluid: HTPB
Orifice diameter: 0.2760 em
Initial height: 4.13 em
Temperature: 25.0oe
Sample wt vessel A: 7.30 g
Sample wt vessel B: 7.38 g
145
146
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa l/s
4.25 19.50 4.00 1:01.83 9.40 1.06 17.59 3.40 22.00 3.45 1:03.17 13.30 1.80 33.23 2.90 26.70 2.85 1:03.18 18.50 2.26 44.31 2.50 30.00 2.50 0:59.96 24.10 2.51 50.28 3.15 33.10 3.15 1:01.44 27.30 1.52 28.04 3.60 35.20 3.60 1:01.86 28.90 0.85 13.92 3.00 36.40 3.00 1:01.90 32.00 1.52 26.96 2.50 38.20 2.50 1:01.06 36.30 2.00 37.91 2.00 40.20 2.00 1:01.93 41.90 2.46 48.68 1.50 43.30 1.50 1:01.62 48.50 2.86 57.66 1.00 47.50 1.00 1:01.39 56.40 3.21 69.27 0.50 52.60 0.50 1:08.81 66.30 3.49 77.45 0.00 58.00 0.00 1:01.20 75.20 3.75 78.28
Binder 2.
Fluid: HTPB
Orifice diameter: 0.2760 em
Initial height: 6.35 em
Temperature: 25.0oe
Sample wt vessel A: 7.40 g
Sample wt vessel B: 7.40 g
147
148
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa lis
0.00 18.00 0.00 2:01.09 30.15 5.00 101.14 0.50 30.30
·0.50 2:04.60 46.14 3.61 69.08 1.10 35.50 1.15 2:04.13 54.92 2.41 38.08 2.15 38.40 2.15 2:09.27 57.57 0.92 11.04 2.30 39.40 2.30 2:04.59 59.58 0.66 8.68 1.75 40.70 1.75 2:01.32 64.30 1.23 20.94 2.00 43.50 2.10 2:07.01 65.30 0.72 4.24 1.75 45.50 1.60 2:01.96 70.53 1.06 23.08 1.00 47.30 1.00 2:01.62 77.42 1.71 30.50 0.00 50.10 0.00 2:04.64 89.63 2.66 52.73 1.50 52.70 1.50 2:00.66 92.14 0.55 11.20 1.15 53.40 1.15 2:03.42 95.92 0.89 16.49 0.50 54.60 0.50 2:01.00 102.85 1.54 30.83 0.00 62.10 0.00 2:01.59 110.90 1.80 35.64
149
B.2.3 Monomodal Mixtures
Suspension 1.
Particles: Intennediate Sample wt vessel A: 19.80 g
% Volume of Solids: 54.5 Sample wt vessel B: 28.64 g
Initial height: 8.64 cm
Temperature: 56.3°c Relative viscosity: 4733.49
Applied Time Sample Shear Shear pressure min:s wt stress rate in. Hg g (xI02
) Pa lis
2.10 20.32 2.10 2:53.29 29.02 43.35 0.68 4.10 20.65 4.10 3:05.60 29.87 77.19 1.43 6.20 21.07 6.20 2:08.63 30.69 112.72 1.99 8.21 23.45 8.21 2:05.91 31.54 146.68 2.10 6.15 23.98 6.05 2:06.35 32.17 110.92 1.55 4.35 24.19 4.20 3:19.05 32.88 80.00 1.11 2.30 24.33 2.20 3:22.94 33.23 45.69 0.54
150
Suspension 2.
Particles: Intermediate Sample wt vessel A: 19.79 g
% Volume of Solids: 50.0 Sample wt vessel B: 28.66 g
Initial height: 10.41 em
Temperature: 57.8°c Relative viscosity: 168.29
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa lis
2.35 21.08 2.20 2:01.16 30.07 9.02 3.76 4.45 21.71 4.30 2:01.90 31.92 11.58 4.90 6.45 22.57 6.40 1:19.16 33.38 14.08 5.96 3.95 23.52 3.70 2:09.61 35.18 10.77 4.49 2.00 24.25 1.70 2:12.03 36.55 8.26 3.35
151
Suspension 3.
Particles: Intennediate Sample wt vessel A: 19.8 g
% Volume of Solids: 45.0 Sample wt vessel B: 28.7 g
Initial height: 10,41 cm
Temperature: 56.7°c Relative viscosity: 36.12
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa l/s
1.00 25.24 0.90 2:00.61 33.50 6.97 13.41 2.70 27.27 2.75 2:25.21 41.18 8.99 17.82 3.30 29.76 3.50 2:14.91 48.69 9.58 18.75 2.20 31.98 2.10 2:16.85 54.75 7.81 14.92 1.10 33.55 0.80 2:16.62 59.33 6.15 11.29
152
Suspension 4.
Particles: Intennediate Sample wt vessel A: 18.09 g
% Volume of Solids: 55.0 Sample wt vessel B: 28.69 g
Initial height: 8.94 cm Note: Stored for 24 hrs
Temperature: 57.9°c Relative viscosity: 8624.80
Applied Time Sample Shear Shear pressure rnin:s wt stress rate in. Hg g (x102
) Pa 1/s
2.25 18.56 2.20 2:21.88 29.16 45.27 1.03 4.15 18.77 4.15 2:15.14 29.67 77.85 1.17 6.20 19.06 6.20 2:08.28 30.35 112.54 1.65 8.35 19.47 8.30 2:10.81 31.20 148.49 2.02 5.95 19.72 5.90 2:05.35 31.74 107.83 1.34 4.00 19.91 4.00 2:47.24 32.27 75.22 0.98 2.05 19.98 2.05 2:57.51 32.52 42.19 0.44
153
Suspension 5.
Particles: Intermediate Sample wt vessel A: 28.73 g
% Volume of Solids: 55.0 Sample wt vessel B: 28.82 g
Initial height: 8.26 cm Note: Stored for 24hrs
Temperature: 56.2°c Relative viscosity: 7178.06
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
11.15 28.99 11.30 2:04.58 28.85 22.09 0.07 16.45 29.24 16.70 2:18.08 29.01 28.75 0.36 25.90 29.45 26.00 2:21.51 29.32 40.42 0.68 30.50 29.65 30.75 2:23.62 29.59 46.23 0.58 58.46 29.85 59.14 2:03.64 30.08 81.30 1.23 84.29 30.10 84.65 2:13.86 30.71 113.26 1.46
115.56 30.51 115.56 2:02.80 31.54 151.95 2.10 54.38 30.82 63.70 2:22.21 31.93 75.31 0.85 31.27 31.03 31.27 2:01.18 32.20 46.93 0.69 10.50 31.20 10.35 2:02.83 32.26 20.96 0.15
154
Suspension 6.
Particles: Coarse Sample wt vessel A: 18.08 g
% Volume of Solids: 65.0 Sample wt vessel B: 29.21 g
Initial height: 9.70 cm
Temperature: 57.11 °c Relative viscosity: 791.97
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
10.10 23.86 9.30 2:13.47 30.14 20.68 1.74
15.00 24.31 15.10 2:28.97 31.84 27.30 2.85 21.30 25.23 31.40 2:12.64 34.25 35.07 4.54 25.70 26.34 26.00 2:17.92 37.18 40.59 5.30 31.00 27.38 31.35 2:07.90 40.45 47.12 6.38 24.75 28.30 24.70 2:21.32 42.96 38.99 4.43 20.05 28.94 19.20 2:36.94 44.93 32.57 3.13 14.40 29.31 14.25 2:19.25 45.78 25.93 1.52 9.60 29.67 9.60 2:38.04 46.38 20.01 0.95
155
Suspension 7.
Particles: Coarse Sample wt vessel A: 18.28 g
% Volume of Solids: 67.0 Sample wt vessel B: 29.00 g
Initial height: 10.87 cm
Temperature: 57.6°c Relative viscosity: 674.09
Applied Time Sample Shear Shear pressure min:s wt stress rate in. Hg g (x102
) Pa lis
3.10 18.49 3.05 2:25.30 31.77 59.79 4.68 5.30 20.44 5.25 1:42.34 40.06 96.86 19.89 3.00 25.00 2.95 2:07.86 44.84 57.65 9.18 2.15 27.32 2.05 2:02.11 47.40 42.68 5.15 1.00 28.22 1.00 2:04.60 48.17 24.00 1.52
156
Suspension 8.
Particles: Coarse Sample wt vessel A: 18.15 g
% Volume of Solids: 62.0 Sample wt vessel B: 29.95 g
Initial height: 9.53 cm
Temperature: 58.3°c Relative viscosity: 576.53
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (xI02
) Pa l/s
2.25 41.80 2.25 2:04.99 30.65 10.97 1.42 5.75 42.09 5.75 2:21.27 31.98 15.30 2.39 8.50 42.71 8.80 2:13.82 33.55 18.86 2.98
11.60 43.44 11.70 2:02.46 35.27 22.54 3.57 15.80 44.27 16.10 2:03.52 37.59 27.83 4.77 11.75 45.15 11.25 2:04.88 38.91 22.22 2.69 7.75 45.45 7.45 2:22.22 39.75 17.33 1.50 4.95 44.66 4.40 2:57.54 40.32 13.69 0.82
157
Suspension 9.
Particles: Coarse Sample wt vessel A: 28.59 g
% Volume of Solids: 67.0 Sample wt vessel B: 28.52 g
Initial height: 9.37 cm
Temperature: 57.4°c Relative viscosity: 1586.68
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
15.60 29.49 15.50 2:51.83 29.25 28.54 1.04 20.85 30.16 21.05 2:37.84 30.02 35.23 0.87 31.05 31.98 31.20 2:11.70 30.93 47.84 1.70 40.75 32.52 40.45 2:57.70 33.90 59.58 4.11 30.00 45.30 30.15 2:00.49 36.25 46.10 4.79 20.00 46.72 20.00 2:27.96 37.69 33.47 2.39 9.70 47.15 9.70 2:00.80 38.11 20.61 0.85
158
Suspension lO.
Particles: Fine Sample wt vessel A: 28.60 g
% Volume of Solids: 50.0 Sample wt vessel B: 31.81 g
Initial height: 10.72 cm
Temperature: 58.2°c Relative viscosity: 45.18
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (xlO2) Pa 1/s
0.00 37.63 0.00 1:13.23 34.37 6.16 10.58 I.lO 39.95 0.90 l:lO.31 37.29 7.28 12.57 2.45 42.81 2.35 1:01.35 40.43 8.89 15.47 0.90 45.40 0.80 1:15.46 43.37 6.82 11.80 0.00 47.76 0.00 1:54.30 47.11 5.62 9.91
159
Suspension 11.
Particles: Fine Sample wt vessel A: 21.48 g
% Volume of Solids: 55.0 Sample wt vessel B: 28.73 g
Initial height: 10.67 cm
Temperature: 56.2°c Relative viscosity: 103.74
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa lis
0.00 28.78 0.00 2:13.62 31.11 6.53 5.16 1.45 30.52 1.15 2:20.74 34.12 8.04 8.04 3.20 32.10 2.80 2:01.88 37.65 10.04 10.04 5.85 34.98 6.65 2:25.56 43.58 13.91 13.91 3.35 37.54 3.35 2:24.42 47.27 10.12 10.12 0.90 38.85 0.90 2:44.75 49.92 6.96 6.96 0.00 39.69 0.00 2:06.66 51.64 5.76 5.76
160
Suspension 12.
Particles: Fine Sample wt vessel A: 28.75 g
% Volume of Solids: 62.0 Sample wt vessel B: 28.75 g
Initial height: lO.87 cm
Temperature: 58.7°c Relative viscosity: 493.82
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (xlO2) Pa lis
2.lO 29.97 2.05 2:09.53 31.00 42.03 4.76 4.15 31.34 4.lO 2:07.09 36.68 76.62 12.24 6.25 36.46 6.20 2:06.04 45.58 111.87 19.34 8.25 40.04 8.35 1:02.74 51.79 146.74 27.11 6.25 43.63 6.25 1:07.05 56.48 111.81 19.16 4.lO 45.64 4.lO 1:05.58 59.09 75.27 lO.90 2.35 46.57 2.35 1:19.33 60.38 45.57 4.45
161
B.2.4 Bimodal Mixtures
Suspension 13.
Coarse/Fine ratio: 70/30 Sample wt vessel A: 28.64 g
% Volume of Solids: 65.0 Sample wt vessel B: 28.69 g
Initial height: 9.65 em
Temperature: 57.8°c Relative viscosity: 30.97
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
0.00 34.37 0.00 1:06.33 33.65 8.41 19.06 2.20 38.75 1.90 1:13.96 40.63 10.72 24.06 4.55 43.61 4.45 1:03.28 48.20 13.47 30.49 1.75 47.88 1.45 1:19.02 54.88 9.59 21.49 0.00 51.94 0.00 1:08.36 59.37 7.36 16.74 1.75 55.65 1.40 1:04.08 64.52 9.11 20.49 4.85 60.46 4.75 1:12.80 73.02 12.85 29.76
162
Suspension 14.
Coarse/Fine ratio: 70/30 Sample wt vessel A: 28.73 g
% Volume of Solids: 70.0 Sample wt vessel B: 28.78 g
Initial height: 10.29 cm
Temperature: 57.5°c Relative viscosity: 73.09
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa lis
2.10 29.92 1.80 1:03.46 31.38 10.71 10.03 4.45 32.52 4.25 1:06.20 34.93 13.56 13.13 6.70 35.37 6.50 1:08.10 39.42 16.20 16.15 8.70 39.97 8.60 1:00.32 43.84 18.53 17.95 5.85 42.95 5.05 1:05.13 47.65 14.38 14.33 3.10 45.21 3.10 1:06.62 50.50 11.31 10.48 1.85 46.88 2.00 1:05.18 52.92 9.75 9.09
163
Susoension 15.
Coarse/Fine ratio: 70/30 Sample wt vessel A: 28.68 g
% Volume of Solids: 75.0 Sample wt vessel B: 28.68 g
Initial height: 9.84 cm
Temperature: 57.4°c Relative viscosity: 310.84
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa l/s
2.10 41.97 2.20 2:07.78 29.99 11.43 2.43 4.20 42.55 4.20 2: 12.32 31.76 13.93 3.16 6.50 43.14 6.35 2:00.53 33.72 16.64 3.85 8.60 44.28 8.75 2:05.42 36.14 19.36 4.56 5.85 45.13 5.70 2:07.24 38.13 15.68 3.70 4.00 45.94 3.80 2:01.67 39.71 13.28 3.07 2.00 46.44 1.85 2:03.45 40.89 10.77 2.26
164
Suspension 16.
Coarse/Fine ratio: 65/35 Sample wt vessel A: 28.7 g
% Volume of Solids: 65.0 Sample wt vessel B: 28.7 g
Initial height: 9.05 em
Temperature: 57.1oc Relative viscosity: 35.87
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
2.65 39.49 2.00 1:01.80 34.43 11.70 23.67 4.40 44.42 4.70 1:15.73 43.40 14.18 30.29 6.50 51.32 6.85 1:06.81 51.87 16.45 32.42 3.95 57.84 3.75 1:11.52 58.57 12.59 23.96 1.45 61.66 1.45 0:58.76 62.42 9.38 16.76
165
Suspension 17.
Coarse/Fine ratio: 65/35 Sample wt vessel A: 28.61 g
% Volume of Solids: 70.0 Sample wt vessel B: 28.57 g
Initial height: 10.95 em
Temperature: 57.0°c Relative viscosity: 75.78
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa 1/s
2.00 45.88 1.30 1:09.48 30.99 9.29 8.55 5.00 48.29 5.15 1:01.92 34.17 13.43 12.60 6.85 51.57 7.05 1:04.96 38.05 15.60 14.66 4.00 54.10 3.00 1:00.59 40.59 11.17 10.29 2.45 56.10 2.45 1:01.19 42.81 9.76 8.90
166
Suspension 18.
Coarse/Fine ratio: 65/35 Sample wt vessel A: 28.74 g
% Volume of Solids: 75.0 Sample wt vessel B: 28.62 g
Initial height: 10.16 cm
Temperature: 57.4°c Relative viscosity: 234.78
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa lis
2.00 29.38 2.15 2:06.29 30.51 11.31 3.54 4.50 30.61 4.30 2:05.63 32.85 14.12 4.41 6.75 32.67 6.70 2:05.34 35.53 16.91 5.06 8.75 33.66 8.95 2:04.59 38.69 19.46 6.00 5.85 34.86 5.75 2:19.84 41.49 15.56 4.74 4.05 36.26 4.00 2:00.35 43.48 13.26 3.91 2.85 38.50 1.80 2:02.80 45.00 11.04 2.93
167
Suspension 19.
CoarselFine ratio: 75/25 Sample wt vessel A: 28.66 g
% Volume of Solids: 65.0 Sample wt vessel B: 28.69 g
Initial height: 11.94 cm
Temperature: 57.4°c Relative viscosity: 29.92
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa lIs
1.90 29.82 1.75 1:00.92 33.25 8.69 19.00 3.50 34.18 3.40 1:04.15 39.57 10.48 25.01 4.50 38.79 4.40 1:23.41 48.59 11.43 27.45 2.95 43.88 2.80 1:30.79 56.38 9.14 21.78 1.95 46.96 1.85 1:04.71 61.09 7.70 18.48 0.90 51.37 0.80 1:35.76 66.51 6.16 14.37
168
Suspension 20.
CoarselFine ratio: 75/25 Sample wt vessel A: 28.75 g
% Volume of Solids: 70.0 Sample wt vessel B: 28.70 g
Initial height: 11.18 cm
Temperature: 57.7°c Relative viscosity: 87.34
Applied Time Sample Shear Shear pressure rnin:s wt stress rate in. H2O g (xI02
) Pa l/s
2.05 29.61 1.90 1:33.07 31.55 9.90 7.48 4.65 32.25 4.80 1:33.80 35.78 13.18 11.01 6.05 35.72 5.25 1:31.51 40.14 14.14 11.64 8.50 39.01 8.95 1:16.39 44.82 17.78 14.96 5.90 42.13 5.90 1:20.48 48.52 14.09 11.23 4.05 43.60 3.90 1:23.15 51.56 11.57 8.93 2.20 47.39 2.10 1 :33.51 54.39 9.14 7.39
169
Suspension 21.
Coarse/Fine ratio: 75/25 Sample wt vessel A: 28.74 g
% Volume of Solids: 75.0 Sample wt vessel B: 28.77 g
Initial height: 10.16 cm
Temperature: 57.6°c Relative viscosity: 545.00
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102) Pa lIs
15.60 30.04 13.40 2:11.47 30.42 26.84 2.95 20.95 32.01 21.30 2:10.79 33.13 34.99 4.87 25.40 33.00 25.90 2:12.64 36.36 40.53 5.72 30.60 34.18 31.00 2:23.14 40.35 46.83 6.55 24.85 36.12 23.20 2:11.23 43.20 38.26 5.10 14.25 36.99 14.60 2:30.08 45.27 26.22 3.24 10.15 37.82 9.95 2:41.92 47.00 20.71 2.51
170
Suspension 22.
Coarse/Fine ratio: 70/30 Sample wt vessel A: 28.89 g
% Volume of Solids: 65.0 Sample wt vessel B: 28.75 g
Initial height: 7.62 em Note: Mix for 2 hrs
Temperature: 54.4 DC Relative viscosity: 35.06
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (x102
) Pa l/s
2.00 33.55 1.65 1:11.47 35.24 12.59 24.77 4.50 39.09 4.50 1:10.79 43.83 15.60 31.23 6.90 46.56 6.80 1:12.64 58.82 18.06 37.08 4.40 51.57 4.10 1:23.14 70.31 14.38 29.12 1.90 55.41 1.80 1:30.08 75.46 11.10 21.78
171
Suspension 23.
CoarselFine ratio: 70/30 Sample wt vessel A: 28.76 g
% Volume of Solids: 65.0 Sample wt vessel B: 28.72 g
Initial height: 9.14 cm Note: Mix for 6 hrs
Temperature: 57.6°c Relative viscosity: 35.29
Applied Time Sample Shear Shear pressure rnin:s wt stress rate in. H2O g (x102
) Pa l/s
1.10 36.10 0.90 1:18.78 34.81 10.08 19.70 2.50 40.75 2.45 1:04.04 40.80 11.66 23.84 3.55 45.80 3.55 1:01.83 45.65 12.73 26.05 2.10 47.12 2.00 1:08.97 48.93 10.64 21.62 1.00 52.97 1.00 1:05.39 52.46 9.12 17.89
57.56
172
Suspension 24.
CoarselFine ratio: 70/30 Sample wt vessel A: 28.85 g
% Volume of Solids: 65.0 Sample wt vessel B: 28.74 g
Initial height: 9.14 cm Note: Stored for 24 hrs
Temperature: 56.6°c Relative viscosity: 35.44
Applied Time Sample Shear Shear pressure min:s wt stress rate in. H2O g (xlOZ) Pa lis
0.80 32.59 0.80 1:11.87 34.18 9.92 19.31 2.30 36.77 2.60 1:17.74 41.30 11.72 23.36 3.20 41.38 2.40 1:06.60 47.57 11.88 24.02 2.70 44.98 3.10 1:03.33 53.32 11.78 23.16 1.90 48.86 2.10 1:06.71 58.68 10.43 20.50 1.05 53.19 0.80 1:20.47 64.02 8.85 16.93
APPENDIX C
COMPUTER PROGRAM FOR THE FLOW SIMULATION
c This program uses a finite difference-based c multi phase mixture model for bimodal solid c size distributed, highly concentrated c suspensions to compute concentration and c velocity flow in a pipe. The model includes c inertial and viscous terms for all the c constituents. c c Definitions of key variables: c VX, VR: axial and radial velocities for mixture. c CONA,VXA,VRA,PA: concentration, axial and radial c velocities, and pressure for constituent A. c CONNA,VXNA,VRNA: new concentration, axial and radial c velocities for constituent A. c DA,ALC,BTA,CVA,CPA,XLA: properties of constituent A. c DA density c ALC concentration distribution modulus c BT A intergranular contact pressure c CV A partial viscosity c CPA partial pressure c XLA bulk viscosity c XMXA,XMRA: axial and radial interaction force terms c of constituent A. c EIGEN: interface pressure. c BDF: body force. c XL,RE,RR,UMAX,VISA,DENA: characteristics properties c used for non-dimensional analysis. c XL ratio of length to radius of pipe. c RE Reynolds number c RR radius of pipe c UMAX velocity at entrance c VISA binder viscosity c DENA binder density c I,J: grid number c NPI,NPJ: number of nodes in I and J direction. c DT,DX,DR: time, axial and radial interval lengths c between nodes.
173
c IT,IP: number of time and pressure iterations. c RFF: relaxation factor for pressure iterations. c ERR: sum errors within the continuity equations of c constituents. c DEIGEN: pressure correction term. c SA,SB: arrays used in solving matrices. c CMAX: maximum concentration for monomodal mixtures. c c Definition of key subroutines. c CON: solves for new concentrations. c ITA: solves for new velocities of constituent A. cARRA YS: defines value of interaction force terms. c VISB: defines value of viscosity for constituent B.
. c INITIAL: sets initial conditions. c PRESSURE: solves for new pressures. c DEIGEN: determines correction to interface pressure. c EIGEN: solves for new interface pressure. c VCA: determines correction to velocities of c constituent A. c ERROR: determines error in continuity equations for c each node. c c In this program, A is considered the fluid constituent, c B is the fine particles and C is the coarse.
PROGRAM MAIN INCLUDE 'MAIN.CMM'
IF=lO JF=1
OPEN(IF,FILE='FIN.DAT' ,STATUS='OLD') OPEN(JF, FILE='FOUT.DAT',STATUS='OLD') CALL GETDATA CLOSE(IF,STATUS='SAVE')
CALL PRINTDA TA CALL NONDIMENSION CALL INITIAL CALL VISB CALL VISC
DO 5 K=l,IT
174
CALL ARRAYS DO 10 L=I,IP CALL CON CALL VISB CALL VISC CALL PRESSURE CALL ITA CALLITB CALL ITC JF(L.EQ.IP.AND.K.EQ.IT)OOTO 15 CALL ERROR JF(L.EQ.l.AND.K.EQ.l)THEN CALL RESID WRITE(JF,II)SUMERM END IF CALL SDEIGEN CALL SEIOEN CALL PRESSURE CALL VCA CALL VCB
10 CALL VCC CALL RESTART
5 CONTINUE
15 CALL RESID WRITE(JF,11)SUMERM
11 FORMAT(1X,E8.2) CALL DIMENSION CALL SVEL CALL PRINTRS
STOP END INCLUDE ' SUBROUTINES.FOR'
************************************************************
c MAIN.CMM contains the primary declaration statements used in all subroutines.
IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NODI NPI,NPJ COMMON ICNSTAI VXA(102,22),VRA(102,22),CONA(102,22),
175
&CPA( 1 02,22),CV A( 1 02,22),XMXA(1 02,22),XMRA( 1 02,22), &DA,ALA,XLA COMMON /CNSTB/ VXB( 1 02,22), VRB( 1 02,22),CONB(1 02,22),
&CPB(102,22),CVB(102,22),XMXB(102,22),XMRB(102,22), &DB,ALB,XLB COMMON /CNSTC/ VXC( 102,22), VRC( 1 02,22),CONC( 1 02,22),
&CPC(102,22),CVC(102,22),XMXC(102,22),XMRC(102,22), &D'C,ALC,XLC COMMON /CNSTNA/ VXNA(102,22),VRNA(102,22),CONNA(102,22) COMMON /CNSTNB/ VXNB( 102,22), VRNB(1 02,22),CONNB( 102,22) COMMON /CNSTNC/ VXNC(102,22),VRNC(102,22),CONNC(102,22) COMMON /SLURI VX(102,22),VR(102,22) COMMON /BC/ CONABC,CONBBC,CONCBC,vXBC,VRBC COMMON /CONST/ BTB,BTC,CMAX COMMON NIS/ VISA,DENB,DENC COMMON /XM/ DRAOB,DRAOC COMMON /ORI BDF COMMON /SP/ DX,DR,DT COMMON /ND/ XL,RE,RR,UMAX,DENA COMMON /OP/ RFF,IF,JF,IT,IP COMMON /PC/ EIOEN (1 02,22),DEIOEN (1 02,22),PA( 1 02,22),
&PB( 1 02,22),PC( 1 02,22),ER( 1 02,22) COMMON /COEF/ A,B,C,D,E,F COMMON /SENS/ AA,AB,AC,AD,AE,AF,BA,BB,BC,BD,BE,BF,
&CA,CB,CC,CD,CE,CF COMMON /SER/ SUMERM COMMON /CHR/ LINE1,LINE2,LINE3 COMMON /SOL/ SA(2000,81),SB(2000) CHARACTER *20 LINE 1 ,LINE2,LINE3 PARAMETER (ORAV=981.0) EXTERNAL DAX,DAR,DAXX,DARR,DAXR
************************************************************
c SUBROUTINES.FOR contains a list of all the subroutines requested by the MAIN program.
INCLUDE 'CON.FOR' INCLUDE'ITA.FOR' INCLUDE'ITB.FOR' INCLUDE 'ITC.FOR' INCLUDE 'ARRAYS.FOR' INCLUDE'INITIAL.FOR'
176
INCLUDE 'PRESSURE.FOR' INCLUDE 'PRINTDATAFOR' INCLUDE 'PRINTRS.FOR' INCLUDE'DEIGEN.FOR' INCLUDE 'EIGEN.FOR' INCLUDE 'COEFPC.FOR' INCLUDE'DIMENSION.FOR' INCLUDE 'NDIMENSION.FOR' INCLUDE 'GETDATAFOR' INCLUDE 'RESTART.FOR' INCLUDE 'VCA.FOR' INCLUDE 'VCB.FOR' INCLUDE'VCC.FOR' INCLUDE 'VEL.FOR' INCLUDE 'COEFCON.FOR' INCLUDE 'COEFVR.FOR' INCLUDE 'COEFVX.FOR' INCLUDE 'ERROR.FOR' INCLUDE 'ERR.FOR' INCLUDE'RESID.FOR' INCLUDE 'DAX.FOR' INCLUDE 'DAR.FOR' INCLUDE 'DAXX.FOR' INCLUDE'DARR.FOR' INCLUDE 'DAXR.FOR' INCLUDE'VISB.FOR' INCLUDE'VISC.FOR' INCLUDE ' SOL VER.FOR'
************************************************************
SUBROUTINE CON INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l CALL COEFCON(VXNC,VRNC,I,J) CONNC(I,J)=(A*CONC(I+l,J)+B*CONC(I-l,J)+
&C*CONC(I,J+ l)+CONC(I,J)+E*CONC(I,J-l »/D CALL COEFCON(VXNB,VRNB,I,J) CONNB(I,J)=(A *CONB(I+ 1,J)+B*CONB(I-l,J)+
&C*CONB(I,J+ l)+CONB(I,J)+E*CONB(I,J-l»)/D CALL COEFCON(VXNA,VRNA,I,J)
177
5 CONNA(I,J)=(A*CONA(I+l,J)+B*CONA(I-l,J)+ &C*CONA(I,J+l)+CONA(I,J)+E*CONA(I,J-l))1D
DO 10I=2,NPI-l DO 10 J=2,NPJ-l CT:.::CONNA(I,J)+CONNB(I,J)+CONNC(I,J) CONNA(I,J)=CONNA(I,J)/CT CONNB (I,J)=CONNB (I,J)/CT
10 CONNC(I,J)=CONNC(I,J)/CT
J=1 DO 15 I=2,NPI-l CONNC(I,J)=CONNC(I,J+ 1) CONNB (I,J)=CONNB (I,J+ 1)
15 CONNA(I,J)=CONNA(I,J+l)
J=NPJ DO 20 I=2,NPI-l CONNC(I,J)=CONNC(I,J-l ) CONNB(I,J)=CONNB(I,J-l )
20 CONNA(I,J)=CONNA(I,J-l)
I=NPI DO 25 J=I,NPJ CONNC(I,J)=CONNC(I-l,J) CONNB(I,J)=CONNB(I-l,J)
25 CONNA(I,J)=CONNA(I-l,J)
RETURN END
************************************************************
SUBROUTINE ITA INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l CALL COEFVR(VXNA,VRNA,CONNA,CPA,
&CV A,XMRA,DA,ALA,I,J) VRNA(I,J)=(A*VRA(I+l,J)+B*VRA(I-l,J)+
&C*VRA(I,J+ 1)+ VRA(I,J)+E*VRA(I,J-l)+F)/D CALL COEFVX(VXNA,VRNA,CONNA,CPA,
178
&CV A,XMXA,DA,ALA,I,J) 5 VXNA(I,J)=(A*VXA(I+l,J)+B*VXA(I-1,J)+
&C*VXA(I,J+l)+VXA(I,J)+E*VXA(I,J-l)+F)1D
J=1 DO 10 I=2,NPI-l
10 VXNA(I,J)=VXNA(I,J+ 1)
I=NPI DO 15 J=1,NPJ-l VRNA(I,J)=VRNA(I-l,J)
15 VXNA(I,J)=VXNA(I-1,J)
RETURN END
************************************************************
SUBROUTINE ITB INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-1 DO 5 J=2,NPJ-1 CALL COEFVR(VXNB,VRNB,CONNB,CPB,
&CVB,XMRB,DB,ALB,I,J) VRNB(I,J)=(A*VRB(I+1,J)+B*VRB(I-I,J)+
&C*VRB(I,J+ 1)+ VRB(I,J)+E*VRB(I,J-1 )+F)ID CALL COEFVX(VXNB, VRNB,CONNB,CPB,
&CVB,XMXB,DB,ALB,I,J) 5 VXNB(I,J)=(A *VXB(I+ 1 ,J)+B*VXB(I-I,J)+
&C*VXB(I,J+ 1)+ VXB(I,J)+E*VXB(I,J-1)+F)1D
J=1 DO 10 I=2,NPI-1
10 VXNB(I,J)=VXNB(I,J+l)
I=NPI DO 15 J=1,NPJ-1 VRNB(I,J)=VRNB(I-1,J)
15 VXNB(I,J)=VXNB(I-1,J)
RETURN END
179
************************************************************
SUBROUTINE ITC INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l CALL COEFVR(VXNC,VRNC,CONNC,CPC,
&CVC,XMRC,DC,ALC,I,J) VRNC(I,J)=(A*VRC(I+l,J)+B*VRC(I-l,J)+
&C*VRC(I,J+l)+VRC(I,J)+E*VRC(I,J-l)+F)1D CALL COEFVX(VXNC,VRNC,CONNC,CPC,
&CVC,XMXC,DC,ALC,I,J) 5 vxnc(I,J)=(A *VXC(I+ 1 ,J)+B*VXC(I-l ,1)+
&C*VXC(I,J+l)+VXC(I,J)+E*VXC(I,J-l)+F)1D
J=1 DO 10 I=2,NPI-l
10 VXNC(I,J)=VXNC(I,J+l)
I=NPI DO 15 J=I,NPJ-l VRNC(I,J)=VRNC(I-l,J)
15 VXNC(I,J)=VXNC(I-l,J)
RETURN END
************************************************************
SUBROUTINE ARRAYS INCLUDE 'MAIN.CMM'
CALL SXMXC CALL SXMXB CALL SXMXA CALL SXMRC CALL SXMRB CALL SXMRA
RETURN END INCLUDE 'XMXC.FOR'
180
INCLUDE 'XM.XB.FOR' INCLUDE , XM.XA. FOR ' INCLUDE 'XMRC.FOR' INCLUDE 'XMRB.FOR' INCLUDE 'XMRA.FOR'
************************************************************
SUBROUTINE SXM.XA INCLUDE 'MAIN.CMM'
DO 5 I=I,NPI DO 5 J=I,NPJ
5 XMXA(I,J)=-XMXB(I,J)-XMXC(I,J)
RETURN END
************************************************************
SUBROUTINE SXM.XB INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l
5 XMXB(I,J)=-DRAGB*CONB(I,J)*(VXB(I,J)-VXA(I,J)) &-EIGEN (I,J)*DAX(CONB,I,J)
RETURN END
************************************************************
SUBROUTINE SXM.XC INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l
5 XMXB(I,J)=-DRAGC*CONB(I,J)*(VXB(I,J)-VXA(I,J)) &-EIGEN (I,J)*DAX(CONB,I,J)
RETURN END
181
************************************************************
SUBROUTINE SXMRA INCLUDE 'MAIN.CMM'
DO 5 I=I,NPI DO 5 J=I,NPJ
5 XMRA(I,J)=-XMRB(I,J)-XMRC(I,J)
RETURN END
************************************************************
SUBROUTINE SXMRB INCLUDE'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l
5 XMRB(I,J)=-DRAGB*CONB(I,J)*(VRB(I,J)-VRA(I,J)) &-EIGEN (I,J)*DAR(CONB,I,J)
RETURN END
************************************************************
SUBROUTINE SXMRC INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l
5 XMRB(I,J)=-DRAGC*CONB(I,J)*(VRB(I,J)-VRA(I,J» &-EIGEN(I,J)*DAR(CONB,I,J)
RETURN END
************************************************************
SUBROUTINE INITIAL INCLUDE 'MAIN.CMM'
182
DO 5 I=I,NPI DO 5 J=I,NPJ CONA(I,J)=CONABC CONB(I,J)=CONBBC CONC(I,J)=CONCBC VRA(I,J)= VRBC VRB(I,J)=VRBC VRC(I,J)= VRBC VXA(I,J)= VXBC VXB(I,J)=VXBC VXC(I,J)=VXBC
5 EIGEN(I,J)=O.
J=NPJ DO 10 I=I,NPI VXA(I,J)=O. VXB(I,J)=O. VXC(I,J)=O. VRA(I,J)=O. VRB (I,J)=O.
10 VRC(I,J)=O.
DO 15 I=I,NPI DO 15 J=I,NPJ CONNA(I,J)=CONA(I,J) CONNB(I,J)=CONB(I,J) CONNC(I,J)=CONC(I,J) VXN A (I,J)= VXA(I,J) VXNB(I,J)= VXB(I,J) VXNC(I,J)= VXC(I,J) VRN A(I,J)= VRA(I,J) VRNB (I,J)=VRB (I,J)
15 VRNC(I,J)=VRC(I,J)
RETURN END
************************************************************
SUBROUTINE PRESSURE INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l
183
DO 5 J=2,NPJ-1 XJ=J-1 PC(I,J)=EIGEN (I,J)+BTC+XLC*CONC(I,J)*
&(DAX(VXC,I,J)+VRC(I,J)J(XJ*DR)+DAR(VRC,I,J» &-2. * ALC*(DAXX(CONC,I,J)+DAR(CONC,I,J)J(XJ*DR) &+DARR(CONC,I,J» PB(I,J)=EIGEN (I,J)+BTB+XLB*CONB(I,J)*
&(DAX(VXB,I,J)+ VRB(I,J)J(XJ*DR)+DAR(VRB,I,J» &-2. * ALB*(DAXX(CONB,I,J)+DAR(CONB,I,J)J(XJ*DR) &+DARR(CONB,I,J» PA(I,J)=EIGEN(I,J)+XLA*CONA(I,J)*
&(DAX(VXA,I,J)+ VRA(I,J)J(XJ*DR)+DAR(VRA,I,J) &-2. * ALA *(DAXX(CONA,I,J)+DAR(CONA,I,J)J(XJ*DR) &+DARR(CONA,I,J» CPC(I,J)=CONNC(I,J)*PC(I,J) CPB(I,J)=CONNB(I,J)*PB(I,J)
5 CPA(I,J)=CONNA(I,J)*PA(I,J)
J=1 DO 10 I=2,NPI-1 PA(I,J)=PA(I,J+1) PB(I,J)=PB(I,J+ 1) PC(I,J)=PC(I,J + 1) CPA(I,J)=CONNA(I,J)*PA(I,J) CPB(I,J)=CONNB(I,J)*PB(I,J)
10 CPC(I,J)=CONNC(I,J)*PC(I,J)
J=NPJ DO 15 I=2,NPI-l P A (I,J)=P A(I,J -1) PB(I,J)=PB(I,J-l) PC(I,J)=PC(I,J -1) CPA(I,J)=CONNA(I,J)*PA(I,J) CPB(I,J)=CONNB(I,J)*PB(I,J)
15 CPC(I,J)=CONNC(I,J)*PC(I,J)
I=NPI DO 20 J=l,NPJ PA(I,J)=PA(I-l,J) PB(I,J)=PB(I-I,J) PC(I,J)=PC(I-l,J) CPA(I,J)=CONNA(I,J)*PA(I,J) CPB(I,J)=CONNB(I,J)*PB(I,J)
184
20 CPC(I,J)=CONNC(I,J)*PC(I,J)
RETURN END
************************************************************
SUBROUTINE PRINTDATA INCLUDE 'MAIN.CMM'
WRlTE(JF,II) LINEI WRlTE(JF,II) LINE2 WRlTE(JF,II) LINE3
WRITE(JF,22) NPI,NPJ WRITE(JF,33) DX,DR,DT WRlTE(JF,44) IT,IP WRITE(JF,55) DENA,DENB,DENC WRITE(JF,66) CMAX,VISA WRITE(JF,66) BTB,BTC WRITE(JF,55) ALA,ALB,ALC WRITE(JF,55) XLA,XLB,XLC WRITE(JF,55) CONABC,CONBBC,CONCBC WRITE(JF,66) VXBC,VRBC WRlTE(JF,66) DRAGB,DRAGC WRITE(JF,77) RFF
11 FORMA T(A) 22 FORMAT(1X,I3,T6,I3) 33 FORMAT(IX,F7.3,TlO,F7.3,TI8,F15.11) 44 FORMAT(lX,I5,T8,I5) 55 FORMAT(lX,F8.3,Tll ,F8.3,T21 ,F8.3) 66 FORMAT(lX,F8.3,Tll,F8.3) 77 FORMAT(1X,F15.4)
RETURN END
************************************************************
SUBROUTINE PRINTRS INCLUDE 'MAIN.CMM'
185
WRITE(JF,II) IT
VFIA=O. VFIB=O. VFIC=O. VFOA=O. VFOB=O. VFOC=O. I=NPI DO 5 J=2,NPJ-l XJ=J-l VFOA=6.2832*XJ*DR **2*CONA(I,J)*VXA(I,J)+ VFOA VFOB=6.2832*XJ*DR **2*CONB(I,J)*VXB(I,J)+ VFOB
5 VFOC=6.2832*XJ*DR **2*CONC(I,J)*VXC(I,J)+ VFOC
1=1 DO 10 J=2,NPJ-l XJ=J-l VFIA=6.2832*XJ*DR **2*CONA(I,J)*VXA(I,J)+ VFIA VFIB=6.2832*XJ*DR **2 *CONB(I,J)*VXB(I,J)+ VFIB
10 VFIC=6.2832*XJ*DR **2*CONC(I,J)*VXC(I,J)+ VFIC
WRITE(JF,22) VFIA,VFIB,VFIC WRlTE(JF,22) VFOA,VFOB,VFOC
DO 15 J=I,NPJ XJ=J-l
15 WRITE(JF,33) XJ,CONA(1,J),CONA(20,J),CONA(40,J), &CONA(60,J),CONA(80,J),CONA(100,J)
DO 20 J=I,NPJ XJ=J-l
20 WRlTE(JF,33) XJ,PA(I,J),PA(20,J),PA(40,J), &PA(60,J),PA(80,J),PA( 1 OO,J)
DO 25 J=I,NPJ XJ=J-l
25 WRITE(JF,33) XJ,VXA(I,J),VXA(20,J),VXA(40,J), &VXA(60,J),VXA(80,J),VXA(100,J)
DO 30 J=I,NPJ XJ=J-l
30 WRlTE(JF,33) XJ,VRA(I,J),VRA(20,J),VRA(40,J),
186
&VRA(60,J),VRA(80,J),VRA(100,J)
DO 35 J=I,NPJ XJ=J-l
35 WRITE(JF,33) XJ,CONB(1,J),CONB(20,J),CONB(40,J), &CONB(60,J),CONB(80,J),CONB(100,J)
DO 40 J=I,NPJ XJ=J-l
40 WRITE(JF,33) XJ,PB(1,J),PB(20,J),PB(40,J), &PB(60,J),PB(80,J),PB(100,J)
DO 45 J=I,NPJ XJ=J-l
45 WRITE(JF,33) XJ,VXB(I,J),VXB(20,J),VXB(40,J), &VXB(60,J),VXB(80,J),VXB(100,J)
DO 50 J=I,NPJ XJ=J-I
50 WRITE(JF,33) XJ,VRB(1,J),VRB(20,J),VRB(40,J), &VRB(60,J),VRB(80,J),VRB(100,J)
DO 55 J=l,NPJ XJ=J-l
55 WRITE(JF,33) XJ,CONC(1,J),CONC(20,J),CONC(40,J), &CONC(60,J),CONC(80,J),CONC(100,J)
DO 60 J=l,NPJ XJ=J-l
60 WRITE(JF,33) XJ,PC(1,J),PC(20,J),PC(40,J), &PC( 60,J),PC(80,J),PC( 100,J)
DO 65 J=l,NPJ XJ=J-l
65 WRITE(JF,33) XJ,VXC(1,J),VXC(20,J),VXC(40,J), & VXC( 60,J), VXC(80,J), VXC( 100,J)
DO 70 J=l,NPJ XJ=J-l
70 WRITE(JF,33) XJ,VRC(1,J),VRC(20,J),VRC(40,J), &VRC(60,J),VRC(80,J),VRC(100,J)
DO 88 J=l,NPJ
187
XJ=J-l 88 WRITE(JF,33) XJ,VX(1,J),VX(20,J),VX(40,J),
&VX(60,J),VX(80,J),VX(100,J)
DO 75 J=I,NPJ XJ=J-l
75 WRITE(JF,33) XJ,ER(1,J),ER(20,J),ER(40,J), &ER(60,J),ER(80,J),ER(100,J)
11 FORMAT(IX,14) 22 FORMA T(2(1X,E 1004)) 33 FORMAT(7 (1 X,E 1004))
RETURN END
************************************************************
SUBROUTINE SDEIGEN INCLUDE 'MAIN.CMM'
M=4*(NPJ-2)+ 1 MD=M/2+1 N=(NPI-2)*(NPJ-2) SP=(2. *DR)**2
DO 5 I=I,NPI DO 5 J=I,NPJ
5 DEIGEN(I,J)=O.
DO 10 I=I,N SB(I)=O. DO 10 J=I,M
10 SA(I,J)=O.
1=2 J=2 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT
188
SA(K,MD)=BC+BE+BB SA(K,MD+ 1)=AB SA(K,MD+2)=AE SA(K,K2)=AA SA(K,K4)=AC
1=3 J=2 CALL COEFPC(l,J) K=(l-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+BC+BE+BB SA(K,MD+ 1 )=AB SA(K,MD+ 2)=AE SA(K,Kl)=BA SA(K,K2)=AA SA(K,K4)=AC
J=2 DO 15 I=4,NPI-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+BC+BE+BB SA(K,MD+ l)=AB SA(K,MD+ 2)=AE SA(K,Kl)=BA SA(K,K2)=AA SA(K,K3)=BD
15 SA(K,K4)=AC
I=NPI-2 J=2 CALL COEFPC(l,J) K=(l-2)*(NPJ-2)+J-l
189
K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+BC+BE+BB SA(K,MD+ 1 )=AB SA(K,MD+2)=AE SA(K,K1)=BA SA(K,K2)=AA+AC SA(K,K3)=BD
I=NPI-1 J=2 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-1 Kl=MD-(NPJ-2) K3=MD-2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+BE+AA+AC+BB SA(K,MD+ l)=AB SA(K,MD+2)=AE SA(K,Kl)=BA+BC SA(K,K3)=BD
1=2 J=3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AF+BC+BE SA(K,MD-l )=BB+BF SA(K,MD+ 1 )=AB SA(K,MD+2)=AE SA(K,K2)=AA SA(K,K4)=AC
1=3 J=3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l
190
K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-1 )=BB+BF SA(K,MD+1)=AB SA(K,MD+2)=AE SA(K,Kl)=BA SA(K,K2)=AA SA(K,K4)=AC
J=3 DO 20 I=4,NPI-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-1 K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-1)=BB+BF SA(K,MD+1)=AB SA(K,MD+ 2)=AE SA(K,K1)=BA SA(K,K2)=AA SA(K,K3)=BD
20 SA(K,K4)=AC
I=NPI-2 J=3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-1 K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AO+AF+BC+BE SA(K,MD-1 )=BB+BF SA(K,MD+ l)=AB
191
SA(K,MD+2)=AE SA(K,Kl)=BA SA(K,K2)=AA+AC SA(K,K3)=BD
I=NPI-l J=3 CALL COEFPC(I,J) K=(1-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ -2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+AA+AC+BE SA(K,MD-l)=BB+BF SA(K,MD+ 1)=AB SA(K,MD+ 2)=AE SA(K,Kl)=BA+BC SA(K,K3)=BD
1=2 DO 30 J=4,NPJ-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l K 1 =MD-(NPJ-2) K2=MD+(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AF+BC+BE SA(K,MD-l )=BB SA(K,MD+ l)=AB SA(K,MD-2)=BF SA(K,MD+ 2)=AE SA(K,K2)=AA
30 SA(K,K4)=AC
1=3 DO 33 J=4,NPJ-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2)
192
K3=MD-2*(NPJ-2) K4=MD +2* (NPJ -2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-1 )=BB SA(K,MD+ 1 )=AB SA(K,MD-2)=BF SA(K,MD+2)=AE SA(K,Kl)=BA SA(K,K2)=AA
33 SA(K,K4)=AC
DO 35 I=4,NPI-3 DO 35 J=4,NPJ-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ -2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-1 )=BB SA(K,MD+ 1)=AB SA(K,MD-2)=BF SA(K,MD+2)=AE SA(K,K1)=BA SA(K,K2)=AA SA(K,K3)=BD
35 SA(K,K4)=AC
I=NPI-2 DO 40 J=4,NPJ-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-l )=BB SA(K,MD+ 1 )=AB
193
SA(K,MD-2)=BF SA(K,MD+2)=AE SA(K,K1)=BA SA(K,K2)=AA+AC
40 SA(K,K3)=BD
I=NPI-1 DO 45 J=4,NPJ-3 CALL COEFPC(I,J) K=(1-2)*(NPJ-2)+J-1 K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+AA+AC+BE SA(K,MD-1 )=BB SA(K,MD+ 1 )=AB SA(K,MD-2)=BF SA(K,MD+2)=AE SA(K,K1)=BA+BC
45 SA(K,K3)=BD
1=2 J=NPJ-2 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l K1=MD-(NPJ-2) K2=MD+(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AF+BC+BE SA(K,MD-l )=BB SA(K,MD+l)=AB+AE SA(K,MD-2)=BF SA(K,K2)=AA SA(K,K4)=AC
1=3 J=NPJ-2 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l K 1 =MD-(NPJ-2)
194
K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-1)=BB SA(K,MD+1)=AB+AE SA(K,MD-2)=BF SA(K,K1)=BA SA(K,K2)=AA SA(K,K4)=AC
J=NPJ-2 DO 50 I=4,NPI-3 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-1 K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-1 )=BB SA(K,MD+ l)=AB+AE SA(K,MD-2)=BF SA(K,K1)=BA SA(K,K2)=AA SA(K,K3)=BD
50 SA(K,K4)=AC
I=NPI-2 J=NPJ-2 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-1 K1=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+BE SA(K,MD-l)=BB SA(K,MD+ 1 )=AB+AE SA(K,MD-2)=BF
195
SA(K,Kl)=BA SA(K,K2)=AA+AC SA(K,K3)=BD
I=NPI-l J=NPJ-2 CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K3=MD-2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BE+AA+AC SA(K,MD-l )=BB SA(K,MD+ 1)=AB+AE SA(K,MD-2)=BF SA(K,Kl)=BA+BC SA(K,K3)=BD
1=2 J=NPJ-l CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K4=MD +2* (NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AF+BC+AB SA(K,MD-l )=BB SA(K,MD-2)=BF SA(K,K2)=AA SA(K,K4)=AC
1=3 J=NPJ-l CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ -2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+AB SA(K,MD-l )=BB
196
SA(K,MD-2)=BF SA(K,Kl)=BA SA(K,K2)=AA SA(K,K4)=AC
J=NPJ-l DO 55 I=4,NPI-3 CALL COEFPC(l,J) K=(l-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(l,J)IDT SA(K,MD)=AD+AF+BC+AB SA(K,MD-l )=BB SA(K,MD-2)=BF SA(K,Kl)=BA SA(K,K2)=AA SA(K,K3)=BD
55 SA(K,K4)=AC
I=NPI-2 J=NPJ-l CALL COEFPC(I,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2) K2=MD+(NPJ-2) K3=MD-2*(NPJ-2) K4=MD+2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+BC+AB SA(K,MD-l)=BB SA(K,MD-2)=BF SA(K,Kl)~BA
SA(K,K2)=AA+AC SA(K,K3)=BD
I=NPI-l J=NPJ-l CALL COEFPC(l,J) K=(I-2)*(NPJ-2)+J-l Kl=MD-(NPJ-2)
197
K3=MD-2*(NPJ-2) SB(K)=-SP*ER(I,J)IDT SA(K,MD)=AD+AF+AA+AC+AB SA(K,MD-1 )=BB SA(K,MD-2)=BF SA(K,K1)=BA+BC SA(K,K3)=BD
CALL SOLVER(SA,SB,M,MD,N)
DO 60 I=2,NPI-1 DO 60 J=2,NPJ-1 K=(I-2)*(NPJ-2)+J-1
60 DEIGEN(I,J)=SB(K)
J=l DO 65 I=2,NPI-1
65 DEIGEN(I,J)=DEIGEN(I,J+1)
J=NPJ DO 70 1=2,NPI-1
70 DEIGEN (I,J)=DEIGEN (I,J -1)
I=NPI DO 75 J=l,NPJ
75 DEIGEN(I,J)=DEIGEN(I-1,J)
DO 80 I=l,NPI DO 80 J=l,NPJ
80 DEIGEN (I,J)=RFF*DEIGEN (I,J)
RETURN END
************************************************************
SUBROUTINE SEIGEN INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-1 DO 5 J=2,NPJ-1
5 EIGEN (I,J)=EIGEN (I,J)+DEIGEN (I,J)
198
RETURN END
************************************************************
SUBROUTINE COEFPC(I,J) INCLUDE'MAIN.CMM'
XJ=J-l DIVXA=1.+4.*CVA(I,J)*DT/
&(CONNA(I,J)*RE*DA *(XL *DX)**2) &+2. *CVA(I,J)*DT/ &(CONNA(I,J)*RE*DA *DR **2) DIVRA=I.+2.*CVA(I,J)*DT/
&(CONNA(I,J)*RE*DA *(XL *DX)**2) &+4. *CV A(I,J)*DT/ &(CONNA(I,J)*RE*DA *DR **2) &+2. *CV A(I,J)*DT/ &(CONN A(I,J)*RE*DA *(DR *XJ)**2) DIVXA1=1.+4.*CVA(I-l,J)*DT/
&(CONNA(I-l ,J)*RE*DA *(XL*DX)**2) &+2.*CVA(I-l,J)*DT/ &(CONNA(I-l,J)*RE*DA*DR**2) DIVRA1=1.+2.*CVA(I,J-l)*DT/
&(CONNA(I,J-l)*RE*DA *(XL*DX)**2) &+4.*CVA(I,J-l)*DT/ &(CONNA(I,J-l)*RE*DA *DR **2) &+2.*CVA(I,J-l)*DT/ &(CONNA(I,J-l)*RE*DA*(DR*XJ)**2) DIVXA2=1.+4.*CVA(I+l,J)*DT/
&(CONNA(I + 1 ,J)*RE*DA *(XL *DX)**2) &+2.*CVA(I+1 ,J)*DT/ &(CONNA(I + 1 ,J)*RE*DA *DR **2) DIVRA2=1.+2.*CVA(I,J+l)*DT/
&(CONNA(I,J+l)*RE*DA*(XL*DX)**2) &+4. *CVA(I,J+ l)*DT/ &(CONNA(I,J+l)*RE*DA*DR**2) &+2. *CVA(I,J+ 1)*DT/ &(CONNA(I,J+ 1)*RE*DA *(DR*XJ)**2) DIVXB= 1. +4. *CVB(I,J)*DT/
&(CONNB(I,J)*RE*DB*(XL *DX)**2) &+2.*CVB(I,J)*DT/ &(CONNB(I,J)*RE*DB *DR **2)
199
DIVRB=1.+2.*CVB(I,J)*DT/ &(CONNB(I,J)*RE*DB*(XL *DX)**2) &+4. *CVB(I,J)*DT/ &(CONNB(I,J)*RE*DB *DR **2) &+2. *CVB(I,J)*DT / &(CONNB(I,J)*RE*DB*(DR *XJ)**2) DIVXBl=1.+4.*CVB(I-l,J)*DT/
&(CONNB(l-l ,J)*RE*DB*(XL *DX)**2) &+2.*CVB(l-1,J)*DT/ &(CONNB(I-l ,J)*RE*DB*DR **2) DIVRBl=1.+2.*CVB(l,J-l)*DT/
&(CONNB(I,J-l)*RE*DB*(XL*DX)**2) &+4. *CVB(l,J-l)*DT/ &(CONNB(I,J-l)*RE*DB*DR **2) &+2.*CVB(l,J-l)*DT/ &(CONNB(I,J-l)*RE*DB*(DR*XJ)**2) DIVXB2=1.+4.*CVB(I+l,J)*DT/
&(CONNB(l+ 1 ,J)*RE*DB*(XL *DX)**2) &+2.*CVB(I+l,J)*DT/ &(CONNB(I+ 1 ,J)*RE*DB*DR **2) DIVRB2=1.+2.*CVB(I,J+l)*DT/
&(CONNB(I,J+l)*RE*DB*(XL*DX)**2) &+4.*CVB(I,J+l)*DT/ &(CONNB(l,J+l)*RE*DB*DR**2) &+2. *CVB(I,J+ l)*DT/ &(CONNB(I,J+l)*RE*DB*(DR*XJ)**2) DIVXC=1.+4.*CVC(l,J)*DT/
&(CONNC(I,J)*RE*DC*(XL *DX)**2) &+2.*CVC(I,J)*DT/ &(CONNC(I,J)*RE*DC*DR**2) DIVRC=1.+2.*CVC(I,J)*DT/
&(CONNC(I,J)*RE*DC*(XL *DX)**2) &+4. *CVC(I,J)*DT/ &(CONNC(I,J)*RE*DC*DR **2) &+2. *CVC(I,J)*DT/ &(CONNC(I,J)*RE*DC*(DR *XJ)**2) DIVXCl=1.+4.*CVC(I-l,J)*DT/
&(CONNC(I-l ,J)*RE*DC*(XL *DX)**2) &+2.*CVC(I-l,J)*DT/ &(CONNC(I-l ,J)*RE*DC*DR **2) DIVRCl=1.+2.*CVC(I,J-l)*DT/
&(CONNC(I,J-l)*RE*DC*(XL*DX)**2) &+4.*CYC(I,J-l)*DT/
200
&(CONNC(I,J-l)*RE*DC*DR**2) &+2. *CVC(I,J-l)*DTI &(CONNC(I,J-1)*RE*DC*(DR *XJ)**2) DIVXC2=1.+4.*CVC(I+1,J)*DTI
&(CONNC(I+ 1 ,J)*RE*DC*(XL*DX)**2) &+2.*CVC(I+1,J)*DTI &(CONNC(I+1,J)*RE*DC*DR**2) DIVRC2=1.+2.*CVC(I,J+1)*DTI
&(CONNC(I,J+1)*RE*DC*(XL*DX)**2) &+4. *CVC(I,J+ l)*DTI &(CONNC(I,J+1)*RE*DC*DR**2) &+2. *CVC(I,J+ l)*DTI &(CONNC(I,J+ 1)*RE*DC*(DR*XJ)**2) PDVXAA=-CONN A(I+ 1 ,J)/(CONNA(I,J)*DIVXA) PDVRAA=-CONNA(I,J+1)/(CONNA(I,J)*DIVRA) PDVXA1A=-CONNA(I,J)/(CONNA(I-1,J)*DIVXA1) IF(I.EQ.NPI-1 )THEN PDVXA2A=PDVXAA ELSE PDVXA2A=-CONN A(I+2,J)/(CONNA(I+ 1 ,J)*DIVXA2) END IF PDVRA1A=-CONNA(I,J)/(CONNA(I,J-1)*DIVRA1) PDVRA2A=-CONNA(I,J+2)/(CONNA(I,J+ 1)*DIVRA2) PDVXAB=CONNA(I-1,J)/(CONNA(I,J)*DIVXA) PDVRAB=CONNA(I,J-l)/(CONNA(I,J)*DIVRA) PDVXAIB=CONNA(I-2,J)/(CONNA(I-1,J)*DIVXA1) IF(I.EQ.NPI-l )THEN PDVXA2B=PDVXAB ELSE PDVXA2B=CONNA(I,J)/(CONNA(I + 1 ,J)*DIVXA2) END IF PDVRA1B=CONNA(I,J-2)/(CONNA(I,J-1)*DIVRA1) PDVRA2B=CONNA(I,J)/(CONNA(I,J+ 1)*DIVRA2) PDVXBA=-CONNB(I+ 1,J)/(CONNB(I,J)*DIVXB) PDVRBA=-CONNB(I,J+ 1 )/(CONNB(I,J)*DIVRB) PDVXB 1A=-CONNB(I,J)/(CONNB(I-1 ,J)*DIVXB 1) IF(I.EQ.NPI-l )THEN PDVXB2A=PDVXBA ELSE PDVXB2A=-CONNB(I+2,J)/(CONNB(I+1,J)*DIVXB2) END IF PDVRB1A=-CONNB(I,J)/(CONNB(I,J-1)*DIVRB1) PDVRB2A=-CONNB(I,J+2)/(CONNB(I,J+1)*DIVRB2)
201
PDVXBB=CONNB(I-1,J)/(CONNB(I,J)*DIVXB) PDVRBB=CONNB(I,J-1)/(CONNB(I,J)*DIVRB) PDVXB1B=CONNB(I-2,J)/(CONNB(I-1,J)*DIVXB1) IF(I.EQ.NPI-l)THEN PDVXB2B=PDVXBB ELSE PDVXB2B=CONNB(I,J)/(CONNB(I+ 1 ,J)*DIVXB2) END IF PDVRB 1B=CONNB(I,J-2)/(CONNB(I,J-1)*DIVRB 1) PDVRB2B=CONNB(I,J)/(CONNB(I,J + 1 )*DIVRB2) PDVXCA=-CONNC(I+ 1 ,J)/(CONNC(I,J)*DIVXC) PDVRCA=-CONNC(I,J + 1 )/(CONNC(I,J)*DIVRC) PDVXC1A=-CONNC(I,J)/(CONNC(I-1 ,J)*DIVXC1) IF(I.EQ .NPI -1 )THEN PDVXC2A=PDVXCA ELSE PDVXC2A=-CONNC(I+2,J)/(CONNC(I+1,J)*DIVXC2) END IF PDVRC1A=-CONNC(I,J)/(CONNC(I,J-1)*DIVRC1) PDVRC2A=-CONNC(I,J+2)/(CONNC(I,J+1)*DIVRC2) PDVXCB=CONNC(I-1,J)/(CONNC(I,J)*DIVXC) PDVRCB=CONNC(I,J-1 )/(CONNC(I,J)*DIVRC) PDVXCIB=CONNC(I-2,J)/(CONNC(I-1,J)*DIVXCl) IF(I.EQ.NPI-1 )THEN PDVXC2B=PDVXCB ELSE PDVXC2B=CONN C(I,J)/(CONNC(I+ 1 ,J)*DIVXC2) END IF PDVRCIB=CONNC(I,J-2)/(CONNC(I,J-1)*DIVRCl) PDVRC2B=CONNC(I,J)/(CONNC(I,J+ 1 )*DIVRC2)
AA=2. *DX*XL*DAX(CONA,I,J)*PDVXAA/DA &+2. *DX*XL *DAX(CONB,I,J)*PDVXBA/DB &+2. *DX*XL*DAX(CONC,I,J)*PDVXCA/DC AB=(2. *DR *DAR(CONA,I,J)+2. *CONNA(I,J)IXJ)
&*PDVRAA/DA &+(2. *DR *DAR(CONB,I,J)+2. *CONNB(I,J)IXJ) &*PDVRBA/DB &+(2.*DR*DAR(CONC,I,J)+2.*CONNC(I,J)IXJ) &*PDVRCA/DC AC=CONNA(I,J)*PDVXA2A/DA
&+CONNB(I,J)*PDVXB2A/DB &+CONNC(I,J)*PDVXC2A/DC
202
AD=-CONNA(I,J)*PDVXAIA/DA &-CONNB(I,J)*PDVXB lA/DB &-CONNC(I,J)*PDVXC1 A/DC AE=CONNA(I,J)*PDVRA2A/DA
&+CONNB(I,J)*PDVRB2A/DB &+CONNC(I,J)*PDVRC2A/DC AF=-CONNA(I,J)*PDVRA 1 A/DA
&-CONNB(I,J)*PDVRB lA/DB &-CONNC(I,J)*PDVRB 1 C/DC BA=2.*DX*XL*DAX(CONA,I,J)*PDVXAB/DA
&+2. *DX*XL *DAX(CONB,I,J)*PDVXBB/DB &+2. *DX*XL*DAX(CONC,I,J)*PDVXCB/DC BB=(2. *DR *DAR(CONA,I,J)+2. *CONNA(I,J)IXJ)
&*PDVRAB/DA &+(2. *DR *DAR(CONB,I,J)+2. *CONNB(I,J)IXJ) &*PDVRBB/DB &+(2. *DR *DAR(CONC,I,J)+2. *CONNC(I,J)IXJ) &*PDVRCB/DC BC=CONNA(I,J)*PDVXA2B/DA
&+CONNB(I,J)*PDVXB2B/DB &+CONNC(I,J)*PDVXC2B/DC BD=-CONNA(I,J)*PDVXA 1 B/DA
&-CONNB(I,J)*PDVXB 1 B/DB &-CONNC(I,J)*PDVXC1 B/DC BE=CONNA(I,J)*PDVRA2B/DA
&+CONNB(I,J)*PDVRB2B/DB &+CONNC(I,J)*PDVRC2B/DC BF=-CONN A(I,J)*PDVRA 1 B/DA
&-CONNB(I,J)*PDVRB IB/DB &-CONNC(I,J)*PDVRCI C/DC
RETURN END
************************************************************
SUBROUTINE DIMENSION INCLUDE 'MAIN.CMM'
DR=DR*RR
DO 5 I=l,NPI DO 5 J=l,NPJ
203
VXA(I,J)= VXA(I,J)*UMAX VRA(I,J)= VRA(I,J)*UMAX VXB(I,J)=VXB(I,J)*UMAX VRB(I,J)= VRB(I,J)*UMAX VXC(I,J)= VXC(I,J)*UMAX VRC(I,J)= VRC(I,J)*UMAX PA(I,J)=PA(I,J)*(DENA *UMAX**2) PB(I,J)=PB(I,J)*(DENA *UMAX**2)
5 PC(I,J)=PC(I,J)*(DENA *UMAX**2)
RETURN END
************************************************************
SUBROUTINE GETDA TA INCLUDE 'MAIN.CMM'
READ(IF,II) LINEI READ(lF,ll) LINE2 READ(lF,II) LINE3
READ(lF,22) NPI,NPJ READ(IF,33) DX,DR,DT READ(lF,44) IT,IP READ(lF,55) DENA,DENB,DENC READ(IF,66) CMAX,VISA READ(lF,66) BTB,BTC READ(IF,55) ALA,ALB,ALC READ(IF,55) XLA,XLB,XLC READ(lF,55) CONABC,CONBBC,CONCBC READ(lF,66) VXBC,VRBC READ(IF,66) DRAGB,DRAGC READ(IF,77) RFF
11 FORMA T(A) 22 FORMAT(IX,I3,T6,I3) 33 FORMAT(1X,F7.3,TlO,F7.3,TI8,FI5.11) 44 FORMAT(1X,I5,T8,I5) 55 FORMAT(1X,F8.3,Tl1,F8.3,T21,F8.3) 66 FORMAT(1X,F8.3,Tl1,F8.3) 77 FORMAT(1X,FI5.4)
204
RETURN END
************************************************************
SUBROUTINE RESTART INCLUDE 'MAIN.CMM'
DO 5 I=l,NPI DO 5 J=l,NPJ CONA(I,J)=CONNA(I,J) CONB (I,J)=CONNB (I,J) CONC(I,J)=CONNC(I,J) VXA(I,J)=VXNA(I,J) VRA(I,J)= VRN A(I,J) VXB(I,J)= VXNB (I,J) VRB (I ,J)= VRNB (I,J) VXC(I,J)= VXNC(I,J)
5 VRC(I,J)=VRNC(I,J)
RETURN END
************************************************************
SUBROUTINE VCA INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l XJ=J-l DIVX=1.+4. *CV A(I,J)*DT/(CONNA(I,J)*RE*DA *(DX*XL)**2)
&+2. *CV A(I,J)*DT/(CONNA(I,J)*RE*DA *DR**2) DIVR=1.+2. *CV A(I,J)*DT/(CONNA(I,J)*RE*DA *(DX*XL)**2)
&+4. *CV A(I,J)*DT/(CONNA(I,J)*RE*DA *DR **2) &+2. *CVA(I,J)*DT/(CONNA(I,J)*RE*DA *(XJ*DR)**2) PDVXAA=-CONN A(I + 1 ,J)*DT/(2. *DX*XL *DA *CONNA(I,J)*DIVX) PDVXAB=CONNA(I-I ,J)*DT/(2. *DX*XL *DA *CONNA(I,J)*DIVX) PDVRAA=-CONNA(I,J+l)*DT/(2.*DR*DA*CONNA(I,J)*DIVR) PDVRAB=CONNA(I,J-l)*DT/(2.*DR*DA*CONNA(I,J)*DIVR) VXNA(I,J)=VXNA(I,J)+DEIGEN(I+l,J)*PDVXAA+DEIGEN(I-l,J)*PDVXAB
5 VRNA(I,J)=VRNA(I,J)+DEIGEN(I,J+l)*PDVRAA+DEIGEN(I,J-l)*PDVRAB
205
J=1 DO 10 I=2,NPI-l
10 VXNA(I,J)=VXNA(I,J+ 1)
I=NPI DO 15 J=I,NPJ-l VXNA(I,J)= VXN A(I -1 ,J)
15 VRNA(I,J)=YRNA(I-l,J)
RETURN END
************************************************************
SUBROUTINE YCB INCLUDE'MAIN.CMM'
DO 5 I=2,NPI-l DO 5 J=2,NPJ-l XJ=J-l DIVX=I.+4.*CYB(I,J)*DT/(CONNB(I,J)*RE*DB*(DX*XL)**2)
&+2. *CYB(I,J)*DT/(CONNB(I,J)*RE*DB *DR **2) DIVR=I.+2.*CVB(I,J)*DT/(CONNB(I,J)*RE*DB*(DX*XL)**2)
&+4. *CYB(I,J)*DT I(CONNB(I,J)*RE*DB *DR **2) &+2.*CVB(I,J)*DT/(CONNB(I,J)*RE*DB*(XJ*DR)**2) PDVXBA=-CONNB(I+ 1 ,J)*DT/(2. *DX*XL *DB*CONNB(I,J)*DIVX) PDVXBB=CONNB(I-l ,J)*DT/(2. *DX*XL *DB*CONNB(I,J)*DIVX) PDVRBA=-CONNB(I,J+ 1)*DT/(2. *DR *DB*CONNB(I,J)*DIVR) PDYRBB=CONNB(I,J-l )*DT/(2. *DR *DB*CONNB(I,J)*DIVR) VXNB(I,J)= VXNB(I,J)+DEIGEN (I + 1 ,J)*PDVXBA+DEIGEN (I-I ,J)*PDYXBB
5 VRNB(I,J)= VRNB(I,J)+DEIGEN (I,J+ 1 )*PDVRBA+DEIGEN (I,J-l )*PDVRBB
J=1 DO 10 I=2,NPI-l
10 VXNB(I,J)=YXNB(I,J+l)
I=NPI DO 15 J=I,NPJ-I VXNB(I,J)= VXNB (I -I ,J)
IS YRNB(I,J)= VRNB(I-I ,J)
RETURN END
206
************************************************************
SUBROUTINE YCC INCLUDE 'MAIN.CMM'
DO 5 I=2,NPI-I DO 5 J=2,NPJ-I XJ=J-I DIYX= 1. +4. *CYC(I,J)*DT/(CONNC(I,J)*RE*DC*(DX*XL)**2)
&+2. *CYC(I,J)*DT/(CONNC(I,J)*RE*DC*DR **2) DIYR= 1. +2. *CYC(I,J)*DT/(CONNC(I,J)*RE*DC*(DX*XL)**2)
&+4. *CVC(I,J)*DT/(CONNC(I,J)*RE*DC*DR **2) &+2. *CVC(I,J)*DT/(CONNC(I,J)*RE*DC*(XJ*DR)**2) PDVXCA=-CONNC(I+ 1 ,J)*DT/(2. *DX*XL *DC*CONNC(I,J)*DIVX) PDVXCB=CONNC(I-I ,J)*DT/(2. *DX*XL *DC*CONNC(I,J)*DIVX) PDVRCA=-CONNC(I,J+I)*DT/(2.*DR*DC*CONNC(I,J)*DIVR) PDYRCB=CONNC(I,J-I)*DT/(2. *DR *DC*CONNC(I,J)*DIVR) YXNC(I,J)=YXNC(I,J)+DEIGEN(I+ 1 ,J)*PDVXCA+DEIGEN(I-I ,J)*PDVXCB
5 YRNC(I,J)=VRNC(I,J)+DEIGEN(I,J+I)*PDVRCA+DEIGEN(I,J-I)*PDVRCB
J=I DO 10 I=2,NPI-I
10 YXN C(I,J)= VXN C(I,J + 1)
I=NPI DO 15 J=I,NPJ-l VXN C(I,J)= YXN C(I -1 ,J)
15 VRNC(I,J)=VRNC(I-l,J)
RETURN END
************************************************************
SUBROUTINE SYEL INCLUDE 'MAIN.CMM'
DO 5 I=I,NPI-l DO 5 J=I,NPJ-I RHA=DA *CONA(I,J) RHB=DB*CONB(I,J) RHC=DC*CONC(I,J) DEN=RHA+RHB+RHC
207
VX(I,J)=(RHA *VXA(I,J)+RHB*VXB(I,J) &+RHC*VXC(I,J»/DEN
5 VR(I,J)=(RHA *VRA(I,J)+RHB*VRB(I,J) &+RHC*VRC(I,J»/DEN
RETURN END
************************************************************
SUBROUTINE COEFCON(VX,VR,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NOD/ NPI,NPJ COMMON /SP/ DX,DR,DT COMMON /ND/ XL,RE,RR,UMAX,DENA COMMON /COEF/ A,B,C,D,E,F DIMENSION VX(NPI,NPJ), VR(NPI,NPJ) EXTERNAL DAX,DAR
XJ=J-l A=-DT*VX(I,J)/(2. *DX*XL) B=DT*VX(I,J)/(2. *DX*XL) C=-DT*VR(I,J)/(2. *DR) D= 1. +DT*(DAX(VX,I,J)+ VR(I,J)/(XJ*DR)+DAR(VR,I,J» E=DT*VR(I,J)/(2. *DR) F=O.
RETURN END
************************************************************
SUBROUTINE COEFVX(VX,VR,CON,CP,CV,XMX,DEN,AL,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NOD/ NPI,NPJ COMMON /SP/ DX,DR,DT COMMON /ND/ XL,RE,RR,UMAX,DENA COMMON /GR/ BDF COMMON /COEF/ A,B,C,D,E,F DIMENSION VX(NPI,NPJ), VR(NPI,NPJ),CON (NPI,NPJ),
&CP(NPI,NPJ),CV (NPI,NPJ),XMX(NPI,NPJ) EXTERNAL DAX,DAR,DAXX,DARR,DAXR
208
XJ=J-1 A=-DT*VX(I,J)/(2. *DX*XL)
&+DT*DAX(CV ,I,J)/(RE*DEN*CON (I,J)*DX*XL) &+2.*DT*CV(I,J)/(RE*DEN*CON(I,J)*(DX*XL)**2) B=DT*VX(I,J)/(2. *DX*XL)
&-DT*DAX(CV,I,J)/(RE*DEN*CON(I,J)*DX*XL) &+2.*DT*CV(I,J)/(RE*DEN*CON(I,J)*(DX*XL)**2) C=-DT*VR(I,J)/(2. *DR)
&+DT*DAR(CV ,1,J)/(2. *RE*DEN*CON (I,J)*DR) &+(1./(2. *XJ)+ l.)*DT*CV(I,J)/(RE*DEN*CON (I,J)*DR **2) D=1.+4.*DT*CV(I,J)/(RE*DEN*CON(I,J)*(DX*XL)**2)
&+2. *DT*CV (I,J)/(RE*DEN*CON (I,J)*DR **2) E=DT*VR(I,J)/(2. *DR)
&-DT*DAR(CV,I,J)/(2. *RE*DEN*CON(I,J)*DR) &+( -1./(2. *XJ)+ 1.)*DT*CV (I,J)/(RE*DEN*CON (I,J)*DR **2) F=-DT*DAX(CP,I,J)/(DEN*CON(I,J»
&+DT*DAR(CV,I,J)*DAX(VR,I,J)/(RE*DEN*CON(I,J» &+DT*CV (I,J)*(DAX(VR,I,J)/(XJ*DR)+DAXR(VR,I,J» &/(RE*DEN*CON(I,J» &+DT*XMX(I,J)/(DEN*CON (I,]) )+DT*BDF &-2. *DT* AL *(2. *DAXX(CON,I,J)*DAX(CON,I,J) &+DARR(CON,I,J)*DAX(CON,I,J) &+DAR(CON,I,J)*DAXR(CON,I,J»/(DEN*CON(I,J»
RETURN END
************************************************************
SUBROUTINE COEFVR(VX,VR,CON,CP,CV,XMR,DEN,AL,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NODI NPI,NPJ COMMON ISPI DX,DR,DT COMMON /NDI XL,RE,RR,UMAX,DENA COMMON ICOEFI A,B,C,D,E,F DlMEN SION VX(NPI,NPJ), VR(NPI,NPJ),CON (NPI,NPJ),
&CP(NPI,NPJ),CV(NPI,NPJ),XMR(NPI,NPJ) EXTERNAL DAX,DAR,DAXX,DARR,DAXR
XJ=J-1 A=-DT*VX(I,J)/(2. *DX*XL)
&+DT*DAX(CV,I,J)/(2. *RE*DEN*CON(I,J)*DX*XL) &+DT*CV(I,J)/(RE*DEN*CON(I,J)*(DX*XL)**2)
209
B=DT*VX(I,J)/(2. *DX*XL) &-DT*DAX(CV ,1,J)/(2. *RE*DEN*CON (I,J)*DX*XL) &+DT*CV(I,J)/(RE*DEN*CON(I,J)*(DX*XL)**2) C=-DT*VR(I,J)/(2. *DR)
&+DT*DAR(CV,I,J)/(RE*DEN*CON(I,J)*DR) &+(1./XJ+2.)*DT*CV(I,J)/(RE*DEN*CON(I,J)*DR**2) D=l. +2. *DT*CV(I,J)/(RE*DEN*CON(I,J)*(DX*XL)**2)
&+4. *DT*CV (I,J)/(RE*DEN*CON(I,J)*DR **2) &+2. *DT*CV(I,J)/(RE*DEN*CON(I,J)*(XJ*DR)**2) E=DT*VR(I,J)/(2. *DR)
&-DT*DAR(CV ,I,J)/(RE*DEN*CON (I,J)*DR) &+(-1./XJ+2.)*DT*CV(I,J)/(RE*DEN*CON(I,J)*DR**2) F=-DT*DAR(CP,I,J)/(DEN*CON (I,J»
&+DT*DAX(CV,I,J)*DAR(VX,I,J)/(RE*DEN*CON(I,J» &+DT*CV (I,J)*DAXR(VX,I,J)/(RE*DEN*CON (I,J» &+DT*XMR(I,J)/(DEN*CON(I,J» &-2.*DT*AL*(DAXX(CON,I,J)*DAR(CON,I,J) &+DAX(CON ,I,J)*DAXR(CON ,I,J) &+2. *DAR(CON ,I,J)*DARR(CON ,I,J»/(RE*DEN*CON (I,J»
RETURN END
************************************************************
SUBROUTINE ERROR INCLUDE 'MAIN.CMM'
DO 5 1=2,NPI-l DO 5 J=2,NPJ-l CALL ERR(VXNC,VRNC,CONNC,CONC,I,J,CONSTERRC) CALL ERR(VXNB,VRNB,CONNB,CONB,I,J,CONSTERRB) CALL ERR(VXNA,VRNA,CONNA,CONA,I,J,CONSTERRA)
5 ER(I,J)=CONSTERRA+CONSTERRB+CONSTERRC
RETURN END
************************************************************
SUBROUTINE ERR(VX,VR,CONN,CON,I,J,CONSTERR) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NODI NPI,NPJ
210
COMMON ISPI DX,DR,DT COMMON /NDI XL,RE,RR,UMAX,DENA DIMEN SION VX(NPI,NPJ), VR(NPI,NPJ),CONN(NPI,NPJ),CON (NPI,NPJ) EXTERNAL DAX,DAR
XJ=J-l CONSTERR=(CONN (I,J)-CON (I,J»IDT
&+ VX(I,J)*DAX(CON ,I,J)+ VR(I,J)*DAR(CON ,I,J) &+CONN(I,J)*(DAX(VX,I,J)+VR(I,J)/(XJ*DR)+DAR(VR,I,J»
RETURN END
************************************************************
SUBROU11NE RESID INCLUDE 'MAIN.CMM'
SUMSQERW=O. V=(NPI-l)*(NPJ-2)**2 DO 5 I=2,NPI-l DO 5 J=2,NPJ-l XJ=J-l
5 SUMSQERW=(ER(I,J)*«l.+ 1./XJ)**2-1.) &*XJ**2N)**2+SUMSQERW
SUMERM=DSQRT(SUMSQERW)*RR/UMAX
RETURN END
************************************************************
DOUBLE PRECISION FUNCTION DAX(A,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NODI NPI,NPJ COMMON ISPI DX,DR,DT COMMON /NDI XL,RE,RR,UMAX,DENA DIMENSION A(NPI,NPJ)
DAX=(A(I+ 1 ,J)-A(I-l ,J»/(2. *DX*XL)
RETURN
211
END
************************************************************
DOUBLE PRECISION FUNCTION DAR(A,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NODI NPI,NPJ COMMON ISPI DX,DR,DT DIMENSION A(NPI,NPJ)
DAR=(A(I,J+ 1)-A(I,J-1»/(2. *DR)
RETURN END
************************************************************
DOUBLE PRECISION FUNCTION DAXX(A,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NODI NPI,NPJ COMMON ISPI DX,DR,DT COMMON /NDI XL,RE,RR,UMAX,DENA DIMENSION A(NPI,NPJ)
DAXX=(A(I+ 1 ,J)-2. * A(I,J)+A(I-l ,J»/(DX*XL)**2
RETURN END
************************************************************
DOUBLE PRECISION FUNCTION DARR(A,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON ISPI DX,DR,DT COMMON /NODI NPI,NPJ DIMENSION A(NPI,NPJ)
DARR=(A(I,J+ 1)-2. * A(I,J)+A(I,J-1»/DR**2
RETURN END
************************************************************
212
DOUBLE PRECISION FUNCTION DAXR(A,I,J) IMPLICIT DOUBLE PRECISION (A-H,O-Z) COMMON /NOD/ NPI,NPJ COMMON /SP/ DX,DR,DT COMMON /ND/ XL,RE,RR,UMAX,DENA DIMENSION A(NPI,NPJ)
DAXR=(A(I+ I,J+ 1)-A(I-l,J+ 1) &-A(I+ I,J-1)+A(I-l,J-l»/(4. *DR*DX*XL)
RETURN END
************************************************************
SUBROUTINE YISB INCLUDE 'MAIN.CMM'
DO 5 I=I,NPI DO 5 J=I,NPJ DF=O.999*CMAX IFCCONNB(I,J).GE.CMAX)CONNB(I,J)=DF C=CONNB(I,J)/( 1.-CONNB (I,J)-CONN C(I,J» YB=(I.+2.5/2.*(C/(l.-C/CMAX»)**2 CYA(I,J)=CONNA(I,J)
5 CYB(I,J)=CONNB(I,J)*YB
RETURN END
************************************************************
SUBROUTINE YISC INCLUDE'MAIN.CMM'
DO 5 l=l,NPI DO 5 J=l,NPJ DF=O.999*CMAX C=CONNC(I,J)+CONNB(I,J) X=CONNC(I,J)/C CA=C*Cl.-X) CB=C*X/(C*X + I.-C) IF(CA.GE.CMAX)CA=DF
213
IF(CB.GE.CMAX)CB=DF VCA=(I.+2.5/2.*(CA/(I.-CA/CMAX»)**2 VCB=(I.+2.5/2.*(CB/(I.-CB/CMAX»)**2 VC=VCA*VCB
5 CVC(I,J)=CONNC(I,J)*VC
RETURN END
************************************************************
SUBROUTINE SOL VER(A,B,M,MD,N) IMPLICIT DOUBLE PRECISION (A-H,O-Z) DIMENSION A(N,M),B(N)
DO 5I=I,N-l PIVOT=A(I,MD) A(I,MD)=1.0 DO 10 J=I,M IF(lEQ.MD)GOTO 10 A(I,J)=A(l,J)IPIVOT
10 CONTINUE B(l)=B(I)IPIVOT DO 15 J=I,MD-l JJ=I+J II=MD-J IF(JlGT.N)GOTO 5 T=A(JJ,II) A(JJ,II)=O. DO 20 K=I,MD-l KK=II+K KKK=MD+K
20 A(JJ,KK)=A(JJ,KK)-A(I,KKK)*T B(JJ)=B(JJ)-B(I)*T
15 CONTINUE 5 CONTINUE
B(N)=B(N)I A(N ,MD) DO 25 I=N-l,I,-1 DO 30 J=I,MD-l JJ=MD+J J11=I+J IF(JJJ.GT.N)GOTO 25
214
30 B(I)=B(I)-A(I,JJ)*B(JJJ) 25 CONTINUE
RETURN END
************************************************************
*************** BI-MODAL FLOW *************** 10222 0.228 0.228 50 5
0.00001
1.0 2.3 2.3 0.90 17.0 0.0 0.0 0.0 0.0 0.0 0.0 0.4 0.18
0.0 0.0 0.42
200. 0.0 9999. 9999. 0.5
************************************************************
*************** BI-MODAL FLOW *************** 10222 0.228 0.228 0.00001000000 50 5 1.000 2.300 2.300 0.900 100.000 0.000 0.000 0.000 0.000 0.000 0.000 0.300 0.210
600.000 0.000 9999.0009999.000
0.5000 0.50E-04 0.4SE-07
0.000 0.000 0.490
215
50 0.1288E+05 0.9014E+04 0.2103E+05 0.1347E+050.9315E+04 0.2115E+05
216
O.OOOOE+OO 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.3000E+OO 0.3000E+OO 0.1000E+01 0.3000E+OO 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.2000E+01 0.3000E+OO 0.3000E+00 0.3000E+00 O.3oooE+OO 0.3000E+00 0.3000E+00 0.3000E+0 1 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.3000E+00 0.3000E+00 0.4000E+01 0.3000E+OO 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.5000E+01 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.3OOOE+00 0.6000E+01 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.7000E+01 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.3000E+00 0.3000E+00 0.8000E+01 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.9000E+01 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.1000E+02 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.3000E+00 0.1100E+02 0.3000E+00 0.3000E+00 0.3000E+OO 0.3000E+00 0.3000E+00 0.3000E+00 O.OOOOE+OO O.OOOOE+OO -.5573E+07 -.1067E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.1000E+01 O.OOOOE+OO -.5573E+07 -.1067E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.2000E+01 O.OOOOE+OO -.5549E+07 -.1067E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.3000E+01 O.OOOOE+OO -.5541E+07 -.1067E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.4000E+Ol O.OOOOE+OO -.5488E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.5000E+Ol O.OOOOE+OO -.5482E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.6000E+01 O.OOOOE+OO -.5405E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.7000E+01 O.OOOOE+OO -.5409E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.8000E+01 O.OOOOE+OO -.5328E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.9000E+Ol O.OOOOE+OO -.5339E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.1000E+02 O.OOOOE+OO -.5289E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 0.1100E+02 O.OOOOE+OO -.5289E+07 -.1066E+08 -.1594E+08 -.2123E+08 -.2651E+08 O.OOOOE+OO 0.6000E+03 0.6417E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.1000E+01 0.6000E+03 0.6417E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.2000E+Ol 0.6000E+03 0.6418E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.3000E+Ol 0.6000E+03 0.6419E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.4000E+Ol 0.6000E+03 0.6421E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.5000E+01 0.6000E+03 0.6423E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.6000E+01 0.6000E+03 0.6426E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.7000E+01 0.6000E+03 0.6427E+03 0.6429E+03 0.6430E+03 0.6430E+03 0.6430E+03 0.8000E+01 0.6000E+03 0.6429E+03 0.6426E+03 0.6426E+03 0.6426E+03 0.6426E+03 0.9000E+Ol 0.6000E+03 0.6372E+03 0.6369E+03 0.6368E+03 0.6368E+03 0.6368E+03 0.1000E+02 0.6000E+03 0.5637E+03 0.5631E+03 0.5631E+03 0.5631E+03 0.5631E+03 0.1100E+02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO 0.1000E+01 O.OOOOE+OO -.7630E+00 -.3284E-01 -.1251E-02 -.3069E-04 0.1923E-04
217
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219
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220
0.1100E+02 O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO O.OOOOE+OO
221
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