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Information Series No. 63 
January, 1 975 



TDA-RCA Capsule 
Pipeline Project 

Phase 3 Report 

Part 1 


edited by 
Erik J. Jensen 

Contributors 
R. A. S. Brown 
H. S. Ellis 
Jan Kruyer 
A. A. Roehl 
L. M. White 


Liberia 

RESEARCH 


Digitized by the Internet Archive 
in 2017 with funding from 
University of Alberta Libraries 


https://archive.org/details/tdarcacapsulepip31 kruy 


TDA-RCA CAPSULE PIPELINE PROJECT 


SEP 1 5 1S75 

c . ' 


PHASE III REPORT 
PART 1 


edited by 
Erik J . Jensen 


contributors 

R. A. S. Brown, H. S. Ellis, Jan Kruyer, A s A. Roehl, L. M. White 


Alberta Research 
Information Series No. 63 


January, 1975 




LIST OF CONTENTS 


PART 1 

Nomenclature 
Chapter I 
Chapter II 
Chapter III 
Chapter IV 

Chapter V 

Chapter VI 
Chapter VII 

PART 2 

Appendix A 
Appendix B 

Appendix C 

Appendix D 

Appendix E 
Appendix F 


Preface and Summary 1 

Designing a Capsule Pipeline 7 

Pressure Data Tabulation 42 

Pressure Gradients of Cylindrical Capsules 

Calculated from the Coefficient of Friction 145 


Correlation of the Pressure Gradients of Cylindrical 
Capsules with Capsule Density, Pipe Diameter 
and Liquid Characteristics 176 

The Hydrodynamics of Spherical Capsules 218 

Calculation of the Liquid Throughput 246 


Capsule Pipelining - The System and Its Potential 

Friction and Surface Roughness Effects in Capsule 
Pipelines 

The Effect of the Density of Cylindrical Capsules 
on the Pressure Gradients in Capsule Pipelines 

Minimizing the Pressure Gradients in Capsule 
Pipelines 

Spherical Capsules in a 102 mm Pipeline 

Computer Program for Operating the 0.5-4 Inch 
Laboratory Pipelines 





NOMENCLATURE 


Except where noted otherwise, the following nomenclature Is used throughout the report. 


A 

B 

Bhp 


C ,C 

1 2 


E b' E m ' E P 


Ehp 

e 

F 

FF 


g 

9 c 

Hhp 

K 

k 

L 

L 


Intercept of a linear least squares regression 
Slope of a linear least squares regression 
Brake horsepower 

Constants in the bulk velocity calculation equations defined in 
Table 7 - 1 

Pipe inside diameter (in.) 

Capsule diameter (in.) 

Bypass, motor and pump efficiencies respectively expressed as 
fractions 

Electric horsepower 
Eccentricity ratio 

The linear fraction of a pipe length that is occupied by capsules 
Friction force (lb.) 

Friction factor for the annulus between the capsule and the 
pipe wall 

Gravitational constant (32.2 ft. /sec. 3 ) 

Gravitational conversion factor (32.2 ft. Sb. /sec. 3 Sb. r ) 

m t 

Hydraulic horsepower 
Constant in Durand equation 
Capsule diameter/pipe diameter 
Length (ft.) 

Capsule length (ft.) 

Pipeline length (mi.) 

Capsule train length (ft.) 


Whole number (pump stations or number of capsules in a train) 
Pressure force (lb.) 

Discharge pressure (psi) 

Suction pressure (psi) 

Maximum allowable working pressure (psi) 

Pressure difference (psi) 

Pressure gradient (psi/ft.) 

Pressure gradient at = 0 (psi/ft.) 

Capsule pressure gradient (psi/ft.) 

Liquid pressure gradient (psi/ft.) 

Pressure gradient due to elevation change (psi/ft.) 

Annular capsule, liquid and total flowrates respectively (ft. 3 /sec.) 

Radius of curvature of a pipe bend (ft.) 

Pipe Reynolds number, C V, D / v 

2 b 

Reynolds number between the capsule and the pipe wall defined 
in equation 2-11 

Pressure ratio = capsule pressure gradient/liquid pressure 
gradient for a smooth pipe 

Specific pressure ratio defined in equation 5-4 
Shear force (lb.) 

Specific gravity 


ss 

T 

V b 

V 

c 

w 

c 

Y 


Pump station spacing (mi.) 

Tension on a pulling cable (lb.) 

Bulk velocity (ft. /sec.) 

Capsule velocity (ft. /sec.) 

Capsule throughput in millions of tons per year (MMTPY) 
Elevation change along a pipeline (ft.) 


Greek 

T1 

1 

*n 

2 

V 

p 

p b 

a 

T 


Buoyed specific gravity. O' ~ p 

Coefficient of lubricated friction based upon the capsule 
pressure gradient 

Coefficient of lubricated friction based upon the increase 
in the pressure gradient due to the presence of the capsules 

Liquid kinematic viscosity (cs.) 

Liquid specific gravity 

Bulk or equivalent specific gravity 

Capsule specific gravity 

Shear stress (lb. /ft. 2 ) 


T and r 
c p 


Average shear stress on the capsule and pipe surfaces (lb. /ft. 2 ) 
as defined in the Addendum to Chapter IV 




CHAPTER ! 


Preface and Summary 


CHAPTER I 


Preface and Summary 


1 . 0 Preface 

The three-year contract on capsule pipeline hydrodynamics signed between the 
Transportation Development Agency (TDA) on behalf of the Ministry of Transport (MOT), 
and the Research Council of Alberta (RCA), which took effect January 1, 1971, expired 
December 31, 1973. It was however extended to March 31, 1974, in order to complete 
the experimental program. 

Reimbursements by the TDA were $8.57 less than the appropriated $535,000. 

The total cost of the project was about $900,000. The major objective of the project, 
which was to establish reliable hydrodynamic design equations, has been attained. 

The Phase 1, Parts 1 and 2, and the Phase 2 reports, describing the work carried 
out until April, 1973, have been published previously. The present Phase 3 report deals 
with results obtained from April, 1973, to the termination of the experimental program. 
The analytical methods derived from the results are also presented. The complex hydro- 
dynamic problems associated with capsule flow in a liquid filled pipeline have been the 
subject of many investigations, but the TDA-RCA reports mark the first time compre- 
hensive methods of predicting the capsule pressure drops and other important variables 
in capsule flow have been presented. 

Over the three years some fifteen papers have been published in the scientific 
literature and two patent applications have been filed. Numerous conferences and 
meetings have been addressed. The project has received extensive coverage in the / 
news media, both nationally and internationally, and individuals as well as delegations 
from many foreign countries have visited the Research Council of Alberta for the 
purpose of learning about capsule pipelining to better assess potential applications. 


3 


In 1973 an additional contract between the TDA and RCA was signed. It 
called for the development of a system analysis of capsule pipelining. This work has 
been completed and reported under the title, "Capsule Pipeline System Analysis," 

RCA Information Series No. 67. This report may be considered a sequel to the three 
TDA-RCA capsule pipeline project reports and should be consulted when further 
development work of the complete system is considered. 

1.1 Major Results 

The major results of the project can be summarized as follows: 

About 67,000 sets of experimental capsule flow measurements have been ac- 
cumulated. These data span a broad range of pipe sizes, capsule sizes, capsule 
densities, capsule surface characteristics and liquid viscosities. They are the most 
accurate and comprehensive data available and design calculations based on them and 
the methods derived from them can be assigned a very high confidence factor. 

A method for calculating the capsule pressure gradient for cylinders directly 
from the coefficient of lubricated friction has been developed and the reliability of it 
confirmed. This calculation relates the pressure gradient to the capsule to pipe dia- 
meter ratio and capsule and liquid densities for all pipe sizes. The accuracy of this 
prediction depends upon the accuracy of the coefficient of lubricated friction. 

A method has been developed for the prediction of the capsule pressure gradient 
for spheres, and the advantages and disadvantages of using spheres as compared with 
cylinders have been established. 

A model has been developed which accurately calculates the liquid throughput 
in a capsule pipeline based on the knowledge of capsule velocity and capsule pressure 


4 


gradient - . This liquid throughput calculation provides the basis for the calculation of 
the liquid pressure gradient and the total required horsepower. 

i ,2 Phase 3 Report Summary 

Chapter 2, "Designing a Capsule Pipeline," presents a systematic approach to 
the calculations required in capsule pipeline design based on methods developed in the 
TDA-RCA project. The procedures are illustrated with examples and detailed calcu- 
lations are given for both cylindrical and spherical capsules. The recommended methods 
may be summarized as follows: 

The size of the pipeline required is determined first from solid throughput. Sine 
fill and capsule velocity. The capsule pressure gradient is then determined with the 
equations presented in Chapters 4 and 5 for cylinders and Chapter 6 for spheres or 
alternatively from the experimental results presented in Chapter 3. The liquid pressure 
gradient is calculated using conventional liquid pipeline equations adjusted for the 
linear fraction of liquid in the pipeline, while the pressure due to elevation change is 
estimated on the basis of the average density of the pipeline contents. The two pressure 
gradients due to flow of capsules and liquid are then combined with the elevation 
pressure change to give the total pipeline pressure drop. The liquid flow rate required 
to move the solids is calculated from the capsule pressure gradient and the capsule 
velocity with the equations developed in Chapter 7. The required horsepower is then 
calculated from the product of total pipeline pressure drop and throughput. Finally, 
the number of pumping stations, the required horsepower per station and the required 
pump pressures are calculated. 

Chapter 3 presents, in graphical form, all the measured pressure gradients. 


5 


These results are presented in groups relating to pipe size, carrier liquid and cylinder 
and sphere surface characteristics. An index is provided to facilitate usage. 

Chapter 4 deals with a method of predicting the pressure gradient of cylindrical 
capsules on the basis of the coefficient of lubricated friction. The force required to 
pull a capsule through a liquid filled pipe is related to the pressure gradient required 
to propel the same capsule through the same liquid filled pipe. Based on this relation- 
ship a mode! for predicting the capsule pressure gradient has been developed. 

Chapter 5 develops a correlation of the pressure gradients of cylindrical cap- 
sules with capsule density, pipe diameter and liquid characteristics. The experimental 
pressure gradients of trains of cylindrical capsules are correlated with capsule and 
liquid specific gravities, bulk (liquid) velocity, pipe diameter and liquid characteristics. 
The correlation can be expressed in a form similar to that of the Durand equation for 
slurries and that of an independent correlation for capsules developed in South Africa. 

The hydrodynamics of spherical capsules are presented in Chapter 6. It has 
been confirmed that with water as the conveying liquid the pressure gradient of spheres 
varies little with specific gravity and comprises a basic pressure gradient due to dis- 
tortion of the liquid velocity profile by the spheres with adjustments to correct for the 
surface and shape characteristics of the spheres. With more viscous liquids the pressure 
gradient rises rapidly with increase of specific gravity and viscosity. On the basis of 
hydrodynamic considerations spheres are preferable to cylinders at low velocities for 
specific gravities down to about 1 .25. The lower power requirement must however be 
weighed against the cost of forming spheres and the cost of the larger pipe required 
for a given throughput. 

The relationship between liquid throughput, capsule velocity and capsule 


6 


pressure gradient is used in Chapter 7 to develop a method of predicting liquid through- 
put. It is based upon the fact that both the capsule pressure gradient and the liquid 
flow rate are affected by the interaction of the pipe and capsule surfaces. The capsule 
pressure gradient is then used as the indicator of this surface interaction to very 
precisely predict the liquid throughput. A tabulation of all the experimental results 
of the TDA-RCA project with capsules run in the 0.5, 2, 4 and 10 inch pipes is also 
presented in Chapter 7 to indicate the accuracy of the method of liquid flow prediction. 

Six appendices form Part 2 of the report. Five of these are published reports 
arising from the project which are included for the convenience of the reader. The 
sixth. Appendix F, is the computer program for operation of the 0.5-4 inch laboratory 
pipelines, the publication of which was deferred when the Phase 1 report was published. 

1.3 Acknowledgements 

At the conclusion of a major undertaking, such as this project has been, it is 
appropriate to extend a word of thanks to persons who have helped bring the work to a 
successful conclusion. The authors wish to express their gratitude to Messrs. Charles 
Halton and Martin Brennan for initiating the negotiations that led to the TDA-RCA 
contract; to Messrs. Peter Eggleton and Ian Gilbert of the TDA for their support during 
the execution of the project; to the industrial members of the Project Advisory 
Committee, Messrs. J. G. Bruce, G. H. Durham and R. Cushing for valuable advice 
whenever needed. We also wish to thank the many RCA staff members who worked 
directly or indirectly on the project. 



CHAPTER II 


Designing a Capsule Pipeline 








/ 



CHAPTER II 


Designing a Capsule Pipeline 

2.0 Summary 

A systematic approach to the calculations necessary for capsule pipeline 
design is presented and the calculations are related to the appropriate results 
developed in the project. The different possible approaches to the design are out- 
lined. Procedures are illustrated for both cylindrical and spherical capsules using 
movement of sulfur from Calgary to Vancouver as the sample case. The effects of 
the density of cylindrical capsules on pumping requirements and of pipe size for 
spherical capsules are calculated. 


9 


2.1 Introduction 

The design of the pipeline is one component of the design of the entire capsule 
pipeline system and includes specification of capsule shape, size, density and velocity, 
pipe diameter, carrier liquid demand and pumping power requirements. Capsule and 
pipe surface characteristics and carrier liquid properties also influence the pipeline 
design through their effect on the required pumping power. The effects of these 
specifications and influences have been examined extensively in this and previous 
studies and the object is to outline the resulting calculation procedures. 

The methods to be considered relate to pipeline design only and not to design 
of other components of the system or to optimization of the capsule pipeline system 
which must include economic considerations. The latter two topics formed the subject 
of a separate study carried out concurrently and published as "Capsule Pipeline 
System Analysis" (Reference 1). 

2.2 Capsule Characteristics 

Capsule characteristics which must be considered in the design of a capsule 
pipeline are: 

a) Shape: there is a choice between spherical, cylindrical or modified 
cylindrical capsules. 

b) Density: this can be modified by inclusion of voids within the capsule 
capsule but at the expense of solid throughput. 

c) Size: the diameter of spheres and cylinders and the length for 
cylinders must be considered. 

d) Surface characteristics: these may be qualitatively smooth or rough. 


10 


Capsule shape will be considered independently, while capsule density and size are 
included as dependent variables in the design equations. Capsule surface character- 
istics will be considered in the relevant sections on pressure gradient. 

2.2.1 Capsule Shape 

The main capsule options in designing a capsule system are spheres, cylinders 
or modified cylinders (e.g. collared cylinders). The hydrodynamic advantage of 
using spheres instead of cylinders is shown in Figure 2 - 1 where the pressure gradients 
for smooth cylinders (vinyl plastic surface) are compared with those for smooth 
machined aluminum and iron spheres. These data illustrate that there is no universal 
choice between the shapes. For capsule densities close to that of the carrier liquid, 
cylinders show lower capsule pressure gradients at medium and high velocities while 
the behavior reverses at higher densities,and spheres give lower pressure gradients at 
all velocities considered. The choice between spherical and cylindrical capsules is 
considered in more detail in Chapter 6. 


2.3 Pipeline Design Calculations 

The design of a capsule pipeline can be illustrated with a flow diagram of the 
calculation processes: 


Input 

Parameters 



In the calculating processes, the size of pipeline required is determined first from the 
solid throughput required, the line fill and the capsule velocity. The capsule pressure 
gradient is then determined with the equations developed in Chapters 4, 5 and 6 or. 


11 


alternatively , from the experimental results presented in Chapter 3. The liquid pressure 
gradient is calculated using conventional liquid pipeline equations adjusted for the 
linear fraction of liquid and the pressure change due to elevation estimated on the basis 
of the average density of the pipeline contents. The two pressure gradients due to 
flow are combined with the elevation pressure change to give the total pipeline 
pressure drop. The liquid flow rate required to move the solids is calculated from the 
capsule pressure gradient and the capsule velocity with the equations developed in 
Chapter 7. The required horsepower is then calculated from the product of total pipe- 
line pressure drop and throughput. Finally, the number of pumping stations, the 
required horsepower per station and the required pump pressures are calculated. The 
procedure is described in detail in the following sections and illustrated with sample 
calculations at the end of this chapter. 

2.4 Input Parameters 

A site-specific case study usually provides as fixed input parameters the pipe- 
line length and the elevation changes. The choice of carrier liquid defines the liquid 
parameters (specific gravity and viscosity) and the commodity stipulates the solids 
specific gravity. The commodity throughput is given by the project terms of reference. 

The variable inputs, which include diameter ratio, capsule shape, capsule 
velocity, actual capsule bulk specific gravity and linear line fill, are subject to 
optimization in the calculations. 

2.5 Pipe Specifications 

Most existing pipelines will initially be unsuitable for capsule transport since 
the tolerances for liquid pipelines are less stringent than for capsule pipelines. A smooth 


12 


internal weld seam is necessary for trouble-free capsule transport and older existing 
pipelines may have weld joints with protruding internal beads or weld icicles. 

Minimum bend radii and out-of-roundness of the pipe at bends have to be considered 
in capsule pipeline design to guard against line blockage. The common practice of 
telescoping, or decreasing the pipe wall thickness by changing the internal pipe 
diameter with decreasing pressure along the pipeline, Wi ile attractive from the view- 
point of economics of capsule pipeline design, may have to be implemented carefully 
to guard against problems of capsule slow-down or speed-up resulting in collisions. 

At present, there are no government regulations or codes specifically applicable 
to capsule transportation in pipelines such as U .S.A.S. Code B 31 .4-1966, or 
C.S.A. Standard Z-l 83-1 967 for liquid petroleum pipelines. However, 
experimentation to date indicates that wear or abrasion of the pipeline will not be 
a problem for practical capsule densities and capsule surfaces; thus the codes can be 
used with minor modifications. 

Steel pipe with yield strengths of 35,000, 42,000, 46,000 , 52,000 and 
60,000 psi, which may be required for capsule pipelining, is presently available. The 
cost of the pipe increases rapidly with increasing strength. Optimization must include 
costs of shipping, welding and installing the pipe. 

2.5.1 Pipe Sizing 

The theoretically required inside pipe diameter based on 350 operating days 
per year can be calculated from the input parameters as follows: 


13 


For cylinders D 


For spheres D 


194.2 W °* 5 

— - — — inches 

oV k 2 F 
c 


291.3 W 

< 

ctV k s F 


0.5 


inches 


( 2 - 1 ) 


*- C -J 

The diameter ratio (k) which is limited by pipe bends and terrain may be larger 
for spheres than for cylinders. For cylinders the maximum allowable diameter ratio 
and the length (L^) of each individual cylinder are related to the pipe diameter (D) 
and the radius of curvature of the pipe bend (R) as follows: 


L c = ji). 667 R D (1 - k)J as ' (2-2) 

A sphere will pass through a pipe bend as long as its diameter is not larger than the 
minimum pipe diameter in the bend, but a diameter ratio of 0.90 is suggested to allow 
for variations in the diameter of manufactured spheres. 

The linear line fill (F) may be controlled by manufacturing and injecting 
techniques, or by the pumping station bypass system employed. Linear line fills of 
100% have been attained on laboratory scale models (5/ 8“ and 1") for cylinders and 
spheres, utilizing a conventional pump and the rotary vane pump bypass system des- 
cribed in Reference 1 . However, this system has not yet been demonstrated on a 
commercial scale. Other suggested systems are unlikely to permit more than a 90% 
linear fill . 

The remaining input parameter is the capsule velocity V^. In commercial 
applications using cylinders the selected capsule velocity will be between 2 and 6 
ft. /sec.; if spheres are used the capsule velocity should be as low as economically 
possible. 


14 


With the input parameters determined, the theoretical inside pipe diameter 
can be calculated using equation 2 - 1 . The value obtained is unlikely to match 
a standard pipe size. In this case the capsule velocity must be recalculated to 
correspond to the nearest standard pipe size which will withstand the expected 
pressures, as follows: 


194.2 W 

For cylinders V = — — 

c a k s F D s 

291.3 W 

c 

For spheres V = 

a k 2 F D 2 


ft. /sec. 


ft. /sec. 


(2-3) 


2.6 Pipeline Pressure Drop 

The total pipeline pressure drop A P is the average pipeline pressure gradient 
multiplied by the pipeline length. 


AP = 5280 L 


„ (" ) f » - « 

( A p \ 

— j— 1 is composed of the capsule pressure gradient multiplied 
by the linear fraction of the line that is occupied by capsules, plus the liquid 
pressure gradient multiplied by the linear fraction of the line that is occupied by 
liquid only, plus the elevation pressure gradient (which may be positive or negative): 


(■¥-) ■ (tV*( 4V- f> * ("1 p,i/ "- 


Thus: 


AP = 5280 L 




psi 


(2-5) 


15 


2.6.1 Capsule Pressure Gradient 

The pressure gradients for homogeneous and uniformly round spheres flowing 
in water, which are discussed in Chapter 6, may be calculated from the relationship: 

k 2 psi/ft. (2 - 6) 

For commercially cast spheres, which tend to be somewhat heterogeneous and out-of- 
round the capsule pressure gradients are higher. For commercial cast spheres of the 
density of aluminum, 40% should be added to the above pressure gradient calculated 
from equation 2-6 and for spheres of the density of iron 150% should be added. For 
intermediate densities the percentage addition equals 24 (a - p). However, the 
additions are based upon the tolerances achieved in the casting process. A description 
of the tolerances for the cast spheres used in the RCA experimental pipelines is given 
in the Phase 2 Report. They were approximately ± 0.5% of D for both the aluminum 
and iron spheres. Pressure gradient data for spheres in liquids other than water are 
discussed in Chapter 6. 

Cylinder transport is recommended when the bulk capsule specific gravity 

approaches that of the carrier liquid (i.e. — < 1.25). The simplest method of 

P 

calculating the capsule pressure gradient for cylinders is from the coefficient of friction 

for the capsule surface sliding over the pipe surface *n the presence of a thin layer of 

carrier liquid which lubricates the two surfaces. Chapter 4 provides a table of these 

coefficients (r] ) for the various combinations tested in water in the pipelines, 
s 

Additional coefficients can be obtained by pulling a block of material through a tank 
filled with carrier liquid as illustrated in Figure 4-8. The capsule pressure gradient 
is calculated from the following relationship: 




16 


m 


0.433 k (o- p ) n + 


(it) psi/ft 


(2 - 7) 


The procedure for calculating capsule pressure gradients is iterative since the 


l!ent (r) 


iquid pressure gradient | ~ J cannot be estimated precisely until the bulk liquid 
velocity is known and this in turn is a function of the capsule pressure gradient. 
Initially in the procedure it is assumed that the bulk liquid velocity is equal to the 
capsule velocity and the liquid pressure gradient is calculated by the conventional 
methods discussed in the next section. The capsule pressure gradient then is calculated 
from equation 2 - 6 or 2 - 7. The liquid bulk velocity is obtained with equations 
2-10 and 2-12 discussed in the following sections and thus a more precise value 
is used to calculate a more precise capsule pressure gradient. Usually only one 
iteration is required to arrive at an answer which is within the accuracy of the reported 
coefficient of friction. 

More precise capsule pressure gradients can be obtained using pressure gradient 
data from experimental pipelines for the conditions matching the proposed commercial 
capsule and pipe surfaces. Chapter 3 shows curves of pressure gradient as a function 
of capsule velocity for all the capsules tested in the RCA experimental pipes. 

Chapter 5 correlates these data on the basis of Froude number and capsule density to 
enable their use for other conditions of pipe and capsule size, capsule velocity and 
liquid properties. 

Prior to construction of the pipeline a full scale loop will probably be built. 


having the capsules and pipe exactly as they would be used in the application, to 
measure actual pressure gradients. This is the procedure commonly used prior to 
building other types of solids pipelines. 


17 


(n 


2.6.2 Liquid Pressure Gradient 

The liquid pressure gradient is calculated by the conventional methods used in 
liquid pipelines from the bulk liquid velocity, which is the total volumetric throughput 
rate divided by the cross-sectional area of the pipej, The bulk I iquid velocity in a cap- 
sule system is calculated from the capsule pressure gradient and this in turn requires a 
knowledge of the liquid pressure gradient. An iterative calculating procedure 
therefore is necessary to arrive at the correct result. Convergence is very rapid 
however so that in most cases only one iteration is required. In the calculations it is 
initially assumed that the bulk liquid velocity equals the capsule velocity. 

There are several empirical equations and charts available for calculating the 
liquid pressure gradient from the bulk liquid velocity. One such equation is derived 
in the hydrodynamics section of the Capsule Pipeline System Analysis (Reference 1), 
by combining the friction factor equation of Drew, Koo and McAdams with the Darcy 
pressure drop equation. For the units employed the equation is: 


(n 


4.53 x 10' 4 pV b 2 0.00230 pV b 168 v 0 - 32 


D 


1.32 


(2-8) 


This is valid for new pipe with a smooth internal surface, and for turbulent flow in 
the pipe, which includes most practical capsule transportation situations. For 
laminar flow (pipe Reynolds Number, Re < 2000) the equation is: 



0.000668 pV, v 
b 

D s 


psi/ft. 


(2-9) 


18 


2.6.3 Bulk Liquid Velocity (V^) 

The bulk velocity is discussed in detail in Chapter 7, with the pertinent 

procedure and equations as follows: 

Assume turbulent flow in the annulus (f Re °' 25 = 0.07) 

c c 


V = kV + (1 - k * 2 ) 
b c 


Check the Reynolds number 


. {¥) K k >] 


1.25 


P v 


0-25 


O .5 71 


(2 - 10 ) 


Re = 
c 


C D(l-k) (V,-kV) 
2 b c 


(2 - 11 ) 


(l-k s ) V 

If Re > 1000, V, is correct 
c b 

If Re < 1000, V. must be recalculated since in laminar flow f Re = 9.6 
c “ b c c 


kV + (1 -k 2 ) 


m 


D 2 (1 - k) 


P v 


(2 - 12 ) 


C , C and C are dimensional constants and have the values 415, 7742 and 2500 

12 3 

respectively for the units used here. Values of these constants for other units are listed 
in Table 7 - 1 . 


2.6.4 Elevation Pressure Gradient 

The effect on the pressure gradient of any elevation change is more readily 

described by first defining a bulk specific gravity as the total weight of liquid plus 

solids in the pipeline divided by the total volume, i.e. 

For cylinders p, = p + k 2 F (a - p) 

2 (2-13) 

For spheres p^ = p + ^ k 2 F(cr-p) 


19 


The elevation pressure gradient has been found to approximate the product of this 
bulk specific gravity and the elevation change (or head) divided by the pipeline 
length. 



62.43 

144 


(± Y )P b 


5280 L 

P 


psi/ft. 


(2 - 14) 


This pressure gradient due to elevation change takes into account the weight compon- 
ent of the capsules acting in the downhill direction of the pipeline as well as the 
elevation head of the liquid. It does not consider the slight friction reduction on 
slopes due to the decreased normal force which would result in a lesser capsule 
pressure gradient. Experimental data from a 4 inch pipeline at an angle of inclination 
of 5° showed the accuracy of this approach to be within 15% for cylindrical capsules 
of a specific gravity of about twice the carrier liquid. The accuracy improves rapidly 
as the specific gravities of the liquid and capsules merge and as the friction coeffi- 
cient decreases; for spheres the accuracy is even better. This approach can therefore 
be considered satisfactory for practical pipeline cases. A more detailed discussion 
of the bulk density approach is given in Reference 2, Chapter 25. 

The total pipeline pressure gradient can now be calculated from its components 
using equation 2-5. 


20 


2.7 Total Throughput 

The total liquid throughput may be calculated from the bulk liquid velocity, 
the capsule velocity, the linear fill, the pipe diameter and the diameter ratio as 
discussed in Chapter 7. 

For cylinders 

irD 2 


Q f 576 


[ V b - V C kS F J 

while for spheres it is equal to 

Q 


, - i2i r 

f 576 L 


V b - 0.667 k 2 F 


The total throughput for cylinders is equal to: 
77 d 2 


Q 


576 


V F 
c 


while for spheres it is equal to 


Q = 1*1 
c 864 


V F 
c 


ft. 3 / 


sec, 


ft. 3 / 


sec. 


ft. 3 / 


sec. 


ft. 3 / 


sec. 


(2 - 15) 


(2 - 16) 


2.8 Power Requirements 

The energy required to move a pipeline full of liquid usually represents a 
significant operating cost for the pipeline system., The addition of capsules to a 
pipeline system will in all but a few cases increase the pressure gradient resulting in 
higher power consumption than in a pure liquid cariying pipeline. 

The theoretical minimum horsepower required is the hydraulic horsepower which 
is a function of the product of the total volumetric flowrate and the pressure drop. For 
the pipeline this becomes: 

77 D 3 V b A? 


Hhp = 


4 x 550 


(2 - 17 ) 


21 


However, the power input to the system will be considerably higher depending on the 

system efficiency. In most cases electrical power would be used but any source of 

power could be employed. The electric horsepower (Ehp) is: 

Eh = Hh p 

" system efficiency 

This is the horsepower used to determine the operating costs of a system. 

The system efficiency may best be described by considering a single pumping 
station. It essentially consists of three sub-systems and may be represented schema- 
tically as follows: 


Power In 



Power Out 


Symbols for the component efficiences are indicated in brackets. 

The "power out" is the power required to move the contents of the pipeline to 
the next pumping station, and is equal to the pipeline hydraulic horsepower described 
earlier. Also, the sum of the power inputs to each station is the total electric horse- 
power which can be shown to be: 

Ehp = Hhp (E b . E p . EJ (2 - 18) 

The pump and motor efficiencies are commonly known and are a function of type, 
size and operating conditions. No information is available on the efficiency of a 
capsule bypass system which will vary with the type of bypass chosen. However, the 
rotary vane bypass device can be expected to give efficiences similar to centrifugal 
or vane type pumps, i.e. in the range of 50% - 80%. 


22 


2.9 Pumping Station Requirements 

The total number of pumping stations required can be calculated directly from 
the total pipeline pressure drop, assuming that the energy which is lost on the inclines 
is regained on the declines, but the position of each pipeline pumping station is a 
function of the topography of the terrain over which the pipeline is constructed. 

The number of pumping stations required, including one at the origin will be: 

N = p^- (2-19) 

d s 

and rounded off to the next higher whole number. The average station spacing is: 

L 

SS = miles (2 - 20) 

and the discharge pressure at each station becomes: 

p d = 4r +p s (2 - 21) 

This value must not exceed the allowable working pressure (P) of the pipe and the 
fittings. 

The horsepower for the system can also be equally divided among the pumping 

stations. The size of the pumping station is often rated on the basis of the installed 

Hh D 

brake horsepower of the motor, and since the hydraulic horsepower per station is — , 
the brake horsepower is: 

Bhp = — ^ (2 - 22) 

E b • E p • N 

Combining equation 2-22 with equation 2-18, the electric horsepower per station 
(N = 1) becomes: 



m 


(2 - 23) 


23 


The brake horsepower is assumed to come from the motor driving the pump, i.e. 
the capsules are not forced into the pipeline by any other power source. If other 
power sources are involved they each contribute according to their volumetric flow- 
rates since the pressure is the same for all. The only time this would be of 
concern is at the first pumping station where a capsule extruder may be inserting 
capsules directly into the pipeline against the pump discharge pressure. In this 
event the pump has to displace liquid only and no bypass will be required. 


Bhp 


144Q f (P d - P s ) 


550 E 

P 


(2 - 24) 


For any bypass system envisaged the pump will always have to displace the 
volume of liquid required to move the capsules, plus an additional volume equal to 
that occupied by the capsules in the bypass system. Thus the pump brake horsepower 
may be expressed in terms of the total flowrate i.e.: 

144 Q (P - P ) 

,h l> = 550 i r 1 12 - 25 > 

p ° 

In capsule pipelining, extra pressure may or may not be required to restart 
a pipeline full of capsules from rest, depending upon the relationship between the 
threshold capsule pressure gradient at zero capsule velocity and the pressure gradient 
for the pipeline with moving capsules. Since horsepower is represented by a product 
of total flow and pressure drop an increase in pressure gradient here usually does not 
require extra horsepower since the bulk velocity required to start a stalled capsule 
pipeline is usually much lower than the normal operating pipeline velocity. 
Experience with experimental pipelines has shown that stalled capsule trains can be 


24 


started readily; the pressure drop at zero capsule velocity however must be checked 
for each installation since it may affect the choice of pipe strength and pump output 
pressure. In liquid pipelines it is common practice to install about 20% more than the 
calculated horsepower. 

2.10 Summary of Design Steps 

1 . Calculate the required inside pipe diameter assuming a capsule velocity 
by means of equation 2 - 1 . 

2. Choose the nearest pipe diameter, and required pipe wall thickness to 
permit the desired operating pressure (P), from a table of standard size 
pipe and allowable working pressures. 

3. Correct the capsule velocity for the new pipe size using equation 
2-3 and assume initially that the bulk liquid velocity is equal to 
this capsule velocity. 

4. Calculate the liquid pressure gradient from the bulk liquid velocity 
using equation 2-8. 

5. Calculate the capsule pressure gradient using equation 2 - 6 for spheres, 
or 2 - 7 (with an appropriate coefficient of friction) for cylinders. 

6. Calculate the bulk liquid velocity using equations 2 - 10, 2-11 
and 2 - 12. 

7. Repeat steps 4, 5 and 6 once. 

8. Calculate the elevation pressure gradient using equation 2 - 14. 

9. Calculate the pipeline pressure drop by means of equation 2-5. 

10. Calculate the total liquid throughput rate using equation 2-15. 


25 


11 . Calculate the pipeline hydraulic horsepower using equation 2-17. 

12. Calculate the pumping station requirements: 

a) number of pumping stations using equation 2-19; 

b) station spacing using equation 2 - 20; 

c) discharge pressure from station using equation 2-21; 

d) brake horsepower to pump using equation 2-22; 

e) electric horsepower per station using equation 2-23. 

13. Repeat for changed input parameters. 


26 


2.11 Examples 

Consider the example of sulfur to be transported as cast capsules from Calgary 
to Vancouver. The capsules will be coated with plastic, and transported in a pipeline 
using water at 39°F as the carrier liquid. Two methods of capsule transport are possible; 
the sulfur can be cast into cylinders or into spheres. Both will be considered in the 
following sample calculations. 

2.11.1 Cylinders 

Cylinder transport is recommended when the solids specific gravity 
approaches that of the carrier liquid. Sulfur, however, which has a specific gravity 
of 1.8 may be foamed so that the bulk specific gravity of the cylindrical capsule can 
be made very close to that of water; it is assumed to be 1.1 in the following example: 

Fixed Input: 


Pipeline length 

L 

P 

= 

560 miles 

Elevation change 

Y 

= 

-3500 feet 

Capsule throughput 

W 

c 

= 

2 MMTPY 

Capsule specific gravity 

a 

= 

1.1 

Water specific gravity 

P 

= 

1.0 

Water viscosity 

V 

= 

1 .486 centistokes 

Coefficient of friction 

V 

2 

= 

0.15 

Efficiencies: Pump 

E 

P 

= 

0.8 

Bypass 

E b 

= 

0.7 

Motor 

E 

= 

0.93 


m 


27 


Assumed Values: 


Diameter ratio 

k 

= 0.89 

Linear line fill 

F 

= 0.8 

Capsule velocity 

V 

c 

= 2.5 ft./s 

Maximum working pressure 

P 

w 

= 1300 psi 

Pump suction pressure 

p 

s 

= 1 00 psi 


Calculations: 

1 . The pipe diameter using equation 2 - 1 is: 


2 . 


3 0 


D 


194.2 W 

c 


0-5 


a V k 2 F 
L c 

= f 194.2x2 I 0 - 5 

[l.l x 2.5 x 0.89 2 x 0.8_ 

= 14.93 inches 

Standard pipe size: in the absence of a code pertaining especially to 
capsule pipelining, U.S.A.S. Code B 31 .4 - 1966 is used and 16.0 
inch O.D. pipe with a wall thickness of 0.344 inches in API 5LX-46 
(i.e. 46,000 psi steel) is acceptable. Thus: 

D = 15.31 inches inside diameter 

Capsule velocity: the assumed capsule velocity is no longer valid for 
the fixed solids throughput and is therefore corrected to match the 
standard pipe size chosen (equation 2-3). 


28 


4. 


5. 


194.2 W 

V = £ 

c a k 2 F D 2 

194.2 x 2 

1.1 x0.89 2 x 0.8 x 15.31 2 

= 2.38 ft. /sec. 

Liquid pressure gradient: the liquid velocity is initially assumed equal 
to the capsule velocity. The liquid pressure gradient is obtained by 
using equation 2-8, since flow of water at this velocity in a 16 inch 
pipe will be turbulent: 



4.53 x 10“ 4 p V b 2 

D 


0.00230 p V L 1,68 i/°' 3S 
b 

D x. 3a 


0.000453x1. 0x2. 38 g 0.00230x 1 .0X2.38 1 - 68 x 1 .486 03 a 

15.31 15.31 L3S 


= 0.000473 psi/ft. 

Capsule pressure gradient: using equation 2-7 gives: 



0.433 k (a - p) ri 

3 



= 0.433 x 0.89 (1.1 - 1.0) 0.15 + 0.000473 


= 0.00625 psi/ft. 


6. 


Bulk liquid velocity: using equation 2-10 the bulk liquid velocity 
is equal to: 


29 


V, = k V + (1 - k : 
b c 


c , ( t ) b -»] 1 


S5 \ 0.571 


p V 


0.35 


= 0.89x2. 38 + (l - 0.89 s ) 


415x0.00625 [15.31 (1-0.89)] 


3-251 0.5 71 


1.0 X 1.486 0 85 


= 2.61 ft. /sec. 

A check on the Reynolds number shows that the annular liquid is 
turbulent so that equation 2 - 10 is the correct one to use. 

Using equation 2-11: 


Re = 
c 


C D (1 - k) (V, - k V ) 

2 be 

(1 - k a ) v 


_ 7742 x 15,31 x (1 - 0.89) (2.61 - 0,89 x 2.38) 
(1-0.89®) 1.486 

= 2.08 x 10 4 

Recalculations: using the above liquid velocity the liquid pressure 
gradient was recalculated to be: 

= 0.000559 psi/ft. 

The capsule pressure gradient: 


(r). 


0.00634 psi 


The bulk liquid velocity: 


V. = 2.61 ft. /sec. 
b 


30 


8 . 


9. 


10 . 


Elevation pressure gradient: using equation 2-13 the bulk specific 
gravity is: 

p b = p + k3 F ^ 

= 1 + 0.89 s x 0.8 (1.1 - 1.0) 

= 1.063 

Equation 2-14 gives the elevation gradient: 




62.43 /+N/ . 

144 - (±Y) 


5280 L 


62.43 

144 


x (-3500) x 1.063 


5280 x 560 
= -0.000546 psi/ft. 

The pipeline pressure drop, using equation 2 - 5 is equal to: 


A P = 5280 L 




(1 - F) + 


(-) 




5280 x 560 0.00634 x 0.8 + 0.000559 x 0.2 - 0.000546 


= 1.37 x IQ 4 


psi 


The total liquid throughput rate is calculated using equation 2 - 15: 
TT D 2 


Q 


576 


[ 


V k s F 
c 


] 


3.1416 x 15.31 s 
576 


£ 2 . 61 - 2.38 x 0.89 s x 0.8^ 


= 1.41 ft. 3 /sec. 


31 


11 . 


12 . 


The pipeline hydraulic power, using equation 2 - 17 is equal to: 

7 r D s V, A? 

Hh P = 4 x 550 

3.1416 x 15.31 s x 2.61 x 1.37 xlO 4 

4 x 550 


= 1.20 x 1 0 4 horsepower 
The pumping station requirements are: 

a) number of pumping stations, using equation 2 - 19 is: 



1 .37 x 10 4 
1300 - 100 

= 11.4 

and rounded to the next integer = 12 

b) the station spacing is obtained using equation 2 - 20: 


L 



560 

12 

= 46.7 miles 

c) using equation 2-21 the discharge pressure at each stations 
becomes: 


A P 
N 


+ P 

s 


1 .37 x 10 4 
— Y1 


= 1 .24 x 10 3 


+ 100 


32 


d) the brake horsepower for each pumping station using equation 
2-22 is: 

Hhp 

e, x e ;tn 

b p 

1.20 x 10 4 
0. 7 x 0.8 x 12 

= 1.78xl0 3 horsepower 

e) the electric horsepower per station is calculated using equation 
2 - 23: 

Eh P = fe 

m 

1.78 x 10 3 
0.93 

= 1.92 x 1 0 3 horsepower 

Calculations were made for several bulk capsule specific gravities 
and these are summarized in the following table for the same 
conditions as the above calculations. 


TABLE 2 - 1 


Results for Cylinders at Various Bulk Specific Gravities 


a 

V 

c 

AP 

Hhp 

1.25 

2.09 

3.41 x 10 4 

3.06 xlO 4 

1.20 

2.17 

2.73 x 10 4 

2.44 xlO 4 

1.10 

2.38 

1.37 x 10 4 

1.20 xlO 4 

1.05 

2.49 

.687 x 10 4 

.591 xlO 4 

1.01 

2.59 

.136 x 10 4 

. 1 13 x 10 4 


33 


The table shows the significant adjustment in power requirement 
that is obtained by changing the cylinder bulk specific gravity. 

2.11.2 Spheres 

The example of sulfur to be transported in the form of spheres is considered next. 
The sulfur is cast into solid spheres with a specific gravity of 1 .8 
Fixed Input: 


Pipeline length 

L 

P 

= 

560 miles 

Elevdtion change 

Y 

= 

-3500 feet 

Capsule throughput 

W 

c 

= 

2MMTPY 

Capsule specific gravity 

a 

= 

1.8 

Water specific gravity 

P 

= 

1.0 

Water viscosity 

V 

= 

1 .486 centistokes 

Efficiencies: Pump 

E 

P 

= 

0.8 

Bypass 

E b 

= 

0.7 

Motor 

E 

m 

= 

0.93 

Assumed Values: 

Diameter ratio 

k 

= 

0.89 

Linear line fill 

F 

= 

0.8 

Capsule velocity 

V 

c 

= 

2 ft. /sec. 

Maximum working pressure 

P 

w 

= 

1300 psi 

Pump suction pressure 

p 

s 

= 

100 psi 


34 


Calculations: 

1 . The pipe diameter using equation 2 - 1 is: 


2 . 


3. 


D 


291.3 W 

c 


0.5 


a V k 2 F 


c 


= | 291.3x2 I 0 * 5 

Ll.8 x 2 x 0.89 2 x 0.8j 

= 15.98 inches 

Standard pipe size: in the absence of a code pertaining especially to 
capsule pipelining, U.S.A.S. Code B 31 .4 - 1966 is used and 16.0 
inch O.D pipe with a wall thickness of 0.344 inches in API 5LX-46 
(i.e. 46,000 psi steel) is acceptable. Thus: 

D = 15.31 inches inside diameter 

Capsule velocity: the assumed capsule velocity is not longer valid for 
the fixed solids throughput and is therefore corrected to match the 
standard size chosen (equation 2-3): 


291.3 W 
_ c 

c a k 2 FD 3 

291.3 x 2 

1.8x0.89 s x 0.8 x 15.31 s 

= 2.18 ft. /sec. 


35 


4 . 


5. 


6 . 


Liquid pressure gradient: the liquid velocity is initially assumed 
equal to the capsule velocity. The liquid pressure gradient is 
obtained by using equation 2-8 since flow of water at this 
velocity in a 16 inch pipe will be turbulent: 
f A? \ 4.53 x 10‘ 4 p V b 2 0.00230 pV^* 68 i/ 0 ' 32 

H = 5 + D 1,3a 

= 0.000453X 1.0 x 2.18 s + 0.00230x 1 .0x2. 18 1 ' 68 x 1 .486 
15.31 15.31 1,38 

= 0.000405 psi/ft. 

Capsule pressure gradient: using equation 2-6 and adding the 
recommended percentage for good commercial cast spheres gives: 


(^’ P ) = [° - 00062 + 2,7 (r)] k2 [ K0 + °- 24(ff_ p)j 




j 0.89 s jj. 


0+0.24(1.8-1.0 


00062 + 2.7 x 0.000405 
= 0.00162 psi/ft. 

Bulk liquid velocity: using equation 2-10 the bulk liquid velocity is 
equal to: 


V = k V + (1 - k s 
b c 


| c .(tU d,, -‘ 

oj 1 - 25 ) 


0.25 

j 

I p 

V 

) 


0.89x2.18 + 0 -0.89 2 ) 


415 x 0.00162 [15.31 (1 -0.89)] 1 - i 


1.0 x 1.486 °* 25 


= 2.17 ft. /sec. 

A check on the Reynolds number shows that the annular liquid is 
turbulent so that equation 2 - 10 is the correct one to use. 


'. 38 


■] 


0.571 


36 


Using equation 2-11: 

C D (1 - k) (V, - k V ) 

Re = k — 

C 0 -k s ) i/ 

7742 x 15.31 x (1 - 0.89) (2. 17 - 0.89 x 2. 18) 

(1 - 0.89 s ) 1.486 

= 9.7 x 10 3 

Recalculations: using the above liquid velocity the liquid pressure 
gradient was recalculated to be: 

0.000401 psi/ft. 

The capsule pressure gradient: 

= 0.00161 psi/ft. 

The bulk liquid velocity: 




V b = 2.17 ft. /sec. 

Recalculations did not significantly alter the results because the 
calculated bulk liquid velocity turned out to be very close to that 
initially assumed. 

When this occurs no recalculations are necessary. 

Elevation pressure gradient: using equation 2-13 the bulk specific 
gravity is: 

p b = p + 1 kS F (<* - p) 


= 1+| (0.89 s ) x 0.8 (1.8- 1) 


1.338 


37 


10 , 


11 


Equation 2-14 gives the elevation gradient: 
62.43 


in- 


w <1Y) Ob 


5280 L 


62.43 

144 


x (-3500) x 1.338 


5280 x 560 
= -0.000687 psi/ft. 

The pipeline pressure drop, using equation 2 - 5 is equal to: 
A P = 5280 L 


(‘ 

-) F + (- 

'A 0- P > + (- 

r 1 ) 

V 

/c V 

L /f V 







= 5280 x 560 10.00161 x 0.8 + 0.000401 x 0.2 - 0.000687 
= 2.01 x 10 3 psi 

The total liquid throughput rate is calculated using equation 2-15 
Q 

t 

3. 1416 x 15.31 s 


= n 

f 576 [ 


V b - 0.667 V c k 3 F 


] 


[2.17 - 0.667 x 2.18 x 0.89 s x 0.8] 


576 

= 1.59 ft. 3 /sec. 

The pipeline hydraulic power, using equation 2 - 17 is equal to: 

it D s V, AP 
Hhp = 4 x 550 

3.1416 x 15.31 s x 2. 17x2006 
4 x 550 

= 1.46 x 10 3 horsepower 


38 


12 . 


The pumping station requirements are: 

a) number of pumping stations, using equation 2 - 19 is: 



2.01 x 10 3 

1300 - 100 

= 1.68 


b) 


c) 


and rounded to the next integer = 2 

the stations spacing is obtained using equation 2 - 20: 



560 

2 


= 280 miles 

using equation 2-21 the discharge pressure at each station 
becomes: 


d) 



+ P 

s 


2.01 x 10 3 

2 


+ 100 


= l.lOxlO 3 


the brake horsepower for each pumping station using equation 
2-22 is: 


Bhp 


Hhp 

E, x E x N 
b P 

1.46 x 10 3 
0.7x 0.8 x 2 


1.30 x 10 3 horsepower 


39 


e) the electric horsepower per station is calculated using equation 
2 - 23: 

ri Bhp 

Ehp = 

m 

1 . 30 x 10 3 
0.93 

= 1.40 x 10 3 horsepower 

13. Calculations were made for several pipe sizes and these are summarized 
in the following table for the same conditions as the above calculations. 

TABLE 2-2 


Results for Spheres in Various Pipe Sizes 


D 

V 

c 

t P 

Hhp 

8.249 

7.51 

4.90 x 10 4 

3.47 xlO 4 

10.374 

4.75 

1.59 x 10 4 

1.13 xlO 4 

12.374 

3.34 

.653 x 10 4 

.466 xlO 4 

13.438 

2.83 

.422 x 10 4 

.303 xlO 4 

15.312 

2.18 

.201 x 10 4 

.146 xlO 4 

17.250 

1.72 

.0894 x 104 

.0660x1a 1 

19.562 

1.33 

. 0237 x 10 4 

,0179x 10 4 


It has been assumed that the topography of the terrain over which the 
pipeline is built has a continuously negative gradient. This is of course 
not true between Calgary and Vancouver. In this case the pipeline 
first increases in elevation before it begins to decrease and this fact 


40 


must be taken into consideration when designing the capsule 
pipeline system in detail. 

The examples show that spheres often are superior to cylinders for the purpose 
of capsule transportation, at least from a hydrodynamic viewpoint. 

REFERENCES 

1. Capsule Pipeline System Analysis, RCA Information Series No. 67. 

2. Capsule Pipelining, Experimental Investigations Performed for SPRDA 

by the RCA, 1967- 68. 


41 



01 23456789 10 

CAPSULE VELOCITY, ft./sec. 


D = 

10.2 in. 

k = 

0.90 


F I GU RE 2-1 



CHAPTER III 


Pressure Data Tabulation 




CHAPTER III 


Pressure Data Tabulation 

The capsule pressure gradient is one of the most important design parameters 
in commercial capsule pipelines. In this chapter plots of the capsule pressure gradients 
are presented as measured with trains of capsules flowing in various liquids in experi- 
mental pipelines from ^ to 10 inches in diameter. The capsule pressure 
gradient is a function of the frictional drag between the capsules and the pipeline, 
lubricated by the carrier liquid. (See Chapter 6 of the Phase 2 Report for a more 
detailed discussion.) For this reason the data are presented according to the capsule 
and pipe surfaces used. The nomenclature used in the figures conforms with that used 
elsewhere in the report, except as indicated in the text and on the figures. 

An index with markers is provided ahead of the figures to simplify locating 
results relating to 1) pipe size; 2) carrier liquid; 3) various kinds of cylinder surfaces, 
and 4) spheres, true and irregular. The figures are grouped as outlined in the follow- 
ing index, which also includes a short summary pertinent to the data presented. 

INDEX 

Figures 3 - 1 to 3 - 17 Metal Cylinders, Flowing in Water in Metal Pipes 

Metal cylinders flowing in water in a metal pipe are 
characterized by high capsule pressure gradients. Data 
are presented for pipes of three diameters and capsules 
of various diameters in each. The data from the 10 inch 
pipeline are shown twice, first to a small scale to include 
the denser capsules and then to a larger scale to show 


44 


better resolution of the results with lighter capsules. 

The shaded areas indicate the range of data recorded due 
to the capsule and pipe surface conditions which normally 
change somewhat between succeeding experimental runs. 

Figures 3 - 18 to 3 - 23 The Difference between Meta! Cylinders and Cylinders of 

Other Materials Flowing in Water in a Metal Pipe 

Non-metal cylinders flowing in water in a metal pipe are 

characterized by low pressure gradients. The value of the 

capsule pressure gradient is closely related to the type of 

cylinder surface. 

In Figures 3 - 18 to 3 - 23 the results from metal capsule 
surfaces are compared with those from non-metal capsule 
surfaces. Polyken tape is a vinyl plastic tape used for 
corrosion protection of conventional oil pipelines. Wear 
tape is a hard plastic tape used as a means of friction 
reduction when surfaces slide past each other. Safety 
walk is a tape used on stairs to prevent accidents due to 
slipping. 

Figures 3 - 24 to 3 - 39 Non-Metal Cylinders Flowing in Water in Metal Pipes 

The following figures present pressure gradient data for non- 
metal cylinders flowing in water in a metal pipe. Data are 
shown for pipe sizes between 2 to 10 inches in diameter. 
Non-metal cylinders are characterized by low capsule 
pressure gradients „ 


45 


Figures 3 - 40 to 3 - 43 The Effect of Cylinder Length 

For a constant train length the effect of the length of each 
cylinder in the train was not found to be significant. 

In the four figures, pressure gradients for a train of 12 inch 

long capsules are compared with pressure gradients 

for a train of equal length of 48 inch long capsules in a 

4 inch stainless steel pipe. 

Figure 3-44 Single Cylinders 

The pressure gradient results with single cylinders more than 

5 pipe diameters in length are representative of results with 
trains of cylinders. 

The effect of the number of cylinders in a train of capsules 
is not significant compared with the effect of changes in 
capsule and pipe surface conditions which normally occur 
between experiments. The results in Figure 3-44 for a 
single cylinder check well with those shown in Figure 3-11 
for four capsules in a train. 

Figures 3 - 45 to 3 - 48 Metal Cylinders Flowing in Water in a Plastic Pipe 

Pressure gradient data for metal cylinders flowing in water 
in a 4 inch plastic pipe are not significantly different from 
data for plastic cylinders flowing in water in a 4 inch metal 
pipe. This is indicated by comparing Figures 3 - 45 to 
3-48 with Figures 3 - 27 to 3 - 32. 


46 


Figures 3 - 49 to 3 - 57 The Effect of Liquid Viscosity on Cylinders Flowing in a 

Metal Pipe 

The effect of liquid viscosity on the capsule pressure gradient 

for cylinders is very complex since the liquid affects both the 

degree of turbulence in the annulus and the lubrication 

between cylinder and pipe wall. Results for each type of 

liquid therefore will give somewhat different results. The 

nine figures in this group show the effects when mixtures of 

water and polyglycol are used as the carrier liquid. 

Some of the data of a particular experimental run show 

significant scatter as indicated by the shading in Figures 

3 - 54, 3 - 55, 3 - 56 and 3 - 57. A smooth curve 

was drawn through the shading in the figures to indicate the 

the general trend of the data. 

Figures 3 - 58 to 3 - 59 The Effect of Pipe Diameter 

The pressure gradients for cylinders of low specific gravity 
decrease significantly as the pipe diameter increases. The 
two figures show this effect for cylinders of two specific 
gravities. 

Figures 3 - 60 to 3 - 62 Effect of Capsule Density and Liquid Viscosity for Banded 

Cylinders 

Placing a band on a metal cylinder as indicated in the figures 

significantly reduces the capsule pressure gradient. Figures 

3 - 60 to 3 - 62 show this effect for two capsule densities and 

two liquid viscosities in a 4 inch stainless steel pipe. 


47 


Figures 3 - 63 to 3 - 66 Effect of Cylinder Diameter for Banded Cylinders 

Figures 3 - 63 to 3 - 66 show capsule pressure gradients for 
a train of cylinders with a 0*03 inch thick, 4 inch wide 
band, 5. 88 inch from the front, run in water in a 4 inch 
stainless steel pipe. Results from four cylinder diameters 
are shown. 

Figures 3 - 67 to 3 - 71 Effect of Band Thickness 

For a given band position the thickness of the band 
influences the capsule pressure gradient. Figures 3-67 
to 3 - 71 show this effect in a 4 inch stainless steel pipe. 
The band configuration shown in Figures 3-68 and 3-71 
gives the lowest pressure gradients. 

Figures 3 - 72 to 3 - 73 Effect of Capsule Density for Banded Cylinders 

The modification of the pressure gradient obtained with 
banded cylinders is dependent upon the capsule density. Th 
is shown when Figures 3-72 and 3-73 are compared with 
Figures 3 - 11, 3 - 12 and 3 - 13. 

Figures 3 - 74 to 3 - 79 Effect of Band Position 

The band position influences the capsule pressure gradient 
significantly. Figures 3 - 74 to 3 - 79 show this for 
cylinders with bands of various thicknesses and widths run 
in a 10 inch pipe. 


48 


The lowest pressure gradient is shown by the s = 8" 
curve in Figure 3 - 76. 

Figures 3 - 80 to 3 - 81 Effect of Band Width 

The effect of band width and thickness is shown in 
Figures 3-80 and 3-81 for cylinders run in a 10 inch 
pipe. 

Figures 3 - 82 to 3 - 83 The Effect of Pipe Diameter for Spheres 

When the commodity to be transported is dense, capsules in 
the form of spheres require much less pressure to move them 
than do cylinders. The capsule pressure gradient with sphere 
trains decreases significantly with increased pipe diameter as 
shown in Figures 3-82 and 3-83 for machined spheres. 

Figures 3 - 84 to 3 - 86 The Effect of Sphere Density and Sphere Diameter 

In a given pipe size the effect of density and diameter on 
the pressure gradient of machined spheres is not very great 
as shown in Figures 3 - 84 to 3 - 86. 

Figures 3 - 87 to 3 - 90 The Effect of Sphere Roughness 

There is a very significant difference between the pressure 
gradients of machined spheres and of cast spheres. The cast 
spheres require much more energy to maintain movement as 
shown in Figures 3 - 87 to 3 - 90. The cast spheres were 
uneven and dented with a variation from spherical up to 


49 


0.25 inch. The machined spheres were turned spherical 
on a lathe to a tolerance of 0.02 inch. 

Figures 3 - 91 to 3 - 93 Effects of Liquid Viscosity on Spheres in a Pipeline 

The liquid viscosity has an important influence on the 
pressure gradient of sphere trains as shown in Figures 
3 - 91 to 3 - 93. 

Figure 3-94 Liquid Pressure Gradient as a Function of Liquid Velocity 

The liquid pressure gradient is shown as a function of the 
liquid velocity for the 2 , 4 and 10 inch pipes. The liquid 
velocity and capsule velocity may be assumed to be equal 
for the purpose of a preliminary comparison between the 
capsule pressure gradient of the preceeding figures with 
the liquid pressure gradient of Figure 3 - 94. 

The liquid pressure gradients were not plotted on the 
preceeding figures since these figures combine data of 
several capsule specific gravities on one plot and the 
liquid pressure gradient as a function of capsule velocity 
varies with capsule specific gravity. 




PIPE SIZE AND MATERIAL 

^ inch stainless steel 

H inch copper 

2 inch steel 

4 inch butyrate plastic 

4 inch stainless steel 

10 inch steel 

LIQUID 

Water 

Poly glycol -water mixture 

CYLINDERS 

Steel 

Stainless steel 

Butyrate plastic 

Coated cylinder surface 

Banded cylinders 

SPHERES 

True spheres 

Irregular spheres 


CAPSULE PRESSURE GRADIENT, psi./ft. 


51 



D= 0.620 in. 

L t = 8.3 ft. 

v= i.o cs. 

k = 0.910 

N = 8 

o 

II 

Qv 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3 - 1 


52 




D = 2.000 m - 

L t = 5.2 ft - 

</) 

o 

o 

IS 

k = 0.880 

N= 5 

II 

o 


Stainless steel cylinders in steel pipe. 


FIGURE 3-2 


CAPSULE PRESSURE GRADIENT, psi./ft. 


53 



D = 4.029 in. 

4= 8.2 ft. 

V - 1.0 cs. 

k = 0.805 

N = 4 

o 

ll 

a. 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3 -3 


54 


5 

CO 

CL 


z 

LU 

Q 

< 

Cd 

o 


LU 

QC 

3 

co 

co 

LU 

C£ 

Q_ 


LU 

3 

cO 

Q_ 

< 

u 



D = 4.029 in. 

L t = 8.2 ft. 

V - 1.0 cs. 

k = 0.805 

N = 4 

O 

II 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3-4 


55 



D = 4.029 in. 

L t = 8.2 ft. 

v= 1.0 cs. 

k = 0.866 

N = 4 

>> 

It 

o 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3 


5 


56 



D= 4.029 in. 

4= 8.2 ft. 

v= 1.0 CS. 

k= 0.866 

N = 4 

O 

II 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3-6 


CAPSULE PRESSURE GRADIENT, psi./ft. 


57 



D = 4.029 in. 

L t = 8.2 ft. 

ii 

o 

o 

c/> 

k= 0.934 

N = 4 

o 

ii 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3-7 


58 


c/) 

Ql 


Q 

< 

o 


Cd 

=> 

CO 

CO 

UJ 

Q_ 


3 

cO 

Q_ 

< 

u 



CAPSULE VELOCITY, ft. /sec. 


D = 4.029 in. 

L t = 8.2 ft. 

i.o cs. 

k = 0.934 

N = 4 

P= 1.0 


Stainless steel cylinders in stainless steel pipe, 


FIGU RE 3 - 8 


59 



D = 10.02 in - 

■t— > 

o 

CM 

_) 

v = 1.0 cs - 

k = 0.85 

N = 5 

o 

II 


Steel cylinders in a steel pipe. 


Effect of capsule S.G. 


FIGURE 3 


9 


CAPSULE PRESSURE GRADIENT, psi./ft. 


60 



D = 10.02 in. 

L t = 20 ft. 

n 

o 

o 

c/> 

k = 0.85 

N= 5 

p= 1.0 


Steel cylinders in steel pipe. 


Effect of capsule S.G . 


FIGURE 3-10 



CAPSULE PRESSURE GRADIENT, psi./ft. 


61 



CAPSULE VELOCITY, ft. /sec. 


D= 10.02 in. 

4= 20 ft. 

v = 1.0 CS. 

k = 0.90 

N = 5 

O 

II 


Steel cylinders in steel pipe 


Effect of capsule S. G . 


FIGURE 3 


11 


CAPSULE PRESSURE GRADIENT, psi./ft. 


62 



Steel cylinders in steel pipe. 


Effect of capsule S.G. 


FIGURE 3-12 


63 



D= 10.02 in. 

4= 20 ft- 

v- 10 CS. 

k = 0.90 

N = 5 

O 

II 


Steel cylinders in steel pipe. 


Effect of capsule S.G. 


FIGURE 3-13 


64 



CAPSULE VELOCITY, ft. /sec. 


D= 10.02 in. 

L t = 20 ft- 

V= 1.0 cs. 

k = 0.95 

N = 5 

II 

o 


Steel cylinders in steel pipe 


Effect of capsule S . G. 


FIGURE 3-14 


65 



D= 10.02 in. 

4= 20 ft. 

to 

o 

o 

II 

k= 0.95 4 

N = 5 

o 

II 


Steel cylinders in steel pipe. Effect of capsule S.G. 


FIGURE 3-15 


66 



D = 10.02 in. 

L t = 20 ft. 

</) 

o 

o 

II 

k = 0.95 

N= 5 

o 

II 

Qs. 


Steel cylinders in steel pipe. Effect of capsule S.G. 


FIGURE 3-16 


67 



D = 10.02 in. 

L t = 20 ft. 

V- 1.0 cs. 

k = Indicated 

N= 5 

O 

II 


Buoyant steel cylinders in steel pipe. Effect of diameter ratio. 


FIGURE 3-17 


68 



CO 

d 


Z 

LU 

O 

< 

CH 

o 


LU 

CH 

3 

cO 

cO 

LU 

oeL 

Q_ 


LU 
l 

3 

CO 

o_ 

< 

U 



D = 4.029 in. 

L t = 8.2 ft. 

ii 

o 

o 

CO 

k = 0.87 

N = 4 

o 

II 


Effect of covering stainless steel cylinders with Polyken tape. 


FIGURE 3-18 


69 


a 

] 

I 



D =4.029 in. 

4= 8 - 2 ft. 

V- 1.0 cs. 

k = 0.87 

N = 4 

O 

II 


The effect of cylinder surface in stainless steel pipe for 
capsule S.G.= 1.25. 


FIGURE 3-19 


70 



D = 4.029 in. 

4= 8.2 ft. 

V = 1.00 cs„ 

k = 0.87 

N = 4 

P= 1.00 


The effect of cylinder surface in stainless steel pipe when 
capsule S . G . = 1.03. 


FIGURE 3-20 


CAPSULE PRESSURE GRADIENT, psi./ft. 


71 



D = 10.02 in. 

L t = 12 ft. 

ii 

o 

° 

■ w 

k = 0.90 

N = 3 

o 

II 


Effect of coating capsule and pipe walls. 

*Residue from running of the Poly ken covered cylinders coated the bottom 
portion of the steel pipe. This produced a marked decrease in the capsule 
pressure gradient. The residue was easily removed with a steel -wire brush 
and plain steel cylinders. FIGURE 3 


21 


CAPSULE PRESSURE GRADIENT, psi./ft. 


72 



D = 10.02 in. 

L t = 20 ft. 

V= 1.0 cs. 

k = 0.89 - 0.90 

N= 5 

O 

II 


Various coatings on steel cylinders in steel pipe. 


FIGURE 3-22 


73 



D = 10.02 in - 

L t = 20 ft- 

v = 1.0 C s - 

k = 0.89 

N = 5 

P= 1.0 


Various coatings on steel cylinders in steel pipe. 


FIGURE 3-23 


74 



co 

CL 


Z 

UJ 

O 

< 

o 

UJ 

C£ 

3 

cO 

cO 

UJ 

CxL 

CL- 


UJ 
I 

3 

cO 

Q_ 

< 

u 



D = 2.000 in. 

L t= 5.4 ft. 

C/5 

o 

o 

II 

> 

k = 0.875 

N= 5 

II 

o 


Butyrate plastic cylinders in steel pipe. 


1 


l 


[ 


FIGURE 3-24 


CAPSULE PRESSURE GRADIENT, psi./ft. 


75 



D = 4.029 in - 

4= 8.2 ft. 

V= 1.0 CS. 

k = 0.806 

N= 4 

O 

• 

F— 

ii 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-25 


76 


+~* 



c n 
Q. 


z 

LU 

Q 

< 

cl 

O 


LU 

CtL 

3 

00 

LO 

LU 

CL 

Q_ 


LU 
— 1 
3 
cO 
CL 
< 
U 



D = 4.029 in. 

L t = 8.2 ft. 

V - i # o cs. 

k= 0.806 

N = 4 

o 

© 

ii 

Qv 


Butyrate plastic cylinders in stainless steel pipe, 


FIGURE 3 - 


26 



77 



0 = 4.029 in. 

4= 8.2 ft. 

v = 1.0 CS. 

k= 0.870 

N= 4 

O 

II 


Butyrate plastic cylinders in stainless steel pipe, 


FIGURE 3-27 



78 



0 


1 2 3 

CAPSULE 


4 5 6 

VELOCITY 


7 8 9 

ft. /sec. 


10 


D = 4.029 in. 

L t = 8.2 ft. 

CO 

o 

o 

II 

k = 0.870 

N = 4 

o 

1—“ 

II 

Ck 


1 

L 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-28 



CAPSULE PRESSURE GRADIENT, psi./ft. 


79 



D= 4.029 in. 

4= 8.2 ft. 

V = 1.0 cs. 

k = 0.902 

N = 4 

o 

il 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-29 



80 



0 


2 3 

CAPSULE 


4 5 6 7 8 9 1© 

VELOCITY, ft. /sec. 


D = 4.029 in. 

4= 8.2 ft. 

v= 1.0 cs. 

k= 0.902 

N= 4 

>5 

II 

O 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-30 


81 



D = 4.029 in. 

4= 8.2 ft. 

< J) 

o 

o 

II 

k = 0.930 

II 

z 

II 

• 

o 


Butyrate plastic cylinders 


in stainless steel 


pipe. 


FIGURE 3-31 


82 



D= 4.029 in. 

L t = 8.2 ft. 

V= 1.0 cs. 

k = 0.930 

N = 4 

IS 

o 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-32 


83 



D = 10.02 in. 

L t = 20 ft. 

V - 1.0 cs. 

k= 0.89 

N = 5 

"0 

II 

o 


Phenolic coated steel cylinders in steel pipe. Effect of 
capsule S.G. 


FIGURE 3-33 


84 



D = 10.02 in. 

L t = 20 ft. 

V- 1.0 cs. 

k= 0.89 

N = 5 

SI 

o 


Phenolic coated steel cylinders in steel pipe. 
Effect of capsule S.G. 


FIGURE 3-34 


85 



D= 10.02 in. 

4= 20 ft- 

v = 1.0 cs. 

k = 0.89 

N = 5 

II 

O 


Polyurethane coated steel cylinders in steel pipe. Effect of 
capsule S. G. 


FIGURE 3-35 



CAPSULE PRESSURE GRADIENT, psi./ft. 


86 



D= 10.02 in. 

L t = 20 ft. 

1.0 cs. 

k= 0.89 

N = 5 

o 

II 

Qs. 


Polyurethane coated steel cylinders in steel pipe. 
Effect of capsule S.G. 


FIGURE 3-36 


87 



c r ) 
CL 


z 

LU 

Q 

< 

a; 

O 


LU 

3 

co 

CO 

LU 

C£ 

Q_ 


LU 

l 

3 

cO 

Q_ 

< 

u 



D= 10.02 in. 

L t = 20 ft. 

V= 1.0 cs. 

k = 0.89 

N = 5 

O 

II 

Ck 


Epoxy coated steel cylinders in steel pipe. Effect of capsule S.G. 


FIGURE 3-37 


88 



c/> 

CL 


Z 

LU 

Q 

< 

o 


LU 

ct: 

3 

CO 

CO 

LU 

O' 

CL- 


UJ 
I 

3 

cO 

Ok 

< 

U 



I 

I 



D = 10.02 in - 

4= 20 ft- 

V= 1.0 cs - 

k = 0.89 

N = 5 

O 

II 


Epoxy coated steel cylinders in steel pipe. 


Effect of capsule S.G. 


FIGURE 3 


38 


CAPSULE PRESSURE GRADIENT, psi./ft. 


89 



D= 10.02 in. 

L t = 20 ft. 

V = 1.0 CS. 

k= 0.90 

N = 5 

o 

ll 


Polyken tape covered cylinders in steel pipe. 


Effect of capsule S.G. 


FIGURE 3-39 


90 



CO 

Q. 


z 

LLJ 

O 

< 

OtL 

o 

LLJ 

CL 

ZD 

CO 

CO 

LU 

CxL 

Q_ 


LU 

3 

cO 

CL 

< 

U 



D= 4.029 in. 

L t = 8.3 ft. 

II 

o 

o 

CO 

k = 0.870 

00 

1! 

Z 

o 

i— • 

II 

a. 


Stainless steel cylinders in stainless steel pipe 


FIGURE 3 - 


40 


91 



D = 4.029 m - 

L t= 8.1 ft - 

v = 1.0 cs. 

k = 0.866 

N = 2 

>> 

II 

*o 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3 


41 


92 



D = 4.029 in. 

L t = 8.3 ft. 

V = l.o cs. 

k = 0.871 

N = 8 

o 

ll 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3-42 


93 



</) 

Cl 


Z 

UJ 

Q 

< 

CsL 

o 

UJ 

CtL 

3 

tO 

to 

LU 

C£ 

Q- 

LU 
i 

3 

tO 

Q_ 

< 

u 



D = 4.029 in. 

4= 8.1 ft. 

v = 1.0 cs - 

k= 0.866 

N = 2 

II 

O 


Stainless steel cylinders in stainless steel pipe. 


FIGURE 3-43 


94 



D= 10.02 in. 

L t = 4 ft. 

V = 1.0 cs. 

k = 0.90 

N = 1 

ii 

o 


Single steel cylinder in steel pipe, 


Effect of capsule S. G . 


L 

I 

[ 


FIGURE 3-44 


95 



D = 4.026 in. 

L t = 8.2 ft. 

v= 1.0 cs. 

k = 0.867 

N = 4 

O 

II 


Stainless steel cylinders in butyrate plastic pipe. 


FIGURE 3-45 


96 



CAPSULE VELOCITY, ft. /sec. 


D = 4.026 in. 

L t = 8.2 ft. 

v= 1.0 cs. 

k = 0.867 

N = 4 

O 

II 

Qv 


Stainless steel cylinders in butyrate plastic pipe. 


FIGURE 3-46 



97 



D = 4.026 in. 

4= 8.2 ft. 

V= 1.0 cs. 

k = 0.935 

N= 4 

O 

II 


Stainless steel cylinders in butyrate plastic pipe. 


FIGURE 3-47 


98 



D = 4.026 in. 

4= 8.2 ft. 

, 

CO 1 

o 1 

o 

p— 

II 

k = 0.935 

N = 4 

P= 1.0 


Stainless steel cylinders In butyrate plastic pipe. 


FIGURE 3-48 


99 



D = o.62 in. 

L t = 8.3 ft. 

^ = Indicated. 

k= 0.910 

N= 8 

P- 1.00-1. 06 


Stainless steel cylinders in stainless steel pipe. Effect of liquid 
viscosity for cylinders of neutral density. 


FIGURE 3-49 


100 



D = 0.620 in. 

L t = 8.3 ft. 

^ = Indicated 

k = 0.910 

N= 8 

yP 3 i. 00 ~ 1.06 


Stainless steel cylinders in stainless steel pipe. Effect of liquid 
viscosity for cylinders with buoyed S.G. =0.03. 


FIGURE 3-50 


101 



D= 0.620 in. 

4= 8.3 ft. 

v 1 

v = Indicated. 

k = 0.910 

CO 

II 

Z 

/ > = 1 .00-1 .06 


Stainless steel cylinders in stainless steel pipe. Effect of 
liquid viscosity for cylinders with buoyed S.G. = 0.25. 


FIGURE 3-51 


102 



D - 2„000 » n - 

4= 5.4 ft. 

i 

V - Indicated. 

k = 0.875 

N = 5 

p = 1.00-1. 06 


Butyrate plastic cylinders in steel pipe. Effect of liquid viscosity 
for buoyed capsule S.G. = 0.00. 


FIGURE 3-52 


103 



D = 2.000 in. 

L t = 5.4 ft. 

V - indicated. 

k= 0.875 

N= 5 

P= 1.00-1. 06 


Butyrate plastic cylinders in steel pipe. Effect of liquid 
viscosity for buoyed capsule S.G. - 1.25. 


FIGURE 3-53 


104 



CO 

CL 


Z 

LU 

Q 

< 

az 

O 

LU 

C£ 

ZD 

cO 

cO 

LU 

C£ 

Q_ 

LU 

ID 

cO 

Q_ 

< 

u 



D= 4.029 in. 

L t = 8.5 ft- 

v= 3.1 cs. 

k = 0.806 

II 

z 

P= 1.03 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-54 


CAPSULE PRESSURE GRADIENT, psi./ft. 


105 



D = 4.029 in. 

L t = 8.5 ft. 

V - 26.2 CS. 

k = 0.806 

N = 4 

P- 1.07 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-55 


106 



in 

Cl 


Z 

LLJ 

Q 

< 

cl 

o 


UJ 

CL 

Z) 

cO 

CO 

LU 

CL 

CL- 


UJ 

3 

uO 

CL 

< 

u 



D= 4.029 in. 

4= 8.5 ft- 

V= 3.1 cs. 

k = 0.930 

N = 4 

P = 1.03 


Butyrate plastic cylinders in stainless steel pipe. 


FIGURE 3-56 


107 



c/) 

CL 


z 

LLi 

Q 

< 

o 


LU 

Cxi 

3 

CO 

cO 

LLi 

Cxi 

CL- 


UJ 

Z> 

cO 

Q_ 


< 

u 



D = 4.029 in. 

4= 8.5 ft. 

v = 25.9 CS. 

k = 0.930 

N = 4 

P = 1.07 


Butyrate plastic cylinders in stainless steel. pipe. 


FIGURE 3-57 


108 



D = Indicated 

4= 5-20 ft. 

V - 1,0 cs. 

k = 0.86-0.91 

N = 4-5 

P= 1.0 


Metal cylinders in metal pipe. Effect of pipe diameter 
for capsul e S . G . = 1.03. 


FIGURE 3-58 


109 



^ ~ Indicated 

L t= 5-20 ft - 

v= 10 cs. 

k = 0.86-0.91 

N= 4 _ 5 

O 

II 


Metal cylinders in metal pipe. Effect of pipe diameter 
for capsule S . G. = 1.13. 


FIGURE 3-59 


no 



D = 4.029 in. 

L t = 8.5 ft. 

V = 3.0 CS. 

k= 0.87 

N = 4 

P= 1.03 


Butyrate plastic cylinders with and without Cadco wear tape 
bands in stainless steel pipe. 


FIGURE 3-60 


Ill 



D= 4.029 in- 

L t = 8.5 ft- 

V = 3.0 cs. 

k = 0.87 

N = 4 

P- 1.03 


Butyrate plastic cylinders with and without Cadco wear tape 
bands in stainless steel pipe. 


FIGURE 3-61 


112 



D = 4.029 in. 

L t = 8.5 ft. 

V = 10.0 CS. 

k = 0.87 

N = 4 

1.06 


Butyrate plastic cylinders with and without Cadco wear tape 
bands in stainless steel pipe. 


FIGURE 3-62 


113 



D= 4.029 in. 

L t = 8.2 ft. 

V= 1.0 CS. 

k = 0.806 

N = 4 

O 

II 


Stainless steel cylinders with Polyken tape bands in stainless steel pipe. 


FIGURE 3-63 


114 



D= 4.029 in. 

L t = 8.2 ft. 

v= 1.0 cs. 

k = 0.870 

N = 4 

O 

II 


Stainless steel cylinders with Polyken tape bands in stainless steel pipe. 


FIGURE 3-64 


115 



c/) 

Cl 


Z 

UJ 

O 

< 

cn 

o 

UJ 

O' 

3 

cO 

CO 

UJ 

C£ 

Q_ 


UJ 
l 

3 

cO 

Q_ 

< 

u 



D = 4.029 in. 

L t = 8.2 ft. 

v = 1.0 cs. 

k = 0.903 

N = 4 

O 

II 


Stainless steel cylinders with Polyken tape bands in stainless steel pipe. 


FIGURE 3-65 


116 



D = 4.029 in. 

L t = 8.2 ft. 

V - 1.0 cs. 

k = 0.929 

N = 4 

ii 

o 


Stainless steel cylinders with Polyken tape bands in stainless steel pipe. 


FIGURE 3-66 


117 



co 

a 


z 

LU 

D 

< 

QL 

o 

LU 

CtL 

3 

CO 

CO 

LU 

C£ 

a. 


LU 

3 

CO 

Q_ 

< 

u 



D= 4.029 in. 

4= 8.2 ft. 

v= 1.0 CS. 

k= 0.903 

N = 4 

O 

II 


Stainless steel cylinders with Polyken tape bands in stainless steel pipe. 


FIGURE 3-67 


118 



D = 4.029 in. 

L t = 8.2 ft. 

V = 1.0 cs. 

k= 0.903 

N= 4 

"0 

II 

o 


Stainless steel cylinders with Polyken tape bands in stainless steel pipe. 


FIGURE 3-68 



119 



0 = 4.029 in. 

L t = 8.2 ft. 

V = 1.0 cs. 

k = 9.03 

N= 4 

II 

O 


Stainless steel cylinders with two Polyken tape bands each in stainless 

steel pipe. 


FIGURE 3-69 


120 



D = 4.029 in. 

L t = 8.2 ft- 

v= 1.0 cs. 

k = 0.903 

N= 4 

O 

P" 

ii 


Stainless steel cylinders with two Polyken tape bands each in stainless 

steel pipe. 


FIGURE 3-70 


121 



D = 4.029 in. 

L t = 8.2 ft. 

V - l.o cs. 

k = 0.903 

N= 4 

II 

o 


Stainless steel cylinders with two Polyken tape bands each in stainless 

steel pipe. 


FIGURE 3-71 


122 



D = 10.02 in. 

L t = 20 ft. 

V - 1.0 cs. 

k = 0.90 

N = 5 

o 

ii 

a. 


Polyken tape covered steel cylinders 
steel pipe. Effect of S 


with Polyken tape bands 
. G. 


in 


FIGURE 3-72 


CAPSULE PRESSURE GRADIENT, psi./ft. 


123 



D = 10.02 in. 

L l= 20 ft. 

V - 1.0 cs. 

k = 0.90 

N = 5 

P- 1.0 


Polyken tape covered steel cylinders with Polyken tape bands in 
steel pipe. Effect of S. G. 


FIGURE 3-73 


124 



D = 

10.02 in - 

L t= 

20 

V - 

1.0 cs. 

k = 

0.90 

N = 

5 

/>= 

1.0 


Effect of band position: 0.25 in. X 2 in. Polyken tape bands positioned 
as shown. Cylinders wrapped with 0.015 in. thick Polyken tape and 

run in steel pipe. 


FIGURE 3-74 


CAPSULE PRESSURE GRADIENT, psi./ft. 


125 



D= 10.02 in. 

L t = 20 ft. 

CO 

o 

o 

II 

k = 0.90 

N = 5 

o 

r— • 

ii 


Effect of band position: 0.063 in. X 2 in. Polyken tape bands 
positioned as shown. Cylinders wrapped with 0.015 in. thick 
Polyken tape and run in steel pipe. 


FIGURE 3-75 


126 



D = 10.02 in. 

4= 20 ft. 

V- 1.0 CS. 

k= 0.90 

N = 5 

O 

II 


Effects of band position: 0. 125 in. X 4 in. Polyken tape bands positioned 
as shown. Cylinders wrapped with 0.015 in. thick Polyken tape and run 

in steel pipe. 


FIGURE 3-76 


CAPSULE PRESSURE GRADIENT, psi./ft. 


127 



CAPSULE VELOCITY, ft. /sec. 


D = 10.02 in. 

4= 20 ft. 

v= 1.0 CS. 

k = 0. 90 

N = 5 

'b 

ii 

• 

O 


Effect's of band position: 0.063 in. X 4 in. Poly ken tape bands 
positioned as shown. Cylinders wrapped with 0.015 in. thick 
Polyken tape and run in steel pipe. 


FIGURE 3-77 


128 



D= 10.02 in. 

L t = 20 ft- 

v= 1.0 CS. 

k = 0.90 

N = 5 

O 

II 


Effect of band position: 0.25 in. X 6 in. Polyken tape bands positioned 
as shown. Cylinders wrapped with 0.015 in. thick Polyken tape and run 

in steel pipe. 


FIGURE 3-78 


CAPSULE PRESSURE GRADIENT, psi./ft. 


129 



D = 10.02 in. 

L t = 20 ft. 

y = i.o cs. 

k = 0.90 

N = 5 

o 

li 


Effects of band position: 0.125 in. X 6 in. Polyken tape bands 
positioned as shown. Cylinders wrapped with 0.015 in. thick 
Polyken tape and run in steel pipe. 


FIGURE 3-79 


130 



D= 10.02 in. 

4= 20 ft- 

v = 1.0 cs. 

k = 0.90 

N= 5 

II 

O 


Effect of band width: 0.062 in. thick Polyken tape bands positioned 
as shown. Cylinders wrapped with 0.015 in. thick Polyken tape and 
run in steel pipe. 


FIGURE 3-80 


131 



c f) 
Cl 


Z 

UJ 

a 

< 

DL 

o 

UJ 

C£ 

3 

cO 

cO 

LLt 

Q£ 

Q_ 

LU 

3 

cO 

Q_ 

< 

u 



01 23456789 10 

CAPSULE VELOCITY, ft. /sec. 


D = 10.02 i n. 

L t = 20 ft. 

V - t .o cs. 

k = 0.90 

N = 5 

o 

ii 


Effect of band thickness: 4 in. wide Polyken bands. Cylinders 
wrapped with 0.015 in. Polyken tape and run in steel pipe. 


FIGURE 3-81 


132 



D = Indicated 

L^= Varies 

CO 

o 

o 

II 

k= 0.90 

N - varies 

P- 1.0 


Trains of machined spheres of the density of aluminum in 
the 10 and 4 inch pipelines. 


FIGURE 3-82 


133 



D = Indicated 

varies 

v = 1.0 cs - 

k = indicated 

N = varies 

II 

*o 


Trains of dense machined spheres in the 10, 4 and 2 inch pipelines. 


FIGURE 3-83 


134 



CO 

CL 


Z 

LU 

Q 

< 

O 

LU 

C£ 

3 

CO 

CO 

LU 

O' 

CL. 

LU 

3 

UO 

Q_ 

< 

U 



D = 2.000 ' n - 

L t = 1.45 ft- 

v= 1.0 CS. 

k= 0.875 

N = 10 

O 

II 


Machined spheres in 2 in. pipe. Effect of capsule 5. G. 


FIGURE 3-84 


CAPSULE PRESSURE GRADIENT, psi./ft. 


135 



D= 10.02 in. 


V = 1.0 cs. 

k= 0.90 

N = indicated 

o 

ii 


Trains of machined spheres in steel pipe. 


FIGURE 3-85 


136 



D = 4.029 in. 

l-^= varies ft. 

o 

n 

> 

k = Indicated 

N = 10 

o 

i— • 

n 


Machined spheres in steel pipe. Effect of diameter ratio. 


FIGURE 3-86 


137 



D = 10.02 in. 


V - l.o cs. 

k= 0.90 

On 

II 

z 

o 

ll 


Trains of cast spheres and machined spheres in steel pipe. 


FIGURE 3-87 


138 


5 

c n 
Cl 


Z 

LU 

Q 

< 

O' 

o 


uu 

O' 

ZD 

cO 

cO 

LU 

O' 

Cl- 


uj 

i 

Z> 

U1 

Q- 

< 

u 



D= 10.02 in. 


V- 1.0 cs. 

k = 0.90 

N = 11 

o 

II 


Trains of cast spheres and machined spheres in steel pipe. 


FIGURE 3-88 


139 



D= 10.02 in - 

4= 20 ft. 

V = 1,0 cs. 

k = Indicated 

N = varies with k 

o 

ii 


Cast aluminum spheres in steel pipe. 


FIGURE 3-89 


140 



D = 10.02 in. 

L t = 20 ft. 

V - i.o cs„ 

k ~ indicated 

N = varies with k 

ii 

o 


Cast iron spheres in steel pipe. 


FIGURE 3-90 


141 



D = 0.62 in. 

L t = 1.12 ft. 

^ = Indicated 

k = 0.905 

N = 24 

P= 1.00-1.06 


Machined steel spheres in various polyglycol -water mixtures. 
Effect of liquid viscosity. 


FIGURE 3 - 91 



D = 2.000 in. 

L t = 1.46 ft. 

V = 10.5 CS. 

k = 0.875 

N = 10 

P- 1.06 


Machined spheres in a polyglycol-water mixture. Effect of 
capsule S. G. 


FIGURE 3-92 


143 



01 23456789 10 

CAPSULE VELOCITY, ft. /sec. 


D = 4.029 in. 

L t = 3.15 ft. 

V - varies CS. 

k = 0.93 

N = 10 

o 

ii 


Machined spheres in steel pipe. Effect of liquid viscosity. 


FIGURE 3-93 


LIQUID PRESSURE GRADIENT, psi./ft. 


144 



01 23456789 10 


LIQUID VELOCITY, ft./sec. 


Liquid pressure gradient for the 2, 4 and 10 
inch pipes as a function of liquid velocity. 


V - 1.0 CS. 
/> = 1.0 


FIGURE 3-94 




CHAPTER IV 


Pressure Gradients of Cylindrical Capsules Calculated 
from the Coefficient of Friction 




CHAPTER IV 


Pressure Gradients of Cylindrical Capsules Calculated 
from the Coefficient of Friction 


4.0 Summary 

The force required to pull a capsule through a liquid-filled pipe is 
compared with the pressure gradient required to propel the same capsule through 
the same liquid-filled pipe. Based on this comparison, a model for predicting the 
capsule pressure gradient has been developed. 

A sample calculation is included to show how the capsule pressure 
gradient can be predicted. 


147 


4.1 introduction 

Prediction of the pressure gradient in capsule pipelines is analogous to 
prediction of the pressure gradient in liquid pipelines except that the number of 
parameters and variables governing capsule pipe! ines is greater. An examination of results 
from atypical friction factopReynolds number plot used for predicting the pressure gradient 
in a liquid pipeline in turbulent flow for a given liquid velocity shows that this pre- 
diction may vary by more than 700% depending upon the relative roughness of the 
pipe used in the calculations. However, over the years,as many measurements for flow 
in pipes accumulated, there emerged guidelines as to which roughness to use in the 
pressure gradient calculations for each condition of pipe surface. And, in many cases, 
these relative roughnesses used in the calculations were based upon values selected to 
give the correct answers and not on actual measurements of the relative roughness of the 
pipe (Reference 1). As a result it is now common practice to use the accepted values of pipe 
roughness for predicting the pressure for the flow of liquid in a pipe and to arrive at a 
calculated value which corresponds to the actually measured values to within 20%, 

In a pipeline with cylindrical capsules this effect of surface roughness is even more 
important since the capsule surface moves past the pipe surface with only a very thin layer 
of liquid between them. Consequently any change in the surface roughness of one of the 
two surfaces will significantly affect the capsule pressure gradient. The coefficient of 
friction between the two solids also greatly influences the capsule pressure gradient. 
However, only limited data are available for various surface conditions in capsule 
pipelines. 

A detailed discussion of the effects of both friction and surface roughness is 


148 


contained in Chapter 4 in the Phase 2 Report. The present chapter discusses the relation- 
ship between the capsule pressure gradient and a coefficient which accounts for both the 
surface roughness and the solid-solid friction . Experimental results are presented giving 
values of this coefficient for various conditions of capsule and pipe surfaces at various 
capsule velocities. A correlation between the capsule pressure gradients as calculated 
from this coefficient and the capsule pressure gradients measured experimentally is also 
presented. A simple method for measuring this coefficient for other conditions of pipe 
and capsule surface and liquid properties is suggested. 

Three approaches for predicting capsule pipeline pressure gradients of cylindrical 
capsules are presented in this report; the use of each approach being dependent upon the 
capsule pipeline system tobe designed. One approach isdetailed in Chapter 3 and consists 
of reading from the graphs provided therein the pressure gradients actually measured during 
the experimental work. Sufficient data are provided in that chapter to give a preliminary 
identification of the expected pressure gradients in a pipeline for which design is planned. 
Another approach is described in this chapter and provides a guide as to what 
pressure gradients to expect for capsule and pipe surfaces not tested in the experiments of 
Chapter 3. The third approach, described in Chapter 5, like the other two, requires 
a knowledge of the effect of the prospective capsule and pipe surface conditions and 
liquid properties on the pressure gradient. However, for a constant capsule - liquid- 
pipe surface interaction it provides correlations to show the effects of the capsule density, 
diameter, velocity and the pipe diameter on the capsule pressure gradient. 


149 


4.2 A Cylinder Pulled by a String 

When a cylinder is pulled through a pipeline (c.f. Figure 4 - 1), and liquid 
is allowed to enter and leave, there will exist a net flow, Q^, in the unobstructed pipe, 
flowing in the same direction as the cylinder and causing a pressure gradient 




(4-1) 


Along the cylinder there will exist a movement of displaced liquid, Q^, flowing in the 
opposite direction and causing a pressure gradient 



Writing a force balance on the cylinder, neglecting end effects but 
including the tension force on the string (T), the liquid and solid friction force (FF) 

between the pipe and the cylinder surface, the shear force on the capsule wall (S) 
and the pressure force (P) due to flow of displaced liquid, yields: 

T -FF-S-P=0 (4-3) 


It is assumed that the shear stresses on any moving cylinder can be separated into 
two components. The first component is due to Couette flow, caused by the 
capsule movement only and the second component is due to the flow of liquid 
caused by the pressure gradient along a stationary cylinder only. The concept of 
separating the shear stress on the cylinder in this manner was suggested by Garg and 
Round (Reference 2) and was shown to be plausible in laminar and in turbulent flow by 
Kruyer and Ellis (Reference 3). In the above force balance the shear stress due to 


150 


Couette flow is arbitrarily included in the friction (FF) and the shear stress due to flow 
caused by the pressure gradient is the source of the shear force (S). 

The tension (T) on the string of a cylinder being pulled through a liquid- 
filled pipe may be readily measured with a device illustrated in Figure 4-2. The 
pressure force on the cylinder is: 

P= | d 3 ^ L c (4-4) 

The shear force on the cylinder may be calculated from the pressure gradient 

1 if the shear forces due to this pressure gradient, on the pipe wall and on 

2 

the cylinder wall for a stationary cylinder can be separated. Snyder and Goldstein 
(Reference 4) presented an analysis of the shear stresses for flow in eccentric annuli for 
fully developed laminar flow and Jonsson and Sparrow (Reference 5) presented experimental 
measurements for turbulent flow for this same case confirming that the ratio of the cylinder to pipe 
shear stresses calculated from laminar flow is almost identical to that measured in turbulent 
flow for diameter ratios of 0.75 and higher. As further developed in the Addendum 

(Reference 6 ) to this chapter, the equations of Snyder and Goldstein provide the following 
solution for the ratio between the average shear stresses on the pipe and cylinder walls: 



T 2 

c _ | ctnh <2 - ctnh 6 + (&- d) / (sinh (4 _ 5 ) 

~f~ ~ ^ ctnh ol - ctnh $ + (a-<9) / ( sin h 3 e ) 

P 


Table 4 - 1 presents values of this ratio for diameter ratios (k) between 0.70 and 
0.95 for various cylinder eccentricity ratios. 


151 


TABLE 4 - 1 


Ratio of Average Shear Stress on the Cylinder Wall to that on the 
Pipe Wall for a Long Stationary Cylinder in a Pipe. 


k 

T 

c 

/ T P 


e =0.900 

e =0.990 

e = 0.995 

0.70 

1.0232763 

1.0023537 

1.0010529 

0.75 

1.0186081 

1.0019588 

1.0007868 

0.80 

1.0143299 

1.0014544 

1.0006971 

0.85 

1.0103884 

1.0011082 

1.0005369 

0.90 

1.0067072 

1.0006905 

1.0003138 

0.95 

1.0033302 

1.0004225 

1.0002661 


As shown by Table 4 - 1 , for practical calculations, the average shear stress on the 
cylinder wall is equal to the average stress on the pipe wall. Writing a force 
balance on the annulus: 


l (D 2 -d 3 ) L,-,^ d ^ Lp _ * D L_= 0 


-c 12 C -c 12 pc 


but 

giving 


T - T 
c p 


T = 3 

c 


("1 


(D-d) 


The shear force is equal to 


- •>-«(£), 


(4-6) 


(4-7) 


(4-8) 


152 


The friction force, which is equal to the normal force due to the buoyed weight of 
the cylinder multiplied by the coefficient of friction, can now be solved using 
equations 4 -3, 4-4 and 4-8: 

F F = 0.433 | d 3 L c (a-p)v,= 1 Dd (^j (4-9) 


4.3 A Cylinder Propelled by a Pressure Gradient in a Capsule Pipeline 

A cylinder moving under the influence of a pressure gradient is retarded 
in its movement by the force of solid-solid and liquid friction. It is propelled by the 
pressure drop between the tail and the nose of the cylinder and by the shear stress on 
the cylinder due to pressure flow of liquid in the annulus. Writing this into a force 
balance for the cylinder: 


* 




L +.^dT L - FF = 0 
c 12 c c 


(4-10) 


The force balance on the liquid in the annulus is: 


(ft£) L C -^. L 


C c VI pc 


D t i =0 (4-11) 


Using the relation T c = T p in pressure flow gives: 




(D-d) 


(4 - 12) 


Substituting this expression for T into equation 4-10, using the definition of FF as 


provided in equation 4-9 gives: 


f 0i (¥) 


L c + 0.433 j- d® (a - p) q L c = 0 (4 - 13) 


153 


This simplifies to: 



0.433 k ( a -p) 


(4-14) 


4.4 The Pulling Force and the Capsule Pressure Gradient Related 


Substituting the above expression for tj into equation 4-9 gives: 



- D d L + T 
4 c 



? D d L = 0 
4 c 


(4 - 15) 


This simplifies to: 



(4-16) 


x 'C v As 

The conventional definition of a pipeline pressure gradient is negative since the pressure 

decreases with distance along the pipe. This is the reason for the negative signs in 

equations 4-14 and 4 - 16. In engineering practice however the pressure gradient 

usually is taken as a positive quantity and in that case the minus signs in these two 
equations disappear. 

Figures 4 - 3 to 4 - 7 show the comparison between results obtained from pulling 

tests and those from the capsule pressure gradients as measured for various capsule 
and pipe conditions. 

The pressure gradients as measured during a regular capsule flow experiment 
in which the capsules are propelled by the pressure gradient are shown by the solid 
symbols in the figures. Solid symbols of various shapes (dots, triangles and diamonds) 
show the repeatability of these measurements between succeeding flow experiments. 


154 


The open circles represent the predicted pressure gradients as calculated with 
equation 4-16 from measurements of the tension on the cable during the pulling tests. 

Figure 4-3 shows data for a train of five heavy 13 inch long stainless steel cylinders 
in a 2 inch steel pipe while Figures 4 - 4, 4-5, 4-6 and 4-7 show data from the 4 
inch pipeline for trains of four 24.5 inch long plastic cylinders of various densities 
in the 4 inch stainless steel pipe. A scale of the capsule pressure gradient 
measurements is presented on each left vertical axis while a scale of the coefficient 
of friction for these data as calculated from equation 4 - 14 is presented on each right 
vertical axis. The horizontal axis shows the capsule velocity. The capsule pressure 
gradient as predicted from the pulling tests is based upon the subtraction of two 
measurements (c.f. equation 4 - 16). Since for light capsules these two may approach 
each other in value, an insert scale is provided at the bottom right of each figure to 
show by a curve the ratio between these two measurements. Errors in the prediction 
obviously increase rapidly as this ratio approaches unity. In these figures for low 
values of this ratio the correlation between the solid symbols and the open circles is 
good but for higher values the prediction deviates considerably from the measured cap- 
sule pressure gradient. This deviation increases with an increase in capsule velocity 
and with a decrease in cylinder specific gravity. 

Equation 4-14 actually replaced the one required measurement (the capsule 
pressure gradient) with two measurements, the pulling force and the restraining pressure 
gradient. However, these measurements may in many cases be easier to obtain than the 
capsule pressure gradient especially if the restraining pressure gradient is reduced below 
the level of significance by the use of a large pipe or tank. 


155 


The coefficient of friction is shown to be a most important parameter in 
determining the behavior of dense capsules. This is verified by the close agreement 

between prediction and measurement using data ranging over more than an order of 
magnitude (c. f . Figures 4-3 and 4-7) for capsules flowing at velocities up to 3 ft. /sec. 


A method such as shown in Figure 4-8 may be used to obtain the coefficient 
of friction. The restraining force due to flow of displaced liquid as represented by 

in equation 4-16 can be reduced below the level of significance by giving 
the block an aerodynamic shape and by selecting a container which is large, relative 
to the size of the block. 


4.5 Predicting the Capsule Pressure Gradient 

Figures 4-9 and 4-10 are replots of Figures 3-11 and 3 - 37, representing 
pressure gradients from trains of cylinders in a 10 inch pipe as a function of the 
capsule velocity. Figure 4 - 9 is for steel cylinders and Figure 4 - 10 is for epoxy coated 
cylinders. Specific gravities of the trains are indicated. The numbers on the curves 
represent the coefficients of friction as calculated from the pressure gradients with 
equation 4-16. These coefficients have values normally expected from a friction measure- 
ment for the materials used. Only at the very low capsule specific gravities do the 
coefficients appear unusually high. At the extreme, not shown in the figures, 
where the cylinders and liquid are equi-dense the coefficients as calculated from 
equation 4-16 become infinite. It therefore becomes necessary to modify equation 
4 - 16 to account for this difference by including the effect of energy loss due to move- 
ment of a cylinder train of neutral density. Ellis discusses this requirement in Reference 7 
which is included as Appendix C in Part 2 of this report. 


156 


In large pipes the pressure gradient to move a cylinder of precisely neutral density 
is not very different from the pressure gradient to move liquid at the same velocity 
(References 8, 9). Rewriting equation 4 - 16 on that basis, and using engineering practice 
in taking the pressure gradients to be positive quantities, gives: 

= 0.433 ri k (a - p) (4-17) 

2 

Figures 4-11 and 4-12 show the data of Figures 4-9 and 4-10, respectively 
replotted on the basis of the difference between the capsule and the liquid pressure gradient, 
i.e. the increase in pressure gradient in a pipeline due to the presence of capsules, as 
a function of the capsule velocity. The numbers on the curves again show the coefficients 
of friction but now as calculated from the increase in pressure gradient with equation 
4-17. The resulting coefficients shown in the latter two figures are more representative 
of lubricated friction than those shown in the former two figures suggesting that equation 
4-17 gives a better description of the phenomenon of friction in capsule flow. 

Table 4-2 gives values of v 2 for various materials of cylinder surfaces tested 
in the experimental pipes. These coefficients were calculated from the measured capsule 
pressure gradient data using equation 4 - 17, for a large number of experimental 
measurements. Cylinder specific gravities up to 1.5 were included in the data except 
for the sulphur capsules which had a specific gravity of 1.9. Average coefficients of 
friction are reported for four ranges of capsule velocity, 0-1, 0-3, 3-6 and greater 
than 6 ft. /sec. The number of experimental data points used for each and the standard 
deviation are also shown. The first velocity range may be used for predicting start-up 
capsule pressure gradients while the other three are for predicting the dynamic pressure 
gradients. 



TABLE 4-2 


Values of r\ Calculated with Equation 4-17 from the 

2 

Measured Pressure Gradients for Flow of Cy liners in Water 


Condition 

ft. /sec. 

Average 

Data 

Points 

Standard 

Deviation 

Steel cylinders 

0 to 1 

0.28 

120 

0.07 

in lOlnch steel pipe 

0 to 3 

0.28 

478 

0.06 


3 to 6 

0.32 

551 

0.07 


above 6 

0.32 

505 

0.07 

Polyken taped cylinders 

0 to 1 

0.25 

7 

0.03 

in 10 inch steel pipe 

0 to 3 

0.20 

38 

0.05 


3 to 6 

0.15 

43 

0.04 


above 6 

0.13 

45 

0.03 

Polyurethane cylinders 

0 to 1 

0.28 

10 

0.03 

in 1 0 inch steel pipe 

0 to 3 

0.27 

32 

0.03 


3 to 6 

0.20 

32 

0.07 


above 6 

0.16 

35 

0.06 

Phenolic cylinders 

0 to 1 

0.29 

16 

0.04 

in 10 inch steel pipe 

0 to 3 

0.27 

62 

0.06 


3 to 6 

0.22 

70 

0.04 

• 

above 6 

0.17 

79 

0.06 

Epoxy cylinders 

0 to 1 

0.29 

4 

0.01 ! 

in 10 inch steel pipe 

0 to 3 

0.24 

24 

0.06 


3 to 6 

0.19 

33 

0.04 


above 6 

0.15 

42 

0.10 

Sulphur cylinders 

0 to 1 

0.34 

3 

0.01 

in 10 inch steel pipe 

0 to 3 

0.29 

9 

0.05 


3 to 6 

0.17 

12 

0.02 


above 6 

0.14 

13 

0.01 

Stainless steel cylinders 

0 to 1 

0.32 

194 

0.08 

in stainless steel 4 inch 

0 to 3 

0.31 

503 

0.08 

pipe 

3 to 6 

0.26 

304 

0.13 


above 6 

0.21 

280 

0.14 

Butyrate plastic cylinders 

0 to 1 

0.21 

65 

0.05 

in stainless steel 4 inch 

0 to 3 

0.21 

173 

0.06 

pipe 

3 to 6 

0.15 

100 

0.06 


above 6 

0.10 

103 

0.06 

Butyrate plastic cylinders 

0 to 1 

0.31 

11 

0.06 

in steel 2 inch pipe 

0 to 3 

0.32 

29 

0.07 


3 to 6 

0.28 

16 

0.10 


cbove 6 

0.14 

11 

0.11 

Stainless steel cylinders 

0 to 1 

0.41 

9 

0.06 

in steel 2 inch pipe 

0 to 3 

0.40 

19 

0.07 


3 to 6 

0.34 

12 

0.08 


above 6 

0.20 

16 

0.14 

Stainless steel cylinders 

0 to 1 

0.36 

23 

0.09 

in stainless steel 0.6 inch 

0 to 3 

0.36 

53 

0.10 

pipe 

3 to 6 

0.25 

37 

0.13 


above 6 

0.13 

18 

0.09 


158 


Table 4-3 gives values of the coefficient of friction for various clean materials 
on steel. These coefficients, which are published in the literature (References 10, 11), 
are generally higher than those measured between a capsule and pipe filled with liquid, 
since the lubricating effect of the carrier liquid reduces the friction coefficient. They 
can however serve to give guidelines in selecting capsule surface coatings. Aluminum for 
example is a much less likely candidate as an encapsulating material than, say, poly- 
ethylene since the friction coefficient of aluminum on steel is much larger than that of 
polyethylene on steel . 


TABLE 4-3 


Published Coefficients of Non-Lubricated Static Friction 


Aluminum on steel 

0.8 to 1.2 

Polymethyl methacrylate on steel 

0.8 

Steel on steel 

0.58 

Glass on steel 

0.5 to 0.7 

Cast iron on steel 

0.4 

Aluminum bronze on steel 

0.45 

Polystyrene on steel 

0.3 to 0.35 

Polyethylene on steel 

0.2 to 0.33 

Polypropylene on steel 

0.12 

Nylon on steel 

0.12 

Polytetrafluoroethylene on steel 

0.04 


159 


Predictions of the pressure gradients in capsule pipelines can thus be based upon the 
lubricated coefficient of friction between the cylinder and pipe surfaces selected. 
The procedure of calculation is detailed in the following example. 


4.6 Example 

An annual throughput of two million tons of sulfur are to be transported in water 
by capsule pipeline „ A 10 inch steel pipe is to be used and the sulfur is to be foamed to 
give 45% voids in order to achieve a capsule density amenable to transport in the form 
of cylinders. A bulk specific gravity of 1 . 1 will result. Assuming that a slab of sulfur, 
pulled at about 5 ft. /sec. over a water wet steel surface of the same texture as the 
proposed pipeline gives a coefficient of friction of 0.15*, the predicted increase in 
pressure gradient for 9 inch diameter cylinders, using equation 4-17, will be: 

(0.433) (0.15) (0.90) (0.1) = 0.0058 psi/ft. 


This is for a linear fill of 100%; however, if the cylinders are injected into the pipeline 
at a rate resulting in a linear fill of 80%, the average increase in pressure gradient due 
to capsules in the pipeline will be 0.0046 psi/ft. (80% of 0.0058). At this concentration 
the required capsule velocity to achieve two million tons of sulfur throughput annually 
will be: 


2,000,000 


tons 

yr. 


2,000 — 

ton 


31.5 x 10' 


yr. 

sec. 


( JL _§L\ ft 

V 4 144 y n 


1.1 (62.4) 


lb. 

W7 


(.80) 


which is equal to: 

127 

(69) (0.44) (0.8) = 5-2 fK//sec 


* 


This value agrees with experimental measurements made with sulfur cylinders flowing 
in a 10 inch steel pipe. 


160 


If it is assumed initially that the bulk water velocity is equal to the capsule velocity, 
the liquid pressure gradient at this velocity can be obtained from a standard pipe handbook 
(Reference 12). For a 10 inch schedule 40 pipe it is: 0.0034 psi/ft. The total pressure 
gradient in the pipeline will therefore be: 

0.0047 + 0.0034 = 0.008 psi/ft. or 43psi/mi. 

The actual bulk velocity and water throughput can subsequently be calculated 
from the model in Chapter 7 for a capsule pressure gradient of 
0.0058 + 0.0034 = 0.009 psi/ft. 

Consulting the pipe handbook once more will then give a precise reading of the liquid 
pressure gradient. 

The above example provides a method for calculating a capsule pressure gradient. 
The lowest capsule pressure gradient can be found by repeating the calculations for various 
conditions of capsule density, throughput and capsule size, as explained in Chapter 2 of 
this report. 

Acknowledgements 

The work of Dr. William T. Snyder, University of Tennessee, in solving for 
the shear stress ratio of equation 4-5 and Table 4 - 1 is gratefully acknowledged. 


REFERENCES 


1. Ackers, P. "Resistance of Fluids Flowing in Channels and Pipes." Hydraulics 

Research Paper No. 1. Dept, of Scientific and Industrial Research, 

Hydraulics Research Station, London, England, 1963. 

2. Garg, V.K 0 and Round, G.F. "Capsule Pipeline Flow - a Theoretical Study." 

Thermodyn. & Fluid Mech. Conv., Inst, of Mech. Eng., London, England. 
Proceedings 1969-70, ]84, Pt. 3 G (1), pp. 89-100. 

3. Kruyer, Jan and Ellis, H.S. "Predicting the Required Liquid Throughput from 

the Capsule Velocity and Capsule Pressure Gradient in Capsule Pipelines." 
Can. J. Chem. Eng., in press. 

4. Snyder, William T. and Goldstein, Gerald A. "An Analysis of Fully Developed 

Laminar Flow in an Eccentric Annulus. " A e I.Ch.E. Journ. 11, No. 3, 
pp. 462-467, May 1965. 

5. Jonsson, V.K. and Sparrow, E.M. "Results of Laminar Flow Analysis and 

Turbulent Flow Experiments for Eccentric Annular Ducts." A. I.Ch.E. 

Journ. 11, No. 6, pp. 1143-1145, November 1965. 

6. Snyder, William T. "Private communication." January - June 1974. 

7. Ellis, H.S. "The Effect of the Density of Cylindrical Capsules on the Pressure 

Gradients in Capsule Pipelines." Hydrotransport 3, Golden, Colorado, 

May 14, 1974. 

8. Charles, M.E. "Pipeline Flow of Capsules: Pt. 2, Theoretical Analysis of the 

Concentric Flow of Cylindrical Forms." Can. J. Chem. Eng. , 41, pp. 46-51, 
1963. ““ 

9. Kennedy, R.J. "Towards an Analysis of Plug Flow Through Pipes. " Can. J. Chem. 

Eng. 44, 354-356, December 1966. 

10. Weast, Robert C. Handbook of Chemistry and Physics. 55th Edition 1974-75, 

CRC Press, Inc., Cleveland, Ohio. 

11. Carmichael, Colin. Machine Design. The Plastic Book Issue, Sept. 1962, 

The Penton Publishing Company, Cleveland, Ohio. 

12. Technical Paper No. 710, "Flow of Fluids Through Valves, Fittings and Pipe," 

Engineering Division, Crane Co., Chicago, 1957. 


162 



Liquid flow and forces in a pipe as a cylinder is 
pulled through it. 


FIGURE 4 - 1 


163 



Apparatus used for pulling capsule trains through a pipe. 


FIGURE 4-2 


164 






Capsule pressure gradients in water for five 13 in. long cylinders 
with a specific gravity of 2.0 in a 2 in. pipe measured during a 
regular run and predicted from the string tension during a pulling 
test. 


FIGURE 4-3 


{aP/1) [psi/ft] 


165 



.50 

.45 

.40 

.35 

.30 

.25 


.6 


.5 


.4 


.3 

.2 


.1 


0 




Capsule pressure gradients in water for four 24.5 in. long cylinders 
with a specific gravity of 2.0 in a 4 in. pipe measured during a reg- 
ular run and predicted from the string tension during a pulling test. 


FIGURE 4-4 


( AP / L ) 2 /(4T/7rDdL) 


(aF/L) [psi/ft] 


166 



.50 

.45 

.40 

.35 

.30 

.25 


7 

6 

5 

4 

3 

2 

1 

0 




Capsule pressure gradients in water for four 24.5 in. long cylinders 
with a specific gravity of 1 .5 in a 4 in. pipe measured during a reg- 
ular run and predicted from the string tension during a pulling test. 


FIGURE 4-5 


(A P/ l | ; ° d [_ ) 


167 



V c [ft /sec] 


Capsule pressure gradients in water for four 24.5 in. long cylinders 
with a specific gravity of 1 .25 in a 4 in. pipe measured during a reg- 
ular run and predicted from the string tension during a pulling test. 


FIGURE 4-6 


( AP / L ) 2 /( 4 T/77'DdL) 


168 



.40 


.35 


.30 


.25 


.8 

.7 

.6 

.5 

.4 

.3 

.2 

.1 

0 




Capsule pressure gradients in water for four 24.5 in. long cylinders 
with a specific gravity of 1 . 125 in a 4 in. pipe measured during a 
regular run and predicted from the string tension during a pulling 
test. 


FIGURE 4 - 7 


( Ap / L ) 2 /(4T/7rDdL) 


169 



Suggested method for measuring lubricated friction. 


FIGURE 4-8 


170 



Steel cylinders in a 10 in. steel pipe. 


Coefficients of friction. 


FIGURE 4 


9 


171 



D = 10.02 in. 

L t = 20 ft. 

V - 1.0 cs. 

k = 0.89 

N= 5 

P- 1.0 


Epoxy coated cylinders in a 10 in. steel pipe. Coefficients of friction. 


FIGURE 4-10 


INCREASE IN PRESSURE GRADIENT, psi/ft. 


172 



D= 10.02 in. 

L t = 20 ft- 

C/> 

O 

o 

II 

> 

k = 0.90 

N= 5 

o 

II 

1 

1 


Steel cylinders in a 10 
relative 


in. steel pipe. Coefficients of friction 
to increase in pressure. 


FIGURE 4-11 


INCREASE IN PRESSURE GRADIENT, psi./ft 


173 



D= 10.02 in. 

4= 20 ft. 

V = 1.0 cs. 

k = 0.89 

N = 5 

P- 1.0 


Epoxy coated cylinders in a 10 in. steel pipe. Coefficients of friction 
relative to increase in pressure. 


FIGURE 4-12 


ADDENDUM 


Derivation of the Wall Shear Stress Ratio 

Referring to the equations of Snyder and Goldstein , and using their nomenclature, 
(Reference 4) the average shear stress at the cylinder wall is: 


' - f /'■ 


d 0 


A transformation from 0 to £ from equation 9a (Reference 4) yields: 
sinh a d $ 


d 0 = 


(cosh OL -cos J) 


The average shear stress at the pipe wall becomes: 

If 

— - ^ f sinh OL T } d £ _ 

T ' if j[j (cosh a - cos*) 
Substituting for T, from equation 8a (Reference 4): 


If 


r = JL. f 

ffr . 4) 


(1 - cosh a c os £) <9u 

(cosh a - cos £ ) cos 0 dv 

But from equation 9a (Reference 4): 

cosh ol cos l - 1 


-]■ 


d £ 


cos 


e = 


cosh OL - COS l 


which gives 




0 

The solution for u from equation 6a (Reference 4) can be written in the form: 

00 

o = a (n) + 2 b (,) cos mf 

m = l m 


2 s| = a 


' tx) + s b 1 (a ) cos m ( 
_ i m s 


m = 1 


- f - 1 

ff J 0 dr, Ja 


df 


a' la) 


175 


This yields: 


— _ . |ia' (a ) 
I f 


and similarly for r 2 



Pa* (J) 


The ratio of the average shears becomes: 

?. 1 o’ (a ) 

\ = ( r , / r 2 f ’ a ' ^ 

From equation 6a (Reference 4): 

°' (a) = E + 2 sinh 2 (aT 


0 W = E + 27Tnh'Mj) 

From the definition of E given by equation 6c (Reference 4), the final form for the shear 
stress ratio becomes: 

Tj [ 1 1 ctnh a - ctnh # + (& - /3)/(sinh 2 CK ) 

— |jyJ ctnh & - ctnh + (a - yS )/ (si nh 2 /3 ) 

r 2 

To conform with the nomenclature of this report, the following conversions are made: 


T 

C 



T 

P 



k = y 


Ol and are constants defined by Snyder and Goldstein. 


CHAPTER V 


Correlation of the Pressure Gradients of Cylindrical Capsules wi 
Capsule Density, Pipe Diameter and Liquid Characteristics 




CHAPTER V 


Correlation of the Pressure Gradients of Cyl indrical Capsules with 
Capsule Density / Pipe Diameter and Liquid Characteristics 


5.0 Summary 

Trains of cylindrical capsules of specific gravities from 0.75 to 3.0 were tested 
in pipes from 0.5 to 10 in. diameter at capsule/pipe diameter ratios from 0.85 to 0.95, 
in liquids with viscosities from 0.7 to 50 cs. and velocities up to 9 ft. /sec. The pipes 
used were of butyrate plastic, case hardened steel, stainless steel and schedule 40 
steel. The capsules covered a similar range of materials, but also included 9 in. 
diameter steel cylinders covered with vinyl, epoxy, polyurethane, phenolic and a 
coarse carborundum tape. The worst combinations tested, including the carborundum 
tape in a steel pipe, were the steel capsules in steel pipes; the lowest pressure gradients 
were yielded by collared* capsules (see Appendix D in Part 2 of this report), closely 
followed by vinyl, epoxy, phenolic or polyurethane-covered cylinders without collars. 

The experimentally measured pressure gradients of trains of cylindrical capsules 
have been correlated with the difference between the capsule and liquid specific 
gravities, the bulk (liquid) velocity, the pipe diameter and the liquid characteristics. 

The significant effects on the pressure gradient of the capsule and pipe materials 
and roughnesses, and the lubricating quality of the liquid, necessitate the use of a 
variable in the correlation to allow for these parameters; this has been related to the 
Froude number for eight different conditions. 

An approximate form of the correlation is given by 


Capsule pressure 
gradient 


r (a - p) + 1 


Liquid pressure gradient 
'here a and p are the specific gravities of the capsules and liquid respectively. 


* or banded 


178 


Values of the above mentioned variable, r /k s , are plotted against Froude number in 

P 

Figures 5-6 to 5-11,5 - 14 and 5 - 15, for different capsule and pipe surfaces 
and liquid viscosities. 

The correlation can be expressed in a form similar to that of the Durand equation 
for slurries and to that of an independent correlation for capsules developed in South 
Africa (Reference 3). 

A very interesting outcome of the work is the technique of expressing the pressure 
gradient of cylindrical capsules as two components, one which represents the pressure 
gradient of capsules of neutral density, and one which is proportional to the buoyed 
density of the capsules tested. Any analysis of the viability of capsule pipelining 
under specific conditions would have to take into account the various methods of 
minimizing pressure gradients described in Chapter 7 of the Phase 2 report and in 
Appendix D in Part 2 of this report, including the use of spherical capsules. (See 
for example Figures 6-11 and 6 - 12 of the report.) 


179 


5.1 Introduction 


In a recent paper based on results from the present project and presented at 

the Hydrotransport 3 Conference (Appendix C) it was shown that if ^k 2 ' s 

plotted against the buoyed capsule specific gravity, j8 = (a - p) at constant velocity, 

linear plots generally result. Figure 5 - 1 illustrates this. It refers to stainless steel 

capsules of diameter ratio 0.94 at four capsule velocities in the 4 in. butyrate test 

section. The division of i~\p\ by k 2 correlates the pressure gradients of capsules of 

' ' c 

different diameter ratios for similar capsule-pipe interface conditions; (for examples see 
Figures 3-4 and 3 - 5 of the Phase 2 report). Figure 5 - 2 is a similar plot for loaded, 
hollow steel capsules, diameter ratio 0.90, in the 10 in. pipe, taken from the same 
paper. 

The intercept on the y axis at $ = 0 in all such plots increases with increase 

of capsule velocity and with decrease of pipe diameter because it is the pressure gradient 

due to the shear stresses on equal density capsules. 

M p\ 

The relationship between l-j-- J and j3 is of the form 

k s + (b/k s )/3 



where | ' s P ressure gradient at $ = 0, b is a constant and (b/k 2 ) is the 
slope of the curve at one capsule velocity. For any one value of k 

tn ■ (n 


+ b/S 


(5-D 


180 


If > s ' n psi/ft. the d imensionless slope p is given by 


3 C 

144 — 


p = 2.31 


(¥), - m 

(4- P ) - r) 


(62.30 ( ,3 - p ) 
1 2 


(P - P) 


(5-2) 


where subscripts 1 and 2 refer to any two data points, and p is the pressure gradient 
increase per unit increase of buoyed capsule density at constant capsule velocity. 
For constant liquid density 
9 


2.31 — 


(t). - (6 


(c t - a 

1 2 


(5-3) 


Plots like Figures 5 - 1 and 5-2 indicate that the capsule pressure gradient 
consists of two components, one of which, is a function only of the 

capsule velocity, the capsule and pipe diameters and liquid viscosity; the other is 
proportional to the buoyed density of the capsules. The latter component is strongly 
affected by the capsule-pipe surface conditions, greater solid-solid friction leading 
to a more rapid increase of pressure gradient with increase of B, i.e. to higher 
values of p. (See Chapter 4 for the effect of friction.) 

(t?\ ! 

This type of plot shows why the simple division l-j- — 1 / generally gives 

an unsatisfactory correlation between and & Phase 2 Report, Figures 

3 - 6 to 3 - 8). 

(n“”l ' w hich is independent of must be subtracted from (— r— ) before 

V L/ ° ( L p\ U p\ / V 

dividing by 0. Frcm equation 5 - 1: b = I-j— 1 - f-j— J / (3 is constant for all 

values of /3 at any one capsule velocity. 


101 


However, values of tn are very small at low velocities in a 10 in. 
pipe, and good results are achieved by using the liquid pressure gradients for ' 


i.e. for the low and medium capsule velocities anticipated for capsule pipelining, 
say up to 6 ft. /sec. , the increase in pressure gradient in a 10 in. pipeline above the 
liquid pressure gradient may be taken as proportional to the buoyed density at a given 
capsule velocity, other conditions being constant. It is anticipated that the same 
remarks would apply to larger pipelines, probably over a greater range of velocities. 
Examples of pressure gradient prediction using equations 5-2 and 5-3 are given in 
Appendix C in Part 2 of this report. 


5.2 Recent Advances 

This work has been further advanced and for given surface conditions: (e.g. 
vinyl -surfaced or steel -surfaced capsules in a steel pipe with water) the pressure 
gradients can now be correlated with the bulk velocity, the pipe diameter and the 
liquid characteristics. 

Figure 5-3 uses data from stainless steel capsules (k = 0.94) in the 4 in. stain- 


less steel pipe. The pressure ratio R^ = ' s c ^ v ' c ^ e< ^ and plotted 

against ft at constant values of the Froude number, V^^/g D/ 1 2 ; the coordinates an< 
parameter are all dimensionless. Since, for a given D, and liquid, an< ^ the 

p\ 

Froude number are constant, each line on the figure is equivalent to a plot of I -j — - J 

' 'c. 

versus ft at constant V^, instead of at constant as previously used. The advantages 
of this change are described below. 


According to Charles (Reference 1) the pressure ratio for an infinite cylinder 
of ft = 0 is independent of the bulk velocity and pipe diameter in both laminar and 


182 


turbulent flow. Kennedy (Reference 2) concludes that in turbulent flow the ratio is 
only weakly dependent on the bulk velocity and pipe diameter. Hence it would 
be expected that the lines of constant Froude number, D/12, in Figure 5-3 


would all intersect the y axis, ft = 0, at approximately the same point. This proves 
to be the case and the plots are linear. The figure is therefore simplified and the 
pressure gradients are related to the pipe diameter and the liquid velocity and 
viscosity through the liquid pressure gradient and the Froude number. 

Figure 5-4 presents similar plots for steel cylinders, k = 0.90, in a 10 in. 
pipe. Again the plots are generally linear and intersect on the y axis. A comparison 
of the present method with the previous one, using the same data, may be made 
between Figures 5-4 and 5-2. 

Figure 5-5 relates to capsules with a very different performance in a 10 in. 
pipe. They are loaded hollow steel capsules wound with Polyken vinyl tape (cf. the 
pressure gradients of Figure 3 - 39). This figure also shows convergence at ft = 0 of 
the lines of constant Froude number, which again are generally linear. Figures 5-3, 

5-4 and 5-5 give for each Froude number: 

R / k® = R A s + (r A s ) P (5-4) 

P po p 

where R is the pressure ratio at ft = Oand r /k 2 is the slope of a constant Froude number line. 

po p 

For any one diameter ratio: 

R = R + r ft 
P Po P 

(5-5) 



or 


r = 
P 


(R ) - (R ) 

Pi P 2 

P - P 

1 2 


(5-6) 


183 


where subscripts 1 and 2 refer to any two data points, r^ may be called ' the pressure 
ratio coefficient at constant Froude number," defined as the increase of pressure ratio 
per unit increase of /3 at constant Froude number. 

If, in addition to the Froude number the liquid is the same for trains of two 
different densities in pipes of different diameter: 

(R ) / k s = (R ) /k 2 - (r ) (a - p ) / k 8 (5-7) 

P 2 2 Pll Pll 1 1 

For a given liquid at a given bulk velocity in one pipe, the Froude number and 
liquid pressure gradient are constant. Flence multiplying equation 5 - 5 by 


The plots of 


(H - (-) 


+ (constant) /3 


(n= 


(5-8) 


k versus /3 (Figures 5 - 1 and 5-2) gave for each capsule 


velocity 


■ (¥). 


+ b (3 


(5-1) 


So the two relationships (equation 5 - 1 and 5-8) are of the same form. The 
original relationship was based on constant capsule velocity, but the use of the pressure 

ratio in place of the pressure gradient requires the Froude number as a para- 


meter, and for any one pipe, constant Froude number (V^ / ^ g D/12) implies 
constant bulk velocity V^. The advantage of this system is that the pressure 
gradients can be correlated with bulk velocity, pipe diameter, and liquid 
characteristics as follows. 


If the slopes, r /k 2 , of the lines of Figure 5-3 are plotted against 
P 


184 


Froude number on log axes, the plot of Figure 5-6 results. This figure and 
Figures 5 - 7 to 5 - 11 may be regarded as performance summaries of the various 
densities of capsules and capsule-pipe surfaces which they represent, relating the 
capsule pressure gradient to the diameter ratio, the liquid pressure gradient (a 
function of bulk velocity, pipe diameter and liquid characteristics) and to the 
Froude number. Also, they allow solution of equation 5-7. 

Similar relationships to those of Figure 5-6 are shown in Figure 5 - 7 for 
the various densities of 9 in. steel capsules of Figure 5 - 4 in a 10 in. pipe; the 
data points for 8.5 in. and 9.5 in. capsules reinforce this line. 

Figures 5 - 8 to 5 - 11 present similar plots for the 9 in. diameter taped 
capsules of Figure 5-5, and for the 9 in. capsules coated respectively with epoxy, 

phenolic and polyurethane. The linearity of the plots of R^/k vs. # at constant 
Froude number, shown by the correlation of the data for the five specific gravities 
in each case, not only permits calculation of the pressure gradients at other specific 
gravities than those measured, but also for other pipe diameters for similar surface 
conditions. The method will be explained at the end of the chapter; in the meantime 
the validity of the correlation for varying surface roughnesses, liquid viscosities and 
pipe diameters will be examined. 

5,3 The Effect of the Capsule and Pipe Surface Roughness 

The high values of pressure ratio and of pressure ratio slopes yielded by the 
steel capsules and pipes (Figures 5-3, 5-4, 5-6 and 5-7) are considerably re- 
duced by the vinyl tape covering (Figures 5-5 and 5 - 8), and by the plastic coatings 


185 


(Figures 5 - 9 to 5 - 1 1). Low specific gravity steel capsules (cr = 1 . 00 to 1 . 12) or 
plastic- or tape-covered capsules of any specific gravity tested, became polished and 
polished the pipe, lowering the pressure gradient after a few runs. Whereas denser 
steel capsules (a = 1.25 or greater) became rougher and roughened the pipe, with 
consequent increase of pressure gradient, even the denser coated capsules maintained 
the pipe and cylinder smoothness, and reduced the slope of the pressure ratio - P 
curves. 

5.4 The Effect of Pipe Diameter 

The value of any correlation which predicts the effect of a change in pipe 
diameter on the capsule pressure gradient depends on the maintenance of the same 
capsule-pipe surface relationship, so that the hydrodynamic effects may be correlated 
without interference from irrelevant frictional effects related to different pipe and 
capsule roughnesses. This requires the maintenance of a constant coefficient of 
friction at all capsule velocities, and not only at the threshold velocity. 

(See Chapter 4.) 

The best comparison that can be made of the effect of change of pipe 
diameter with the available data is for the various steel capsule and pipe surfaces; 
these are steel capsules in a 10 in. steel pipe, and stainless steel capsules both in 
a 4 in. stainless steel pipe and in a 2 in. surface-hardened plain steel pipe. Flow 
in the annuli between the stainless steel capsules and the 0.5 in. stainless steel pipe 
was laminar, even with water, but the data will be compared with those from turbulent 
flow in the other pipes. 

Comparison of Figures 5-3 and 5 - 4 or 5 - 6 and 5 - 7 for 4 in. and 10 in. 


186 


pipes shows that values of Rp/k (the capsule pressure gradients related to the 
liquid pressure gradients and diameter ratios) are very similar for the same Froude 
number. Capsules with other diameter ratios, but with similar surfaces, confirm the 
validity of the correlation for predicting the effect on the pressure gradient of a 
change in pipe diameter. 

In Figure 5 - 12, R / k 2 is plotted against for the stainless steel capsules 
P 

in the hardened and polished 2 in. pipe, at Froude numbers of 0.5, 1.0, 

1 .5, 2.0 and 3.0. The liquid was water, but only in the runs with buoyed specific 
gravity of 0.5 or more was the flow in the annulus fully turbulent as indicated by 
the Reynolds number (Re^ > 4,000 calculated as in Chapter 7). For the lower 
specific gravities the flow is transitional (1 000 < Re < 4,000), and although there 
are changes of slope as laminar flow is approached, the intercepts on the y axis of the 
linear turbulent plots are again coincident. The plots of Figure 5-12 correlate well 
with those of 10 in. and 4 in. pipes at high Froude numbers and specific gravities, 
where flow is turbulent in the 2 in. pipe. 

Figure 5 - 13 is a similar diagram for stainless steel capsules in the 0.5 in. 
pipe with water. Flow in the annulus was laminar throughout except for the lower 
velocities at & = 2. However, the linearity of the plots is maintained, though there 
is some divergence of the intercepts on the y axis. The pressure ratios are understand- 
ably lower than those from the 2 in. pipe with transitional and turbulent flow in the 
annulus, since the flow in the annulus of the 0.5 in. pipe is laminar, while that in the 
pipe remains turbulent. The suppression of turbulence in the annulus leads to a 
reduced capsule pressure gradient, while that of the liquid in the pipe remains high. 


187 


5.5 The Effect of Liquid Viscosity 

Although there is generally an increase of capsule pressure gradient with 
increased liquid viscosity (e.g. Figures 3 - 49 to 3 - 53), the pressure ratio generally 
remains rather constant or even decreases, since the liquid pressure gradient also 
increases. If the increased viscosity causes turbulent flow in the annulus to become 
laminar there is a reduction of capsule pressure gradient and hence a considerable 
drop in pressure ratio (Reference 1). Under these conditions, if the flow in the pipe 
remains turbulent the pressure ratio may fall to less than unity, i.e. , the capsules 
and liquid are transported with less energy than the liquid alone because the energy 
expended in the annulus in laminar flow is less than that expended in the pipe in 
turbulent flow. This condition has been substantiated in all the pipes with capsule 
densities close to that of the liquid. 

The validity of the correlation can be tested for different liquid viscosities 
with turbulent flow in the annulus with data from the 10 in. pipe, using water at 60°F 
and 100°F, and from the butyrate capsules in the 4 in. stainless steel pipe with water 
and with a polyglycol mixture giving a mean viscosity of 3. 15 cs.; at higher viscosities 
the flow became laminar. The stainless steel capsules in the 4 in. pipe were only 

tested with water. In the 2 in. and 0.5 in. pipes, flow in the annulus was laminar with 
the polyglycol mixtures. The effect of variation of liquid viscosity in laminar flow is 
discussed in section 5.6. 

In Figure 5-14 the values of r /k 2 for 9 in. capsules with water at a 

P 

viscosity of 0.70 cs. (symbols) are compared with those for the same capsules with a 
water viscosity of 0.98 cs. (line) as plotted in Figure 5-7. The values for the lower 
viscosity are slightly higher in the mid-velocity region, but this is known to occur 


188 


with somewhat increased capsule roughness, which is likely to be the cause rather than 
the lower viscosity. 

Figure 5 - 15 compares the values of r ^/k 2 f° r butyrate capsules, k = 0.90, 
in the 4 in. stainless steel pipe for v - 3.15 cs. (symbols) and for water, V - 1.08 cs. 
(line). Except for a scattered rise in pressure ratio at the highest velocity the mean of 
the data is indistinguishable from the mean line for water. The two other diameter ratios 
tested give very similar results, but do not show the rise in pressure ratio at the highest 
velocity. 

5.6 Viscosity Correlation in Laminar Flow 

In laminar flow, obtained with large additions of polyglycol to increase the 

viscosity, variations of capsule-pipe interaction due to differences of capsules roughness 

and of lubrication were observed to have more effect than in water. Consequently more 

scatter is evident in the results, but the correlation is still effective in bringing the 

values of r^/k 2 close to a single line for different specific gravities at each viscosity 

in the 0.5 in. pipe over the buoyed specific gravity range from 0.03 to 3.00 at 

viscosities from 1.0 to about 50. However, the correlation is not recommended for 

changes of viscosity in laminar flow below a Reynolds number of 200 since the values 

of r change systematically as the Reynolds number is reduced below this value, the 
P 

slope of the r^/k 2 , Froude number curve becoming progressively flatter and the 

values of r /k 2 becoming lower. 

P 


189 


5.7 The Procedure for Calculating the Pressure Gradient Using the Correlations 
The prediction of the capsule pressure gradient with the correlations is 
illustrated with three examples. This method may be used if prior knowledge of the 
pipe-liquid-capsule interaction exists. 

In the Addendum to this chapter sample calculations are presented. 


Example 1 

Suppose the pressure gradient in a 10 in. or larger pipeline is known at one 
capsule specific gravity and is requred at another for similar conditions. 

As explained in section 5 - 1 the increase of the capsule pressure gradient over 
the liquid pressure gradient is proportional to jS at the same capsule velocity in the 
range of 6 to 8 ft. /sec. The liquid pressure gradient is for a smooth pipe. 


(-L ■ [(-1 • N * + &), 


V, and 
b 




(5-9) 


may be calculated as in example 3 below, or assumed to be 


equal to as an approximation. 

For epoxy-, vinyl-, polyurethane-, phenolic-, or steel -surfaced capsules in a 

( £ p\ 

schedule 40, 10 in. pipeline I— j— J may be obtained from the figures in Chapter 3 


and the pressure gradient calculated from equation 5-9. 

The Phase 2 Report indicated that the capsule pressure gradient divided by the 
Froude number gave rather constant values at a given value of bulk velocity for 4 in. 
and 10 in. pipes (Phase 2, Figure 3 - 12), so that for a given bulk velocity the pressure 
gradient may be taken as inversely proportional to the square root of the pipe diameter 
over a limited range of pipe diameters. 


190 


Example 2 

Predict the change in pressure gradient due to a change in liquid viscosity or 
capsule/pipe diameter ratio at the same bulk velocity and in the same pipe. (If the 
diameter ratio is changed the new capsules must have a surface identical to the original 
capsules.) 

The Froude number is unaltered, so that plots like Figures 5 - 3 to 5 - 5 remain 
relevant. The pressure ratio therefore is unchanged for a different liquid viscosity at 
the same diameter ratio (Figures 5-14 and 5 - 15), and will change inversely with 
k 2 for different values of k at the same viscosity. Therefore for a change in either 
liquid viscosity or diameter ratio: 

/ A P\ AP\ 

(5 - 10) 


where 




smooth pipe, from wh 


may be found by inserting 


the known values of conditions 1 and 2. E.g 0 , if k is unaltered: 

("l ■ (-1 (vl/tel 


(5-11) 


'C 3 

If p is constant 


("1. - (-! fc) 


0.85 


(5 - 12) 


Example 3 

Predict the pressure gradient for a given solid throughput in turbulent or 
transitional flow (Reynolds number > 1000; see Chapter 7). 

In the present state of knowledge previous data taken under similar surface 
conditions are necessary, and plots such as Figures 5-6 to 5 - 11, 5 - 14 and 5-15 


191 


used. From the capsule diameter, linear fill and throughput, V is readily obtained 
(cf. Chapter 2).* If the pipe diameter is not known, it must be obtained as explained 
in Chapter 2. In order to calculate the bulk velocity, V^, (cf. Chapter 7) a value 
of the capsule pressure gradient must be assumed, e.g. constant pressure ratio might be 
postulated for a change of viscosity, or the pressure gradient could be taken as 
inversely proportional to yJ~D for a change of pipe diameter. For a change in bulk 
velocity, data from curves of similar surface and specific gravity can be used (cf. 
Chapter 3). 


can also be calculated from equations 7-3 and 7-4 and the liquid pressure 

gradient is then calculated from 

.0027 p V b x - 75 i/- S5 


(v), 


D 


l.S 5 


psi/ft. 


(turb) 

using the Blasius equation for smooth pipes at Reynolds numbers between 2 x 10 3 

and 10 5 , or one of the alternative equations for flow in smooth pipes. 

The value of r^/k 2 is rea d the figure appropriate to the capsules and pipe 

at the relevant Froude number. Since r /k s =(R - R )/k s fi, r /k 3 is 

P P P° P 

multiplied by k 2 jS to give the increase in pressure ratio (R - R ) due to . 

P po 

(See Figures 5 - 3 to 5 - 5.) For the 4 in. and 10 in. pipes, R was 0.5 to 0.8 and 

po 

may be taken conservatively as 1.0. The pressure gradient is then given by 



(r p /k 2 ) k 2 /3 + 1 



(5-13) 


* 


Lazarus (Reference 3) gives a nomogram from which V 
values of d, L^, a and capsule spacing. 


can be obtained for different 


192 


This value is compared with the assumed value. If it is not within, say, 

10% of this, an intermediate value is assumed and the procedure repeated until the 
assumed and calculated values are sufficiently close. If the required pipe diameter is 
in doubt, the calculation may be repeated for different diameters to secure the minimum 
pumping and pipe costs as explained in Chapter 2. 


5.8 A Comparison of Slurry and Capsule Pressure Predictions 

A comparison of the expressions for predicting pressure gradients in slurry and 
capsule pipelines is of interest, for it turns out that in spite of the different modes of 
transport, the predictions have much in common. 

The classic Durand equation (Reference 8) is still widely used (e.g. References 
5, 6) as a criterion for new attempts to improve the prediction of pressure gradients. 
The Durand equation may be expressed as: 


c i 


= 4> = K 


gD 

sr* 


m 


v^r 


1.5 


(5-14) 


where i is the pressure gradient of the slurry (feet of water/foot) and i that of the 

w 

carrier water. C is the volumetric solids concentration, K is a constant, V is the 

m 

mean slurry velocity and c^ is the mean drag coefficient of the particles. 

d 2 

If the equation is considered for capsule flow, C = = k 2 for 100% linear 

fill of capsules (on which the pressure gradient measurements are based) = V^, and 
1 3 replaces the buoyed density of the slurry particles (a- 1). So, using the capsule 
symbols for slurries where applicable, equation 5-14 becomes 



(5 - 15) 


193 


By comparison, a linear log plot of r /k 2 against Froude number with slope -n gives: 



Multiplying through by yS : 


(R - R )/k 2 = 
p po 




P 


(5 - 16) 


(n 


Equation 5 - 16 is of similar form to equation 5-15 


' but (r) 


replaces 


c . has become constant for cylindrical capsules, and $ is not raised to the power 
d 


( £ p \ 

of 1.5; n is usually 0. 8 to 1.2 for capsules. I -j- — 1 in equation 5 - 16, if applied to 

' 'Q 

slurries would correspond to the pressure gradient of a slurry with particles of the same 
specific gravity as the liquid which, for small particles, would be the liquid pressure 


gradient 


(r) f 


of equation 5 - 15. The left hand sides of the two equations therefore 


correspond closely, the increase of pressure ratio due to capsule specific gravity, 

divided by k s , corresponding to Durand's 4> . 

The value for n of 0.8 to 1 .2 for capsules, compared with 1.5 for slurries 

according to the Durand equation, is of interest. It indicates that 4> [=(R - R )/k s ] 

p po 

rises much less rapidly for capsules than for slurries with decrease of Froude number, 

i.e. it rises much less rapidly with decrease of bulk velocity or increase of pipe diameter. 

However, the actual values of <t> depend on K and c^ for slurries and A for capsules. 

As would be expected. Figures 5 - 6 to 5 - 11 show that A, the value of r when the 

P 

Froude number is unity, depends on the capsule-pipe interaction. 


194 


The fact that ft has an exponent of unity for capsules, in place of 1 .5 for 
slurries, indicates a lower rate of increase of pressure ratio with increase of ft , but 
the absolute values again depend on other factors, such as particle size and shape and 
the flow pattern for slurries, and capsule-pipe interaction for capsules. 


5.9 Comparison with the Lazarus-Kil ner Method for the Prediction of Capsule 
Pressure Gradients 

Lazarus and Kilner (Reference 9) at the University of Cape Town obtained the 

equation (expressed in the present nomenclature): 


(n - m, 

k - m, 


25 


gD 


p 


(5- 17) 


for the pressure gradient requirements of a capsule system, optimized it for design 
purposes and presented design charts. 

Comparison of equation 5-17 with equation 5-16 derived from the present 
work shows that Lazarus uses the liquid pressure gradient to obtain the increase due to 
the capsules, as Durand does for slurries, rather than the pressure gradient of a capsule 
with ft = 0. However, the difference between the two methods is small, especially 
for large pipelines (cf. Appendix C). 

A more significant difference is that the factor A and the exponent n in equation 
5-16 are made constant in equation 5-17. This implies a single capsule-pipe surface 
interaction and limits the applicability of the expression, though a correlation for a 
larger pipe is shown in Reference 3 with the constant equal to 29 instead of 25. For 
examples of the effects of different capsule and pipe surfaces see section 5.3 and the 


195 


differences in Figures 5-6 to 5 - 11, 5 - 14 and 5-15, where the values of A range 
from 21 to 65. However, the independent solutions to capsule pressure gradient pre- 
diction comprised in equations 5-16 and 5-17 give confidence in the Froude number 
as a parameter, and in the fact that the excess pressure gradient term is proportional to 
the buoyed specific gravity. 


REFERENCES 


1 0 Charles, M.E. The Pipeline Flow of Capsules, Part 2: Theoretical Analysis of 
the Concentric Flow of Cylindrical Forms. Can. J .Chem. Eng. , 41, 46-51 
(1963). “ 

2. Kennedy, R.J 0 Towards an Analysis of Plug Flow Through Pipes. Can. J. Chem. 

Eng., 44, 354-6 (1966). 

3. Lazarus, J.H. Power Optimization of Hydraulic Transportation of Solid Capsules. 

Paper G.4. Hydrotransport 3, Golden, Colo. 1974. 

4. Experimental Studies in Solids Pipelining of Canadian Commodities for the Canadian 

Transport Commission and the Transportation Development Agency. Saskatchewan 
Research Council , Saskatoon, Saskatchewan, Canada, 1973. 

5. Bonapace, A.C. Conditions of Particle Equilibrium at the Boundary of a Stream. 

Paper E. 3, Hydrotransport 3, Golden, Colo. 1974. 

6. Okude, T. and Yagi, T. The Spatial Solid Concentration and the Critical 

Velocity. Paper E.2, Hydrotransport 3, Golden, Colo. 1974. 

7. Hrbek, J. and Gibian, E. Hydraulic Transport of Pb Zn Ore from 450 m Depth 

to Surface. Paper A. 4, Hydrotransport 3, Golden, Colo. 1974. 

8. Durand, R. The Hydraulic Transportation of Coal and Solid Material in Pipes. 

Colloquium of the National Coal Board, London 1952. National Coal 
Board Scientific Dept., England. 

9. Lazarus, J.H. and Kilner, F.A. Incipient Motion of Solid Capsules in Pipelines. 

Paper C. 3, Hydrotransport 1, Coventry, England, 1970. BHRA Fluid 
Engineering, Cranfield, Bedford, England. 


ADDENDUM TO CHAPTER V 


Calculated Examples of the Use of the Data to Obtain Pressure Gradients Under 
Other Conditions than those Tested 

N. B. The liquid pressure gradients are those in a smooth pipe. 

1 . Finding the pressure gradient for capsule specific gravities different from those 

tested, but with other conditions unaltered. 

Suppose the pressure gradient is required for epoxy-coated cylinders of specific 
gravity 1.18, diameter ratio 0.90, running in water in a 10 in. steel pipeline, V = 5 ft. /sec. 
From Figure 3-37 the pressure gradient of epoxy-coated cylinders of specific gravity 
1 .5 and diameter ratio 0.89 in a 10 in. pipe at V c = 5 ft. /sec. is 0.042 psi/ft. 

Since the slope of the pressure ratio, curve is better defined over a large range of 
specific gravities, it is usually most accurate to use the pressure gradients for the largest 
specific gravity available when making specific gravity calculations. 

The difference in diameter ratio between the required 0.90 and the measured 

O. 89 can be neglected. From equation 5-9 the required pressure gradient would be: 



Assuming that V. = V , the liquid pressure gradient is found to be 0.0032 psi/ft. 

b c — 

= 0.36 (0.042) + 0.64 (0.0032) = 0.017 psi/ft. 

C2 

Note: Data for steel -surfaced cylinders should not be extrapolated to higher 
specific gravities than 1.5, since considerable wear and consequent 
rise in pressure gradient then occurs. The coated capsules have been 
tested up to a specific gravity of 1.5; at this specific gravity only 
polishing of the pipe and capsule surfaces occurred and pressure gradients 
remained comparatively low. 



198 


2. Finding the pressure gradient when the liquid viscosity is changed from 1 cs. to 
10 cs. at constant density, otherwise with the above conditions. 

Equation 5-12 gives: 



= 0.042 



0,25 


= 0.075 psi/ft. 


3. Finding the effect on the pressure gradient of changing the diameter ratio in example 

1 above from 0.89 to 0.95 at the same bulk velocity: 


Condition 1: 


Condition 2: 



k a = 0.95 
k * * 3 =0.90 

3 

The change of diameter ratio at the same bulk velocity will cause a small 
change in V^, but since the pressure gradient curves are rather flat with variation 
of V the effect may be neglected. 

The liquid pressure gradient is unaltered between the two conditions. There- 
fore from equation 5-10: 


199 



s 

= 0 . 0 48 psi/ft. 


4. Calculating the capsule and pipe diameters, the pressure gradient and the capsule 
and bulk velocities from the throughput and density of the material to be conveyed. 

Suppose a million tons per year of material of specific gravity 1 .25 are to be 
transported as capsules with a polyurethane surface in a schedule 40 steel pipeline; 
water is to be the conveying liquid and the linear line fill is estimated to be 75%. 

As a first trial a, capsule diameter of 9 in. is assumed. 


Throughput (millions 
of tons/year) 


0.75 TJ d s /4 x V c x a x 62.34 x 3600 x 24 x 365 
144 x 2000 x 10® 


4.02 x 10" 3 d s V a = 1 

C 

V = 2.46 ft/sec. 

c 


Since this is a rather small capsule velocity the capsule diameter could be 
decreased to give a diameter ratio of 0.90, say, in the next smaller standard pipe, 
i.e. 0.90 x 8 in. =7.2 in. 

3 

The new V c = 2 . 46 x = 3.84 ft/sec. 


200 


According to the method of calculating explained in Chapter 7, the 
pressure gradient must be inserted. From Figure 3 - 36, polyurethane-coated 
cylinders of cr = 1 .25, k = 0.89 in a 10 in. schedule 40 steel pipe at V c = 3.84 ft. /sec. 
would have a pressure gradient of 0.026 psi/ft. Using the approximate inverse square 
root rule for changing to a new pipe diameter. 


in an 8 in. pipe 




= 0.029 psi/ft. 


According to equation 7-2 (and Table 7-1, column 1 for C ), assuming 

i 


V - p = 1 for water 


V, = 0.90 x 3.84 + 0.19 
b 


4. 13 ft. /sec, 


415 x 0 


125 v 0.571 


.029 [8 (■ 1 0)] | 

xl j 


The next step is to check if the flow in the annulus is turbulent by calculating 
the Reynolds number. Equation 7-3 gives 

7742 x 0.8 (4.13 - .9 x 3.84) 


Re 


19 x 1 


2.20 x 10 


The flow is therefore turbulent. If the Reynolds number had been below 
1000, would have had to have been recalculated according to equation 7-4, 
The Froude number V^^/yg D/12 = 4.1 l>j y^32.2 x 8/12 


0.89 


201 


At this Froude number: 

r /k 8 = 49 
P 

according to Figure 5 - 1 1 for polyurethane-coated cylinders. 

The liquid pressure gradient for water in an 8 in. pipe at =4.13 ft. /sec. 
is 0.30 psi/100 ft. 

Therefore from equation 5-13 

= j49 x .81 x 0.25 + l] 0.30/100 

= 0.033 psi/ft. 

The difference between the assumed pressure gradient, 0.029 psi/ft., calculated 
from the approximate inverse square root rule for change of pipe diameter and the value 
calculated by the more accurate method, 0.033 psi/ft., is about 12%. A still more 
accurate value could be found by assuming an intermediate value, e.g. 0.031 psi/ft. 

in equation 6-2, since too low a trial value of pressure gradient produces too high 
a value of calculated pressure gradient. However, 0.031 psi/ft. is probably as close as 
is necessary. Hence the required values are: 

pipe diameter: 8 in., capsule diameter 7.2 in. 
pressure gradient 0.031 psi/ft. 

V Q = 3.84 ft. /sec., V b = 4.13 ft. /sec. 

The effect of the choice of capsule and pipe diameters may be studied by 
considering the effect of leaving at 2.46 ft. /sec. in a 10 in. pipe as first 
calculated. Figure 3-36 showed that the pressure gradient would be 0.026 psi/ft. 

The change to the 8 in. pipe therefore resulted in an increased pressure gradient of 



202 


about 27%. However, the effect on the power requirements must also be considered. 

Since the power requirements are proportional to D s V b (cf. Chapter 2), 

the proportional decrease in power requirements by using the smaller pipe would be 

/ 0.026 x 1Q S x 4.09 - 0.031 x 8 a x4.13 

\ 0.026 x 10 3 x 4.09 

The use of the smaller pipe has thus increased the pressure gradient, but 
reduced the power requirements and, of course, the cost of pipe. Since the increased 
pressure gradient may mean that more pumping stations are required, the choice between 
the two sizes of pipe requires careful economic evaluation. 


) 


x 100% - 23% 


203 



(psi/ft) 



4 in. butyrate pipe 


Stainless steel capsules at four velocities. 


FIGURE 5 - 1 


204 



10 in. steel pipe 

Hollow steel capsules at five velocities. 


FIGURE 5-2 


205 



P 

4 in. stainless steel pipe 
Stainless steel capsules ( k = 0.94) 


FIGURE 5 - 3 


206 



0 


10 in. steel pipe 


Hollow steel capsules, k = 0.90. 


FIGURE 5-4 


207 



10 in. steel pipe 

Hollow 9 in. cylinders covered with Polyken tape. 


FIGURE 5-5 


208 



4 in. stainless steel pipe k = .94 


Specific pressure ratio/ (diameter ratio) 3 versus Froude 
number. Stainless steel capsules of Figure 5-3, specific 
gravities 1.03 to 1.50. 


FIGURE 5-6 


209 



10 in. steel pipe k = 0.90 


Specific pressure ratio/ (diameter ratio) 2 versus Froude 
number. Hollow steel capsules of Figure 5-4, specific 
gravities 1.03 to 1.50. 


FIGURE 5-7 


210 



10 in. steel pipe 


Specific pressure ratio/ (diameter ratio) 2 versus Froude 
number. 9 in. Polyken taped cylinders. 


FIGURE 5-8 


211 



Specific pressure ratio/ (diameter ratio) 2 versus Froude 
number. 9 in. epoxy coated cylinders. 


FIGURE 5-9 


212 



10 in. steel pipe 

Specific pressure ratio/ (diameter ratio) 2 versus Froude 
number. 9 in. phenolic coated cylinders. 


FIGURE 5-10 


213 



Specific pressure ratio/ (diameter ratio) 2 versus Froude 
number. 9 in. polyurethane coated cylinders. 


FIGURL 5-11 


214 





2 in. hardened and polished steel pipe 

Stainless steel capsules, k = 0.88. 


FIGURE 5-12 


215 



P 


0.5 in. stainless steel pipe 


Stainless steel capsules, k = 0.91. 


FIGURF 5-13 


216 



10 in. steel pipe 


Specific pressure ratio/ (diameter ratio) 8 versus Froude 
number. 9 in. steel capsules, V = 0.70 cs. 


FIGURE 5-U 


217 



Specific pressure ratio/ (diameter ratio) 3 versus Froude 
number. Butyrate capsules, k = 0.90, v - 3.15 cs. 


FIGURE 5-15 



CHAPTER VI 


The Hydrodynamics of Spherical Capsules 







CHAPTER VI 


The Hydrodynamics of Spherical Capsules 


6.0 Summary 

It has been confirmed that with water as the conveying liquid the pressure gradient 
of spheres varies little with specific gravity and is comprised of a basic pressure gradient 
due to distortion of the liquid velocity profile by the spheres, to which has to be added a 
very small density-dependent increment due to solid friction, and a further increment due 
to any irregularity of the spheres. For practical purposes the increase due to friction can 
be neglected and the problem is how much to add to the basic pressure gradient of perfect 
spheres to allow for irregularity. For cast spheres in the 10 in. pipe the increment for out- 
of-roundness was from 40% to over 100% depending on the density. A method of calculating 
the power requirements of spheres is outlined. 

The statements in the previous paragraph apply to spheres transported in water. 

With viscous liquids the pressure gradient generally rises rapidly with increase of specific 
gravity and viscosity. 

The power requirements of spheres and cylinders of specific gravity 1 .25 and 1 .50 
transported by water are compared in the last two figures, 6-11 and 6 - 12, and it is 
evident that solely from hydrodynamic considerations spheres are preferable to cylinders 
at even the low specific gravity of 1.25, at low velocities. However, the Sower power 
requirement has to be weighed against the possibly greater cost of forming spheres, and 
the cost of the larger pipe that would be necessary to secure a given throughput, 
particularly if the spheres were run at low velocities to secure very low pressure gradients. 
This calculation can only be made with knowledge of the relative costs of sphere and 
cylinder fabrication for a particular commodity, and any other advantages and disadvan- 


220 


tages of using spheres. Among these considerations would be the relative ease of injection, 
bypassing and extraction of spherical and cylindrical capsules and their relative strengths 
to resist pressure, wear and impact. 


221 


6 . 1 The effects of sphere irregularity and specific gravity 

In Chapter 4 of the Phase 2 Report on spherical capsules it was noted that the pressure 
gradients for such capsules are generally smaller than for cylindrical capsules for the 
same mass throughput, and are increasingly attractive hydrodynamical ly as the 
specific gravity of the transported material increases. It was also noted that whereas 
the pressure gradients of the spherical capsules in the 4 in. pipe showed little effect 
of varying specific gravity, the pressure gradients of aluminum spheres in the 10 in. 
pipe were much lower than those of iron spheres. This was ascribed to the spheres 
in the 4 in. pipe being machined to a close tolerance, while those in the 10 in. pipe 
were cast, and were much less spherical. It was pointed out that as spheres move 
through the pipeline, any small bumps on the spheres involve the expenditure of energy 
proportional to the height of the bump and the buoyed weight of the sphere. It was 
suggested that for accurate sphericity there is a basic pressure gradient due to the 
distortion of the velocity profile, and that the pressure gradient due to any irregularities 
of density or shape must be added to this basic pressure gradient. 

For commercial practice it is important to know the effect of sphere irregularity 
on the pressure gradient, and in order to investigate this parameter the 9.5 in. cast iron 
and aluminum spheres were machined on a lathe to 9 in. diameter. In the course of 
this work air and sand pockets were revealed in a number of the spheres, indicating that 
the finished spheres would very probably not be of uniform density. The maximum 
tolerance on the diameter was ± 0.020 in. , and 95% of the thirteen measurements made 
on each sphere showed variations from the 9 in. mean of no more than 0.005 in. 

The pressure gradients of the machined and original cast spheres are plotted against 


222 


capsule velocity in Figures 3-87 and 3 - 88. As anticipated there was a substantial 
decrease of pressure gradient when the capsules were made more spherical, especially in 
the case of the iron spheres. The pressure gradients of the machined iron and aluminum 
spheres were much more similar than those of the original cast spheres , supporting 
the hypothesis that most of the difference of pressure gradient between the original 
aluminum and iron spheres was due to the lack of sphericity. 

Figure 4 - 3 of the Phase 2 report showed that with water as the conveying 
liquid there was little increase of pressure gradient in the 4 in. pipe as the specific 
gravity of the spheres increased from 1.55 to 4.33. Similar remarks can be made 
about spheres tested in the 2 in. pipe in water at S.G.'s from 1.3 to 7.8 and about 
the 9 in. machined aluminum and iron spheres except at the highest velocities. Because of 
the effect of liquid viscosity discussed in a later section it is important to note that all 
these spheres were conveyed in water. The concept is therefore confirmed that with 
water as the conveying liquid there is a basic pressure gradient due to distortion of 
the liquid velocity profile, to which has to be added a very small density-dependent 
increment due to solid friction, and a further increment due to irregularity which is 
a function of both the irregularity and the density. It is possible that these two 
increments could be combined into one, which would then be a function of the capsule 
and pipe roughnesses, capsule irregularity and capsule density. 


223 


6.2 The Effect of Change of Pipe Diameter 

Figure 6 - 1 compares the pressure gradients of the aluminum and iron machined 
spheres in the 10 in. pipe on the basis of liquid pressure gradient instead of capsule 
velocity. A comparison between the sphere and liquid pressure gradients is readily 
made from this type of plot. It is also the means of correlating the effect of change 
of pipe diameter from 4 in. to 10 in. when water is the liquid. This is shown by the 
line on the figure which is the mean line of the spheres of specific gravities from 
1 .59 to 4.72 in the 4 in. pipe, (cf. Figure 4 - 3 of the Phase 2 report); the diameter 
ratio, 0.90, was the same as for spheres in the 10 in. pipe and the liquid in each 
pipe was water. The data for aluminum spheres in the 10 in. pipe lie close to the 
line for the 4 in. pipe spheres, and so does the data for the iron spheres at high veloci- 
ties. Since the specific gravity of aluminum is within the range covered by the tests in 
the 4 in. pipe,the data from the two pipes correlate well. The effect of the combined 
high specific gravity and lack of homogeneity of the iron spheres can be seen in the 
higher pressure gradients at low and medium velocities. At high velocities the effect 
disappears, probably due to partial sliding of the spheres, as discussed in Chapter 4 
of the Phase 2 Report. 

Figure 6-2 compares the pressure gradients of steel spheres in the 0.5 in. and 2 in. 

pipes, PVC spheres loaded to S.G. 4.33 in the 4 in. pipe and iron spheres in 

2 

the 10 in. pipe. The pressure gradients have been divided by k to allow for changes 
in the diameter ratio in the four pipes (see next section) and are plotted against the 
liquid pressure gradients. The 0.5 in. data are higher than the 2 in. data even at the 
same liquid pressure gradient, and the latter are higher than the 4 in. and 10 in. data. 


224 


The same data are plotted to a larger scale in Figure 6-3. The 4 in. and 10 in. 
data fall close to the same line, cis noted in Figure 6-1, but the pressure gradient rises 
rather uniformly with decrease of pipe diameter below 4 in. Changes of flow regime and 
of turbulence are certainly involved here, but friction factor-Reynolds number plots are 
of little help in deciding when turbulence occurs, since the correct velocity and dimension 
terms for f and Re are not known for spheres as they are for cylinders (cf. Chapter 7). 

Sphere train velocity, V , is plotted against bulk velocity, V^, in Figure 6-4 
for steel spheres in the 0.5 in. and 2 in. pipes, PVC spheres loaded to specific gravity 4.72 
in the 4 in. pipe and turned iron spheres in the 10 in. pipe, all with water as the liquid. 

All the data lie very close to a single straight line passing through the origin with a slope 
of 1.030. 

Calculation of velocity ratio, V^/V^, which is very sensitive to small variations 
of V c and V^, showed that the 2 in. pipe generally yielded the highest velocity ratios and 
the 10 in. the lowest, but nearly all the values fell in the range between 1.02 and 1.04; 
this is the approximate range that would be covered by measurement differences of ± 1% 
of or in the pipes of different diameters, based on a mean velocity ratio of 1 .030 
as indicated by the slope of Figure 6-4. Since such discrepancies (particularly in V^), 
cannot be ruled out, the velocity of perfect dense spheres in water at a given bulk velocity 
may be taken to be independent of pipe diameter. However, variations do occur with more 
viscous liquids or when using spheres of very low specific gravity at high velocities 
(Reference 1). 

Considering the less dense spheres in tests, all the spheres in the 4 in. pipe with 
specific gravities from 3 0 35 to 1 .59 (the minimum) showed velocity ratios of 1 .04; in the 


225 


10 in. pipe the machined aluminum spheres averaged 1.035 and the original aluminum 
spheres ranged from 1.00 at low velocities to 1.04 at high velocities. 

6.3 The Effect of Capsule/Pipe Diameter Ratio 

For the higher sphere densities in the 4 in. pipe, the diameter ratio of true 

spheres is correlated in Figure 6 - 5 by dividing the capsule pressure gradient by k s , 

the square of the diameter ratio. Figure 6 - 6 is a similar plot for the lower densities 
and also shows good correlation of the three diameter ratios. 

In the other pipes only one diameter of true spheres was tested. The tests 
of the three diameter ratios of rough aluminum and iron spheres in the 10 in. pipe 
showed that a change of diameter ratio had much more effect than for true spheres, 
especially with the iron spheres (Figures 3-89 and 3 - 90). The reason for this is the 
increase of the moment of any out-of-balance masses as the diameter increases, and 
the diameter ratio effect can be expected to depend on the irregularity of spheres and 
on their density. 

Figure 6 - 7 is a plot of horsepower per mile against solid throughput for the 
original cast spheres in the 10 in. pipe and shows that the power requirements of the 
aluminum spheres are little affected by diameter ratio, and only for the largest dia- 
meter of iron spheres is there a very significant increase of power requirements. For 
a given bulk velocity the larger spheres generally compensate in greater throughput 
for their greater power requirements. The expressions for power and throughput are 
given in section 6-5 where the power requirement of spheres is discussed. 


226 


6.4 The Effect of Liquid Viscosity 

Comparison of Figures 3-84 and 3-92 shows clearly the change brought about 
by an increase of liquid viscosity from 1.0 to 10.5 cs. in the 2 in. pipe. With water 
the pressure gradient is only slightly dependent on S.G., but at the higher viscosity 
the pressure gradient rises rapidly with S.G. , particularly at low values, and at medium 
and high velocities. 

The pressure gradients for spheres of specific gravity 1.3 were much higher 
at low sphere velocities when using the 10 cs. liquid than with water; however, as 
the sphere velocity rose above about 2 ft. /sec. the pressure gradients with the 10 cs. 
liquid levelled out and subsequently became increasingly less than with water. At 
specific gravity 2 the pressure gradients with the more viscous liquid were generally 
much higher than for even the steel spheres in water at the same velocity and as the 
specific gravity of the spheres was increased the pressure gradients continued to rise. 

Figure 6-8 plots the pressure gradients of Figure 3-92 against liquid pressure 
gradient. As in the original figure, the data for all the specific gravities lie on one 
line at the lowest liquid pressure gradients, but branch off in order of specific gravity 
as the liquid velocity increases. 

The pressure gradient can generally be expected to increase at a given specific 
gravity when using the more viscous liquid, even when some allowance is made for the 
increase of viscosity by comparing the capsule pressure gradients at the same liquid 
pressure gradient since the spheres incur greater losses than the liquid with increase of 
viscosity because of their rotation through the liquid. The reason for the lower pressure 
gradients of the spheres of specific gravity 1 .3 in the more viscous liquid at higher 
velocities is probably that these light spheres slide as well as roll at higher velocities. 


227 


Sliding begins when the pressure force overcomes the sum of the solid-solid 
friction and the horizontal resistive shear force (Reference 1). Increasing the specific 
gravity of the spheres increases friction and therefore delays the onset of sliding. This 
explains the sequential branching of the curves as specific gravity increases. 

An increase of liquid viscosity would also be expected to increase the velocity 
at which sliding occurs because it increases the horizontal resistive force due to shear, 
which acts with solid friction to oppose the pressure gradient force and is similar to 
friction in this regard. Figure 6-9 plots the pressure gradients due to steel spheres, 
diameter ratio 0.91, in the 0.5 in. pipe with liquids of viscosity from 1 .0 to 45 cs. 

(Figure 3-91 provides an interesting comparison for part of the same data plotted against 
V .) Figure 6 - 9 is of similar form to Figure 6-8 and shows that the pressure gradient is 
little influenced by viscosity at very low velocities when plotted against the liquid 
pressure gradient. However, as the latter increases, so does the capsule pressure gradient, 
due to increased shear, and the curves branch off sequentially with increasing viscosity. 
The figure supports the suggestion that the sequential branching off in Figures 6-8 and 
6-9 may be due to the spheres beginning to slide, the liquid pressure gradient at which 
this occurs being increased by an increase of either of the forces resisting sliding, viz. 
solid friction or liquid viscosity. 

The effect of viscosity is complicated by changes in the flow pattern around the 
spheres as they begin to slide or as the viscosity is altered. As mentioned in connection 
with the effect of pipe diameter, plots of friction factor against Reynolds number based 
on the velocity model for cylinders do not help in resolving this aspect. 

Figure 6-10 presents similar data for spheres of specific gravity 4.33, diameter 
ratio 0.93, in the 4 in. stainless steel pipe with different viscosity liquids. It corresponds 


228 


to Figure 3-93 with as the abscissa, and shows the same sequential branching-off 
as Figure 6-9 with increase of viscosity. 

6,5 Predicting the Pressure Gradients of Spheres 

It was pointed out in the first section of this chapter that the pressure gradient of 
spheres consists of a basic pressure gradient for perfect spheres to which has to be added 
a very small density-dependent increment due to solid friction, and a further increment 
due to any irregularities. For commercial purposes the small effect of density on the 
basic pressure gradient can be neglected, and the problem becomes one of estimating 
how much to add to the latter in order to take irregularity into account for different 
sphere densities. Commercial spheres are unlikely to be regular unless new manufacturing 
and finishing techniques are evolved to take advantage of the reduction of pressure 
gradient offered by good sphericity. 

The expression for calculating the diameter of standard pipe to carry the required 
throughput of spheres is equation 2 - 1 . A diameter ratio of 0.90, rather than a larger one, 
will allow for out-of-roundness or damage to the pipeline at bends, and avoid possible 
increase of power requirement due to a higher diameter ratio, (cf. Figure 6-7). The 
sphere velocity should be as low as possible to take advantage of the very small pressure 
gradients at low velocities, (cf. Figures 3 - 88 to 3 - 93), and 2 to 3 ft. /sec. is suggested 
for a first trial. The liquid velocity and pressure gradient are calculated as explained in 
Chapter 2. The capsule pressure gradient for spheres transported in water, may then 

be calculated from the equation of the line of Figure 6 - 1 for turned spheres of diameter 
ratio 0.90, i .e. : 

= 0.0005 + 2.2 


229 


There was so little effect of sphere density in the 4 in. pipe that the higher pressure 
gradients of the turned iron spheres of Figure 6 - 1 were certainly due largely to the lack of 
homogeneity mentioned in section 6. 1 . For other diameter ratios k, the pressure gradient 

/ k \ 2 

is multiplied by ( q — 90 ) ' s ' nce P ressure gradient is proportional to the square of the 

diameter ratio, i.e. ■ 


(¥) 


0.00062 +2.7 


m, 


(6-1) 


A penalty will have to be paid for any out-of-roundness, and in the case of the 
cast spheres tested in the 10 in. pipe this amounted to about 40% of the pressure gradUnt 
of machined aluminum spheres and 150% of that of the machined iron spheres (Figures 3 - 
87 and 3 - 88). The tolerances of these spheres are given in Chapter 4 of the Phase 2 
report and can be considered good commercial tolerances with present casting methods. 

With the capsule and liquid pressure gradients, the bulk velocity and the pipe 
diameter determined, the power requirement can be calculated. An example of the 
procedure is given in Chapter 2. The effect on the economics of a smaller pipe and higher 
velocity should then be considered. 


6.6 Comparison of Spheres with Cylinders. When Should Spheres Be Used? 

Due to the geometry, the throughput of spheres is only two-thirds that of cylinders 
at the same capsule velocity if the pipeline is completely linearly filled in each case; 
in addition, spheres require some space between them to allow rotation. However, since 
spheres are suggested mainly for denser materials, where the cylinder velocity is 
considerably less than that of spheres at the same bulk velocity, the difference in throughput 
is reduced. Further, it pays hydrodynamical ly to run spheres at low bulk velocities to take 


230 


advantage of the very low pressure gradients and power requirements at these velocities 
(cf. Figure 6-7). The question is therefore more a matter of deciding on the economics 
of a larger pipe to produce the lower sphere velocities versus the possibly higher power 
requirements and throughput of cylinders in smaller and less expensive pipe. 

The Phase 2 report suggested that spheres should be considered instead cif cylinders 
if the specific gravity of the material to be conveyed were greater than ca. 1 .25. No 
spheres below the specific gravity of aluminum were run in the 10 in. pipe, so in order 
to compare the power requirements of spheres and cylinders the approximate pressure 
gradients of rough spheres of specific gravity 1 .50 were calculated by assuming a linear 
relationship between the pressure gradient and the sphere density at a given sphere train 
velocity. The pressure gradients obtained by this method might be expected to be 
rather low since the "basic pressure gradient" is reduced in proportion to the specific 
gravity as well as the component due to this factor. However, as Table 6 - 1 shows, 
the differences between the calculated pressure gradients and those of the cast aluminum 
spheres are in any case not large for the low velocities (1 to 3 ft. /sec.) at which spheres 
should preferably be run, and the error in the calculation in this region is not more than 


5 %. 


231 


TABLE 6 - 1 


The pressure gradients of the original 9 in. cast 
aluminum spheres compared with the calculated 
values for specific gravity 1.50. 


Pressure gradient of Calculated pressure gradient 

V cast spheres (psi/ft.) for specific gravity 1.50 (psi/ft.) 


2 

0.0032 

0.0030 

3 

0.0050 

0.0044 

4 

0.0070 

0.0058 

5 

0.0090 

0.0072 


The power requirements for spheres is given by 


horsepower per mile 


(t- P ) c f f (t), 11 - F) (^T 1 )' 52801 v l 


550 


= 7.54 


(¥) e F F (¥) <• - F > 


D 2 V, (6-2) 


The throughput (millions of tons/year of 350 days) 
W 


2 t t d 2 x/ a x 62,3 x 3600 x 24 x 350 F 
c “ 3 4 144 c 


2000 x 10 s 


W = 3.43 x 10' 3 d 2 V a F 
c c 


(6-3) 


where F is the fraction of the pipe length filled by capsules. 


The power requirements and throughputs of out-of-round spheres of specific gravities 
1 .50 and 1 .25 were calculated from equations 6-2 and 6 - 3 for F = 1 . The equations 
are the same for cylinders, except that the throughput is 1 .5 times as great. 


232 


Figure 6-11 plots horsepower per mile against throughput for the calculated out- 
of-round 9 in. spheres of specific gravity 1 .25 and for 9 in. cylindrical capsules of the 
same specific gravity covered with vinyl tape, which gave the lowest pressure gradients 
of all plain cylinders. In order to be sure that the pressure gradient values of 
spheres for specific gravity 1.25 were not too low the same values were used to calculate 
the power requirements as for specific gravity 1 .50, but the throughputs were based on 
specific gravity 1.25. 

Allowance has been made for the reduced throughput of the spheres due to their 
geometry, but 100% linear fill has been assumed in both cases. The power requirements 
for the cylinders and spheres are the same at about the mid-velocity range of the cylinders 
and near the top of that of the spheres. The advantage of the spheres increases for a 
given throughput as the velocity decreases. 

Figure 6-12 contains similar plots for the out-of-round spheres and taped cylinders 
with specific gravity 1.50. Here the spheres have a decided advantage over the whole 
velocity range, again especially at the low velocities. 

The two figures for spheres and smooth cylinders in water indicate that spheres 
with a fairly close tolerance surpass smooth cylinders from a hydrodynamic point of view 
even at the low specific gravity of 1.25 at the lower throughputs, and the preference 
for spheres increases rapidly with increased specific gravity. But the specific gravity at 
which spheres would be selected depends largely on the economics of the two methods of 
capsule fabrication and on the balance between power requirements and pipe size referred 
to above. Examples of pipeline design using spheres and cylinders are given in Chapter 2. 


REFERENCE 


Ellis, H C S. The Pipeline Flow of Capsules, Part 5. An Experimental 
Investigation of the Transport by Water of Single Spherical Capsules 
with Density Greater than the Water. Can. J .Chem. Eng. , 42, 155 (1964). 


234 


(n 

(psi/ft. ) 



Pressure gradients of 9 in. turned aluminum and iron spheres plotted 
against liquid pressure gradient. 


FIGURE 6 - 1 


235 



k 2 

(psi/ft.) 



Pressure gradients of spheres in four diameters of pipe plotted against 
liquid pressure gradient. 


FIGURE 6-2 


236 



Part of Figure 6 - 2 to a larger scale. 


FIGURE 6-3 


237 


V 

c 

(ft/sec) 



V b (ft/sec) 


The variation of capsule velocity with bulk velocity. The densest 
spheres in water in the 0.5 in., 2 in., 4 in. and 10 in. diameter 

pipes. 


FIGURE 6-4 


238 



k 2 

(psi/ft.) 



{¥) f (psi/fK) 

Correlation of data from spheres of three diameter ratios in the 4 in. 

pipe. O’ = 4.33 to 5.07. 


FIGURE 6-5 


239 



k 2 

(psi/ft.) 



pipe . a = 1 . 55 to 1 .59. 


FIGURE 6-6 


240 



Power requirements of the original cast spheres in the 10 in. 
pipe versus throughput of metal . 


FIGURE 6-7 


241 



Pressure gradients in the 2 in. pipe with different values of s.g., and liquid 

viscosity 10.5 cs. 


FIGURE 6-8 


242 



k 2 

(psi/ft.) 



Pressure gradients of steel spheres in the 0.5 in. pipe with different 

liquid viscosities. 


FIGURE 6-9 


243 



k 2 

(psi/ft.) 



(fc) f ( p s!/ft - } 

Pressure gradients of spheres of specific gravity 4.33 in the 4 in. 
pipe with different liquid viscosities. 


FIGURE 6-10 


244 



Solid Throughput (Millions of tons/yr) 


Power requirements of smooth cylinders and estimated requirements 
of out-of-round spheres, both of s.g. 1 .25, in the 10 in. pipe. 


FIGURE 6-11 


Horsepower Per Mile 


245 



Power requirements of smooth cylinders and estimated requirements 
of out-of-round spheres, both of s.g. 1.50, in the 10 in. pipe. 


FIGURE 6-12 



CHAPTER VII 


Calculation of the Liquid Throughput 





CHAPTER VII 


Calculation of the Liquid Throughput 

7.0 Summary 

A method for predicting the liquid throughput in a capsule pipeline from the 
capsule velocity and the capsule pressure gradient is described. 

A feature of this method is that the predicted capsule pressure gradient, used for 
the liquid flow prediction, may deviate substantially from the measured capsule pressure 
gradient before the predicted liquid throughput deviates appreciably from the measured 
liquid throughput. 

A tabulation of all the experimental results of the TDA-RCA project with capsules 
run in the 0.5, 2, 4 and 10 inch pipes is presented at the end of the chapter to show for each 
run the accuracy of the method of liquid flow prediction. The tabulation comprises ca. 

1700 runs or a total of 67,000 passes. 

The measured data are compared with the predicted data and values of the slope 
(B) and intercept (A) of a linear least squares fit are presented for each run. A perfect 
fit between measurement and prediction would gives values of B = 1.0 and A =0.0 and 
most of the experimental runs approach this ideal closely. Histograms showing the agree- 
ment between measurement and prediction are also presented. 


248 


7. 1 Calculation of the Total Liquid Flow 

A knowledge of the bulk liquid velocity is required to estimate the quantity 
of liquid required for a capsule pipeline. The total liquid flow is 


Q f ■ 5 tT v b 


n d' 


576 


V F ft. 3 /sec. 
c 


(7-1) 


when D and d are expressed in inches and V in ft. /sec. F is the fractional linear fill 
of capsules in a pipeline. 

As discussed in Chapter 2 the capsule velocity is determined from the required 
solids throughput, the capsule diameter and the linear fill. The bulk liquid velocity can 
be calculated from the capsule pressure gradient by means of the model developed in the 
Phase 2 Report. The method can be summarized as follows. 


7.2 Calculation of the Bulk Liquid Velocity 

Assume turbulent flow in the annulus so that f Re °’ 35 = 0.07. The bulk 

c c 

velocity then becomes 


V, = k V + (1 - k 8 ) 
b c 



(7-2) 


A check of the Reynolds number is needed to verify that the flow is turbulent. 


C D (1 - k) (V - k V ) 

Re = -2 (7 - 3) 

c (1 -k a ) v 


If Re > 1000 V, is correct 
c b 

If Re ^ 1000 V, must be recalculated since in laminar flow f Re = 9.6 
c b c c 


249 


The bulk velocity in that case is: 

C 3 (i^) DS 0 " k)8 

V, = k V + (1 + k s ) — — - 2 (7 - 4) 

b P p 

The constants C 1 , C 2 and C 3 used in these equations depend upon the units used 
and are detailed in Table 7 - 1 . 

7. 2 Accuracy of the Method 

Linear least squares regressions for each capsule run as well as histograms using 
the individual data points were used to check the validity of this model. 

Figure 7 - 1 presents data of a representative run in the 10 inch pipeline showing 
the bulk liquid velocity as calculated from equations 7 - 2 to 7 - 4 plotted as a function 
of the measured bulk liquid velocity. The solid line represents the linear least square 
fit for these data and it is of the form: Calculated = A + B (measured V^) . For a 
perfect correlation the solid line would coincide with the dashed line of unity 
slope through the origin. Values of A and B for each capsule tested in the experimental 
pipeline are tabulated at the end of this chapter showing how well the data of each run 
approach the ideal of A = 0 and B = 1 . The column labelled X in the tables refers 
to the number of measured bulk velocity data points which deviated more than 0.5 
ft. /sec. from the calculated values due to experimental errors and hence were not included. 


250 


TABLE 7 - 1 

Values of C , C , C for various units 

12 3 


D 

inches 

inches 

inches 

inches 

V 

ft/sec. 

ft/sec. 

ft/sec. 

ft/sec. 

(A PA) 

psi/ft. 

psi/ft. 

psi/ft. 

psi/mile 

P 

S.G. 

S.G. 

ib. /ft . 3 

lb. /ft. 3 

V 

centistokes 

ft. 2 /sec. 

ft. 2 /sec. 

ft. 2 /sec. 

C 

i 

415 

24 

1480 

0.28 

c 

2 

7742 

0.083 

0.083 

0.083 

c 

3 

2500 

0.027 

1.67 

0.00032 


251 


The tables show that the correlation is good for trains of capsules where the ratio of train 
length to pipe diameter is large. For very small values of this ratio the end effects at 
the tail and nose of the capsules are largely responsible for the deviation between 
prediction and measurement. The model was developed for cylindrical capsules but, 
as the tables show, the correlation is just as good for trains of spheres. The sphere data 
are indicated in the tables by a large dot preceding each run number. 

Histograms of the data are presented in Figures 7-2, 7-3, 7-4 and 7-5 

for the 2, 4 and 10 inch pipelines respectively. The histogram data were limited to 

trains of cylinders. The histograms are presented on the basis of calculated V^/ 

measured V, on the X axis and sample fraction on the Y axis. The bar graphs in the figures 
b 

show the frequency distribution. Any samples for which the ratio fell below 0.90 are 
included in the bar graphs between 0.88 and 0.90 and any samples for which the ratio 
fel I above 1.10 are included in bar graphs between 1.10 and 1.12 in the figures. The 
average ratio between calculated and measured bulk liquid velocity is 1.01 for all 
four pipes. This indicates that the predicted averages 1% too high, and multiplying 
the result by 0.99 therefore would give a slight improvement for very precise calcula- 
tions of the liquid throughput. The model as presented does however provide a solution 
of sufficient accuracy for all practical calculations of the bulk liquid velocity and hence 
the liquid throughput. 

7.3 Sensitivity of the Results to the Capsule Pressure Gradient 

Since the prediction of the bulk velocity is partially based upon the capsule 
pressure gradient, its sensitivity to errors in the capsule pressure gradient is an important 


252 


consideration. It appears that the sensitivity varies with varying conditions but may be 
readily checked in each case. An example is presented to demonstrate this point. 

Run 744 in the 10 inch pipeline represents a 20 ft. long train of 0.90 diameter 
ratio epoxy surface cylindrical capsules with a specific gravity of 1.13 flowing in 
water. Data selected at a practical velocity range in that run are: 

Capsule velocity: 6.69 ft. /sec. 

Bulk liquid velocity: 6.43 ft. /sec. 

Capsule pressure gradient: 0.0113 psi/ft . 

The predicted bulk velocity is calculated using equation 7 - 2 to be 6.49 ft. /sec. showing 
an error of 1%. A check on the Reynolds number, using equation 7-3, shows that the 
flow is turbulent, confirming that the use of equation 7 - 2 is correct. If, however, in the 
preliminary design equations of a capsule pipeline the assumed capsule pressure gradient is 
double the measured pressure gradient, the predicted bulk liquid velocity is calculated to be 

6.72 ft. /sec. , showing an error of 4.5%. When the assumed capsule pressure gradient 
is only half of that measured the predicted bulk liquid velocity is calculated to be 
6.33 ft. /sec., showing an error of 1.6%. Hence the model will predict the bulk liquid 
velocity accurately even if the pressure gradient used for the calculations is not given 
precisely. 

7.4 Calculating Capsule Velocity Rather than Bulk Liquid Velocity 

As presented, the model allows the calculation of the bulk liquid velocity from 
the capsule velocity and the capsule pressure gradient. The capsule velocity is normally 
provided as a direct consequence of the desired throughput of solids and the capsule 
pressure gradient is determined either from the tabulated graphs of experimental data of 


253 


Chapter 3, from a pulling test described in Chapter 4, from the correlation of Chapters 5 
and 6, or from a combination of these. The bulk liquid velocity is then calculated from 
the model . 

However, the model may also be used to calculate the capsule velocity from 
the bulk liquid velocity and the capsule pressure gradient. This method is particularly 
applicable for the capsule pressure gradient correlation in Chapter 5. This is achieved 
by rearranging equations 7-2 and 7 - 4 as follows: 

Assume turbulent flow in the annulus 


V 

c 



0.571 


(7-5) 


Check the Reynolds number using equation 7-3, and if it is below 1000 recalculate: 


V, 


(1 ~k s ) 




D 2 (1 - k) 


P v 


(7-6) 


7,5 Conclusion 

The model described above makes use of the fact that the capsule pressure gradient 
is a very good indicator of the degree of surface interaction between the capsule, the 
carrier liquid and the pipe wall. The liquid velocity which is very much influenced by 
this interaction can therefore be predicted very precisely using the capsule pressure 
gradient as one of the inputs in the calculations. 


254 



Calculated versus measured bulk liquid velocity for run 798. 


FIGURE 7- 1 


255 



CALCULATED V / 

b / MEASURED V b 

Histogram showing the agreement between calculated and meas- 
ured bulk liquid velocity in a 0.5 in. capsule pipeline. 

D = Pipe Diameter: N = Number of Measurements: 

AV. = Mean Value: S.D. = Standard Deviation. 


FIGURE 7-2 


256 



CALCULATED 


MEASURED V, 


Histogram showing the agreement between calculated and meas- 
ured bulk liquid velocity in a 2 in. capsule pipeline. 


FIGURE 7-3 


257 


-8 _ 


.7 _ 


.6 _ 


Z - 5 - 

O 


u 

< 


.4 _ 


I- 3 L 
< 
oO 

.2 . 


.1 _ 


D = 4.03 
N = 18140 


.92 


.96 


AV.= 1.009 
S.D.= 0.019 


1.00 


1.04 


1.08 


CALCULATED V, 


MEASURED V L 


Histogram showing the agreement between calculated and meas- 
ured bulk liquid velocity in a 4 in. capsule pipeline. 


FIGURE 7 -4 


258 



CALCULATED V, 


MEASURED V, 


Histogram showing the agreement between calculated and meas- 
ured bulk liquid velocity in a 10 in. capsule pipeline. 


FIGURE 7 -5 


TABULATION OF EXPERIMENTAL RUNS 


A tabulation of all the capsule test runs during the project in the 0.5, 2, 4 and 
10 inch pipes is presented on the following pages. There are ca. 1700 runs with from 
20 to 60 data points in each run. Shown for each are the run number, the pipe dia- 
meter (D), the diameter ratio (k), the total length of the capsule train used (L^), the 
capsule specific gravity (a) , the liquid viscosity (v) , the values of A and B of the 
linear least square fit and the number of data points excluded from these fits (X) by 
reason of deviation between prediction and measurement of more than 0.5 ft/sec., 
due to experimental errors. 

A dot ( • ) in front of a run indicates that spheres were used. 


260 


Run 

D 

k 

Lt 

c r 

V 

A 

B 

X 

Run D 

k 

Lt 

0" 

■y 

A 

B 

X 

, 

10.020 

0.848 

4.000 

1.000 

0.966 

-0.017 

0.986 

1 

525 

10.020 

0.898 

20.000 

1 .000 

0.974 

0.026 

1.004 

0 

2 

10.020 

0.848 

20.000 

1.000 

0.952 

0.024 

1.020 

1 

-525 

10.020 

0.898 

20.000 

1.000 

0.977 

0.017 

1.010 

0 

-2 

10.020 

0.848 

20.000 

1 .000 

0,959 

-0.001 

1.027 

0 

1016 

10.020 

0.948 

4.000 

1.120 

0.958 

0.009 

1.003 

0 

3 

10.020 

0.848 

16.000 

1.000 

0.946 

-0.001 

1.029 

0 

-1016 

10.020 

0.948 

4.000 

1.120 

0.962 

0.010 

1.004 

0 

-3 

10.020 

0.848 

16.000 

1.000 

0.953 

0.021 

1 .020 

0 

1017 

10.020 

0.948 

12.000 

1.120 

0.945 

-0.027 

1.009 

0 

4 

10.020 

0.848 

20.000 

1.500 

1.039 

0.812 

0.909 

0 

-1017 

10.020 

0.948 

12.000 

1.120 

0.964 

-0.034 

1.011 

0 

-4 

10.020 

0.848 

20.000 

1.500 

1.012 

0.728 

0.934 

0 

1018 

10.020 

0.948 

20.000 

1.120 

0.971 

-o.o4o 

1 .012 

0 

5 

10.020 

0.848 

4.000 

1.500 

1.048 

0.815 

0.978 

0 

-1018 

10.020 

0.948 

20.000 

1.120 

0.949 

-0.036 

1 .012 

0 

-5 

10.020 

0.848 

4.000 

1.500 

1.018 

0.886 

0.977 

0 

1019 

10.020 

0.948 

20.000 

1.120 

0.943 

-0.028 

1 .013 

0 

-6 

10.020 

0.848 

4.000 

1 .500 

1.012 

0.848 

0.985 

0 

1020 

10.020 

0.948 

4.000 

1.000 

0.974 

0.015 

0.996 

0 

-7 

10.020 

0.848 

4.000 

1.500 

0.987 

0.851 

0.988 

0 

-1020 

10.020 

0.948 

4.000 

1.000 

0.956 

0.006 

1.001 

0 

8 

10.020 

0.848 

8.000 

1.500 

0.989 

0.594 

0.983 

0 

1021 

10.020 

0.948 

20.000 

1.000 

1.028 

0.002 

1 .003 

0 

-8 

10.020 

0.848 

8.000 

1.500 

0.962 

0.616 

0.985 

0 

-1021 

10.020 

0.948 

20.000 

1.000 

0.992 

0.010 

1.002 

0 

9 

10.020 

0.848 

12.000 

1.500 

0.974 

0.645 

0.961 

0 

1022 

10.020 

0.948 

4.000 

1.030 

0.963 

-0.005 

1 .004 

0 

-9 

10.020 

0.848 

12.000 

1.500 

0.951 

0.469 

0.984 

0 

-1022 

10.020 

0.948 

4.000 

1.030 

0.949 

o.oi4 

1 .001 

0 

10 

10.020 

0.848 

16.000 

1.500 

1.029 

0.592 

0.962 

0 

1023 

10.020 

0.948 

12.000 

1.030 

0.938 

0.005 

1.006 

0 

-10 

10.020 

0.848 

16.000 

1.500 

1.025 

0.665 

0.946 

0 

-1023 

10.020 

0.948 

12.000 

1.030 

0.956 

-0.013 

1 .008 

0 

500 

10.020 

0.898 

20.000 

1.000 

0.919 

0.002 

1 .021 

0 

1024 

10.020 

0.948 

20.000 

1.030 

0.966 

0.016 

1.002 

0 

-500 

10.020 

0.898 

20.000 

1.000 

0.92' 

0.022 

1.017 

0 

-1024 

10.020 

0.948 

20.000 

1 .030 

0.955 

0.014 

1.003 

0 

501 

10.020 

0.898 

20.000 

1.500 

0.984 

0.016 

1.019 

0 

30 

10.020 

0.848 

20.000 

1.030 

0.932 

0.050 

1.005 

0 

-501 

10.020 

0.898 

20.000 

1.500 

0.962 

-0.005 

1.017 

0 

526 

10.020 

°-|9| 

4.000 

1.030 

0.970 

0.077 

0.987 

0 

1000 

10.020 

0.948 

8.000 

1.000 

1.001 

-0.000 

1.017 

0 

-526 

10.020 

°.898 

4.000 

1.030 

0.971 

0.039 

0.997 

0 

1001 

10.020 

0.948 

20.000 

1.500 

0.952 

-C.029 

1.030 

0 

527 

10.020 

0.898 

12.000 

1.030 

0.992 

0.039 

1 .000 

0 

-1001 

10.020 

0.948 

20.000 

1.500 

0.942 

-0.035 

1.030 

1 

-527 

10.020 

0.898 

12.000 

1.030 

0.947 

-0.011 

1.012 

0 

5 

10.020 

0.848 

4.000 

1.500 

'.035 

0.769 

K019 

0 

528 

10.020 

0.898 

20.000 

1.030 

0.969 

-0.000 

1.007 

0 

-5 

10.020 

0.848 

4.000 

1.500 

0.999 

0.852 

0.984 

0 

-528 

10.020 

0.898 

20.000 

1.030 

0.966 

0.005 

1.006 

0 

501 

10.020 

0.898 

20.000 

1.500 

0.973 

0.000 

0.000 

1 

529 

10.020 

0.898 

20.000 

1.030 

1 .044 

0.018 

1.002 

0 

-501 

10.020 

0.898 

20.000 

1.500 

0.944 

0.030 

1 .01 1 

0 

22 

10.020 

0.848 

20.000 

1.120 

0.893 

0.065 

I.OO3 

0 

1001 

10.020 

0.948 

20.000 

1 .500 

0.937 

-0.030 

1.006 

0 

19 

10.020 

0.848 

12.000 

1.060 

0.944 

0.017 

1.012 

0 

-1001 

10.020 

0.948 

20.000 

1.500 

0.941 

-0.059 

1 .028 

0 

20 

10.020 

0.848 

4.000 

1.120 

0.890 

0.289 

0.988 

0 

11 

10.020 

0.848 

20.000 

1.500 

1.023 

0.824 

0.914 

0 

-20 

10.020 

0.848 

4.000 

1.120 

0.900 

0.322 

0.980 

0 

-11 

10.020 

0.848 

20.000 

1.500 

1.023 

0.473 

0.969 

0 

21 

10.020 

0.848 

12.000 

1.120 

0.905 

0.070 

1.004 

0 

502 

10.020 

0.898 

4.000 

1 .500 

1.039 

0.368 

0.984 

0 

-21 

10.020 

0.848 

12.000 

1.120 

0.888 

0.112 

1.002 

0 

-502 

10.020 

0.898 

4.000 

1.500 

1 .016 

0.396 

0.980 

0 

31 

10.020 

0.848 

4.000 

0.980 

1 .029 

0.085 

0.992 

0 

503 

10.020 

0.898 

8.000 

1.500 

1.001 

0.258 

0.982 

0 

-31 

10.020 

0.848 

4.000 

0.980 

1.023 

0.101 

0.997 

0 

-503 

10.020 

0.898 

8.000 

1.500 

1.003 

0.195 

0.991 

0 

32 

10.020 

0.848 

12.000 

0.980 

0.994 

0.049 

1.003 

0 

504 

10.020 

0.898 

12.000 

1.500 

O.963 

0.091 

1.000 

0 

-32 

10.020 

0.848 

12.000 

0.980 

0.994 

0.046 

1.007 

0 

-504 

10.020 

0.898 

12.000 

1.500 

1 .004 

O.O87 

1.000 

0 

33 

10.020 

0.848 

20.000 

0.980 

1.031 

0.030 

1.008 

0 

505 

10.020 

0.898 

16.000 

1 .500 

0.961 

O.136 

0.984 

0 

-33 

10.020 

0.848 

20.000 

O.98O 

1.007 

O.034 

1.009 

0 

-505 

10.020 

0.898 

16.000 

1.500 

0.972 

0.087 

0.993 

0 

34 

10.020 

0.848 

20.000 

0.980 

0.979 

0.044 

1.005 

0 

506 

10.020 

O.898 

20.000 

1.500 

0.940 

-0.018 

1 .027 

0 

514 10.020 

0.898 

12.000 

1.060 

0.912 

0.068 

1.000 

0 

-506 

10.020 

0.898 

20.000 

1.500 

0.963 

0.023 

0.997 

0 

530 

10.020 

0.898 

4.000 

0.980 

1 .002 

0.005 

1.004 

0 

1002 

10.020 

0.948 

4.000 

1.500 

0.973 

0.099 

1.004 

0 

-530 

10.020 

0.898 

4.000 

0.980 

0.991 

-0.017 

1.010 

0 

-1002 

10.020 

0.948 

4.000 

1.500 

0.967 

0.010 


0 

531 

10.020 

O.898 

12.000 

O.98O 

0.991 

0.025 

1.009 

0 

1003 

10.020 

0.948 

8.000 

1.500 

0.93 1 * 

-0.028 

1 .018 

0 

-531 

10.020 

0.898 

12.000 

0.980 

0.971 

0.005 

1.010 

0 

-1003 

10.020 

0.948 

3.000 

1.500 

0.933 

-0.040 

1.021 

0 

532 

10.020 

0.898 

20.000 

0.980 

0.968 

0.064 

0.998 

0 

1004 

10.020 

0.948 

12.000 

1.500 

0.917 

-0.037 

1.025 

0 

*532 

10.020 

0.898 

20.000 

0.980 

0.975 

0.061 

0.997 

0 

-1004 

10.020 

0.948 

12.000 

1 .500 

0.895 

-o.o46 

1.027 

0 

533 

10.020 

0.898 

20.000 

0.980 

0.987 

0.031 

1.007 

0 

1005 

10.020 

0.948 

16.000 

1.500 

0.930 

-0.069 

1.027 

0 

ioi4 

10.020 

0.948 

12.000 

1.060 

0.914 

0.029 

0.999 

0 

-1005 

10.020 

0.948 

16.000 

1.500 

0.939 

-0.051 

1 .028 

0 

1025 

10.020 

0.948 

20.000 

1.030 

1.059 

0.019 

1.004 

0 

1006 

10.020 

0.948 

20.000 

1.500 

0.927 

-0.065 

1.018 

0 

1026 

10.020 

0.948 

4.000 

0.980 

0.962 

0.007 

I.OO3 

0 

-1006 

10.020 

0.948 

20.000 

1.500 

0.908 

-0.028 

1 .019 

0 

-1026 

10.020 

0.948 

4.000 

0.980 

0.964 

-0.002 

1.004 

0 

12 

10.020 

0.848 

4.000 

1.250 

0.932 

0.480 

0.990 

0 

1027 

10.020 

0.948 

12.000 

0.980 

1 .019 

0.012 

1.007 

0 

-12 

10.020 

0.848 

4.000 

1.250 

0.913 

0.459 

0.994 

0 

-1027 

10.020 

0.948 

12. COO 

0.980 

0.941 

0.004 

1 .007 

0 

13 

10.020 

0.848 

12.000 

1.250 

0.909 

0.161 

1.019 

0 

1028 

10.020 

0.948 

20.000 

0.980 

1.008 

0.023 

1 .003 

0 

-13 

10.020 

0.848 

12.000 

1.250 

0.880 

0.190 

1 .007 

0 

-1028 

10.020 

0.948 

20.000 

O.98O 

1.004 

0.022 

1.002 

0 

507 

10.020 

0.898 

4.000 

1.250 

0.949 

0.179 

0.993 

0 

1029 

10.020 

0.948 

20.000 

0.980 

0.964 

0.01 1 

1.008 

0 

-507 

10.020 

0.898 

4.000 

1.250 

0.927 

0.166 

1.001 

0 

534 

10.020 

0.898 

20.000 

1.250 

1.062 

0.007 

1.015 

11 

14 

10.020 

0.848 

20.000 

1.250 

0.875 

0.169 

1.000 

0 

-534 

10.020 

0.898 

20.000 

1.250 

1 .002 

-0.061 

1.028 

0 

-14 

10.020 

0.848 

20.000 

1.250 

0.888 

0.209 

0.991 

0 

535 

10.020 

0.898 

20.000 

1.500 

1.000 

0.082 

0.986 

0 

508 

10.020 

0.898 

12.000 

1 .250 

0.939 

-0.02' 

1.012 

0 

-535 

10.020 

0.898 

20.000 

1.500 

0.973 

0.084 

0.987 

0 

-508 

10.020 

0.898 

12.000 

1.250 

0.925 

-0.028 

1 .019 

0 

536 

'0.020 

0.898 

12.000 

1.250 

1.033 

0.018 

1 .013 

0 

509 

10.020 

0.898 

20.000 

1.250 

0.933 

-0.085 

1.028 

0 

-536 

10.020 

0.898 

12.000 

1.250 

1 .042 

0.013 

1.007 

0 

-509 

10.020 

0.898 

20.000 

1.250 

0.953 

-0.075 

1.016 

0 

537 

10.020 

0.898 

12.000 

1.500 

1.093 

0.115 

0.993 

0 

1007 

10.020 

G.9h8 

4.000 

1.250 

0.915 

-0.023 

1.011 

0 

-537 

10.020 

0.898 

12.000 

1 .500 

1.054 

0.090 

0.998 

0 

-1007 

10.020 

0.948 

4.000 

1.250 

0.931 

-0.010 

1.008 

0 

-538 

10.020 

0.898 

12.000 

1.500 

1.052 

0.126 

0.990 

0 

1008 

10.020 

0.948 

12.000 

1.250 

0.907 

-0.067 

1.015 

0 

35 

10.020 

0.848 

20.000 

1.250 

'.015 

0.252 

0.972 

0 

-1008 

10.020 

0.948 

12.000 

1.250 

0.904 

-0.054 

1.014 

0 

-35 

10.020 

0.848 

20.000 

1.250 

1.036 

0.248 

0.973 

0 

1009 

10.020 

0.948 

20.000 

1.250 

0.948 

-0.041 

1.014 

0 

36 

10.020 

0.848 

20.000 

1.500 

0.955 

0.692 

0.928 

0 

-1009 

10.020 

0.948 

20.000 

1.250 

0.877 

-0.019 

1.010 

0 

-36 

10.020 

0.848 

20.000 

1.500 

0.981 

0.656 

0.929 

0 

15 

10.020 

0.848 

4.000 

1.250 

0.930 

0.405 

0.996 

0 

1030 

10.020 

0.948 

20.000 

1.250 

1.003 

-0.041 

1.017 

0 

510 

10.020 

0.898 

4.000 

1.250 

0.920 

0.172 

0.996 

0 

-1030 

10.020 

0.948 

20.000 

1.250 

0.987 

-0.053 

1.015 

0 

1010 

10.020 

0.948 

4.000 

1.250 

0.901 

-0.007 

1.007 

0 

1031 

10.020 

0.948 

20.000 

1.500 

1.003 

-0.066 

1.010 

0 

16 

10.020 

0,848 

20.000 

1.060 

0.914 

-0.024 

1.019 

0 

-1031 

10.020 

0.948 

20.000 

1 .500 

1.029 

-0.066 

1.009 

0 

-16 

10.020 

0.848 

20.000 

1.060 

0.887 

0.019 

1.013 

0 

• 1500 

10.020 

0.848 

4.250 

2.700 

1.106 

0.193 

0.991 

0 

511 

10.020 

0.898 

20.000 

1.060 

0.904 

0.037 

i.oo4 

0 

• 1501 

10.020 

0.848 

8.500 

2.700 

1.076 

0.204 

0.980 

0 

-511 

10.020 

0.898 

20.000 

1.060 

0.888 

0.010 

1.008 

0 

® 1502 

10.020 

0.848 

12.042 

2.700 

1.073 

0.119 

0.995 

0 

17 

10.020 

0.848 

4.000 

1.060 

0.907 

0.080 

0.999 

0 

• 1503 

10.020 

0.848 

16.292 

2.700 

1.O66 

0.114 

1.004 

0 

-17 

10.020 

0.848 

4.000 

1.060 

0.889 

0.109 

1.001 

0 

• 1504 

10.020 

0.848 

19.833 

2.700 

1.038 

0.103 

0.998 

0 

18 

10.020 

0.848 

12.000 

1.060 

0.932 

-0.001 

1.015 

0 

• 1600 

10.020 

0.898 

3.75° 

2.650 

1.055 

0.064 

0.994 

0 

-18 

10.020 

0.848 

12.000 

1.060 

0.937 

0.047 

1 .006 

0 

® 1601 

10.020 

0.898 

8.250 

2.650 

1 .038 

0.064 

0.987 

0 

512 

10.020 

0.898 

4.000 

1.060 

0.919 

0.042 

1.000 

0 

• 1602 

10.020 

0-898 

12.000 

2.650 

1.047 

0.044 

0.996 

0 

-512 

10.020 

0.898 

4.000 

1.060 

0.918 

0.027 

1.000 

0 

• 1603 

10.020 

0.898 

15.750 

2.650 

1.033 

0.059 

0.992 

0 

513 

10.020 

0.898 

12.000 

1.060 

0.903 

0.065 

0.999 

0 

• 1505 

10.020 

0.848 

19.833 

2.700 

1.048 

O.131 

0.986 

0 

-513 

10.020 

0.898 

12.000 

1.060 

0.899 

0.037 

1.006 

0 

• 1604 

10.020 

0.898 

20.250 

2.650 

1.053 

0.060 

0.994 

0 

1011 

10.020 

0.948 

4.000 

1.060 

0.897 

-0.005 

0.999 

0 

• 1605 

10.020 

0.898 

20.250 

2.700 

1.077 

0.074 

0.990 

0 

-1011 

10.020 

0.948 

4.000 

1.060 

0.875 

0.020 

0.997 

0 

• 1700 

10.020 

0.948 

4.750 

2.700 

1.020 

0.009 

0.998 

c 

1012 

10.020 

0.948 

12.000 

1.060 

0.873 

0.004 

1.003 

0 

• 1701 

10.020 

0.948 

7.917 

2.700 

1.044 

0.022 

0.992 

0 

-1012 

10.020 

0.948 

12.000 

1.060 

0.878 

0.006 

1.003 

0 

® 1702 

10.020 

0.948 

7.917 

2.700 

1.033 

O.OO3 

0.996 

0 

1013 

10.020 

0.948 

20.000 

1.060 

0.887 

0.044 

1.000 

0 

• 1703 

10.020 

0.948 

11.875 

2.700 

1.046 

o.oi4 

0.996 

0 

-1013 

10.020 

0.948 

20.000 

1.060 

0.952 

0.014 

1.001 

0 

• 1704 

10.020 

0.948 

15.833 

2.700 

1.035 

0.027 

0.994 

1 

-22 

10.020 

0.848 

20.000 

1.120 

0.873 

0.011 

1.015 

0 

• 1705 

10.020 

0.948 

19.792 

2.700 

1.030 

0.035 

0.993 

0 

23 

10.020 

0.848 

4.000 

1.500 

0.889 

0.983 

0.956 

0 

• 1706 

10.020 

0.948 

19*792 

2.700 

I.039 

0.020 

0.996 

0 

-23 

10.020 

0.848 

4.000 

1.500 

0.871 

1.039 

o.94i 

0 

• 1800 

10.020 

0.848 

4.250 

7.250 

'.049 

0.396 

0.979 

0 

1015 

10.020 

0.948 

4.000 

1.500 

0.896 

0.055 

0.997 

0 

• 1801 

10.020 

0.848 

7.792 

7.250 

1.028 

0.294 

1.001 

0 

-1015 

10.020 

0.948 

4.000 

1.500 

0.876 

0.080 

0.994 

0 

• 1802 

10.020 

0.848 

12.042 

7.250 

1.082 

0.279 

1.002 

0 

515 

10.020 

0.898 

4.000 

1.500 

0.897 

0.362 

0.986 

0 

• 1803 

10.020 

0.848 

16.292 

7.250 

1.098 

O.189 

1 .014 

1 

-515 

10.020 

0.898 

4.000 

1.500 

0.876 

0.468 

0.962 

0 

• 1804 

10.020 

0.848 

19.833 

7.250 

1.125 

0.242 

0.995 

0 

-516 

10.020 

0.898 

4.000 

1.500 

0.958 

0.402 

0.968 

0 

• 1900 

10.020 

0.898 

3.750 

7.250 

1.097 

0.128 

0.998 

0 

*517 

10.020 

0.898 

4.000 

1.500 

0.944 

0.391 

0.967 

0 

• 1901 

10.020 

0.898 

8.250 

7.250 

1.094 

0.105 

' ,005 

0 

-518 

10.020 

0.898 

4.000 

1.500 

0.932 

0.399 

0.970 

0 

• 1805 

10.020 

0.848 

19.833 

7.250 

1.049 

0.194 

1.008 

0 

-519 

10.020 

0.898 

4.000 

1.500 

0.975 

0.389 

0.969 

0 

• 1902 

10.020 

0.898 

12.000 

7.250 

1.069 

0.119 

1.000 

0 

24 

10.020 

0.848 

20.000 

1.120 

0.970 

0.099 

1 .oo4 

0 

• 1903 

10.020 

0.898 

15.750 

7.250 

1.122 

0.091 

1.006 

0 

25 

10.020 

0.848 

4.000 

1.000 

1.029 

0.026 

1.011 

0 

• 1904 

10.020 

0.898 

20.250 

7.250 

1.106 

0.095 

1.002 

0 

-25 

10.C20 

0.848 

4.000 

1 .000 

0.998 

-0.006 

1.015 

0 

• 1905 

10.020 

0.898 

20.250 

7.200 

1.049 

0.167 

0.992 

0 

26 

10.020 

0.848 

20.000 

1.000 

0.985 

-0.002 

1.011 

0 

• 2000 

10.020 

0.948 

3-958 

7.250 

1.076 

0.006 

1.003 

0 

-26 

10.020 

0.848 

20.000 

1.000 

0.989 

0.032 

1 .007 

0 

• 2001 

10.020 

0.948 

7.916 

7.250 

1.069 

0.043 

0.997 

0 

27 

10.020 

0.848 

4.000 

1.030 

1.011 

0.098 

0.990 

0 

• 2002 

10.020 

0.948 

11.875 

7.250 

1.038 

0.007 

1.006 

0 

-27 

10.020 

0.848 

4.000 

1.030 

1.012 

0.115 

0.984 

0 

• 2003 

10.020 

0.948 

15.833 

7.200 

1.030 

0.006 

1.002 

0 

28 

10.020 

0.848 

12.000 

1 .030 

I.005 

0.050 

1.002 

0 

• 2004 

10.020 

0.948 

19.792 

7.200 

1.029 ■ 

■0.012 

1.008 

0 

-28 

10.020 

0.848 

12.000 

1.030 

0.984 

0.022 

1 .007 

0 

• 2005 

10.020 

0.948 

19.792 

7-190 

1.057 

0.051 

1.002 

0 

29 

10.020 

0.848 

20.000 

1.030 

0.981 

0.037 

1.007 

0 

900 

10.020 

0.898 

4.000 

1.500 

1.038 

0.491 

0.959 

0 

-29 

10.020 

0.848 

20.000 

1.030 

0.983 

0.009 

1.010 

0 

901 

10.020 

0.898 

4.000 

1.500 

1.029 

0.324 

0.979 

0 

520 

10.020 

0.898 

4.000 

1.120 

1.019 

0.126 

0.992 

0 

902 

10.020 

0.898 

4.000 

1.500 

1 .009 

0.433 

0.968 

0 

-520 

10.020 

0.898 

4.000 

1.120 

0.988 

0.100 

0.996 

0 

903 

10.020 

0.898 

4.000 

1.500 

1.014 

0.498 

0.960 

0 

521 

10.020 

0.898 

12.000 

1.120 

0.976 

-0.016 

1.014 

0 

904 

10.020 

0.898 

4.000 

1 .500 

0.995 

0.365 

0.972 

0 

-521 

10.020 

0.898 

12.000 

1.120 

1.003 

-0.026 

1.013 

0 

905 

10.020 

0.898 

4.000 

1.500 

1.090 

0.298 

0.982 

0 

522 

10.020 

0.898 

20.000 

1.120 

0.940 

-0.049 

1.021 

1 

906 

10.020 

0.898 

4.000 

1.500 

1.071 

0.463 

0.961 

0 

-522 

10.020 

0.898 

20.000 

1.120 

0.969 

-0.062 

1 .021 

0 

907 

10.020 

0.898 

4.000 

1 .500 

1 .067 

0.682 

0.936 

0 

523 

10.020 

0.898 

20.000 

1.120 

0.959 

-0.096 

1.025 

0 

908 

10.020 

0.898 

12.000 

1.500 

1.047 

O.183 

0.986 

0 

524 

10.020 

0.898 

4.000 

1 .000 

0.964 

O.OO5 

0.999 

0 

909 

10.020 

0.898 

20.000 

1.500 

1.O63 ■ 

•0.045 

1.02' 

0 

-524 

10.020 

0.898 

4.000 

1.000 

0.999 

0.014 

1.002 

0 

910 

10.020 

0.898 

12.000 

'•500 

1.035 

0.240 

0.989 

0 


261 


Run 

D 

k 

U 

c r 

V 

A 

B 

X 

Run 

-539 

D 

10.C20 

k 

O.898 

it 

20.000 

b ? 

V A 

0.907 0.074 

B 

• .992 

X 

c 

911 

10.020 

0.898 

20.000 

1.500 

1.050 

0.217 

0.992 

0 

540 

10.020 

O.898 

20.000 

1 .500 

0.96k -0.C17 

1 .001 

0 

912 

10.020 

0.898 

12.000 

1.500 

1.02' 

0.465 

0.963 

0 

541 

10.020 

0.898 

20.000 

1.250 

0.904 -0.101 

1.024 

c 

913 

10.020 

0.898 

20.000 

1.500 

'.037 

0.51? 

0.954 

0 

542 

10.020 

0.898 

20.000 

1.130 

.909 -0.067 

1.029 

0 

919 

10.020 

0.898 

4.000 

1.500 

1 .02 * 

0.39k 

0.965 

0 

543 

10.020 

0.898 

20.000 

1 .060 

0.945 1 .013 

1.011 

c 

915 

10.020 

0.898 

12.000 

1.500 

1 .005 

0. 165 

0.996 

0 

IK 

10.020 

0.898 

20.000 

1.060 

.937 C.019 

1 .Cl 1 

0 

916 

10.020 

0.898 

20.000 

1.500 

1.064 

0.109 

1 .001 

0 

10.020 

0.898 

20.000 

1 .060 

0.925 0.009 

1.012 

0 

917 

10.020 

0.898 

4.000 

1 .500 

1.044 

0.552 

0.946 

0 

545 

10.020 

0.898 

20.000 

1.030 

0.957 0.043 

1 .003 

0 

918 

10.020 

0.898 

12.000 

1 .500 

1 .052 

0.389 

0.968 

0 

546 

10.020 

0.898 

20.000 

1.000 

0.911 -0.004 

1 .013 

c 

919 

10.020 

0.898 

20.000 

1 .500 

1 .060 

0.393 

0.966 

0 

547 

10.120 

0.898 

20.000 

0.980 

0.907 0.044 

1.002 

c 

920 

10.020 

0.898 

4.000 

1 .500 

1.040 

0.391 

0.960 

0 

548 

10.020 

0.898 

20.000 

1 .060 

0.912 -0.025 

1.017 

0 

921 

10.020 

0.898 

12.000 

1.500 

1.070 

0. ’55 

0.993 

0 

600 

10.020 

0.898 

20.000 

1 .500 

0.062 0.000 

C.000 

17 

922 

10.020 

0.898 

20.000 

1 .500 

1.048 

0.115 

1 .001 

0 

601 

10.020 

O.898 

20.000 

1 .250 

0.147 0.463 

1 .007 

0 

923 

10.020 

0.898 

4.000 

1 .500 

1.030 

0.545 

0.934 

0 

602 

10.020 

0.898 

20.000 

1.130 

0.668 -0.037 

1.024 

71 

924 

10.020 

0.898 

12.000 

1.500 

1.051 

0.425 

0.957 

0 

603 

10.020 

0.898 

20.000 

1.060 

0.700 C.C10 

1 .012 

0 

925 

10.020 

0.898 

20.000 

1.500 

1.035 

0.412 

0.955 

0 

604 

10.020 

0.898 

20.000 

1.030 

C.626 -0.021 

1.021 

0 

926 

10.020 

0.898 

4.000 

1 .500 

1 .065 

0.350 

0.969 

0 

605 

10.020 

0.898 

20.000 

1 .000 

0.021 0.015 

1 .043 

c 

III 

10.020 

0.898 

12.000 

1.500 

1 .044 

0.166 

0.995 

0 

606 

10.020 

0.898 

20.000 

0.980 

0.700 0.055 

1 .001 

0 

10.020 

0.898 

20.000 

1.500 

1.027 

0.22' 

0.988 

0 

100 

10.020 

0.848 

20.000 

1.500 

0.524 2.C55 

r .753 

6 

929 

10.020 

0.898 

4.000 

1.500 

1 .124 

0.499 

0.952 

0 

101 

10.020 

0.848 

20.000 

1.250 

0.084 1.301 

1 .002 

1 

930 

10.020 

0.898 

12.000 

1 .500 

1 .101 

0.481 

0.951 

0 

102 

10.020 

0.848 

20.000 

1.130 

0.632 -0.454 

1.184 

3 

931 

10.020 

0.898 

20.000 

1.500 

1 .085 

0.437 

0.958 

0 

103 

10.020 

0.848 

20. coo 

1 .060 

0.706 -0.020 

1.026 

0 

932 

10.020 

0.898 

4.000 

1.500 

I.116 

0.319 

0.972 

0 

104 

10.020 

0.848 

20.000 

1.030 

0.077 0.336 

1.006 

0 

III 

10.020 

0.898 

12 .000 

1 .500 

1.097 

0. 156 

0.996 

0 

105 

10.020 

0.848 

20.000 

1 .000 

O.182 C.068 

1 .044 

0 

10.020 

0.898 

20.000 

1 .500 

1.122 

0.131 

1 .000 

0 

106 

10.020 

0.848 

20.000 

C.980 

0.707 -0.C16 

1 .021 

c 

935 

10.020 

0.898 

4.000 

1.500 

1.097 

0.544 

0.951 

0 

1100 

10.020 

0.948 

20.000 

1.500 

O.215 -C.C32 

1.055 

c 

936 

10.020 

0.898 

12.000 

1 .500 

1.092 

0.437 

0.961 

0 

1101 

10.020 

0.948 

20.000 

1 .250 

O.C23 0.168 

1 .019 

0 

937 

10.120 

0.898 

20.000 

1.500 

1.090 

0.460 

0.956 

0 

1102 

10.020 

0.948 

20.000 

1.130 

i .561 -C .002 

1 .014 

0 

938 

10.020 

0.898 

4.000 

1.500 

1 .068 

0.327 

0.973 

0 

1103 

10.C20 

0.948 

20. LOO 

1 .060 

0.703 0.006 

1.008 

0 

939 

10.020 

0.898 

12.000 

1.500 

1.062 

0.145 

0.998 

0 

1104 

10.020 

i .948 

20.000 

1.030 

0.296 0.009 

1 .014 

0 

990 

10.020 

0.898 

20.000 

1 .500 

1 .030 

0.125 

1.001 

0 

1105 

10.020 

0.948 

20. COO 

1.000 

0.264 -0.015 

1.019 

c 

991 

10.020 

0.898 

4.000 

1 .500 

1.009 

0.417 

0.964 

0 

1106 

10.020 

0.948 

20.000 

0.980 

C.707 0.009 

1 .C07 

0 

992 

10.020 

0.898 

12.000 

1 .500 

1 .043 

0.401 

0.968 

0 

109 

10.020 

0.848 

20.000 

1.000 

0.036 C.000 

0.000 

1 

IK 

10.020 

0.898 

20.000 

1.500 

1 .022 

0.386 

0.966 

c 

1111 

10.020 

0.948 

20.000 

1 .060 

0.694 -0.135 

1.022 

0 

10.020 

0.898 

4.000 

1 .500 

1.004 

0.291 

0.975 

0 

1112 

10.020 

0.948 

20. COO 

1 .000 

0.069 0.000 

0.000 

1 

995 

10.020 

0.898 

12.000 

1.500 

1 .011 

0.124 

1.000 

0 

1113 

10.020 

0.948 

20.000 

0.980 

0.700 C .020 

0.998 

0 

946 

10.020 

0.898 

20.000 

1 .500 

0.992 

0.088 

1.006 

0 

107 

10.020 

0.848 

20. COO 

1 .coo 

0.704 -0.016 

1 .018 

0 

9 l 7 

10.020 

0.898 

4.000 

1.500 

0.976 

0.424 

0.964 

0 

108 

10.020 

0.848 

20.000 

1.000 

0.703 0 .02 1 

1 .009 

0 

998 

10.020 

0.898 

12.000 

1.500 

0.990 

0.359 

0.970 

0 

607 

10.020 

O.898 

20.000 

1.000 

0.596 -0.017 

1 .027 

0 

999 

10.020 

0.898 

20.000 

1 .500 

0.972 

0.358 

0.967 

0 

608 

10.020 

0.898 

20. COO 

1 .000 

0.706 C.C14 

1 .010 

0 

950 

10.020 

0.898 

4.000 

1.500 

0.957 

0.291 

0.975 

0 

609 

10.020 

0.898 

20.000 

1.500 

0.703 -0.003 

1.004 

c 

951 

10.020 

0.898 

12.000 

1.500 

0.974 

0.103 

1 .003 

0 

610 

10.020 

0.898 

20.000 

1 .000 

0.705 -0.017 

1.01 1 

0 

952 

10.020 

0.898 

20.000 

1.500 

0.959 

0.067 

1 .009 

0 

1107 

10.020 

0.948 

20.000 

1 .000 

0.703 -0.010 

1 .01 1 

0 

953 

10.020 

0.898 

4.000 

1.500 

0.941 

0.390 

0.968 

0 

1108 

10.020 

0.948 

20.000 

1 .000 

0.698 -0.012 

1.008 

0 

959 

10.020 

0.898 

12.000 

1 .500 

0.958 

0.332 

0.971 

0 

1109 

10.020 

0.948 

20.000 

1.030 

0.706 0.008 

1 .006 

0 

955 

10.020 

0.898 

20.000 

1.500 

0.945 

0.309 

0.975 

0 

1110 

10.020 

0.948 

20.000 

1 .060 

0.702 0.004 

1 .007 

0 

956 

10.020 

0.898 

4.000 

1.500 

0.934 

0.291 

0.973 

0 

1114 

10.020 

0.948 

20.000 

1 .coo 

0.697 0.003 

1.006 

c 

957 

10.020 

0.898 

12.000 

1.500 

0.Q21 

0.105 

1 .002 

0 

700 

10.020 

0.898 

4.000 

1.C00 

0.907 0.007 

1.015 

0 

958 

10.020 

0.898 

20.000 

1 .500 

0.91' 

0.073 

1 .006 

0 

701 

10.020 

0.898 

20.000 

1.000 

0.897 0.042 

0.997 

0 

959 

10.020 

0.898 

4.000 

1 .500 

0.964 

0.441 

0.956 

0 

702 

10.020 

c .898 

20.000 

1.000 

0.885 C.006 

1.005 

0 

960 

10.020 

0.898 

12.000 

1.500 

0.950 

0.346 

0.967 

0 

703 

10.020 

0.898 

4.000 

0.980 

C.903 - n .044 

1.016 

0 

961 

10.020 

0.898 

20.000 

1.500 

0.911 

0.000 

0.000 

15 

704 

10.020 

0.898 

20.000 

c.980 

0.899 -0.016 

1.009 

0. 

99 

10.020 

0.898 

4.000 

1.500 

0.985 

0.364 

0.971 

0 

705 

10.020 

0.898 

4.000 

1.030 

0.927 0.C39 

0.999 

0 

962 

10.020 

0.898 

4.000 

1.500 

0.996 

0.325 

0.968 

0 

706 

10.020 

0.898 

20.000 

1 .030 

0.894 -C.014 

1.009 

0 

963 

10.020 

0.898 

12.000 

1 .500 

0.984 

0.118 

0.997 

0 

707 

10.C20 

0.898 

20.000 

1 .030 

0.892 -0.010 

1.008 

c 

969 

10.020 

0.898 

20.000 

1.500 

0.971 

0.097 

1.002 

0 

708 

10.020 

-.898 

4.000 

1.060 

0.911 0.039 

1.001 

0 

965 

10.020 

0.898 

4.000 

1.500 

0.957 

0.489 

0.939 

0 

709 

10.020 

0.898 

20.000 

1 .060 

0.922 -0.080 

1.025 

c 

966 

10.020 

0.898 

12.000 

1.500 

0.980 

0.366 

0.958 

0 

710 

10.020 

0.898 

20. COO 

1 .060 

0.919 -0.079 

1.024 

c 

967 

10.020 

0.898 

20.000 

1 .500 

0.963 

0.298 

0.969 

0 

711 

10.020 

0.898 

4.000 

1.130 

0.895 0.066 

0.996 

0 

968 

10.020 

0.898 

20.000 

1.500 

0.952 

0.045 

1.011 

0 

712 

10.020 

0.898 

20.000 

1.130 

0.908 -0.186 

1.037 

0 

969 

10.020 

0.898 

20.000 

1 .500 

0.974 

0.002 

1 .016 

0 

71 3 

10.020 

0.898 

20.000 

1.130 

0.889 -0.173 

1.037 

c 

970 

10.020 

0.898 

20.000 

1.500 

0.958 

-0.012 

1 .019 

0 

71k 

10.020 

0.898 

4.000 

1.250 

0.900 0.160 

0.986 

0 

971 

10.020 

0.898 

20.000 

1.500 

1 .003 
0.988 

-0.032 

1 .022 

0 

715 

10.020 

0.898 

20.000 

1.250 

C.887 -0.246 

1.038 

0 

972 

10.020 

0.898 

20.000 

1.500 

0.001 

1.015 

0 

716 

10.020 

0.898 

20.000 

1.250 

0.973 -0.262 

1.044 

c 

973 

10.020 

0.898 

12.000 

1 .500 

0.999 

0.035 

1 .01 1 

0 

717 

10.020 

0.898 

4.000 

1 .500 

C.953 0.209 

0.978 

c 

979 

10.020 

0.898 

20.000 

1.500 

0.986 

0.016 

1 .016 

0 

718 

10.020 

0.898 

20.000 

1.500 

0.951 -0.275 

1.027 

0 

975 

10.020 

0.898 

20.000 

1 .500 

0.994 

-0.010 

1.018 

0 

719 

10.020 

0.898 

20.000 

1 .500 

0.937 -0.289 

1.029 

0 

976 

10.020 

0.898 

20.000 

1.500 

0.975 

0.006 

1.017 

0 

720 

10.020 

0.898 

20.000 

1 .500 

0.908 -0.293 

1.034 

0 

977 

10.020 

0.898 

20.000 

1 .500 

0.989 

-0.041 

1.022 

0 

730 

10.020 

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4.000 

1 .000 

0.891 0.C01 

1 .009 

0 

978 

10.020 

0.898 

20.000 

1.500 

0.975 

-0.035 

1.023 

0 

731 

10.020 

0.898 

20.000 

1 .coo 

0.916 C.C41 

0.993 

0 

979 

10.020 

0.898 

20.000 

1.500 

0.988 

0.028 

1 .016 

0 

732 

10.020 

0.898 

20.000 

1 .000 

0.900 0.016 

1.000 

c 

980 

10.020 

0.898 

20.000 

1.500 

1.037 

0.409 

0.963 

0 

III 

10.020 

0.898 

4.000 

c.980 

0.898 -0.001 

1 .009 

1 

981 

10.020 

0.898 

20.000 

1 .500 

1 .030 

0.074 

1 .006 

0 

10.020 

0.898 

20.000 

0.980 

0.888 O.O31 

0.998 

0 

982 

10.020 

0.898 

20.000 

1 .500 

1.009 

0.281 

0.979 

0 

735 

10.020 

0.898 

20. COO 

c.980 

C.910 0.C41 

C.995 

0 

983 

10.020 

0.898 

20.000 

1 .500 

1 .012 

-0.007 

1.018 

0 

736 

10.020 

0.898 

4.000 

1.C30 

0.904 0.049 

0.998 

0 

989 

10.020 

0.898 

20.000 

1.500 

0.990 

0.072 

1 .010 

0 

737 

10.020 

0.898 

20.000 

1 .030 

0.934 0.040 

0.995 

0 

985 

10.020 

0.898 

20.000 

1.500 

0.996 

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1 .019 

1 

738 

10.020 

' -§5§ 

20.000 

1.030 

0.917 0.031 

0.997 

0 

986 

10.020 

0.898 

20.000 

1.500 

0.978 

0.046 

1 .010 

0 

739 

10.020 

0.898 

4.000 

1.060 

0.892 0.046 

0.999 

0 

-915 

10.020 

0.898 

12.000 

1.500 

0.973 

0.156 

1.002 

0 

740 

10.020 

t.898 

20.000 

1.060 

0.907 -0.019 

1 .010 

0 

-915 

10.020 

0.898 

12.000 

1.500 

0.998 

0.175 

0.998 

0 

741 

10.020 

0.898 

20.000 

1.060 

C.889 -0.C46 

1.013 

0 

998 

10.020 

0.898 

12.000 

1 .500 

1.001 

0.183 

0.997 

0 

742 

10.020 

O.898 

4.000 

1.130 

0.961 O.135 

0.984 

0 

997 

10.020 

0.898 

12.000 

1 .500 

1.028 

0.161 

1.001 

0 

743 

10.020 

0.898 

20. COO 

1.130 

0.938 -O.119 

1.027 

c 

996 

10.020 

0.898 

12.000 

1.500 

0.967 

0.216 

0.996 

0 

* 744 

10.020 

0.898 

20.000 

1.130 

0.917 -0.154 

1.036 

0 

-919 

10.020 

0.898 

4.000 

1 .500 

0.930 

0.360 

0.979 

0 

745 

10.020 

0.898 

4.000 

1.250 

0.928 0.101 

0.991 

0 

919 

10.020 

0.898 

4.000 

1 .500 

0.931 

0.389 

0.974 

0 

746 

10.020 

0.898 

20.000 

1 .250 

0.899 -0.199 

1.033 

0 

590 

10.020 

0.898 

12.000 

1 .500 

1 .033 

0.103 

0.999 

0 

747 

10.020 

0.898 

20.000 

1.250 

0.904 -0.240 

1 .041 

0 

1 

10.020 

0.898 

12.000 

1 .500 

0.991 

0.169 

0.988 

0 

748 

10.020 

0.898 

4.000 

1.500 

0.887 0.306 

0.971 

0 

0 

10.020 

0.898 

12.000 

1.500 

0.954 

0.077 

0.999 

0 

749 

10.020 

0.898 

20.000 

1 .500 

0.871 -0.268 

1 .031 

0 

37 

10.020 

0.848 

20.000 

1 .500 

0.938 

0.432 

0.969 

0 

750 

10.020 

0.898 

20.000 

1.500 

C.901 -0.256 

1.030 

0 

-37 

10.020 

0.848 

20.000 

1.500 

0.917 

0.579 

0.946 

0 

760 

10.020 

0.898 

4.000 

1 .000 

C.944 0.018 

1.004 

0 

-38 

10.020 

0.848 

20.000 

1 .250 

0.930 

0. 146 

0.997 

0 

761 

10.020 

0.898 

20.000 

1.000 

0.955 0.007 

1 .000 

0 

38 

10.020 

0.848 

20.000 

1.250 

0.920 

0.178 

0.994 

0 

762 

10.020 

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20. COO 

1 .000 

C.938 0.028 

0.997 

0 

39 

10.020 

0.848 

20.000 

1 . 120 

0.883 

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1.032 

0 

763 

10.020 

0.898 

4.000 

0.980 

0.934 -0.002 

1.008 

0 

90 

10.020 

0.848 

20.000 

1 .060 

0.939 

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1.024 

0 

764 

10.020 

0.898 

20.000 

0.980 

C.912 0.037 

0.996 

0 

-90 

10.020 

0.848 

20.000 

1 .060 

0.971 

-0.100 

1.041 

0 

765 

10.020 

0.898 

20. COO 

0.980 

0.930 O.C27 

1.001 

0 

91 

10.020 

0.848 

20.000 

1 .030 

0.886 

0.026 

1.011 

0 

766 

10.020 

0,898 

4.000 

1 .030 

0.915 0.061 

0.997 

0 

92 

10.020 

0.848 

20.000 

1 .000 

0.934 

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1.019 

0 

767 

10.020 

0.898 

20.000 

1.030 

0.930 0.035 

0.999 

0 

93 

10.020 

0.848 

20.000 

0.980 

0.896 

0.060 

1 .010 

0 

768 

10.020 

0.898 

20.000 

1 .030 

0.919 0.027 

1.000 

0 

99 

10.020 

0.848 

20.000 

1 .060 

0.883 

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1 .028 

0 

769 

10.020 

0.898 

4.000 

1.060 

c.903 0.100 

0.989 

0 

1032 

10.020 

0.948 

20.000 

1.500 

0.945 

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1.017 

0 

770 

10.020 

0.898 

20.000 

1.060 

0.960 -0.016 

1.013 

0 

1033 

10.020 

0.948 

20.000 

1.250 

0.907 

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1.013 

0 

771 

10.020 

0.898 

20.000 

1 .060 

0.962 -0.012 

1.012 

0 

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10.020 

0.948 

20.000 

1.250 

0.924 

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1 .014 

0 

772 

10.020 

0.898 

4.000 

1.130 

0.934 0.C93 

0.996 

0 

1039 

10.020 

0.948 

20.000 

1.130 

0.895 

0.924 

-0.002 

1 .01 1 

0 

773 

10.020 

0.898 

20. COO 

1.130 

C.931 -0.106 

1.027 

0 

-1039 

10.020 

0.948 

20.000 

1.130 

-0.026 

1.012 

0 

774 

10.020 

0.898 

20.000 

1 . ’ 30 

0.904 -0.132 

1.03 1 

0 

1035 

10.020 

0.948 

20.000 

1 .060 

0.901 

0.013 

1 .006 

0 

775 

10.020 

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4.000 

1.250 

0.912 0.173 

0.989 

0 

-1035 

10.020 

0.948 

20.000 

1.060 

0.912 

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1.012 

0 

776 

10.020 

0.898 

20.000 

1.250 

O.889 -0.162 

1.030 

0 

1036 

10.020 

0.948 

20.000 

1 .030 

0.945 

0.030 

1.003 

0 

777 

10.020 

0.898 

20. COO 

1.250 

0.942 -O.188 

1 .032 

0 

1037 

10.020 

0.948 

20.000 

1 .000 

0.919 

0.007 

1.007 

0 

778 

10.020 

0.898 

4.000 

1.500 

C.922 0.234 

0.983 

0 

1038 

10.020 

0.948 

20.000 

0.980 

0.894 

0.019 

1.006 

0 

779 

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1.500 

0.901 -0.190 

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0 

1039 

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0 

780 

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1.500 

0.913 -C.159 

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0 

987 

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12.000 

1.500 

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0.033 

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0 

795 

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0.893 0.345 

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0 

988 

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12.000 

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0 

796 

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0.877 0.257 

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0 

988 

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12.000 

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0.967 

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1.013 

0 

797 

10.020 

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0.903 0.384 

0.975 

0 

989 

10.020 

0.898 

12.000 

1 .500 

1.015 

0.034 

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0 

798 

10.020 

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0.972 0.039 

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c 

990 

10.020 

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1.500 

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550 

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993 

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551 

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554 

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C.981 -C.164 

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0 

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555 

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0 

539 

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0.931 

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0 

556 

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1.500 

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0.990 

0 


262 


Run D 

k 

If 

O' 

> 

A 

B 

X 

557 

10.020 

0.898 

20.000 

1.500 

0.956 

c.008 

0.992 

0 

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10.020 

0.898 

20.000 

1.500 

0.928 

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10.020 

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f .915 

-0.234 

1.021 

0 

723 

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1.500 

0.927 

-0.327 

1.035 

0 

724 

10.020 

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1.500 

c .951 

-0.296 

1.038 

0 

725 

10.020 

0.898 

20.000 

1.500 

0.946 

-0.259 

1.024 

0 

726 

10.020 

0.898 

4.000 

1.060 

0.914 

0.090 

0.992 

0 

727 

10.020 

0.898 

4.000 

1.060 

0.913 

0.094 

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0 

726 

10.020 

C.898 

20.000 

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0.896 

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1.027 

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9 

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1.250 

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0.166 

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0 

782 

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0 

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0 

784 

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C.898 

20.000 

1.500 

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0 

785 

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0 

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0 

788 

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0.982 

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0 

792 

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4.000 

1.500 

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0.244 

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0 

793 

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0.978 

0.241 

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0 

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752 

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0.109 

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0 

753 

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20.000 

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0.242 

7 

754 

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0.126 

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806 

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0 

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0 

808 

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1.060 

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0 

809 

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1.060 

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0 

810 

10.020 

0.898 

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1.060 

0.968 

0.001 

1.014 

0 

811 

10.020 

0.898 

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1.060 

0.934 

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1.016 

0 

812 

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1.060 

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0.020 

1.011 

0 

813 

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1.060 

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0.006 

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0 

814 

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1.060 

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0.020 

1.012 

0 

815 

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0 

816 

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0.959 

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0 

817 

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0 

818 

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0 

819 

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821 

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822 

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1.060 

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0 

876 

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20.000 

1.130 

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0 

877 

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0 

878 

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0 

880 

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827 

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0 

829 

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0 

830 

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831 

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832 

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263 


Run 

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k 

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0 

1504 

4.029 

0.909 

2.042 

2.000 

1.095 

0.030 1.011 

0 

1383 

4.029 

0.929 

8.175 

1.125 

1.086 

0.027 

0.992 

0 

1505 

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0.903 

0.870 

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0 

1381* 

4.029 

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0 

1385 

4.029 

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1.086 

0.013 

1.002 

0 

1507 

4.029 

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8.187 

1.500 

1.093 

-0.065 1.021 

0 

1386 

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2.047 

1.125 

1.086 

0.061 

0.992 

0 

1508 

4.029 

0.870 

8.188 

1.500 

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-0.069 1.025 

0 

1387 

4.029 

0.870 

8.188 

1.125 

1 .086 

0.008 

1.007 

0 

1509 

4.029 

0.903 

8.187 

1.250 

1.102 

-0.040 1.019 

0 

1388 

4.029 

0.805 

2.046 

1.125 

1 .084 

0.140 

0.980 

0 

1510 

4.029 

0.870 

8.188 

1.250 

1.089 

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0 

1389 

4.029 

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1.125 

1.085 

0.006 

1.016 

0 

1511 

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0.903 

0.870 

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1.125 

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0 

1390 

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0 

1512 

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8.188 

1.125 

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1391 

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0 

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0.002 1.014 

0 

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4.029 

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0 

1515 

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0 

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0 

1395 

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0 

7517 

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1.500 

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0 

1396 

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1.500 

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0.092 

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1 

1518 

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0 

1397 

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0 

1519 

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1.250 

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0 

1398 

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0 

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0 

1399 

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1.500 

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0 

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1.125 

1.117 

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2.046 

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0 

1522 

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1401 

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1.062 

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1525 

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1.548 

1.100 

0.015 O.983 

0 

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2.000 

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0 

1526 

4.029 

0.935 

1.575 

1.548 

1,089 

0.022 0.981 

0 

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0.806 

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0 

1527 

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0 

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0 

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0 

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0.023 0.980 

0 

1408 

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0 

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4.029 

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C.001 0.983 

0 

1409 

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0.806 

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1 .068 

0.062 

1.002 

0 

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4.029 

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0.024 0.979 

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a 1532 

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265 


Run 

D 

k 

Lt 

c r 

V 

A 

B 

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Run 

D 

k 

Lt 

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V 

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• 1533 

4.029 

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2.663 

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4.029 

0.935 

0.630 

2.663 

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0.016 

0.980 

0 

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0.935 

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3.114 

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0 

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0.020 

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0 

1787 

4.029 

0.806 

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1.559 

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4.029 

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3.497 

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1788 

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4.029 

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1789 

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0.017 0.992 

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4.029 

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1791 

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1.309 

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4.C29 

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0.315 

4.331 

1 .093 

0.033 

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1792 

4.029 

0.903 

8.500 

1.309 

9.013 

-0.034 1 .031 

0 

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4.029 

0.935 

O.630 

4.33' 

1 .060 

0.043 

0.975 

0 

1793 

4.029 

0.903 

2.125 

1.309 

9.061 

0.010 1.002 

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• '547 

4.029 

0.935 

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4.33' 

1.098 

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0.984 

0 

1794 

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8.500 

1.309 

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0.016 1.014 

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4.029 

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1.309 

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4.029 

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0.313 

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0 

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4.029 

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1.066 

0.092 

1.002 

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1797 

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2.125 

1.309 

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0.080 1.013 

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4.029 

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1.084 

0.124 

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0 

1798 

4.029 

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2.125 

1.309 

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0.148 0.998 

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0 

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0.165 0.995 

0 

1678 

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24.504 

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0 

1803 

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0 

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0 

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0.036 

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0 

1806 

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0 

1682 

4.029 

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1807 

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0 

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0 

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24.583 

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0 

1809 

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8.500 

1.184 

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0.006 1.027 

0 

1685 

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2.125 

2.068 

24.669 

0.004 

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0 

1810 

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8.500 

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0 

1686 

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24.829 

0.001 

0.998 

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1811 

4.029 

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0 

1687 

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0.903 

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0 

1812 

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8.500 

1.121 

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1.002 

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4.029 

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0 

1689 

4.029 

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0.070 

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0 

1814 

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1.121 

9.741 

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0 

1690 

4.029 

0.806 

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2.068 

25.507 

0.390 

0.966 

0 

1815 

4.029 

0.929 

8.500 

1.121 

9.703 

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0 

169' 

4.029 

0.807 

2.125 

2.068 

24.566 

0.646 

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1816 

4.029 

0.903 

8.500 

1.12’ 

9.798 

-0.016 1.012 

0 

1692 

4.029 

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8.500 

1 .568 

25.423 

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1.003 

0 

1817 

4.029 

0.903 

2.125 

1.121 

9.823 

0.046 1.012 

0 

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4.029 

0.903 

8.500 

1 .568 

26.000 

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1.011 

0 

1818 

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2.125 

1.121 

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G.027 0.992 

1 

1694 

4.029 

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0 

1819 

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8.500 

1.121 

9.725 

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0 

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4.029 

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26.516 

0.017 

0.994 

0 

1820 

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0 

1696 

4.029 

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1 .568 

24.832 

0.162 

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0 

1821 

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2.125 

1.121 

9.864 

0.067 1.018 

0 

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4.029 

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25.305 

0.007 

0.997 

0 

1822 

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2.125 

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9.821 

0.089 1.007 

0 

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4.029 

0.929 

8.500 

1 .3'8 

25.567 

0.001 

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1823 

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8.500 

1.121 

9.757 

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0 

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4.029 

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25.561 

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0 

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0.806 

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1.318 

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0 

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0 

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4.029 

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1.193 

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1826 

4.029 

0.806 

8.500 

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10.3" 

0.015 1.028 

0 

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4.029 

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26.015 

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0 

1827 

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0.262 0.991 

0 

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1.193 

26.618 

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8.500 

1.090 

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0 

1704 

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0 

1705 

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27.000 

0.006 

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1830 

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1 .090 

10.336 

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0 

1706 

4.029 

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26.913 

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0 

1831 

4.029 

0.903 

8.500 

1 .090 

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0 

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4.029 

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1.130 

26.433 

0.016 

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0 

1832 

4.029 

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8.500 

1 .090 

10.368 

-0.035 1.040 

0 

1708 

4.029 

0.806 

8.500 

1. '30 

27.044 

0.038 

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0 

1833 

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8.500 

1.090 

10.481 

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0 

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4.029 

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8.500 

1.099 

26.149 

0.002 

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0 

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4.029 

0.806 

8.500 

1.090 

10.346 

0.078 1.008 

0 

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4.029 

0.903 

8.500 

1.099 

25.383 

0.022 

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0 

1835 

4.029 

0.806 

8.500 

1.090 

10.370 

0.055 1.015 

G 

17" 

4.029 

0.869 

8.500 

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26.251 

0.025 

0.984 

0 

1836 

4.029 

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8.500 

1.059 

10.322 

0.019 0.994 

0 

1712 

4.029 

0.806 

8.500 

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26.484 

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0 

1837 

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2.125 

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10.330 

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0 

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26.892 

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0 

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2.125 

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10.352 

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0 

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4.029 

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4.33' 

25.826 

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0 

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17.812 

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0 

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1721 

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16.574 

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0 

1846 

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2.125 

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10.367 

0.104 1 .002 

0 

1722 

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8.500 

2.068 

16.222 

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0 

1847 

4.029 

0.869 

8.500 

1.059 

10.355 

0.071 1.007 

0 

'723 

4.029 

0.869 

2.125 

2.068 

16.370 

0.166 

1 .019 

0 

1848 

4.029 

0.806 

8.500 

1.059 

10.347 

0.095 '.012 

0 

'724 

4.C29 

0.806 

8.500 

2.068 

17.878 

0.043 

1 .006 

0 

1849 

4.029 

0.807 

2.125 

1.059 

10.284 

0.346 0.946 

0 

'725 

4.029 

0.807 

2.125 

2.068 

17.474 

0.369 

0.980 

0 

1850 

4.029 

0.807 

2.125 

1.059 

10.652 

0.321 0.953 

0 

'726 

4.029 

0.929 

8.500 

1.568 

16.931 

-0.001 

1 .003 

0 

1851 

4.029 

0.806 

8.500 

1.059 

10.631 

0.076 1.010 

0 

'727 

4.029 

0.903 

8.500 

1.568 

18.033 

0.011 

1.008 

0 

• 1852 

4.029 

C.935 

3.150 

4.331 

10.539 

0.005 0.986 

0 

'728 

4.029 

0.869 

8.500 

1.568 

17.359 

0.005 

1.041 

0 

• 1853 

4.029 

0.935 

3.150 

4.331 

10.439 

-0.001 G.990 

0 

'729 

4.029 

0.806 

8.500 

1.568 

17.560 

0.112 

1.000 

0 

1854 

4.029 

0.866 

4.043 

1.559 

10.523 

C.091 1.003 

0 

1730 

4.029 

0.895 

8.500 

1.318 

17.848 

0.029 

0.997 

0 

1855 

4.029 

0.866 

4.043 

1.559 

10.509 

0.087 1.003 

0 

'73' 

4.029 

0.929 

8.500 

1.318 

17.139 

0.013 

0.994 

0 

1856 

4.029 

0.866 

4.043 

1.559 

10.590 

0.074 1.002 

0 

1732 

4.029 

0.869 

8.500 

1 .318 

17.219 

0.039 

1.010 

0 

1857 

4.029 

0.866 

4.043 

1.559 

10.721 

0.065 1.005 

0 

1733 

4.029 

0.806 

8.500 

1.318 

17.221 

0.247 

0.998 

0 

1858 

4.029 

0.866 

4.043 

1.559 

10.632 

0.042 1.012 

0 

'734 

4.029 

0.929 

8.500 

1.193 

16.537 

0.011 

0.997 

2 

1859 

4.029 

0.866 

4.043 

1.559 

10.590 

0.101 0.998 

1 

1735 

4.029 

0.903 

8.500 

1.193 

17.1 '2 

-0.020 

1.025 

0 

i860 

4.029 

0.866 

4.043 

1.559 

10.640 

0.060 1.004 

0 

'736 

4.029 

0.869 

8.500 

1.193 

17.181 

0.008 

1 .014 

0 

1861 

4.029 

0.870 

2.047 

1.559 

10.690 

0.280 0.985 

0 

1737 

4.029 

0.806 

8.500 

1.193 

17.385 

0.124 

1.023 

0 

1862 

4.029 

0.870 

2.047 

1.559 

10.512 

0.218 0.984 

0 

1738 

4.029 

0.929 

8.500 

1.130 

17.7" 

0.009 

0.997 

0 

1863 

4.029 

0.870 

2.047 

1.559 

11.170 

0.274 0.974 

0 

1739 

4.029 

0.903 

8.500 

1.130 

17.710 

0.007 

1.008 

0 

1864 

4.029 

0.870 

2.047 

1.559 

10.783 

0.202 0.980 

0 

1740 

4.029 

0.869 

8.500 

1.130 

17.015 

0.006 

1.013 

1 

I865 

4.029 

0.870 

2.047 

1.559 

11.005 

0.245 0.982 

0 

1741 

4.029 

0.806 

8.500 

1.130 

17.314 

-0.004 

1.034 

0 

1866 

4.029 

0.870 

2.047 

1.559 

10.745 

0.309 0.968 

0 

1742 

4.029 

0.929 

8.500 

1.099 

17.354 

C.003 

0.994 

0 

1867 

4.029 

0.870 

2.047 

1.559 

10.783 

-0.014 I.O36 

0 

1743 

4.029 

0.903 

8.500 

1.099 

17.190 

0.030 

0.985 

0 

1868 

4.029 

0.870 

2.047 

1.559 

10.971 

0.008 1.013 

0 

'744 

4.029 

0.869 

8.500 

1.099 

17.944 

0.021 

0.994 

0 

1869 

4.029 

0.870 

2.047 

1.559 

11.135 

0.011 1.010 

0 

1745 

4.029 

0.806 

8.500 

1.099 

17.160 

- 0.01 1 

1.056 

0 

1870 

4.029 

0.870 

2.047 

1.559 

1 1 .065 

0.017 1.012 

0 

1746 

4.029 

0.929 

8.500 

1.068 

17.481 

0.01 1 

0.993 

0 

1871 

4.029 

0.869 

8.500 

1.559 

10.635 

-0.028 1.029 

0 

'747 

4.029 

0.903 

8.500 

1 .068 

17.221 

0.022 

0.989 

0 

1872 

4.029 

0.869 

8.500 

1.559 

10.675 

-0.023 1.029 

0 

'748 

4.029 

0.869 

8.500 

1 .068 

17.866 

0.042 

0.999 

0 

1873 

4.029 

0.869 

8.500 

1.530 

2.795 

-0.022 1.028 

0 

1749 

4.029 

0.806 

8.500 

1.068 

18.293 

0.016 

1.005 

0 

1874 

4.029 

0.869 

8.500 

1.530 

2.757 

-0.035 1.030 

0 

• 1750 

4.029 

0.935 

3.150 

4.331 

17.913 

- 0.001 

0.986 

0 

1875 

4.029 

0.869 

8.500 

1.530 

2.755 

-0.031 1.030 

1 

'75' 

4.029 

0.929 

8.500 

2.059 

8.710 

-0.027 

1 .014 

0 

1876 

4.029 

0.869 

2.125 

1.530 

2.776 

0.083 0.997 

0 

1752 

4.029 

0.929 

2.125 

2.059 

8.464 

-0.002 

1.007 

0 

1887 

4.029 

0.869 

8.500 

1.530 

2.976 

-0.073 1.027 

0 

'753 

4.029 

0.903 

8.500 

2.059 

8.458 

0.096 

1.004 

0 

1878 

4.029 

0.869 

8.500 

1.155 

2.784 

0.019 1.016 

0 

1754 

4.029 

0.903 

2.125 

2.059 

8.441 

0.097 

1.016 

1 

1879 

4.029 

0.869 

8.500 

1.155 

2.779 

0.006 1.022 

1 

'755 

4.029 

0.869 

8.500 

2.059 

8.466 

-0.038 

1 .032 

0 

1880 

4.029 

0.869 

2.125 

1.115 

2.786 

0.030 1.OO8 

0 

'757 

4.029 

0.806 

8.500 

2.059 

8.438 

0.069 

1 .010 

0 

1881 

4.029 

0.869 

2.125 

1.155 

2.776 

0.037 1.008 

0 

'759 

4.029 

0.806 

8. 167 

2.059 

8.464 

-0.014 

1.024 

0 

1882 

4.029 

0.929 

8.500 

1.530 

2.785 

-0.028 1.017 

0 

'760 

4.029 

0.807 

2.042 

2.059 

8.441 

0.33 1 * 

0.989 

1 

1883 

4.029 

0.929 

2.125 

1.530 

2.789 

0.010 1.007 

0 

1761 

4.029 

0.806 

8.167 

2.059 

8.467 

-0.129 

1.044 

0 

1884 

4.029 

O.903 

8.500 

1.530 

2.971 

-0.045 1.017 

0 

1762 

4.029 

0.807 

2.042 

2.059 

8.431 

0.269 

0.991 

0 

1885 

4.029 

0.903 

2.125 

1.530 

2.966 

0.033 1-012 

0 

'763 

4.029 

O.869 

2.042 

2.059 

8.418 

0.156 

0.995 

0 

1886 

4.029 

0.869 

8.500 

1.530 

3.048 

-0.050 1.017 

0 

'764 

4.029 

0.869 

2.042 

2.059 

8.485 

0.160 

0.996. 

0 

1887 

4.029 

0.869 

8.500 

1.530 

2.976 

-0.073 1.027 

0 

1765 

4.029 

0.869 

8.167 

2.059 

8.727 

-0.130 

1 .061 

0 

1888 

4.029 

O.869 

2.125 

1.530 

2.977 

0.122 1.003 

0 

1766 

4.029 

O.869 

8. 167 

2.059 

8.739 

-0.129 

1 .047 

0 

1889 

4.029 

0.869 

2.125 

1-530 

2.988 

0.117 '.002 

0 

'767 

4.029 

0.903 

8.167 

2.059 

8.750 

-0.030 

1.046 

0 

1890 

4.029 

0.806 

8.5OO 

1.530 

2.967 

-0.012 1.006 

0 

1768 

4.029 

0.903 

2.042 

2.059 

8.722 

0.236 

0.981 

0 

1891 

4.029 

0.807 

2.125 

1.530 

3.145 

0.275 0.986 

0 

1769 

4.029 

0.903 

8.167 

2.059 

8.719 

0.090 

1 .021 

1 

1892 

4.029 

0.929 

8.500 

1 .280 

3.163 

0.013 1.015 

0 

'770 

4.029 

0.903 

2.042 

2.059 

8.546 

0.178 

0.998 

0 

1893 

4.029 

0.903 

8.500 

1.280 

3.168 

-0.037 1.025 

0 

'771 

4.029 

0.903 

2.042 

2.059 

8.632 

0.239 

0.983 

0 

1894 

4.029 

0.869 

8.500 

1.280 

3.143 

-0.039 1.028 

0 

'772 

4.029 

0.929 

8. 167 

2.059 

8.686 

0.055 

0.998 

0 

1895 

4.029 

0.806 

8.500 

1.280 

3.145 

-0.050 1.026 

0 

'773 

4.029 

0.929 

2.042 

2.059 

8.703 

0.H6 

0.991 

0 

1896 

4.029 

0.929 

8.500 

1.155 

3-147 

0.011 1.016 

0 

'774 

4.029 

0.929 

2.042 

2.059 

8.633 

0.030 

0.997 

0 

1897 

4.029 

0.929 

2.125 

1.155 

3.171 

0.038 1.005 

0 

'775 

4.029 

0.806 

8.167 

1.559 

9.041 

-0.066 

1.028 

0 

1898 

4.029 

0.903 

8.500 

1.155 

3.'51 

-0.026 1.024 

0 

'776 

4.029 

0.806 

8.167 

1.559 

9.084 

-0.072 

1 .034 

0 

1899 

4.029 

0.903 

2.125 

1.155 

3.148 

0.027 1.007 

0 

'777 

4.029 

0.869 

8.167 

1.559 

9.118 

-O.O83 

1.054 

0 

1900 

4.029 

0.869 

8.500 

1.155 

3.139 

-0.023 1.023 

0 

'778 

4.029 

0.869 

8.167 

1.559 

9.105 

0.003 

1 .030 

0 

1901 

4.029 

0.869 

2.125 

'.155 

3. 144 

0.045 1.011 

G 

1779 

4.029 

0.869 

8.167 

1.559 

9.079 

0.148 

0.991 

0 

1902 

4.029 

0.806 

8.500 

1.155 

3.171 

-0.062 1.036 

0 


266 


Run 

D 

k 

Lt 

O' 

V 

A 

B 

X 

1903 

1904 

4.029 

4.029 

0.807 

0.929 

2.125 

8.500 

1.155 

1.091 

3.162 

3.198 

0.117 

0.006 

0.995 

1.014 

1 

0 

1905 

4.029 

0.903 

8.500 

1.091 

3.133 

0.029 

I.007 

0 

1906 

4.029 

0.869 

8.500 

1.091 

3.143 

-0.006 

1.024 

0 

1907 

4.029 

0.806 

8.500 

1.091 

3.148 

-0.042 

1.031 

0 

1908 

4.029 

0.929 

8.500 

1.060 

3-155 

0.015 

1.008 

0 

1909 

4.029 

0.929 

2.125 

1.060 

3.141 

0.030 

1.006 

0 

1910 

4.029 

0.903 

8.500 

1.060 

3.195 

3.204 

0.028 

1 .006 

0 

1911 

4.029 

0.903 

2.125 

1.060 

0.039 

1.007 

0 

1912 

4.029 

O.869 

8.500 

1.060 

3.214 

0.021 

1.011 

0 

1913 

1914 

4.029 

0.869 

2.125 

1.060 

3.206 

0.047 

1.003 

0 

4.029 

0.806 

8.500 

1.060 

3.179 

0.02’ 

1.013 

0 

1915 

4.029 

0.807 

2.125 

1.060 

3.185 

0.077 

1.010 

0 

1916 

4.029 

0.929 

8.500 

1 .030 

3.253 

0.037 

1.001 

0 

1917 

1918 

4.029 

0.903 

8.500 

1.030 

3.181 

0.024 

1 .006 

0 

4.029 

0.869 

8.500 

1 .030 

3.185 

0.029 

1.002 

0 

1919 

4.029 

0.806 

8.500 

1.030 

3.180 

0.074 

0.993 

0 

• 1920 

4.029 

0.935 

3.150 

4.331 

3.186 

-0.008 

1.000 

1 

1921 

4.029 

0.929 

8.500 

1.500 

1 .088 

-0.023 

’ .011 

0 

1922 

4.029 

0.929 

2.125 

1.500 

1.088 

0.012 

1.013 

0 

1923 

1924 

4.029 

0.903 

8.500 

1.500 

1.103 

-O.O63 

1.017 

3 

4.029 

0.903 

2.125 

1.500 

1.095 

0.030 

1 .011 

0 

1925 

4.029 

0.869 

8.500 

1.500 

1.092 

-0.051 

1.022 

0 

1926 

4.029 

0.869 

2.125 

1.500 

1.092 

0.169 

0.998 

0 

1927 

4.029 

0.806 

8.500 

1.500 

1.093 

0.077 

1 .005 

0 

1928 

4.029 

0.807 

2.125 

1 .500 

1.085 

0.538 

0.964 

0 

1929 

4.029 

0.929 

8.500 

1.250 

1.102 

-0.042 

1 .024 

0 

1930 

4.029 

0.903 

8.500 

1.250 

1 .105 

-0.052 

1 .028 

0 

1931 

4.029 

O.869 

8.500 

1.250 

1.111 

-0.040 

1.024 

1 

1932 

4.029 

0.806 

8.500 

1.250 

1.085 

0.021 

1 .015 

0 

1933 

4.029 

0.929 

8.500 

1.125 

1 .087 

-0.017 

1.017 

0 

1934 

4.029 

0.929 

2.125 

1.125 

1.087 

0.007 

1 .008 

0 

1935 

4.029 

0.903 

8.500 

1.125 

1.085 

-0.047 

1 .029 

0 

1936 

4.029 

0.903 

2.125 

1.125 

1.109 

0.017 

1.010 

0 

1937 

4.029 

0.869 

8.500 

1.125 

1.061 

-0.044 

1 .028 

0 

1938 

4.029 

0.869 

2. 125 

1.125 

1.060 

0.081 

1.000 

0 

1939 

1940 

4.029 

0.806 

8.500 

1.125 

1.060 

-0.066 

1.041 

0 

4.029 

C.807 

2.125 

1.125 

1.062 

0.199 

0.986 

0 

1941 

4.029 

0.929 

8.500 

1 .062 

1.060 

0.014 

1.005 

0 

1942 

4.029 

0.903 

8.500 

1.062 

1.060 

0.004 

1.013 

0 

1943 

4.029 

0.869 

8.500 

1.062 

1.060 

-0.006 

1.017 

0 

1944 

4.029 

0.806 

8.500 

1.062 

1.062 

-0.014 

1.020 

0 

1945 

4.029 

0.929 

8.500 

1.031 

1.060 

0.017 

1.002 

0 

1946 

4.029 

0.929 

2.125 

1 .031 

1.056 

0.031 

0.998 

1 

1947 

4.029 

0.903 

8.500 

1.031 

1.059 

0.003 

1.012 

0 

1948 

4.029 

0.903 

2.125 

1.031 

1 .056 

0.020 

1.004 

0 

1949 

4.029 

0.869 

8.500 

1.031 

1.058 

0.002 

1.016 

0 

1950 

4.029 

0.869 

2.125 

1.031 

'.057 

0.037 

1 .008 

0 

1951 

4.029 

0.806 

8.500 

1.031 

1.058 

-C .024 

1.036 

5 

1952 

4.029 

0.807 

2.125 

1.031 

1.057 

0.067 

1.013 

0 

1953 

4.029 

0.929 

8.500 

1 .000 

1.055 

0.022 

0.998 

0 

1954 

4.029 

0.903 

8.500 

1.000 

1 .056 

0.016 

1 .002 

0 

1955 

4.029 

0.869 

8.500 

1.000 

1.054 

0.019 

1.005 

0 

1956 

4.029 

0.806 

8.500 

1 .000 

1.056 

0.030 

1 .009 

0 

• 1957 

4.029 

0.935 

3-150 

4.331 

1.061 

0.012 

0.990 

0 

1958 

4.029 

0.903 

8.500 

1.500 

I.O63 

-0.059 

1.012 

0 

1959 

4.029 

0.903 

8.500 

1.500 

1 .064 

-0.064 

1.014 

0 

i960 

4.029 

0.903 

8.500 

1.250 

1 .063 

-0.049 

1 .027 

0 

1961 

4.029 

0.903 

8.500 

1.250 

1 .063 

-0.050 

1.027 

1 

1962 

4.029 

0.903 

8.500 

1.125 

1.070 

-0.001 

1 .018 

0 

1963 

1964 

4.029 

0.903 

8.500 

1.125 

1.071 

-0.006 

1.022 

0 

4.029 

0.903 

8.500 

0.367 

1 .061 

-0.067 

1 .022 

0 

ill! 

4.029 

0.903 

8.500 

0.367 

1.064 

-0.071 

1.023 

0 

4.029 

0.935 

3.150 

1.548 

'.060 

0.01 1 

0.989 

0 

1967 

4.029 

0.935 

3.150 

2.141 

1.065 

0.006 

0.991 

0 

1968 

4.029 

0.935 

3.150 

2.663 

1 .071 

0.005 

0.993 

0 

1969 

4.029 

0.935 

3.150 

3.114 

1.071 

0.010 

0.990 

0 

1970 

4.029 

0.935 

3.150 

3.497 

1 .061 

0.008 

0.991 

0 

1971 

4.029 

0.935 

3.150 

4.331 

1.059 

0.012 

0.989 

0 

2000 

4.029 

0.903 

8.500 

1.062 

1.057 

0.041 

’ .002 

18 

2001 

4.029 

0.903 

8.500 

1.250 

1.061 

-0.041 

1.020 

4 

2002 

4.029 

0.903 

8.500 

10.250 

1.062 

-0.046 

1.024 

0 

• 2003 

4.029 

0.917 

3.079 

1.570 

1.057 

O.O38 

0.994 

0 

• 2004 

4.029 

0.917 

3.079 

1.570 

1.050 

0.030 

0.995 

0 

• 2005 

4.029 

0.917 

3-079 

4.500 

1.065 

0.200 

0.989 

0 

• 2006 

4.029 

0.917 

3.079 

4.509 

1 .044 

0.245 

0.987 

0 

• 2007 

4.029 

0.917 

3.079 

4.556 

1.053 

0.116 

0.987 

0 

• 2008 

4.029 

0.917 

3.079 

4.556 

1.053 

0.123 

0.994 

0 

• 2009 

4.029 

0.917 

3.079 

3.662 

1.052 

0.093 

0.991 

0 

• 2010 

4.029 

0.917 

3.079 

3.662 

1.047 

0.160 

0.973 

0 

• 2011 

4.029 

0.917 

3.079 

3.252 

1.053 

0.098 

0.982 

0 

• 2012 

4.029 

0.917 

3-079 

3.252 

1.054 

0. ’ 14 

0.976 

0 

• 2013 

4.029 

0.917 

3.079 

2.768 

1.073 

0.196 

0.965 

0 

• 2014 

4.029 

0.917 

3.079 

2.768 

1.077 

0.017 

1.004 

0 

• 2015 

4.029 

0.917 

3.079 

2.768 

1 .078 

0.091 

0.978 

0 

• 2016 

4.029 

0.917 

3.079 

2.207 

1.072 

0.056 

0.992 

1 

• 2017 

4.029 

0.917 

3.079 

2.207 

1.071 

0.155 

O.98O 

0 

• 2018 

4.029 

0.917 

3.079 

1.572 

1.065 

0.042 

0.987 

0 

• 2019 

4.029 

0.917 

3.079 

1.572 

1.053 

0.046 

0.989 

0 

• 2020 

4.029 

0.917 

3.079 

1.572 

1.046 

0.048 

0.985 

0 

• 2021 

4.029 

0.906 

3.042 

4.649 

1.062 

0.075 

0.986 

0 

• 2022 

4.029 

0.906 

3.042 

4.649 

1.063 

0.069 

0.987 

0 

• 2023 

4.029 

0.906 

3.042 

3-727 

1 .061 

0.042 

0.987 

0 

• 2024 

4.029 

0.906 

3.042 

3.727 

1.059 

0.042 

0.988 

0 

• 2025 

4.029 

0.906 

3.042 

3-305 

1 .061 

0.038 

0.987 

0 

• 2026 

4.029 

0.906 

3.042 

3.3°5 

2.807 

1 .062 

0.038 

0.986 

0 

• 2027 

4.029 

0.906 

3.042 

1 .062 

0.028 

0.989 

0 

• 2028 

4.029 

0.906 

3.042 

2.807 

1.062 

0.023 

0.991 

0 

• 2029 

4.029 

0.906 

3.042 

2.228 

1.061 

0.029 

0.988 

0 

• 2030 

4.029 

0.906 

3.042 

2.228 

1.075 

0.018 

0.992 

0 

• 2031 

4.029 

0.906 

3.042 

1.575 

1.059 

0.023 

0.988 

0 

• 2032 

4.029 

0.906 

3.042 

1.575 

1.061 

0.027 

0.987 

0 

3033 

4.029 

0.869 

8.500 

1.500 

1.069 

-0.091 

1.033 

0 

2034 

4.029 

0.869 

8.500 

1.500 

1.070 

-0.072 

1.025 

0 

2036 

4.029 

0.869 

8.500 

1.250 

1.068 

-0.059 

1 .029 

0 

2038 

4.029 

0.869 

8.500 

1.250 

1 .068 

-o.o47 

1.021 

0 

2039 

4.029 

0.869 

8.500 

1.125 

1.040 

-0.025 

1.021 

0 

2041 

4.029 

0.869 

8.500 

1.125 

1.041 

-0.026 

1.019 

0 

• 2042 

4.029 

0.900 

3.017 

4.726 

1.C25 

0.026 

0.991 

0 

• 2043 

4.029 

0.900 

3.017 

4.726 

1.028 

0.023 

0.992 

0 

• 2044 

4.029 

0.900 

3.017 

3.787 

1.026 

0.022 

0.992 

0 

• 2045 

4.029 

0.900 

3.017 

3.787 

1.024 

0.022 

0.992 

0 

• 2046 

4.029 

0.900 

3.017 

3.357 

1 .023 

0.024 

0.991 

0 

• 2047 

4.029 

0.900 

3.017 

Ull 

1.022 

0.024 

0.990 

0 

• 2048 

4.029 

0.900 

3.017 

1 .015 

0.022 

0.990 

0 

• 2049 

4.029 

0.900 

3.017 

2.850 

1 .021 

0.011 

0.994 

0 

• 2050 

4.029 

0.900 

3.017 

2.261 

1 .036 

0.017 

0.990 

0 

• 2051 

4.029 

0.900 

3.017 

2.261 

1.022 

0.013 

0.992 

0 

• 2052 

4.029 

0.900 

3.017 

1.596 

1.022 

0.012 

0.994 

0 

• 2053 

• 2054 

4.029 

0.900 

3.017 

1.596 

1.026 

0.010 

0.994 

0 

4.C29 

0.900 

3.017 

4.716 

1.021 

0.027 

0.991 

0 

• 2055 

4.029 

0.900 

3.017 

4.716 

1.022 

0.023 

0.991 

0 

• 2056 

4.029 

0.900 

3.017 

3-778 

1.022 

0.019 

0.991 

0 


Run 

D 

k 

It 

O' 

V 

A 

B 

X 

• 2057 

4.029 

C.900 

3.017 

3.778 

1.024 

0.02’ 

0.990 

0 

• 2058 

4.029 

0.900 

3.017 

3.348 

1.025 

C .020 

0.990 

0 

• 2059 

4.029 

0.900 

3-017 

3.348 

1.023 

0.015 

0.992 

1 

• 2060 

4.029 

0.900 

3.017 

2.841 

1.023 

0.020 

0.988 

0 

• 2061 

4.029 

0.900 

3.017 

2.841 

1.024 

0.015 

0.992 

0 

• 2062 

4.029 

0.900 

3.017 

2.253 

1.021 

0.018 

0.991 

0 

• 2063 

4.029 

0.900 

3.017 

2.253 

1.020 

0.018 

0.991 

0 

• 2064 

4.029 

0.900 

3.017 

1.587 

1.021 

0.019 

0.990 

0 

• 2065 

4.029 

0.900 

3.017 

1.587 

1.022 

0.024 

0.988 

0 

• 2066 

4.029 

0.868 

2.913 

5. 068 

1.024 

0.027 

0.995 

0 

• 2067 

4.029 

0.868 

2.913 

5.068 

1.023 

0.029 

0.996 

0 

• 2068 

4.029 

0.868 

2.913 

4.027 

1.023 

0.028 

0.996 

0 

• 2069 

4.029 

0.868 

2.913 

4.027 

1.039 

0.031 

0.994 

0 

• 2070 

4.029 

0.868 

2.913 

3.549 

1.022 

0.035 

0.992 

0 

• 2071 

4.029 

0.868 

2.913 

3.549 

1.027 

0.032 

0.995 

0 

• 2072 

4.029 

0.868 

2.913 

2.986 

1.022 

0.033 

0.994 

c 

• 2073 

4.029 

0.868 

2.913 

2.986 

1.024 

0.029 

0.995 

0 

• 2074 

4.029 

0.868 

2.913 

2.333 

1.024 

0.015 

1.002 

0 

• 2075 

4.029 

0.868 

2.913 

?:» 

1.022 

0.016 

1.001 

0 

• 2076 

4.029 

0.868 

2.913 

1.022 

0.027 

0.993 

0 

• 2077 

4.029 

0.868 

2.913 

1.594 

1.023 

0.023 

0.997 

0 

• 2078 

4.029 

0.869 

8.500 

2.000 

1.043 

-0.037 

1.013 

0 

2080 

4.029 

0.869 

0.868 

8.500 

2.000 

1.038 

-0.022 

1.006 

0 

• 2081 

4.029 

2.913 

2.986 

1.019 

0.027 

0.996 

0 


267 


Run 

D 

k 

Lt 

cr 

V 

A 

B 

X 

Run 

D 

k 

Lt 

cr 

V A 

b ; 

X 







1.065 



1556 

0.620 

0.910 

8.333 

1.031 

1.029 -0.030 

1 .C4l 

0 

2100 

2.000 

0.875 

5.617 

0.991 

1.032 

1 .029 

0 

1557 

0.620 

0.910 

8.333 

0.969 

1.008 -0.024 

1.038 

0 

2101 

2.000 

0.879 

5.200 

1.011 

1.030 

0.065 

1.040 

0 

1558 

0.620 

0.910 

8.333 

1.062 

1 .01 3 -0 .025 

1 .038 

c 

2102 

2.000 

C.880 

5.200 

1.013 

1.045 

0.013 

1 .009 

0 

1559 

C.620 

0.910 

8-333 

1.125 

1.C21 -0.026 

1 .036 

0 

2103 

2.000 

0.875 

5.617 

1 . 162 

1.035 

0.035 

o.oi4 

1.002 

0 

1560 

0.620 

c .910 

8.333 

1.250 

1.022 -0.035 

1.037 

0 

2104 

2.000 

0.879 

5.200 

1.166 

1.035 

1 .015 

0 

1561 

C.620 

0.910 

8.333 

i.coo 

1.028 -0.C15 

1 -035 

0 

2105 

2.000 

0.880 

5.200 

1 . 166 

1.033 

c.016 

1 .012 

0 

1562 

l .620 

C.910 

8.333 

1 .500 

1.017 -0.045 

1 .041 

0 

2 106 

2.000 

0.875 

5.617 

1.193 

1.035 

-c.003 

1 .019 

0 

lit: 

0.620 

0.910 

7.292 

2.000 

1.027 -0.032 

1 .047 

0 

2107 

2.000 

0.879 

5.200 

1.197 

1.032 

-0.010 

1 .014 

0 

0.620 

0.910 

8.333 

3.000 

1.039 C.013 

1 .043 

0 

2 ' 06 

2.000 

0.880 

5.200 

1 . 198 

1.031 

0.012 

1.012 

0 

1565 

0.620 

c.910 

8.333 

5.000 

1 . C16 -0. 180 

1.060 

0 

2 '09 

2.000 

0.875 

5.617 

1.256 

1.031 

0.023 

1 .010 

c 

1566 

0.620 

c.910 

8.333 

1 .031 

1.OI8 -0.043 

i.037 

0 

2M0 

2.000 

0.879 

5.200 

1.260 

1.030 

-0.039 

1.022 

1 

1567 

0.620 

0.910 

8.333 

0.969 

1 .020 -0.035 

1.037 

c 

21 11 

2,000 

0.880 

5.200 

1.125 

1 .031 

0.010 

' .009 

c 

1568 

0.620 

0.910 

8.333 

1.062 

1.017 -0.051 

1 .040 

0 

2112 

2.000 

0.875 

5.617 

1.031 

1.017 

0.047 

0.994 

0 

1569 

0.620 

0.910 

8.333 

1.125 

1.031 -0.025 

1.034 

0 

2113 

2.000 

0.979 

5.200 

1.031 

1.026 

0 .023 

1.007 

0 

1570 

0.620 

0.9 10 

6.250 

1.000 

1.015 -0.032 

i .034 

0 

2114 

2.000 

0.880 

5.200 

1.031 

1.024 

C.026 

1.005 

c 

1571 

0.620 

0.910 

8.333 

1 .250 

1.030 -0.029 

1 .034 

0 

2M5 

2.000 

0.875 

5.617 

1 .062 

1 .027 

0.050 

0.996 

0 

1572 

0.620 

0.910 

8.333 

1 .500 

1.013 -0.049 

1 .038 

0 

2116 

2.000 

0.879 

5.200 

1 .062 

1 .019 

0.018 

1.012 

0 

1574 

0.620 

0.910 

8.333 

2.000 

1.008 -0.067 

1.044 

0 

2117 

2.000 

0.880 

5.200 

1 .062 

1 .020 

0.012 

1.009 

0 

C.620 

0.910 

8.333 

3.000 

1.002 -0.052 

1.055 

0 

2118 

2.000 

0.875 

5.617 

1.125 

1.023 

C.051 

0.999 

0 

1575 

0.620 

0.910 

8.333 

5.000 

1.010 -0.205 

1 .056 

0 

2119 

2.000 

0.879 

5.200 

1.125 

-.02 3 

-0.017 

1.018 

0 

1576 

0.620 

c.910 

8.333 

3.068 

42.946 0.051 

" .962 

0 

2120 

2.000 

0.875 

5.617 

1.250 

1.024 

0.012 

1 . C 1 4 

0 

1577 

0.620 

c.910 

8.333 

3.C68 45.272 0.014 

C.972 

0 

2121 

2.000 

0.879 

5.200 

1.250 

1.023 

-0.038 

1 .023 

c 

1578 

C.620 

l .910 

7.792 

3.005 

43.606 -0.C06 

0.977 

0 

2122 

2.000 

0.880 

5.200 

1 .250 

1.025 

-C.033 

1.023 

0 

1579 

0.620 

0.910 

8.333 

2.068 

44.231 -0.010 

0.972 

0 

2123 

2.000 

0.875 

5.617 

1 .500 

' .020 

-0.066 

1.026 

0 

1580 

0.620 

0.910 

8.333 

1.568 44.031 -0.021 

0.996 

0 

2126 

2.00c 

C.879 

5.200 

1 .500 

1 .021 

-0.045 

1.026 

0 

1581 

C.620 

c.910 

8.333 

1 .318 43.698 -0.027 

1.003 

0 

2125 

2.000 

0.880 

5.200 

1.500 

1 .023 

-0.073 

1 .030 

c 

1582 

0.620 

0.910 

8.333 

1.193 

45.795 0.025 

0.997 

0 

2126 

2.000 

0.875 

5.617 

2.000 

1 .022 

-0.135 

1.039 

0 

• 1583 

0.620 

0.905 

1.122 

7.800 

46.708 -O.O33 

0.998 

0 

2127 

2.000 

0.879 

5.200 

2.000 

1.023 

-0.092 

1.031 

0 

1584 

0.620 

c.910 

8.333 

1.130 

48.694 C.C48 

0.997 

c 

2128 

2.000 

0.880 

5.20^ 

2.000 

1.023 

0.039 

1 .02 1 

c 

1585 

0.620 

0.910 

8.333 

1.099 

45.290 0.038 

0.994 

c 

2129 

2.000 

0.879 

5.200 

2.000 

1.023 

-0.169 

1.033 

0 

1586 

0.620 

0.910 

8.333 

1.068 44.803 0.051 

0.998 

0 

2130 

2.000 

0.880 

5.200 

3.000 

1.024 

-0.C62 

1.019 

0 

1587 

0.620 

0.910 

8.333 

1.068 

44.974 -0.019 

1 .008 

0 

2131 

2.000 

0.880 

5.200 

1 .061 

9.698 

C.027 

1.005 

0 

1588 

0.620 

c.910 

8.333 

1.099 45.434 -0.019 

1 .008 

0 

2132 

2.000 

0.879 

5.200 

1.061 

9.689 

0.038 

1 .014 

0 

1589 

0.620 

0.910 

8.333 

1.130 45.635 -0.022 

1.011 

0 

2133 

2.000 

0.875 

5.617 

1 .061 

9.699 

0.074 

C.993 

0 

1590 

0.620 

0.910 

8.333 

1.193 45.809 -0.023 

1.008 

0 

2136 

2.000 

0.880 

5.200 

1 .092 

9-735 

-0.013 

1.015 

0 

1591 

0.620 

0.910 

8.333 

1 .318 46. ’76 -0.017 

1.004 

0 

2135 

2.000 

0.879 

5.200 

1.092 

9.686 

0.039 

1.002 

0 

1592 

0.620 

0.910 

8.333 

1 .568 46.8U -C.018 

1.OO6 

0 

2136 

2.000 

0.879 

5.200 

1.092 

9.686 

-0.005 

1 .013 

0 

1593 

1594 

0.620 

c.910 

8.333 

2.068 

46.291 -0.020 

1 .005 

0 

2 '37 

2.000 

0.875 

5.617 

1.092 

9.690 

0.088 

0.996 

0 

0.620 

0.910 

8.333 

3.068 

46.194 -0.020 

1.003 

0 

2138 

2.000 

0.880 

5.200 

1.i23 

9.723 

-0.009 

1.012 

c 

1595 

0 .620 

c.910 

7.792 

3.005 

46.493 -0.020 

1.002 

0 

2139 

2.000 

0.879 

5.200 

1.-23 

9.708 

-0.005 

1.016 

0 

• 1596 

0.620 

n .905 

1.122 

7.800 

45.694 -0.074 

1.000 

0 

2160 

2.000 

0.875 

5.617 

1.123 

10.653 

0.029 

0.993 

c 

1597 

0.620 

c .910 

8.333 

3.068 

27.351 -0.016 

0.984 

0 

2161 

2.000 

0.880 

5.200 

1.186 

10.561 

0.003 

1.005 

0 

1598 

C.620 

0.910 

7.792 

3.005 

28.037 -0.015 

0.987 

0 

2162 

2.000 

0.879 

5.200 

1.186 

10.622 

-0.002 

1 .007 

0 

1599 

0.620 

c.910 

8.333 

2.068 

28.114 -0.018 

0.990 

0 

2163 

2.000 

0.875 

5.617 

1.186 

10.617 

0.006 

1 .007 

0 

1600 

0.620 

0.910 

8.333 

1.568 27.988 -0.018 

0.990 

0 

21 66 

2.000 

0.880 

5.200 

1.311 

10.585 

-0.037 

1 .019 

0 

1601 

0.620 

0.910 

8.333 

1 .318 27.985 -0.014 

0.990 

c 

2165 

2.000 

C.879 

5.200 

1.311 

10.946 

-0.052 

1 .022 

0 

1602 

0.620 

0.910 

8.333 

1 • '93 

27.849 -0.C14 

0.990 

0 

2166 

2.000 

0.879 

5.200 

1.3H 

10.976 

-0.033 

i .014 

0 

1603 

0.620 

c.910 

8.333 

1.-30 

27.939 -0.012 

0.990 

0 

2 ’ 67 

2.000 

0.875 

5.617 

1.311 

10.934 

-0.016 

1.009 

0 

1604 

0.620 

0.910 

8.333 

1.099 

28.800 -0.017 

0.990 

0 

2168 

2.000 

0.880 

5.200 

1.561 

10.495 

-0.040 

1.016 

c 

1605 

0.620 

c.910 

8.333 

1.068 

28.915 -0.016 

0.990 

c 

2169 

2.000 

0.879 

5.200 

1.561 

10.339 

-0.041 

1.027 

0 

• 1606 

0.620 

C.905 

1 . 122 

7.800 

29.049 -0.079 

0.988 

0 

2150 

2.000 

0.875 

5.617 

1.561 

10.665 

-0.008 

1.010 

0 

1607 

0.620 

C.806 

1.875 

7.780 

29.494 -0.002 

0.972 

0 

2151 

2.000 

0.880 

5.200 

2.061 

10.451 

-0.047 

1.022 

0 

1608 

0.620 

0.806 

1.875 

7.780 

29.373 0.007 

0.969 

0 

2152 

2.000 

0.879 

5.200 

2.061 

'0.567 

-0.038 

1.021 

0 

1609 

0.620 

0.806 

1.875 

7.780 

18.919 -0.020 

0.983 

0 

2153 

2.000 

C.880 

5.200 

3.061 

10.603 

-0.092 

1 .049 

0 

1610 

C.620 

0.806 

1.875 

7.780 

19.'03 0.003 

0.974 

0 

2156 

2.000 

0.879 

5.200 

3.061 

10.601 

-0.023 

1.023 

0 

• 1611 

0.620 

C.905 

1 .122 

7.800 

19.017 -0.096 

0.994 

0 

• 2’55 

2.000 

0.875 

1 .658 

1.276 

10.467 

0.031 

0.989 

0 

1612 

0.620 

C.910 

8.333 

1 .068 

18.492 -0.016 

C.989 

0 

• 2156 

2.000 

0.875 

1.658 

1.276 

10.362 

0.035 

C.989 

c 

1613 

0.620 

0.910 

8.333 

1.099 

18.208 -0.016 

0.990 

0 

• 2157 

2.000 

0.875 

1.658 

1.950 

10.453 

0.044 

0.987 

0 

1614 

0.620 

0.910 

8.333 

1.13O 

18.336 -0.019 

0.991 

0 

• 2158 

2.000 

0.875 

1.658 

1.950 

10.523 

0.060 

0.987 

0 

1615 

0.620 

0.910 

8.333 

1.193 

18.520 -0.017 

0.990 

0 

• 2159 

2.000 

0.875 

1.658 

2.396 

10.789 

0.028 

0.993 

0 

1616 

C.620 

0.910 

8.333 

1.318 

18.603 -0.018 

0.990 

0 

0 2 ’60 

2.000 

0.875 

1.658 

2.396 

10.799 

0.022 

C .997 

0 

1617 

C.620 

0.910 

8.333 

1.568 

18.426 -C .02 1 

0.990 

c 

• 2161 

2.000 

0.875 

1.658 

3.065 

10.798 

0.025 

0.999 

0 

1618 

C.620 

C.910 

8.333 

2.068 

18.469 -0.018 

0.988 

0 

• 2 ’62 

2.000 

0.875 

1.658 

3.065 

10.873 

0.025 

0.999 

0 

1619 

0.620 

0.910 

8.333 

3.068 

18.590 -0.017 

0.986 

0 

• 2’63 

2.000 

0.875 

1.658 

3.608 

10.843 

C.031 

0.999 

0 

1620 

C.620 

0.910 

7.792 

3.005 

18.648 -C.014 

0.988 

0 

• 2166 

2.000 

0.875 

1.658 

3.608 

10.738 

C.035 

0.994 

0 

1621 

C.620 

C.8O6 

1.875 

7.780 

12.971 -0.066 

0.986 

0 

• 2165 

2.000 

0.875 

1 .658 

7.790 

10.705 

0.043 

0.999 

0 

• 1622 

0.620 

0.905 

1.122 

7.800 

12.565 -0.090 

0.989 

0 

• 2166 

2. COO 

C .875 

1.658 

7.790 

10.782 

0.041 

i.OOO 

0 

1623 

0.620 

0.910 

8.333 

3.068 

12.496 -0.016 

0.983 

0 

• 2167 

2.000 

0.875 

3.500 

7.790 

10.892 

0.035 

0.999 

0 

1624 

0.620 

0.910 

7.792 

3.005 

12.719 -0.038 

0.994 

0 

• 2168 

2.000 

0.875 

3-500 

7.790 

10.908 

C.030 

0.999 

1 

1625 

0.620 

0.910 

8.333 

2.068 

12.528 -0.037 

0.991 

0 

• 2169 

2.000 

0.875 

1.658 

3.608 

1.034 

0.C49 

0.995 

0 

1626 

0.620 

0.910 

8.333 

1.568 

12.480 -0.036 

0.993 

0 

• 2170 

2.000 

0.875 

1.658 

3.608 

1.036 

0.048 

0.998 

0 

1627 

0.620 

0.910 

8.333 

1.318 

12.586 -0.032 

0.993 

0 

• 2171 

2.000 

0.875 

1.658 

3.065 

1.037 

C.C48 

0.995 

0 

1628 

0.620 

0.910 

8.333 

1.193 

’2.622 -C.029 

0.994 

0 

• 2172 

2.000 

0.875 

1.658 

3. 065 

1.035 

0.047 

0.996 

0 

1629 

C.620 

0.910 

8.333 

i.’30 

12.917 -0.063 

1.010 

0 

• 2173 

2.000 

0.875 

1 .658 

2.396 

1.031 

0.034 

1.001 

0 

I63O 

C.620 

0.910 

8.333 

1.099 

12.550 -0.030 

0.994 

0 

• 2176 

2.000 

0.875 

1.658 

2.396 

1.043 

0.028 

1.002 

0 

1631 

0.620 

0.910 

8.333 

1.068 

12.504 -0.027 

0.993 

0 

• 2175 

2.000 

0.875 

1.658 

i.950 

’.036 

0.023 

1 .003 

0 

1632 

0.620 

C.806 

1.875 

7.780 

7.120 C.123 

0.978 

0 

• 2176 

2.000 

0.875 

1.658 

1.950 

1.050 

0.034 

1 .001 

0 

• 1633 

0.620 

C.905 

1.122 

7.800 

7.177 -0.057 

0.989 

0 

• 2' 77 

2.000 

0.875 

1.658 

1.276 

1.032 

0.008 

1 .006 

0 

1634 

C.620 

0.910 

8.333 

3.058 

7,355 -0.047 

1.001 

0 

• 2178 

2. COO 

0.875 

1.658 

1.276 

1 .038 

0.016 

1.005 

0 

1635 

0.620 

0.910 

7.792 

3.050 

7.257 -O.C37 

1.002 

0 

• 2179 

2.000 

0.875 

1.658 

7.790 

1 .036 

0.069 

0.995 

0 

1636 

0.620 

O.910 

8.333 

2.058 

7.224 -0.C42 

1 .001 

0 

• 2180 

2.000 

0.875 

1 .658 

7.790 

1 .036 

0.067 

0.997 

0 

1637 

0.620 

0.910 

8.333 

1.558 

7.207 -0.041 

1 .003 

0 

• 2181 

2.000 

0.875 

3.500 

7.790 

1.051 

0.039 

1 .003 

0 

1638 

0.620 

0.910 

8.333 

1.308 

7.177 -0.039 

1 .005 

0 

• 2182 

2.000 

0.875 

3.500 

7.790 

1 .058 

0.047 

1 .001 

0 

1639 

0.620 

C.910 

8.333 

1.183 

7.121 -C.O36 

1 .005 

0 

2183 

2. COO 

l.879 

5.200 

2.000 

1.037 

0.000 

0.000 

29 

i640 

0.620 

0.910 

8.333 

1.120 

7.'51 -0.038 

1.005 

0 










1641 

0.620 

C.910 

8.333 

1 .089 

7.'32 -0.037 

1 .005 

0 










1642 

0.620 

0.910 

8.333 

1.058 

7.268 -0.034 

1 .003 

0 










1643 

0.620 

c .806 

1.875 

7.780 

3.738 -0.325 

'.012 

1 










• 1644 

0.620 

0.905 

1 .122 

7.800 

3.725 -0.031 

0.985 

0 










1645 

0.620 

0.910 

8.333 

3.041 

3.707 -0.050 

1 .002 

0 










1646 

0.620 

0.910 

7.792 

3.005 

3.733 -O.C33 

1 .002 

0 










1647 

0.620 

0.910 

8.313 

0,041 

1,721 -C.043 

1 .001 

0 










1648 

0.620 

0.910 

8.333 

1 .541 

3.746 -0.036 

1.003 

0 










1649 

0.620 

0.910 

8.333 

1.291 

3.754 -0.035 

1.005 

0 










1650 

0.620 

0.910 

8.333 

I.166 

3.766 -0.028 

1.004 

0 










1651 

0.620 

0.910 

8.333 

1.104 

3.784 -0.027 

1 .003 

0 










1652 

0.620 

0.910 

8.333 

1 .072 

3.783 -0.029 

1.005 

0 










1653 

0,620 

0.910 

8, 11 1 

1 , 041 

1,790 -0,008 

1,004 

0 










1444 

0,620 

0.806 

1.855 

5,580 

1,040 -0,494 

1,040 

0 










• 1665 

0.620 

0.905 

1.122 

7.800 

1.057 -0.053 

1.011 

0 










1666 

0.620 

0.910 

8.333 

3.000 

1.048 -0.087 

1.036 

0 










1667 

1668 

0.620 

0.910 

7.792 

3.006 

1.041 C.082 

1 .001 

c 










0.620 

C.910 

8.333 

2.000 

1 .046 -C.090 

1 .034 

c 










1669 

0.620 

0.910 

8.333 

1.500 

I.051 -0.070 

1.024 

0 










1670 

0.620 

0.910 

8.333 

1.250 

1.048 -0.051 

1.018 

0 










1671 

0.620 

0.910 

8.333 

1.’25 

1.048 -0.051 

1.017 

0 










1670 

0,420 

0.910 

8.333 

1.062 

1.043 -0.055 

1.017 

0 










1673 

0.620 

0.910 

8.333 

1.031 

1 .053 -0.053 

1.017 

0 










1674 

0.620 

C.910 

8.333 

1 .000 

1 .055 -0.056 

1 .017 

0