aerodynamic drag coefficients of

AERODYNAMIC DRAG COEFFICIENTS OF

A VARIETY OF ELECTRICAL CONDUCTORS

by

JEFFREY C. STROMAN, B.S.M.E.

A THESIS

IN

MECHANICAL ENGINEERING

Submitted to the Graduate Facul ty of Texas Tech University in

Par t ia l Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

MECHANICAL ENGINEERING

Approved

May, 1997

/^ / A /

TABLE OF CONTENTS

LIST OF TABLES iii

LIST OF FIGURES iv

CHAPTER

1. INTRODUCTION 1

1.1. Objective 1

1.2. Application of Results 1

1.3. Methodology Overview 1

2. BACKGROUND 3

2.1. Theoretical Considerations 3

2.2. Literature Survey 8

3. EXPERIMENTAL APPARATUS 13

3.1. End Plate Design 13

3.2. Towtank Apparatus 17

3.3. Wind Tunnel Apparatus 20

3.4. Uncertainty Analysis 22

4. RESULTS AND DISCUSSION 26

4.1. Conductors in Normal Crossflow 26

4.2. Conductors in Yawed Flow 42

5. CONCLUSIONS AND RECOMMENDATIONS 51

SOURCES 53

APPENDICES

A TABULATED DATA 54

B CONDUCTOR INFORMATION 69

11

LIST OF TABLES

3.1. Towtank Uncertainty Analysis Results 24

3.2. Wind Tunnel Uncertainty Analysis Results 25

A.l. Smooth Cylinder Data 55

A.2. Conductor 100-58600-3 Data 56










A. 12. Conductor 100-61200-4 Yaw Data 65



A.15. Conductor 100-58200-8 Yaw Data 68

B.l. Conductor Information 70

111

LIST OF FIGURES

2.1. Regimes of the Drag Coefficient versus Reynolds Number Plot 4

2.2. Laminar (a) and Turbulent (b) Boundary Layer Separation Points 5

2.3. Surface Pressure Distribution on a Cylinder in Crossflow 6

2.4. Variation of Drag Coefficient with Surface Roughness 7

2.5. Conductor, as Seen from Above, in Yawed Flow 8

2.6. Cross-Sections of Round Wire (a) and Trapezoidal Wire (b) Conductors 10

2.7. Comparison of Drag Coefficient Data Collected in Wind Tunnels

and Field 10

3.1. Saddle Clamp Block 13

3.2. End Plate 14

3.3. Saddle Block, Unflexed (a) Flexed (b) 14

3.4. End Plates in Untensioned (a) and Tensioned (b) Configurations 15

3.5. End Plate vAth Shroud Installed 16

3.6. End Plate Slot 16

3.7. End Plate Slot, Taped for Tare (a) and for Conductors (b) 17

3.8. Texas Tech Towtank Facility 18

3.9. Load Cells in Their Upright (a) and Testing (b) Positions 18

3.10. End Plate Installation in the Wind Tunnel 21

3.11. End Plate with Conductor in Yaw, Unshrouded (a), and Shrouded (b) 22

4.1. CD versus Re for Conductor 100-58600-3 27








iv



4.11. Comparison of Round and Trapezoidal Strand Conductors 40

4.12. CD versus Re for a Smooth Cylinder 41

4.13. CD,yaw versus Re for Conductor 100-61200-4 43




4.17. CD, yaw/cosV versus Re for Conductor 100-61200-4 47

4.18. CD, yaw/cosV versus Re for Conductor 100-58650-0 48

2 4.19. CD,yaw/cos vj/versus Re for Conductor 100-58600-3 49 .2 4.20. CD,yaw /cos> versus Re for Conductor 100-58200-8 50

CHAPTER 1

INTRODUCTION

1.1 Objective

The objective of this study is to determine accurately the aerodynamic drag

coefficients of several different conductors, also called electric transmission lines The

study was handed by Florida Power & Light in hopes of acquiring better data with which

to design transmission line towers. The reason that Florida Power & Light has funded this

research is that recent legislation changes in Florida require a higher tolerance to hurricane

wind loading for structures. This includes transmission line towers, and thus the wind's

drag force on the conductors is paramount to design.

1.2 Application of Results

The author hopes that the results contained in this paper may be used to design and

construct safer electric transmission lines and structures, not only in Florida, but in any

region with high storm wind speeds. The data presented are applicable to any

transmission line loading design problem involving the conductors tested in this study.

1.3 Methodology Overview

The drag coefficient data were collected using two methods. First was the Low

Speed Wind Tunnel Facility operated by Texas A&M University's Aerospace Engineering

Department. Second was the Water Towtank Facihty at the Mechanical Engineering

Department of Texas Tech University. In both tests, actual conductor samples were used

instead of models of conductors. Also, in opposition to the traditional wind tunnel testing

technique for conductors, the samples were not mounted through the walls of the wind

tunnel to external balances but were mounted between two end plates in the center of the

wind tunnel test section on the facihty's internal force balance. Mounting the conductors

in the center of the free stream circumvented the traditional problem of boundary layer

effects along the wind tunnel walls and also provided the flexibility required for testing

under yawed flow conditions.

The drag forces encountered in the wind tunnel and the water towtank tests were

measured, and the drag coefficients calculated for a range of Reynolds numbers. In the

wind tunnel, drag coefficients were obtained for normal and yawed wind conditions. In

both v^nd tunnel and towtank experiments, the drag force on the end plates alone was

measured and subtracted from the total drag force of the conductor and end supports.

This was done in order to isolate the drag due to the conductors. This subtracted quantity

is referred to as the tare drag.

CHAPTER 2

BACKGROUND

2.1 Theoretical Considerations

Many factors affect the drag coefficient of a conductor. Several important factors

are Reynolds number, roughness, turbulence, and wind direction. The equation defining

the drag coefficient may be found in any standard fluids or aerodynamics reference. The

equation states:

C D = - J — ^ 2 (2.1)

In Equation 2.1, FD is the drag force, p is the density of the fluid, Voo is the free

stream velocity, and Af is the frontal area. This equation may be used to calculate drag

coefficients for any object subjected to a fluid flow, including conductors. Notice that the

equation is nondimensional, i.e., unitless. A second nondimensional parameter of some

importance is Reynolds number. It is calculated in the following fashion:

V D Re = - ^ - . (2.2)

V

In Equation 2.2, Voo once again represents the free stream velocity. The

characteristic dimension, D, for this study is the outer diameter of the conductor. And

finally, the kinematic viscosity of the fluid is represented by v. The major resuhs of this

study are plots of the drag coefficient, CD, as a fiinction of the Reynolds number. Re.

Reynolds number represents a balance of the relative magnitudes of inertial and

viscous forces on an object subjected to a fluid flow. Low Reynolds numbers indicate that

viscous effects, along with stable, laminar flow are dominant. High Reynolds numbers

indicate that inertial (pressure) forces, along with unstable, titrbulent flow are dominant.

Several regimes of the drag coefficient versus Reynolds number plot may be defined by the

relative magnitude of Reynolds number. Low Reynolds numbers refer to the subcritical

regime, while high Reynolds numbers refer to the supercritical regime. Intermediate

values of Reynolds number refer to the critical regime. These subcritical, critical, and

supercritical regimes are illustrated in Figure 2.1.

CD

Subcritical Critical Supercritical

Re

Figure 2.1 Regimes of the Drag Coefficient versus Reynolds Number Plot

The sharp drop of the drag coefficient in the critical regime occurs due to a

transition from laminar to turbulent boundary layer separation. Laminar and turbulent

boundary layer separation are illustrated on the next page in Figure 2 2 The subcritical

regime is characterized by a broad downstream wake associated with early separation of

the laminar boundary layer at approximately 82° on the front side of the cylinder. The

supercritical regime is characterized by a narrow downstream wake associated with

delayed turbulent boundary layer separation at approximately 120° on the aft side of the

cyhnder.

The high drag coefficient of the cyhnder with laminar boundary layer separation

and subcritical flow is caused by early separation and the subsequent broad downstream

wake. The region of separated flow behind the cylinder is at a much lower relative

pressure than the stagnation point (6 = 0°) on the fi-ont of the cylinder. Laminar boundary

(a)

~ ^

Broad Wake

J> (b)

Figure 2.2. Laminar (a) and Turbulent (b) Boundary Layer Separation Points

Narrow ^ o Wake r ^ 9

layer flow is vulnerable to separation in the presence of this adverse pressure gradient

Figure 2.3 on the next page shows the pressure distribution on a cylinder in cross flow as a

function of circular angle. On the front of the cylinder (6 < 90°), there is a steep drop in

surface pressure as the flow progresses through approximately 80°. The laminar boundary

layer, being very susceptible to separation in this adverse pressure gradient, diverges from

the surface at an angular position of 82°, just aft of the point with the lowest surface

pressure.

The lower drag coefficient of the cylinder with turbulent boundary layer separation

and supercritical flow occurs because turbulent flow is more resistant to separation in the

presence of the adverse pressure gradient. The turbulent separation point is farther

downstream (9 = 120°) than the separation point for laminar flow. The delayed separation

results in a narrower wake, providing less area for the wake's low pressure to act upon.

The drag coefficient is not only affected by the Reynolds number; it is also affected

by surface roughness of the cyhnder and turbulence in the free stream. The Reynolds

number at which the critical regime transition occurs is affected by the surface roughness

of the cylinder or conductor. The roughness parameter is defined by:

8 r = —.

D (2.3)

The roughness, r, is defined in Equation 2.3 by the outside or overall diameter of the

conductor, D, and the wall roughness, e. The critical regime transition occurs earlier (at

8

Ii

90

i9

135°

Figure 2.3. Surface Pressure Distribution on a Cylinder in Crossflow (White 414)

180°

lower Reynolds number) for conductors with large roughness values (see Fig. 2.4, next

page). Conductors with small values of roughness behave more like smooth cylinders.

Note in Figure 2.4 that the drag coefficient of rough cylinders does not decrease as much

as that of smooth ones. This early transition behavior for rough cylinders occurs because

the roughness element trips the boundary layer. In other words, the roughness element

perturbs the flow, causing enough turbulence for the boundary layer to make the early

transition from laminar to turbulent. The more rough a cylinder is, the earlier this

transition occurs. This concept is illustrated on the next page in Figure 2.4.

6

D 0.7 i f

0 .5 - 0.004 ojore

Figure 2.4. Variation of Drag Coefficient with Surface Roughness (White 227)

Also of importance to the drag coefficient is the turbulence of the fluid moving

around the conductor. The information presented above concerning critical regimes,

boundary layer separation, etc., is for smooth laminar flow in the free stream. However, if

the free stream is very turbulent, the critical regime is shifted to lower Reynolds numbers.

In this sense, turbulence has the same effect as surface roughness: high turbulence in the

free stream (or surface roughness of the cylinder) causes an early transition. The same

boundary layer transition trends occur with increasing turbulence as with increasing

surface roughness. The general trend of earlier transhion for increasmg surface roughness

shown in Figure 2.4 apphes for increasing free stream turbulence as well. A high quahty

wind tunnel will keep free stream turbulence to a minimum in order to prevent this early

turbulent transition phenomenon from affectmg drag force measurements.

The last consideration in the drag of a circular cylinder is the angular difference

between the longitudinal axis and the direction normal to the flow. This angle is caUed the

yaw angle, \\f, and is illustrated in Figure 2.5 on the next page. The drag force on the

cylmder can be expected to vary with the normal velocity component, or the inverse of the

cosine of psi, squared. Equation 2.4 shows this dependence.

'D , normal 'D.yaw

cos \\J

(2.4)

In Equation 2.4, CD,nonnai represents the drag coefficient aX\\f = 0°, while CD,yaw represents

the drag coefficient calculated from measurements of the drag force on a conductor at the

Normal velocity component

V

Flow direction

Direction A normal to flow t

Conductor's longitudinal axis

Figure 2.5. Conductor, as Seen from Above, in Yawed Flow

yaw angle, v|/. Note that the measured drag force, FD,yaw is the drag force in the direction

perpendicular to the conductor's longitudinal axis. Yaw angle may be critical in

determining the actual force on a conductor, as compared to the predicted force calculated

from drag coefficients for normal flow.

2.2 Literature Survey

Much research in the area of conductor wind loading has been done. The majority

of the information available is focused on characterization of the random wind loads

8

generated during storm conditions. However, the focus of this paper is characterization of

the aerodynamic drag coefficients of conductors. The majority of pubhshed research has

been performed by the power industry in hopes of better understanding wind loads on

transmission lines. Contributions in the field of aerodynamic drag on conductors have

been made by Krishnasamy of Ontario Hydro, Shan of Sverdrup Technology Incorporated

for the Electric Power Research Institute, and Wardlaw of the National Research Council

Canada.

The Ontario Hydro Research Division has conducted wind tunnel and field tests of

round wire and trapezoidal wire conductors (Krishnasamy). Due to their supposed lower

drag coefficients at high speeds, trapezoidal wire conductors were considered as a

replacement for aging round wire conductors. The difference between trapezoidal and

round wire conductors is shown in Figure 2.6 on the next page.

Wind tunnel tests were performed on four Aluminum Conductor Steel Reinforced

(ACSR) conductor samples at wind speeds between 20 and 190 km/hr. One important

conclusion was drawn by Ontario Hydro concerning the results of wind tunnel testing.

Trapezoidal wire conductors may have significantly lower drag coefficients than round

wire conductors at supercritical Reynolds numbers. Transition for the trapezoidal

conductor occurred at a Reynolds number of approximately 8 x lO' .

Sverdrup Technology, Incorporated of Haslet, Texas, has conducted research on

conductor drag coefficients for the Electric Power Research Institute's Transmission Line

Mechanical Research Center (Potter and Boylan, Shan). Shan used a "free-air-wind-

tunnel" to obtain drag coefficients of several conductor samples in an attempt to rectify

the marked differences between field data and wind tunnel data noted by Potter and

Boylan. This difference is illustrated in Figure 2.7 on the next page.

The difference between wind tunnel and field measurements is large, and the exact

cause of the discrepancy is unknovm. However, Potter and Boylan list and discuss several

possible explanations for the differences. Among these are conductor roughness,

atmospheric turbulence, spanwise variations in wind speed and direction, unsteady flow,

and wmd tunnel end effects and blockage. They conclude that two of these factors are

Core with circular steel strands

Outer layers with circular aluminum strands

Core with circular steel strands

Outer layers with trapezoidal aluminum strands

(a) (b)

Figure 2.6. Cross-Sections of Round Wire (a) and Trapezoidal Wire (b) Conductors (Krishnasamy 9)

O

1.4

1.2

1

0.8

0.6

0.4

0.2U

»

-

-

-

-

-

."A

'

\ »•—

\ \ - ^

1V»

\

Wind TVinnel Data

^ -

' • • • 1

1 -

FUid T«sts

'

Counlhan Imp Coll Sci & T«ch

Pr ic* Bristol Unlv

NRC Roport LA-110 (20 OL Strands)

MRC/FFL/ACPC 6' X 9' TUnnol

NRC/FPL/ACPC 30' X 30' Tiinnsl

Ont Hydro Fiold Data

Harrison Fiold Data

2 3 4 5 6 7 Reynolds Number xio^

Figure 2.7. Comparison of Drag Coefficient Data Collected in Wind Tunnels and Field (Shan 8)

paramount. First is the roughness and its associated early boundary layer transition, which

has been thoroughly studied and reported in the literature. The second, more important

factor is accurate wind definition. In field studies which use the maximum wind gust

10

speed to calculate drag, an apparent 40% drop in the drag coefficient may arise because

the maximum wind speed does not exist at all points along a span. Using averaging

procedures for wind speed between spans does not accurately characterize the wind either.

Wind direction, or yaw angle, and wind turbulence are also included in the difficulty of

wind definition. Additionally, in the field there are vertical and horizontal components of

the wind. If the vertical component of the field wind has a yaw angle, (j), then the drag

coefficient is affected by two yaw angles, one for the horizontal component, and one for

the vertical component. The overall drag coefficient of the conductor in the field could

then be described by the following equation.

CD = T - F - ^ T^^ : T— • (2.5) 1 2^

2 2 2

V cos (\|/)cos ((p) A,

In Equation 2.5 the wind speed's absolute magnitude is represented by V with an overbar.

The yaw angle of the horizontal component of wind is \\J, while that of the vertical wind

component is ^. Notice that the original frontal area, Af, is used. This double dependence

on yaw angles further comphcates the issue of resolving differences between wind tunnel

and field studies, because in normal wind tunnel tests the vertical component of wind is

not studied

Potter and Boylan conclude by writing that "accuracy and uncertainty bands of the

field-measured Ca associated with ... variable wind conditions may deserve additional

study as part of the effort to resolve discrepancies between field and wind tunnel data"

(Potter and Boylan 37). Therefore, apphcation of wind tunnel measurements of the drag

coefficient to actual field structures is difficult because wind speed and direction in the

field are not regular or simple to quantify.

Sverdrup's "free-air-wmd-tunnel," used to collect data for Shan's paper, consisted

of a 66 ft tower with a rotatable frame attached to the top. The 12 ft conductor models

were attached at either end of the frame to load cells for data collection. Shan concludes,

by comparison of "free-air-wmd-tunnel" data, field data, and wind tunnel data, that

"quality wind tunnel drag coefficients are correct and can be used to calculate the drag

11

force on transmission line conductors if the wind velocity is accurately known" (Shan 26).

This statement is not encouraging to the engineer faced with the problem of designing

transmission line towers, due to the difficulty in accurately characterizing the wind

velocity.

The National Research Council Canada conducted tests in a wind tunnel in order

to "determine the significance of some of the errors associated with the wind tunnel"

(Wardlaw, Phase I). They assessed, among other things, the effect of the wind tunnel wall

boundary layer, the end condition effects, and the aspect ratio (length divided by diameter)

of the conductor models. The end conditions used in their testing were open holes through

the walls of the wind tunnel test section. This allowed air to leak through the small gap

between the cable and the wind tunnel wall at both ends. They also added rough materials

to the walls of the wind tunnel in order to determine the effect of the boundary layer on

measurements of the drag coefficient. They concluded that the wall boundary layer and

end conditions play large roles in the measurement of the drag coefficient. Their data also

seem to imply that the aft centerhne pressure is most constant on conductor sections near

the center of the wind tunnel and that there is more variation near the walls.

In a second National Research Council Canada paper, the effect of tension on the

measurement of the drag coefficient was studied (Wardlaw, Phase II). The data suggest

that the amount of tension in the conductor is immaterial to the measurement of drag

coefficient m the wind tunnel, especially at supercritical Reynolds nimiber. This is to be

expected, as the shape and diameter of the conductor do not change significantly when

tension is applied.

12

CHAPTER 3

EXPERIMENTAL APPARATUS

3.1 End Plate Design

The same pair of end plates was used in both the wind tunnel and towtank phases

of the study. The end plates were constructed of aluminum in the Mechanical Engineering

Shops of Texas Tech University by the author. The abihty of the conductor to flex in the

direction of the wind force was determined to be important in obtaining high quality

measurements of the drag force. This flexure was accomphshed with a saddle clamp block

in each end plate, shown in Figure 3.1. One of the pair of end plates is shovm in Figure

3.2. The unflexed and flexed positions of the saddle block are shown, as mstalled in the

end plate, in Figure 3.3.

Figure 3.1. Saddle Clamp Block 13

Figure 3.2. End Plate

(a) (b)

Figure 3.3. Saddle Block, Unflexed (a) and Flexed (b)

14

Also important in the design of the end plates was the abihty to apply tension to

the conductor. Although the tension applied in the laboratory was very small compared to

tension used in actual applications, h was necessary in order to prevent oscillation and

vibration of the conductor during experiments. The tension was apphed by separating the

"saddle" block clamps from the end plates after the conductor was clamped. Each plate

allowed for approximately 0.5 inch elongation of the conductor. The tensiomng

mechanism's operation is shown in Figure 3.4.

^^Httji j i^X

wooaoo^>MgFy J^H^^^^^^^^^^^W

«^V/>. m

(a) (b)

Figure 3.4. End Plates in Untensioned (a) and Tensioned (b) Configurations

The tensioning mechanism and saddle clamps were covered during experiments

with semi-spherical fiberglass shrouds. The shape of the end plate shrouds was chosen to

reduce the overall drag force on the end plate assembly. An end plate with the shroud

installed is shown in Figure 3.5. Also important to the design of the end plates was

interference-free motion of the conductors, especially during experiments with yaw. This

was accomphshed with slots in the end plates which, when combined with the saddle block

clamp's capability to rotate, allowed for mterference-free motion of the conductor. The

slot in one of the end plates is shown in Figure 3.6. During tests, the portion of the slot

not occupied by conductor was covered with adhesive-backed aluminum tape. Durmg

15

measurements of the tare force on the end plates alone, the entire slot was covered with

the aluminum tape. The plate is shown, as taped during tests, in Figure 3.7 (next page)

Figure 3.5. End Plate with Shroud Installed

Figure 3.6. End Plate Slot

16

(a) (b) Figure 3.7. End Plate Slot, Taped for

Tare (a) and for Conductors (b)

3.2 Towtank Apparatus

The water towtank in the Mechanical Engineering Department at Texas Tech

University was used for the towtank phase of the experiment. The towtank is 80 ft long,

15 ft wide, and 10 ft deep. Unlike wmd tunnel and water tunnel arrangements, the water

in the towtank is static while the experiment model is moved through the water. A cart

spanning the width of the towtank moves on steel tracks along the top edges of the

longest dimension of the towtank. A panoramic photograph of the towtank facility is

included in Figure 3.8. Experimental apparatus are suspended from the cart, which has a

maximum speed of 5.0 ft/s. Using the dimensionless Reynolds number, this speed

corresponds to approximately 50 mph in standard air for the same size model.

The drag force on the conductors was measured using two load cells, which were

suspended from the front of the cart. In order to facilitate model changes, the load cells

were buih with the capacity to be lifted out of the water with a hinge mechanism. The two

load cells were bound together with a steel member parallel to the conductor in order to

prevent side force contamination of drag force measurement resuhs. The two load cells

are shown in their upright (for model changes) and testing positions in Figure 3.9.

17

Figure 3.8. Texas Tech Towtank Facility

(a)

Figure 3.9. Load Cells in Their Upright (a) and Testing (b) Positions

(b)

18

Both load cells had four strain gages mounted longitudinally on their surfaces. The

four strain gages on each load cell were connected in a full Wheatstone bridge

configuration and were used to measure the drag force. The strain gage configuration

filtered out the effects of temperature variation. Each full bridge circuit, also called a

channel, was connected to a strain gage balancing and amphfying device. The output of

the amplifier was connected to an analog to digital circuit board in a personal computer.

Data were recorded by the computer during experiments in the towtank.

The loadcells were calibrated in the following way. Known weights were hung at

three positions on a rigid bar in the saddle clamp blocks. This was done while the

loadceUs were in their upright, or model change, position so that the weight force

direction would be the same as the drag force direction. While each individual weight was

apphed, the voltage output of the strain gage amplifier was recorded for both loadcells.

Since the loadcells were connected with a steel member during conductor tests, the

calibration was also done with the member connecting the loadceUs. While the member

prevented side force contamination of the drag force, it also had the effect of coupling the

individual forces apphed to the loadcells. That is, a force applied to one loadcell caused a

voltage output from both loadcells. This was not a problem, however, as the calibration

procedure was designed to account for this coupling.

The calibration resuhed in a 2 x 2 calibration matrix, relating the known forces to

measured voltages. The diagonal terms were dominant, indicating that there was little

coupling between the loadcells. Voltages were recorded during tests, averaged over the

duration, and then muUiplied by the calibration matrix to calculate the drag force. The

tare drag associated with the test speed was subtracted from the total drag force, and the

drag coefficient calculated. The loadcell's force resolution was approximately 0.1 lb.

Each test was executed with the foUowing steps. Fhst, the cart's speed was

gradually increased to the test speed. Second, the cart ran for 10 ft before any data was

taken to ensure that any transients and residual vibration had been damped. Third, one

thousand data points were coUected at a samplmg frequency proportional to the speed.

Fourth, the speed was gradually reduced to rest. Fifth, the cart retumed to the front of the

19

tank at 1.0 ft/s in order to avoid causing extra turbulence in the water. The cart remained

stationary for two minutes before performing the next test, to allow any residual

turbulence in the water to abate. The range of speeds tested in the towtank was 2.0 to 5.0

ft/s, with tests done at each 0.5 ft/s. The uncertainty in the velocity of the towtank cart

was approximately ±0.1 ft/s.

3.3 Wind Tunnel Apparatus

The Low Speed Wind Tunnel Facility at Texas A&M University in CoUege

Station, Texas, was used for the wind tunnel phase of this experiment. The experimental

test section of the wind tunnel had a 10 ft by 7 ft cross section. The approximate

maximum wmd velocity used in this study was 200 mph. Drag force was measured using

a six component pyramidal balance located directly under the test section. The drag force

resolution of the facUity was ± 0.10 lbs with repeatability to within 0.10%. The average

dynamic pressure variation in the test section was no greater than ± 0.4%. The flow

angularity was no more than ± 0.25% from straight. These statistics lead one to believe

that the results from the wind tunnel are very accurate. However, the statistics hsted

above concerning the wind tunnel do not take into account the individual setup for this

study.

The conductors were mounted in the center of the wind tunnel's test section. As

described previously, the conductor samples were mounted between two shrouded end

plates. The end plates were in turn connected to the facUity's mtemal force balance,

located under the floor of the wind tunnel test section. The end plates are shown in Figure

3.10 on the next page as they were installed in the wmd tunneL

During tests at nonzero yaw angles, the internal force balance was rotated 10°,

20°, 30°, and 40°. For each yaw angle, a new tare force measurement was taken. The

end plates were also rotated so that their flat surface was parallel to the wind direction.

During yaw tests, the conductors were stUl connected to the end plates in the same manner

as normal tests, with the exception of the saddle block flexure joint. During this portion of

the testing the saddle clamp block flexure was most hnportant in order to prevent a large

20

Figure 3.10. End Plate InstaUation in the Wind Tunnel

bending moment from being applied at the ends of the conductor samples. An end plate

with a conductor installed in yaw is shown on the next page in Figure 3.11.

The wind tunnel setup for this study also circumvented all three of the error

sources studied by the National Research Council Canada (Wardlaw, Phase I). The

conductors in this study were not mounted through the walls of the wind tunnel, so

boundary layer thickness on the test section waU did not affect the measurements. The

end condition in which air leaks through gaps between wind tunnel walls and conductors

also did not exist since the conductor samples were mounted in the center of the tunnel

between end plates. This mounting also seemed to take advantage of a near-constant aft

centerhne pressure for the conductor near the center of the wind tunnel. The amount of

tension applied to the conductor in the wind tunnel was insignificant compared to actual

field conditions, being on the order of 50 lb. However, the low tension is unimportant

because drag coefficient is independent of cable tension according to Wardlaw (Phase II).

21

(a) (b)

Figure 3.11. End Plate with Conductor in Yaw, Unshrouded (a), and Shrouded (b)

3.4 Uncertainty Analysis

The uncertainty of the drag coefficient and Reynolds number will be calculated in

the standard manner. In the following equations, the uncertainty of a variable x is Ux. In

Equation 3.1, y is a function of n independent variables. Its uncertainty is represented in

Equation 3.2.

y "~ l ( X i ? ^ 2 ' ' " ' ^ n / • (3.1)

f U y H

ay v5x,

^ •

u, f

+ U. +...+ y

y J

(3.2)

Using these general expressions which define the uncertainty of a fijnction, the uncertainty

for the drag coefficient calculation can be derived. In reference to Equation 2.1, the drag

coefficient is dependent upon drag force, free stream velocity, fluid density, and frontal

area. The drag force for this study is defined by the difference between the drag of the

conductor with end plates and the drag of the end plates alone. The second quanthy, the

22

drag of the end supports alone, is the tare drag. The frontal area is defined by the product

of the conductor's outer diameter and the conductor's length. Equation 3.3 shows these

modifications of the drag coefficient from Equation 2.1.

c . = ^ D

pV/A,

F _ ^ total

tare

^PV.^LD (3.3)

In Equation 3.3, L represents the length of the conductor under test, and D represents the

outer diameter of the conductor. Other notation is famUiar, with the exception of Ftotai and

Ftare- These represent respectively the total drag of conductor and end plates, and the tare

drag of the end plates alone, respectively. As per Equation 2.2, the Reynolds number is

dependent upon the velocity, diameter, and kinematic viscosity. The uncertainties in the

drag coefficient and Reynolds number are as foUows:

UCD=CD ^total ftare

V (F, total Ftare) J +

r

V

u. ^' ru„V f^^\'• v . ; + +

V p ; \ \ . ) +

UD.^

^ D J (3.4)

URe = Re V J + ru„^ .f""^ V D y V V y

(3.5)

The first two terms of the bracketed multipher in Equation 3.4 are notable. The

tare force, Fure, is subtracted from the total force, Ftotai, and thus their uncertainties are

added in a root mean square sense. The uncertainty in the velocity measurement appears

twice due to its power of two m Equations 2.1 and 3.3. The other terms in Equations 3.4

and 3.5 are uncertainty approximations of the usual sort because the variables are not

additive and do not have exponents.

For the towtank resuhs, a complete uncertainty analysis was performed for two

cases: the smaUest diameter cable and one of the largest diameter cables. These two

analyses yield, respectively, upper and lower boimd uncertamties for the study. The

resuhs are listed with approxunately a 90% confidence level m Table 3.1 on the next page.

Uncertainty in the towtank for small diameters and low speeds is obviously a major

23

Table 3.1

Speed

2.0 2.5 3.0 3.5 4.0 4.5 5.0

. Towtank Uncertainty Analysis Resuhs 142-14100-0 (smallest diameter) Re

2380 2975 3570 4165 4760 5355 5950

%URe

5.74 4.89 4.37 4.02 3.77 3.59 3.46

CD

1.35 2.16 2.16 2.37 2.57 2.60 2.19

%UcD

146.24 202.51 41.97 32.52 19.53 13.41 18.77

100-57900-7 (larger diameter) Re

9287 11608 13930 16252 18573 20895 23217

%URe

5.05 4.06 3.41 2.95 2.60 2.34 2.13

CD

1.10 1.13 1.19 1.26 1.28 1.26 1.07

%UcD

56.66 17.31 13.57 8.99 7.30 7.51 9.21

concern. The large uncertainties at these condictions are caused because the magnitude of

the drag force measurement is of the same order as the resolution of the data acquisition

equipment.

Also a large component of uncertainty exists because two "large" numbers (total

drag and tare drag) are being subtracted to calculate a "small" number (conductor drag

alone). Another component of the uncertamty that is magnified at low speed is the

uncertainty in the speed of the cart. At low speeds the 0.1 fl/s uncertainty is a larger

portion of the overall speed, and thus uncertainty is mcreased. It should also be noted that

constant values for water density and kinematic viscosity were used throughout the

calculations, due to the mability in the towtank facihty to measure these quantities for each

test. The density used was 1.937 slug/ft with an uncertainty of 0.005 slug/ft^ The

kmematic viscosity used was 2.5 x 10" ft^/s with an uncertainty of 1 x 10" ft^/s.

The towtank data show a consistent trend in the range of Reynolds numbers

tested. The trend is "bell" shaped and may be observed in figures in the next chapter.

This "bell" shaped trend is not thought to accurately represent the drag coefficient or any

physical phenomenon of the drag on a conductor. The trend is thought to occur due to

some systematic uncertainty in the velocity of the towtank cart, and the towtank data

should be considered only as a luniting case for the drag coefficient. In other words, it is

thought that the true drag coefficient lies somewhere between the upper and lower bounds

ofthe "bell" trend.

24

Uncertainty analysis was also completed for the same two conductors in the wind

tunnel. The results of this analysis are shown with approximately a 95% confidence level

on the next page in Table 3.2. As with the towtank, there is a concern with large

uncertainties at low speeds. This uncertainty is due to the small forces involved at low

speed. The difference of two "large" measurements to obtain a "small" resuh is also

applicable for the wind tuimel uncertainty. The uncertainties in dynamic pressure and drag

force measurement described in the wind tunnel apparatus section were used to calculate

the drag coefficient uncertainty. These uncertainties are ± 0.4% for the dynamic pressure

and ±0.1 lb for the drag force.

Table 3.2. Wind Tunnel Uncertainty Analysis Resuhs 142-14100-0 (smallest diameter)

V wind, free stream

29.29 43.59 58.08 72.61 87.04 101.78 116.54 131.04 145.91 160.29 174.71 189.38 203.97 218.56

%UcD

471.41 428.56 83.23 78.18 101.05 106.37 56.63 27.86 44.55 31.13 34.60 22.26 15.18 15.22

100-57900-7 (larger diameter)

V windj free stream

29.4 43.64 57.91 72.52 87.03 101.61 116.28 130.66 145.52 159.85 174.43 188.96 203.38 218.02

%UcD

50.51 88.39 196.42 543.93 404.06 211.08 64.29 28.64 43.79 23.31 17.18 16.26 11.95 10.71

25

CHAPTER 4

RESULTS AND DISCUSSION

4.1 Conductors in Normal Crossflow

The drag coefficient data resulting from the experimental force measurement

results and subsequent calculations show some interesting trends. The conductors may be

divided into two general categories for study: trapezoidal wire conductors and round wire

conductors. Trapezoidal and round wire conductors are discussed below at length.

The conductors with trapezoidal stranding have serial numbers: 100-58600-3, 100-

58604-6, and 100-58650-0. The last of these three trapezoidal strand conductors, 100-

58650-0, was manufactured with dimples at regular intervals on each strand. The dimples

were most likely added in hopes of decreasing the drag coefficient in the same manner as

dimpled golf balls do. By increasing the roughness ofthe surface shghtiy, the

manufacturer hoped to cause an earlier critical regime transition for the conductor and

thus reduce its drag coefficient and drag force at high wind speeds. Noting earlier

discussion in Chapter 2, this is a reasonable notion for conductor design. Drag

coefficients of dimpled and undimpled trapezoidal conductors should differ at equal Avind

speed and Reynolds number, with dimpled models having lower drag due to slightly

increased roughness.

Plots ofthe drag coefficient as a function of Reynolds number for the three

trapezoidal strand conductors are included on the next three pages in Figures 4.1, 4.2, and

4.3. Inspection of Figures 4.1 and 4.2 yields some interesting information about

conductors 100-58600-3 and 100-58604-6. These two conductors behave similarly in

most respects. The critical regime transition appears to begin at a Reynolds number of

approximately 3.5 x lO' for both conductors. The minimum drag coefficient at the

juncture ofthe critical and supercritical regimes appears to be approximately 0.8 at a

Reynolds number of approximately 9 x lO' . Also, the maximum drag coefficients of these

conductors appear to have a near constant value of 1.2 in the subcritical region.

26

—

1

! 1

o T

owta

nk

o W

ind

Tunn

el

• : 1

.. ! \

: , i i

1

j i

'<

1

1

!

1

1

J

i

1

^

o

o

o

oO° o o o

/^ \J

o o

I

o

]

i

&

o o o >

-^--^

,

i

1

i

i :

1

1

1

i : : ,

+

I

o

s " '"' o o

o c o

+

1/5

> Q

+

^3

27

- T

—

1 1

] o

Tow

tank

o

Win

d Tu

nnel

i

1 1

1

i

! i ! •

1 '

^

1

, ' I

I 1

i

"i '— i I 1

i

— 1 — \ —

!

- _- ,

:

<

o

o

o

. < ? o o o o

o

>

1 o 0 < o

&

• ! ' 1

i i i : ! 1

' i

I

1 —

i

:

! 1

-

\

+

I

W ^ ^ O

o I—I

U I

o o 3

-O c o

c2

3 U i

> Q

u

3

^3

28

o T

owta

nk

o W

ind

Tunn

el

i 1

— - - +

1

•

- - - ' - -

• 1

\

i

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1 , j

1 1 ' ! 1

1

i

— \—' i

i

i i

'— - L, y

<

o

o

o » o o o

o o

o 0

^ I 0

o o >

0

j..-—

--I 1 ! ; ! ; !

1

1

i

' ;

J

! j

i I ' :

VO

+

o IT) VO + 00 W «^ ^ o

o

o 3

T3 C o

3 t/i

>

W <u

^3 29

Now consider Figure 4.3, in which the drag coefficient is plotted as a function of

Reynolds number for the dimpled trapezoidal conductor, 100-58650-0. The critical

regime for this conductor appears to start at a much lower Reynolds number than that of

the other trapezoidal conductors: approximately 2 x 10" . The drag coefficient also does

not appear to drop as low in the critical regime as the undimpled conductors, reaching a

minimum value of approximately 0.85 at a Reynolds number of approximately 6 x 10"*.

Comparison ofthe two trapezoidal undimpled conductors to the trapezoidal

dimpled conductor is particularly interesting. The critical regime ofthe dimpled conductor

appears to occur at a lower Reynolds number than the undimpled conductors. This is to

be expected because the dimpled conductor is slightly rougher than the undimpled

conductors, and the literature suggests that h will have an earlier transhion. Another

interesting comparison is the minimum drag coefficient in the critical/supercritical regime.

According to the literature, the drag coefficient ofthe rougher conductor wiU not be

reduced in the critical regime as much as the smoother conductors. This is verified by

inspection ofthe minimum drag coefficients. The rougher, dimpled conductor's drag

coefficient appears to drop to approximately 0.9, while the smoother, undimpled

conductors' drag appears to drop to approximately 0.8. This concludes discussion ofthe

trapezoidal conductors.

The remaining conductors have round wire stranding. Their serial numbers are:

100-58104-4, 100-61200-4, 142-15026-2, 100-57900-7, 100-58200-8, 100-58300-4, and

142-14100-0. Plots ofthe drag coefficient as a function of Reynolds number are shown

for these conductors in Figures 4.4 through 4.10. All of these conductors, whh the

exception of 142-14100-0, have either steel or aluminum cores with aluminum outside

strands. Conductor 142-14100-0 is a seven-strand galvanized steel conductor. The round

conductors can be divided into two groups for discussion. The first of these groups

contains round wire conductors whose drag coefficient does not appear to drop below a

value of 1.0. This group includes conductors 100-61200-4, 142-15026-2, and 142-14100-

0. The second group consists of round wire conductors whose drag coefficient does

30

o

c c 3

-o c

c

O

H

o o

- I — I -

-Q-

+

+ Ui

I o o

VO I o o

o 3 -o c o U u.

c< Vi 3 Vi u. >

U ' ^ '^'

3 00

+

^3 31

c 3

c

O H O

4

O—

O

UJ

+

I VO

o I

r f

3 -o c o

cS

3 c« u.

>

U

+ rr UJ <y

3 00

+ UJ

Q'

32

c c 3 H c

C BJ

o

o

T ~0~ o o

o o

! o

o

-o-

o o

, S + rr

O 3

T3 C

o

a: Vi

3 O)

u.

>

c ^, rr UJ <u — u.

3 00

UJ

^3

33

1

o T

owta

nk

o W

ind

Tunn

el

r 1''' 1

'

•

1

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OO

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1 i /i

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) 1 •

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•

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1

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j 1 1

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1 j T

j

-

<

o o o o o o

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>

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t

- - - - - - -

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I

: I

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- -

-

VO + UJ

+

I o o

UJ «A)

"~ o o

o 3 C o

U a Vi 3 Vi

>

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+ rr UJ CD '—' u-

3 00

+

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34

— 1 1 i- , 1

1 h

rtl o

Win

d Tu

nn(

o T

owta

nk

1 I

i i

1 I

1 1 \

, I i

1 :

o o o O^ o o o

i t , o • 1 ! • i , .

o

o o o o

o s

\J

i ; ]

" 

<

'

j

,

' '

i

'•

— -

! ;

i j i

1 :

L _ J ^ \ , :

i

-

1 ' ' '

-

VO + UJ

i n + UJ

00 I o o

<N OO

I o o

o 3

-o a o

fl) WH

Vi

3 Vi 0)

+ UJ

00 '^'

3 00

UH

+ UJ

^3

35

—

I

1 r o

Tow

tank

o

Win

d Tu

nnel

1

i

- - - -

- - - - - -

oooo

L .

.

V O ' '

" ' '" • - - - ' -- - - - - v r— - -

1

i 1

. j

i

:

i

1 i i

i ;

i 1

H

i' i

J

6

<t <•

<

o

o o o o

o

1

1

1

I , , , 1 ,

:

i : • '.

i i ,

1 i

i

__J

;

i

I

1 ;

1

i

, . - . 1

VO + UJ

«/) + w )

(1> p<

r f

+ w '—'

rr <-)

00 t n

1

o o — UH

o • » - »

CJ 3

-o c o U

V-i

o <^

 Q

u OS Tt a> u. 3 00

b

+ UJ

^3

36

UJ

c

O

rr I

o o

+ 00 UJ '^' -' o

o

^

o o o

o 3 c o

c2

Vi

3

> U

T 2 UJ rr — (U

u, 3 00

1 + " UJ

^3

37

appear to drop below a value of 1.0. The second group includes conductors 100-57900-7,

100-58200-8, 100-58104-4, and 100-58300-4.

The plots of drag coefficient as a function of Reynolds number for the first group

described above, containing conductors 100-61200-4, 142-15026-2 and 142-14100-0, are

shown in Figures 4.4 through 4.6. In Figure 4.4, conductor 100-61200-4 appears to have

a drag coefficient of approximately 1.2 in the subcritical critical regime with a critical

Reynolds number of approximately 2 x 10" . The second conductor in the first group is

conductor 142-15026-2. In Figure 4.5, hs drag coefficient appears to have a subcritical

value of approximately 1.3 and a critical Reynolds number of approximately 1.5 x 10" .

The last conductor in the first group, shown in Figure 4.6, is 142-14100-0. This is the

smallest conductor tested. It appears that this conductor's roughness is so large that h has

no transition and does not behave as a circular cylinder.

The most conspicuous trend observed in all three ofthe plots for conductors in the

first group is that the drag coefficient decreases slightly with Reynolds number. This trend

is not characteristic ofthe literature concerning drag on circular cyhnders. The literature

states that the drag coefficient should begin to rise at the junction ofthe critical and

subcritical regimes. There are no other trends in the data for these conductors that can be

stated conclusively. This concludes discussion ofthe first group of round wire

conductors.

The drag coefficients ofthe second group of round wire conductors to be studied

appear to drop below a value of 1.0 in the critical regime. Figures 4.7 through 4.10 show

plots ofthe drag coefficient as a function of Reynolds number for conductors 100-57900-

7, 100-58200-8, 100-58104-4, and 100-58300-4, respectively. Note that wind tunnel data

for conductor 100-58300-4 was not coUected and therefore is not included in Figure 4.10.

Conductor 100-57900-7 appears to have a subcritical regime drag coefficient of 0.95 and

a critical Reynolds number of approximately 2.5 x 10" . Conductor 100-58200-8 appears

to have a drag coefficient in the subcritical regime of 0.9 and a critical Reynolds number of

approximately 3.0 x 10" . The third conductor, 100-58104-4, appears to have a subcritical

drag coefficient of 1.2 and a critical Reynolds number of approximately 1.9 x 10" . The

38

drag coefficient for this conductor appears to reach a minimum of 1.0 in the critical

regime. The last conductor in the second group is 100-58300-4,

The conductors in the second group exhibh an interesting trend. The trend is that,

unlike the first group of round wire conductors, the drag coefficients of conductors 100-

57900-7 and 100-58200-8 increase slightly with Reynolds number in the supercritical

range. Conductor 100-58104-4 appears to behave similarly to the first group of

conductors in that ks drag coefficient decreases slightly with Reynolds number in the

supercritical regime. The behavior of conductor 100-58300-4 in the supercritical regime

is unknown due to the absence of wind tunnel data.

A comparison of trapezoidal strand conductors to round strand conductors is the

next logical step m the study ofthe conductors. Figure 4.11 on the next page is a

compilation of two representative conductors, one trapezoidal, the other round. Although

neither representative exactly portrays all the characteristics of its classification, some

trends may be observed. Generally, the drag coefficient of trapezoidally stranded

conductors drops more than round stranded models in the critical region. Not well

depicted in Figure 4.11 is another trend suggesting that the critical Reynolds numbers of

trapezoidally stranded conductors are larger than those of round stranded conductors.

The last tests in normal crossflow were done on a smooth cyhnder in order to

verify the accuracy of drag coefficient data taken in the wind tunnel and towtank. Results

from tests ofthe conductors may be verified by comparing smooth cylinder resuhs from

the wind tunnel and towtank to the classical values reported in the literature. Figure 4.12

is a plot ofthe drag coefficient as a fiinction of Reynolds number for a smooth cylinder.

The drag coefficient data from the wmd tunnel is relatively constant throughout the range

of Reynolds numbers tested. The towtank data appears to have much more drift. The

classical drag coefficient value for subcritical flow (10"* < Re < 3 x 10 ) around a smooth

circular cyhnder is approximately 1.2. The resuhs ofthe testing on a smooth cyhnder

verify that the wind tunnel resuhs are accurate because the wind tunnel resuhs show a

constant value of 1.2 for the drag coefficient ofthe smooth cylinder. Note that the critical

39

^

-B-T

rape

zoid

al S

tran

d R

epre

sent

ativ

e (1

00-5

8650

-0)

-&-R

ound

Str

and

Rep

rese

ntat

ive

(100

-581

04-4

)

i I

i

j t

1 1 1 1 1 !

• 1

1 1

i

I i i

1 ^ • ,'

1 —

1

( () <) \ 1

() (

<

<

\ LJ 1

m

u i • !

/ f i 1 1

1 ^

U l

V

i

i

1

1

-

!

1 i

^ • - J

i

: 1 •

1 i i

1.

i

VO +

Vi

o 3

o

+ " UJ S " ^ CO

'o N 0)

a-cd

c u cd

3 O

a: o c o Vi

cd

" a, + S UJ o

u 3 00

tin

+ UJ

^3

40

1 r -

0 T

owta

nk

o W

ind

Tunn

el

—1

i •'

1

1 i 1 '

1 ;

! 1 1

i

i

I

: <> ' 0

1 ; V ! ' • ' ; 1

'

\ i

i

i

i

- - -

- —

o o o

o

oO o o

o

o

o

i 1 1 ' I

i : • 1

i

\

>^: : i 1

,

i '

,

-

1 1 1 . . \

VO + UJ

IT) + UJ

Pi

-o c * > . U

o o S cd

c2 

CNJ

''^

+ UJ '""'

-^ <u 3 00

tin

+ UJ

^3

41

Reynolds number was not reached in the tests performed in the wind tunnel. The towtank

data appear to hover near the correct value, but are inconsistent.

When considering the plots in Figures 4.1 through 4.12, the uncertainty ofthe drag

coefficient calculation should be taken into account. In several ofthe figures, outlying

data points are obvious. The outhers have been plotted for completeness, but should be

disregarded as means for serious design work.

4.2 Conductors in Yawed Flow

Four conductors were tested in the wind tunnel in yawed flow conditions. They

are conductors 100-61200-4, 100-58650-0, 100-58600-3, and 100-58200-8. It is

expected from the theory discussed in Chapter 2 that the drag coefficients measured m

yawed flow conditions will be equal to the drag coefficient for normal flow muhiplied by

the square ofthe cosine ofthe yaw angle. Plots ofthe drag coefficient measured in yawed

flow for the four conductors tested are shown in Figures 4.13 through 4.16. Plots ofthe

yawed flow drag coefficient normalized by cos \\f are shown in Figures 4.17 through 4.20.

The most important resuh of Figures 4.13 through 4.16 is that the drag coefficient

is strongly affected by the angle ofthe wind (yaw angle). Consideration of this factor in

calculating the drag force on a conductor is critical. As yaw angle increases for each

conductor tested, the overall drag force is reduced by an amount proportional to the

wind's normal velocity component, squared. The drag coefficient is similarly reduced.

Figures 4.17 through 4.20 serve mostly as a check on previous resuhs obtained for

conductors in normal flow. The overlapping ofthe points for each yaw angle tested

shows that the cosine squared phenomenon for the drag in yawed flow conditions holds

for conductors as well as smooth cyhnders. The overlap also verifies the accuracy ofthe

wind tunnel measurements.

42

o o o o o o o o o o ^1 (N m - ^ D O < O X

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!

j : '

j i • -

i i 1 1 '

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i • • i

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>

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m

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tin

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MBX 'Q^

43

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47

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CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

The results of this study lead to several conclusions. The obvious first conclusion

is that trapezoidal wire conductors can achieve lower drag coefficients than round wire

conductors. However, this lower drag coefficient is not realized until a higher critical

Reynolds number than that of round wire conductors. This has unphcations for the design

of structures that carry conductors. Trapezoidal conductors may not be practical for an

area in which the wind does not produce flow with a Reynolds number high enough to

reach the critical point. In these areas, round wire conductors may be more practical and

economical.

Another conclusion that can be reached from the hterature surveyed is that the

results of quahty wind tunnel studies ofthe drag coefficient are correct for the wind

condhions in which they are obtained. In the field, the exact wind velocity, direction, and

turbulence are difficult to characterize over an entire span of conductor. The marked

difference between field- and laboratory-collected data has not been resolved by this study

due to its limited scope. In the ideal conditions ofthe laboratory, drag coefficient resuhs

acquired are correct. However, this is not to say that field results are incorrect. Both

types of studies produce correct, yet different results for different conditions. The difficuh

problem faced by engineers in this field is apphcation of laboratory data to design

problems m the field, where the same ideal laboratory condhions do not exist.

Resuhs ofthe wind tunnel tests in this study are accurate. This is confirmed best

by the test ofthe smooth cyhnder. The drag coefficient ofthe smooth cylinder tested in

the wind tunnel matched with the data quoted in White's reference The resuhs ofthe

towtank tests are not highly accurate. The uncertainties of mdividual data points ranged

from approximately 5%) to 200%. The "bell" shape of aU ofthe data acquired in the

towtank is thought to be the resuh of a systematic uncertamty in the velocity, and not the

true representation of physical phenomena. OveraU, the resuhs are quite good, and h is

hoped that the resuhs obtained in this study can be used to design safer transmission lines.

51

Two recommendations for further study are in order. The first recommendation is

to conduct a field study in which the wind's characteristics are measured at several points

along a span. If wind speed, direction, and turbulence are measured at regular intervals

along a span of conductor in the field, a reconcihation between field and laboratory resuhs

may be forthcoming. The second recommendation is to repeat the towtank tests on

conductors. Several steps could be taken to improve the accuracy ofthe resuhs. The use

of a professionally buih load ceU with high accuracy could decrease the uncertainty.

Flatter end plates would reduce the tare drag appreciably and thus decrease uncertainty.

Finally, a longer waiting time between consecutive runs in the towtank would allow the

water to settle and avoid the precision problem associated whh residual turbulence in the

water.

52

SOURCES

Krishnasamy, S. G. "Response of Smooth Body, Trapezoidal Wire Compact Conductors to Wind Loading." Ontario Hydro Research Division Report No. B91-44-K, Progress Report No. 1, File 825.151. October 10, 1991.

Potter, J. Lehh, and David E. Boylan. "Aerodynamic Force Coefficients for the Prediction of Wind Loads on Electrical Transmission Lines." Final Report of Sverdrup Corporation to The University of Oklahoma and the Electric Power Research Institute. March, 1981.

Shan, L. "Measurement of Electrical Conductor Drag Coefficients in a 'Free-Air-Wind-Tunnel.'" Draft Report of Sverdrup Technology, Inc., of Haslet, Texas, to the Transmission Line Mechanical Research Center, Electric Power Research Institute. October, 1991.

Wardlaw, R. L. "Determination ofthe Aerodynamic Drag of Overhead Electric Power Cables : Phase I." Report from National Research CouncU Canada to the Aluminum Company of America, LTR - LA - 313, ALCOA Purchase Order #MB 213520 MB, Phase I. April, 1990.

Wardlaw, R. L. "Determination ofthe Aerodynamic Drag of Overhead Electric Power Cables : Phase II." Report from National Research CouncU Canada to the Aluminum Company of America, LTR - LA - 314, ALCOA Purchase Order #MB 213520 MB, Phase H. April, 1990.

White, Frank M. Fluid Mechanics. 3rd ed. New York: McGraw-HUl, 1994.

53

APPENDIX A

TABULATED DATA

54

Table A. 1

Re

7995

15860

23380

31330

39680

47250

55150

63090

70940

78830

86710

94460

102300

110000

117900

125600

133300

141000

148600

156000

Re

10820

13525

16230

18935

21640

24345

27050

Smooth Cylinder Data

Smooth, 0'

Wind Tunnel: D =

CD 1 V 0.644

1.151

1.191

1.180

1.216

1.207

1.198

1.202

1.215

1.204

1.194

1.198

1.199

1.200

1.199

1.199

1.202

1.205

1.200

1.205

Towta

CD

1.19 1.07

1.12

1.22

1.07

1.04

0.85

10.330

20.393

30.027

40.186

50.925

60.661

70.834

81.164

91.336

101.516

111.702

121.820

132.027

142.139

152.352

162.559

172.732

183.075

193.445

203.680

n k : D = l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0.0833 ft

q

0.27

1.06

2.31

4.13

6.63

9.41

12.83

16.83

21.30

26.31

31.85

37.86

44.45

51.48

59.13

67.27

75.90

85.18

94.97

105.10

.623 in

q 3.87

6.05

8.72

11.86

15.50

19.61

24.21

FD/L

0.014

0.102

0.229

0.406

0.671

0.946

1.280

1.685

2.155

2.638

3.169

3.779

4.438

5.146

5.904

6.720

7.599

8.547

9.496

10.545

FD/L

0.62

0.87

1.32

1.95

2.25

2.76

2.80

Note : V is free stream velocity (miles / hour)

q is dynamic pressure = 0.5 * p ' V (lb / ft ) FD/L is drag force per length conductor (lb / ft)

55

Table A.2

Re

7785

16830

25480

33820

42370

50700

59400

67530

76080

84360

92780

101200

109500

117900

126200

134600

142900

151200

159300

167500

Re

7180

8975

10770

12565

14360

16155

17950

Conductor 100-58600-3 Data

10(

Wind Tui

CD

1.494

1.300

1.180

1.150

1.136

1.053

0.948

0.873

0.794

0.760

0.757

0.760

0.767

0.776

0.789

0.800

0.816

0.823

0.836

0.851

Towta

CD

1.17 1.36

1.47

1.53

1.62

1.48

1.25

)-58600-3,

nnel: D =

V

9.355

20.127

30.327

40.159

50.277

60.150

70.520

80.284

90.586

100.541

110.625

120.716

130.814

140.905

150.989

161.161

171.239

181.377

191.489

201.757

n k : D = l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.0875 ft

q

0.23

1.05

2.39

4.20

6.58

9.42

12.94

16.75

21.31

26.24

31.76

37.80

44.37

51.46

59.06

67.26

75.91

85.10

94.74

105.06

.077 in

q

3.87

6.05

8.72

11.86

15.50

19.61

24.21

I

FD/L

0.030

0.119

0.247

0.423

0.654

0.868

1.073

1.279

1.480

1.745

2.103

2.513

2.977

3.495

4.077

4.706

5.421

6.131

6.931

7.822

FD/L

0.41

0.74

1.15

1.63

2.26

2.61

2.71

56

Table A.3. Conductor 100-58604-6 Data

Re

21460

32660

43170

54050

64370

75480

86020

97010

107600

118400

129000

139700

150300

160900

Re

9100

11375

13650

15925

18200

20475

22750

10(

Wind Tu]

CD

1.125

1.205

1.159

1.117

1.001

0.875

0.823

0.815

0.818

0.833

0.849

0.861

0.868

0.874

Towta

CD

1.07

1.29

1.33

1.36

1.36

1.28

1.08

)-58604-6,

nnel: D =

V

20.175

30.655

40.459

50.639

60.327

70.786

80.700

91.070

100.936

111.150

121.214

131.284

141.436

151.541

n k : D = l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.1125 ft

q

1.05

2.43

4.23

6.63

9.40

12.94

16.82

21.41

26.31

31.89

37.91

44.47

51.58

59.18

.365 in

q 3.87

6.05

8.72

11.86

15.50

19.61

24.21

FD/L

0.133

0.329

0.552

0.833

1.058

1.273

1.558

1.963

2.422

2.988

3.619

4.307

5.036

5.818

FD/L

0.47

0.89

1.32

1.84

2.39

2.85

2.98

57

Table A.4

Re

8339

16850

24890

33270

41550

49830

58130

66430

74750

83020

91190

99350

107600

115900

124100

132300

140500

148500

156400

164200

Re

7047

8808

10570

12332

14093

15855

17617


100-58650-0,

Wind Tunnel : D =

CD 1 V 0.827

1.250

1.222

1.078

0.980

0.894

0.844

0.861

0.868

0.874

0.883

0.896

0.904

0.912

0.923

0.932

0.946

0.954

0.960

0.973

Towta

CD

1.61 1.53

1.60

1.69

1.67

1.52

1.26

10.200

20.530

30.211

40.343

50.373

60.443

70.541

80.720

90.968

101.059

111.027

121.084

131.195

141.395

151.493

161.645

171.893

182.011

192.239

202.582

n k : D = l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.0867 ft

q 0.27

1.09

2.36

4.21

6.56

9.44

12.85

16.82

21.34

26.33

31.78

37.78

44.33

51.48

59.07

67.22

75.98

85.10

94.78

105.02

.057 in

q

3.87

6.05

8.72

11.86

15.50

19.61

24.21

I

FD/L

0.019

0.118

0.250

0.394

0.557

0.732

0.940

1.256

1.607

1.994

2.434

2.936

3.476

4.069

4.728

5.432

6.232

7.035

7.891

8.855

FD/L

0.55

0.82

1.23

1.77

2.28

2.62

2.68

58

Table A. 5.

Re 1 14770

21970

29180

36560

43870

51130

58540

65790

73090

80370

87560

94870

102100

109200

Re

5873

7342

8810

10278

11747

13215

14683


100-61200-4,

Wind Tunnel: D =

CD 1 V 1.142

1.166

1.103

1.090

1.056

1.049

1.052

1.052

1.050

1.054

1.046

1.047

1.046

1.046

Towta

CD

1.36

1.56

1.60

1.68

1.76

1.61

1.36

20.175

30.041

39.955

50.039

60.000

69.989

80.120

90.075

100.111

110.080

120.027

130.132

140.168

150.109

nk:D = 0

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.0750 ft

q

1.07

2.37

4.19

6.57

9.46

12.86

16.86

21.30

26.30

31.81

37.80

44.41

51.49

59.02

.881 in

q

3.87

6.05

8.72

11.86

15.50

19.61

24.21

L

FD/L

0.092

0.207

0.346

0.537

0.749

1.011

1.330

1.680

2.072

2.515

2.965

3.489

4.038

4.630

FD/L

0.39

0.69

1.02

1.46

2.00

2.32

2.42

59

Table A.6

Re 1 9986

15020

19950

25030

29990

35180

40080

45080

50100

55060

60030

65000

69980

74840

Re

4080

5100

6120

7140

8160

9180

10200


142-15026-2,

Wind Tunnel : D =

CD 1 V 1.191

1.284

1.222

1.213

1.203

1.159

1.141

1.123

1.121

1.107

1.097

1.091

1.082

1.081

Towta

CD

1.34

1.72

1.68

1.88

1.88

1.77

1.51

20.114

29.939

39.457

49.343

59.168

69.464

79.180

89.107

99.055

108.968

118.950

128.898

138.880

148.616

nk:D = 0

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.0500 ft

q

1.07

2.39

4.16

6.52

9.37

12.91

16.77

21.23

26.24

31.73

37.78

44.35

51.47

58.91

.612 in

q 3.87

6.05

8.72

11.86

15.50

19.61

24.21

L

FD/L

0.064

0.153

0.254

0.395

0.564

0.748

0.957

1.192

1.470

1.756

2.072

2.418

2.783

3.184

FD/I.

0.26

0.53

0.75

1.14

1.49

1.77

1.86

60

Table A.7.

Re J 6409

9645

12910

16160

19370

22640

25900

29090

32370

35520

38700

41920

45110

48300

Re

2380

2975

3570

4165

4760

5355

5950

Concuctor 142-14100-0 Data

142-14100-0,

Wind Tunnel: D =

CD 1 V 1.260

1.184

1.169

1.166

1.180

1.158

1.148

1.165

1.144

1.155

1.148

1.153

1.153

1.144

Towta

CD

1.35

2.16

2.16

2.37

2.57

2.60

2.19

19.970

29.720

39.600

49.507

59.345

69.395

79.459

89.345

99.484

109.289

119.120

129.123

139.070

149.018

nk:D = 0

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.0325 ft

q

1.05

2.34

4.17

6.53

9.38

12.82

16.80

21.23

26.31

31.73

37.68

44.26

51.32

58.90

.357 in

q 3.87

6.05

8.72

11.86

15.50

19.61

24.21

F D L

0.043

0.090

0.158

0.248

0.360

0.482

0.627

0.804

0.979

1.191

1.406

1.659

1.923

2.190

FD/L

0.16

0.39

0.56

0.84

1.18

1.52

1.57

61

Table A. 8.

Re 1 23480

35220

46890

58880

70740

82590

94470

106100

118100

129600

141400

153100

164600

176300

Re

9287

11608

13930

16252

18573

20895

23217


100-57900-7,

Wind Tunnel: D =

CD 1 V 1 0.951

0.945

0.998

1.036

1.030

1.034

1.040

1.048

1.045

1.046

1.049

1.056

1.053

1.054

Towta

CD

1.10 1.13 1.19 1.26

1.28

1.26

1.07

20.045

29.755

39.484

49.445

59.339

69.280

79.282

89.086

99.218

108.989

118.930

128.836

138.668

148.650

n k : D = l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

0.1167ft

q 1.07

2.37

4.18

6.57

9.47

12.90

16.89

21.33

26.44

31.89

37.97

44.55

51.58

59.24

.393 in

q 3.87

6.05

8.72

11.86

15.50

19.61

24.21

I

FD/L

0.119

0.261

0.487

0.794

1.138

1.557

2.050

2.609

3.224

3.894

4.647

5.488

6.340

7.288

FD/L

0.49

0.80

1.21

1.73

2.30

2.87

3.00

62

Table A.9. Conductor 100-58200-8 Data

Re

9926

19940

29410

39130

48900

58550

68400

78280

87940

97590

107300

116800

126600

136400

146100

155500

165300

174500

184100

193100

Re

7980

9975

11970

13965

15960

17955

19950

101

Wind Tu

CD

1.000

0.903

0.914

0.864

0.934

0.966

0.977

0.992

1.006

1.004

1.005

1.016

1.018

1.018

1.021

1.025

1.030

1.030

1.035

1.040

Towta

CD

1.09

1.29

1.33

1.39

1.46

1.34

1.04

3-58200-8

nnel: D =

V

10.268

20.509

30.143

40.084

50.059

59.966

70.057

80.257

90.293

100.289

110.298

120.232

130.364

140.516

150.620

160.548

170.768

180.750

191.141

201.136

nk:D=l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

,0°

0.1000 ft

q 1 FDL 0.27

1.10

2.38

4.21

6.57

9.43

12.87

16.88

21.35

26.32

31.83

37.80

44.43

51.60

59.26

67.27

76.09

85.13

95.08

105.10

.197 in

q

3.87

6.05

8.72

11.86

15.50

19.61

24.21

0.027

0.099

0.218

0.364

0.614

0.911

1.258

1.675

2.149

2.642

3.200

3.842

4.522

5.250

6.047

6.898

7.837

8.768

9.838

10.935

FDL

0.42

0.78

1.15

1.65

2.26

2.61

2.52

63

Table A. 10. Conductor 100-58104-4 Data 100-58104-4, 0°

Wind Tunnel : D = 0.0933 ft

Re 1 CD 1 V 1 q 17960

27150

36040

44640

53570

62790

71480

80410

89350

98260

107100

116000

124900

133600

Re

7460

9325

11190

13055

14920

16785

18650

1.212

1.039

0.998

1.021

1.014

1.025

1.036

1.041

1.037

1.037

1.036

1.036

1.033

1.030

Towta

CD

1.54

1.58

1.54

1.62

1.56

1.49

1.25

20.332

30.655

40.636

50.264

60.361

70.814

80.639

90.743

100.834

110.932

120.941

131.093

141.327

151.377

nk:D=l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

1.07

2.43

4.27

6.54

9.43

12.97

16.82

21.30

26.30

31.82

37.81

44.41

51.57

59.14

.119m

q 3.87

6.05

8.72

11.86

15.50

19.61

24.21

FD/L

0.121

0.235

0.397

0.623

0.892

1.240

1.626

2.069

2.544

3.079

3.654

4.292

4.969

5.682

FD/L

0.55

0.89

1.25

1.79

2.25

2.72

2.83

Table A l l . Conductor 100-58300-4 Data

Re

9540

11925

14310

16695

19080

21465

23850

10(

Towta

CD

1.06

1.30

1.21

1.25

1.19

1.08

0.89

)-58300-4,

nk:D=l

V

1.36

1.70

2.05

2.39

2.73

3.07

3.41

0°

431 in

q 1 FD/L 3.87

6.05

8.72

11.86

15.50

19.61

24.21

0.49

0.94

1.26

1.77

2.20

2.53

2.57

64

Table A. 12. Conductor 100-61200-4 Yaw Data

0°

Ke [ CD,yaw

14770

21970

29180

36560

43870

51130

58540

65790

73090

80370

87560

94870

102100

109200

1.142

1.166

1.103

1.090

1.056

1.049

1.052

1.052

1.050

1.054

1.046

1.047

1.046

1.046

10°

Rp 1 C

7054

14440

21490

28560

35640

42880

50230

57100

64280

71420

78550

85580

92650

99900

106800

113900

121000

128000

135000

141800

1.052

1.124

1.070

1.050

1.057

1.060

1.041

1.053

1.054

1.053

1.065

1.061

1.055

1.058

1.056

1.057

1.061

1.063

1.064

1.061

100-61200-4

20°

Re 1 CDJ^HW

7214

13910

21080

27630

34290

41190

48270

55030

61810

68820

75740

82520

89410

96140

103000

109800

116600

123300

130000

136400

1.606

1.118

1.051

1.011

1.009

1.021

1.007

1.007

1.012

1.008

1.006

1.006

0.999

0.996

0.993

0.990

0.988

0.986

0.987

0.982

30

Re

7536

13640

20620

33960

47510

54390

61000

67890

74720

81450

88040

94810

101700

108300

115000

121400

127900

134300

^D.yau

0.649

0.957

0.844

0.837

0.834

0.845

0.842

0.836

0.837

0.839

0.824

0.818

0.815

0.816

0.823

0.814

0.802

0.799

40

Re

7995

13860

20910

27890

34680

41550

48570

55270

62130

69170

75930

82880

89750

96660

103300

110300

117100

123900

130400

137100

^D.vaw

1.263

0.803

0.709

0.629

0.644

0.644

0.647

0.630

0.649

0.634

0.646

0.628

0.621

0.623

0.618

0.617

0.604

0.608

0.616

0.606

65


0'

Re

8339

16850

24890

33270

41550

49830

58130

66430

74750

83020

91190

99350

107600

115900

124100

132300

140500

148500

156400

164200

3

^D,yaw

0.827

1.250

1.222

1.078

0.980

0.894

0.844

0.861

0.868

0.874

0.883

0.896

0.904

0.912

0.923

0.932

0.946

0.954

0.960

0.973

10

Re

9092

17090

25420

33910

42220

50930

59330

67680

76160

84490

92880

101200

109500

118000

126300

134700

142900

151100

159200

167100

^D.yaw

0.716

1.151

1.117

1.118

0.997

0.907

0.861

0.876

0.884

0.893

0.913

0.915

0.926

0.935

0.941

0.945

0.953

0.957

0.963

0.973

100-58

20

Re

16800

25510

34150

42590

51220

59920

68370

76970

85420

93820

102200

110800

119100

127700

136100

144500

152800

161100

168900

650-0

°

^D,yaw

0.721

0.946

0.951

0.872

0.835

0.793

0.806

0.819

0.830

0.834

0.848

0.851

0.857

0.861

0.868

0.869

0.868

0.879

0.888

30

Re

8717

16430

24380

32290

40170

48370

56800

64560

72610

80640

88810

96820

104700

112700

120600

128800

136700

144400

152200

159900

^D,yaw

0.804

0.978

0.921

0.841

0.769

0.726

0.704

0.704

0.702

0.710

0.719

0.727

0.720

0.724

0.729

0.738

0.740

0.745

0.742

0.750

4C

Re

7898

16320

24540

32830

40940

49500

57680

65860

74090

82090

90360

98450

106700

114800

123000

131100

139100

147000

155000

162900

^D.vau

1.102

0.679

0.699

0.633

0.580

0.518

0.502

0.508

0.513

0.502

0.524

0.517

0.516

0.518

0.522

0.521

0.521

0.530

0.536

0.537

66


0

Re

7785

16830

25480

33820

42370

50700

59400

67530

76080

84360

92780

101200

109500

117900

126200

134600

142900

151200

159300

167500

3

^D,yaw

1.494

1.300

1.180

1.150

1.136

1.053

0.948

0.873

0.794

0.760

0.757

0.760

0.767

0.776

0.789

0.800

0.816

0.823

0.836

0.851

10

Re

8371

16890

25450

33490

42020

50430

58970

67350

75780

84090

92540

100900

109200

117500

125800

134300

142600

150800

158900

167000

'-'Djyaw

0.597

1.067

1.144

1.147

1.134

1.058

0.980

0.944

0.900

0.872

0.858

0.847

0.846

0.851

0.856

0.862

0.872

0.878

0.886

0.894

100-58

20

Re

8628

15800

23850

31740

39820

47560

55760

63560

71560

79370

87360

95330

103200

111100

119100

126900

134700

142600

150400

157900

600-3

p ^^D,yaw 0.996

0.984

1.044

1.059

1.037

0.982

0.916

0.872

0.842

0.817

0.799

0.785

0.780

0.779

0.779

0.780

0.787

0.792

0.796

0.803

30

Re

8702

16290

24210

32530

40550

48380

56550

64850

72800

81010

88770

96820

104900

112900

119700

128700

136700

144400

152100

159700

° ^D,yaw

0.881

1.009

0.949

0.937

0.899

0.868

0.786

0.722

0.691

0.674

0.672

0.674

0.661

0.670

0.671

0.679

0.681

0.682

0.688

0.690

4C

Re

8475

16590

24410

32590

41000

49350

57520

65490

73670

81760

89990

97990

106200

114400

122400

130500

138600

146500

154300

162100

1°

^D,yaw

1.030

0.782

0.734

0.745

0.735

0.718

0.657

0.605

0.575

0.552

0.551

0.535

0.529

0.523

0.518

0.525

0.519

0.516

0.522

0.525

67


0

R e I 9926

19940

29410

39130

48900

58550

68400

78280

87940

97590

107300

116800

126600

136400

146100

155500

165300

174500

184100

193100

3

^^D.yaw

1.000

0.903

0.914

0.864

0.934

0.966

0.977

0.992

1.006

1.004

1.005

1.016

1.018

1.018

1.021

1.025

1.030

1.030

1.035

1.040

10

Re

9629

19530

28970

38660

48140

57580

67260

76860

86350

95830

105500

114900

124500

134000

143500

152800

162400

171800

181000

190200

'^D.yaw

0.661

0.826

0.829

0.856

0.913

0.945

0.957

0.976

0.995

0.996

1.002

1.004

1.010

1.015

1.018

1.018

1.021

1.023

1.026

1.035

100-58200-8

20°

Re 1 CD^THW

9982

18530

27520

36170

45640

54500

63820

72910

81960

91110

100100

109200

118400

127400

136500

145600

154400

163100

172100

180800

1.245

0.803

0.807

0.809

0.820

0.843

0.841

0.861

0.881

0.882

0.891

0.898

0.892

0.899

0.896

0.902

0.903

0.902

0.905

0.908

30

Re

10110

17820

27660

36650

45800

54690

64070

73240

82470

91750

100700

109700

118700

127800

136500

145800

154600

163400

172100

180700

^D.vavv

0.497

0.786

0.643

0.678

0.700

0.703

0.697

0.724

0.726

0.737

0.739

0.750

0.733

0.743

0.743

0.743

0.751

0.750

0.747

0.751

40

Re

10090

19040

28010

37030

46470

55810

65230

74200

83760

92930

102200

111300

120500

129800

139000

148100

157100

166200

175000

184000

1°

^D.vaw

0.574

0.615

0.578

0.554

0.562

0.586

0.574

0.585

0.600

0.592

0.605

0.592

0.591

0.596

0.597

0.596

0.586

0.587

0.591

0.590

68

APPENDIX B

CONDUCTOR INFORMATION

69

c o

nfor

mat

] T

able

B.l

. C

ondu

ctor

1

ex

Stre

ngth

(lb

) D

iam

eter

(in

) St

rand

ing

Wei

ght

(lb/

ft)

kcm

il FP

&L

Ser

ial N

umbe

r 1

AC

SR/A

W

30,5

00

1.10

8 1.

042

26/7

79

5 10

0-58

104-

4 A

CSR

/AW

37

,600

1.

427

1.57

3 45

/7

1431

10

0-58

300-

4 A

CSR

/AW

32

,900

VO ON r-H

1.17

8 54

/7

954

100-

5820

0-8

OH

GW

15

,400

0.

375

0.27

3 7x

0.12

0"

1

142-

1410

0-0

AC

SR/T

W/A

W/L

DX

30

,800

1 — <

o

1.04

2 26

/7 T

rap.

79

5 10

0-58

650-

0 A

CSR

/TW

/AW

38

,100

1.

286

1.57

2 45

/7 T

rap.

14

31

100-

5860

4-6

AC

AR

12

,200

0.

879

0.55

1 15

/4

568.

3 10

0-61

200-

4 A

CSR

/TW

/AW

1

1.07

7

1

26/7

Tra

p.

795

100-

5860

0-3

AC

SR/A

W

1

1.39

3

1

54/1

9 12

72

100-

5790

0-7

OPG

W

'

0.60

2

•

9x0.

1496

"

1

142-

1502

6-2

70

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It is understood that any copying or publication of this thesis for financial gain

shall not be allowed without my further written permission and that any user

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Agree (Permission is granted.)

.^

(. student's Signature / / j Date

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71

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