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This is to certify that the

thesis entitled
TOWER MOVEMENT EFFECT ON THE DISTRIBUTION
UNIFORMITY ALONG THE PATH OF TRAVEL IN
CENTER-PIVOT IRRIGATION SYSTEMS
presented by

MARIO FUSCO JR.

has been accepted towards fulfillment
of the requirements for

14.8. degree in AGR. ENGR.

 

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Major professor

Date 1995

 

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TOWER MOVEMENT EFFECT ON THE DISTRIBUTION-
UNIFORMITY ALONG THE PATH OF TRAVEL IN

CENTER-PIVOT IRRIGATION SYSTEMS

BY

MARIO FUSCO JR.

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Agricultural Engineering

1995

 

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TOWER MOVEMENT EFFECT ON THE DISTRIBUTION
UNIFORMITY ALONG THE PATH OF TRAVEL IN

CENTER-PIVOT IRRIGATION SYSTEMS

BY

Mario Fusco Jr.

Because of the intermittent movement of the support
towers on electrically driven center-pivot systems, they are
believed to produce less uniform water distribution along
the path of travel than hydraulically driven systems.
Uniformity along the path of travel may be an important
factor during chemigation, especially for low water
applications.

A computer model of a Center-Pivot System was developed
and validated with field data. The model was used to run
simulations of both traditional and LEPA (Low Energy
Precision Application) systems. Simulations were run with
the systems towers moving both continuously and
intermittently. The importance of other parameters (design
and management) were also addressed.

The results showed that for all practical purposes the
uniformity coefficients (Wilcox-Swaile Uniformity

Coefficient, UCW) along the path of travel with the towers

 

 

 

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moving continuously were equal to 100%. With the towers
moving intermittently uniformity coefficient as low as 82.9
and 15.3% were found for traditional and LEPA systems
respectively . Among other parameters studied, the
magnitude of the wetted radius and alignment angle affected
the uniformity the most. Generally, the smaller the
alignment angle the higher UCW values. However, the
alignment effect was more obvious for patterns with smaller
wetted radius as in LEPA Systems.

These findings point to the importance of considering
the uniformity along the path of travel as well as radially
from the pivot-point as a measure of center-pivot water
application distribution, specially when evaluating LEPA

Systems.

Approved \Z‘é/ KM 74/75..
Major Professor Date

Approved QL‘VAT“Ay<E‘U"é{E4“" fiééé/Zar’

Department Chairperson Date

In memory of my mother Nylza and my brother Reinaldo

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ACKNOWL-GEMENTS

First I would like to thank my wife Mimi and.my wonderful
sons Alex and Erik for all their love, support, encouragement
and patience, for without them this work would never be
finished. I would also like to thank my father and mother-
in-law for their never ending support. Special thanks to Dr.
T. Loudon for the opportunity he has given me and for his
patience throughout this journey. Thanks to my committee
members, Dr. Von Bernuth and Dr. Alocilja for their comments
and suggestions. Last but not least, my special gratitude to
Dan Rajzer for his help in the field data collection and to
his brother Cris Rajzer for allowing me to conduct the field

experiment in his farm.

GHUELEIIIF CKMMTTDEP

LIST OF TABLES ....................................... ix
LIST OF FIGURES ...................................... xi
I. INTRODUCTION ................................... 1

A. Irrigation Systems ........................ 2

1. Surface Irrigation ................... 2

2. Sub-surface Irrigation ............... 3

3. Microirrigation ...................... 3

4. Sprinkler Irrigation ................. 4

B. NEED OF STUDY ............................. 5

C. OBJECTIVES ................................ 7

II. LITERATURE REVIEW ............................. 8

A. System Description ....................... 8

B. Hydraulics of Center—Pivot Systems ....... 9

1. Sprinkler Hydraulics ................. 11

2. Lateral Hydraulics ................... 12

a. Pipe Energy Loss Equations ...... 12

b. Lateral Pressure Distribution ... 16
C. Center-Pivot Design Considerations ........ 22
D. Uniformity Coefficients ................... 42

E. Uniformity Coefficients for
Center-Pivot Systems ........... 56

vi

F. Center-Pivot Evaluation ................... 62
G. Center—Pivot Simulation ................... 65

1. Theoretical Development

and System Evaluation ........ 65
2. Uniformity Along the
Path of Travel ............... 75
III . METHODOLOGY ................................... 80
A. Field Experimentation ..................... 80
B. Computer Model ............................ 82
1. Model Development .................... 82
2. Model Description .................... 84
a. Input Requirements .............. 84
b. Model Execution Phase ........... 87
c. Sprinklers Patterns ............. 88
1- Geometrical Patterns .......... 88
2- Actual Patterns .............. 92
3. Model AT Sensitivity ................. 94
4. Model Validation ..................... 95
a. Continuous Mode Operation ....... 98
b. Intermittent Mode Operation ..... 98
5. Model Application ................... 104
IV. RESULTS AND DISCUSSION ........................ 129
A. Traditional Center-Pivot Systems ......... 129

vii

v..-

 

 

 

 

 

1. Analysis of Variance Results ........

2. Main effects ........................
a.

b.

Alignment Angle ................
Guide Tower Timer Setting ......
Sprinkler Pattern ..............

1- Geometrical Patterns .......

2- Actual Sprinkler Patterns...

. Wetted Radius ..................

B. LEPA Systems .............................

V. CONCLUSION AND RECOMMENDATIONS ...............

APPENDIX A. Coefficient values of the high order
polynomials used to represent the
actual sprinkler patterns used in
the simulations .........................

APPENDIX B. Center-Pivot Model Computer Code ..........

BIBLIOGRAPHY .........

viii

142
150

151

180

 

 

Tabl

Table

10.

11.

LIST OF TABLES

Friction Loss Coefficient for some

common devices .................................

Equation II—17 Coefficients for different

SCS Soil Intake Families .......................

Tower no.2 Starting times for different AT’s,
guide tower timer setting equal to 100% and

alignment angle equal to 1.0 degrees ...........

Number of cycles/h as function of Time Step,
guide tower timer setting equal to 100% and

alignment angle equal to 1.0 degrees ...........

Depth of Application, Mathematical Solution
and Simulation Results With model Operating
in a continuous mode for Triangular and

Elliptical Patterns ...........................

Alignment Angle between Towers, field

measurements ..................................

Flowrates and Wetted Radius for sprinklers
contributing to the Depth of Water applied at
different Distances from the Pivot-point for a

traditional Center-pivot System ................

Flowrates and Wetted Radius for sprinklers
contributing to the Depth of Water applied at
different Distances from the Pivot-point for a

LEPA System ....................................

Tower Velocities used in Center-Pivot

simulations 0 O O O D O O O O I C O O O O O O O O O O OOOOOOOOOOOOOO

Uniformity Coefficient (UCW) and Average Depths
(mm) for Triangular Pattern Sprinklers with

Guide Tower Timer set at 100% .................

ix

Page

16

37

96

97

127

127

 

 

l3.

14.

16.

l7,

l9.

21

Tablo

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

Page

Uniformity Coefficient (UCW) and Average Depths
(mm) for Triangular Pattern Sprinklers with
Guide Tower Timer set at 50% ................. 132

Uniformity Coefficient (UCW) and Average Depths

(mm) for Triangular Pattern Sprinklers and

Lateral moving continuously with Guide Tower

Timer set at 100% and 50% .................... 133

Uniformity Coefficient (UCW) and Average Depths
(mm) for Elliptical Pattern Sprinklers with
Guide Tower Timer set at 100% ................. 134

Uniformity Coefficient (UCW) and Average Depths
(mm) for Elliptical Pattern Sprinklers with
Guide Tower Timer set at 50% .................. 135

Uniformity Coefficient (UCW) and Average Depths
(mm) for Polygonal Pattern Sprinklers with
Guide Tower Timer set at 100% ................. 136

Uniformity Coefficient (UCW) and Average Depths
(mm) for Polygonal Pattern Sprinklers with
Guide Tower Timer set at 50% .................. 137

Uniformity Coefficient (UCW) and Average Depths
(mm) for Actual Pattern Sprinklers with
Guide Tower Timer set at 100% ................. 138

Uniformity Coefficient (UCW) and Average Depths
(mm) for Actual Pattern Sprinklers with
Guide Tower Timer set at 50% .................. 139

Uniformity Coefficient Averages for different
Alignment Angles and Guide Tower Timer set at
different distances from the Pivot-Point ........ 152

Uniformity Coefficient (UCW) and Average Depths

(mm) for Triangular, Elliptical and Polygonal

Pattern LEPA Sprinklers moving intermittently

and continuously with Guide Tower Timer set

at 100% ....................................... 157

FIGURE
1. Typical Center-pivot System Layout ...............
2. The Moody Diagram ................................
3. Typical Distribution Pattern of Impact Sprinklers
operating under Low, Normal and High pressure
4. Lateral Section Lengths for Uniform
Water Distribution ...............................
5. Different Sprinkler Arrangements along
the Lateral ......................................
6. (a) LEPA Head operating in Bubble and Aerated
Bubble Mode ..................................
6. (b) LEPA Head operating in Spray Mode ............
6. (c) LEPA Head operating in Chemigation Mode ......
6. (d) Schematic of a Furrow-drop used in
LEPA Systems .................................
7. Potential Run-off for Elliptical Pattern
Sprinkler at different distances from the
pivot-point ......................................
8. Modified Intake Function .........................
9. Maximum Application Rate of Elliptical
Application Pattern by super-imposing the
Application Distribution Pattern on the Soil
Intake Curve ....................................
10. Linear Regression Fit of Non-Dimensional

LIST OF FIGURES

distribution Curve for Sprinkler Patterns .......

xi

PAGE

10
15

24

31
32

FIGURE PAGE
11. Computer model Flowchart .......................... 85
12. Geometrical Sprinkler Patterns .................... 88
13. Nelson R-30 U4 No.20 3RN Approximated Pattern
operating at 30.0 psi ............................. 93
14. Linear Regression Fit, Model vs. Kincaid’s
System F Results for Triangular Pattern
Sprinklers ........................................ 101
15. Linear Regression Fit, Model vs. Kincaid's
System F Results for Elliptical Pattern
Sprinklers ........................................ 102
16. Depth of Application on transect No.1, guide
tower timer set at 25%. ........................... 105
17. Depth of Application on transect No.1, guide
tower timer set at 37%. ........................... 106
18. Depth of Application on transect No.1, guide
tower timer set at 50%. ........................... 107
19. Depth of Application on transect No.2, guide
tower timer set at 25%. ........................... 108
:20. Depth of Application on transect No.2, guide
tower timer set at 37%. ........................... 109
21. Depth of Application on transect No.2, guide
tower timer set at 50%. ........................... 110
22!. Depth of Application on transect No.3, guide
tower timer set at 25%. ........................... 111
23 . Depth of Application on transect No.3, guide
tower timer set at 37%. ........................... 112
24“, Depth of Application on transect No.3, guide
tower timer set at 50%. ........................... 113
25. Tower No.1 On-Times, guide tower timer
set at 25%. ....................................... 114

xii

FIGURE PAGE

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

37.

38.

Tower No.1 Off-Times, guide tower timer
set at 25%. ....................................... 115

Tower No.1 On—Times, guide tower timer
set at 37%. ....................................... 116

Tower No.1 Off-Times, guide tower timer
set at 37%. ....................................... 117

Tower No.1 On-Times, guide tower timer
set at 50%. ....................................... 118

Tower No.1 Off—Times, guide tower timer
set at 50%. ....................................... 119

Tower No.2 On-Times, guide tower timer
set at 25%. ....................................... 120

Tower No.2 Off-Times, guide tower timer
set at 25%. ....................................... 121

Tower No.2 On-Times, guide tower timer
set at 37%. ....................................... 122

Tower No.2 Off-Times, guide tower timer
set at 37%. ....................................... 123

Tower No.2 On-Times, guide tower timer
set at 50%. ....................................... 124

Tower No.2 Off—Times, guide tower timer
set at 50%. ....................................... 125

Span No.3 Position and Off-Times and Depth

of Application on Transect located at 292.7 m

from pivot-point For Triangular Pattern

Sprinklers (WR=9.8 m, guide tower timer = 100%)

with Alignment Angle equal to 0.5° ................ 144

Span No.3 Position and Off-Times and Depth

of Application on Transect located at 292.7 m

from pivot-point For Triangular Pattern

Sprinklers (WR=9.8 m, guide tower timer = 100%)

with Alignment Angle equal to 1.0° ................ 145

xiii

FIGURE

39.

40.

41.

42.

Span No.3 Position and Off-Times and Depth

of Application on Transect located at 292.7 m
from pivot-point For Triangular Pattern
Sprinklers (WR=9.8 m, guide tower timer = 100%)

with Alignment Angle equal to 2.0° ...............

Span No.5 Position and Off-Times and Depth

of Application on Transect located at 170.8 m
from pivot—point For Triangular Pattern
Sprinklers (WR=9.4 m, guide tower timer = 100%)

with Alignment Angle equal to O.5°. ..............

Span No.5 Position and Off-Times and Depth

of Application on Transect located at 170.8 m
from pivot-point For Triangular Pattern
Sprinklers (WR=9.4 m, guide tower timer = 100%)

with Alignment Angle equal to 1.0°. ..............

Span No.5 Position and Off-Times and Depth

of Application on Transect located at 170.8 m
from pivot-point For Triangular Pattern
Sprinklers (WR=9.4 m, guide tower timer = 100%)

with Alignment Angle equal to 2.0°. .............

xiv

PAGE

146

I- INTRODUCTION

Irrigation, the science and art of artificially
applying water to plants, has been credited with the
flourish and decay of early civilizations. It is believed
that the increased stability of food resources brought about
through irrigation has allowed the shift from nomadic food
gathering groups to societies with semi or permanent
dwellings (Cuenca, 1989). Ancient irrigation works, some at
least 4000 years old can still be found in Egypt, Iraq,
India and China. In the western hemisphere most of the
early irrigation works are found in Peru, Mexico and in the
southwest of the United States of America. In Arizona,
traces of old canal distribution systems are still visible
today (Taylor and Ashcroft, 1972).

Today the importance and the economic impact of
’irrigation can be appraised by knowing that the total global
.irrigated area, 223 million ha (FAO, 1977), representing
only'13% of the total global arable land is responsible for

34% of the total crop production (FAO, 1979).

A. IRRIGATION SYSTEMS

Irrigation systems can be classified as:

1. Surface Irrigation
2. Sub-surface Irrigation
3. Microirrigation

4. Sprinkler Irrigation

1. SURFACE IRRIGATION

Surface irrigation systems are the systems that deliver
and spread water over a field by gravity, and for that
reason they are also termed gravity systems. They were the
first irrigation systems used by mankind when water was
allowed to spill over the banks of rivers and flood adjacent
lower land. Today many other surface methods are used which
include Contour Ditch Irrigation (flooding or wild
flooding), Border Irrigation, Contour Levee Irrigation and
IFurrow Irrigation. The average irrigation efficiency of
surface systems is usually low but such systems are the
predominant irrigation systems in the world. The 1987 U.S.

Census reported that about 58% of the total nation's

irrigation was accomplished by surface methods, a large

percentage, but a decrease from the 75% reported in the 1974

U.S. Census.

2. SUBSURFACE IRRIGATION

Subsurface irrigation or subirrigation is any
irrigation method in which the water is applied below the

surface of the soil. There are two ways this can be

accomplished. Most commonly the level of a shallow water

table is controlled allowing the capillary water to reach
the root zone. A second method, subsurface drip, uses
underground lines and applicators to apply water below the

soil surface. The 1987 U.S. Census reported that 581,940
acres were being irrigated by subirrigation systems, a 6.6%

«decrease from 1984, representing only 1.2% of the total

irrigated acreage.
3. MICROIRRIGATION
Microirrigation Irrigation is the method of frequent

auui slow application of water to the soil near the plant.

The water is applied through low rate outlet devices called

emitters, placed along selected points on the distribution
lines. Microirrigation research began in Germany about
1869. The development of economical plastic pipe
manufacturing and of emitters in Israel in the 1950’s, made
its field use practical. The 1987 U.S. Census reported an
area of 866,731 acres under drip irrigation, mainly in
California, Florida, and Texas, a 3.5% increase from 1984.
Despite the increase, the area under drip irrigation

represents only about 2% of the total irrigated area.

4. SPRINKLER IRRIGATION

Sprinkler Irrigation Systems consist basically of a
pumping unit, a network of tubes or pipes to convey the
‘water (main and 1atera1(s)), and sprinkler heads or nozzles
attached to the lateral(s) for sprinkling water over the
land surface. A sprinkler irrigation system is commonly
classified by the movement of its lateral. Systems are
classified as Solid Set, Set-Move Irrigation Systems (hand—
rmqve, tow-move, side-roll and big-gun), and Continuous-Move
Systems (center-pivot systems, linear-move systems and
traveler sprinkler systems). There was a 9% increase from
11984 to 1987 in the acreage irrigated by sprinkler methods

(1987 U.S. Census), representing roughly 40% of the total

irrigated area. This increase is due to the increase of
area under center pivot systems. Land under other methods

of sprinkler irrigation has decreased.

3. NEED OF STUDY

It is no surprise that with escalating labor costs,
systems suitable to automation are increasing in popularity.
Suitability to automation and to application of chemicals
through the irrigation water (Chemigation) in addition to
high efficiency and uniformity of application, explain the
increase acreage under microirrigation and center pivot
sprinkler irrigation.In center-pivot systems, the uniformity
of application becomes very critical when doing chemigation,
especially in light application of pesticides (fungicides
and insecticides). Traditionally, in the evaluation of
center-pivot systems the uniformity of application is
:measured radially (along the lateral). The measurement of
the radial uniformity assesses the adequacy of the design
(sprinkler types, flowrates and spacing) but tells us little
about the effect of the lateral movement on the uniformity
of application. Hanson and Wallender (1986) were the first
to study the uniformity along the path of travel and related

it in part to the start and stop movement of the towers in

electrically driven systems. It is believed by many that
because of the stop and go lateral movement (intermittent
movement), electrically driven center-pivot systems produce
less uniform application along the path of travel than
hydraulically driven systems. If such a belief is proven it
will increase the popularity of hydraulic systems among
chemigators. In Michigan, more than 300,000 acres are being
irrigated (U.S.Census, 1987). Over half of the irrigated
area is under center pivot systems, mainly in corn for grain
or seed, vegetables, irish potatoes and soybeans. Of 816
irrigated farms growing corn only 137 farms applied
fertilizers and only 15 applied pesticides in the irrigation
'water. For other crops the proportion is still lower. The
survey data from U.S. Census (1987) tells us that the
majority of Michigan farmers are not making use of the
Chemigation technology available to them. For some, the
time required in handling a Chemigation operation is a
distraction from other management responsibilities. For
(others the lack of knowledge of how their actual system

would perform is an impediment in the adoption of

chemigat ion .

C. OBJECTIVES

The overall goal of this study is to determine how the

lateral movement of electrically driven center—pivot systems

affects the uniformity along the path of travel as compared

to hydraulic systems. The objectives of this research are:

l)

2)

.3)

4)

to develop a program to assist the average farmer
and/or dealer in his decision in choosing among

the different sprinkler packages currently available
in the market;

to evaluate the distribution uniformity along the
path of travel of center-pivot and LEPA systems for
different sprinkler packages commonly used in
Michigan;

to determine the suitability of these sprinkler
packages in the application of chemicals through the
irrigation water (Chemigation);

to determine how different design parameters, i.e.,
alignment angle between towers, guide—tower timer
setting, wetted radius, and sprinkler pattern shape,
affect the uniformity of water application along the

path of travel.

II- LITERATURE REVIEW

The evaluation of sprinkler packages used in center-
pivot systems requires some knowledge of the system as well
as the parameters studied to make their performance
evaluation. Among all the parameters, the uniformity of
application is undoubtedly the most important and will
receive most attention. In this section the equations,
coefficients and conversion factors will be written as they
are found in the literature, predominantly in English units.

In other sections they will be expressed in the SI units.

A. SYSTEM DESCRIPTION

A center-pivot system consists of a single sprinkler
lateral with one end fixed, the pivot point, and the other
end moving in a circle about the pivot. Pressurized water
is supplied at the pivot-point, and the lateral is supported
by wheel-towers and trusses and/or cables. In general the
towers are driven by electric motors, but some systems are
driven by hydraulic oil motors or water driven rachets.
Lateral length varies. It can be as short as 60 m or as

long as 800 m, but the most common size is about 400 m (1/4

9
mile long). The span between towers is usually dictated by
the slope of the field; it is shorter in fields with higher
slopes. Therefore, more towers, or drive units, are
necessary to support the system under increasing slopes.
The pivoting lateral is kept aligned by controlling the
movement of the towers. Each tower is equipped with a
control box which turns the tower motor ON or OFF depending
on the position of adjacent towers. To irrigate the corner
areas that normally would be left unwetted, some of the
center-pivot systems have a large sprinkler (end gun) at the
end of the lateral. Other systems have a swing anm to
accomplish the same task and these are usually called
corner-systems. Figure 1 shows a typical layout of a

center-pivot system.

B. HYDRAULICS OE CENTER-PIVOT SYSTEMS

Knowledge of lateral and sprinkler hydraulics is
necessary to understand the design criteria of center-pivot
systems. Proper design is critical in having a system with
high water application uniformity. The determination of the
jpressure distribution along the lateral will assist the
designer engineer in the selection of appropriate sprinklers
for a desired discharge. A sprinkler working in the right

;pressure range will have an adequate distribution pattern.

10

 

 

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Figure 1. Typical Center-Pivot System layout.

ll

1- SPRINKLER HYDRAULICS

The theoretical discharge of a nozzle can be computed

from the orifice flow equation:

Q = AC ,/ 2gH 11-1

where:

discharge
A = area of the orifice

Cd:= the coefficient of discharge

the acceleration of gravity

m
II

the pressure head.

This equation can be written in general form as:

_ 05 _
Q - KCdAP II 2
‘where:
P = the pressure at the nozzle
K = a constant of proportionality that depends of

the units being used.

In cases where the sprinkler has more than one nozzle the

<discharge of an individual sprinkler can be computed as:

12

0:; chi A, P,” 11-3
i=1

2- LATERAL HYDRAULICS

In a center-pivot system, water is introduced at the
pivot point and flows through the lateral toward the outer
end supplying individual sprinklers. The energy at the
pivot point must be equal to or greater than the energy
requirement of the last sprinkler plus the total energy lost
due to friction along the pipe and components in the

lateral.

a. PIPE ENERGY LOSS EQUATIONS

Many empirical equations such as Darcy—Weisbach, Hazen—
Vflilliams, Scobey, etc., can be used to compute the friction
jloss in a pipe. The Darcy-Weisbach is probably more
:nationally based than the others, since it can be derived by
dimensional analysis, and has received wide acceptance

<hJeppson, 1982). The Darcy-Weisbach equation is given by:

LY:
D 2g

hf = f 11-4

13

where:

In = the head loss

f = the friction factor
L = the pipe length

D = the pipe diameter

V = the average velocity

the acceleration of gravity.

LO
II

For most commercial pipes, the friction factor, f, can be
(determined directly from the Moody diagram (Figure 2) with
corresponding relative roughness, e/D, and the Reynolds
number (Re) . For laminar flow ( Re < 2100) the friction

factor is given by:

II-S

-91
f—R.

For the transition region the friction factor can be

determined by implicitly solving the Colebrook equation,

x/f RM“

1 = 1,14 - 2 1.0+; + 133. ”—6

where:
e = pipe equivalent sand grain size
D = pipe diameter
e/D = pipe relative roughness.

14

This equation is the basis for the Moody diagram. A table
with values of e (equivalent sand grain size) for different
pipe materials and the Moody diagram can be found in
Streeter (1985) and in others fluid mechanics text books.

Swamee and Jain (1978) developed an explicit equation
for the friction factor, which facilitates computer

calculations:

1325 [H

e +5.74
" 3.7D R29

 

 

This equation was shown to be valid within ranges of:

0.01 2 e/D 2 10'6 and 108 2 R. 2 5000

The energy losses in the system.components generally
«denoted as minor losses, are usually small compared to total
pipe friction losses and usually are accounted for by adding
110% to the total pipe friction loss. The component losses

can also be computed using the equation:

[2)

2:

vfliere:

In = the component head loss

g'ssauuonou mnau
O
8

,§ “mag; é,

035
OI

\nvn
occS

REYNOLDS uuuaen, n = 39

 

l 'UOLOVi HOLLOW!

Figure 2. The Moody diagram.

 

 

 

w"_
.33.:

 

 

 

 

//

'fi, ‘h

.
‘ c

16

K = the component friction loss coefficient

Table 1 gives the values of K for the most common components.

Table 1. Friction Loss Coefficients for some common devices.

 

 

 

 

 

 

 

 

 

DEVICE K H
TEE, Through Side Outlet 1.8
Globe Valve (fully open) 10.0
Rounded Entrance 0.05
Square Entrance 0.5
Gradual contraction 0.04
Abrupt contraction 0.5

 

 

b. LATERAL PRESSURE DISTRIBUTION

Heermann and Kincaid (1970),

branching flow done by Vennard and Dentoni

referring to work in

(1954) I used

numerical techniques to determine the pressure distribution in

a center-pivot lateral. They used the ratios:

dp/D s 0.2

where :

 

 

.
.A .u C.» TL
r « nu O
a UN an;

 

 

17

(1,: the diameter of the branching line (riser)
D = diameter of the main line (lateral).
and
q / Q < 0.3
where:
q = the branch discharge (sprinkler)

the discharge in the upstream main line section.

For the range of the second ratio, the total head loss for
the flow continuing along the main line (lateral) is
negligible. Their results showed that the head loss for the

flow into the riser is given approximately by the equation:

1:, = 3’: EXP (9.21) 11-9
28 0

They concluded that the head loss into the risers would
affect the sprinkler-head pressures but would not affect the
total loss in the lateral. The procedure used to calculate
the pressure distribution along the lateral was (starting at
the outer end): evaluate the sprinkler discharge (in gpm)
'using any of the equations derived by Heermann and Hein
(1968) (Equation II-10 and 11); compute the discharge and
'the head loss in each lateral section (between adjacent
sprinklers) moving toward the pivot point. The sprinkler

discharge is given by:

 

 

whe

 

 

 

Di.

Cor

18

q, = d WR,(R‘*‘I;R'“) ”-10

where:

Ch = the discharge of the ith sprinkler, in gpm

d = the desired depth, in inches

W = the angular velocity of the sprinkler line,
in radian/hour

IQ = the distance from the ith sprinkler to the
pivot-point, in feet (ft),

K = 192

(gpm.h/in.ft2), constant that includes

the conversion factor.

or by specifying the total system discharge rather than the

desired depth and angular velocity by the relation:

qi ___ Q R1(R'+lR—R‘-l)

”.lT-ll

‘where:

Q = the total system discharge, in gpm

IQ — the radial distance from the pivot point to

the last sprinkler on the line.

In their analysis they found that for a 0.152 m (6 in)
pipe-diameter center-pivot system with 396 m (1300 ft)

lateral, a pressure drop of 103 - 138 Kpa (15 - 20 psi) is

commonly found. This results in high pumping costs and a

19

higher pressure than the required for the smaller sprinklers
close to the pivot and lower pressure for the larger
sprinklers toward the outer end of the lateral. Lower
pressures result in large water droplet sizes and a
reduction of the soil intake rat. Their suggestion to
reduce pumping cost and provide a more uniform pressure
distribution was to reduce the pressure loss by increasing
the pipe diameter.

Chu and Moe (1972) derived an analytical solution for
the total pressure loss and for the distribution of the head
loss along the lateral. Starting with the energy equation,

they found that the total pressure loss is given by the

 

equation:
ho—hR = hm 3(m*21’°'5) 11-12
where:
1% = the pressure head at the pivot point, in feet
In = the pressure head at the end of the lateral,
in feet
1% = the friction head loss of the lateral
operating as a supply line, in feet
B(HH1, 0.5) = the beta function

The friction head loss of the lateral operating as a

supply line, hm is given by:

20

where:

(h = the total system discharge, in gpm

C = the roughness factor of the lateral pipe
D = the diameter of the lateral, in feet
R = the radius of the irrigated area
m, n = constants in the slope of the energy grade

line, Sf, as given by Christiansen (1942).

The distribution of the pressure head loss along the

lateral is given by the equation:

___(hr'hk) ... 1 _(_12)( -E‘LE] 11-14
(ho-bk) 8 3 5
where:
X = the ratio r/R
r = the distance from the sprinkler to the pivot-
point
R = the radius of the irrigated area
In = the pressure head at r
In = the pressure head at the end of the

lateral

é"
II

the pressure head at the pivot-point.

21

They also derived an equation for the distribution of the

discharge:

1 ’2
Q - Q0 “165 11-15

where:
00:: the total system discharge

Q = the discharge at distance r from the pivot

The validity of these equations was verified by comparison
with field data.

Iterative solutions used in flow analysis in pipe
network such as Hardy—Cross, Newton-Raphson (Wood and Rayes,
1981), and the Linear Theory Method (Wood and Charles,
1972), can also be used in the analysis of sprinkler
systems. However, the discussion of these methods is beyond
the scope of this work. A complete discussion of these
methods can be found in Jeppson (1982).

Saldivia (1988) did an analysis of sprinkler irrigation
systems, including minor losses, using the Finite Element
:Method (FEM). His results were compatible with results
obtained using the Linear Theory Method. The advantages of
using the FEM as he stated, are: less computer memory used
for storage, and that it is not necessary to change the

computer code when analyzing different systems. Pandey

22

(1989) also used the FEM in the analysis of center-pivot
systems in which auxiliary sprinklers were added to the
lateral. These auxiliary sprinklers would start operating at
the time the end-gun was off. By doing so, the pump
operating point remained constant, resulting not only in a

saving of energy but also in a better distribution

uniformity.

C. CENTER-PIVOT DESIGN CONSIDERATIONS

Christiansen (1941) was one of the first researchers to
do extensive tests on stationary agricultural sprinklers to
determine the most important factors affecting the water
distribution. He found out that for a stationary sprinkler,
the water is applied at a relatively constant rate (at any
particular point) and that the rate of application decreased
‘with the distance from the sprinkler. The pattern of this
‘variation is often called sprinkler distribution pattern or
;profile. Christiansen also determined that among the many
factors influencing a sprinkler distribution pattern, the
(operating pressure, wind and sprinkler speed of rotation
(were the most important ones. At high pressures most of the

Inater is deposited near the sprinkler while at low pressures

Inost of the water was deposited in a ring on the edge of the

23

pattern, a pattern normally called doughnut shape. The
sprinkler pattern produced by unfavorable wind conditions
(4.5 - 6.7 m/s or 10 - 14 mph) were asymmetrical with most
of the water deposited near the sprinkler. A high sprinkler
rotation speed also had the effect of reducing the wetted
area and therefore increasing the application rate.

Kohl (1974) studying the water droplet size
distribution from medium size agricultural sprinklers also
determined the importance of the operating pressure in
affecting the shape of the sprinkler pattern. He referred
to work done by Merrington and Richardson (1947) stating
that the mean diameter of droplets formed from jet breakup
is inversely dependent on the jet’s velocity relative to the
surrounding air. Since jet velocity is proportional to
water pressure, higher pressure produce smaller droplets.
Also, since the speed of the droplet decreases faster with
the decrease of the droplet diameter, smaller droplets will
fall closer to the nozzle. Kohl also studied the effect of
the nozzle size. He concluded that although smaller nozzles
produce a smaller mean drop size than larger nozzles, its
effect is less than the pressure effect. Operating smaller
nozzles at low pressure can produce larger mean drop sizes
than larger nozzles operating at higher pressure. Figure 3
shows the typical distribution patterns of an impact

sprinkler operating under low, normal and high pressures.

24

 

7N\ / /
K. +—/

30 20 IO 0 10 20 30
A. PRESSURE TOO LOW

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

30 20 IO 0 IO 20 30
B. PRESSURE SATISFACTORY

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

// ‘\\
/ j
// ' N
30 20 10 O 10 20 30

C. PRESSURE TOO HIGH

Figure 3. Typical distribution pattern of impact
sprinklers operating under low, normal and
high pressure.

25

In the case of continuous move systems, the water
application rate at a given point is not constant. It is
zero until the moment the first sprinkles start to wet the
point, it increases as the system approaches and reaches a
maximum as the system is directly over the point and starts
decreasing as the system moves away. Finally it becomes
zero after the last droplets wet the particular point. The
application rate is a function of the nozzle size, nozzle
operating pressure, sprinkler spacing, sprinkler type and
distance from the pivot point. For a center—pivot system,
the average application rate varies from a low value close
to the pivot—point to higher values toward the end of the
lateral. Pair (1978) illustrates with a figure (Figure 4
redrawn here) why the application rate must increase toward
the outer end of the lateral. In a 402 m (1320 feet)
system, the same volume of water must be applied through 54
m (177 feet) of lateral covering the outside one-fourth of
the area irrigated as through the 201 m (660 feet) of
lateral over the inner one-fourth of the area. He also
stated "... water application rate varies along a center-
pivot because of the time water is applied per unit length
of lateral decreases from the center pivot to the outer end

" . For that reason the area being irrigated by the end
of the lateral is the one more subject to potential runoff.

The spacing between sprinklers as well as their

 

26

WATER APPLICATION ALONG
LATERAL

  

25% of area

  
 

25% of area

 
 
    
 

25% of area

  
 

25% of area

 

A

 

Figure 4. Lateral section lengths for uniform
water distribution.

27

distribution pattern, as mentioned before, is of great

importance in affecting the water application. The most

common sprinkler type and spacing arrangements used by the

industry are:

II-

Using increasingly larger sprinklers (conventional
impact sprinklers) toward the end of the lateral and
maintaining the sprinkler spacing constant. One
disadvantage of this arrangement is the use of a large
sprinkler at the end of the lateral. Although these
sprinklers have a lower average application rate, since
they wet a large area, their instantaneous application
rate (i.e. the rate at which water is applied to a
given point on the soil surface during an instant in
time (James and Stillmunkes, 1980) is higher. In some
soils, high instantaneous application rates can cause
soil dislodgement and increase the potential for runoff
and erosion. One advantage of a larger sprinkler is
the potential for better overlapping of the patterns

from different sprinklers.

Using all medium size conventional impact sprinklers
with increasing nozzle size toward the outer end and

with decreasing spacings toward the end of the lateral.

 

28

III- Using spray nozzles with increasing nozzle sizes toward
the outer end with a constant spacing between them.
This arrangement is the one commonly used in low

pressure center pivot systems.

IV- Using spray nozzles mounted on spray booms which are
themselves mounted across the center-pivot lateral with
the objective of increasing the area of application and

therefore reducing the application rate.

V— Using Low Energy Precision Application (LEPA)
packages in which drop tubes are used to lower the
sprinkler discharge head or a furrow drop (Fangmeier et
al. , 1990). Besides having a low energy requirement,
this arrangement has the advantage of a more precise
application, and a smaller drifting loss resulting in

an increased water application efficiency.

Figure 5 illustrates the first three arrangements and
figures 6a-6d show the different heads used in LEPA
systems.

When designing a center-pivot system it is desired to
match the application rates to the soil intake rate to avoid
3potential runoff. Potential runoff is the portion of the

‘water applied at rates exceeding the soil intake rate.

29

CONSTANT SPACING SPRINKLERS

"‘V”’T'TVWVVV§T'"“V‘Vv"VP'VfirVV‘Qyu.§f-‘”
.44.3.7.3333“?.9M'm'AYAYmI‘TSxAQ‘flxf‘QX“
‘1".‘.l.!‘g!‘!‘.1‘1‘1‘!‘2‘!‘!£'Q1919w1k‘1‘!‘”

4L

       
 

   
  
   
  

A A A A A -

.‘-‘ - g ‘.

53m

   
   

VA RIABLE SPACING SPRIN K LE RS

        

A'A'A'Ila’la'laffififfir‘.9701916fr(ctrz'i'birlrzrzfi'igm;35$???“\
V‘WL‘VJIIRLQIQLQIQI!191933419192?:32153932332333:529:329.:21’1/

 
   
      

SPRAY NOZZLES i

 

Figure 5. Different sprinkler arrangements along
the lateral.

31

 

 

 

 

“ 7 I“

ivy/120$: ‘
O] \

if?” ,,,.....
/

.... ~\T‘L\
”777/ ’l\\\\\\\§\\\\\

///

Figure 6b. LEPA head operating in spray mode.

OUTLET \ fl

 

 

 

 

 

 

 

\

LATERAL

DROP TUBE

VALVE
NOZZLE ( + HOUSING )Ԥ

FURROW DROP

SOCK
\

 

 

 

 

FURROW
CHECK

Figure 6d. Schematic of a furrow drop used in a LEPA
systems.

34

The empirical formula:

I = kt" ”-16
where:
I = the intake rate
t =

the time, in hours

k and n are constants for the given soil.

has been widely used to represent the soil intake rate
because it provides good fit with experimental data. An
analysis by Heermann and Hein (1968), assuming a constant
soil intake rate, found that the potential runoff was
proportional to the distance from the pivot point. Kincaid
et a1. (1969) using equation II—17 showed that potential
runoff can be substantial near the pivot point. They
explained " ... even though peak application rates are
lower, these rates are reached at a time when intake rates
are also lower.". Figure 7 illustrate this concept. They
continued questioning the validity of using the soil intake
function given by equation II-17, since it assumes that the
intake rate is independent of the application rate during
the initial period of application (period prior to
saturation) , which is not valid for a moving sprinkler
irrigation system. Under moving sprinkler irrigation

systems the soil is gradually saturated rather than suddenly

APPLICATION - INTAKE RATE (UT)

35

 
 
  
 
 

APPLICATION RATE AT
DISTANCE EOUAL 2X

CONSTANT INTAKE
RATE

 

 

      

APPLICATION RATE AT 0
DISTANCE EQUAL X I - KT

(SOILINTAKERATE)

    

 

TIME (T)

Figure 7. Potential run-off for Elliptical Pattern
Sprinklers at different distances from the

pivot-point.

36
flooded. Assuming that the intake at any time depends on
the volume infiltrated up to that time, they proposed the

use of a modified intake rate for the period prior to when

potential runoff begins, given by:

I... = I/Z ...11-17
where:
I = the intake rate function given
by equation (II-16)
z = D, / Dp
where:
II = the depth applied up to the time potential

runoff begins

I% = maximum.depth that could be applied before

potential runoff begins

JEor the period after potential runoff begins the intake
function is given by the same equation (equation II-17) ,

Imit using a new time, t - at. The intake function in this

case is given by:

I = k ( t-At )" II-18

where:

At = the amount of time by which the intake
function given by equation II-16 must be

delayed so that it will pass through the

37

intersection between the application rate

curve and the modified intake rate (Im).

Figure 8 helps clarifying this concept. Their results
showed that using the modified intake rate function had the
effect of reducing the predicted potential runoff and a
better prediction of the actual runoff could be made.

Dillon et al. (1972) proposed a design procedure that

would not only match the center-pivot system to the soil

with no potential runoff occurring, but also to the crop.
Soil sprinkler intake curves developed by the State Office
of the Soil Conservation Service, Temple, Texas (Vittetoe,
1970) were used. These soil curves were grouped into soil
intake families and the coefficients of equation II-16 for

each family is presented in table 2. Assuming that the

Table 2. Equation II-l7 coefficients for different SCS
soil intake families.

 

INTAKE FAMILIES

 

PULLMAN AND LIKE SOILS 0.810 -0.69
0.5 INTAKE SOIL 0.880 -0.615
0.3 INTAKE SOIL 0.520 -0.860

 

 

 

 

 

 

 

 

38

   
 

 

uwmumm
\ RATE
m AT
‘2
m l-KHWAU"
‘2
...
.2.
'2.
Q
'2
9.
a‘.
G. I-KTn
<
flME(T)

Figure 8. Modified Intake Function.

 

39

water is being applied uniformly and in an elliptical
pattern the maximum application rate occurs somewhere at the
end of the system. They proposed an equation to determine
the approximation of the average maximum application rate

(in./h) of the last few sprinklers given as:

h = 1225—9— 11-19
Rr

where:

the system capacity, in gpm

the radius of coverage of the system, in ft.

21
II

the radius of coverage of the last few

H
II

sprinklers, in ft.

With knowledge of the maximum application rate and surface
storage allowance (Shockley, 1968) , the maximum time
required for the elliptical pattern to pass a point can be
determined by superimposing the elliptical pattern on the
particular soil's sprinkler intake curve, See figure 9

The minimum design speed can be determined by:

V = L 11-20
30:

where:

r = the radius of the last few sprinklers, in ft.

APPLICATION RATE OR INTAKE RATE (LIT)

 

 

 

Figure 9.

40

SOIL SPRINKLER INTAKE
CURVE

ELLIPTICAL APPLICATION
PATTERN

MAXIMUM APPLICATION
RATE

I

I

I

V
TIME(T)

MAXIMUM TIME FOR PATTERN‘ J
TO PASS A POINT

 

Maximum application rate of Elliptical
application pattern by super-imposing the
application distribution pattern on the
soil intake curve.

41

t = the maximum time required for the elliptical

application pattern to pass a point.

The minimum design time, in hours, to complete one

revolution can be easily determined by:

ZnRL

T = ...II-21

 

'where:
R.L = the distance from the pivot to the last
tower, in ft.
V = the minimum design speed determined by the

equation II-20.

Finally, the average net depth, in inches, can be calculated

by:
PT
24
vflmre:
P = the peak water use, in in./day. It varies
for different crops and climates.
T = the minimum design time.

42

D. UNIFORMITY COEFFICIENTS

Over the past many researchers have proposed different
uniformity coefficients in order to express the uniformity
of water application in a field. Different sprinkler
irrigation systems have been compared on the basis of such
coefficients.

Christiansen (1941), in order to compare different
sprinkler patterns and to determine how different spacings
would affect the distribution of water, introduced an

expression which he called "uniformity coefficient” given

as:
Cu = 100 ( 1 my) II—23
Inn
uflnare:
m = the mean of the observations
n = the number of observations
d = the absolute value of the deviation of

individual observations from the mean value.

As can be seen, an absolute uniform distribution (2d = 0)
will be represented by an uniformity coefficient of 100%.
Today this coefficient is usually referred to as the
"curristiansen Uniformity Coefficient", and in this present

study it will be denoted by UCC.

43

Wilcox and Swailes (1947) instead of using the sum of
the absolute deviation from the mean, have used the sum of
the squares of the deviation from the mean to define a new

coefficient, given as:

S

LKTW’==100 1.-1: ".17-24
X’
where:
S = the standard deviation of the observations
§'= the mean of the observations.

The relation S/i is known as the coefficient of variability.
The USDA (Cridle et al., 1956) introduced the following
parameter for the evaluation of overlapped sprinkler

patterns:

PE” = 11-25

 

‘where:

PEU = the USDA Pattern Efficiency

3 = the sum of the lowest 25 percent of the
observations

If = the number of observations used in computing
2x3

K’ = the average of all observations in the pattern

44

Hart (1961), when testing the overlap of patterns of a
wide range of sprinkler sizes (4 to 300 gpm) operating at
normal pressures and at reasonable spacings, showed that the
distribution of the observations approximated a normal
(Gaussian) distribution. Assuming a normal distribution
curve, the mean of the absolute deviation from the mean
equals approximately 0.798 S, where S is the standard
deviation of the sample. Substituting in the equation of
UCC (Christiansen Uniformity Coefficient), a new uniformity

coefficient was obtained:

UCH = 1 - 0.798 g ...II-26

X

'Ehis coefficient is known as Hawaiian Sugar Planters
.Association (HSPA) Uniformity Coefficient . Hart also
stated that assuming a normal distribution the USDA Pattern
Efficiency could be written in terms of the coefficient of
variability. He showed that the value of point ZX', / n'
corresponds to a value 3? - 1.27s . Substituting in

equation II-25, an equation relating pattern efficiency and

cv can be written as:

PE” = 1 - 1.27% II-27
x

45

Benami and Hore (1964) questioned the adequacy of using
both UCC and UCW coefficients. They showed that the same
UCC values could be found for different distributions even
though one distribution was far superior than the other.
They pointed out the weakness of the UCC as in not giving
more emphasis to deviations below the mean, which are
usually considered to be more critical than deviations above
the mean. With respect to the UCW they stated that although
it does give added weight to the extreme readings it still
does not adequately differentiate between satisfactory and
unsatisfactory distribution patterns. For that reason they
proposed a new coefficient known as Benami and Hore
Uniformity Coefficient. This coefficient considers the
deviations from the mean of the group of observations above

and below the general mean, and it is given by the

re]. a t i onship:

Cl

A = — II-28

C2

Where 3
c - M - 2' X I" II 29
l — b —_——— 000 .—
Nb

46

 

23IXI.
C2 - Ma + ...11-30
a
vflieezren
EL = the mean of the group of readings above the
general mean.
N5 = the mean of the group of readings bellow the
general mean.
EL = the number of observations above the general
mean.
Kg = the number of observations bellow the general
mean.
|X| = the absolute deviation from the group mean.
Bexaeamni and Hore stated ”... by considering the ratio of

deviations below to deviations above the general mean, the
new coefficient lays particular stress on deviations below
the general mean. At the same time, the coefficient takes
intr: reaccount that deviations near the general.mean are not
as inzportant as deviations further from the general mean." .
The eacpression given above is sufficient for pattern
comparison purposes, for recommendation purposes they
suggeSted using an absolute value of A, referenced to a

value of one hundred, given by:

 

 

9.4.

NS

47

Cl
A = 166 — 11—31
C2

The coefficient A, can also be computed more readily using

the equation:

 

A =166-1Y3- [2Tb +D’M"] ...11—32
N. m. + D.M.J
where:

T. = the sum of the readings above M,

Tb = the sum of the readings below Mb

Da = the difference between the number of
observations below and above M,, for the
group above the general mean.

Db = the difference between the number of

observations above and below Mb, for the

group below the general mean.

Benami and Hore claimed two important advantages in using
their new coefficient instead of UCC and UCW. The first
advantage is that the coefficient A is a better index of the
degree of pattern uniformity, that is, different values of A
were found when comparing two different pattern
distributions with same UCC values. The second advantage is

that the differences between different distribution patterns

48

are more pronounced due to the wider range in values of A.
Values of A greater than one hundred can be found.

Beale and Howel (1966) showed that the USDA Pattern
Efficiency was of the same form as UCC, but only accounting
for the mean deviation of the lowest 25% precipitation
values. They also proposed a new coefficient which they

called High Pattern Efficiency (HPE), expressed as:

(m
TD

HPE = 100 l - II-33

 

where:

CKP)h== the mean deviation of the highest 25
percent precipitation values.

'P = the mean of the precipitation values.

They pointed out that if a normal distribution is assumed,
<due to its symmetrical shape the value of HPE and PE are the
sanmn

Karmeli (1978) proposed using the linear regression fit
based upon the dimensionless cumulative frequency curve of
the infiltration depth (Y) and fractional area (X) to
«describe sprinkler distribution patterns (see figure 10).
He demonstrated that the linear regression model may be as
«good as the normal model for highly uniform patterns (small

S/§:values), as most of the distribution curve would tend to

 

 

49

)

.1
'9'
E

 

 

 

 

DIMENSIONLESS PRECIPTATION DEPTH (Y

 

 

I
0 0.5 1.0
FRACTION OF AREA (X)

Figure 10. Linear Regression Fit of Non-Dimensional
Distribution Curve for Sprinkler Patterns.

50

concentrate around the mean. However, for less uniform
patterns (larger S/§ values), the linear fit would be better
than the normal fit as the magnitude of the errors at the
extremes of the curve (of actual data) would be smaller.
The linear regression model has some basic properties where
for Y = 1.0 (average precipitation depth infiltrating the
soil = depth designed to replenish soil moisture
deficiency), X = 0.5 (half of the irrigated area). The
regression coefficient b (the slope of the regression line)
is equal to Y“Ax - Y"m (the difference between maximal and
minimal wetting zones of the field). The regression
coefficient a (the intercept) is equal to the estimated
minimal precipitation depth (Ymm). ‘With respect to figure
10, some related formulations were also written as:

1) Deficit area
AD = —— ...11—34

where:

Y5 = the maximal depth in the deficit area.

2) Surplus area,

”-35

where:

51
Y; = the minimal depth in the surplus area.

3) Adequate irrigated area,

AA =1 -(AD +215) I...I-36

4) Average depth in the deficient area,

YD+l--P-

2
= ...II-37
D 2

 

5) Average depth in the surplus area,

I} + 1 +2?
I; = ...II-38
2

 

.A uniformity coefficient based on the linear regression

model was then derived, and is given by:

UCL = l - 0.512 ...II-39

In tests done, this coefficient showed great similarity with
UCC (r2 = 0.998).

Howell (1964), assuming a polynomial yield function and

 

 

52

that the same yield per area versus water depth curve would
apply from irrigation to irrigation, demonstrated that the
characteristics of nonuniformity necessary to completely
determine its effects on yield were the moments of the
distribution (of water application) to the same order as the
polynomial yield function.

Varlev (1976) pointed out that the equation proposed by
Howell (1964), within the set of his assumptions, is only
valid for situations where irrigation water provides all the
water received by the crop. Considering usable rainfall and
available soil water content at planting time as the total
depth of water available to the plants, he proposed the

following equation for subhumid and humid zones:

II=AO+A13I+aJP- luizlgfowxfidx a2? ...II—40

where:

A0=ao+a17+a272

A1==ah + 2afy
where

7 = the sum of usable rainfall and available soil
water at planting time

anal, and a2 = constants

and

w = the area integrated

 

 

 

 

 

((2

53

§'= the average depth infiltrated
X

the depth infiltrated by irrigation.

Varlev noticed that the absolute yield loss, AY, is
given by the last expression of equation 40, and that it
depends only on the nonuniformity of distributed water
infiltrated by irrigation. Then, a new coefficient was
proposed, which he called ”Coefficient of Nonuniformity",

expressed as:

Fm: 1-? — lwowf deX ...11-41

In this way, the absolute yield loss could be written as,
.AY= ajame.' The integrated form of me given by equation
II-41 is appropriate when the equation of the distribution
of infiltrated depth of water along the length of the
furrows is known. For sprinkler or trickle irrigation,
‘Varlev suggested the use of me expressed as:
X‘ 2

1 N
Fm,“= — E — - 1 ”-42
N I=1 X

Varlev explained that because of the linear relationship

between absolute yield loss and.me , the later had a

 

['1

u:
(I,

I"
‘ ,
«IC-

 

54

definite economic significance. A reduction in me implies
an equal reduction in the absolute yield loss. He also
suggested the use of me not only for the comparison of
different sprinkler irrigation equipment,i.e. spacing
between lateral and sprinklers, but also when comparing
among different irrigation methods.

Solomon (1984) demonstrated that for different
hypothetical water—yield functions, the relationship between
traditional irrigation uniformity and efficiency measures
with expected yield, were special cases of the general yield

prediction equation:

Y = f;Y(pw) 11(wa II-43

where:
w = W / u, a dimensionless irrigation variable
f(w) = the distribution function for the
dimensionless irrigation depths,
p = u / W' , a water management parameter.
where:
u = the average (over space) of the non-

uniform seasonal irrigation application,

the seasonal irrigation depth,

t

W
VI = the seasonal application depth which

correspond a relative yield of Y = l.

 

A:
Dc

 

 

 

 

 

 

55

In this way, the calculation of UCC gives the same result as
calculating the expected yield using the equation II-64,

with the water-yield function given by
y(w) = l - I w - 1|

when the irrigator causes )1 to be equal to W', that is p =
1. Therefore, a yield related uniformity coefficient ( U; )

and efficiency measure ( EL ) were defined as:

Uy = f; y (pw) f(w) dw II-44

and

Ey = (l) f. y (pw) f(w) dw ”-45
p .

Solomon argued that although the use of Uy or 15‘.Y requires
an estimate of the appropriate water—yield function to the
particular crop and site of interest, both measures assess
the significance of irrigation decisions as they would
affect the crop at that particular site.

With the exception of the Linear Uniformity Coefficient

proposed by Karmeli (1978), all the other statistical based

56

coefficients reviewed above were developed assuming a normal
function distribution. However Chaudry (1978), Elliot et
a1. (1980),and Warrick (1983) have demonstrated that
sprinkler application distributions can also be represented
by other statistical function distributions such as log—
normal, specialized power, beta and gamma distributions.
Furthermore, Warrick (1983) also derived expressions for UCC
and Distribution Uniformity Coefficient for all the

distributions above.

E. UNIFORMITY COEFFICIENT FOR CENTER-PIVOT SYSTEMS

Heermann and Hein (1968) were the first to propose a
uniformity coefficient for center-pivot systems. Based upon
the fact that, the depth observed by each catch can is

representative of the subarea expressed as:

A, = ZuSSAS II-46

where:

88:: the distance from the catch can to the
pivot point.

AS = the spacing between the catch can,

and assuming that each measured observation remained

57

constant for one revolution of the system, they modified the
UCC (Christiansen uniformity coefficient) utilizing the
summation of the absolute deviation of observed volumes for

the subareas from the mean volume. This coefficient is
expressed as:

2| V. -V.|

UCHH = l - S V II-47

 

where:

\h= the volume for the subarea A,.
Ti: the mean volume for the subarea A,.

V = the total volume for the system.

The subarea volumes is expressed by:

V, = DA, = 2110,5215 II-48

where:

[A = the observed depth at point 5.

The mean volume for the subarea A3 is given by:

(N I
2v,
7, =54, = :1 A .II..-49
2.1,
(..I I

 

 

 

where:

 

58

'D = the average depth for the system.

Substituting equations 48 and 49 in equation 47, noting that

V =12 Egg , and simplifying, the UCHH can be expressed as:

 

 

20.5.
ES D - ‘
3 S 3 XS,
UCHH = 1.0 - , " : ”-50
20.5.

5'

In the same way as for a stationary sprinkler systems
the USDA recommends the use of the Soil Conservation Service
Pattern Efficiency (1982), adapted for center-pivot systems,
to evaluate the uniformity of water distribution. This
coefficient compares the water application on the low one-
fourth of the wetted area to the average application on the

total wetted area. It is expressed by:

WEIGHIED LOW 25 95 AVG APPLICATION
PE“: 100 II-Sl
WEIGHIED SYSTEM AVG APPLICATION

where, WEIGHT LOW 25% AVG APPLICATION (WGTLZS) and WEIGHTED

SYSTEM AVG APPLICATION (WGTSYSI:

_ 2 OF LOW 25% CATCH*FACIUR.S'
2 OF LOW 25% FACTORS

WGIZZS II-52

 

59

and

WGTSYS = L CAWHWACIDR [HS

2 FACTORS

 

where :

FACTORS = the number of the catch can.

Marek et al. (1986) proposed the use of a new area-
weighted uniformity coefficient based on the coefficient of
variability (CV). This uniformity coefficient is based on
the square of deviations of the volumes from the mean volume
associated with each subarea. One of the advantages claimed
for their coefficient is an increased sensitivity in
depicting nozzle discharge deviations when compared with
UCHH. This increased sensitivity is explained by the fact
that the mean deviation does not emphasize the observations
that are significantly different from the mean. A second
advantage is the implication of better expected yield
inferences when using a uniformity coefficient based on the
coefficient of variability (Solomon, 1984) . This

co . G .
effitment .‘LS expressed as:

60

 

 

N _
21V. - V3
i=0

UCM = 100 1 — N ' 1 II-54

 

 

VI

where :

the volume associated with subarea A1.

Vi = the mean volume for subarea A1.

the number of catch cans.

<:|
u

the average volume for the system.

The area represented by each catch can (subarea) is

exPreS sed as:

A, = 21: r, Ar ”-55
where ;

r1 = the distance from the pivot point to catch
can i

Ar = the spacing between catch can.

ASSUming x1, the depth observed at a distance r, from the
pivot, constant for one revolution,

the volume associated
Wit-h subarea A, is:

61

K = 23x, r, Ar II-56

The mean volume for the subarea is given by:

V, = I A, II-57

where :
i =the average depth for the system is given by:

_ i
x - t1 II-58

 

The average volume for the system is:

N
EA, 1:,
T! = 121—— II-59
N

substituting equations II-55 to II-59 in equation II-54 and

$1“"pJ-ifying, this coefficient may be expressed by:

62

 

 

 

 

 

 

 

 

 

 

 

 

r N )2
N 5:0"er
2 ’er ' r, N
I=1
2r,
V 1 i=0 1.
N -l
N
Erix,
UCM = 100 1 — ”N ...II-ss

F. CENTER-PIVOT EVALUATION

In the past, different test procedures for water
application data collection have been recommended with the
Objective of evaluating center-pivot performance. For the
i1’1teli‘e'é‘»1::ed reader, a short review is given by Ring and
Heernauun. (1978). The field test procedures recommended by
the AmeEricanSociety of Agricultural Engineers (ASAE) and by
the Soil Conservation Service (SCS) have been the most
accepted and used procedures in performing field evaluations
Of Canter-pivot irrigation systems. A brief summary of both
procfidures is given below.

In 1983, the ASAE adopted a standard (ASAE S436) test

63

procedure for determining the uniformity of water
distribution of center-pivot and other self—moving machines.
This standard was approved by the American National
Standard Institute (ANSI) in 1989. Specifically for center-
pivot machines, among other recommendations, the ASAE
standard recommends that the catch cans must be placed in at
leas 1: two radial lines with their outer end no more than 50
m ( 1165 ft) apart. The catch cans must be spaced with a
maximum spacing of 30% of the average sprinkler wetted

diameter, and never more than 4.5 m (14.8 ft). The entire

tes t shall be conducted in an representative area and during
periods that the effect of evaporation will be minimized,
with a wind velocity of less than 1.0 m/s (2.2 mile/h),
measured at a minimum height of 2m (6.6 ft) and within 200 m
of the test site. For such reasons the ASAE recommends the
test to be conducted at night. For the calculation of the
water application uniformity it recommends the use of UCHH I
Heerrnann and Hein coefficient) given in previous sections as
equation II-47 and II-50. In its calculation the ASAE
reCOIrIInends the elimination of the data on the inner 20% of
the total length of the system, as well as any obviously
incorrect data points that may be explained for. Other
observations which are unreasonably high or low may also be
eliminated from the analysis, but eliminated values should

be less than 0.5% of the total number of data points.

64

In its evaluation test procedure, the SCS recommends

the catch cans to be placed in a single radial line

extending from the pivot-point to a point beyond the wetted

Usually the spacing is of

area , at any uniform interval.

9.1 In (30 ft). When evaluating the adequacy of the

irrigation system design (at ideal conditions) the SCS

recommends the use of the Pattern Efficiency coefficient

defined previously as equation II—Sl. This coefficient has

been also referred as Distribution Uniformity as well as

Distribution Efficiency. For the evaluation of the

irrigation application it recommends the use of the

Application Efficiency Coefficient defined as:

 

 

ECAICHakFACTOR
. . . ZFACIDR
ApplrcaaonEflicrency = II—61
D
8
where:
CATCH = depth measured in the catch can,
FACTOR = the number of the can,
D = the gross application, the area times the

average depth .

This coefficient compares the amount of water pumped to the

amount of water applied in the field; therefore it is a

measure of the losses due to evaporation, wind drift and

l .
aaks in the system.

65

G. CENTER-PIVOT SIMULATION

1- THEORETICAL DEVELOPMENT AND SYSTEM EVALUATION:

Bittinger and Logenbaugh (1962) established the

theoretical basis for water distribution from moving

sprinkler irrigation systems. They derived equations for

the application rates and total depth of precipitation for a

sing 1e sprinkler having triangular and elliptical patterns
moving in a straight line and in a circular path at a

constant velocity. Assuming a straight line movement, the

equations for the application rate and total depth for

sprinklers with triangular distribution pattern are given

respectively by:

 

 

l
_ 2 2
AR.=h{' ("”2“”) .262
r
and
1 1
_ _ 2 2
0,21". (1_m2)2-m21n(1 m) *1 ...II-63
m

66

And for the elliptical pattern sprinklers by:

 

 

1
hv r2 - mzr2 "
ARE=— -t2 2 ...II—64
r v2
anti
2
TIE -
DE h (1 m) ...II-65
2v
wherxe:
h = the application rate at the center of the

sprinkler pattern,

r = the radius of sprinkler pattern,

m = the ratio of the perpendicular distance
between point p and the line of travel of the
sprinkler to the sprinkler pattern radius,

v = the linear velocity of the sprinkler,

t = time.

F - . . . .
or trfiLEirigular pattern sprinklers movxng in a Circular path.

a . . , .
pplication rate is given by:

l
ARTE“):- h — %{(R + mr)2 — 2R ( R + mr )cosa + 1122}2 II-66

67

where :
R = the distance from the sprinkler to the pivot-
point
a = the angle of the rotation about the pivot-
point.

and the total depth of precipitation on point p with a

complete pass of the sprinkler is given in the form of an

elliptic integral of the second kind:

... -31 . i - 15-21: ..
Dam.) ZhT c”(2:1 m){E(k, 2) 50:, 2 2)}...11 67

where:
u.) = angular velocity of sprinkler (radian/T)
T = time required for one-half of sprinkler
pattern to pass point p

n = the ratio of radius of rotation to pattern

radius , R/ r

K = (4n(m + n)/ (m +2n)2)“2

o
E . . . .
”(AM e I ( 1 - k2 sm2 4))“2 d4) , the elliptic integral.
0
For elliptical pattern sprinklers, also moving in a

circu . . o I
lat path, the equation for the application rate is

Given by .

68

NIT-

Mam) — II—68

[rz—Sz-R2+2RScosa]

wl:r

The total depth is also given in form of an elliptical
integral, which is not readily evaluated, except by

numerical means:

 

1 0T
Dam) 4—h(1 - mz)E [7 1 - 4”“ + "9 ° 20 2 db II—69
0° ° 1 - m2
where:
0 = n/2 — a/z
and
a = is the angle of rotation about the pivot-point.

Bittinger and Logenbaugh noticed that the sprinkler moving
in a circular path has a skewed application rate pattern
near the pivot point, and that at a distance equal to 5
times the sprinkler radius from the pivot point the pattern
was nearly symmetrical. For this situation, for all
practical purposes, the equations for a straight-line travel
path could be used. Studying the overlapping effect of the
sprinklers, they concluded that the triangular pattern

sprinkler produced a mare uniform.distribution than the

69

elliptical pattern sprinklers, with the best distribution

being when the sprinklers were spaced a distance equal to

the pattern radius. The best distribution for elliptical

pattern sprinklers occurred at a spacing equal to 1.4r.
Heermann and Hein (1968) , based on the work done by

Bi ttinger and Longenbaugh (1962) , wrote the equations for

the total depth at any distance from the pivot point for one

pass of the system for the triangular and elliptical pattern

spr‘inklers. The equation for a triangular pattern sprinkler

moving in a straight—line path travel is given by:

1
.(l-m12)2+1|

 

I"hr 1
ID = —‘—‘ 1-m22-m21n ...II-70
T ngI( i) i I "’1 I

and for the elliptical pattern sprinkler by:

N h
" "(l-m2) ...II—71

Where:

N = is the total number of sprinklers contributing to

the depth at the given point.

I“Oil? sprinklers moving in a circular path, the equations for

IZhe triangular and elliptic pattern sprinklers respectively

70

are:

N h l
_ _ 1 ”T: 2 2 _ 2 _
112nm)— 22‘:1 ,1} Th) 0 [S + R, 2R,S cosa] dc II 72

and

4’v £325 4n,(n,+m,).,%
222%,"):th — m,) [o 1- 2 31nd) 44> II-73
I=l

l-m,

Us ing the sprinkler discharges and the pattern radius taken
from the manufacturer’s handbook corresponding to the
sprinkler orifice size and pressure, Heermann and Hein
performed simulations of two different center-pivot systems.
The theoretical depth distribution compared favorably with
field data, validating the adequacy of the mathematical
calculated from

It'ICDoL'iel. The uniformities expressed by UCHH ,

tfile theoretical distribution and field data were, for all

Practical purposes, the same.
James (1982) developed a model combining generalized
Versions of Heermann and Hein (1968) and Kincaid and
Heermann (1970) models to study the performance of center-
pivot irrigation systems operating on variable topography.
In this model, radially symmetric individual sprinkler

distribution patterns as well as a part circle pattern for a

71

sprinkler mounted in the end of the lateral could also be
simulated. When comparing model predictions with field
data, James concluded that, for the systems and conditions
considered, the model was accurate and precise in predicting
the pressure along the lateral. However, while the model
accurately predicted the depth of application along the
lateral, average depth of application, flow rate at the
pivot and uniformity of application, its precision was not
as good. He also noticed that the accuracy and precision of
model predictions were not significantly improved by the use
of a radially symmetric individual sprinkler pattern rather
than a triangular pattern.

James (1984) also used computer simulations to study
the effect of pump selection and terrain on center-pivot
system performance. His results showed that the amount of
water applied, energy use and adequacy of irrigation (the %
of irrigated area receiving at least 90% of design
application depth) were significantly influenced by the
selection of the proper size rather than type of pump
(centrifugal or turbine). However, when constant discharge
nozzle sprinklers (or sprinklers with pressure regulators)
were used, pump size had little effect on the adequacy of
irrigation. The type or the pump size did not influence
significantly the uniformity of application. The effect of

terrain (zero to + 5%) was more pronounced in center-pivot

72

systems with conventional fixed nozzle impact sprinklers
rather than on systems with constant discharge nozzle
sprinklers. The effect of terrain on uniformity of
application was small and practically insignificant when
changing pump types.

James and Blair (1984) used computer simulations to
evaluate the theoretical performance of different low
pressure center-pivot configurations (conventional
sprinklers, low pressure impact sprinklers, fixed head spray
sprinklers mounted above the lateral, on drop tubes, and on
booms). The model was also used to study the effect of
terrain (slope) and sprinkler spacing on the performance of
the systems. The performance variables considered were:
uniformity of application, adequacy of irrigation (as
defined as in James, 1984), and energy use. Results from
simulations (288) showed that the systems with conventional
sprinklers, spray nozzle mounted on the lateral, and low
impact sprinklers had the highest overall uniformities.
Systems with conventional and low pressure impact sprinklers
had the highest uniformities and adequacies (of irrigation)
when spaced 12 m along the lateral. Systems with fixed head
spray nozzles had the highest uniformities and adequacies
when spaced 1.5 mm Terrain affected system energy use more
than uniformity or adequacy of irrigation. Systems with low

pressure impact sprinklers used 82% of the energy used by

73

systems with conventional sprinklers, while systems with
fixed head spray nozzles used only 68%.

Heermann and Swedensky (1984) derived equations for the
application rate of a polynomial pattern sprinkler, which
was believed to be more representative of the distribution
of fixed head spray nozzles. These equations are expressed

as:

AR, = 1: C1 + 7:": (1 - c1) 11-74
2

for m 5 C2

and

 

g

l - m II—75
l-C2 ( )

for m.> C2

where:
C; = the fraction of the maximum.application rate
at the sprinkler location,
C; = the fraction of the pattern radius from.the

sprinkler to the radius at which the maximum
application rate occurs,
m = the ratio of the perpendicular distance

between point p and the sprinkler line of

74

travel to the sprinkler pattern radius.

h = the maximum application rate

They also modified the Heermann and Hein (1968) model to
allow the simulation of part circle pattern sprinklers,
which is necessary for modeling systems with an end gun.

The computer model was then used to evaluate the effect of
the catch can spacing on the calculated unifonmity and
application efficiency of both high pressure (impact
sprinkler) and low pressure (spray nozzle) center-pivot
systems. The catch can spacing used varied from 15 cm to

12 m for the low pressure system.and from 30 cm.to 27 m, for
the high pressure systems. Application efficiency was
defined as the ratio of the integrated discharge for the
selected can spacing to the integrated discharge with the
smallest can spacing for the system” Simulations of systems
with high pressure sprinklers were performed assuming both a
triangular and elliptical distribution pattern for the
sprinklers. The change from triangular to a elliptical
pattern had very little effect on the uniformdty until the
catch can spacing approached the SCS recommended spacing of
9.1 m (30 ft). Variation in collector spacing showed little
influence in the uniformity on low pressure systems.
However, water application efficiencies, for both systems,

were quite variable when the catch can spacing changed.

75

2- UNIFORMITY ALONG THE PATH OF TRAVEL

Traditionally, when evaluating center-pivot irrigation
systems, the uniformity of water distribution is measured
radially from the pivot point extending to the end of the

wetted area as discussed in the previous section.

T _

Hanson and Wallender (1986) were one of the first
researchers to study the uniformity along the travel path
for mpving sprinkler machines. Because these machines
(center-pivot and linear-move systems) move in a series of
start and stop sequences and not continuously, tests were
performed with the objective of determining the uniformity
along the path of travel, and also of any possible
uniformity-movement relationship. In their test procedure
for center—pivot systems, the catch can were placed in
transects along the travel path installed inside towers
No.10 (the guide tower) and No.5. The spacing chosen was
0.3 mm Both transects were installed at a position
approximately underneath a nozzle. Additional transects
along the lateral and across the span of towers No.2, 5 and
9 were also installed. The cans were installed with a
spacing of 3 m for the transect along the lateral and 0.6 m
for the transects across the towers' span. Besides the

catch can data, the distance per move, the on/off-times of

76

the tower nearest to the transects, and all nozzles
pressures and discharges were recorded. Their results
showed that the uniformity along the travel path for the
transect inside tower No.10 span was much higher than for
the transect inside tower No.5 span (UCC=90% and 75%
respectively). Results of distance per move, on-times and
off-times were almost constant for the transect inside tower
No.10 span, but the results for the span inside tower No.5
were more variable. Spectral Analysis (analysis in the
frequency domain) was used in the analysis of the catch can
data. The analysis showed that a direct correlation existed
between tower No.5 movement and catch can variance. Their
explanation for such behavior was that the distances per
move of tower No.5 were large compared to the sprinklers’
wetted diameter, resulting in a reduced overlapping.
However, no similar behavior was identified for the other
towers, and as stated by the authors ".... the difficulty in
clearly relating nonuniformity in the can data with the
tower movement is apparently caused.by a complex interaction
between the tower movement and overlapping of the spray
.patterns along both the travel path and lateral ... ". Can
data of the transect along the lateral showed strong
periodic behavior starting at about 174 m from the pivot
point, however this behavior appeared to be independent of

the nozzle discharge. For such reason the authors

(. infill iii‘. \

 

77

hypothesized that this long range variability along the
lateral is also related to the tower movement. Sample
spectrums of the transects across the span of towers No.2, 5
and 9 showed that much of the nonuniformity in the catch can
data was related to the nozzle spacing. Also of great
importance was the finding that, in contrast with a linear-
move system, the nonuniformity along the path of travel of
the center-pivot occurred over large distances, therefore
with the potential of reducing yield.

Heermann and Stahl (1986) modified the Heermann and
Swedensky (1984) model to be able to simulate the start-stop
movement of the towers with the objective of evaluating the
uniformity along the travel direction of a high and low
pressure center-pivot systems. For the simulation of the
high-pressure system a triangular distribution pattern was
assumed for the sprinklers. For the simulation of the low-
pressure system, equations for the application rate for a
"doughnut” distribution pattern were derived. This
distribution pattern is similar to the polynomial pattern
developed by Heermann and Swedensky (1984) differing only in
the fact that the shape of the pattern from points of
maximum.application rate to the end of the pattern is
parabolic instead of linear. The two systems were simulated
first assuming a constant velocity (1.22 m/min) for all

towers, and then by decreasing the tower velocities toward

78

the pivot point. Simulations of the lower-pressure system
were performed for two different settings of the guide
tower, 60% and 100% , the later being the common setting
when doing Chemigation. Results showed that the radial
uniformity (alignment angle 1 1°) did not vary from the one
calculated when the system moved continuously. However,
results for the low-pressure system showed an almost 10%
reduction (from UCC: 98.9 to 89.0%) in the radial
uniformity. A slight improvement in the radial uniformity
was observed for the variable speed low-pressure system.as
compared with the constant speed system. The UCC along the
travel path was calculated from.depth measurements at a
radius of 230 m.and with the spaced 2 m. The uniformity was
calculated for simulations performed with alignment angles
of 2, 1, 0.5 and 0.25°. Although the UCC increased with a
decrease of the alignment angles for both systems, more
pronounced changes were observed in the low-pressure system.
Increasing the setting of the guide tower from 60% to 100%
increased the uniformity of the lowepressure system
(alignment angle of 1°) from‘UCC= 82.0 to 95.6%. Reducing
the time step from 1.2 to 0.6 sec in generating the start-
stop time series also resulted in a slight increase of the
UCC. Such increase was due to the more randomness of
nonuniformity for time step equal to 1.2 sec, and not to the

magnitude of deviations from the average depth, which to the

his... _ .3 ....

 

79

authors’ surprise were not reduced with the smaller time
interval. After performing the spectral analysis of the
results in a similar way as Hanson and Wallender (1986),
Heermann and Stahl concluded that the UCC along the travel
path is more a function of the sprinkler pattern radius and
of the magnitude of the arc lengths (the relative trajectory
of the can position on the sprinkler pattern) than the

magnitude of the alignment angle.

3"

I I I - METHODOLOGY

A. FIELD EXPERIDENTATION

Field experiments to generate data for validation of
the computer model presented in the next section were
conducted during the summers of 1991 and 1992, at the farm
property owned by Chris Rajzer located at 76301 M-51,
Decatur, Michigan. The center-pivot evaluated was a Valley
6000 three tower electrical system with a total length of
192.3 m (631.0 ft) and a capacity of 1514 L/min (400 gpm) at
a pivot-point pressure of 262 kPa (38 psi). All three spans
had the same length of 55.7 m (182.9 ft). The system also
had an overhang of 25.1 m (82.4 ft) beyond the last tower
adding to its total length. The system was equipped with
pressure regulating nozzles and with Valmont Spray
sprinklers mounted at the top of the lateral and at a
uniform.spacing of 9.1 ft. An end-gun (model SR100) was
mounted at the end of the lateral.

The field procedure adopted was kept very simple so it
could be performed by anyone. The procedure consisted of
placing a series of catch cans spaced 0.6 m.(2 ft) apart in

the travel path (in a circular arc) of the last sprinkler of

80

81

each lateral span. The last sprinkler of each span was
chosen because it was believed that any influence of the
intermittent movement of a tower would be more pronounced
closer to it. Every catch can had the same shape (section
of cone), same dimensions and conformed with ANSI/ASAE
$330.1 standard. Each catch can was mounted on stakes made
of 1/2 in PVC pipe at a height of 1.7 m (5.5 ft). The
position of each tower at each stop point was marked with
wire flags from the moment the first droplets started
falling in the first catch can of each transect until the
moment water stopped wetting the last can of the same
transect. The time a tower was moving (on-times) and the
time a tower stopped (off—times) in a given position were
measured using a stop watch. Immediately after the water
stopped wetting the last can in each transect the volume
(ma) of water deposited in each catch can was measured using
a 100 ml ( t 0.5 m1) volumetric cylinder. At last, the
alignment angles between tower No.1 (the guiding tower) and
tower No.2 and between tower No.2 and tower No.3 were
measured using a theodolite. The procedure was, to mark
with a wire flag the position of tower No. X each time tower
No. X+1 stopped and.when it resumed movement. In cases when
tower No. x+2 moved within the time interval, the
measurements were discarded. Every time tower No. x+2 moved

‘within the time interval, it would move the lateral span

82

between it and tower No. X+1 which would cause an error in
the angle measurement. It would cause the angle measured to
be larger or smaller depending on whether tower No. X was
lagging or leading tower No. X+1. The results of the field
experimentation is presented in the Computer Validation

section.

B . COMPUTER MODEL
1 . MODEL DEVELOPMENT

A computer model was developed to perform the
simulation of an electrical driven center-pivot system
considering its intermittent movement (the series of starts
and stops of the towers) with the towers operating in a
level terrain at constant velocities. That is, the velocity
of each tower is constant when the tower is moving.

However, the velocities of the towers may or may not be
different. The computer model was based on the mathematical
model developed by Bittinger and Logenbaugh (1962)

presented in chapter II as equations II-62 to equations II—
69. With exception of equation II-67 all other equations
can be readily evaluated when the assumption of continuous

movement and constant velocities is made. However, when the

83

intermittent movement of a center-pivot system is considered
and with sprinkler pattern profiles other than triangular or
elliptical, a mathematical model does not exist. When
considering the intermittent movement of a center-pivot
system, the contribution of an individual sprinkler to the
total depth at any given point p, is equal to the integral
of the sprinkler application rate over the interval of time
necessary for the sprinkler pattern to pass through point

p, and is given by:

tiff: A(t)dt III-l

where:
dg = depth of application at point p by an
individual sprinkler,

A(t) = the application rate function.

Considering the overlapping of different sprinklers

contributing to the total depth at any point p, we have:

T.
_ Ta _
DP - g]: [o A,(t) dt III 2

.In order to carry out the evaluation of the above

84

equation it is necessary to know not only the individual
time it takes for each sprinkler profile to move through
point p but also to know at each instant in time (or at each
time iteration) the exact distance from each contributing
sprinkler to the point being wetted. These calculations as
well as the solution of equations above are performed by the
computer model developed in this study. The flowchart of
the computer model is given below and the computer code is

presented in the appendix.

2. MODEL DESCRIPTION

a. INPUT REQUIREMENTS

The required inputs to initiate the simulation are:

1) the system’s total number of towers,

2) the length of the spans between towers (it is
assumed that all spans are of the same length), ft

3) the sprinkler spacing (assumed the same throughout
the system), ft

4) the angle of alignment between towers (different
angles between different towers are allowed),
degrees

5) the number of rows of catch cans

V

85

 

 

 

 

 

 

CALL SUBROUTINE
CANPOS

FORI-I

    

 

    
 

TO NUMBER OF
TOWERS

  
  

  
 

IF
TOWER IS
MOVING

      
 

 

CALCULATE TOWER
COORDINATES

4

CALL SUBROUTINE
SPRPOS

 

 

 

 

 

  
 
 
 
 

FOR I I I TO
NUMBER OF
SPRINKLERS

 

 

 

 

 

 

 
 

 

IF

 
 
 
 
 
 
 
 

 
 
 
 
 

FOR II n I
TO NUMBER
OF CANS

  

 

CALC. DISTANCE
BETWEEN SPR. AND

 

 

 

 

 

  

DIST ‘ VETRAD

  

l m

 

ADD
CONTRIBUTION TO
CAN DEPTH

 

 

 

 

 

 

 

CALL SUBROUTINE TO
CALC. DEPTN
CONTRIBUTION

 

 

 

UPDATE TIME

W

 

 

 

 

Figure 11.

Computer Model Flowchart

 

 

 

6)

7)

8)

9)

10)

11)

12)

l3)

14)

15)

16)

17)

18)

86

the number of catch cans per row

the polar coordinates of the first can of
farthest transect from the pivot—point, radian
the distance from each row of catch cans to the
pivot-point, ft

the spacing between catch cans of the farthest
row from the pivot-point, ft

the setting of the guide tower timer, in %

the duration of the simulation, hours

the time between iterations, AT, seconds

choice of lateral spans where the sprinkler that
contribute to the depth in cans are located

the number of sprinklers and the sprinkler number
to be initialized in a chosen lateral span

the sprinkler pattern, triangular, elliptical,
polygonal or any of the actual patterns built in
the flow rate (gpm) and wetted radius (ft) if
triangular or elliptical pattern sprinklers were
chosen

the peak instantaneous application rate (in/h), the
distance (ft) from the sprinkler to the point of
maximum application rate and the application rate
directly under the sprinkler (in/h) if polygonal
pattern sprinkler were chosen

the wetted radius (ft) for each sprinkler if a

87

actual pattern sprinkler were chosen.

b. MODEL EXECUTION PHASE

After the inputs are entered, the coordinates of the
catch cans are computed (SUBROUTINE CANPOS) in each
transect. The spacings between the cans are computed in
such way that the cans of same number in different transect
are positioned radially. If spacing between cans of
different rows are desired to be equal it is necessary to
run different simulations for each row. The coordinates of
the sprinklers chosen to perform the simulation are computed
(SUBROUTINE SPRPOS). At the beginning of the simulation all
towers with the exception of the guide tower are not moving,
"OFF“ state. The state of the guide tower is controlled by
its setting while the state of the other towers is
controlled by their alignment angle. A tower state remains
”ON" or “OFF" as long the angle formed by the adjacent
lateral spans do not equal or exceed the pre—set alignment
angle for that tower. At each time iteration the positions
0f the towers are updated if their state is "ON", and the
new sprinklers coordinates are calculated along the lateral
sPan(s) (SUBROUTINE SPRPOS) chosen, keeping the

predetermined space between them. The distances between

 

88

every sprinkler and all the catch can is computed at each
iteration and if it is smaller or equal to the wetted radius
of the sprinkler considered, the sprinkler contribution to
the total depth caught in that can is calculated and added.
The computation to the depth of water caught in a can by a
contributing sprinkler is done according to its pattern

(subroutines DEPTRIPAT, POLIPAT, ACTPAT).

c. SPRINKLER PATTERNS

The model allows a choice of different sprinkler
profiles to perform the simulation. The profiles available
are:

1— Triangular profile

2- Elliptical profile

3- Polygonal profile

4— Actual profile from distribution can data

1- GEOMETRICAL PATTERNS

The triangular and elliptical patterns are more

appropriate when simulating high and low impact sprinklers,

While the polygonal pattern is believed to give a better

89

representation of spray nozzles.

For the triangular and elliptical profiles the maximum
application rate occurs underneath the sprinkler position in
the middle of the profile. The application rate decreases
linearly towards the end of the triangular profile and it
follows the ellipse equation for the elliptical profile.
Figure 11 illustrates these profiles. The polygonal profile
available in the model is the profile derived by Heermann
and Swedensky (1984) and presented in chapter II by

equations II—74 and II—75 and rewritten here as:

 

A=h cl+fl(l-cl) III-3
62
for m.$ c2
and
_ h
A- (l -m) III-4
(1 -c2)

for m > c2
where:
c1 = the ratio between the application rate
underneath the sprinkler and the maximum
application rate,
c2 = the ratio between the distance from the
sprinkler to the point of maximum application

rate and the pattern wetted radius

APPLICATION RATE (L/I')

 

9O

 

\
I \

SPRINKLER POSITION

f
7

Figure 12. Geometrical Sprinkler Patterns.

+R

91

m = the ratio between the perpendicular distance

between point p and the sprinkler line of

travel to the sprinkler pattern radius.

h = the maximum application rate

However, the inputs required in the model are h,

cfircaR therefore being necessary to modify the above

equations, as:

... _h__ _
A R_CZR(R L)

for RZL>C2R

h1-
= _(___c.l_)L + CI];

for L .<_ czR

III-5

III-6

where:
R = the pattern radius,
ch = the distance from the sprinkler to the point
of maximum application rate,
Clh =

the application rate underneath the sprinkler,

L = the distance between the catch can and the

sprinkler.

92

2- ACTUAL PATTERNS

The actual profiles available in the computer model
are:
1) Nelson R30 - U4 nozzle #20 (5/32) 3RN at 30 PSI
2) Nelson R30 - U4 nozzle #30 (15/64) 3RN at 30 PSI
3) Nelson R30 - U4 nozzle #40 (5/16) 3RN at 30 PSI
4) Nelson R30 - D6 nozzle #20 (5/32) 3RN at 30 PSI
5) Nelson R30 - D6 nozzle #30 (15/64) 3RN at 30 PSI
6) Nelson R30 - D6 nozzle #40 (5/16) 3RN at 30 PSI
7) Nelson R30 - D6C nozzle #18 (9/64) 3RN at 30 PSI
8) Nelson R30 — D6C nozzle #32 (1/4) 3RN at 30 PSI
9) Nelson R30 - D4 nozzle #20 (5/32) 3RN at 30 PSI
10) Nelson R30 - D4 nozzle #30 (15/64) 3RN at 30 PSI

11) Nelson R30 - D4 nozzle #40 (5/16) 3RN at 30 PSI

The actual profiles were obtained directly from.the
IT‘acnufacturerl and were expressed by one or more higher
C>erder polynomials fitted to the total or partial data by

1.1.3; :lng nonlinear regression (procedure NLIN - SAS/STAT ver.

6 ~ 03 by SAS Institute Inc., Cary, NC USA.) . Figure 13 shows

‘tllblse actual profile, the profile recreated using the higher

\

J.
inplie'l‘he exclusive use of NELSON sprinklers does not necessary
other S any superiority of their products over the products of
Eunufacturers.

93

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order polynomials as well as their R-Squared values for
spray Nelson R-30-U4 nozzle #20 3RN operating at 30.0 psi.
The figures for the other actual pattern sprays as well as
the order and coefficients of each polynomial used to

reconstruct the pattern are presented in the appendix.

3. Mbdel AT SENSITIVITY

When performing a computer simulation, there is

gyenerally a trade-off between accuracy and computation time.
Its the time step shrinks greater accuracy is obtained until

a point that further shrinking results in no more evident

gain in accuracy. Usually the choice of the most adequate

t:iJue step to perform any computer simulation depends on the

E“Englication, that is, how important the accuracy of the

results will be. Studying the effect of different time

Steps in the proposed model showed that the choice of the
t:jLIne step, due to the nature of the model, influenced the

£3IEFIuChronization of the towers as well as the number of

<2ly'cles per time interval and the times the towers changed

Es‘tléate. Tables 3 and 4 illustrate the last two concepts. It
ij‘:33 clear by examining these two tables that when using the
7:7‘5353u1ts of the simulation in applications where accurate

E“7’"DE-Sithons of the towers are required, a smaller time step is

95

recommended. Other parameters (design or management
variables) should also be considered when choosing the time
step for the simulation. The choice of the time step is
sensitive to the magnitude of the angle of alignment. By
increasing the angle of alignment from 1.0 to 2.0 degrees in
a simulation of a three tower system, the time step could be
increased four fold (from 0.0125 sec. to 0.05 sec) without
affecting the accuracy of the results (towers positions, on
and off-times) . The choice of the time step, however, was
not sensitive to the magnitude of the velocity of the
t7-(3Vovers. An increase of 50% in the velocity of the interior
tZOVArers (tower No.2 and No.3) of the same three tower
SYS tems, did not affect the accuracy of the results. The
Inag'r-iitude of the time step, however, had a much smaller
effect on the simulation of the depth of application
especially for systems equipped with larger wetted radius

Spr inkler patterns .

4. MODEL VALIDATION

The computer model developed was validated with the
lateJi‘al operating in both a continuous and intermittent

mode - Continuous mode is defined as the idealized state of

96

Tadolxe 3. Tower no.2 starting times for different AT’s,
guide tower timer setting equal to 100% and
alignment angle equal to 1.0 degrees.

             
      
   
 

TIME 'ON', SEC. (TOWER NO. 2)

  

TIME STEP, AT (sec.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.0125' 0.025 0.05 0;;

16.19 16.20 16.20 16.200

88.14 88.15 88.10 88.000

188.67 188.65 188.63 188.55 188.500
253.41 253.40 253.38 253.35 242.100

“ 343.91 343.93 343.83 343.80 341.300
“445.62 445.64 445.50 445.45 406.000
510.36 510.38 510.25 510.25 496.300
600.87 600.88 I 600.75 600.75 597.900
702.55 702.55 702.45 702.40 662.900
767.27 767.26 767.15 767.10 753.200
857.77 857.74 857.63 857.55 854.900
959.46 959.40 959.32 959.25 908.600

 

1024.19 1024.14 1024.10 1024.10 1007.800
1114.69 1114.63 1114.60 1114.65 1072.500

 

 

1216.40 1216.33 1216.28 1216.50 1162.700
1281.14 1281.05 1281.05 1270.05 1264.500
1
1371.65 1371.55 1371.55 1369.30 1318.200
l
|

 

 

 

 

1473.34 1473.20 1473.15 1434.05 1417.300

 

1538.06 1537.90 1537.85 1524.35 1482.100 ‘
1628.54 1628.38 1628.33 1626.00 1572.700
‘ZFIPDZO' 0.00 1.05 2.14 70.72 528.57

 

 

 

 

 

 

\

2
'32:; <2limulative time difference of the first twenty time "On"
e:E‘IIL‘ed to AT = 0.01 sec..

97

Table 4. Number of cycles/hr as function of time step,
guide tower timer setting equal to 100% and
alignment angle equal to 1.0 degrees.

 

" NUMBER OF CYCLES/HOUR "
" TIME STEP, AT (SEC) “

 

TOWER N0 .

 

 

 

 

 

 

 

 

 

 

all lateral spans moving continuously at the same angular
velocity, resulting in the perfect alignment of the lateral.
It should not be confused with the lateral movement when the
guide tower timer is set at 100%. Performing the validation
in a continuous mode had the objective of testing the
adequacy of the model, that is the adequacy of the
integration formula (Euler Integration) used by the
subroutines in computing the depth of application. The
simulation of a center-pivot system operating in a
continuous mode can be accomplished simply by choosing the
number of the towers of the system to be equal to one, i.e.
a long span. The objective of the validation of the model
operating intermittently was to test the adequacy of the
model in simulating the movement of the towers (on—times and

off-times) as well as the depth of application.

98

a. CONTINUOUS MODE OPERATION

The simulation of the depth of water application by a
center-pivot system (system F, Kincaid (1968)) operating in
a continuous mode was performed. The system had a 3470
L/min (917 gpm) capacity covering an area of 54.4 ha (134.3
acres) with an angular velocity of 0.118 rad/hr.

Simulations were performed for triangular and elliptical
sprinkler patterns. The simulation results were then
compared with the mathematical solution obtained by Kincaid
using equations II-70 and II—71. The simulation results
and the respective results found by Kincaid are presented in
Table 5 . A Linear Regression Analysis of these results was

also performed and it is summarized in Figures 14 and 15.

b. INTERMITTENT MODE OPERATION

The simulation of the three tower electrical center-
pivot system described in a previous section (field
experimentation) was performed. The angle of alignment
between towers used in the simulation was the average value

of the angles measured in the field experimentation. These

99

Table 5. Depth of application, mathematical solution
and simulation results with model operating in
a continuous mode for triangular and elliptical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

patterns.
DEPTH (IN)
DISTANCE MATHEMATICAL SOL. SIMULATION RESULTS
FROM
PIVOT'FT TRIANGULAR ELLIPTICAL TRIANGULAR ELLIPTICAL
440 0.945 0.873 0.925 0.889
450 0.916 0.886 0.900 0.911
460 0.907 0.866 0.893 0.898
470 0.917 0.856 0.907 0.876
480 0.890 0.882 0.984 0.911
490 0.884 0.881 0.878 0.917
500 0.893 0.843 0.889 0.866
540 0.900 0.842 0.888 0.861
550 0.890 0.876 0.893 0.904
560 0.884 0.841 0.876 0.872
570 0.891 0.848 0.879 0.868
580 0.893 0.891 0.879 0.917
590 0.888 0.866 0.873 0.893
600 0.892 0.922 0.872 0.833
740 0.931 0.943 0.942 0.949
750 0.925 0.939 0.937 0.948
760 0.909 0.870 0.917 0.882
770 0.909 0.913 0.914 0.917
780 0.901 0.922 0.906 0.927
790 0.882 0.866 0.884 0.873
900 0.876 0.883 0.873 0.880
910 0.868 0.885 0.865 0.883
J=====.

 

 

 

 

 

 

 

100

Table 5 (cont’d)

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0.
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0.
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values are presented below in table 6. The velocity of the
towers in a system is dependent on the speed of the motor,
the gear box ratio and the type of tires. The system
simulated was equipped with a 1.5 hp motor (high speed
drive) on towers No.1 (the guide tower) and tower No.2 and
with a 1 hp motor (normal speed drive) on tower No.3. All

three towers had the same type of tires, retread 11 X 24.5

101

 

    

 

 

 

DEPTH (in), Model Prediction
1
. 0.8 -
0.6 -
0.4 -
R-SOUARED - 0.988
02- ”255523333
0 I I I I
0 0.2 0.4 0.6 0.0 1
' MODEL PREDICTION DEPTH (in), Kincaicls systemF

Figure 14. Linear Regression Fit, model vs. Kincaid's system F
results for triangular pattern sprinklers.

102

 

    

 

 

 

DEPTH (in.), Model Prediction
1
0.8 -
0.6 —
0.4 —
R-SOUARED a 0.985
INTERCEPT : -0.025
0.2 ‘ SLOPE = 1.04
0 I I I I
o o.2 0.4 0.5 0.8 1

MODEL PREDICTION DEPTH (in.), l0ncaid’s SystemF

Figure 15. Linear Regression Fit, model vs. Kincaid’s system F
results for elliptical pattern sprinklers.

Table 6. Alignment angle between towers,

103

measurements .

 

 

field

ALIGNMENT ANGLE (DEGREES): "

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TOWER No.1 & TOWER No.2 & "
No.2 No.3
0.285 0.250
0.237 0.318
0.280 0.422
0.256 0.387
0.225 0.365 “
0.269 0.418
0.279 0.385
0.303 0.398 “
0.227 0.420
0.224 0.351
0.280 0.330
0.226 0.410
----- 0.380
----- 0.363
AVERAGE 0.258 0.371 “

 

 

Treble 7. Tower velocities used in the simulations.

 

 

 

 

 

 

F77 ANGULAR VELOCITY, rad/sec
TOWER No.1 0.0003595
TOWER No.2 0.0005395
TOWER No.3 0.0005767

 

 

104
tires. The tower velocities used in the simulation were the
velocities given by the manufacturer (Valmont Industries,
Inc - ) for the description above and are given in Table 7.
Simulations were performed for three different guide tower
percentage timer settings, 50, 37 and 25%. The results of
the simulations as well as the field data (on and off-times
for tower No.1 and No.2 only) and the water depth applied
in transect inside span No.1, 2 and 3, for the three
percentage timer settings of the guide tower are given in
figures 16 through 36. As can be seen, the simulations
resulted in periodic behavior similar to the field data.
The differences found could be due to the influence of
SeVeral factors, the most obvious being: slippage of tires,
non alignment of towers on beginning of the field
e’cperimentation, experimental error, small variations on the
alignment angle depending on if tower is leading or lagging

the adjacent outer tower and terrain effects.

5. MODEL APPLICATION

Simulations using the center-pivot computer model
de\re loped in this work were performed to make the analysis
of a traditional center-pivot and a LEPA system. Simulations

We:IE‘e performed with the objective of determining how the

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126

sprinkler application pattern, the magnitude of the
sprinkler wetted diameter, the magnitude of the tower
alignment angle and the guide tower timer percentage setting
affect the uniformity along the path of travel.

Triangular, elliptical and polygonal pattern sprinklers
as well as "actual" pattern sprinklers were compared. The
wetted diameters chosen to perform the simulations were the
wetted diameter of the "actual" pattern sprinklers and a
typical value for spray nozzles used for Chemigation with a
LEPA system (Buchleiter, 1992). The simulations were
performed with the towers moving continuously (for
triangular pattern sprinklers and LEPA spray nozzles) as
'well as moving intermittently (all patterns). In the later
case three different tower alignment angles were used: 2, 1
and 0.5 degrees. Simulations of the LEPA system were only
performed with a guide tower timer set of 100%. All other
simulations were performed with the guide tower timer set at
100% and 50%. The simulations generated application data at
different distances from the pivot-point as shown in tables
8 and 9. In all simulations the depth of application was
Idetermined at 40 positions along the travel path with a 0.3
In (1 ft) spacing. The flowrate of the sprinklers
contributing to the depth at each distance from the pivot-
;point.for the traditional center-pivot and LEPA system are

also given in the tables 8 and 9.

Table 8.

127

Flowrates and wetted radii for sprinklers

contributing to the depth of water applied at
different distances from the pivot—point for a
traditional center—pivot system.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE FROM SPRINKLER SPRINKLER
PIVOT-POINT (H0 FLOWRATE (L/min) ‘WETTED RADIUS (m)
56.0 10.7 “
292.7
56.0 9.8
56.0 8.8
189.1 36.3 10.1
32.2 11.0
170.8
32.2 9.4
32.2 8.8
14.4 10.7
.4 .5
.4 .6
.7 .4w

 

 

Table 9.

 

  

Flowrates and wetted radii for sprinklers

contributing to the depth of water applied at
different distances from.the pivot-point for a
LEPA system.

r————"—‘ - __n—. __. .nn —__—~_.._

( DISTANCE FROM
' PIVOT- POINT (m)

FLOWRATE (L/min)

SPRINKLER

WETTED RADIUS (m)

 

 

SPRINKLER
l
t

 

 

 

J 196.7 9.35 1. 0
( 140.3 6.67 1. o
. 85.4 4.05 1.0 J

 

 

128

All complementary information needed to perform the

Simulations, are presented in the summary of the system

specifications given below and in table 10.

NUMBER OF TOWERS: 7

SPAN LENGTH: 55.7 m (182.9 ft)

TOTAL SYSTEM LENGTH: 390.2 m (1280.3 ft)
SYSTEM CAPACITY: 2270.0 L/min (600 gpm)

SPRINKLER OPERATING PRESSURE: 206.8 Kpa (30 psi))

Table 10. Tower velocities used in center-pivot
simulation.

 

TOWER NUMBER ANGULAR VELOCITY (RAD/SEC)
.000153

.000179

.000215 I
.000268

.000179

.000268
I) “011537

 

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The system was designed to apply a net depth of

application equal to 3.8 mm (0.15 in) per revolution with

the guide tower timer set at 100%. Each revolution of the

system is completed in a 12 hr period at a guide tower timer

set at 100%

VI- RESULTS AND DISCUSSION:

The results and discussion of the data generated by the
simulations performed in the previous section, Model

Application, are presented in this section.

A. TRADITIONAL CENTER-PIVOT SYSTEMS

The uniformity Coefficient (UCW) and depth of
application averages of simulations performed are presented
in this section. The Swaile-Wilcox Uniformity Coefficient
(UCW) was chosen over the widely used Christiansen
Uniformity Coefficient (UCC) for reasons presented in
chapter II, that is it gives greater weight to the deviates
far from the average.

Simulations were performed with different sprinkler
jpatterns: triangular, elliptical, polygonal and actual
patterns for R—30 Nelson Spray Nozzles equipped with U4, D4,
D6, and D6C rotary plates. Application data were generated
at 5 different distances from the pivot-point, with the
system towers moving intermittently but for the triangular
sprinkler pattern it was also performed with the lateral

moving continuously. Simulations were performed for three

129

130

different angles of alignment, 2, 1, and 0.5 degrees. All
simulations were performed with the guide tower set at 100%
and at 50%. The results of the simulations are given in
tables 11 through 19. The simulated UCW ranged between
76.3% to 99.7% when the towers moved intermittently. UCW
for simulations with the lateral moving continuously (only
for the triangular sprinkler pattern) were for all practical
purposes equal to 100%. For that reason, simulations with
the lateral moving continuously and using other sprinkler
patterns were not performed. The same response would be
found. It is important to remember that these are the
highest values possible for each set of parameters. In
field conditions, the two most important factors responsible
for these values not being reached are probably distortions
in the sprinkler pattern caused by wind and slippage of the
tower tires.

Contrary to results found by other researchers
(Wallender and Hansen, 1986; Heermann and Stahl, 1988), the
uniformity coefficient did not always decrease toward the
center of the system, the area believed to present the most
irregular movement pattern. This finding is of extreme
importance because it raises the hypothesis that such
irregularity can be particular to the systems studied, and
that it may be a function of the towers’ velocities and the

alignment angles between towers.

131

TABLE 11. Uniformity Coefficient (UCW)' and Average Depths
(mm)" for Triangular Pattern Sprinklers with
guide tower set at 100%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE WETTED ALIGNMENT ANGLE
FROM RADIUS (m)
PIVOT-
POINT(m) 2.0 1.0 0.5
10,7 88.6' 96.5 98.7
3.78“ 3.43 3.53
292,7 9,3 85.9 95.4 98.5
3.87 3.47 3.57 n
t 8.8 82.9 II
t 3.91 3.45 3.56
I 189.0 10,1 99.1 99.4 98.3
‘ 3.21 3.33 3.52 n
| 1],,0 94.9 99.0 97.6 n
I 3.16 3.31 3.55 “
170.8 9,4 94.0 99.0 96.7 II
' 3.21 3.37 3.64 n
8.8 93.6 98.7 96.4 n
3.19 3.36 3.64 I
10,7 98.7 98.9 99.5
3.39 3.62 3.59
73,2 3,5 98.3 98.7 99.2
3.38 3.65 3.61
'7.6 97.9 98.6 98.8 n
3.31 3.63 3.54 I
55,0 7,0 97.0 98.4 98.3 II
_1§,;‘ J

 

132

TABLE '12. Uniformity Coefficient (UCW)' and Average Depths
(mm)" for Triangular Pattern Sprinklers with
guide tower set at 50%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE WETTED ALIGNMENT ANGLE
FROM RADIUS (m)
PIVOT-
POINT(m) 2.0 1.0 0.5
10,}7 88.8 99.3 99.7
6.65 7.16 7.01
292,7 9,3 85.8 98.4 99.7
6.88 7.26 7.09
3,3 83.2 96.3 99.7
6.88 7.26 7.09
189,0 10,1 95.6 96.5 98.1
7.92- 6.68 6.71
1],,0 89.7 92.7 97.6
7.57 5.36 6.07
170.8 9,4 ' 89.6 91.4 95.6
7.95 5.28 6.02
3,3 89.7 90.8 94.5
8.03 5.21 5.94
10,7 91.5 99.2 99.3
6.15 6.96 7.09
73,2 3,5 89.8 97.4 98.3
6.12 7.06 7.16
'7,5 88.9 97.3 97.8
6.20 7.16 7.11
55,0 7,0 85.8 93.0 97.6
6.81 6.35 7.39

 

 

 

 

133

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 13. Uniformity Eoefficients (UCW)' and Average
Depths (mm) for Triangular Pattern Sprin-
klers and lateral moving continuously with
guide tower timer set at 100 and 50%.

DISTANCE WETTED TIMER SET PERCENTAGE, %
FROM RADIUS
PIVOT- (M)
POINT (m) 100 50
10,7 100.0 99.95
3.51 7.04
292.7 9.8 100.0 99.92
3.56 7.11
8.8 100.0 99.90
3.56 7.09
189.0 10,1 100.0 99.88
3.56 6.86
11,0 100.0 99.37
3.43 6.93
170.8 9.4 100.0 99.44
3.51 7.24
8,8 100.0 99.50
3.48 7.34
10,7 100.0 99.85
2.79 6.83 '
73.2 8.5 100.0 99.94
3.02 7.01
7,6 100.0 99.91
3.12 7.09
55.0 7,0 100.0 99.79
3.45 7.47

 

 

 

 

 

 

 

134

TABLE 14. Uniformity Coefficients (UCW)' and Average Depths
(mm)" for Elliptical Pattern Sprinklers with
guide tower timer set at 100%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE WETTED ALIGNMENT ANGLE (DEGREES)
FROM RADIUS (m)
PIVOT-
POINT(m) 2.0 L0 0.5
10,7 94.3 ' 98.5 98.8
3.61 " 3.38 3.45
292,7 9,8 91.2 96.3 98.6
3.78 3.51 3.58
8,8 87.8 94.7 98.3
3.91 3.56 3.63
189,0 10,1 98.1 99.2 99.0
3.15 3.23 3.40
11,0 95.6 98.3 98.4
3.09 3.22 3.41
170.8 9,4 94.6 98.4 97.6
3.25 3.39 3.64
8,8 94.1 98.7 97.2
3.28 3.42 3.69
10,7 97.9 98.3 99.0
3.39 3.62 3.55
73,2 8,5 97.9 98.4 99.2
3.55 3.84 3.74
7,6 96.4 98.8 98.6
H 3.46 3.75 3.64
55,0 7,0 95.8 97.9 98.1
3.40 3.45 3.68 =

 

 

 

 

135

TABLE 15 . Uniformity coefficients (UCW)* and Average Depths
(mm)" for Elliptical Pattern Sprinklers with
guide tower set at 50%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE WETTED ALIGNMENT ANGLE (DEGREES)

FROM RADIUS (m)
PIVOT-

POINT(m) 210 1.0 0.5

10,7 93.2 ' 96.5 99.5

6.68 “ 7.01 6.88

292,7 9,8 90.3 97.1 99.3

6.86 7.24 7.11

8,8 86.4 98.4 98.7

6.99 7.39 7.24

189,0 10,1 91.1 96.7 97.9

7.72 6.43 6.63

11,0 92.5 96.6 98.9

7.25 6.64 6.60

170.8 9,4 89.6 96.7 98.9

7.82 6.99 6.93

8,8 89.7 97.4 98.7

8.00 7.06 6.99

10,7 94.2 99.3 99.3

6.40 7.15 7.12

73,2 8,5 91.0 98.8 98.9

6.62 7.53 7.51

7,6 89.3 98.4 98.9

6.47 7.36 7.33

55,0 7,0 88.0 93.4 97.4

6.76 6.76 7.39

 

 

 

 

 

 

 

 

136

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TABLE 16 . Uniformity Coefficients (UCW)' and Average Depths
(mm) for Polygonal Pattern Sprinklers with
guide tower timer set at 100%
DISTANCE WETTED ALIGNMENT ANGLE (DEGREES)
FROM RADIUS (m)
PIVOT-
POINT(m) 2.0 41;0 0.5
10,7 92.1 ‘ 96.5 98.7
3.40 “ 3.18 3.23
292,7 9,3 89.5 94.8 98.5
3.43 3.15 3.20
3,3 86.7 96.7 98.2
3.35 3.05 3.10
189,0 10,1 97.5 98.4 98.8
2.92 3.05 3.20
11,0 95.1 98.2 98.2
2.91 3.02 3.23
170,8 9,4 94.0 98.3 97.5
2.87 2.99 3.23
3,3 93.5 98.7 97.1
2.79 2.91 3.15
10,7 97.9 98.1 99.3
3.15 3.38 3.30
73.2 8.5 98.3 98.2 98.8
2.94 3.19 3.10
'7.6 97.4 98.6 98.2
2.72 2.96 2.86
55,0 7,0 96.1 97.2 97.2
2.97 3.18 3.07

 

 

 

 

 

 

 

137

TABLE 17 . Uniformity Coefficients (UCW)* and Average Depths
(mm)" for polygonal pattern sprinklers with
guide tower timer set at 50%

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE ‘WETTED ALIGNMENT ANGLE (DEGREES)
FROM RADIUS (m)
PIVOT—
POINT(m) 2.0 110 0.5
10,7 91.2 ' 96.7 99.2
6.22 “ 6.55 6.43
292,7 9,8 88.1 97.6 98.5
6.15 6.50 6.38
8,8 84.7 98.4 98.0
5.92 6.30 6.17
189,0 10,1 90.0 97.2 98.1
7.14 6.10 6.20
11,0 90.2 96.3 99.0
6.88 6.23 6.20
170.8 9,4 88.2 96.9 98.8
6.95 6.16 6.12
8,8 88.4 97.3 98.6
6.83 6.01 5.95
10,7 93.9 99.2 99.0
5.91 6.65 6.63
73.2 8,5 90.1 98.6 98.6
5.48 6.26 6.24
7,6 88.7 98.7 98.7
5.06 5.79 5.77
55,0 7,0 92.2 94.9 98.0
6.48 6.45 6.63

 

 

 

 

 

 

138

TABLE 18. Uniformity Coefficients (UCW)' and Average Depths
(mm)' for Actual Pattern Sprinklers with guide
tower timer set at 100%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE SPRINKLER ALIGNMENT ANGLE (DEGREES)

FROM TYPE
PIVOT-

POINTQR) 2.0 1.0 0.5

1230-114 94.6‘ 97.3 98.6

NOZZLE #40 4.01“ 3.81 3.86

292,7 1230-134 90.7 93.1 96.6

NOZZLE #40 3.51 3.43 3.43

330.95 86.8 92.3 97.5

NOZZLE #40 3.96 3.63 3.68

139,0 R30-D6C 97.8 99.0 98.2

NOZZLE #32 3.61 3.75 4.00

R30-U4 95.9 97.5 98.6

NOZZLE #30 3.61 3.73 3.91

170,3 1230-134 94.0 94.0 97.3

- NOZZLE #30 3.35 3.42 3.64

R30-D6 93.7 98.5 97.2

NOZZLE #30 3.40 3.54 3.82

R30-U4 95.3 97.1 97.5

NOZZLE #20 3.45 3.63 3.57

73,2 R30-D4 93.0 95.9 97.9

NOZZLE #20 4.05 4.37 4.27

1130-135 97.3 98.8 98.3

NOZZLE #20 3.22 3.51 3.40

55,0 R30-D6C 92.3 95.4 97.9

NOZZLE #18 3.67 3.98 3.85

 

 

 

 

 

 

 

139

TABLE 19. Uniformity Coefficients (UCW)' and Average Depths
(mm)" for Actual Pattern Sprinklers with guide
tower timer set at 50%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE SPRINKLER ALIGNMENT ANGLE (DEGREES)
FROM TYPE
PIVOT-
POINT(m) 2.0 1.0 0.5
R30—U4 94.1 ' 94.4 99.2
NOZZLE #40 7.44 " 7.82 7.70
292.7 R30-D4 92.5 90.7 97.2
NOZZLE #40 6.53 6.91 6.83
R30-D6 83.4 95.9 96.6
NOZZLE #40 6.99 7.49 7.32
189,0 R30-D6C 95.6 96.9 98.9
NOZZLE #32 8.69 7.82 7.62
R30-U4 91.2 95.3 98.0
NOZZLE #30 8.14 7.72 7.68
170,8 R30—D4 76.3 90.7 93.0
NOZZLE #30 7.52 7.04 7.06
R30-D6 88.3 96.9 98.4
NOZZLE #30 8.27 7.30 7.24
R30-U4 91.9 97.3 98.7
NOZZLE #20 6.55 7.20 7.16
73.2 R30-D4 89.4 96.3 97.1
NOZZLE #20 7.55 8.57 8.55
R30—D6 88.6 98.3 98.8
NOZZLE #20 6.02 6.87 6.85
55,0 R30-D6C 83.1 995.4 98.0
NOZZLE #18 6.73 7.76 7.79

 

 

 

 

 

 

".

140

1- ANALYSIS OF VARIANCE RESULTS

An Analysis of Variance (ANOVA) was done for the
results of the geometrical sprinkler patterns (triangular,
elliptical and polygonal) at distances equal to 292.7, 170.8
and 73.2 m from the pivot-point. Results of simulations
using the actual sprinkler pattern were not included in the
ANOVA because the sprinkler pattern is not the same at
different distances from the pivot-point. A split-plot
design was used with the distances from the pivot—point as
blocks, and the other parameters: alignment angle (AA);
guide tower setting (GT); sprinkler profile (SS); and wetted
radius (WR) as factors. The level of each factor is given
in the ANOVA results below. The dependent variable was the

coefficient of uniformity, UCW.

ANOVA RESULTS

Analysis of Variance Procedure
Class Level Information

Class Levels Values
BLOCK 3 1 2 3
AA 3 1 2 3
SS 3 1 2 3
WR 3 1 2 3
GT 2 1 2

number of observations in data set = 162

 

141

ANOVA RESULTS (cont’d)

Dependent Variable: UCW

Sum of Mean

Source DF Squares Square P Value Pr > F
Model 97 2482.358704 25.591327 69.86 0 0001
Error 64 23.446296 0.366348
Corrected
Total 161 2505.805000

R-Square C.V. Root MSE UCW Mean

0.990643 0.630304 0.605267 96.0277778

Analysis of Variance Procedure

Dependent Variable: UCW
Source DF Anova SS Mean Square F Value Pr > F
BLOCK 2 185.300370 92.650185 252.90 0.0001
.AA 2 1539.917037 769.958519 2101.71 0.0001
BLOCK*AA 4 181.745926 45.436481 124.03 0.0001
SS 2 7.002593 3.501296 9.56 0.0002
BLOCK*SS 4 17.747037 4.436759 12.11 0.0001
AA*SS 4 15.319259 3.829815 10.45 0.0001
BLOCK*AA*SS 8 39.891111 4.986389 13.61 0.0001
EH! 2 56.934444 28.467222 77.71 0.0001
BLOCK*WR 4 17.292963 4.323241 11.80 0.0001
.AA*WR 4 56.505185 14.126296 38.56 0.0001
BLOCK*AA*WR 8 20.950741 2.618843 7.15 0.0001
SS*WR 4 1.202963 0.300741 0.82 0.5166
BLOCK*SS*WR 8 1.452963 0.181620 0.50 0.8548
.AA*SS*WR 8 4.348519 0.543565 1.48 0.1808
GT 1 39.803025 39.803025 108.65 0.0001
BLOCK*GT 2 47.743086 23.871543 65.16 0.0001
.AA*GT 2 165.751605 82.875802 226.22 0.0001
BLOCK*AA*GT 4 57.641728 14.410432 39.34 0.0001
SS*GT 2 3.857160 1.928580 5.26 0.0076
BLOCK*SS*GT 4 6.229506 1.557377 4.25 0.0041
.AA*SS*GT 4 1.418765 0.354691 0.97 0.4312
'WR*GT 2 0.686790 0.343395 0.94 0.3970
BLOCK*WR*GT 4 6.342099 1.585525 4.33 0.0037
.AA*WR*GT 4 6.933580 1.733395 4.73 0.0021
SS*WR*GT 4 0.340247 0.085062 0.23 0.9193

 

142

Among the factors included in this experiment, the
alignment angle and the guide tower timer setting affect the
tower movement which in turn affects the uniformity of water
distribution along the path of travel. Sprinkler pattern
and magnitude of wetted radius each affect the uniformity
along the path of travel directly by determining the
instantaneous application rate at any given point and the
time required for the sprinkler pattern to move through the
transect. The ANOVA showed all two and three factor
interactions to be significant at a = 0.01, with the
exception of the interactions where sprinkler pattern and
wetted radius factor appear together. The reason for the
interactions being significant is due to the response of one
factor in the presence of another factor (0r combination of
factors) being not constant for all factor levels. Although
meaningful observations on the main effects can be difficult
to make in face of the interactions among factors being

significant, an attempt is made in the following pages.

2- MAIN EFFECTS

a. ALIGNMENT ANGLE:

If no interactions involving the alignment angle were

 

143

existent, it would be logical to assume that the smaller the
alignment angle between towers, the higher the coefficient
of uniformity along the path of travel. Because the
interactions were significant the results showed this was
not always the case. At 292.7 m from the pivot-point all
results but one, the UCW increased when decreasing the
alignment angle from 2 to 0.5 degrees, independent of the
guide-tower setting, sprinkler pattern and magnitude of
wetted radius. Figures 37 to 39 show the relative position
of the lateral, time-off of closest tower and depth of
application for the transect at 292.7 m from the pivot-point
for triangular pattern sprinklers. However, at 170.8 m from
the pivot-point and particularly when the guide-tower was
set at 100% and for a smaller wetted radius, the coefficient
of uniformity was in many instances higher for the
distributions generated with the alignment angle equal to 1
degree than for 0.5 degrees. It is hypothesized that due to
interactions, the movement of the lateral over the transect
‘was mbre irregular (irregular start-stop cycles and variable
off—times) when the alignment angle was equal to 0.5
degrees. The movement of the lateral (or sprinkler) over any
given.point is a function of the state of its adjacent
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150

tower is stopped and inner tower is moving (Heermann and
Sthal, 1986). Unless a sprinkler is mounted exactly at a
tower position, the sprinkler is always moving except for

the first combination above. It is also clear that the

 

closer a sprinkler is to a tower, the more its movement and

'h

velocity will be influenced by the state and velocity of the

tower. Figures 40 through 42 show the relative lateral

}

position, off-times of closest tower at each of its stops
and the depth of application distributions for some cases
'where the coefficient of uniformity was higher for an

alignment angle equal to 1 degree than 0.5 degrees.

b. GUIDE-TOWER SETTING:

Decreasing the guide-tower setting from.100% to 50% had
the effect of doubling the depth of application as expected.
Theoretically, the time for the lateral to move over a set
of catch cans at any given transect would also be doubled.
Usually, the longer time to complete a pass, the higher the
number of start—stop cycles, resulting in better overlapping
of the sprinkler patterns and a more uniform distribution.
No general trend was identified by examining the data
presented. However, when examining the averages (see

table No.20) for each sprinkler pattern and alignment angle

151

at each.transect some observations can be made. For any
sprinkler pattern and alignment angle equal to 2 degrees,
the average of the coefficient of uniformity at any distance
from the pivot-point decreased when the guide-tower setting
‘was changed from 100% to 50%. For alignment angles equal to
l and 0.5 degrees, with one exception (see table 21), the
averages of the coefficient of uniformity increased when the
guide-tower setting was changed from 100% to 50%. This
confirms the importance of the interaction involving the

alignment angle, guide-tower setting and distance from.the

pivot-point.

C. SPRINKLER PATTERN:
1- GEOMETRICAL PATTERNS

Even though the overall UCW average for the elliptical
sprinkler pattern (96.3%) was slightly higher than for the
triangular (95.8%) and polygonal (95.9%) sprinkler patterns,
at smaller alignment angles the averages were practically
the same (98.3, 98.2, and 98.1%). The lowest and highest UCW
averages were for the triangular sprinkler pattern with
alignment angles equal to 2 and 0.5 degrees respectively. A

possible reason for the similarity in the results is

I

152

TABLE 20. Uniformity Coefficient Averages for different
Alignment angles and guide tower settings at
different distances from the pivot-point.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE FROM PIVOT-POINT
(m)
SPRINKLER GUIDE A.ANGLE
PATTERN TOWER % (DEGREEs)
292.7 170.8 73.2
2.0 85.8 94.2 97.7
1.0 95.5 98.9 98.7
100
0.5 98.5 96.9 98.8
TRIANGULAR 2.0 85.9 92.2 90.0
1.0 98.0 97.6 98.8
50
0.5 99.7 98.7 99.2
2.0 91.1 94.8 97.4
1.0 96.5 98.5 98.5
100
0.5 98.6 97.7 98.9
ELLIPTICAL 2.0 90.0 90.6 91.5
1.0 97.3 96.8 98.8
50
0.5 99.2 98.8 99.0
2.0 89.4 94.2 97.9
1.0 96.0 98.4 98.3
100
0.5 98.5 97.6 98.8
POLYGONAL 2.0 88.0 88.9 90.9
1 0 97.6 96.8 98.8
50
0.5 98.6 98.8 98.8

 

 

 

 

 

 

 

 

 

 

153

probably due to the equalizing effect of the overlapping of

different sprinklers. More will be said with respect to the

geometrical Sprinkler pattern when discussing the results of

simulations of the LEPA center-pivot system.

2- ACTUAL SPRINKLER PATTERNS

The UCW results of simulations performed using the

actual sprinkler patterns of R-30 NELSON SPRAY NOZZLES

equipped with U4, D4, D6, and D6C rotary plates operating at

207 kPa (30 psi) were in most cases lower than those

obtained with geometrical sprinkler patterns. They ranged

between 76.3 and 99.2%. Different rotary plates showed

different responses, i.e, the results of the D6 pattern

were lower than the geometrical patterns only at 292.7 m

from the pivot-point while the results of U4 and D4 patterns

were lower at all distances from it. With few exceptions,

results for D6C patterns were more like those obtained with

geometrical patterns. The actual sprinkler pattern results

can also be compared among themselves; however, when doing

it is necessary to remember that the wetted radius

in this

so,
varies from one pattern to another. Therefore,

sprinkler pattern comparison an implicit comparison of the

 

 

154

wetted radii is also being made. When comparing different

Sprinkler patterns it would be possible for one to be
superior to another, based solely in its shape, but it might

not produce a better distribution along the path of travel

because of its smaller wetted radius. In tables 18 and 19

at distances of 73.2 m and 170.8 m from the pivot-point, the
results of D6 patterns were higher than those of U4 and D4

patterns specially at smaller alignment angles (1 and 0.5
at 292.7 m from the pivot-point, with

- 50%, alignment angle

However ,

degrees).
one exception (guide tower setting

the highest values were for the U4 pattern.

1.0 degree),
Despite the higher uniformities obtained by simulation with

U4 rotary plates as compared to the ones with D4 plates,
Spray nozzles

under field conditions this may not be so.
equipped with U4 rotary plates are mounted on the lateral at

a height approximately equal to 3.7 m (12 ft) while spray

nozzles with D4 plates are mounted on drop tubes much closer

to the ground making their pattern less susceptible to wind

distortion.

(3. ENEHIEED RADIUS:

In general, the results found showed that larger wetted

radii sprinkler patterns resulted in more overlapping (with

 

p

 

155

itself in the direction of travel and with other sprinklers)

and more uniform distribution along the path of travel.
Exceptions were found where UCW decreased with an increase
in the magnitude of the wetted radius when the alignment

angle was equal to 1.0 degree and at distances equal to

170.8 and 73.2 m from the pivot-point.

e. DISTANCE FROM PIVOT-POINT:

The data did not show a trend in UCW values with

However, higher values of

distance from the pivot—point.

UCW were found for the transect located at 189.0 m from the

pivot-point than for the one at 170.8 m, specially with the

guide tower timer set at 100% and alignment angles equal to

2 and 1 degrees. One possible explanation is that the

transect located at 189.0 m from the pivot-point is

positioned about the middle of the span between towers no.4

and no.5. The movement of the lateral going through this
transect would be equally affected by the state and movement

of both towers. The lateral would be moving, unless both
The transect at 170.8 m from the pivot-

towers were off.
and the

point is positioned only at 11.6 m from tower no.5,

movement of the lateral at that point is largely influenced

by the state of that tower The lateral would be stopped

 

"h

 

156

or moving very slowly if the tower no.5 were off, depending

on the state of tower no.6.

B . LEPA SYSTEMS

The simulation results using LEPA sprinkler packages
are presented in table 21. The spacing between sprinklers
was kept constant along the lateral and equal to 1.52 m (5
ft). The uniformity of water distribution was determined at
three distances from the pivot-point, 196.7, 140.3 and 85.4
m (645.3, 460.3 and 280.3 ft), just underneath a sprinkler.
Each transect contained 40 collector cans spaced at 0.305 m
(1 ft) in the same way as the simulations for the
traditional center—pivot system. The wetted radii of the
sprinklers at the three different distances from the pivot-
point were equal to 1.0 m (3.3 ft). Simulations were
performed with the lateral moving continuously and
intermittently. The alignment angles used when the lateral
moved intermittently were, 2, l, and 0.5 degrees. The
simulations were only performed with the guide-tower set at
100%, which is the case when application of chemicals
(Chemigation) is done with irrigation water.
The results showed clearly that the difference between

the continuous and intermittent moving systems becomes much

 

h.

 

157

TABLE 21. Uniformity Coefficients (UCW)' and Average Depths
(mm)" for Triangular, Elliptical and Polygonal
Pattern LEPA sprinklers moving intermittently and
continuously with guide tower timer set at 100%.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

DISTANCE SPRINKLER ALIGNMENT ANGLE (DEGREES )
FROM PATTERN
PIVOT-
POINT(m) 2.0 1.0 0.5 CONT.
TRIANGULAR 52.2 ‘ 70.2 80.1 100.0
4.76 “ 4.77 4.76 4.45
54.2 74.5 83.9 100.0
ELLIPTICAL 3.74 3.76 3.76 3.73
195.7
54.0 73.9 82.5 100.0
POLYCCNAL 3.40 3.43 3.43 3.51
74.5 67.8 75.2 100.0
TRIANCULAR 4.06 4.63 5.19 4.90
75.7 71.9 80.0 100.0
ELLIPTICAL 3.19 3.63 4.06 3.86
140.3
75.6 71.3 78.7 100.0
PCLYC'CNAL 2.91 3.31 3.70 3.51
15.3 48.3 74.6 100.0
TRIANCULAR 4.58 4.60 4.77 4.90
25.3 58.0 78.6 100.0
ELLIPTICAL 3.60 3.62 3.74 3.84
85.4
22.9 55.3 77.6 100.0
POLYCONAL 3.27 3.34 3.43 3.51

 

 

 

 

 

 

 

158

more striking as the sprinkler wetted radius is reduced and
no overlapping between different sprinklers occurs. As

mentioned before, the lateral of the continuous moving

system moves at a constant angular velocity, which is not
In such systems the

the case with hydraulic moving systems.

velocity of any given tower varies according to its

alignment with adjacent towers (proportional control).

The results for the continuous moving system showed a

perfect distribution (UCW = 100%) along the path of travel

independent of the sprinkler pattern. In

for all transects,
the same way as the traditional center-pivot systems the

results of the LEPA system moving intermittently showed
dependence on the distance from the pivot-point, magnitude

of the alignment angle and on the shape of the sprinkler

pattern.
The magnitude of the alignment angle was of greater

importance for the LEPA system than for traditional systems.

The range of the UCW values varied from 15.3% to 83.9%, with

the smallest results obtained when the alignment angle were

equal to 2 degrees. The UCW increased with a decrease in
the alignment angle at distances of 196.7 and 85.4 m (645.6
and 280.3 ft) from the pivot—point. However, at 140.3 m
from the pivot-point the coefficient of

( 4 60 . 3 ft)
uniformity of the distributions generated with alignment

angle equal to 2.0 degrees were higher than the ones with

 

1

 

 

159

1.0 degree, with no exception. This confirms the importance

of the interactions existent between the distance from the

pivot-point and the alignment angle.

With respect to the shape of the sprinkler pattern, the

UCW of the distributions obtained with the elliptical
pattern sprinkler were the highest. They were followed by

the ones obtained with the polygonal pattern and then by the
triangular pattern. Such findings should cause no big
surprise since these patterns are more like the uniform
The lower the pattern maximum application rate the

pattern.
less it will be its influence on the depth of application of

collector cans closer to it, at each lateral stop.

V- CONCLUSION AND RECOMMENDATIONS

The objectives stated in the introductory chapter were

An easy to use computer model for the

 

fully addressed .

simulation of center-pivot systems was developed. The

computer model developed in this study differs from the

model proposed by Heermann & Sthal (1986) for not having a

The sequences of “on-time" and "off-

I

modular structure.
time" of the towers are not necessary prior running the

model. This feature makes it simpler to run, and more

importantly makes it suitable for optimization. Another

advantage is that it also allows the user to run simulations

using actual sprinkler profiles. The results of the

simulations ("on-times" and "off-times") performed using the

model showed a periodic behavior similar to the results

obtained in the field. The model also showed good accuracy

in predicting the depth of water application. However, good

accuracy in predicting tower position was not found, mainly

because of tower velocity variability due to tire slippage

in field conditions.

The uniformity coefficient for the distributions

obtained with the lateral moving continuously were higher
For both,

than for the lateral moving intermittently.

traditional and LEPA systems the Wilcox and Swailes

160

\

 

161

Uniformity Coefficient (UCW) , were for all practical
purposes equal to 100%, when the lateral moved continuously.
With the lateral moving intermittently, UCW values as low as
82.9% and 15.3% were found for traditional and LEPA system
respectively. These results showed the necessity of
considering the uniformity along the path of travel when
determining a center-pivot system uniformity, specially for
LEPA systems. Among the factors that influence the
uniformity along the path of travel in an intermittently
moving lateral systems, the magnitudes of the wetted radius
and of the alignment angle are the most inportant. In
general, smaller alignment angles and larger wetted radii
reflect a higher uniformity. The sprinkler pattern shape
proved to be of little importance when in combination with
large wetted radii. However, in LEPA systems where the
magnitude of the wetted radius issmaller than in
traditional system the differences among the pattern shapes
were more evident. Distributions generated with sprinkler
pattern shapes approximating uniform distribution resulted
in higher uniformity. The inportance of the sprinkler
pattern should not be overlooked when designing the system.
In general, systems operating at reasonable alignment
theoretically produced high distribution uniformities making
them suitable to Chemigation. Therefore, the choice of the

sprinkler should be made based on their field performance.

 

 

 

162

RECOMNDATIONS
Recommendations for further research include:

1) Field Tests of hydraulically driven center-pivot

 

systems to determine how well the lateral movement

of these systems approximates a truly continuous

movement .

Use of the model developed to perform optimization

2)
of the system, that is, find the set of design and
management parameters that will maximize the
uniformity of water application.

3) Development of new procedures to determine an

overall system uniformity coefficient considering
both the uniformity of application along the lateral

and along the path of travel, mainly in LEPA

systems .

4) Field work to assess the inportance of uniformity of
application along the path of travel when performing

Chemigation .

APPENDIX A

Coefficient values of the high order polynomials used
to represent the actual sprinkler patterns used in the

simulations .

163

164

Each pattern of the spray nozzle R-30 used in the

simulations were represented by a high order polynomial of the

fonm

Y=C1+C2X+C3X2+...

where: Y

+ CnX (II-1)

the application rate (in/hr) at distance X from

the sprinkler position ( X S wetted radius)

C1 to C11

sprinkler

type and height,

operating pressure.

COEFFICIENTS

A- NELSON R-3O

IA

FTHI O S XI

FOR 21 < X S

_B- NELSON R— 3 0

INDFI (D 53.x <

coefficients which values are function of the

nozzle size and

VALUES OF SPRINKLERS USED IN SIMULATIONS:

U4 / NOZZLE #20 (5/32") 3RN - 30 PSI

21

35

U4 /

20

NOZZLE #30

l I
UILAJQU'IIb

5.
-4.
1.

.7638212E-02
.1125734E-02
.4459446E-03
.5068430E-O4
.1924000E-06

.878694OE-02
.2022900E-02
.0463800E-03
.4360890E-04
.9898665E-06
.6626120E-08

l36180E-02
l77242E—03
610852E-02

(15/64") 3RN - 30 PSI

 

FOR 20 S x < 28

IA
00
U1

FOR 28 S X

I
C}
p
\

C- NELSON R-30

A
.h

FOR 0 S X

IA
KO

FOR 4 S X

FOR 9 < X S 35

D- NELSON R-30 - D6 /

FTHZ O S X < 4

PIE? 4 S X S 9

FOR9<X535

165

—1.699431E-03
-7.972334E-06
.439168E-06
-l.866352E—O7

II II II II
m

2.514501
-9.311897OE-02
-1.1540330E-02

7.6955680E—04

-l.210340E-05

8.467767
-1.013946

4.093127E-02
-5.397946E-04

NOZZLE #40 (5/16") 3RN - 30 PSI

0.23
-4.0E-03

1.20E-03
5.32E-02

1.498498

-2.312978E-01
.727711E-02
—4.834677E-04
4.060102E-06

II II II II II
H

NOZZLE #20 (5/32") 3RN - 30 PSI

0.23
-4.0E—03

1.20E-03
5.32E-02

1.498498
—2.312978E-01

FOR (IS X S

NELSON R-30 - D6 /

2

FOR 2 < X S 29

F-

FOR 0 S X S

FOR 4 < X S

FOR 13 < X S

FOR 19 < XLS

.NELSON R-30 - D6 /

4

13

19

NOZZLE #30 (15/64") 3RN - 30 PSI

NOZZLE #40 (5/16") 3RN - 30 PSI

166

1.727711E-02
-4.834677E-04
4.060102E-06

6.0E-01
—1.08E-01

3.698215E-01
.576273E-03
1.659467E-O3
.919641E-04
8.080694E-05
.l68070E-06

2.35E-01
5.73E-02

1.008953E—01
1.237290E-01
.829177E-03
2.585948E—04

~28.43676
5.672495

-3.554213E-01
7.221505E-03

-20.75859
2.5904332

-1.014985E-Ol
1.269538E-03

167

G- NELSON R-30 — D6C / NOZZLE #18 (9/64") 3RN - 30 PSI

FOR 0 S X < 2

c12= 3.00E-Ol
c2): 1.81E-01
FOR 2 s x s 12
c1:= 7.498E—Ol
c2:= -4.39OE-02
FOR 12 < x s 23
c1:= 4.66276E—01
c2 = -2.0273OE-02

H- NELSON R-30 D6C / NOZZLE #32 (1/4") 3RN - 30 PSI

FOR 0 s x s 15
c; = 8.199914E-01
c2:= 1.498167E—Ol
cg = -3.699838E—02
c,:= 2.809597E-03
cg = -7.711797E-05

FOR 15 < X S 33

c; = 2.243668E-04
c,:= 1.721536E-Ol
cg = —1.946300E-02
c; = 7.877457E-04
cg = -1.079675E-05

I- NELSON R-30 D4 / NOZZLE #20 (5/32") 3RN - 30 PSI

FOR 0 S X S 16

c12= 1.638221E—01
cg = -4.99514OE-02
c5 = 1.498470E-02
cg = —1.701006E-03
c5== 7.898698E-05
c6 = -l.260777E-06

FOR 16 < X S 28

c1 = —29.17034
c2 = 5.167176

cg = -3.043008E-Ol
c, = 4.963067E-03
cg = 1.1708258-04

-3.361771E-06

 

168

J- NELSON R-30 - D4 / NOZZLE #30 (15/64") 3RN - 30 PSI

FOR 0 S X S 7

C1== 6.828395E-02
(5 = 1.954686E—01
cg = -l.883152E-01
C4z= 6.517420E-02
C5== -9.101001E-O3
C64: 4.505710E-04
FOR 7 < X S 16
C1 = 1.033247E-03
C2:= 1.922724E-02
C3:= 5.280385E-03
(L = -3.770389E—O4

FOR 16 < X S 26

c1 = -2.822429E-04
c2 = 1.191840E-01
c3 = -1.211325E-02
c, = 3.244428E-04

FOR 26 < X S 31

c1 = -1.035154E-03
c2 = 1.072518
c3 = -6.707106E-02
c, = 1.04697OE-O3
K- NELSON R-30 - D4 / NOZZLE #40 (5/15") 3RN - 30 PSI

FOR 0 S X < 16

c1 = 2.739339E-01
c2 = -8.035340E—02
c3 = 1.476606E—02
c, = -4.742017E-O4
FOR 16 s x s 26

c1 = 2.128959E-O3
c2 = 1.2881754

c3 = -l.608466E-01
c, = 6.713171E-03

-9.210217E-05

169

FOR 26 < X S 32

c1 = -188.0355
c2 = 19.50391
c3 = -6.668486E-—01

7.530886E-03

170

 

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APPENDIX B

Center-Pivot Model Computer Code

180

181

MAIN PROGRAM

REM
REM Program to simulate the intermittent movement of the towers of
REM a Center pivot system, and compute the depth of application

for System Equipped with Geometrical Sprinkler Patterns or
REM Nelson R-30 series operating at 30.0 psi.

REM WRITTEN BY: MARIO FUSCO JR.

REM LAST REVIEW: JAN 12, 1993.

REM LAST USE: JAN 13,1993.

REM

DEFDBL N ,

REM declare arrays dynamic so they can be bigger than 64k

REM
REM $DYNAMIC
REM
DIM RADTW(20), NANGTW(20), NXTW(20), NYTW(20), TOWER(20),
W(20), DIST(20)
DIM DANGTW(20), ACLENIZO),
XCAN(20, 200)
DIM YCAN(20, 200), YSPR(10, 100), XSPR(10, 100),
WETRADIZO, 100)
DIM FLOWRATE(20, 100), DEPTH(100, 100), TOWINI(40),
TNUMSPR(200)
DIM NINISPR(100), TSPRNUM(20, 100), LINANG(10), ALPHA(10)
DIM MAXRATIZO, 100), C1H(20, 100), C2RAD(20, 100)

SPACCAN(10) , RADCAN(100) ,

REM

REM OPEN OUTPUT FILES

REM
OPEN "A:\STATE.DAT" FOR OUTPUT AS #1
OPEN 'A:\ANGALIGN.DAT" FOR OUTPUT AS #2
OPEN 'A:\CORDTOW.DAT' FOR OUTPUT AS #3
OPEN "A:\COORDSPR.DAT" FOR OUTPUT AS #4
OPEN 'A:\CANPOS.DAT" FOR OUTPUT AS #5
OPEN "A:\DEPTH.DAT" FOR OUTPUT AS #6

REM

PRINT #1, “TIME", ”TOWER NUMBER", "TOWER STATE"

PRINT #1, '----", " ------------ ", " ----------- "

PRINT #2, TAB(4); "T"; TAB(14); ”ANGTW(1)"; TAB(24);

”ANGTW(2)"; TAB(34); "ANGTWI3I'; TAB(44); "

BETA(1)"; TAB(54); "BETA(2)"

PRINT #2, TAB(2);" ----- ",' TAB(14);"----"; TAB(24); " ——-- ";
TAB(34); ' ---- “; TAB(44); ' ---- '; TAB(54);

PRINT #3, “TIME", I'TOWER NUMBER",

PRINT #3 , " ---- , ,
PRINT #4 , "TOWER" , " SPRINKLER“ , " XSPR" ,
PRINT #4 , " ----- " , " --------- " , ,
PRINT #5 , "NUMROW" , "NUMCAN" , " XCAN" , " YCAN"

PRINT #5, u ------ "I I I
PRINT #6, "NUMROW", "NUMCAN", " DEPTH(in) "

" X-COORD " , ”Y-COORD "
" YSPR'

REM
REM
REM

REM

REMC

182

PRINT #6, .. ------ -, ~ ...... n, -- ___________ ..
INPUTS

CLS
SCREEN 9
COLOR 2, 1

PRINT "ENTER NUMBER OF TOWERS“
INPUT NTOWERS
PRINT "ENTER THE LENGTH OF THE TOWERS"

INPUT LENG
PRINT "ENTER THE SPACING BETWEEN SPRINKLERS"

INPUT SPRSPAC

FOR I = 1 TO NTOWERS - 1

PRINT I'ENTER THE ANGLE OF ALIGMENT BETWEEN TOWER #”, I,
"AND TOWER #", I + 1

INPUT LINANG(I)

NEXT I
PRINT "ENTER THE NUMBER OF ROWS ALONG THE PATH OF TRAVEL"

INPUT NUMROW
PRINT "ENTER THE NUMBER OF CANS“

INPUT NUMCAN
PRINT "ENTER THE ANGLE OF THE RADIAL LINE WHERE FIRST CAN

WILL BE"

INPUT CANANG
PRINT "ENTER THE LINEAR SPACING OF CANS OF THE OUT MOST

ROW"
PRINT "THE COMPUTER WILL COMPUTE SPACING FOR OTHER ROWS'I

INPUT SPACCAN(1)
PRINT ”ENTER THE % SETTING TIME GUIDING TOWER WILL BE

ON, PER CYCLE."

INPUT SETTG
PRINT “ENTER THE DURATION OF THE SIMULATION, IN HOURS”

INPUT DUR
PRINT "ENTER THE TIME STEP, DT, THE SIMULATION WILL BE

RUN, IN SECONDS"
INPUT DT

REM TRANSECT DISTANCES FROM THE PIVOT-POINT

REM'

REE!
R534

I = 2

WHILE I <= NUMROW + 1
PRINT ”ENTER THE DISTANCE FROM ROW", I - 1, "TO THE

PIVOT POINT"

INPUT RADCAN(I ‘ 1)
:EF I > 2 AND RADCAN(I - 1) >= RADCAN(I - 2) THEN'I = I:
PRINT "ERROR, RESTART THE PROGRAM!"
I = I + 1

“EH“?

CLS
SCREEN 9

183

COLOR 2, 1
PRINT TAB(18); "MENU”
PRINT

PRINT
PRINT TAB(lO); "CHOOSE HOW MANY TOWERS YOU WANT THE

SPRINKLERS TO BE INITIALIZED'

PRINT : PRINT TAB(10); '1. INITIALIZE SPRINKLERS ON

EVERY TOWER”

PRINT : PRINT TAB(lO); "2. INITIALIZE SPRINKLERS ON

CHOSEN TOWERS"

REM

REM

REM

REM

REM

REM

REM
REM

REE!
RED!

PRINT
PRINT
INPUT "ENTER THE NUMBER OF YOUR CHOICE”, CHOICE

IF CHOICE = 1 THEN

FOR I = 1 TO NTOWERS

TOWINI(I) = 1
NEXT I
END IF
IF CHOICE = 2 THEN
PRINT
INPUT "ENTER THE NUMBER OF TOWERS TO BE INITIALIZED",

NUTWINI
FOR I = 1 TO NTOWERS
TOWINI(I) = 0

NEXT I

FOR I = 1 TO NUTWINI
INPUT "ENTER THE TOWER NUMBER", II
TOWINI(II) = 1
NEXT I
END IF
FOR I = 1 TO NTOWERS
PRINT TOWINI(I)
NEXT I

CALL SUBROUTINE TO INITIALIZE CAN POSITIONS.

CALL CANPOS(NUMROW, NUMCAN. SPACCAN(). RADCAN(). XCAN(I,
YCAN(). CANANG)

PRINT THE COORDINATES OF THE CAN POSITIONS
FOR I = 1 TO NUMROW
FOR J = 1 TO NUMCAN
PRINT #5, I, J, XCAN(I, J), YCAN(I, J)
NEXT J

BHDCP I
COMPUTE THE TOWERS DISTANCES FROM THE PIVOT-POINT.

FOR I = 0 TO NTOWERS - 1
RADTW(I + 1) = (NTOWERS - I) * LENG
INEXH'I

REM
REM

REM

REM

REM

REM

REM
REM

REM

REM
REM
REM

REM

184

INITIALIZE THE TOWER POSITIONS (POLAR COORDINATES)

First the angles and then the “X's"
and finally the state (on/off) of each tower to be off.

FOR I = 1 TO NTOWERS
NANGTWII) = 3.1415927# / 2

NXTW(I) = 0!
NYTW(I) = (NTOWERS - (I - 1)) * LENG
TOWER(I) = 2 ' 1 = ON; 2: OFF '
ALENG(I) = LENG
NEXT I

CALL SUBROUTINE SPRPOS TO INITIALIZE SPRIKLER POSITIONS
AT T = 0

CALL SPRPOS(LENG, SPRSPAC, NTOWERS, TNUMSPR(), NYTW(),
NXTW(I. YSPR(I, XSPR”. TOWER”. ‘1‘)

FOR I = 1 TO NTOWERS
PRINT 'TOWER=', I, ”NUMBER OF SPRINKLERS=", TNUMSPR(I)

FOR J = 1 TO TNUMSPR(I)
PRINT #6, I, J, XSPR(I, J), YSPR(I, J)
NEXT J
NEXT I

CHOICE OF SPRINKLER PROFILES.

CLS .
PRINT TAB(lO); "CHOOSE THE BEST CHOICE FOR THE SPRINKLER

PATTERN IN THE SYSTEM"
PRINT
PRINT TAB (10) ; " 1 . TRIANGULAR"
PRINT TAB(lO) ; '2 . ELLIPTICAL"
PRINT TAB(10) ; "3 . POLIGONAL "
PRINT TAB(lO); "4. ACTUAL PROFILE"

PRINT
INPUT ”ENTER THE NUMBER OF YOUR CHOICE", PATCHOICE

PRINT

IF PATCHOICE = 1 THEN
PATSPR$ = ”TRIANGULAR”
ELSEIF PATCHOICE = 2 THEN
PATSPR$ = ”ELLIPTICAL”
ELSEIF PATCHOICE = 3 THEN
PATSPR$ = "POLIGONAL"
ELSEIF PATCHOICE = 4 THEN
PATSPR$ = "ACTUAL"

END IF
ENTER WETTED RADIUS AND FLOWRATES.

FOR I = 1 TO NTOWERS
SPRNUM = 1

185

IF TOWINI(I) = 1 THEN
PRINT ”ENTER THE NUMBER OF SPRINKLERS TO BE
INITIALIZED IN TOWER"
PRINT ”NUMBER", I, “.THESE ARE THE SPRINKLERS THAT
WILL CONTRIBUTE“
PRINT "TO THE DEPTH IN THE ROW OF CANS"
INPUT NINISPR(I)
FOR K = 1 TO NINISPR(I)
PRINT "ENTER THE SPRINKLER NUMBERS OF THE
NUMBER", K, "SPRINKLER“
INPUT TSPRNUMII, K)

NEXT K
REM
FOR II = 1 TO TNUMSPR(I)
FOR JJ = 1 TO NINISPR(I)
IF II = TSPRNUM(I, JJ) THEN
IF PATSPR$ = "TRIANGULAR" OR PATSPR$ =”ELLIPTICAL"
THEN

PRINT "ENTER THE WETTED RADIUS IN (ft)
AND THE FLOW RATE (GPM)'
PRINT 'FOR THE SPRINKLER NUMBER",
TSPRNUM(I, JJ), "STARTING AT OUT MOSTI
PRINT “END OF THE TOWER", I
INPUT WETRAD(I, II), FLOWRATEII, II)
ELSEIF PATSPR$ = ”POLIGONAL“ THEN
PRINT “ENTER THE WETTED RADIUS IN FT,
THEN THE MAXIMUM APPLICATION RATE”
PRINT "AND APPLICATION RATE UNDERNEATH
THE SPRINKLER AND FINALY THE DISTANCE FROM THE POINT OF
MAXIMUM APPLICATION RATE TO THE SPRINKLER“
INPUT WETRAD(I, II), MAXRAT(I, II), C1H(I,II)
, C2RAD(I, II)
ELSEIF PATSPR$ = "ACTUAL" THEN
CLS
PRINT TAB(lO); "ENTER THE WETTED RADIUS, IN FT"
INPUT WETRAD(I, II)
PRINT TAB(lO); "SPRINKLER LIBRARY"
PRINT
PRINT TAB(lO); " 1. R30 - U4 /NOZZLE #20 (5/32) 3RN - 30 PSI“

PRINT TAB(lO); " 2. R30 - U4 /NOZZLE #30 (15/64) 3RN - 30 PSI"
PRINT TAB(lO); " 3. R30 - U4 /NOZZLE #40 (5/16) 3RN - 30 PSI"
PRINT TAB(lO); " 4. R30 - D6 /NOZZLE #20 (5/32) 3RN - 30 PSI"
PRINT TAB(lO); " 5. R30 - D6 /NOZZLE #30 (15/64) 3RN - 30 PSI"
PRINT TAB(lO); " 6. R30 - D6 /NOZZLE #40 (5/32) 3RN - 30 PSI"
PRINT TAB(lO); " 7. R30 - D6C /NOZZLE #18 (9/64) 3RN - 30 PSI”
PRINT TAB(lO); " 8. R30 - D6C /NOZZLE #32 (1/4) 3RN - 30 PSI”
PRINT TAB(lO); ” 9 R30 - D4 /NOZZLE #20 (5/32) 3RN - 30 PSI"

PRINT TAB(lO); . R30 - D4 /NOZZLE #30 (15/64) 3RN - 30 PSI”
PRINT TAB(lO); "11. R30 - D4 /NOZZLE #40 (5/32) 3RN - 30 PSI"
‘PRIEH?

INPUT ”ENTER THE NUMBER OF YOUR CHOICE", SPRTYPE(I, II)

END IF

REM

a
H
0

END IF

186

NEXT JJ
SPRNUM = SPRNUM + 1
INEXT II
END IF
INEXT I
REM

IUflH'TO complete initialization phase all other variables
REM should be initialized here

REM
PI = 3.1415927#
DUR = 30/3600 'duration of the simulation, in hours
DT = 1 ’time increment in seconds
T = 0! 'initial time
NIT = (DUR * 3600 / DT) ’number of iterations
NIPP = 1 ’number of iterations per print
NIOL = NIT / NIPP ’number of iterations on outside loop
REM
FOR I = 1 TO NTOWERS - 1
ALPHA(I) = LINANG(I) * 3.1415927# / 180 'alignment angle in
radians
NEXT I
REM
REM also initialize angular velocities
REM
FOR J = 1 TO NTOWERS
PRINT"ENTER THE ANGULAR VELOCITY OF TOWER".J.'IN RAD/SEC“
INPUT W(J)
NEXT J
REM
REM print initial values
REM
PRINT #1, T, TOWER(1), TOWER(Z), TOWER(B)
PRINT #2, USING '###.### "; NANGTW(1); NANGTW(2);
NANGTWI3); BETAIl); BETA(2)
PRINT #3, USING "###.#### "; NXTW(l); NYTW(l); NXTW(Z);
NYTW(Z); NXTW(3); NYTW(3)
REM
REM Print the time of begining of simulation
REM

T1$ = TIMES
PRINT "THE TIME AT THE BEGINING OF THE SIMULATION IS", T1$
REM
REM START EXECUTION PHASE
REM
FOR M = 1 TO NIOL
FOR N = 1 TO NIPP
REM
REM initialize the distance moved and increment angle to zero
REM like a default value that will change if tower is "on"
REM
FOR I = 1 TO NTOWERS
DIST(I) = O!
DANGTWII) = 0!
NEXT I

187

T = T + DT
COUNT = COUNT + 1
REM
REM
REM Set towerl ON depending on value of COUNT
REM
IF COUNT <= (SETTG * 60! / (100 * DT)) THEN
TOWER(1) = 1
ELSE TOWER(1) = 2
END IF
REM

REM Print the time tower #1 changes state, also the tower
REM coordinates. First when tower #1 becomes on, and then when it

REM becomes off.

REM
TWNl = 1
IF COUNT = CINT(SETTG * 60 / (100 * DT)) + 1 THEN
PRINT #1, T, TWNl, TOWER(1)
END IF
REM
IF TOWER(1) = 1 THEN
DIST(l) = W(1) * NTOWERS * LENG * DT
XIN = ((DIST(1) / 2) / (NTOWERS * LENGH
DANGTW(1) = 2 * ASIN(XIN)
NANGTW(1) = NANGTW(1) + DANGTW(1)
NXTW(l) = NTOWERS * LENG * COS(NANGTW(1))
NYTW(I) = NTOWERS * LENG * SIN(NANGTW(1))
END IF
REM
REM Compute new coordinates if towers are 'ON"
REM

FOR I = 2 TO NTOWERS
IF TOWER(1) = 1 THEN
DIST(I) = W(I) * (NTOWERS - (I - 1)) * LENG * DT
XIN = ((DIST(I) / 2)/((NTOWERS -(I - 1)) * LENG))
DANGTW(I) = 2 * ASIN(XIN)
NANGTW(I) = NANGTW(I) + DANGTW(I)

NXTW(I) = (NTOWERS - (I - 1))*LENG * COS(NANGTW(I))

NYTW(I) = (NTOWERS - (I - 1)) * LENG * SIN(NANGTW(I))
ELSE

NXTW(I) = (NTOWERS - (I - 1)) * LENG * COS(NANGTW(I))

NYTW(I) = (NTOWERS - (I - 1)) * LENG * SIN(NANGTW(I))
END IF

NEXT I
REM Determine the angle between towers and compare with

REM alignment angle

REE!
FOR I = 1 TO NTOWERS - 1
Rim!
SlMINOR = ABS(NXTW(I + 1) - NXTW(I + 2))
S2MINOR = ABS(NYTW(I + 1) - NYTW(I + 2))

TETA = ATN(52MINOR / SlMINOR)
IF NANGTWKI + 1) > PI/2 ANDINANGTWKI + l) < 3 * PI/2

188

THEN
SlMINOR = -SlMINOR
END IF
NXDUM = 2 * SlMINOR + NXTW(I + 2)
IF NANGTW(I + 1) > PI AND NANGTW(I + 1) < 2 *PI THEN
SZMINOR = -SZMINOR
END IF
NYDUM = NYTW(I + 2) + 2 * SZMINOR
DISPT = SQR((NYTW(I) - NYDUM)“2 + ((NXTW(I) - NXDUM) A 2))
SIDUM = SQR((LENG) A 2 - (DISPT / 2) A 2)
BETA(I) = 2 * ATN((DISPT / 2) / SIDUM)
REM
IF BETA(I) >= ALPHA(I) THEN
IF (TOWER(I + 1) = 1) .AND CNANGTWKI + 1) >
NANGTW(I)) THEN
TOWER(I + 1) = 2
PRINT #1, T, I + 1, TOWER(I + 1)

PRINT #3, USING ”###.#### "; T; I + 1;
NXTW(I + 1); NYTW(I + 1)
PRINT #2, USING "###.#### '; T; NANGTW(1);

NANGTW(2); NANGTW(3); BETA(l); BETA(2)
ELSEIF (TOWER(I + 1) = 2) AND (NANGTW(I + 1) <
NANGTW(I)) THEN
TOWER(I + 1) = 1
PRINT #1, T, I + 1, TOWER(I + l)

PRINT #3, USING '###.#### "; T; I + 1;
NXTW(I + l); NYTW(I + 1)
PRINT #2, USING '###.#### ": T; NANGTW(1);
NANGTW(2); NANGTW(3); BETA(l); BETA(Z)
ELSE
END IF
END IF
NEXT I
REM
REM
REM Reset COUNT after one minute
REM
IF COUNT >= (60! / DT) THEN COUNT = 0!
REM

REM CALL SUBROUTINE SPRPOS TO DETERMINE THE COORDINATES OF THE
SPRINKLERS
REM ALONG THE TOWERS’ SPANS.
REM
CALL SPRPOS(LENG, SPRSPAC, NTOWERS, TNUMSPR(), NYTW(), NXTW(),
YSPR():XSPR(), TOWER(), T)
REM
REM Print the coordinates of the sprinkler positions
FOR I = 1 TO NTOWERS
FOR J = 1 TO TNUMSPR(I)
PRINT #4, I, J, XSPR(I, J), YSPR(I, J)
NEXT J
NEXT I
REM
REM.CALL SUBROUTINE TO COMPUTE THE TOTAL DEPTH APPLIED IN EACH CAN.

 

189

REM
IF PATSPR$ = “TRIANGULAR“ OR PATSPR$ = “ELLIPTICAL“ THEN

CALL DEPTRIPAT(DT, NUMROW, NUMCAN, TNUMSPR(), NTOWERS,
XCANI). YCANI), XSPR(), YSPR(), WETRAD(), FLOWRATE():
DEPTH(), TOWINI(), PATSPR$, NINISPR(), TSPRNUM())

REM
ELSEIF PATSPR$: "POLIGONAL” THEN
CALL POLIPAT(DT, NUMROW, NUMCAN, TNUMSPR(), NTOWERS.
XCAN(), YCANH, XSPR(), YSPR(), WETRADU. DEPTH(),
TOWINI(). NINISPR(), TSPRNUM(), MAXRAT(). C1H(),
C2RAD())
REM
ELSEIF PATSPR$ = "ACTUAL" THEN
CALL ACTPAT(DT, NUMROW, NUMCAN, TNUMSPR(), NTOWERS,
XCAN(), YCANU, XSPR(), YSPR(), WETRADU. DEPTH(),
TOWINI(), NINISPR(), TSPRNUM(). SPRTYPE())
END IF
REM
NEXT N
REM
NEXT M
REM

REM Print the time at the end of the simulation
T2$ = TIME$
PRINT I'THE TIME AT THE END OF THE SIMULATIONS IS", T2$

REM
REM Print the total depth of water in each can

REM
FOR I = 1 TO NUMROW
FOR J = 1 TO NUMCAN
PRINT #6, I, J, DEPTH(I, J)
NEXT J
NEXT I
REM
REM Sound alarm
SOUND 100, 100

SUBROUTINE ACTPAT

REM $STATIC
SUB ACTPAT (DT, NUMROW, NUMCAN, TNUMSPR(), NTOWERS, XCAN(),

YCAN() , XSPR() , YSPRH , WETRADH , DEPTH” , TOWINI() ,
NINISPR(), TSPRNUM(). SPRTYPE())
REM Subroutine to compute the depth of. water in a catch can
REM from the contribution of different sprinklers.
REM
DIM DISTA(20, 100), HBAR(15, 100), H(15, 100)
REM

REM
FOR I = 1 TO NUMROW

 

 

190

FOR J = 1 TO NUMCAN

FOR II = 1 TO NTOWERS
IF TOWINI(II) = 1! THEN
FOR JJ = 1 TO TNUMSPR(II)
FOR KK = 1 TO NINISPR(II)

IF JJ

REM

+ (YSPR(II, JJ)
REM

+(FUNC1(DISTA():

+ (FUNC2(DISTA():

+ (FUNC3(DISTA():

+ (FUNC4(DISTA():

+ (FUNC5(DISTA():

+ (FUNC7(DISTA():

+ (FUNC8(DISTA()I

+ (FUNC9(DISTA():

+ (FUNC10(DISTA()

+ (FUNC11(DISTA()
REM

DISTA(I, J) = SQR((XSPR(II, JJ)
- YCAN(I, J))

IF DISTA(I, J) <= WETRAD(II, JJ) THEN
IF SPRTYPE(II, JJ)

= TSPRNUM(II, KK)

A2)

DEPTH(I, J) =

I, J)) * DT

I.

I,

ELSEIF SPRTYPE(II,

DEPTH(I, J) =

J)) * DT

ELSEIF SPRTYPE(II,
DEPTH(I, J) =

J)) * DT

ELSEIF SPRTYPE(II,
DEPTH(I, J) =

J)) * DT

ELSEIF SPRTYPE(II,

DEPTH(I, J) =
J)) * DT

ELSEIF SPRTYPE(II,

DEPTH(I, J) =
J)) * DT

ELSEIF SPRTYPE(II,

DEPTH(I, J) =
J)) * DT

ELSEIF SPRTYPE(II,

DEPTH(I, J) =

J)) * DT

ELSEIF SPRTYPE(II,
DEPTH(I, J) =

J)) * DT

ELSEIF SPRTYPE(II,

DEPTH(I, J) =

. I. J)) * DT

, I. J))

ELSEIF SPRTYPE(II,

DEPTH(I, J) =
* DT

END IF

NEXT JJ

END IF
NEXT II
NEXT J
NEXT I
END SUB

END IF
END IF
NEXT KK

THEN

= 1 THEN
DEPTH(I, J) +

JJ) = 2 THEN
DEPTH(I, J) +

JJ) = 3 THEN
DEPTH(I, J) +

JJ) = 4 THEN
DEPTH(I, J) +

JJ) = 5 THEN
DEPTH(I, J) +

JJ)'= 6 THEN
DEPTH(I, J) +

JJ) = 7 THEN
DEPTH(I, J) +

JJ) = 8 THEN
DEPTH(I, J) +

JJ) = 9 THEN
DEPTH(I, J) +

JJ) = 10 THEN
DEPTH(I, J) +

JJ) = 11 THEN
DEPTH(I, J) +

- XCAN(I,J))

A

2

191

SUBROUTINE CAMPOS

DEFDBL N
SUB CANPOS (NUMROW, NUMCAN, SPACCAN(), RADCAN(). XCAN(), YCAN():
CANANG)
REM SUBROUTINE TO INITIALIZE THE POSITION OF THE CANS FOR A CENTER
PIVOT EVALUATION
REM Compute the radial angle
REM
COSA = (2 * (RADCAN(I)) A 2 - SPACCAN(1) A 2) / (2 *
(RADCAN(1)) A 2)
SINA = SQR(1 - COSA A 2)
TANA = SINA / COSA
A = ATN(TANA)

REM
REM Initialize can positions
REM
FOR I = 1 TO NUMROW
FOR J = 1 TO NUMCAN
IF J = 1 THEN
XCAN(I, J) = RADCAN(I) * COS(CANANG)
YCAN(I, J) = RADCAN(I) * SIN(CANANG)
ELSE
XCAN(I, J) = RADCAN(I) * COS(CANANG + (J * A))
YCAN(I, J) = RADCAN(I) * SIN(CANANG + (J * A))
END IF
NEXT J
NEXT I
END SUB

SUBROUTINE DEPTRIPAT

SUB DEPTRIPAT (DT, NUMROW, NUMCAN, TNUMSPR(), NTOWERS, XCAN(),

YCAN(). XSPR(), YSPR(), WETRADU, FLOWRATE”,

DEPTH(), TOWINI(), PATSPR$, NINISPR(), TSPRNUM())
REM Subroutine to compute the total depth of water in a catch can
REM after one pass from the contribution of different sprinklers.
DIM DISTA(20, 100), HBAR(IS, 100), H(15, 100)
REM

FOR I = 1 TO NUMROW
FOR J = 1 TO NUMCAN
FOR II = 1 TO NTOWERS
IF TOWINI(II) = 1! THEN
FOR JJ = 1 TO TNUMSPR(II)
FOR KK = 1 TO NINISPR(II)
IF JJ = TSPRNUM(II, KK) THEN
IF FLOWRATE(II, JJ) = 0! OR WETRAD(II, JJ) = 0! THEN
HBAR(II, JJ) = 0
ELSE
HBAR(II, JJ) = (.02674 * FLOWRATE(II, JJ)) /

(3.1459 * (WETRAD(II, JJ) A 2))

END IF

192

REM
DISTA(I, J) = SQR((XSPR(II, JJ) - XCAN(I, J))A 2
+ (YSPR(II, JJ) - YCAN(I, J)) A 2)
REM
IF DISTA(I, J) <= WETRAD(II, JJ) THEN
IF PATSPR$ "TRIANGULAR” THEN
H(II, JJ) 3 * HBAR(II, JJ)
DEPTH(I, J) = DEPTH(I, J) + H(II,JJ) * (1
- (DISTA(I, J) / WETRAD(II,JJ)))*DT
ELSEIF PATSPR$ = "ELLIPTICAL" THEN
H(II, JJ) = 1.5 * HBAR(II, JJ)
DEPTH(I, J) = DEPTH(I, J) + H(II, JJ) * SQR(1
-((DISTA(I, J) A 2)/(WETRAD(II,JJ)A2)))*DT
END IF
END IF
END IF
NEXT KK
NEXT JJ
END IF
NEXT II
NEXT J
NEXT I
END SUB

SUBROUTINE POLIPAT

SUB POLIPAT (DT, NUMROW, NUMCAN, TNUMSPR(), NTOWERS, XCAN(),
YCAN(), XSPR(), YSPR(), WETRAD(), DEPTH(), TOWINI().
NINISPR(), TSPRNUM(). MAXRAT(), C1H(), C2RAD())

REM Subroutine to compute the total depth of water in a catch can

REM after one pass from the contribution of different sprinklers.

DIM DISTA(ZO, 100), HBAR(lS, 100), H(15, 100)

REM
REM
FOR I = 1 TO NUMROW
FOR J = 1 TO NUMCAN
FOR II = 1 TO NTOWERS
IF TOWINI(II) = 1! THEN
FOR JJ = 1 TO TNUMSPR(II)
FOR KK = 1 TO NINISPR(II)
IF JJ = TSPRNUM(II, KK) THEN
REM

DISTA(I, J) = SQR((XSPR(II, JJ) - XCAN(I, J))A2
+ (YSPR(II, JJ) - YCAN(I, J))A 2)
REM .
IF DISTA(I, J) <= WETRAD(II, JJ) THEN
IF DISTA(I, J) <= C2RAD(II, JJ) THEN
DEPTH(I,‘J)==DEPTH(I,.J)+(((MAXRAT(II,.IJ)
- C1H(II, JJ))/(C2RAD(II,JJ)))*DISTA(I, J)+ C1H(II, JJ)) * DT
REM

193

ELSEIF C2RAD(II, JJ) < DISTA(I, J) <=

WETRAD(II, JJ) THEN
DEPTH(I, J) = DEPTH(I, J) + (MAXRAT(II, JJ)

*C2RAD(II, JJ) - DISTA(I, J))/(WETRAD(II, JJ) - C2RAD(II, JJ)))* DT
REM

END IF

END IF

END IF
NEXT KK
NEXT JJ
END IF
NEXT II
NEXT J
NEXT I

END SUB

SUBROUTINE SPRPOS

SUB SPRPOS (LENG, SPRSPAC, NTOWERS, TNUMSPR(), NYTW(L NXTW(),
YSPR(), XSPR(), TOWER(), T)

REM Subroutine to compute the rectangular coordinates of the

REM sprinkler positions along the towers keeping the same spacing

REM on the whole system.

REM

REM DATE : JANUARY 21,1993

REM BY : MARIO FUSCO JUNIOR

REM
DIM FSPRDIS(200)
REM
FOR I = 1 TO NTOWERS
J = 1
WHILE J <> 9999
IF I = 1 THEN
IF J = 1 THEN
YSPR(I, J) = NYTW(I)
XSPR(I, J) = NXTW(I)
J = J + 1
ELSE
YSPR(I, J) = (LENG - (J - l) * SPRSPAC) * (NYTW(I)
- NYTW(I + 1)) / LENG + NYTW(I + 1)
XSPR(I, J) = (LENG - (J - l) * SPRSPAC) * (NXTW(I)

- NXTW(I + 1)) / LENG + NXTW(I + 1)
IF (LENG - (J - 1) SPRSPAC) <= SPRSPAC THEN

I-

J = 9999
ELSE
J = J + 1
TNUMSPR(I) = J
END IF
END IF

REM

194

ELSE

FSPRDIS(I) = SPRSPAC - (LENG - ((TNUMSPR(I - 1) - 1) *
SPRSPAC + FSPRDIS(I ~ 1)))

YSPR(I, J) = (LENG - (FSPRDIS(I) + (J - 1) * SPRSPAC))
*(NYTW(I) - NYTW(I + 1)) / LENG + NYTW(I + 1)

XSPR(I, J) = (LENG - (FSPRDIS(I) + (J - 1) * SPRSPAC))
*(NX'I'W(I) - NXTW(I + 1)) / LENG + NXTW(I + 1)

IF (LENG - ((J - 1) * SPRSPAC + FSPRDIS(I)) <= SPRSPAC)

THEN
J = 9999
ELSE
J = J + 1: TNUMSPR(I) = J
END IF
END IF
WEND
NEXT I
END SUB
FUNCTION ASIN ( ARC SINE)
DEFSNG N

FUNCTION ASIN (XIN) STATIC

ASIN = XIN + XIN'A 3 / 6 + 3 * (XIN A 5) / 40 + 15 * (XIN A 7) / 336
+ 105 * (XIN A 9) / 3456 + 945 * (XIN A 11) / 42240

END FUNCTION

FUNCTION FUNCl:

FUNCTION FUNCl (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) <= 21 THEN
INOVHOUR = .047638212 + .0511257336# * DISTA(I, J)
- .0074459446 * (DISTA(I, J) A2) + 3.506843E-04 * (DISTA(I, J) A 3)
- 5.1924E-06 * (DISTA(I, J) A 4)
ELSEIF DISTA(I, J) > 21 AND DISTA(I, J) <= 35 THEN
INOVHOUR = 4.878694E-02 + .0420229# * DISTA(I, J)
- 5.04638E-03 * (DISTA(I, J) A 2) + 1.436089E-04 *(DISTA(I, J) A 3)
+ 1.9898665E-06 *(DISTA(I, J) A 4) - 8.662612E-08 *(DISTA(I, J) A5)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUEl = INOVHOUR / 3600
IF VALUEl <= 0! THEN

FUNCl = O!
ELSE

FUNCl = VALUEl
END IF

END FUNCTION

 

 

195

FUNCTION FUNC2:

FUNCTION FUNC2 (DISTA(). I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) < 20 THEN
INOVHOUR = .0513618 - 4.177242E—03 * DISTA(I, J)
.610852E-02 * (DISTA(I, J) A 2)- 1.699431E-03 * (DISTA(I, J)A3)
.972334E-06 * (DISTA(I, J)A 4)+ 6.439168E-06 * (DISTA(I, J) A 5)
.866352E-07 * (DISTA(I, J) A 6)
ELSEIF DISTA(I, J) >= 20 AND DISTA(I, J) < 28 THEN
INOVHOUR = 2.514501 - 9.311897E-02 * DISTA(I, J)
.154033E-02 * (DISTA(I, J) A 2)+ 7.695568E-04 *(DISTA(I, J) A 3)
.21034E-05 * (DISTA(I, J) A 4)
ELSEIF DISTA(I, J) >= 28 AND DISTA(I, J) <= 35 THEN
INOVHOUR = 8.467767 - 1.013946 * DISTA(I, J)
+ 4.093127E-02 * (DISTA(I, J) A 2)- 5.397946E-04 *(DISTA(I, J) A 3)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE2 = INOVHOUR / 3600
IF VALUE2 <= 0! THEN

l+
Hxll-J

l
kaH

FUNC2 = 0!
ELSE

FUNC2 = VALUE2
END IF

END FUNCTION

FUNCTION FUNC3

FUNCTION FUNC3 (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0 AND DISTA(I, J) < 4 THEN
INOVHOUR = .23 - .004 * DISTA(I, J)
ELSEIF DISTA(I, J) >= 4 AND DISTA(I, J) <= 9 THEN
INOVHOUR = .0012 + .0532 * DISTA(I, J)
ELSEIF DISTA(I, J) > 9 AND DISTA(I, J) <= 35 THEN
INOVHOUR = 1.498498 - .2312978 * DISTA(I, J)
+1.727711E-02 *(DISTA(I, J) A 2) - 4.834677E-04 * (DISTA(I, J) A 3)
+ 4.060102E-06 * (DISTA(I, J) A 4)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE3 = INOVHOUR / 3600
IF VALUE3 <= 0! THEN

FUNC3 = 0!
ELSE

FUNC3 = VALUE3
END IF

END FUNCTION

 

 

196

FUNCTION FUNC4

FUNCTION FUNC4 (DISTA(). I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) < 3! THEN
INOVHOUR = .35 - .01933 * DISTA(I, J)
ELSEIF DISTA(I, J) >= 3! AND DISTA(I, J) <= 5! THEN
INOVHOUR = .1525 + .0465 * DISTA(I, J)
ELSEIF DISTA(I, J) > 5! AND DISTA(I, J) <= 25! THEN
INOVHOUR = .2641389 + 3.626893E-02 * DISTA(I, J)
- .003135126 * (DISTA(I, J) A 2)+ 4.942799E-05 * (DISTA(I, J) A 3)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE4 = INOVHOUR / 3600
IF VALUE4 < 0! THEN

FUNC4 = 0!
ELSE

FUNC4 = VALUE4
END IF

END FUNCTION

FUNCTION FUNC5

FUNCTION FUNC5 (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) <= 2! THEN
INOVHOUR = .6 - .108 * DISTA(I, J)
ELSEIF DISTA(I, J) > 2! AND DISTA(I, J) <= 29! THEN
INOVHOUR = .3698215 - 2.576273E-02 * DISTA(I, J)
+ 1.659467E—02 * (DISTA(I, J) A 2)- 1.919641E-03 *(DISTA(I, J) A 3)
+ 8.080694E-05 * (DISTA(I, J) A 4)- 1.16807E-06 * (DISTA(I, J) A 5)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUES = INOVHOUR / 3600
IF VALUES < 0! THEN

FUNC5 = 0!
ELSE

FUNC5 = VALUES
END IF

END FUNCTION

FUNCTION FUNC6

FUNCTION FUNC6 (DISTA(), I, J) STATIC
Rfml COMPUTE THE APPLICATION RATE IN INCHES PER HOUR

 

 

 

197

IF DISTA(I, J) >= 0! AND DISTA(I, J) <= 4! THEN
INOVHOUR = .235 + .0573 * DISTA(I, J)

ELSEIF DISTA(I, J) > 4! AND DISTA(I, J) <= 13! THEN
INOVHOUR = .1008953 + .1237291 * DISTA(I, J)

- 6.829177E-03 *(DISTA(I, J) A 2)+ 2.585948E-04 * (DISTA(I, J) A 3)
ELSEIF DISTA(I, J) > 13! AND DISTA(I, J) <= 19! THEN
INOVHOUR = -28.43676 + 5.672495 * DISTA(I, J)

- .3554213 * (DISTA(I, J) A 2)+ 7.221505E-03 * (DISTA(I, J) A 3)
ELSEIF DISTA(I, J) > 19! AND DISTA(I, J) <= 29! THEN

INOVHOUR = -20.75859 + 2.5904332 * DISTA(I, J)
- .1014985 * (DISTA(I, J) A 2)+ 1.269538E-03 * (DISTA(I, J) A 3)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE6 = INOVHOUR / 3600
IF VALUE6 < 0! THEN

FUNC6 = 0!
ELSE

FUNC6 = VALUE6
END IF

END FUNCTION

FUNCTION FUNC7

FUNCTION FUNC7 (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) < 2! THEN
INOVHOUR = .3 + .181 * DISTA(I, J)
ELSEIF DISTA(I, J) >= 2! AND DISTA(I, J) <= 12! THEN
INOVHOUR = .7498 - .0439 * DISTA(I, J)
ELSEIF DISTA(I, J) > 12! AND DISTA(I, J) <= 23! THEN
INOVHOUR = .466276 - .020273 * DISTA(I, J)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE7 = INOVHOUR / 3600
IF VALUE7 < 0! THEN

FUNC7 = 0!
ELSE

FUNC7 = VALUE7
END IF

END FUNCTION

FUNCTION FUNC8

FUNCTION FUNC8 (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) <= 15! THEN

 

 

198

INOVHOUR = .8199914 + .1498167# * DISTA(I, J)
3.699838E-02 *(DISTA(I, J) A 2)+ 2.809597E-03 *(DISTA(I, J) A 3)
7.711797E-05 * (DISTA(I, J) A 4)

ELSEIF DISTA(I, J) > 15! AND DISTA(I, J) <= 33! THEN

INOVHOUR = 2.243668E-04 + .1721536# * DISTA(I, J)
.019463 * (DISTA(I, J) A 2)+ 7.8774S7E-04 * (DISTA(I, J) A 3)
1.079675E-05 * (DISTA(I, J) A 4)

END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE8 = INOVHOUR / 3600
IF VALUE8 < 0! THEN

FUNC8 = 0!
ELSE

FUNC8 = VALUE8
END IF

END FUNCTION

FUNCTION FUNC9

FUNCTION FUNC9 (DISTA(), I, J)
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) <= 16 THEN
INOVHOUR = .1638221 - .0499514 * DISTA(I, J)
+ 1.498472E-02 *(DISTA(I, J) A 2)- 1.701006E-03 *(DISTA(I, J) A 3)
+ 7.898698E-05 *(DISTA(I, J) A 4) - 1.260777E-06 *(DISTA(I, J) A 5)
ELSEIF DISTA(I, J) > 16 AND DISTA(I, J) <= 28 THEN
INOVHOUR = -29.17034 + 5.167176 * DISTA(I, J)
- .3043008 * (DISTA(I, J) A 2)+ 4.963067E-03 * (DISTA(I, J) A 3)
+ 1.170825E-04 *(DISTA(I, J) A 4)- 3.361771E-06 *(DISTA(I, J) A 5)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE9 = INOVHOUR / 3600
IF VALUE9 < 0! THEN

FUNC9 = 0!
ELSE

FUNC9 = VALUE9
END IF

END FUNCTION

FUNCTION FUNClO

FUNCTION FUNClO (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) <= 7 THEN
INOVHOUR = 6.82839SE-02 + .1954686 * DISTA(I, J)
— .1883152 * (DISTA(I, J)) A 2 + .0651742 * (DISTA(I, J)) A 3
- 9.101001E-03 * (DISTA(I, J) A 4)+ 4.50571E-04 *(DISTA(I, J) A 5)

 

199

ELSEIF DISTA(I, J) > 7 AND DISTA(I, J) <= 16 THEN
INOVHOUR = 1.033247E-03 + 1.922724E-02 * DISTA(I, J)
+ 5.280385E-03 *(DISTA(I, J)) A 2 - 3.770389E-04 *(DISTA(I, J))A 3
ELSEIF DISTA(I, J) > 16 AND DISTA(I, J) <= 26 THEN
INOVHOUR = -2.822429E—04 + .1191841# * DISTA(I, J)
- 1.211325E-02 *(DISTA(I, J))A 2 + 3.244428E-04 *(DISTA(I, J))A 3
ELSEIF DISTA(I, J) > 26 AND DISTA(I, J) <= 31 THEN
INOVHOUR = -1.035154E-03 + 1.072518# * DISTA(I, J)
- 6.707106E-02 *(DISTA(I, J))A 2 + 1.04697E-03 * (DISTA(I, J))A 3
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE10 = INOVHOUR / 3600
IF VALUE10 < 0! THEN

FUNC10 = 0!
ELSE

FUNC10 = VALUE10
END IF

END FUNCTION

FUNCTION FUNC11

FUNCTION FUNC11 (DISTA(), I, J) STATIC
REM COMPUTE THE APPLICATION RATE IN INCHES PER HOUR
IF DISTA(I, J) >= 0! AND DISTA(I, J) < 16 THEN
INOVHOUR = .2739339 - .0803534 * DISTA(I, J)
+ 1.476606E-02 *((DISTA(I, J))A2)- 4.742017E-04 *((DISTA(I, J))A 3)
ELSEIF DISTA(I, J) >= 16 AND DISTA(I, J) <= 26 THEN
INOVHOUR = 2.128959E-03 + 1.288174S# * DISTA(I, J)
- .1608466 * (DISTA(I, J) A 2)+ 6.713171E-03 * (DISTA(I, J) A 3)
- 9.210217E-05 * (DISTA(I, J) A 4)
ELSEIF DISTA(I, J) > 26 AND DISTA(I, J) <= 32 THEN
INOVHOUR = -188.0355 + 19.50391 * DISTA(I, J)
- .6668486 * (DISTA(I, J) A 2) + 7.530886E-03 * (DISTA(I, J) A 3)
END IF
REM RETURN THE FUNCTION VALUE IN INCHES PER SEC
VALUE11 = INOVHOUR / 3600
IF VALUE11 < 0! THEN

FUNC11 = 0!
ELSE

FUNC11 = VALUE11
END IF

END FUNCTION

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