Chapter XIV: Design of Lateral Lines

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Chapter XIV Design of Lateral Lines
V. A. Gillespie, A. L. Phillips and I.P. Wu
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(No Transcript)
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Chapter 14 Design of Lateral
LinesIntroduction

• 1. In a drip irrigation system, a major design
  criteria is the minimization of the discharge (or
  emitter flow) variation along a drip irrigation
  line, either a lateral or a submain.
• 2. The discharge variation can be kept within
  acceptable limits in laterals or submains of a
  fixed diameter by designing a proper length for a
  given operating pressure.

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Capítulo 14 Design of Lateral
LinesIntroduction

• The discharge (or emitter flow) variation is
  controlled by the pressure variation along the
  line which results from the combined effect of
  friction drop and slope of line.
• 4. When the kinetic energy is considered to be
  small and neglected in a drip irrigation line,
  the pressure variation will be simply a linear
  combination of the friction drop and energy gain
  or loss due to slopes.

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Chapter 14 Design of Lateral LinesIntroduction

• A lateral length (or submain) can be designed by
  using a step by step calculation.
• The computer program can be used to simulate
  different situations to develop design charts.

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Chapter 14 Design of Lateral LinesIntroduction

• 7. Simplified design procedures were developed by
  using a general shape of the energy gradient line
  and line slopes.
• 8. Design charts for lateral line design were
  introduced by Wu and Gitlin, however trial and
  error techniques are required in the design
  procedure.

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Chapter 14 Design of Lateral LinesObjectives

• To derive mathematical expressions for lateral
  lines (or sub mains) which will simplify design
  techniques.
• Apply to different types of uniform slope
  conditions, where there is no change in land
  slope along the length of the emitting line.

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Chapter 14 Design of Lateral LinesObjectives

• 3. Relate design length to the total pressure
  head.
• 4. The calculations can be done by digital
  computer, or may be done using a pocket
  calculator.

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Chapter 14 Design of Lateral LinesObjetives

• 5. The adaptability of these design equations to
  computerized solutions should represent a
  significant challenge in drip irrigation design.

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Chapter 14 Design of Lateral
LinesWilliams-Hazens Equation

• Hf K1 V1.852 L - - - - - - - - - - - - -
  - - - - - /1/
• C1.852 D1.167
• where
• K1 3.023 for English Units.
• K1 0.0837 for SI Units.
• Hf Friction Drop, feet (meters).
• L Pipe Length, feet (meters).
• D Inside Diameter, feet(meters).
• C Roughness coefficient.

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Chapter 14 Design of Lateral LinesWilliams-Haze
ns Equation

• For a total discharge
• Hf K2Q1.852L - - - - - - - - - - - - - - -
  - - - - - - - - - - - - - /2/
• C1.852 D4.871
• where
• K2 10.45 for English Units.
• K2 2.264 x 107 for the SI Units.
• Q Expressed in gallons per minute (liters per
  second)
• D Expressed in inches (centimeters).

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Chapter 14 Design of Lateral LinesWilliams-Hazen
s Equation

• Equation /2/ calculates the friction drop using
  total discharge, which is constant in the pipe.
  For lateral line or submain, the discharge in the
  line decrease with respect to the length of the
  line. The total friction drop at the end of the
  line can be calculated by applying a correction
  factor which is determined as, 1 / 2.852, by Wu
  and Gitlin

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• The total energy drop due to friction at the end
  of a lateral line or submain can be expressed as
  
• ?H K3 Q1.852 L - - - - - - - - - - - - -
  - - - - - - - - - - - - -/3/
• C1.852 D4.871
• where
• K3 3.6642 for English Units.
• K3 7.94 x 106 for SI Units.
• ?H The total friction drop at the end of a
  lateral line or submain, in feet (meters).

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• Assuming the emitter flow q is uniform or is
  designed with a certain varation, one can
  rearrage equation /3/ into
• ?H K3 q1.852 L2.852 - - - - - - - - -
  - - - - - - - - - - - - - /4/
• C1.852 Sp1.852 D4.871
• where
• q Average emitter flow, in gallons per
  minute (liters per second)
• Sp The emitter spacing, in feet (meters).

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• In a drip irrigation design, the terms q, Sp,
  and D are usually known, therefore
• ?H KL2.852
• where
• K K3 q1.852 Constant - -
  - - /5/
• C1.852 Sp1.852 D4.871

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• If the total friction drop, ?H, is divided by
  total length L. It is considered as a
  dimensionless term S, equation /5/ can be
  expressed as
• S K L1.852 - - - - - - - - - - - - - - - -
  - - - - - - /6/
• The total friction drop shown in equation /5/ is
  the total friction drop over the full length of
  the line.

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• The friction drop along the line can be
  determined from a dimensionless energy gradient
  line as derived by Wu and Gitlin. It can be
  expressed as follows
• ?Hp 1- 1- P/L2.852 ?H - - - - - - - -
  - - - - - /7/
• where
• ?Hp The total friction drop at a distance
  P, from the inlet.

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• When a lateral line or submain is laid on uniform
  slopes, the total energy gain (down slope
  situation) or loss (up slope situation) or loss
  (up slope situation) due to change in elevation
  can be expressed as
• ?H' SO L - - - - - - -- - - - - - - - - - -
  - - - - - - - - /8/
• where
• ?H' The total energy gain or loss due to
  uniform slope at the end of the line, in feet
  (meter)
• SO The line slope.

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Chapter 14 Design of Lateral LinesSOME BASIC
EQUATIONS

• The energy gain or loss at a point along the
  line due to uniform slopes can be shown as
• ?H'p So p - - - - - - - - - - - - - - - - -
  - - - - - - - - - - - -/9/
• ?H'p P/L ?H' - - - - - - - - - - - - - - -
  - - - - - - - - - - /10/
• where
• ?H'p The energy gain or loss due to slopes
  at a length P measured from the inlet
• SO The land slope
• ?H' The energy gain or loss due to slope
  over the total length of the line.

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Chapter 14 Design of Lateral
Lines PRESSURE PROFILES

• The pressure head profile along the lateral or
  submain can be determined from the inlet
  pressure, friction drop and energy change due to
  slopes.
• Hp H ?Hp ?H'p - - - - - - - - - - - - - -
  - - - /11/
• where
• H The inlet pressure or operating pressure
  expressed as pressure head, in feet (meters), the
  plus sign mean down slope and the minus sign
  means up slope.

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Chapter 14 Design of Lateral
Lines Profiles of Pressure

• Substituting equations /7/ and /10/ into equation
  /11/, we have
• Hp H 1 1 P/L2.852 ?H P/L?H' - -
  - - - - - - - - /12/
• The equation /12/ describes pressure profiles
  along a lateral line or submain.

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Chapter 14 Design of Lateral Lines Profiles of
Pressure

• The shape of profiles will depend on the inlet
  pressure (initial pressure), total friction drip
  and total energy change by slopes.
• There are five typical pressure profiles as shown
  in the figure 1 and these can be explained as
  follows

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Chapter 14 Design of Lateral LinesProfiles of
PressureProfile Type I

• 1. This occurs when the lateral line (or submain)
  is on zero or uphill slope.
• 2. Energy is lost by both elevation change due to
  upslope and friction.
• 3. The pressure decreases with respect to the
  length of the line and the maximum pressure. Hmax
  is at the inlet and minimum pressure,Hmin is at
  the downstream end of the line.

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Chapter 14 Design of Lateral LinesProfile
Type II Type a

• 1. This occurs when the lateral line (or submain)
  is on downslope situation, where a gain of energy
  by slopes at downstream points is greater than
  the energy drop by friction but the pressure at
  the end of the line is still less than the inlet
  pressure.
• 2. The maximum pressure, Hmax is at the inlet and
  a minimum pressure is located somewhere along the
  line.

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Chapter 14 Design of Lateral LinesProfile
Type II Type b

• 1. This is similar to Type IIa but the profile is
  such that the end pressure is equal to the inlet
  pressure. The maximum pressure, Hmax is at the
  inlet and the end of the line. The minimum
  pressure, Hmin is located somewhere near the
  middle section of the line.

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Chapter 14 Design of Lateral LinesProfile
Type II Type c

• 1. This occurs when the line slope is even
  steeper so the pressure at the end of line is
  higher than the inlet pressure.
• 2. In this condition, the maximum pressure, Hmax
  is at the downstream end of the line and the
  minimum pressure is located somewhere along the
  line.

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Chapter 14 Design of Lateral Lines
Profile Type III

• 3. This occurs when the lateral line (or submain)
  is on steep down slope conditions where the
  energy gain by slopes is larger than the friction
  drop for all sections along the line.
• 4. In this condition, the maximum pressure is at
  the downstream end of the line and minimum
  pressure is at the inlet.

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Chapter 14 Design of Lateral Lines Profile
Type III

• The location of the minimum pressure along the
  pressure profile II-a-b-c, can be determined by
  differentiating equation /12/ with respect to the
  length P and setting the derivative equal to
  zero.
• 2.852 ( 1 P/L )1.852 ?H/L ?H'/L 0 - -
  - - - - /13/

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Chapter 14 Design of Lateral Lines Profile Type
III

• If the term ?H/L, the ratio of total friction
  drop to length, is set as energy slope S,
  equation /13/ becomes
• 2.852 ( 1 P/L )1.852 S So 0 - - - -
  - - - - - - - /14/

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Chapter 14 Design of Lateral Lines Profile
Type III

• Simplifying
• P/L 1 0.3506 So /S 0.54 - - - - - - - - -
  - -/15/
• Equation /15/ shows the location of the point of
  minimum pressure when both So and S are known.

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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONS

• 1. Since the five pressure profiles are smooth
  curves as shown in figure 1, pressure variation
  can be used as a design criteria.

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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONS

• 2.The pressure variation is defined as
• Hvar Hmax Hmin - - - - - - - - - - - - - -
  - - - - - - - /16/
• Hmax
• where,
• Hvar Pressure Variation.
• Hmax Maximum Pressure.
• Hmin Minimum Pressure along the line.

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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONS

• 3. La variación del caudal del emisor también
  puede expresarse en forma similar
• qvar qmax - qmin - - - - - - - - - - - - - -
  - - - - - /17/
• qmax
• where, qmax y qmin are the maximum and
  minimum flows along the emitting line produced by
  Hmax y Hmin.

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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONS

• 4. For the orifice type of emitter flow, the
  relationship between qvar and Hvar is given by
  equation /18/
• Hvar 1 1 qvar 2 - - - - - - - - - - -
  - - - - - /18/
• 5. The design equations using pressure variation
  as a design criterion for each pressure profile
  type are derived as follows.

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Chapter 14 Design of Lateral Lines Profiles of
Pressure
Figure1a. Profiles of pressure head
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Chapter 14 Design of Lateral LinesProfiles of
Pressure
Figure 1b. Profiles of pressure head
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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONSProfile Type I

• The inlet pressure is the maximum pressure
  and the minimum pressure is at the end of the
  line
• Hmin H (?H ?H') - - - - - - - - - - - - - -
  - - ----------- - /19/
• The pressure variation can be expressed as
• Hvar H H ( ?H ?H') - - - - - - - - -
  - - - - - - - - - - /20/
• H
• Hvar ?H ?H' - - - - - - - - - - - - - - - -
  - - - - - - - - - - - - -/21/
• H

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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONS

• Both sides of equation /21/, can be multiplied
  by H/L, to obtain
• Hvar H ?H ?H' - - - - - - - - - - - - - -
  - - -/22/ L L L
• Hvar H ( S So ) L - - - - - - - - - - -
  - - - - /23/
• L Hvar H - - - - - - - - - - - - - - - -
  - - - - - /24/
• S So

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Chapter 14 Design of Lateral LinesDESIGN
EQUATIONS

• The values of Hvar and H are selected by the
  designer.
• So can usually be obtained from field
  measurements S and L are unknown S is as a
  function of L (equation /6/), it is possible to
  substitute equation /6/ for S and derive a
  computational form of equation /24/ that contains
  only one unknown variable, L.
• It is the same equation as given by Howell and
  Hiler 2.
• L Hvar H - - - - - - - - - - - - -
  - - - - - - - - - - - -/25/
• K L1.852 So

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Chapter 14 Design of Lateral LinesProfile
IIType a

• The inlet pressure is the maximum pressure and
  the minimum pressure is somewhere along the line.
  The line slope is downhill and there is energy
  gain due to slope.
• The pressure variation can be expressed as
• Hvar H H (?H'p ?Hp) - - - - - - - -
  - -/26/
• H

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Chapter 14 Design of Lateral LinesProfile
Type IIType a

• Hvar ?Hp ?H'p - - - - - - - - - - - /27/
• H
• Hvar H ?Hp ?H'p - - - - - - - - - /28/
• L L L

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Chapter 14 Design of Lateral LinesProfile
Type IIType a

• Substituing equations /7/ and /10/ into /28/ and
  simplifying, we obtain
• L Hvar H - - - - - - - -
  /29/
• 1 (1- P/L)2.852 . P . So
• 
  L
• Substituting equation /15/ into /29/ and
  simplifying, we obtain
• L Hvar H - - - - /30/
• S So 0.3687 . (So /S)0.54 1

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Chapter 14 Design of Lateral LinesProfile
Type IIType a

• The computerized form of equation /30/ is
  obtained by substituting the equation /6/ into
  /30/
• L Hvar H /31/
• KL1.852 So 0.3687(So/ KL1.852 )0.54
  1

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Chapter 14 Design of Lateral LinesProfile Type
IIType b

• This is similar to Type II-a, the only
  difference is that S and So are equal as defined
  by equations /6/ and /8/. It is therefore
  possible to substitute So for S in both equations
  /30/ and /31/.
• Equation /30/ can be shown as
• L Hvar H - - - - - ---------- /32/
• S So 0.3687 . (So /So )0.54 1
• and simplifying
• L Hvar H - - - - - - - - - - - - - - -
  - - - - - - - - - - - -/33/
  0.3687So

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Chapter 14 Design of Lateral LinesProfile
Type IIType c

• The maximum pressure is located at the
  downstream end of line and the minimum pressure
  is somewhere along the line.
• The pressure variation can be expressed as
• Hvar H ( ?H' ?H ) H (?Hp ?Hp) -
  - - - - - - - /34/
• H (?H' ?H )
• Hvar ?H ?H ?Hp ?Hp - - - - - - - - - -
  - - - - - - - - -/35/
• H (?H' ?H )

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Chapter 14 Design of Lateral LinesProfile
Type IIType c

• Hvar H ?H'/L ?H/L ?Hp /L ?Hp/L
  Hvar /L ?H'/L ?H/L - - - - - - - - - - - -
  - - - - - - - - - - - - - - - - - - - - - /36/
• Substituting equations /7/, /10/, /15/ and
  simplifying, we obtain
• L Hvar H - - - - - -/37/
• So 0.3687 (So /S)0.54 (So S) .
  Hvar
• The computational form of this equation is
  expressed as
• L Hvar H /38/
• So 0.3687(So/ KL1.852 )0.54
  Hvar( So KL1.852 )

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Chapter 14 Design of Lateral LinesProfile
type III

• The derivation of the Type III profile is simpler
  than the other down slope situations because
  there is no minimum point along the pressure
  profile.
• The value Hmin is H at the head of the emitting
  line, and Hmax is at the end of the emitter line.

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Chapter 14 Design of Lateral LinesProfile
Type III

• The pressure variation can be expressed as
• Hvar H ( ?H' ?H ) H - - - - - - - - -
  - - - - - - - - - - -/39/
• H (?H' ?H )
• Hvar H Hvar (?H' ?H ) ( ?H' ?H ) - - -
  - - - - - - - /40/
• Hvar H ( So S )Hvar ( So S ) - - - -
  - - - - - - - - - - -/41/
• L
• L Hvar H - - - - - - - - - -
  - - - - - - /42/
• ( So S ) . ( 1 - Hvar )

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Chapter 14 Design of Lateral LinesProfile
Type III

• The design length can be expressed as
• L Hvar H - - - - - - - -
  -/43/
• (So KL1.852 )( 1 Hvar)

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Chapter 14 Design of Lateral LinesCRITERIA FOR
THE SELECTION OF THE APPROPRIATE DESIGN EQUATION

• The criteria for selecting which of the five
  design equations to use for a given land slope
  and flow situations are dependent on the
  relationship between S and So.
• The criteria for the Type I profile is simplest,
  equation /25/ is used when there is zero slope or
  for uphill slopes.

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Chapter 14 Design of Lateral LinesCRITERIA FOR
THE SELECTION OF THE APPROPRIATE DESIGN EQUATION

• 3. The criteria for choosing which of the four
  down slope design equations to use are based on
  the magnitude of S and So and on equation /15/.

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Chapter 14 Design of Lateral LinesCRITERIA FOR
THE SELECTION OF THE APPROPRIATE DESIGN EQUATION

• 4. The Type II-a profile is characterized by S
  being greater than So
• S gt So S/So gt 1 KL1.852 gt 1 - - - - - - - -
  - - /44/
• So
• 5. The profile type II-b is characterized because
  S is equal to So
• S So S/S 1 KL1.852 1 - - - - - - - - -
  /45/
• So

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Chapter 14 Design of Lateral LinesCRITERIA FOR
THE SELECTION OF THE APPROPRIATE DESIGN EQUATION

• 6. The profiles II-c and type III are
  characterized, because S is smaller than So
• S lt So S/So lt 1 KL1.852 lt 1 - - - - - - - - -
  - - - /46/
• So
• 7. If the land slope and flow conditions satisfy
  this inequality, it is possible to use equation
  /15/ to determine which design equation to use
  for the Type II-c pressure profile.

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Chapter 14 Design of Lateral LinesCRITERIA FOR
THE SELECTION OF THE APPROPRIATE DESIGN EQUATION

• 8. The minimum point occurs at P/L greater than
  zero and less than 1. This occurs if the
  following inequality holds true.

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Chapter 14 Design of Lateral
LinesDESIGN EXAMPLES

• In the developed design equations, the design
  length cannot be solved directly.
• One can use a calculator and use a trial and
  error method to determine the length, or use
  Newtons method of approximation iteratively to
  determine the length using a computer program.
• Two design examples are shown as follows

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Chapter 14 Design of Lateral LinesExample 1

• Lateral line on a 1 uphill slope.
• The following data are given, and it is necessary
  to determine the maximum L for the land slope and
  flow conditions using Type I profile design
  equation
• Sp 2.0 feet (0.61 m) Emitter spacing.
• d 0.010 inches (0.2540) Emitters
  diameter.
• q 0.0047 gpm (2.1 x 10-5 lps) Design
  emitter flow.
• D 0.56 inches (1.42 cm) Lateral line
  diameter.
• H 10.4 feet (3.17 m ) Inlet pressure.
• Hvar 0.10 Pressure variation magnitude.
• So 0.10 Landslope uphill.

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Chapter 14 Design of lateral linesExample
1

• c 150 Roughness Coefficient.
• The equation /5/ is written as
• K 7.94 x 106 q1.852 - - - - - - - - - -
  - - - - - - - - - -/51a/
• C1.852 Sp1.852D4.871
• K 7.94 x 106 ( 2.1 x 10-5 )1.852 - - - - -
  - - - - - - - - - /51b/
• (150)1.852 (0.61)1.852 (1.42)4.871
• K 7.82 x 10-8 - - - - - - - - - - - - - - - -
  - - - - - - - - - - - - /51c/

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Chapter 14 Design of Lateral LinesExample 1

• From the equation /25/
• L (0.19) (3.17) - - - - - - - - - -
  - - - - - /52a/
• 7.82 x 10-8 L1.852 0.010
• L 178 ft (54.27 m ) - - - - - - - - - - - - - -
  - - - - - - - - - - - /52b/
• We can also obtain a graphical solution solving H
  for various L and tracing a graph to determine
  L.
• The particular type of line can extend
  approximately to 180 feet (55 m) over an up slope
  of 1, before Hvar will exceed 19, that
  corresponds to a qvar of 10.

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Chapter 14 Design of Lateral LinesExample 2

• Lateral line on 1.5 downhill slope.
• The first design equation used is that for the
  Type II-a profile.
• Sp 8 feet (2.4 m) Emitter spacing.
• d 0.019 inches (0.48 mm) Emitter diamter.
  
• q 0.026 inches (1.55 cm) Design emitter
  flow.
• H 28.37 feet (8.65mm) Inlet pressure.
• Hvar 0.19 Pressure variation magnitude.
• So 0.015 Land slope downhill.
• C 137 Roughness Coefficient.

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Chapter 14 Design of Lateral LinesExample 2

• Using these values, we obtain
• K 7.98 x 106 (1.14 x 10-14 )1.852 - - - - - - -
  - - - - - - - - - - /53a/
• K 9.93 x 107 - - - - - - - - - - - - - - - - -
  - - - - - - - - - - - - /53b/
• From equation /31/
• L 9.93 x 10-7L1.852 0.015 0.3680.015/9.93
  x 10-7 L1.852)0.54 -1 - - - - - - - - - - - - -
  - - - - - - - - - - - - - - - - - /54a/
• L 201 meters - - - - - - - - - - - - - - - - -
  - - - - - - - - - - - - /54b/

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Chapter 14 Design of Lateral LinesExample 2

• The solution can also be obtained, graphing L and
  H as shown in example 1. The answer need to be
  verified to determine, if the right equation was
  used.
• S/So 9.93 x 10-7 (201)1.852 1.22 - - -
  - - - - - - /55/
• 0.015

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Chapter 14 Design of Lateral LinesSummary

• 1. Five pressure profiles were presented and
  considered.
• 2. These represent design conditions which
  result from a lateral line (or submain) laid on
  uniform slopes.

63
Chapter 14 Design of Lateral LinesSummary

• 3. Procedures were developed to identify
  pressure profiles by land slope and total
  friction drop at the end of the line.
• 4. Equations for designing lateral length (or
  sub main) based on a given criteria, pressure
  variation, were derived.

64
Chapter 14 Design of Lateral LinesSummary

• 5. These equations cannot be solved directly,
  but solutions can be obtained using trial and
  error technique on a pocket calculator or by
  using Newtons method of approximation in a
  computer program.
• 6. The developed mathematical equations can be
  useful in the future development of computerized
  drip irrigation system design.

65
Chapter 14 Design of Lateral LinesList of
Symbols
66
Chapter 14 Design of Lateral LinesList of
Symbols
67
Chapter 14 Design of Lateral LinesList of
Symbols
68
Chapter 14 Design of Lateral LinesList of
Symbols
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Chapter 14 Design of Lateral LinesList of
Symbols
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Chapter 14 Design of Lateral Lines
BIBLIOGRAPHY

• Gillespie, V. A., A. L. Phillips and I. P. Wu,
  1979. Drip irrigation design equations. Journal
  of the Irrigation and drainage Division, ASCE,
  105 (IR 3) 247-57, Proc. Paper 14819.
• Howell, T. A. and E. A. Hiler, 1974. Trickle
  irrigation lateral design. Transactions American
  Society of Agricultural Engineers, 17(S) 902-08.
• 3. Howell, T. A. and E. A. Hiler, 1974. Designing
  trickle irrigation laterals for uniformity.
  Journal of the Irrigation and Drainage Division,
  ASCE, 100 (IR 4) 443-54, Proc. Paper 10983.

71
Chapter 14 Design of Lateral Lines BIBLIOGRAPHY

• Williams, G.S. and A. Hazen, 1960. Hydraulic
  Tables 3rd edition. New York, John Wiley and
  Sons.
• Wu, I.P. and D.C. Fangmeir, 1974. Hydraulic
  Design of Twin-chamber Trickle
• 6. Irrigation Laterals, Technical Bulletin No.
  216, The Agricultural Experiment Station,
  University of Arizona, Tucson, Ariz., Dec.
• 7. Wu, I.P. and H.M. Gitlin, 1973, Hydraulics and
  Uniform for Drip Irrigation.

72
Chapter 14 Design of Lateral Lines BIBLIOGRAPHY

• Journal of the Irrigation and Drainage
  Division, ASCE, Vol.
• 99, (IR3) 157-168. Paper 9786, June.
• Wu, I.P. and H.M. Gitlin, 1974, Design of Drip
  Irrigation
• Lines. Technical
• 10. Bulletin No. 96, Hawaii Agricultural
  Experiment Station,
• University of Hawaii, Honolulu, Hawaii,
  June.
• 11. Wu, I.P. and H.M. Gitlin, 1975, Energy
  Gradient Line for
• Drip Irrigation Laterals,
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