Over the past decade in the United States, irrigation as a whole, and drip irrigation in particular, has experienced unprecedented growth. In. 1985, total area ...
DRIP IRRIGATION DESIGN AND EVALUATION BASED ON THE STATISTICAL UNIFORMITY CONCEPT V. F. Bralts and D. M. Edwards Department of Agricultural Engineering Michigan State University East Lansing, Michigan 48824
l-Pai Wu Department of Agricultural Engineering University of Hawaii Honolulu, Hawaii 96822
I. Introduction Over the past decade in the United States, irrigation as a whole, and drip irrigation in particular, has experienced unprecedented growth. In 1985, total area under drip irrigation in the United States was over 300,000 ha (750,000 acres). This compares with only 50,000 ha (125,000 acres) in 1970. A drip irrigation system, Fig. 1, consists of a water supply and pump followed by a network of mainlines and submains, laterals, and emitters. The mainline is the primary artery for delivery of water to the various irrigation zones. Within each zone there is usually a number of submain units. Submain units can consist of 1-5 ha (2.5-12.5 acres), while a zone consists of 20-50 ha (50-125 acres). Hydraulic design is probably the most important factor in the ultimate success or failure of a drip irrigation system. Over the past decade a significant amount of research has been done in this area. To assist in the improved design of drip irrigation systems, Keller and Karmeli (1974) developed the Emission Uniformity Concept, Wu and Gitlin (1974) developed the Emitter Flow Variation Concept, and Bralts et al. (1981a,b) developed the Statistical Uniformity Concept. The design of a drip irrigation submain unit for optimum emitter uniformity is very important, because once the emitters and lateral and submain components have been 67 A D V A N C E S IN IRRIGATION, V O L U M E 4
Copyright © 1987, by Academic Press Inc. All rights of reproduction in any form reserved.
68
V. F . BRALTS et
F I G . 1.
al.
Drip irrigation system components.
chosen, very little additional flow control is possible. Thus, the engineer making the design decisions regarding pipe size and emitter selection must have a method of determining submain unit emitter flow uniformity at the design stage. In addition to the delivery of water, a drip irrigation system can be an effective means for the application of fertilizer and other chemicals to plants. It has been shown that a savings of fertilizer can be attained using drip irrigation systems (Kesner et al., 1985). In addition, drip irrigation systems can reduce labor and the energy cost of fertilizer application. Despite these advantages, the use of fertilizer injection through drip irrigation systems has not been fully realized. An important reason for the lack of development has been that irrigators were not certain that their drip irrigation systems were performing at an acceptable level of uniformity for fertilizer injection. The problem has been a lack of the simple useroriented field evaluation tools. The field evaluation of drip irrigation submain units is important to the design engineer to confirm the successful design of a submain unit to the desired emitter flow uniformity specifications. Second, field evaluation is important to an irrigator in deciding if the submain unit performance is acceptable for fertilizer injection. Third, field evaluation is important as a diagnostic tool for problem submain units.
DRIP IRRIGATION DESIGN AND EVALUATION
69
In this article drip irrigation design and evaluation procedures based on the statistical uniformity concept will be developed. The coefficient of variation and the constant odds uncertainty formulation will be used to relate hydraulic design and field evaluation procedures for drip irrigation submain units. In particular, hydraulic variation, manufacturer's variation, emitter plugging variation, and the number of emitters per plant will be related to submain unit performance. This work is the first comprehensive presentation of a statistically based approach to both design and evaluation of drip irrigation systems. In addition, the relationships of the statistical uniformity to irrigation and application efficiency and schedules will be discussed. Based on the above, simplified graphical techniques for hydraulic design and field evaluation of drip irrigation systems will be presented.
II. Basic Hydraulics The hydraulic principles of drip irrigation, based upon the classical equations of continuity and energy, have been developed by various researchers, (Wu and Gitlin, 1974; Howell and Hiler, 1974a,b; and Keller and Karmeli, 1974). The following development will closely follow the theory and nomenclature used by Wu et al. (1979). A . P I P E F L O W EQUATIONS
The flow in drip irrigation lateral or manifold pipe can be considered to be hydraulically steady, spatially varied pipe flow. This means that the total flow through the pipe is changing, usually decreasing, with respect to length. The pressure distribution or energy gradient line along the pipe is also changing, usually decreasing, as it is affected by friction and elevation. Figure 2 represents the flow and pressure distribution along a drip irrigation lateral line. By considering drip irrigation pipes as hydraulically smooth, any one of several empirical equations can be used to calculate head loss due to friction. In this article only two such equations will be discussed. The first equation, which is based on the Darcy-Weisbach equation, begins with h =f(LVVD2g) f
(1)
where h = head loss due to friction,/ = dimensionless friction factor, L = length of pipe, V = velocity of water in the pipe, D = diameter of the pipe, and g = acceleration of gravity. Since drip irrigation lateral lines can be assumed to be hydraulically f
70
V. F . BRALTS et
al.
L A T E R A L LINE PRESSURE H
h!
G
I
I
h
h 2
n-2
I
h -1 n
!
h
r»
!
EMITTER FLOW
F I G . 2.
Water and pressure distribution along a drip irrigation lateral line (Wu and
Gitlin, 1974).
smooth and their flow fully turbulent, the Blasius empirical formula for turbulent flow in a smooth pipe can be substituted f o r / ( W u and Gitlin, 1974; Howell et al., 1981). The Blasius formula is / - 0.3164//?°
25
(4,000 < R < 100,000) Q
(2)
where / = friction coefficient and R = Reynolds number. Watters and Keller (1978) combined Eqs. (1) and (2) at 20°C and found e
hf = 7.89 x l O ^ 5
1 7 5
/D
4 75
)L
(3)
where hf = head loss in meters, Q = flow rate in liters/second, D = pipe diameter in millimeters, and L = pipe length in meters. The second empirical equation which is commonly used in hydraulic design is the Hazen-Williams formula (Keller and Karmeli, 1975; Jeppson, 1982). In equation form, h = 1.22 x l O ^ 1 0
1
8 5 2
{
/^-
8 5 2
^ 4
8 7 1
)^
(4)
were C = the pipe roughness coefficient and all other variables are as previously defined. If a C value of 150 for smooth pipe is substituted into Eq. (4), we obtain the following empirical equation: hf= 11.38 x 10 (e 5
, 852
/Z)
4 871
)L
(5)
Both Eqs. (3) and (5) are in the same units and result in very similar solutions to drip irrigation hydraulic problems. The major difference be-
71
DRIP IRRIGATION DESIGN A N D EVALUATION
tween the two equations is that the Darcy-Weisbach equation can be corrected for viscosity through the Reynolds number term in the Blasius equation while the Hazen-Williams equation cannot. Figure 3 is a comparison of the two equations for 27-mm (1-in.) pipe as taken from Hughes and Jeppson (1978). Howell et al. (1981) reviewed the work done by Hughes and Jeppson (1978) and commented that C values of 130 and 150 in the Hazen-Williams equation are clearly dependent upon R . Furthermore, when comparing the two equations at the same velocity, the C value of the HazenWilliams equation seemed dependent upon pipe diameter. Howell et al. (1981) found C equal to 130 for 14- to 15-mm (0.58-in.) plastic pipe, C equal to 140 for 18- to 19-mm (0.75-in.) plastic pipe, and C equal to 150 for 25- to 27-mm (1-in.) plastic pipe. Generally, underestimating the C value results in a more conservative design. Both Eqs. (3) and (5) are generalizable into the form used by Wu and Gitlin (1975) and Wu et al. (1979), as e
(6)
AH = -aQ L
FRICTION FACTOR, f
m
REYNOLDS NUMBER, R
F I G . 3.
e
M o o d y diagram with friction factors (Hughes and Jeppson, 1978).
72
V. F . BRALTS et
al.
where AH = head loss due to friction (-/i ), Q = total lateral line flow, a = a pipe constant, L = lateral line length, and m = the pipe flow exponent. In the next section, Eq. (6) will be used to approximate the energy gradeline due to friction required. f
B. E M I T T E R F L O W EQUATIONS
Drip irrigation emitters vary in their design from elaborate pressurecompensating devices to long-flow-path and simple orifice-type emitters. In general, the flow characteristics of emitters been shown by Karmeli (1977) and Wu et al. (1979) to be (7)
q = kh
x
where q = emitter flow rate, k = constant of proportionality, h = pressure head at the emitter, and x = emitter discharge exponent. Equation (7) can be derived from a combination of the Bernoulli energy and continuity equations. The constant of proportionality, k, in the emitter flow equation contains the variables such as the coefficient of discharge, emitter geometry, and the acceleration of gravity. The value of x in Eq. (7) characterizes the type of emitter and/or the flow regime in a long-flow-path emitter. For example, orifice-type emitters have an emitter discharge exponent of 0.5. In long-flow-path emitters x = 0.5 for fully turbulent flow and x = 1.0 for laminar flow (Karmeli, 1977). An emitter with an x value of less than 0.5 would be pressure compensating in nature (Wu et al., 1979). The ideal value of x is zero, which would make the emitter fully pressure compensating. A special form of the emitter flow equation when considering emitter plugging is q = (1 - a)kh
x
(8)
where a = the degree of plugging and all other variables are as previously defined. Equation (8) will be used when considering design and evaluation procedures including emitter plugging. C . HYDRAULIC ANALYSIS TECHNIQUES
Since the flow in the lateral line is spatially varied with decreasing discharge, the energy gradeline is an exponential curve rather than a straight line. The solution of the energy gradient in drip irrigation submain units can be determined by the approximation or by iterative procedures. In this section the approximation procedure developed by Wu and Gitlin (1975) and the finite element method developed by Bralts and Segerlind (1985) for microcomputers will be presented.
73
DRIP IRRIGATION DESIGN A N D EVALUATION
1. Dimensionless
Energy Gradient Line
The dimensionless energy gradient procedure developed by Wu and Gitlin (1975) assumes that all emitters along a lateral line discharge the same flow. This approximation results in a curve which can be used to directly calculate the head loss at any point along any lateral line. When assuming constant emitter flow, along the lateral line, the shape of the energy gradeline can be expressed dimensionlessly to be the energy drop ratio (Ri), as shown by Wu and Gitlin (1975), as Rt = AHi/AH = 1 - (1 -
i)
(9)
m+l
where / = ratio l/L; /, L, A///, and AH are as defined in Fig. 4; and m is as defined in Eq. (6). If the Hazen-Williams equation is used, then, Eq. (9) becomes Ri
=
\
-
(I
-
(10)
/)2.«52
Figure 5 illustrates the theoretical shape of the dimensionless energy gradient curves for various flow conditions (Wu and Gitlin, 1975). The total pressure variation along a lateral line can be expressed as a combination of the original pressure and the variation due to energy slope and terrain. Expressed summarily, using the dimensionless energy gradient line concept and uniform slope, Wu et al. (1979) developed the equation hi = Ho-
RiAH ± R'iAH'
(11)
where hi = pressure head for a given length ratio, H = pressure head at the origin, R AH = pressure head loss due to friction, and R\AH = pressure head loss or gain due to elevation. Figure 6 shows schematically 0
f
(
o
1
LATERAL LINE LENGTH
o I
PRESSURE HEAD
L
DISTANCE
0 F I G . 4.
Lateral line notation.
74
V. F . BRALTS et al. C O M P L E T E T U R B U L E N C E , R O U G H PIPE TURBULENT
PRESSURE DROP RATIO, Rj (AHj/AH)
LAMINAR
FLOW IN SMOOTH
FLOW
HAZEN—Wl LLIAMS
0
0.1
0.2
0.3
0.4
0.5
LENGTH FIG. 5 .
0.6
PIPE
EQUATION
0.7
0.8
0.9
1.0
RATIO^/L
D i m e n s i o n l e s s energy gradient curve (Wu and Gitlin, 1 9 7 4 ) .
the pressure distributions due to various situations and the resulting energy gradient line. Since the emitter flow is related to pressure (h), as shown in Eq. (7), the emitter flow rate at any point along the lateral line will be equal to a combination of Eqs. (7) and (11), or, combining, we obtain q = k(hi) = k(H - RAH ± RlAH')* x
t
0
(12)
Furthermore, if Eq. (12) is divided by the emitter flow equation for the first emitter q (q = kHo), then the resulting equation becomes independent of the coefficient k. In simplified form Eq. (12) becomes 0
0
q i
= q [l ~ Ri(AH/H ) 0
0
± R'iiAH'lHvW
where all variables are as previously defined.
(13)
P R E S S U R E LOSS B Y F R I C T I O N F I N A L PRESSURE D I S T R I B U T I O N P R E S S U R E G A I N B Y SLOPE
a
PRESSURE HEAD
AH'
0
0.1
JL 0.2
X 0.3
X
0.4
LENGTH
0.5
0.6
0.7
0.8
0.9
1.0
RATIO,//L
P R E S S U R E LOSS B Y F R I C T I O N F I N A L PRESSURE D I S T R I B U T I O N . P R E S S U R E LOSS B Y S L O P E
b
H h 0
< UJ
X
LU QC
D
CO CO UJ
cc Q-
X 0.1
JL 0.2
X
X
X
X
X
0.3
0.4
0.5
0.6
0.7
LENGTH RATIO, F I G . 6.
0.8
0.9
1.0
f/L
(a) Pressure distribution for d o w n s l o p e c o n d i t i o n s ; (b) pressure distribution for
upslope c o n d i t i o n s (Wu et al.,
1979).
76
V. F . BRALTS et
al.
With Eq. (13) the emitter flow can be calculated at various points along a lateral line once the emitter flow rate at the original pressure is known. The dimensionless energy gradient line concept can also be used for drip irrigation submain manifolds, where lateral lines are considered similar to uniformly spaced emitters. Using this concept, Eq. (11) can be rewritten as (14)
hj = H - RjAH ± RjAH' s
where hj = pressure head for a given submain manifold length ratio, H = original pressure at the head of the submain unit, R AH = submain manifold pressure head loss due to friction, and RjAH' = submain manifold pressure head loss or gain due to elevation. Combining Eqs. (11) and (14) to determine the pressure at any point in a submain unit results in s
(
hjt = H s
(15)
RjAH - R AHi ± RjAH' ± RlAHi s
t
s
where h = pressure head at the submain manifold length ratio (j) and the lateral line length ratio (0, RjAH = submain manifold head loss due to friction, R AHi = lateral line head loss due to friction, RjAH = submain head loss or gain due to elevation, and RlAH'i = lateral line head loss or gain due to elevation. Equation (15) can be modified to calculate the emitter flow relative to the emitter flow at the original pressure by the equation jt
s
f
t
qji = q [l ~ s
s
RJ(AHJH ) s
- Ri(AHi/H ) s
± Rj(AH:/H ) s
± R[{AH[IH )Y (16) %
where all variables are as previously defined. Thus using Eq. (16) the approximate emitter flow at any point in the submain unit can be determined once the basic parameters of the submain unit hydraulic system are known. 2. Network Analysis
Techniques
Hydraulic network analysis techniques can be implemented on digital computers. The implementation of these techniques by hydraulic engineers has brought improved speed and accuracy to the analysis of steadystate hydraulic networks. The equations, which are used to describe hydraulic phenomena, are basically nonlinear in nature and, thus, cannot be solved directly. For this reason, numerous algorithms have been written to solve for unknown pipe flow rates and/or junction pressures in an iterative manner. Solutions obtained using the Hardy Cross, NewtonRaphson, and linear theory techniques are the mainstay of present-day hydraulic network analysis.
DRIP IRRIGATION DESIGN A N D EVALUATION
77
Historically speaking, one of the first methods devised for the analysis of hydraulic pipe networks was the Hardy Cross technique (Cross, 1936). This method was popular in the days of hand calculations. The Hardy Cross technique is still taught in most beginning hydraulic courses and has been incorporated into numerous computer algorithms for the solution of hydraulic network problems. The original Hardy Cross method is a flow corrective technique which uses assumed pipe flow rates, based on continuity, to solve the energybased loop equations of a hydraulic network. The resulting loop equations are a set of nonlinear simultaneous equations which cannot be solved directly. The Hardy Cross method uses a combination of the assumed flow and a corrective flow to solve the loop equations. Once the corrective flow is known, a new assumed flow is determined and another solution is calculated. The solution process stops when the corrective flows are within a specified tolerance. A second method proposed by Hardy Cross, described by Chenoweth and Crawford (1974) and Jeppson (1977), solves the equations by adjusting the heads of each node until continuity is obtained. The resulting junction equations are also nonlinear and an iterative solution is necessary. Corrective pressures are calculated in this approach. The fundamental drawback of the Hardy Cross method is slow convergence primarily due to the independent solution of loop and nodal equations. The Newton-Raphson method overcomes this handicap by simultaneously determining all of the corrective flow and head values. Convergence in this case is quadratic. Each subsequent error reduction is proportional to the square of the previous error (Jeppson, 1977). The convergence of the Newton-Raphson method, however, is highly dependent upon a reasonable first approximation (Jeppson, 1977). When the first approximation is in the immediate neighborhood of the solution, it is one of the best methods available. On the other hand, when the initial estimate is quite far off, the Newton-Raphson method has a tendency to overshoot. A disadvantage of the.Newton-Raphson method is the need to evaluate, either analytically or numerically, the first derivative of each flow equation with respect to each corrective flow. Since the flow equations are not related to the corrective flows by simple equations, the derivatives are sometimes tedious to evaluate. The linear theory method of network analysis was first proposed by Wood and Charles (1972). This method has several advantages, the most significant of which is that convergence to the final result is very rapid. Other advantages are that method does not require initial flow estimates or complicated differential equations for the solutions.
78
V. F . BRALTS et
al.
Network analysis by the linear theory method is based on the continuity and energy equations similar to the Hardy Cross and NewtonRaphson methods. Both loop and nodal equations can be solved. The basic theory transforms the loop or nodal equations into linear equations. A set of simultaneous linear equations result and a solution is easily determined. An initial approximation of the flow rates is obtained by assuming laminar flow exits. The calculated flow rates are used to determine the coefficients in the equation, one each successive iteration. Wood and Rayes (1981) reported that when computer solutions based on the nodal equations were successful, a highly accurate solution was obtained in relatively few trials. In some cases, however, convergence was never obtained. The most common application of the linear theory method is with the loop equations. A special case of the linear theory method, which can be solved using the finite element method, was presented by Norrie and deVries (1978). The special case exists when laminar flow is present throughout the hydraulic network. Under such circumstances, the friction drop is already a linear function of flow velocity and can be analyzed using the nodal equations and the finite element method. The primary advantages of the finite element method were a banded and symmetric solution matrix which minimized computer storage requirements. The implementation of computer-based network analysis techniques by irrigation engineers has been sparse at best. Edwards and Spencer (1972) presented design criteria for computer-based analysis of sprinkler irrigation systems using the Hardy Cross method. Solomon and Keller (1974), Wu and Fangmeier (1974), and Perold (1977) have used iterative techniques to solve for flow rates and pressures in individual drip irrigation lateral lines based on assumed end line pressures. These techniques were expanded to submains but were considered too cumbersome for practical sprinkler and drip irrigation design procedures. The layout of conventional sprinkler and drip irrigation systems is such that the development of efficient algorithms for their analysis is possible. Wood (1979) included sprinkler irrigation systems in his hydraulic network analysis computer programs. The finite element method is a systematic numerical procedure for solving complex engineering problems. The method can be used for the solution of discrete element problems such as occur in structural analysis, or for approximate solutions to continuum element problems such as groundwater movement (Segerlind, 1984). In general, the classical finite element method uses an integral formulation and a set of piecewise smooth equations to approximate a quantity. The use of the finite element method for the solution of hydraulic network problems is a simple exten-
79
DRIP IRRIGATION DESIGN A N D EVALUATION
sion of the original development for structural assemblages. The finite element method is presented here as an example, one of many network analysis techniques available for drip irrigation analysis. The nomenclature follows that was used by Bralts and Segerlind (1985). The following development is based on a physical analysis of what a single pipe (element) contributes to the continuity equation. Friction losses in tees and elbows have been neglected. Bernoulli's equation for a straight pipe, Fig. 7 (neglecting the velocity component), is Zi + Pt/y = Zj + Pj/y + hf
(17)
Where Z, and Zj are the downstream and upstream elevation, respectively; P( and Pj are the downstream and upstream pressures, respectively; y is the specific weight of water; and hf is the head loss due to friction. It should be noted that for this network development the subscripts / and j represent any two points along the submain manifold pipe or lateral line. Equation (17) also has the form Zi + Hi = Zj + Hj + kQ
(18)
m
where H and Hj are the downstream and upstream static pressure heads, respectively; Q is the flow through the pipe element; and k and m are the friction loss coefficient and exponent such as those defined by the DarcyWiesbach or Hazen-Williams equations. Equation (18) can be rearranged into t
*
1 / m
e = m
+ Hd - (Zj + Hj)]
Vm
(19)
or Q = C (Hi - Hj) + C (Z - Zj) p
F I G . 7.
p
t
Straight pipe element.
(20)
80
V. F . BRALTS et al.
where C
= [|(Zf + Hd - (Zj + Hj)\V- ]lK m)lm
p
(21)
xlm
is the coefficient for the straight pipe. The absolute value signs are needed to allow for the optimal numbering of the network grid. The pressure at node / is not necessarily greater than the pressure at node j in an optimally numbered grid. Historically the finite element method utilizes the concept of an element stiffness matrix and an element force vector to construct the system of equations (Segerlind, 1984). The element matrices for a single pipe (element) will be developed here. Consider the sequence of nodes r, s, t, and u shown in Fig. 8. These nodes are separated by elements (e - 1), (e), and (e + 1). Element (e) touches nodes s and t; therefore, its contribution to the final system of equations is limited to the equations for nodes s and t. Assuming flow into a node is negative and flow away from a node is positive, the nodal equations are + Qi = 0
~Q r {
X)
(22)
e)
and _Q(E)
+
(E l)
Q
+
j,
=
(
The contribution of element (e) to Eqs. (22) and (23) is simply Q or
( e) s
= C (H - H ) + C (Z - Z ) P
S
= -C (H P
P
t
S
S
t
P
- Z)
)
and Q[
e)
(25)
t
S
3
(24)
t
- H ) - C (Z
2
The element matrices, by definition, are the contribution of an element to the nodal equations that it touches, C AZ
en
P
.en
(26)
CpAZ
Cp.
where AZ = Z - Z . Equation (26) has the standard finite element form t
s
{R^} = WW*}
(27)
-
(e)
(e-1)
s
V
F I G . 8.
(e+1)
t
Three successive pipe elements.
U
81
DRIP IRRIGATION DESIGN A N D EVALUATION
where c p.
-C
p
(28)
is the element stiffness matrix,
is the vector containing the element nodal valves, and
-£}-{'! is the element force vector. The variable g is defined as the product of CpAZ and all other variables are as previously defined. The element matrices are assembled using a direct stiffness algorithm (Segerlind, 1984) and yield a system of equations which have the general matrix form [K]{H} - {F} = {0}
(31)
The vector {//} contains the nodal pressure values for the network. The emitter in a drip irrigation system can be considered a separate component (or element). In this case Bernoulli's Eq. (17) reduces to Pi/y
= PJy + h
(32)
because the elevation difference across the emitting device can be neglected. Equation (32) can be rearranged into Q = C (H - Hj) e
t
(33)
where C is the linearized coefficient for the general form of the emitter flow equation (Q = kh ) as presented by Wu et al. (1979). In this case the friction loss coefficient k and the exponent x can be experimentally determined. The flow equation for an emitter junction, Fig. 9, is e
x
-Q r {
X)
+ G/ (
+
1)
= 0
(34)
once Eq. (33) is substituted for Q[ . An emitter head connected to node s is incorporated into the system of equations as follows: (1) Add the value of C directly to the diagonal value in [K]. (2) Add the value of C H directly to row s of {F}. Implementation of the emitting device in this fashion eliminates the need to number the t node of the emitter and thus reduces the computer e)
e
Q
t
82
V. F . BRALTS et
al.
t (e)
(e-1)
( e + 1)
r
s
F I G . 9.
u
Emitter or sprinkler junction.
storage requirements. This implementation procedure is analogous to the way a spring support is incorporated in a structural analysis. See Fig. 10 for an example submain unit layout (a) and the associated stiffness matrix (b). One advantage of using the matrix form given in Eq. (31) is that many of the existing finite element computer programs can then be used. Another advantage is that the final system of equations is symmetric and banded. Proper numbering of the pipe junctions and emitters produces a relatively small bandwidth. A large drip irrigation design problem can be stored and solved in a relatively small computer.
III. The Statistical Uniformity Concept The statistical uniformity concept consists of a statistical approach to the uniformity of emitter flow and irrigation application efficiency based on the coefficient of variation. Statistically speaking the coefficient of a 1
5
,
(1)
9
(5)
(9)
6
2
10
(2)
7
11
(
(7)
(3) (4)
4
(10)
(6)
3
—I
»
90
1
0.02.04.05.06.07.08
1
1
.09
HYDRAULIC 88
1
VARIATION
.11
.12
OR S T A T I S T I C A L 86
UNIFORMITY 84
82
80
1
1
1
1
1
1
1
.13
.14
.15
.16
.17
.18
.19
1
F I G . 1 6 . N o m o g r a p h to determine the statistical uniformity given the hydraulic and manufacturer's coefficients of variation.
Design example Step 2 is a continuation of the design example begun in the previous section and has been written to illustrate the effect of manufacturer's variation or system uniformity. Design Example: Given:
Step 2
The hydraulic variation or statistical uniformity from Step 1 is 92%. The emitter manufacturer's variation as reported by the manufacturer is 10%. Determine the statistical uniformity including manufacturer's variation. 1. Using the nomograph in Fig. 16, the combined statistical uniformity can be determined. Answer: U = 87%. The manufacturer's variation reduced the statistical uniformity from 92 to 87%. This is still an acceptable uniformity for the drip submain unit based on the criteria given in Section II.
Required: Solution:
s
Comments:
3. Emitter Plugging
Variation
The variation of emitter flow due to emitter plugging (emitter plugging variation, V ) has been researched by numerous individuals. The approach presented by Bralts et al. (1981b) will be used here. From a design standpoint emitter plugging is presumed to occur at some point after installation of the system and its effects should be estimated as part of the design procedure. There are at least five possible cases of emitter plugging. The specific cases are as follows: p
Case I: Case II:
All plugging is partial and affects all emitters equally. All plugging is partial and affects a specific proportion of the emitters equally.
1 .20
V.
96
V. F . BRALTS et
Case III: Case IV:
al.
All plugging is complete and affects all emitters equally. All plugging is complete and affects a specific proportion of the emitters equally. All plugging is a combination of complete plugging and partial plugging or Cases I, II, and IV above.
Case V:
Considering Cases I through IV above, it can be shown that each of these conditions is a special case deriving from the general Case V. For this reason Case V will be developed further. The authors acknowledge that the case of varying degrees of partial plugging within the submain unit exists; however, the solution for that case is beyond the scope of this presentation. Bralts et al. (1981b) developed an equation for Case V where all plugging is a combination of complete and partial plugging. The equation for Case V is
a) )/(0 2
V = {[n( + p'(l P
+ p'U - a) )] - 1} 2
(50)
,/2
where V = the coefficient of variation of emitter flow due to emitter plugging, n = the total number of emitters, = the number of openly flowing emitters, p' = the percentage of emitters partially plugged, and a = the degree of partial plugging. Thus given the varying degrees of complete and partial plugging, Eq. (50) can be used to determine the coefficient of emitter flow variation due to emitter plugging. Once the coefficient of variation due to emitter plugging is determined it can be combined with hydraulic and manufacturer's variation using Eq. (44) p
v = (vl + vi + * n ) 2
(5i)
,/2
q
where all variables are as previously defined. To simplify the design procedure being presented in this chapter only the complete emitter plugging case will be considered. The nomograph presented in Fig. 17 will combine the emitter flow variation or the statisti-
E M I T T E R
1
100
1
In i i t i—i 0 u
. 0 4
s
. 0 6
.1
.08
. 1 2
9 0
88
8 6
0 . 0 4 (
%
.08
.1
I
. 1 2
•
. 1 4
. 1 4
84
1
1
82
1
I
. 1 6
9 6
9 4
9 2
0 . 0 2 . 0 4 . 0 5 . 0 6 . 0 7 . 0 8
1 . 0 9
9 0
1
i
.1
. 1 1
1—i
. 1 6
/
(%)
88
4
1
1
f+
76
1
^ . 2 2
FLOW V A R I A T I O N
1
r
/
8 6
. 1 2
. 1 3
. 1 4
1
1
1
1
1
.22
v
. 2 4
UNIFORMITY
78
1
5
1 .2
. 1 8
STATIS/lCAL
1 .2
E M I T T E R
)
m i i i — i — i — i
1
80
I
. 1 8
s
100
VARIATION
3
1
H
COMBINED 92
i
u
1
(*) 9 6
100
PLUGGING
2
1
i
1
74
1
1
. 2 4
WITHOUT
1 . 1 5
72
1
. 2 6
70
1
1
1
1
V
q
.3
.28
PLUGGING 84
1
.16
1 . 1 7
82
1
. 1 8
1 . 1 9
F I G . 17. Nomograph for calculating the statistical uniformity given the percentage of emitter plugging.
80
1 v
.2
h
97
DRIP IRRIGATION DESIGN A N D EVALUATION
cal uniformity due to hydraulics and manufacturing with the coefficient of variation due to complete emitter plugging. To use the nomograph, identify the percentage of complete emitter plugging on the top line and the statistical uniformity for hydraulic plus manufacturer's variation on the lower line. Connect the two points, and the combined statistical uniformity including emitter plugging can be read on the center line. Design example Step 3 is a continuation of the design example begun in the previous section and has been written to illustrate the efforts of emitter plugging on the statistical uniformity. Design Example: Given: Required: Solution:
Step 3
The statistical uniformity due to hydraulics and manufacturer's variation as determined from the previous design step is 87%. The emitter plugging rate is 3%. Determine the statistical uniformity including emitter plugging. Using the nomograph in Fig. 17, the statistical uniformity including emitter plugging can be determined. Answer: U = 78%. The statistical uniformity including 3% emitter plugging is less than 80% and thus this design would not be acceptable if the emitter plugging is allowed to occur. Either a strict water quality control program should be considered or additional emitters per plant should be installed. s
Comment:
4. Number of Emitters per Plant Statistically speaking, if there are several emitters per plant, the variations of emitter flows tend to cancel one another. The result is that the total variation of emitter flow to any one plant is less than the variation of emitter flow from the individual emitters (Solomon, 1979). If we assume that each emitter is independent then the variation of flow is additive across emitters. This condition of orthogonality allows the emitter flow variation to be combined with the coefficient of variation as in Eq. (45) V = e~ (Vl m
q
+ V\ + x V ) 2
m
h
(52)
where all variables are as previously defined. Equation (52) illustrates the effect of emitter spacing or grouping on the coefficient of variation. If more than one emitter can affect the amount of moisture available to the plant due to such factors as plant spacing, emitter spacing, and/or translocation of water in the soil, the coefficient of variation of emitter flow
98
V. F . BRALTS et
al.
ORIGINAL UNIFORMITY U (%) S
0 10
I
30
I I
1 . 9
50
I .7
60
1
I
70
1
.5
.4
(
%
)
0 20
50
Inni 1 .8
70
80
/ 9 0
i i—\/
.5
.3 / . 2
/
.1
| |
| II 3
I
.1
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95
96
I I I .05.04
97
98
1
1
.03
.02
99
1
98
99
i i—i—i
.05
.03
.02
4 5
Mil 7
10
1 20
.01
I I I I MIL 50
99
F I G . 1 8 . Nomograph for determining the statistical uniformity given the number of emitters per plant.
should be adjusted in this fashion at the design phase. Figure 18 consists of a nomograph to solve Eq. (52). Design example Step 4 is a continuation of the design example begun in the previous section and has been written to illustrate the effects of emitter number per plant or grouping on the statistical uniformity. Design Example: Given: Required: Solution:
Step 4
The statistical uniformity including emitter plugging is 78% as determined in the previous design step and the fact that there are two emitters per plant. Determine the statistical uniformity when considering two emitters per plant. Using the nomograph in Fig. 18, the statistical uniformity considering the number of emitters per plant can be determined. Answer: U = 84%. The resulting design uniformity is accepted based on the criteria given in Section II. s
Comments:
B.
APPLICATIONS TO EVALUATION
The evaluation of a drip irrigation system encompasses significantly more than the uniformity of emitter flow. However, the accurate estimation of system uniformity is probably the single most important indicator of system performance. Field uniformity estimation of drip irrigation systems is important from several perspectives. First, from the engineer's perspective, field uniformity estimation is important in confirming
L
.01
NUMBER OF EMITTERS PER GROUP
/ \ 2
93
UNIFORMITY CONSIDERING EMITTER GROUPS 97
y
e
.2
02
I I I
/
95
luni
90
7-1
.3
/ U
80
1
V
Q
DRIP IRRIGATION DESIGN A N D EVALUATION
99
whether a design was satisfactory. Second, from the purchaser's perspective, field uniformity estimation is important in confirming product performance. Third, from the irrigators's perspective, field uniformity estimation is important when considering irrigation application efficiency and schedules. To date several methods have been proposed for field uniformity estimation. Keller and Karmeli ( 1 9 7 4 ) proposed a modified form of the Soil Conservation Service irrigation system evaluation equation as the absolute emission uniformity (EU ). The field evaluation procedure of this method consisted of collecting emitter flows from each of 1 6 areas in a submain unit and calculating the emission uniformity. The emission uniformity concept has been presented in numerous publications, including Keller and Karmeli ( 1 9 7 5 ) , Merriam and Keller ( 1 9 7 8 ) , and the Soil Conservation Service National Engineering Handbook ( 1 9 8 4 ) . General criteria for E U values are 9 0 % or greater, excellent; 8 0 - 9 0 % , good; 7 0 - 8 0 % , fair; less than 7 0 % , poor. The primary disadvantage of this method is its nonstatistical basis. For this reason, confidence limits and a further breakdown of the components of emitter flow variation are not possible. A second method of field evaluation based upon the design procedures developed by Wu and Gitlin ( 1 9 7 4 , 1 9 7 5 ) would be an estimation of emitter flow variation based on submain pressures. In this case, the field evaluation procedure consists of finding the minimum and maximum pressures in the submain unit and then calculating the emitter flow variation (
