# Dual-Diameter Laterals in Center-Pivot Irrigation System

## Abstract

**:**

_{m}, equal to its minimum value, the optimal pressure head tolerance of the outer lateral, δ

_{I}, amounting to about 2%, with the RE in pressure heads being less than 0.4%, which makes the suggested procedure very accurate. Several applications were performed, compared, and discussed.

## 1. Introduction

## 2. Theory

_{0}(m), Baiamonte et al. [26] considered a lateral where N sprinklers, j = 1, 2, … N, with gradually decreased sprinkler spacing, s

_{1}, s

_{2}, … s

_{N}, are installed (Figure 1).

_{0}, w

_{1}, … w

_{N-}

_{1}. The annulus area irrigated by each sprinkler j, A, was imposed equal for all the sprinklers, and it was expressed by the ratio between the design sprinkler flow rate, q

_{n}, and the desired water application rate, i:

_{n}is fixed, Equation (1) states that for a uniform water application rate, i, a constant irrigated annulus area A, for each sprinkler installed with variable spacing (Figure 1), needs to be imposed. One parameter, A

_{*}

_{,}i.e., the annulus area, A, normalized with respect to the pivot irrigated area, ${A}_{0}=\pi {r}_{0}^{2}$, can be used to fully describe the CPIS geometric and the sprinkler flow rate characteristics:

_{0}and q

_{n}parameters, Figure 2 shows A

_{*}as a function of the water application rate, i, indicating the high variability of A

_{*}(four orders of magnitude). The figure also reports the A

_{*}values corresponding to the applications, run #1 and run #2, that will be discussed later.

_{*}parameter (Equation (2)). For the first annulus, where the first outer sprinkler, j = 1, is installed (Figure 1), the width, w

_{0}, normalized with respect the CPIS radius, r

_{0}, can be expressed by using A

_{*}(Equation (2)), which Baiamonte et al. [26] did not consider:

_{*0}is established, the normalized annulus width of the sprinklers after the first one (j > 1, Figure 1) was determined by imposing for each sprinkler a constant annulus area (A

_{*}), according to a recurrence relation that recursively defines a sequence of gradually decreasing annulus widths, associated with A

_{*}. The distance between the edge of each width and the center-pivot, r

_{j}(Figure 1), normalized with respect to r

_{0}, is denoted as r

_{*j}. Therefore, for the first annulus width, wide w

_{0}, the initial terms, r

_{*0}= r

_{0}/r

_{0}= 1 and w

_{*0}(and A

_{*}, Equation (3)), are established, and the w

_{*j}sequence can be derived:

_{*j}, w

_{*j}, r

_{*j−1}, and w

_{*j−1}are the radius and annulus width corresponding to the sprinkler j and j−1, normalized with respect to lateral length, r

_{0}. Thus, each further term of the w

_{*j}sequence is defined as a function of the preceding terms and corresponds to a sequence of annuli characterized by a constant area, A, irrigated by each sprinkler, thus assuring a uniform water application rate.

_{*j}is the sprinkler spacing between the sprinkler j and j−1, normalized with respect to lateral length, r

_{0}. For N sprinklers, the corresponding normalized lateral length, L

_{*}, which is referred to the distal end of the lateral, starting from the outer sprinkler (Figure 1), equals to:

_{*}values, the cumulated w

_{*j}, ∑w

_{*j}and the cumulated s

_{*j}, ∑s

_{*j}are graphed in Figure 3a and in Figure 3b, respectively, as a function of the sprinkler number j, illustrating an expected monotonically increasing trend. Moreover, for a fixed A

_{*}, because of the radicand constrain in Equation (4), depending on the geometric and hydraulic CPIS characteristics, a lateral length fully covering the CPIS radius (∑w

_{*j}= 1) cannot always be achieved. Indeed, to fully cover the lateral length, different sprinkler characteristics, i.e., q

_{n}, should be considered. However, the latter issue is beyond the purpose of this paper, which aims to design CPIS laterals by only using different inside diameters.

_{*}. For a high number of simulations performed for reasonable values of the triplet (i, r

_{0}, q

_{n}), N is graphed in Figure 4 versus A

_{*}, together with (see Figure 5): (i) the normalized width irrigated by the first sprinkler w

_{*0}(Equation (3)), (ii) the normalized width irrigated by the last sprinkler w

_{*N}, and (iii) the normalized width irrigated by the total number of sprinklers, W

_{*N}:

_{*}(solid line) is fitted by a power law well, and, of course, so is w

_{*0}(Equation (3)). Contrarily, W

_{*N}and w

_{*N}show a worse power-law fitting than N and w

_{*0}, which is due to the sequence annuli randomness in fully covering or not fully covering the entire pivot area, already discussed for Figure 3. However, it seems that the lower A

_{*}is, the more the total annuli width, W

_{*N}(blue circles), approaches the unity (fully coverage).

_{*}values, greater than those displayed in Figure 4, could be not recommended because the corresponding high widths, w

_{*0}and w

_{*N}, even for high spray distance (Figure 1b), might not be covered by one sprinkler, and thus, they should be preliminarily checked [26]. Figure 4 also illustrates the values corresponding to the applications run #1 and run #2 that will be performed later.

_{I}< N, according to two or more than two sectors. For a clear legibility of the considered sketch, for the case of two sectors, i.e., dual-diameter lateral, Figure 5 illustrates the aforementioned parameters, w

_{*0}, w

_{*N}, W

_{*N}, where the subscript I refers to the sector I (outer lateral), whereas the subscript II refers to the sector II (inner lateral). The last sprinkler width, w

_{*N}, which belongs to a common w

_{*j}sequence of the two sectors (green circles in Figure 4), is associated with the total number of sprinklers, N = N

_{I}+ N

_{II}.

_{I}= 28, was imposed (Table 1), Figure 6 shows the corresponding two sectors that could be geometrically designed with a dual-diameter lateral.

#### Hydraulic Design

_{n}, along the lateral [29,30] and local losses, as generally assumed for the pivot laterals [31], which contrarily have to be considered in the drip laterals design.

_{n}, for sector I and for sector II, i.e., for the outer lateral and for the inner laterals (Figure 5 and Figure 6), the PHD line can be expressed as:

_{*j}is the pressure head at the sprinkler j; h

_{*min}is the minimum pressure head to be imposed at the first sprinkler j = 1, both normalized with respect to r

_{0}, h

_{j}/r

_{0}and h

_{min}/r

_{0}, respectively; and K

_{I}and K

_{II}denote the parameters of sector I and of sector II, accounting for the friction losses (friction head loss gradient) and according to the well-accepted Hazen–Williams’s resistance equation [5,32], also recently considered for CPIS [33]:

_{I}(m) and D

_{II}(m) are the inside diameters of the outer lateral and of the inner laterals (sector I and II), respectively, and C is a pipe smoothness factor, which is a function of the pipe material’s characteristics [32,34].

_{*min}, the maximum pressure head at j = N

_{I}, normalized with respect to r

_{0}and denoted as h

_{*n}(Figure 7a), can be imposed to delimit the PHD line in the range h

_{*min}≤ h

_{*j}≤ h*

_{n}:

_{I}the pressure head tolerance assumed for sector I, the normalized average pressure h

_{*n,I}(Figure 7a) can be used to express h

_{*min}and h

_{*n}, respectively:

_{*n}can be expressed as a function of h

_{*min}:

_{*max}and δ

_{ΙΙ}, the maximum normalized pressure and the pressure head tolerance for the inner lateral, a similar relationship to Equation (16) can be derived, if considering that the maximum normalized pressure of sector I, h

_{*n}, corresponds the minimum normalized pressure head of sector II (Figure 7a), providing:

_{*n,II}is the average normalized pressure head for the inner lateral (Figure 7a). Using Equations (17) and (18), h

_{*max}can be expressed as a function of h

_{*n}:

_{*min}by using Equation (15):

_{I}= 0.02 was imposed, h

_{*min}, h

_{*nI}, h

_{*n}, h

_{*nII}, and h

_{*max}values are also reported in Figure 7a, together with their average h

_{*av}= 0.5 (h

_{*max}+ h

_{*min}), which will be considered later to evaluate the corresponding coefficient of variation of the pressure heads around h

_{av}, CV

_{av}.

_{*n}≤ h

_{*j}≤ h

_{*max}, Figure 7a), and thus the sprinkler flow rate variations, as for sector I, h

_{*max}can be imposed at the distal end of the lateral (j = N = N

_{I}+ N

_{II}) in the corresponding energy balance equation (Equation (9)), which by using Equation (12) provides:

_{II}relationship:

_{II}parameter as a function of h

_{*min}can be obtained:

_{*max}also represents the normalized pressure head that the pump system has to provide at the inlet, h

_{*max}≡ h

_{*min}, by using Equation (20), it is also useful to express K

_{I}and K

_{II}as a function of h

_{*in}:

_{I}and δ

_{II}, the K values for the inner and the outer lateral, K

_{I}and K

_{II}, can be expressed according to any fixed h

_{*min}value (Equations (16) and (23)) or to any fixed h

_{*in}value (Equations (24) and (25)), once A

_{*}(Equation (2)) and any pair of pressure head tolerances (δ

_{I}, δ

_{II}), are assumed. Of course, the CPIS pressure head tolerance, δ, is equal to δ

_{I}+ δ

_{II}. Importantly, for any input data, in the K relationships, the mathematical formulation of the principle of the conservation of energy for the lateral of a Center-Pivot is satisfied, so that the pressure head variation is balanced by the sum of friction losses in between the sprinklers.

_{I}and K

_{II}values allows for designing the inner and the outer lateral diameters, by inverting Equations (10) and (11):

_{n}, when deriving K relationships, agrees with the assumption to impose a pressure head tolerance δ

_{I}and δ

_{II}. In fact, along a lateral where sprinklers are installed, it is under limited pressure head variations established by the pressure head tolerances that the ratio between the sprinkler flow rate variation (q

_{max}–q

_{min}), and its average, q

_{n}, is low (for x = 0.5, it equals to 5%, when δ = 10 %), providing a good approximation of this assumption [38,39].

## 3. Applications

_{*}= 0.0149 and r

_{0}= 400 m (Table 1) were assumed; for fixed pressure heads tolerances δ

_{I}= 0.02 and δ

_{II}= 0.08, so that the CPIS pressure head tolerance δ = δ

_{I}+ δ

_{II}= 10%, as it is usual; for a fixed h

_{min}= 13.5 m (h*

_{min}= 0.0338); and for N

_{I}= 28, Table 2 reports all the design variables previously introduced, in particular the cumulated sprinkler spacing ∑s

_{*j,I}and ∑s

_{*j,II}and the corresponding terms ∑s

_{*j,I}(j − 1)

^{1.852}and ∑s

_{*j,II}(j − 1)

^{1.852}, which are required into the K relationships (Equations (24) and (25)).

_{I}and D

_{II}(Table 2), by Equations (26) and (27), which resulted in 82.79 mm and 120.58 mm, respectively. Thus, as it is commonly accepted in practice, this dual-diameter case involves using a larger diameter pipe at the beginning of the lateral and then a lower diameter, as the flow rate decreases starting from the pivot axes.

_{m}, which could be considered as an index of the investment costs, is also reported in Table 2. D

_{m}was calculated as:

_{*,I}= L

_{I}/L and L

_{*II}= L

_{II}/L. Of course, for a CPIS irrigated area fully covered by the sprinklers, the denominator of Equation (28) is equal to the unity.

_{m}) laterals, RE

_{D}, was calculated as:

_{m}is 4.5% less than the diameter, D, corresponding to a one-diameter lateral design. Of course, D

_{m}could be used to roughly detect the most convenient choice of the diameter pair to be considered in the design by an economic point of view, since the lower D

_{m}is, the lower the telescoping pipe cost will be [40,41]. However, a deeper economic analysis should be recommended, since the cost pipe does not necessary vary linearly with the diameter and for the best choice of the lateral diameters, the cost of reducing coupling or adapter should also be considered.

#### 3.1. Numerical Validation

_{I}and K

_{II}relationships requires the corresponding PHD lines to be determined, by considering that, for the outer lateral, the PHD line can be expressed by (Equation (8)), whereas the inner lateral requires the application of Equation (9). For run #1, Figure 7 shows the corresponding PHD lines for both one-diameter and dual-diameter laterals in dimensionless (Figure 7a) and dimensional terms (Figure 7b), together with the PHD derived by the commonly used step-by-step (SBS) procedure (dots), which is rigorous since it considers the continuity and the motion equations repetitively applied to the consecutive sprinkler outlets and the actual sprinkler flow rate–pressure head relationship. The comparison showed in Figure 7 indicates the reliability of the described procedure.

_{*min}= 0. 0338, the PHD line achieves the normalized pressure head in the changing section (h

_{*n}= 0.0351) calculated by Equation (16), and then the maximum normalized pressure, h

_{*max}= 0.0412, is achieved at the distal sprinkler of the lateral.

_{e}= q

_{n}/√h

_{n}= 200.08 l h

^{−1}m

^{−0.5}, which was set equal for both dual- and one-diameter laterals (Table 2). Of course, for x = 0, the solutions provided by the suggested procedure matches that provided by SBS, and it has no sense to be compared. Figure 7 also reports the PHD derived by the suggested procedure and by the SBS method, in the case of a one-lateral diameter.

_{0}value.

#### 3.2. Varying the Pressure Head Tolerances

_{I}(and δ

_{II}) and of the number of sprinklers installed in the first (N

_{I}) and in the second sectors (N

_{II}), for run #1, further applications have been performed. By varying the pressure tolerances, δ

_{I}= 0.02, 0.04, 0.05, 0.06, and 0.08, and N

_{I}(and N

_{II}), Figure 8a–e show the corresponding PHDs, all laying in the admitted range (h

_{*min}≤ h

_{*j}≤ h

_{*max}), indicating that different pairs of the inside diameters could be selected, providing suitable behaviors in terms of sprinkler pressure head distribution.

_{I}= 0.05, i.e., h

_{*n}≡ h

_{*av}, which could be a good choice for constant interspace sprinklers, Figure 8c shows that for an equally distributed number of sprinklers (e.g., N

_{I}= 35), the PHD is not uniform. This because of the high variability in the sprinklers’ interspace, s (Figure 3), which needs to be imposed in CPISs for a uniform water application rate.

_{*n})

_{dual}, and at the inlet, RE(h

_{*max})

_{dual}, whereas for one-diameter laterals, REs were calculated at the maximum pressure section, RE(h

_{*max})

_{one}:

_{*PS}is the normalized pressure head according to the present solution, and h

_{*SBS}is the corresponding value according to the SBS procedure, where x = 0.5 was imposed.

_{I}and δ

_{I}, Table 3 reports RE(h

_{*n})

_{dual}, RE(h

_{*max})

_{dual}, and RE(h

_{*max})

_{one}, which, of course, does not depend on N

_{I}and δ

_{I}. The results show that for dual-diameter, REs are almost slight (generally lower than for one-diameter), and for any N

_{I}, REs decrease at decreasing δ

_{I}. Bold values refer to the maximum RE, which was less than 2.28%, thus demonstrating a good approximation of the exact SBS procedure.

_{I}and D

_{II}values together with the mean weight diameters D

_{m}(Equation (28)), which generally resulted in being lower than those corresponding to one lateral diameter (D = 116.45 mm).

_{m}values that resulted in being higher than D (116.45 mm, Table 2) are in bold, whereas the lowest D

_{m}value (D

_{m,min}= 111.26 mm, bold and underline) is obtained for N

_{I,min}= 28 and δ

_{I,min}= 0.02, indicating that the application illustrated in Figure 7 could be the most convenient choice from an investment cost point of view. For a few cases, for high N

_{I}values, uncommon D

_{I}< D

_{II}conditions occur, and the corresponding D

_{I}values are highlighted by an asterisk. These cases can be observed in the corresponding PHD slopes (Figure 8). For example, compare the case δ

_{I}= 0.06 and N

_{I}= 65 with the corresponding PHD line reported in Figure 8d.

_{m,min}, can also be observed in Figure 9a, where D

_{m}is plotted as a function of N

_{I}, for different δ

_{I}values. Figure 9a shows that for some of the considered N

_{I}and δ

_{I}values, D

_{m}is higher than D (116.45 mm, dashed line), and that the higher D

_{m}values are associated with low and high N

_{I}values. Thus, N

_{I}and δ

_{I}need to be accurately selected to obtain a desirable design. The latter could also be analyzed in terms of the coefficient of variation of the pressure heads, as it is described in the following.

_{*j}), indicating that, contrarily to Figure 9a, the lowest CV values, which are lower than the CV = 0.0498 obtained for one lateral diameter (dashed line), occur for low and high N

_{I}values. However, it needs considering that for dual-diameter laterals, CV values that are higher than for a one-diameter lateral is an expected issue, since it is associated with the abrupt modification of the PHD due to the changing diameter (Figure 7 and Figure 8). A more suitable comparison in terms of the coefficient of variation could be performed with respect to the average pressure head, h

_{*av}, calculated according to the minimum and maximum values, h

_{*av}= 0.5 (h

_{*max}+ h

_{*min}):

_{av}is the variation coefficient calculated with respect to h

_{av}. This because h

_{av}refers to a linear PHD providing a uniform distribution of sprinkler flow rate and can be selected as a reference to evidence the benefit of the dual-diameter laterals.

_{av}as a function of N

_{I}for the same δ

_{I}previously considered, evidencing the benefit of a dual-lateral diameter in terms of PHD around the average h

_{*av}value. Indeed, with exception of few cases, the CV

_{av}calculated for dual-diameter lay below that for one-diameter lateral (dashed line).

_{*}value selected (0.0149), and further applications are needed to investigate the suggested procedure for a wider range of the input parameters.

_{*}parameter, which describes the geometry of the CPIS. The triplet of values (q

_{n}, r

_{0}, i) was selected according to their common ranges also considered in Baiamonte et al. [26] in order to arrange a whole reliable dataset. For each simulation, the D

_{m,min}value was calculated according to an objective function by minimizing D

_{m}and by varying N

_{I}and δ

_{I}:

_{*}, a slight increase in δ

_{min,I}occurs, and that δ

_{I}is close to 0.02 (Figure 8a), which is reasonable, if considering that the outer laterals are characterized by larger areas to irrigate than the inner laterals. Thus, δ

_{I}= 0.02 could be a good approximation of the recommended value to obtain the lowest D

_{m}. The number of sprinklers to be installed in the first sector, N

_{I,min}(and in the second sector, N

_{II,min}), are power-laws, as illustrated in Figure 10b; thus, they could be used for an easy CPIS design.

_{*}(different CPIS geometries), RE

_{D}varies in a narrow range (0.043–0.047). Thus, D

_{m}results in being almost 4.5 % lower than D, thereby confirming the result obtained for run #1 (Table 2).

_{*n})

_{dual}and RE(h

_{*max})

_{dual}(Equation (30)) versus A

_{*}, which makes it possible to conclude that the previous RE range (<2.28%, Table 3) is much lower and almost negligible (<0.3 %), if referred to the considered D

_{m,min}(Equation (32)). For the maximum pressure head in the case of one-diameter lateral, RE(h

_{*max})

_{one}, a bit higher RE values were obtained (Figure 11b).

_{D}were observed (Figure 11a), a more significant effect of the dual-diameter can be observed in terms of CV

_{av}(Equation (31)). For both one and dual-diameters and for the whole dataset, in Figure 11c, CV

_{av}is plotted versus A

_{*}. The figure shows that for one-diameter laterals, CV

_{av}is higher (≅0.083) than for dual-diameters laterals (≅0.068), indicating more suitable PHDs for the latter. Figure 11 also indicates the dots corresponding to the applications run #1 and run #2 that will be performed in Section 3.4.

#### 3.3. Varying the Inlet Pressure Head

_{*in}, which was set equal for the whole dataset (h

_{*in}= 0.0412, Table 2), also plays an important role in the suggested CPIS design procedure. Using Equations (24) and (25), it can be easily observed that h

_{*in}is a scale factor of the K relationships. For run #1 (A

_{*}= 0.0149), Figure 12a,b plot the diameter of sector I, D

_{I}, and sector II, D

_{II}, by varying h

_{*in}with q

_{n}as a parameter. As expected, lower D

_{I}and D

_{II}values could be selected for normalized inlet pressures higher than that considered for run #1 (dot circles), and these values further decrease with decreasing q

_{n}.

#### 3.4. Application for Run #2

_{0}= 700 m) than run #1 (r

_{0}= 400 m), an inlet pressure, h

_{in}= 25 m, higher than 16.5 m (fixed for run #1) was set. For the selected q

_{n}= 300 l/h and i = 0.15 mm/h, A

_{*}equals 0.0013 (Equation (2), Table 1). First, for A

_{*}= 0.0013, the annulus width sequence irrigated by each sprinkler (Equations (3) and (4)) was derived. Second, the number of sprinklers in the first sector N

_{I}has to be determined by the equation displayed in Figure 10b, and approximating to the integer, 316 sprinklers were obtained.

_{I}= 0.02 and δ

_{II}= 0.08 were selected, in order to set the pressure head tolerance of the dual-diameter lateral δ = δ

_{I}+ δ

_{II}= 0.1. Once the parameter ∑s

_{*j,I}(j − 1)

^{1.852}and ∑s

_{*j,II}(j − 1)

^{1.852}, which are required in K relationships are calculated (Table 2), for C = 135, the corresponding sectors’ diameters were determined by Equations (26) and (27), providing D

_{I}= 152.1 mm and D

_{II}= 224.8 mm. Of course, these diameter values do not match the available commercial ones; however, commercial pipe diameters immediately larger than the design ones could be selected, which will provide δ < 0.1. Since the latter is beyond the purpose of this study, in the following, the design diameter values (D

_{I}= 152.1 mm and D

_{II}= 224.8 mm) are considered to compare the suggested procedure and the exact one provided by the numerical step-by-step (SBS) method.

_{I}and δ

_{II}, neglecting the flow variation in the approximate design procedure determines very moderate errors in PHD lines and makes it possible to achieve the established pressure head extremes (h

_{min}and h

_{max}, Table 2). The suggested procedure also provides convenient solutions from an energy-saving point of view. Indeed, it was observed that confining the sprinkler pressure heads into an admitted range, and thus the sprinkler flow rates, favors high emission uniformity and, as for drip laterals, allows one to also save energy [6].

## 4. Conclusions

_{I}, equals 2%, which is reasonable if considering that the outer laterals are characterized by larger areas to irrigate than the inner lateral. For δ

_{I}= 0.02, an error analysis showed that the relative error, RE, between the sprinkler pressure heads, evaluated according to the suggested procedure and those evaluated by the numerical and exact step-by-step procedure are lower than 0.4 %, thus validating the suggested approach. For some practical cases, applications of the proposed procedure were performed and discussed. Although the procedure was presented, applied, and tested for dual-diameter laterals, it could be easily extended to three or more pipe diameters.

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## List of Symbols

A [L^{2}] | = area irrigated by each sprinkler |

A_{0} [L^{2}] | = pivot irrigated area |

A_{*} | = area irrigated by each sprinkler normalized with respect to the pivot irrigated area |

C | = pipe smoothness factor |

CV | = variation coefficient of pressure heads |

CV_{av} | = variation coefficient calculated with respect to h_{av}. |

D [L] | = inside diameter of the one-diameter lateral |

D_{m} [L] | = mean weighted diameter |

D_{m,min} [L] | = minimum value of the mean weighted diameter |

D_{I} [L] | = inside diameter of the lateral of the outer lateral |

D_{II} [L] | = inside diameter of the lateral of the inner lateral |

h_{j} [L] | = pressure head at the j-th sprinkler |

h_{av} [L] | = average pressure head |

h_{min} [L] | = minimum admitted pressure head |

h_{max} [L] | = maximum admitted pressure head |

h_{n} [L] | = nominal pressure head |

h_{*av} | = normalized average pressure head |

h_{*in} | = inlet pressure head normalized with respect to r_{0} (equals to h*_{max}) |

h_{*j} | = pressure head at the j-th sprinkler normalized with respect to r_{0} |

h_{*max} | = maximum allowable pressure head normalized with respect to r_{0} |

h_{*min} | = minimum allowable pressure head normalized with respect to r_{0} |

h_{*n} | = average pressure head normalized with respect to r_{0} (it matches the normalized pressure head in the changing section, for dual-diameter laterals) |

h_{*n,I} | = average pressure head normalized with respect to r_{0} of the outer lateral (sector I) |

h_{*n,II} | = average pressure head normalized with respect to r_{0} of the inner lateral (sector II) |

h_{*PS} | = normalized pressure head according to the present solution |

h_{*SBS} | = normalized pressure head according to the SBS procedure |

I [L T^{−1}] | = water application rate |

k_{e} [L^{3} T^{−1} L^{−0.5}] | = coefficient of the sprinkler flow rate–pressure head relationship |

K_{I} | = friction head loss gradient of the outer lateral |

K_{II} | = friction head loss gradient of the inner lateral |

L [L] | = length of the Center-pivot lateral |

L_{*} | = length of the Center-pivot lateral normalized with respect to r_{0} |

N | = number of sprinklers |

N_{I} | = number of sprinklers of the outer lateral of the telescoping pipe |

N_{I,min} | = number of sprinklers of the outer lateral corresponding to D_{m,min} |

N_{II} | = number of sprinklers of the inner lateral of the telescoping pipe |

N_{II,min} | = number of sprinklers of the inner lateral corresponding to D_{m,min} |

q_{n} [L^{3} s^{−1}] | = design flow rate of the sprinkler |

r_{0} [L] | = radius of the Center-pivot |

r_{j} [L] | = radius corresponding to the sprinkler j |

r_{*j} | = radius corresponding to the sprinkler j normalized with respect to r_{0} |

s_{1} [L] | = sprinkler spacing between the sprinkler j and the sprinkler j − 1 |

s_{*j} | = sprinkler spacing between the sprinkler j and the sprinkler j − 1 normalized with respect to r_{0} |

w_{j} [L] | = annulus width of the sprinkler j |

w_{*j} | = annulus width of the sprinkler j normalized with respect to r_{0} |

w_{0} [L] | = annulus width of the first sprinkler (j = 1) |

w_{N} [L] | = annulus width of the last sprinkler N |

w_{*0} | = annulus width of the first sprinkler (j = 1) normalized with respect to r_{0} |

W_{*N} | = normalized width irrigated by the total number of sprinklers |

x | = exponent of the sprinkler flow rate–pressure head relationship |

δ | = pressure head tolerance |

δ_{I} | = pressure head tolerance of the outer lateral (sector I) |

δ_{I,min} | = pressure head tolerance corresponding to D_{m,min} |

δ_{II} | = pressure head tolerance of the inner lateral (sector II) |

ω rad [T^{−1}] | = angular velocity |

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**Figure 1.**Sketch of sprinklers (axis in black dots) installed along the lateral of a center-pivot of length, r

_{0}, according to Baiamonte et al. [26]. Sprinkler spacing, s, and fractions of the irrigated annulus with different width, w, are indicated: (

**a**) plane view for four sprinklers and (

**b**) cross-section view for two sprinklers. Adapted with permission from Ref. [26]. 2021, ASCE.

**Figure 2.**Relationships between the A

_{*}parameter as a function of the water application rate i (mm/h), for different pairs (r

_{0}, q

_{n}). Dots refer to the application performed (run #1 and run #2).

**Figure 3.**For different values of the A

_{*}parameter, (

**a**) normalized cumulated annulus width, Σw

_{*j}, associated with the sprinklers and (

**b**) normalized cumulated sprinkler spacing, Σs

_{j}, versus the number of the sprinkler, j. Dots indicate the corresponding maximum values Σw

_{*jmax}and Σs

_{*jmax}.

**Figure 4.**Relationships between the number of sprinklers, N, and dimensionless widths, w

_{*0}, w

_{*N}, and W

_{*N}, as a function of the A

_{*}parameter. Dots refer to the application performed (run #1, circles, and run #2, squares).

**Figure 5.**For a dual-diameter lateral, geometric sketch of sector I (with N

_{I}sprinklers) and sector II (with N

_{II}sprinklers), where the geometric parameters are indicated.

**Figure 6.**For a dual-diameter lateral, corresponding to run #1 (Table 1), plan view of the sectors I and II, where the geometric parameters are indicated.

**Figure 7.**For run #1 (Table 1 and Table 2), for a dual-diameter lateral and for a one-diameter lateral, comparison between the PHD line is obtained by the suggested procedure (solid line) with that obtained by the step-by-step procedure (dots) (

**a**) in dimensionless terms and (

**b**) in dimensional terms. The characteristics pressure heads are indicated.

**Figure 8.**For run #1 (Table 2), PHD lines obtained by the suggested procedure by varying N

_{I}and N

_{II}(see the legend in Figure 8c) for different values of the pressure head tolerance of sector I, δ

_{I}: (

**a**) δ

_{I}= 0.02, (

**b**) δ

_{I}= 0.04, (

**c**) δ

_{I}= 0.05, (

**d**) δ

_{I}= 0.06, and (

**e**) δ

_{I}= 0.08.

**Figure 9.**For run #1, relationships between (

**a**) the mean weight diameter, D

_{m}, (

**b**) the variation coefficient, CV, and (

**c**) the variation coefficient with respect to the average pressure h

_{av}, CV

_{av}, as a function of N

_{I}, with δ

_{I}as a parameter. Dashed line refers to the corresponding values obtained for one-diameter laterals.

**Figure 10.**For the whole investigated dataset, the relationship between (

**a**) δ

_{I}value, δ

_{I,min}, corresponding to the minimum D

_{m}, D

_{m,min}, and (

**b**) related number of sprinklers in the first sector, N

_{I,min}, in the second sector, N

_{II,min}, and their sum N

_{min}= N

_{I,min}+ N

_{II,min}, versus A

_{*}. Dots refer to the applications performed (run #1 and run #2).

**Figure 11.**For the whole investigated dataset, the relationship between (

**a**) the relative difference (D − D

_{m})/D; (

**b**) the relative errors RE for dual-diameter lateral, RE(h

_{*n})

_{dual}and RE(h

_{*max})

_{dual}, and for one-diameter lateral, RE(h

_{*max})

_{one}; and (

**c**) the variation coefficient CV

_{av}, with respect to the average pressure h

_{av}, as a function of A

_{*}. Dots refer to the application performed (run #1 and run #2).

**Figure 12.**For run #1, and for different q

_{n}values, the relationship between (

**a**) the sector I and (

**b**) sector II lateral inside diameter, associated with D

_{m,min}, D

_{I,min}, and D

_{II,min}, respectively, versus h

_{*max}. Dot refers to the run #1 application performed.

**Figure 13.**For run #2, and for a dual-diameter lateral and for a one-diameter lateral, comparison between the PHD line obtained by the suggested procedure (solid line) with that obtained by the step-by-step procedure (

**a**) in dimensionless terms and (

**b**) in dimensional terms. The characteristics pressure heads are indicated.

Parameter | Run #1 | Run #2 | Parameter | Run #1 | Run #2 |
---|---|---|---|---|---|

i (mm/h) | 0.1 | 0.15 | N_{I} | 28 | 275 |

r_{0} (m) | 400 | 700 | N_{II} | 39 | 494 |

q_{n} (L/h) | 750 | 300 | W_{*I} | 0.2370 | 0.1983 |

A_{*} | 0.0149 | 0.0013 | W_{*II} | 0.7455 | 0.7717 |

w_{*0} | 0.00749 | 0.00065 | W_{*N} | 0.9824 | 0.9701 |

N = j_{max} | 67 | 769 | |||

w_{*N} | 0.10585 | 0.01692 |

Parameter | Run #1 | Run #2 | Parameter | Run #1 | Run #2 |
---|---|---|---|---|---|

i (mm/h) | 0.1 | 0.15 | L_{*,I} | 0.2284 | 0.2315 |

r_{0} (m) | 400 | 700 | L_{*,II} | 0.6974 | 0.7298 |

q_{n} (l/h) | 750 | 300 | L_{I} (m) | 91.4 | 162.1 |

A_{*} | 0.0149 | 0.0013 | L_{II} (m) | 279.0 | 510.9 |

A (m) | 7500 | 2000 | K_{I} | 3.435 × 10^{−5} | 3.252 × 10^{−7} |

w_{*0} | 0.00750 | 0.00065 | D_{I} (mm) | 82.79 | 152.10 |

N_{I} | 28 | 316 | K_{II} | 5.501 × 10^{−6} | 4.852 × 10^{−8} |

∑ s_{*}_{j} (j − 1)^{1.852} | 40.10 | 3675.30 | D_{II} (mm) | 120.58 | 224.79 |

N_{II} | 39 | 452 | k_{e} (L h^{−1} m^{−0.5}) | 200.08 | 60.50 |

∑ s_{*}_{j} (j − 1)^{1.852} | 1110.46 | 109264.40 | D_{m} (mm) | 111.26 | 207.28 |

δ_{I} | 0.02 | 0.02 | N | 67 | 769 |

δ_{II} | 0.08 | 0.08 | ∑ s_{*}_{j} (j − 1)^{1.852} | 1150.57 | 112,939.70 |

h_{*max} ≡ h_{*in} | 0.0412 | 0.0357 | δ | 0.1 | 0.1 |

h_{max} ≡ h_{in} (m) | 16.5 | 25.0 | L_{*} | 0.9258 | 0.9613 |

h_{*n} | 0.0351 | 0.0305 | L (m) | 370.3 | 672.9 |

h_{*min} | 0.0338 | 0.0293 | K | 6.52 × 10^{−6} | 5.76 × 10^{−8} |

h_{min} (m) | 13.5 | 20.5 | D (mm) | 116.45 | 216.99 |

h_{*n,I} | 0.0344 | 0.0299 | k_{e} (l h^{−1} m^{−0.5}) | 200.08 | 60.50 |

h_{*n,II} | 0.0382 | 0.0331 | RE_{D} = (D − D_{m})/D | 0.0446 | 0.0447 |

**Table 3.**For run #1, and by varying N

_{I}and δ

_{I}, relative errors (RE) for dual-diameter lateral at the changing diameter section, RE(h

_{*n})

_{dual}, and at the maximum pressure section, RE(h

_{*max})

_{dual}, and for one-diameter lateral at the maximum pressure section, RE(h

_{*max})

_{one}.

N_{I} | δ_{I} = 0.02 | δ_{I} = 0.04 | δ_{I} = 0.05 | δ_{I} = 0.06 | δ_{I} = 0.08 | δ_{I} = 0.02 | δ_{I} = 0.04 | δ_{I} = 0.05 | δ_{I} = 0.06 | δ_{I} = 0.08 |
---|---|---|---|---|---|---|---|---|---|---|

RE(h_{*n})_{dual} | RE(h_{*max})_{one} | |||||||||

5 | 0.13% | 0.49% | 0.76% | 1.08% | 1.87% | −0.07% | 0.41% | 0.72% | 1.06% | 1.89% |

10 | 0.13% | 0.49% | 0.75% | 1.07% | 1.86% | −0.03% | 0.46% | 0.77% | 1.11% | 1.91% |

15 | 0.13% | 0.49% | 0.75% | 1.07% | 1.85% | 0.01% | 0.52% | 0.83% | 1.16% | 1.94% |

20 | 0.13% | 0.49% | 0.75% | 1.07% | 1.85% | 0.05% | 0.58% | 0.88% | 1.22% | 1.97% |

25 | 0.13% | 0.49% | 0.75% | 1.07% | 1.86% | 0.10% | 0.64% | 0.94% | 1.27% | 2.01% |

28 | 0.13% | 0.49% | 0.75% | 1.07% | 1.86% | 0.13% | 0.67% | 0.98% | 1.30% | 2.03% |

30 | 0.13% | 0.49% | 0.75% | 1.07% | 1.86% | 0.15% | 0.69% | 1.00% | 1.33% | 2.04% |

35 | 0.13% | 0.49% | 0.75% | 1.07% | 1.86% | 0.20% | 0.75% | 1.05% | 1.38% | 2.08% |

40 | 0.13% | 0.49% | 0.75% | 1.07% | 1.86% | 0.25% | 0.81% | 1.11% | 1.43% | 2.11% |

45 | 0.13% | 0.49% | 0.76% | 1.07% | 1.87% | 0.31% | 0.86% | 1.16% | 1.48% | 2.14% |

50 | 0.13% | 0.49% | 0.76% | 1.08% | 1.87% | 0.36% | 0.92% | 1.22% | 1.53% | 2.17% |

55 | 0.13% | 0.49% | 0.76% | 1.08% | 1.87% | 0.42% | 0.98% | 1.27% | 1.57% | 2.20% |

60 | 0.13% | 0.49% | 0.76% | 1.08% | 1.88% | 0.47% | 1.04% | 1.33% | 1.62% | 2.24% |

65 | 0.13% | 0.50% | 0.77% | 1.09% | 1.89% | 0.54% | 1.10% | 1.39% | 1.68% | 2.28% |

N | RE(h_{*max})_{one} | |||||||||

67 | 0.29% | 0.95% | 1.27% | 1.60% | 2.26% |

**Table 4.**For run #1, and by varying N

_{I}(and N

_{II}) and δ

_{I}, lateral diameters corresponding to sector I and sector II and D

_{I}and D

_{II}, respectively, and the associated mean weight diameter, D

_{m}(Equation (28)).

N_{I} | N_{II} | δ_{I} = 0.02 | δ_{I} = 0.04 | δ_{I} = 0.05 | δ_{I} = 0.06 | δ_{I} = 0.08 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

D_{I} (mm) | D_{II} (mm) | D_{m} (mm) | D_{I} (mm) | D_{II} (mm) | D_{m} (mm) | D_{I} (mm) | D_{II} (mm) | D_{m} (mm) | D_{I} (mm) | D_{II} (mm) | D_{m} (mm) | D_{I} (mm) | D_{II} (mm) | D_{m} (mm) | ||

5 | 62 | 27.7 | 121.5 | 118.4 | 23.9 | 128.4 | 124.9 | 22.8 | 133.0 | 129.4 | 21.9 | 139.0 | 135.1 | 20.6 | 159.6 | 155.0 |

10 | 57 | 43.2 | 121.4 | 115.5 | 37.3 | 128.3 | 121.5 | 35.6 | 133.0 | 125.6 | 34.2 | 138.9 | 131.0 | 32.1 | 159.5 | 149.9 |

15 | 52 | 55.8 | 121.3 | 113.5 | 48.2 | 128.2 | 118.6 | 45.9 | 132.9 | 122.4 | 44.1 | 138.8 | 127.5 | 41.4 | 159.4 | 145.2 |

20 | 47 | 66.8 | 121.2 | 112.1 | 57.7 | 128.0 | 116.3 | 55.0 | 132.7 | 119.7 | 52.9 | 138.6 | 124.3 | 49.6 | 159.2 | 140.9 |

25 | 42 | 77.0 | 120.8 | 111.4 | 66.5 | 127.7 | 114.5 | 63.4 | 132.3 | 117.4 | 60.9 | 138.3 | 121.6 | 57.2 | 158.8 | 136.8 |

28 | 39 | 82.8 | 120.6 | 111.3 | 71.5 | 127.4 | 113.6 | 68.2 | 132.0 | 116.3 | 65.5 | 137.9 | 120.1 | 61.5 | 158.4 | 134.5 |

30 | 37 | 86.6 | 120.4 | 111.3 | 74.8 | 127.2 | 113.1 | 71.3 | 131.8 | 115.6 | 68.5 | 137.7 | 119.2 | 64.3 | 158.1 | 133.0 |

35 | 32 | 95.8 | 119.6 | 111.9 | 82.7 | 126.4 | 112.3 | 78.9 | 131.0 | 114.1 | 75.8 | 136.9 | 117.1 | 71.1 | 157.2 | 129.3 |

40 | 27 | 104.8 | 118.6 | 113.3 | 90.5 | 125.3 | 112.0 | 86.3 | 129.9 | 113.1 | 82.9 | 135.7 | 115.4 | 77.8 | 155.8 | 125.9 |

45 | 22 | 113.8 | 117.1 | 115.6 | 98.3 | 123.7 | 112.3 | 93.7 | 128.2 | 112.7 | 90.0 | 133.9 | 114.2 | 84.5 | 153.8 | 122.6 |

50 | 17 | 122.9 * | 114.9 | 119.0 | 106.1 | 121.4 | 113.4 | 101.1 | 125.8 | 112.9 | 97.2 | 131.4 | 113.5 | 91.2 | 150.9 | 119.6 |

55 | 12 | 132.3 * | 111.4 | 124.2 | 114.3 | 117.8 | 115.7 | 108.9 | 122.0 | 114.1 | 104.7 | 127.5 | 113.6 | 98.3 | 146.4 | 117.1 |

60 | 7 | 142.7 * | 105.6 | 132.1 | 123.2 * | 111.6 | 119.9 | 117.5 * | 115.6 | 116.9 | 112.9 | 120.8 | 115.2 | 106.0 | 138.7 | 115.3 |

65 | 2 | 155.6 * | 91.1 | 147.1 | 134.4 * | 96.3 | 129.3 | 128.1 * | 99.8 | 124.3 | 123.1 * | 104.3 | 120.6 | 115.5 | 119.7 | 116.1 |

_{I}> D

_{II}, bold values when D

_{m}> D = 116.45 mm (with D the one-diameter lateral, Table 2). Bold and underlined value indicates the lowest D

_{m}value (111.3 mm), occurring for N

_{I}= 28, and δ

_{I}= 0.02.

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**MDPI and ACS Style**

Baiamonte, G. Dual-Diameter Laterals in Center-Pivot Irrigation System. *Water* **2022**, *14*, 2292.
https://doi.org/10.3390/w14152292

**AMA Style**

Baiamonte G. Dual-Diameter Laterals in Center-Pivot Irrigation System. *Water*. 2022; 14(15):2292.
https://doi.org/10.3390/w14152292

**Chicago/Turabian Style**

Baiamonte, Giorgio. 2022. "Dual-Diameter Laterals in Center-Pivot Irrigation System" *Water* 14, no. 15: 2292.
https://doi.org/10.3390/w14152292