Increasing the Lateral Line Length of Drip Irrigation Systems

Aims: The hypothesis of this research is that it is possible to increase the drip irrigation lateral line length by using a larger spacing between emitters at the beginning of the lateral line and a smaller one after a certain distance, which would allow for a higher pressure variation along the lateral line under an acceptable value of distribution uniformity. Study Design: Non-pressure compensating drip hose is widely utilized for vegetables and orchards irrigation. Though there is a limitation, which is the lateral line length must be short to maintain uniformity due to head loss and slope, any procedure to increase the length is appropriate because it represents low initial cost of the irrigation system. Place and Duration of Study: This study was conducted at the College of Agricultural Sciences of Sao Paulo State University in Botucatu, SP, during the year 2011. Methodology: To evaluate this hypothesis, a nonlinear programming model (NLP) was developed. The input data were: diameter, roughness coefficient, pressure variation, emitter operational pressure, relationship between emitter discharge and pressure. The output data were: line length, discharge and length of the each section with different spacing between drippers, total discharge in the lateral line, multiple outlet adjustment coefficient, head losses, localized head loss, pressure variation, number of emitters, spacing between emitters, discharge in each emitter, and discharge per linear meter. Research Article British Journal of Environment & Climate Change, 3(3): 499-509, 2013 500 Results: The mathematical model developed was compared with the lateral line length obtained with the algebraic solution generated by the Darcy-Weisbach equation. The NLP model showed the best results since it generated a greater gain in the lateral line length, maintaining the uniformity and the flow variation under acceptable standards. It also had lower flow variation. Conclusion: NLP model showed the best results when compared with the conventional procedure, generating gain in the lateral line length, keeping the uniformity and flow variation under acceptable standards.


INTRODUCTION
In drip irrigation systems water is applied directly in the root system region, with high efficiency, but this system has the disadvantage of possible emitters clogging and its installation cost is high [1].Basically, the emitters can be compensating or nonpressure compensating.The compensating drippers provide constant flow rate under pressure variations along the lateral line, allowing longer lengths but, they are more expensive.Using non-pressure compensating emitters, the flow rate decreases as the pressure is reduced, resulting in shorter lateral lines in order to obtain the desired uniformity.Non-pressure compensating drip hose is widely used for vegetables and orchards irrigation.The limitation of this emitter is that the lateral line length must be short to maintain uniformity due to head loss and slope.
It is important to study procedures and criteria to obtain longer lateral lines when using nonpressure compensating emitters.It is possible to extend the lateral line length using two emitters spacing in different sections [2].In this case, the system design consists in the determination of the two emitters spacing utilized and the changing point between spacing.It is assumed that the spacing changing point would be at 40% of the total length, because this is approximately the average location [3].For practical purposes, the average pressure is located at 40% of the lateral line length and at this point 75% of total head loss (hf) has already been consumed [4].However, this arbitrary criterion does not necessarily ensure the best solution.
The use of a 30% flow variation (Δq) was proposed [5] and it was found that this value resulted in distribution uniformity over 80%.With the non-pressure compensating emitters, the design usually adopted a flow variation (Δq) of 10% and a corresponding pressure variation (∆H) of 20%, allowing uniformity distribution between 95 and 98% [5,4].To evaluate the irrigation uniformity two indicators can be used: distribution uniformity (DU) which is the ratio between the average of lower 25% of flow values and the average, expressed as a percentage [6,7] and the emission uniformity (EU), which considers the emitters characteristics and the hydraulic configuration of the drip irrigation subunit [8].
Longer lateral lines in drip irrigation systems using conventional drippers provide cost reduction, but it is necessary to obtain irrigation uniformity [2].Utilizing higher Δq levels can provide longer lateral lines.
The design should be optimized and it can be obtained with the use of mathematical optimization models based on operations research techniques, as it is the case of Nonlinear Programming (NLP).
Maximizing the lateral line length with two spacing and defining the spacing changing point are typically an optimization problem, but it can be characterized and solved by a nonlinear programming model.This study aimed to evaluate the possibility of increasing the lateral line length of an irrigation system using non-pressure compensating drip hose with different spacing between emitters but maintaining irrigation uniformity at appropriate levels.For this, a comparison was carried out between the NLP model and the conventional design procedure.

METHODOLOGIES
A mathematical model using Nonlinear Programming was developed for comparison with the conventional methodology to determine the lateral line maximum length, which is based on the Darcy-Weisbach equation.

Objective Function of Model
The developed model objective function is the total lateral length maximization using two spacing in different sections as described in the objective function (eq.1).MAX where: L = lateral line total length (m); L 1 = first section length (m); L 2 = second section length (m).

Pressure head variation
For each section the model provides a pressure variation (eq. 2 and 3) and the sum (eq.4) of them must be equal to the Δh informed on the input data.
hf H H   (3) where: hf 1 = head loss in the section 1; hf 2 = head loss in the section 2; ΔH 1 = pressure variation in the section 1; ΔH 2 = pressure variation in the section 2; ΔH = pressure variation informed on the input data.
Head loss in each section is also estimated using Hazen-Williams equation and multiple outlet adjustment coefficients, as showed by equations 5 and 6.
where: Christiansen multiple outlet adjustment coefficients (F) is given by: where: m = discharge exponent in the friction loss equation; N = total number of outlets.
From equation 8 it was possible to calculate the Christiansen adjustment factor to the total length of lateral line (F t ) and to the second section (F 2 ) using the total number of emitters in the lateral line and the emitters number in the second section, respectively.

Discharge
The emitter flow can be characterized empirically as a function of the operational pressure, according to equation 9 [10 and 11].(9) where: q = emitter flow (L h -1 ); K = proportionality factor; H = emitter pressure, water column in meters (m.c.a); X = exponent of flow which characterizes the flow regime.
x q K H  The emitter discharge was calculated at three different points using the equation 9 and the respective pressure in each point.The first one is located at lateral line inlet (eq.10), the second is at the transition point (eq.11) between the spacing and the third is at the end of line (eq.12).The average discharge in each section is given by equations 13 e 14.
x e q K H  where: q e = discharge at the lateral line inlet q int = discharge at the transition point between sections with different the spacings q f = discharge at the end of the lateral line q L1 = average discharge in the section 1 q L2 = average discharge in the section 2

Discharge per linear meter
The discharge per meter linear is given by equations 15 and 16.
where: q unit1 = discharge per meter in the section 1 q unit2 = discharge per meter in the section 2 N 1 = number of emitters in the section 1 N 2 = number of emitters in the section 2 Se 1 = emitter spacing in section 1, m Se 2 = emitter spacing in section 2, m

Input Data
As an example a commercial non-pressure compensating drip hose was adopted, the characteristics are shown in Table 1.The data used in the calculations are shown in Table 2.As the model in GAMS ® uses nonlinear equations, the definition of the allowable variation of some variables is necessary in order to avoid division by zero during the calculations (Table 3).The model provides several output data: lateral line total length, flow and length of each lateral line section with different spacing between emitters, total number of emitters, emitter spacing, flow rate per linear meter, head loss, pressure variation, multiple outlet adjustment coefficient, discharge in each emitter, and emitter average discharge.
The NLP model was compared with the conventional procedure for lateral line length estimation from the head loss equation of head loss.In the case of non-compensating emitters used in orchards, the lateral lines works on level.Thus, any pressure variation is due to the total head loss (in the pipeline and located in the emitters).The lateral line diameter adopted in this study is 16 mm, the most used commercially.Thus, the lateral line length becomes the only variable to be defined.
For the conventional procedure, the lateral line length was calculated by equation 26, which is basically the combination of Darcy-Weisbach equation [3] and Blasius equation (eq.24).To compare the NLP model and the conventional procedure two pressure variations were used: 20 and 40%.

RESULTS AND DISCUSSION
The NLP model allowed the lateral line design to use two sections with different spacing.The conventional method calculated the lateral line to a single spacing between drippers (0.4 m).
The emitter average flow was located at 37.85 and 38.64% of the lateral line length, with head loss until this point of 73.61 and 75.20% of the total lateral line head loss for the pressure variation of 20 and 40%, respectively (Table 4).These values are consistent with those found by [4].The ∆H and the found Δq values were the same as those obtained by [12]; it was suggested that Δq can be related to ∆H, to obtain ∆H of 20 and 40%, Δq must be of 10 and 22%, respectively (Table 5).The use of higher pressure variation (40%) when compared to lower pressure variation (20%) in the lateral line, showed a lower distribution uniformity, a lower emission uniformity, a higher emitters flow variation, and a greater discharge variation per linear meter (Table 5).However, for the NLP model the length gain was of 42.39 m, compared to the ∆H 20%, and keeping the Δq and the DU values within the limits offered by [5].
Comparing the NLP model to the conventional procedure, it was found that the DU had difference of 2% (for 40% ∆H) and 0.52% (for 20% ∆H).The difference in the EU was 1.97% (for 40% ∆H) and 0.5% (for 20% ∆H).The variation in flow rate per meter showed a difference of 0.31% (20% ∆H) and 1.68% (40% ∆H).The lowest flow variation per linear meter was found using GAMS ® and ∆H of 20%.
Comparing the uniformity indexes, the EU showed lower values compared to DU, however, the values were higher than 88% (Table 5).The reason the EU is more restrictive is because it considers the lowest discharge value while the DU considers the average 25% lower flow values in the lateral line or subunit.
The values obtained for the emitter number, lateral line length, and total evaluated flow for the ∆H are shown in Table 6.The conventional procedure showed a lower lateral line length than the NLP model.Emitter spacing obtained in the NLP model was between 0.39 and 0.47 m (Table 6).In all situations the obtained spacing in the initial section was higher than in the one in the final section, because this is necessary to maintain the flow rate per meter.As the pressure decreases along the lateral line, the emitter discharge also decreases and the model select a smaller spacing for the second section aiming to keep the flow rate per meter uniform in the lateral line.
In the NLP model, the first section were at 44.51 and 46.11% of L, and at these points 80.55 and 82.61% of the total head loss were consumed, considering the ∆H equal to 20 and 40% respectively.
The NLP model was developed to design the lateral line while optimizing the length using different spacing and assuring a uniform discharge per linear meter.This can be seen in Figs. 1 and 2 for the ∆H equal to 20 and 40%, respectively.In the spacing changing point, the discharge per linear meter modified from 2.87 to 3.03 L. h -1 m -1 and from 2.7 to 3.07 L. h -1 m -1 for the pressure variation of 20 and 40%, respectively.The highest discharge per linear meter occurred in the beginning of the lateral line, 3.14 (to 20% ∆H) and 3.29 L. h -1 m -1 (to 40% ∆H) and the lowest values occurred at the spacing changing point.At the end of lateral line, the discharges per linear meter were of 2.97 and 2.92 L. h -1 m -1 for a ∆H of 20 and 40%, respectively.The longest lateral line was obtained with the NLP model.The use of higher pressure variation resulted in longer lateral lines and maintained the uniformity under acceptable standards.The adoption of two spacing in the same lateral line showed advantages compare to a single one.
In NLP model even with the lateral line showing higher Δq and Δql, the DU was under acceptable standards (over 93%).The system low cost implementation was the result of the lateral line increase.

CONCLUSIONS
The initial hypothesis that the adoption of two emitters spacing would increase the lateral line length was confirmed.For 20 and 40% pressure variations it was obtained a length gain of 16.2 and 15.7%, respectively, in the NLP model compared to the conventional method that uses a single spacing.
The spacing changing ideal location was approximately 45% of the total lateral line length.Using high flow variations under acceptable uniformity standards allowed the best results.NLP model showed the best results when compared to the conventional procedure by increasing the lateral line length and keeping the uniformity plus flow variation under acceptable standards.
multiple outlet adjustment coefficient for the total lateral line; F 2 = multiple outlet adjustment coefficient for the second section; C = Friction coefficient (Hazen-Williams equation); D = internal diameter of pipe, m

Fig. 1 .Fig. 2 .
Fig. 1.Discharge per linear meter as a function of the lateral line length, for the 20% pressure variation