A MODEL FOR HYDRAULIC DESIGN OF IRRIGATION LATERAL LINES

There are several models for hydraulic designs and optimization of lateral lines depending on the existing pressure head profile and flow which allows designing longer lateral lines, therefore decreasing the cost of the system implementation. A model has been developed to calculate the pressure head and required flow rate at the inlet of lateral line using the back step method. A set of equations was implemented in an algorithm in the R language. For the calculations, the following variables must be provided: pressure head at the end of the lateral line (Hend), coefficients K and x of the characteristic equation (flow-pressure) of the emitter, pipe diameter (D), emitter spacing (Se) and number of emitters (Ne). For the evaluation of the model, the pressure head at the end of the lateral line, the pipe diameter and the number of emitters were varied within the established limits. Relationships between these variables were established by regression analysis using the least-squares method. The model shown in the study was suitable for the calculation of the pressure head and flow rate profile along the lateral line. The power, plateau, exponential and linear equations were adjusted to describe these relationships. These equations can help in the design of irrigation systems by simplifying the procedures in order to meet the design criteria. Also, the proposed equations allow evaluation of the systems still in the design phase.


INTRODUCTION
Irrigated agriculture has been increasing in Brazil. However, with the high availability of water resources in the country, there is no concern with the rational use of water, leading to low efficiency of irrigation systems. Such fact can be attributed to inadequate management and erroneous hydraulic design (AYARS, 2007). According to IICA (2008), in Brazil, the design of the irrigation hydraulic system is constantly neglected, causing 36% of water to be lost in the system. These losses are attributed to poor distribution in hydraulic structures.
The good development of a crop is associated with adequate water availability through rainfall or irrigation. Irrigation water is distributed to plants through the lateral lines spaced along the derivation lines (Bernardo et al., 2011). The laterals must be designed to meet the water distribution uniformity criteria , defined as the ability of an irrigation system to deliver the same amount of water over an irrigated area. (Sokol et al., 2019).
The main tasks of the hydraulic design of a drip irrigation system are to determine the geometric characteristics (diameter and length) of the lateral lines; the pressure at the beginning of the lines; head losses along the lateral lines and the flow of the emitters . The performance of irrigation systems depends greatly on proper hydraulic design (CLARK et al., 2007). According to , determining the correct pressure is of great importance so not to compromise the system, as high pressures can result in failures in water application or result in injury to irrigation pipes. Thus, the use of software is important for the hydraulic design of an irrigation system, whose benefit is to reduce the time to perform the calculations and quickly analyze different situations (CASTIBLANCO, 2013).
To improve the design of lateral irrigation lines, hydraulic models can be used to obtain the flow and pressure along the lateral line based on iterative calculations. Many authors have described mathematical models to describe the pressure and flow along a lateral line (Kang & Nishiyama, 1996;Vallesquino & Luque-Escamilla, 2001;Mizyed, 2002;Zella et al., 2006;Yildirim, 2009Yildirim, , 2010Sadeghi et al., 2012Sadeghi et al., , 2015Perea et al., 2013;Pandey, 2016) The aim of this work was to develop a model to calculate the pressure and the required flow at the inlet of the lateral line and the pressure and flow profile along the lateral line using the back step method. Also, to fit the relationships between the design variables that are necessary to establish the laterals with greater efficiency.

MATERIAL AND METHODS
The pressure and flow profile along the lateral line was calculated through the back step method. In this method, the pressure head is set at the end of the lateral line and the pressure head and the flow rate profile is calculated iteratively up to the beginning of the lateral line (Jain et al., 2002). The backstep methods are a combination of mass conservation and energy conservation equations.
For the last emitter in lateral line (n th emitter) the pressure head was initially set (Equation 1). The flow rate of the last emitter was calculated through the characteristic equation of the emitter using the coefficients K and x (Equation 2). The flow rate of the last element of the lateral was considered equal to zero (no flow) (Equation 3). (1) Where, H n = pressure head in the last emitter, m; H e = pressure head in the end of lateral, m; q n = flow rate in the last emitter, m³.s -1 ; K = emission coefficient; x = emission exponent; and Q n = flow rate in the last lateral section, m³ s -1 .
The iterative calculation begins in the penultimate emitter (i-1) and continues up to the first emitter of the lateral. The flow rate of the i-1 element of the lateral and the pressure head and flow rate of the i-1 emitter were calculated using equations 4-6.
(4) (5) Where, Q i-1 = flow rate in the lateral section i-1, m³ s -1 ; Q i = flow rate in the lateral section i, m³ s -1 ; q i = flow rate in the emitter i, m³.s -1 ; H i-1 = pressure head in the emitter i-1, m; H i = pressure head in the emitter i, m; hf (Q i-1 ) = head loss in the lateral section i-1, m; and q i-1 = flow rate in the emitter i-1, m³.s -1 .
The inlet flow rate and inlet pressure head of the lateral line was calculated through Equations 7 and 8.
Where, Q in = flow rate at inlet of the lateral, m³ s -1 ; Q 1 = flow rate in the first lateral section, m³.s -1 ; q 1 = flow rate in the first emitter, m³.s -1 ; H in = pressure head at inlet of lateral, m; H 1 = pressure head in the first emitter, m; and hf (Q in ) = head loss at the inlet section of the lateral, m.
The head loss caused by friction was calculated through the Darcy-Weisbach (equation 9).  The described equations were implemented in an algorithm in R language (Ihaka & Gentleman, 1996). In addition, the following variables must be provided: pressure head at the end of the lateral line (H end ), coefficients K and x of the characteristic equation (flow-pressure) of the emitter, pipe diameter (D), emitter spacing (Se) and number of emitters (Ne).
The model shown in here was calculated with the following variables: H end = 17 m; K = 1.05e-6; x = 0.5; D = 0.016 m; Se = 1 m; and Ne = 100.
For the evaluation of the model, the pressure head at the end of the lateral, the pipe diameter and the number of emitters were varied at the intervals shown below.
Pressure at the end of the lateral line ranging from 1 to 100 m, with an increment of 0.1.
Lateral line diameter ranging from 0.01 to 0.1 m, with an increment of 0.001.
Number of emitters ranging from 50 to 300 units, with an increment of 1.
These ranges allowed to calculate the pressure head at the inlet of the lateral line, the flow at the inlet of the lateral line and the coefficient of variation of the flow emitters. Relationships between these variables were established by regression analysis using the least-squares method.

RESULTS AND DISCUSSION
Regression was used to determine the relationship between the variables. The relationship between the pressure head at the end and at the inlet of the lateral line was a linear-type equation whose coefficient of determination was quite high (R 2 ≈ 1) (Figure 1). In the studied situation, the pressure at the start of the lateral line was 2.12 times greater than the pressure at the end of the lateral. The relationship between flow at the inlet of lateral line and pressure at the inlet and at the end of the lateral line were equations of power type whose coefficient of determination were quite high (R 2 ≈ 1) (Figures 2 and 3).  Also, there was a good relation between the coefficient of variation of discharge the emitters and the inlet pressure head and inlet flow rate of the lateral (Figures 4 and 5). The best fit was obtained with the power equation type whose determination coefficients were high (R 2 > 0.93).  The coefficient of variation (CV) is used to classify the emission uniformity. Coefficients of variation greater than 90% are classified as excellent, between 80% and 90% are very good, 70% and 80% are regular, 60% to 70% are very bad and less than 60% are unacceptable. In general, low uniformity of irrigation means excess water at certain points and deficit in other points of the field (Bralts et al., 1987). The proposed equations allow the evaluation of the systems still in the design phase.
For the diameter variation, a good fit was found with the plateau equation type. For this equation type, from a determined diameter value, there is no change in the response. In the example studied, the inlet pressure head of the lateral to maintain a given flow rate remains constant for diameters above 0.0124 m ( Figure 6). The inlet flow rate of the lateral for a given pressure head at the end of the line remains constant for diameters above 0.0151 m (Figure 7). The coefficient of variation of discharge remained unchanged for diameters above 0.018 m (Figure 8).   In relation to the change in the number of emitters, which is directly related to the length of the lateral line, exponential equations were adjusted for the relationship between the inlet pressure head and the number of emitters ( Figure 9) and for the relationship between inlet flow rate and the number of transmitters (Figure 10). For the relationship between the coefficient of variation of discharge and number of emitters, the best fit was obtained by an equation of the linear type ( Figure 11).

CONCLUSIONS
• The back step method was suitable for the calculation of the pressure head and flow rate at the beginning and along a lateral line.
• Several regression equations were presented relating the most important variables in the irrigation design. The power, plateau, exponential and linear equation were considered adequate to describe these relationships.
• These equations can help in the design of irrigation systems, simplifying the procedures in order to meet the design criteria. Also, the proposed equations allow evaluation of the systems still in the design phase.