MODELLING HEAD LOSS ALONG EMITTING PIPES USING DIMENSIONAL ANALYSIS

Local head losses must be considered in estimating properly the maximum length of drip irrigation laterals. The aim of this work was to develop a model based on dimensional analysis for calculating head loss along laterals accounting for inline drippers. Several measurements were performed with 12 models of emitters to obtain the experimental data required for developing and assessing the model. Based on the Camargo & Sentelhas coefficient, the model presented an excellent result in terms of precision and accuracy on estimating head loss. The deviation between estimated and observed values of head loss increased according to the head loss and the maximum deviation reached 0.17 m. The maximum relative error was 33.75% and only 15% of the data set presented relative errors higher than 20%. Neglecting local head losses incurred a higher than estimated maximum lateral length of 19.48% for pressure-compensating drippers and 16.48% for non pressure-compensating drippers.


INTRODUCTION
The main objective of the trickle irrigation design is the uniform distribution of water delivered through the emitters (ZHU et al., 2010).Although drip irrigation systems have several advantages over other irrigation systems, the ideal water distribution along the lateral cannot be achieved due to variations in emitter discharge.These variations are influenced by operating pressure and water temperature (DOGAN & KIRNAK, 2010;BORSSOI et al., 2012); emitter manufacturing process; emitter clogging (TARCHITZKY et al., 2013;ZHOU et al., 2013) and pressure variations caused by slope (ZHU et al., 2010) and friction losses (RETTORE NETO et al., 2009;GOMES et al., 2010;VEKARIYA et al., 2011).
Total energy loss along laterals can be divided into two parts: major and minor losses.Major losses are associated with energy loss along the pipe due to frictional effects, which depend on fluid viscosity, wall roughness, internal diameter of the pipe, pipe length, and flow velocity.Although many equations are available for determining friction losses along laterals, the Darcy-Weisbach equation seems to be the most accepted for small diameter polyethylene pipes (YILDIRIM, 2009).
The introduction of the Blasius friction factor (equation 1) into the Darcy-Weisbach equation provides an accurate estimate of the frictional losses produced by turbulent flow inside uniform pipes with low wall roughness and when the Reynolds number (R) falls within the range 3,000-10 5 (CARRIÓN et al., 2013). (1) For laminar flow regime ( ), the friction factor is given by [eq.( 2)]. (2) Low-density polyethylene is usually used in the manufacture of micro irrigation laterals.The internal diameter of polyethylene pipes is affected by operating pressure (JUANA et al., 2002), which can change the hydraulic conditions of an irrigation system.RETTORE NETO (2011) presented a model for determining continuous head loss that considers the modulus of elasticity of material, wall thickness of pipe, pressure inside the pipe, and consequently internal diameter variation due to pressure effects (equation 3).According to the author, the equation was an improvement on the Darcy-Weisbach formula and presented excellent results in estimating continuous head loss of polyethylene pipes.Dimensional analysis is a simple, clear and intuitive method for determining the functional dependence of physical quantities that influence a process (VEKARIYA et al., 2011).The aim of this work was to develop a model based on dimensional analysis for calculating head loss along laterals accounting for in-line drippers.

Model development
Dimensional analysis is a useful tool for developing predictive equations, which reduces the physical quantities to dimensionless groups called Π terms (VEKARIYA et al., 2011).The Π theorem enables the organisation of experimental runs and the analysis of measurements by dimensionless groups.Using dimensionless groups permits a reduction in the number of runs and the testing of the global effects of the variables that occur in each group rather than the effect o f each singular variable (FERRO, 2010).
In this study the term was used to express the total head loss between two consecutive emitters.Using this approach, data analysis is easier to perform and the model becomes easier to operate.The proposed model considers that is the sum of continuous and local head loss (equation 6).Continuous head loss between two consecutive emitters was expressed by the Darcy-Weisbach formula, in which the pipe length ( ) was replaced with the distance between two consecutive emitters ( ).Local head loss was calculated based on [eq.( 4)], considering the mean cross-sectional area of flow where an emitter is located ( ).
( 6) Equation ( 7) resulted from [eq. ( 6)] and it presents a theoretical model for estimating total head loss between two consecutive emitters, which considers the friction coefficient given by the Blasius equation (equation 1)] and the change in internal diameter due to elasticity and pressure inside the pipe (equation 3).( 7) The following relationship can therefore be defined: (8) in which, is a functional symbol.
Table 1 shows eleven variables, where is the dependent variable and the remainder are independent variables ( ).The physical process involves three dimensions (M, L, and T) and eight Π terms.The basic selected variables were , and because they are dimensionally independent.The Π terms shown in Table 2 were obtained by applying the Vaschy-Buckingham theorem.(9) Rewriting [eq.( 9)] as a dimensional equation: (10) The exponents of each dimension in [eq.( 10)] are: , , , and consequently the first Π term (equation 11) was obtained by substituting these values into [eq.( 9)]. (11) The second Π term was determined by combining with the basic variables and the other dimensionless terms were also obtained by a similar process.
Equation ( 12) was determined considering .The terms and were grouped into a single term ( ).

(12)
Since Π terms were defined, 816 measurements were performed with 12 models of cylindrical in-line drippers to obtain the experimental data required for developing and assessing the model.Seventy percent of the collected data was applied to model developing and 30% was applied to model assessment and validation.

Model assessment and validation
The

Testing conditions
The research was carried out at the Irrigation Laboratory of the Department of Biosystems Engineering, University of São Paulo, Piracicaba, Brazil.The tests were performed in a closed circuit system (FIGURE 1) consisting of: a water tank; a centrifugal pump; three valves for controlling pressure and flow rate; an electromagnetic flowmeter (range 0 to 1500 Lh -1 , accuracy ±0.5%); a digital manometer (range 0 to 1470 kPa, accuracy ±0.25%) and a mercury differential manometer able to measure a maximum value of 266.65 kPa.
The differential manometer was used to determine total head loss along the lateral.Emitting pipes/emitters were totally plugged before carrying out the experiments and four repetitions were performed on each model.The pressure at the lateral inlet during all the tests was 196 kPa (±0.98 kPa) and the flow rate ranged from 160 to 1420 Lh -1 in increments of 80 Lh -1 .FIGURE 1. Diagram of the facility used to perform the tests.
For each emitting pipe model, the internal diameter ( ) and wall thickness ( ) of eight samples were measured using a horizontal benchtop optical comparator Starret HB400.The crosssectional area of the pipe ( ) and the flow velocity at the pipe section ( ) were determined based on measured values of .
A digital thermometer (resolution 0.01ºC) was used to measure water temperature during the tests in order to estimate the kinematic viscosity and density of the water.
Following the procedure described by RETTORE NETO (2011), a tensile test machine (FIGURE 2) was used to perform uniaxial tensile tests in order to determine the modulus of elasticity of the pipes.Three repetitions were performed for each model of emitting pipe.The tested samples were 25-cm lengths and the machine was configured to apply tension at a speed of 10 mm/minute.indirectly based on the volume of distilled water required to fill up a cylinder of pipe in which the emitter was assembled.Eight samples (cylinders) were extracted from each model of emitting pipe.
The length of each sample was exactly the length occupied by an emitter inside the pipe.The samples were sealed at one side so that they could be filled with water.Empty and water-filled samples were weighed using a digital balance (resolution 0.01 g).The value of each sample was obtained by dividing the water volume inside each cylinder by its length.A digital caliper (resolution 0.01 mm) was used to measure the cylinder length.The standard deviation values of shown in Table 5 represent manufacturing differences between samples of the same model of emitter.The highest coefficient of variation among the tested models was 2.02% (emitter 6) and such a low value seems to be sufficient to support the feasibility of using this methodology.Moreover, this methodology allows easy and rapid determination of and requires no sophisticated equipment.

Validation and assessment of the model
As mentioned previously, 70% of the gathered data was applied to fitting the model coefficients by multiple linear regression.The statistical results related to this procedure are shown in Table 6.The calculated value of F was higher than the critical value and therefore the hypothesis of regression existence was accepted.The adjusted coefficients were evaluated by t-test and all of them differed from zero considering a significance level of 1% (Table 7).Since the Reynolds number and Froude number presented collinearity, the Reynolds number was removed from the analysis because this term presented low statistical correlation with the dependent variable.With a Reynolds number higher than 10,000, BAGARELLO et al. (1997) observed that local head losses are not influenced by the Reynolds number and only depend on emitter geometry.JUANA et al. (2002) also mentioned that, in practice, the effects of viscous forces on the coefficient of local head loss are negligible beyond a limiting value of Reynolds number.The term was also removed because it is only meaningful within the significance level of 45.85%.The testing pressure during all experiments was 196 kPa (±0.98 kPa).Perhaps if other testing pressures were evaluated during experiments, this term could have presented a higher significance level.Nevertheless, both removed terms presented low values of partial R 2 and have practically no effect on the model (Table 7).
According to Figure 3A, the spread of data increased proportionally to the head loss.DEMIR et al. ( 2007) observed the same behavior studying drip irrigation laterals equipped with in-line and on-line emitters.The maximum error (ϵ) between observed and estimated values of head loss was 0.17 m (Figure 3A).Considering 816 measurements, Figure 3B shows that only 15% of the data set presented a relative error higher than 20%, whereas the maximum value was 33.73%.Since only a single equation was used to estimate head losses in several models of emitting pipes, the relative errors varied between emitting pipes due to their inherent differences.PROVENZANO & PUMO (2004) also reported significant differences among investigated dripline models.Relative errors increased according to head loss for all emitting pipes except pipe 9. Differences in relative errors between repetitions were observed and might be caused by experimental errors.Emitting pipes 5, 9 and 12 presented the highest values of relative error (Figure 3C).

Applying the developed model on designing late rals
An example of how to apply the developed model on des igning the maximum length of laterals is presented and it assumes the following criteria: a) the maximum flow velocity along the lateral is 1.5 m s -1 ; b) the allowable variation in emitters' flow rate along the lateral is 5% and c) pressure at the lateral inlet is 120 and 200 kPa for non pressure-compensating (NPC) and pressurecompensating (PC) emitters, respectively.
The maximum length of 12 models of emitting pipes was simulated by two methods.The first one was a step-by-step method based on the Darcy-Weisbach equation that does not take into account local head loss effects (a), whereas the second refers to the model developed in this study (b).The maximum length of the laterals and the difference between the values obtained from both methods are summarised in Table 8.
of the pipe (m); = mean water velocity at uniform pipe sections (m s -1 ); = gravitational acceleration (m s -2 ); = pressure inside the pipe (Pa); = wall thickness of pipe (m), = modulus of elasticity of material (Pa).The insertion of emitters along a lateral line modifies the flow streamlines inducing additional pressure losses, which must be taken into consideration in order to accurately evaluate total energy losses along laterals (YILDIRIM, 2009).The Borda-Carnot equation (equation 4) enables quantification of head losses caused by a sudden expansion or contraction along a pipeline: (4) in which, = local head loss (m); = water velocity at the section of flow contraction (m s -1 ); = cross-sectional area of flow where an emitter is located (m 2 ), = cross-sectional area of the pipe (m 2 ).DEMIR et al. (2007) presented a model based on dimensional analysis to calculate the friction head loss in drip irrigation laterals equipped with in-line emitters (equation 5), which is valid for the following conditions Relationship between head loss and pipe diameter Inverse of Reynolds number Relationship between distance of emitters and pipe diameter Inverse of Froude number Relationship between cross-sectional area of the pipe and the square of the pipe diameter Relationship between mean cross-sectional area of flow where an emitter is located and the square of the pipe diameter Relationship between wall thickness and pipe diameter Relationship between modulus of elasticity and pressure inside the pipe The first dimensionless term was obtained by combining with the three basic variables (equation 9).
Camargo & Sentelhas coefficient (CAMARGO & SENTELHAS, 1997) was used to indicate the model performance at estimating head loss.This coefficient combines the accuracy and precision of a model in a single value (equation 15) that can be interpreted based on Table 3.The Camargo & Sentelhas coefficient is the product of Pearson's correlation coefficient (equation 13) and Willmott's index of agreement (equation 14).s index of agreement, = the Camargo & Sentelhas coefficient.TABLE 3. Criteria for interpreting model performance based on the Camargo & Sentelhas coefficient (CAMARGO & SENTELHAS, 1997).

FIGURE 2 .
FIGURE 2. Tensile test machine used to determine modulus of elasticity of polyethylene pipes.

=
Number of emitters along the lateral; and = coefficients of the pressure-flow rate function; * = non pressure-compensating drippers.
FIGURE 3. A) Estimated versus observed values of head loss; B) Cumulative frequency versus relative error on estimating head loss ( ); C) Relative Figure 4 groups charts presenting observed hf Se , estimated hf Se by the proposed model and estimated hf Se using the model presented by DEMIR et al. (2007).Comparing both models, the proposed model presented estimated hf Se closer to the observed values for all emitting pipes, except pipe 5.

TABLE 4 .
Table4and 5 describe the geometric properties of the pipes and emitters evaluated during the experiments.Properties of the pipes evaluated during the experiments.

TABLE 5 .
Properties of the emitters evaluated during the experiments.

TABLE 6 .
Statistical analysis.= degrees of freedom; SQ = Sum of squares; QM = M ean of squares; F = Calculated value of F-Test statistic; F tab = Critical value of the F distribution at α = 0.01; F sig = Significance level of acceptance of the null hypothesis. DF

TABLE 7 -
Regression coefficients.Value of P = Significance level of acceptance of the hypothesis that the regression exponent is equal to zero.