Method for determining friction head loss along elastic pipes (PDF ...

Irrig Sci DOI 10.1007/s00271-014-0431-7

Original Paper

Method for determining friction head loss along elastic pipes O. Rettore Neto · T. A. Botrel · J. A. Frizzone · A. P. Camargo 

Received: 27 November 2012 / Accepted: 10 February 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract  Pipes made of plastic materials are generally used in pipelines and the laterals of irrigation systems. Plastic materials such as polyethylene allow significant changes in pipe cross section due to operating pressure, but traditional equations used for determining head loss do not account for this effect. The purpose of this research was to develop an equation for determining friction head loss along elastic pipes. The equation developed is based on the Darcy–Weisbach equation and focuses on pipe crosssectional variations caused by pressure effects, hence the name pressure-dependent head loss equation (PDHLE). In addition to the parameters required by the Darcy–Weisbach equation, the PDHLE also considers the modulus of elasticity of the pipe material, the pipe wall thickness, and the internal diameter variation due to operating pressure. The PDHLE resulted in high accuracy in determining the friction head loss of elastic pipes.

Introduction In micro-irrigation systems, water is distributed through emission devices installed on polyethylene pipes, i.e., laterals Communicated by G. Merkley. O. Rettore Neto (*)  Departamento de Engenharia Rural, Universidade Federal de Pelotas, Campus Universitario, CP 354, Pelotas, Rio Grande do Sul CEP 96010‑900, Brazil e-mail: [email protected] T. A. Botrel · J. A. Frizzone · A. P. Camargo  Departamento de Engenharia de Biossistemas (LEB), Escola Superior de Agricultura Luiz de Queiroz (ESALQ), Universidade de São Paulo, Av. Padua Dias, 11, Piracicaba, SP CEP 13418‑900, Brazil

(Yildirim 2010). Irrigation systems are usually designed to maximize water distribution uniformity, which is mainly affected by variations in emitter discharge along the lateral. Changes in emitter discharge are caused by operating pressure, water temperature, emitter manufacturing variations, emitter clogging, elevation change, and friction losses (Demir et al. 2007; Vekariya et al. 2011). The design of maximum lateral length is intended to ensure water distribution uniformity and is generally determined based on the criterion of emitter flow varying from 10 to 20 % along the lateral (Yildirim 2009). The assessment of head loss along pipelines and laterals is an important factor that affects both the overall cost and hydraulic balance of the network (Kamand 1988). Total head loss may be divided into two parts: major and minor losses. Major losses are associated with energy loss along the pipe due to frictional effects, which depends on the fluid viscosity, wall roughness and internal diameter of the pipe, lateral length, and flow velocity. Minor losses represent additional energy dissipation caused by secondary flows that are induced by entries, exits, fittings, and valves. Each emitter in a lateral determines local pressure loss or minor loss, which is often expressed as a fraction of the kinetic head (Yildirim 2010). Many irrigation system designers prefer to use empirical equations such as Hazen‐Williams, Manning, and Scobey to determine head loss, rather than the theoretical equation of Darcy–Weisbach. Although empirical equations are simpler to use, they consider a single roughness factor independently of pipe diameters and flow velocities. Such a critical limitation may result in values of head loss significantly different from those obtained by the Darcy–Weisbach equation (Kamand 1988; Bombardelli and García 2003; Rettore Neto et al. 2009). Low-density polyethylene is usually used in the manufacture of micro-irrigation laterals due to low flow rates along

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these lines and because polyethylene is cheaper than other available pipe materials. In addition, polyethylene material has a number of advantages, such as high performance-tocost, lightweight, thermal stability, chemical resistance, easy installation, and durability (Weon 2010). On the other hand, operating pressure affects the pipe cross section due to the elasticity of polyethylene (Juana et al. 2002), which is not taken into account by the equations generally used for determining head loss, this may change the hydraulic conditions of irrigation systems (Vilela et al. 2003). The total head loss, pressure head, and power required by the pumping system can be overestimated by traditional equations due to the increase in the cross section when pipes are pressurized. An oversized pumping system consumes more energy, and the extra power it requires can increase the inlet pressure of laterals, which affect emitter discharge and irrigation depths. Andrade (1990) analyzed the hydraulic features of perforated polyethylene pipes with a wall thickness of 200 μm. He observed a 10.67 % rise in internal diameter due to a 90 % increase in the pressure head. The pressure range specified by the pipe’s manufacturer was not exceeded during the tests. Assuming a constant flow rate, the head losses measured during the tests were lower than the values calculated by the Darcy–Weisbach equation and the deviations from theoretical values ranged from 15 to 60.24 %. Based on the reported effects, the hydraulic and electric conditions of a micro-irrigation system could diverge from the values estimated during the design stage. Frizzone et al. (1998) evaluated polyethylene emitting pipes of 225-μm wall thickness and assessed internal diameter variation due to pressure effects. They carried out an analysis of variance and verified the meaningful variation in internal diameter under various testing pressures. Vilela et al. (2003) worked on polyethylene pipes with diameter of 12 mm (A) and 20 mm (B), wall thickness of 1,050 and 1,325 μm, respectively. The internal diameter of the pipes presented a rise of 3.9 % (A) and 7.3 % (B) changing the testing pressure from 50 kPa to 400 kPa. Besides, the wall thickness decreased in 2.0 % (A) and 3.3 % (B) varying the testing pressure from 50 to 300 kPa. They also reported that internal diameter variations due to operating pressure are liable to result in changes of around 20 % in head loss. Thompson et al. (2011) reported results of friction loss tests for lay-flat laterals with wall thickness of 125, 200, 250, and 500 μm. The laterals have inflated even at low pressures; thus for each wall thickness, the authors defined effective diameters according to pressure head thresholds. They observed that samples of 500-μm wall thickness did not follow the same pattern exhibited in the thinner walled laterals. In addition, the effects of temperature on the pipe cross section were the major difficulty in accurately estimating friction loss in the 500-μm samples.

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Irrig Sci

The main purpose of this research was to develop an equation for determining friction head loss along elastic pipes. The equation was based on the Darcy–Weisbach equation and takes account of pipe cross-sectional variations caused by pressure effects, or pressure-dependent head loss equation (PDHLE). The equation also considers the modulus of elasticity of the pipe material, the pipe wall thickness, the pressure inside the pipe, and the internal diameter variation due to pressure. Equations of friction head loss The effects of hydraulic resistance and energy dissipation are always present in real fluid flow. The energy dissipation represented by the head loss in the turbulent flow of real fluids through cylindrical tubes can be calculated by equations presented in the basic literature of hydraulics (Porto 1998). Although many equations are available for determining friction loss, the Darcy–Weisbach equation (Eq. 1) is the most important (Kamand 1988; Von Bernuth 1990; Bagarello et al. 1995; Romeo et al. 2002; Sonnad and Goudar 2006).

hf = f

L V2 D 2g

(1)

where hf  = head loss (m); f  = friction coefficient of the Darcy–Weisbach formula (−); L = pipe length (m); D = internal diameter (m); g  = gravitational acceleration (m s−2); V = mean water velocity at uniform pipe sections (m s−1). Equation (1) also can be written in terms of head loss per unit of length (J) or ‘friction slope’ (Swamee and Rathie 2007), resulting in Eq. (2).

J=f

1 V2 D 2g

(2)

The hydraulic resistance, expressed as the friction coefficient (f), constitutes the basic information to the hydraulic project. For conditions of laminar flow (R