In the design of irrigation systems there are many equations widely used in order to calculate the friction losses that occur in pipelines. In general, empirical formulas such as that of Hazen-Williams are frequently used, which can be expressed as follows:
(1)
where J is the water flow rate, Q the flow rate[L3 T-¹] expressed in l / s, D is the internal diameter of the pipe [L] expressed in mm, and with C is the roughness coefficient which also depends on the diameter. According to the experiences of Howell et al. (1981) carried out on PE pipes, it is appropriate to attribute to C the values of 130, 140 and 150 respectively for the diameters of 14-15mm 18-19mm and 25-27mm.
The piezometric falling can also be calculated using the Darcy-Weisbach equation:
(2)
as a function of the Darcy-Weisbach resistance index λ, of the average velocity of the current V[L T-¹] , of the acceleration of gravity g[L T-²] and the internal diameter D [L] of the pipeline. The resistance index λ is evaluated as a function of the flow rate of the current in pipeline A. According to whether the motion is laminar, transitional or turbulent, three different expressions are used.
In the case of laminar flow regime (values of R< 2000) the resistance index can be evaluated using the Hagen-Poiseuille formula:
(3)
For Reynolds number values in the range 2000< King< 100,000 the Blasius equation can be used:
(4)
Finally for 100,000< King< 10,000,000 the resistance index can be evaluated through the relationship suggested by Watters and Keller [ASAE technical study n. 78-2015]:
(5)
For smooth pipes the resistance index can also be evaluated using the Prandtl-Kàrmàn formula
(6)
In the field of purely turbulent motion inside rough pipes, the resistance index can be expressed only as a function of the roughness of the pipe, canceling the influence of the viscosity of the current so that the resistance index can be expressed through the Prandtl formula:
(7)
in this equation ε [L] represents the absolute roughness of the material that constitutes the pipe.
Colebrook[1939] combining the two equations obtained for rough tube, he obtained the following semi-empirical expression, valid both in the transition regime and in the purely turbulent regime
(8)
The use of these equations is not easy as it is not explicit with respect to the resistance index λ, and therefore involves some application difficulties. The resistance index can however be obtained in any case through an iterative procedure, or graphically, by means of the Moody abacus.
The problems connected to the use of the Colebrook equation can however be overcome if one uses, for the computation of λ, the explicit equation proposed by Swamee and Jainn[1976] :
(9)
Along a delivery wing the flow rate varies between a maximum value at the upstream end equal to the sum of the flow rates actually delivered by the drippers arranged along the wing, to a zero value at the downstream end. Consequently, the speed of the current and the Reynolds number (Re) also vary between a maximum value and zero. Generally, conditions of turbulent motion occur in the initial portion of the delivery wing, transition in the intermediate portion and laminar in the terminal portion. Therefore, for each section the head losses should be calculated as a function of the flow rate of the current depending on whether the flow rate is laminar, transitional or turbulent.
In practice, taking into account that in laminar flow conditions the current velocities are small and that the load losses (which linearly depend on the velocities themselves) are also small, it is assumed without committing appreciable errors that the motion is turbulent in the entire pipeline and therefore, for the calculation of the head losses, the expressions (2) and (4) (5) (6) can be used.
