Microirrigation Hydraulics Review
Conservation principles
Computations of pressure, discharge and flow in microirrigation systems are based on well known principles of fluid mechanics: energy and mass conservation. For the conditions found in microirrigation systems, energy conservation is given by Bernoulli’s equation.
(1)
Where:
Z vertical elevation above an arbitrary frame
of reference, m
h pressure head, m,
α dynamic head adjustment coefficient,
v mean velocity, m/s
Hf friction loss in a pipe section of constant diameter, m
Hk localized head losses, m
He head loss through the emitter, m.
Mass conservation is given by continuity
(2)
Where:
Qin inflow to a pipe section, m3/s
Qout outflow from a pipe section, m3/s
n number of inflows,
m number of outflows.
Hydraulic friction
For a pipe section of constant diameter, the head loss can be calculated by the Darcy-Weisbach equation:
(3)
Where:
f hydraulic friction factor,
L length of the pipe, m
D Internal pipe diameter, m.
The friction factor can be calculated by the equations shown in Table 1 as a function of the Reynolds number.
(4)
Where 
is the kinematic viscosity of water.
Table 1. Friction Factor Equations For Smooth Pipe |
||
Author |
Equation |
Range |
Poiseulle |
|
0 < Re ≤ 1200 |
Blasius |
|
1200 < Re ≤ 105 |
Watters y Keller (1978) |
|
105 < Re ≤ 107 |
Other equations applicable to rough pipes were developed by Karman-Prandtl and Colebrook-White (Sotelo, 1974).
Local head losses.
Localized losses can be calculated by the equivalent length method or the following equation:
(5)
The localized head loss coefficient, K, is determined experimentally for any device that introduces turbulence into the flow stream. Dependence on the type and number of microirrigation connections was studied by several researchers (Urbina, 1976; Al-Amoud, 1995; Howel and Barinas, 1980; Juana et al., 2002ª, 2002b). Their results were presented as tables and graphs.
Head loss through emitters
Microirrigation emitters completely dissipate the pressure head they are subjected to. Thus the head loss through an emitter can be obtained from the emitter’s discharge equation.
(6)
Where:
q emitter discharge, m3/s
C emitter constant, and
x discharge exponent of the emitter.
Hydraulics of multiple outlet pipes.
Calculation methods for multiple outlet pipes were over time subjected to a series of physical and mathematical simplifications that made their use feasible with available computing machinery.
Christiansen (1942) developed a simple equation for head loss computations in pipes with equally spaced emitters under the assumptions that the friction coefficient and emitter discharges are constant.
Under these assumptions, it is possible to factor the head loss for each section of pipe and present the equation as the product of a coefficient depending solely on the number of outlets and the head loss of in a pipe of equivalent length, diameter, and flow rate.
(7)
Where:
Hfn head loss in a pipe with n equally spaced outlets,
F(n) Christiansen coefficient, and
Hf head loss in a pipe of equal length and diameter with
one outlet at the end, m.
Chirstiansen’s simplifications established the principles that governed hydraulic calculation in pressurized irrigation for more than 50 years. Over this period many improvements were made to the F(n) coefficient. Stemberg (1967) presented a design methodology based on this principle. Benami (1968) presented a collection of tables useful to carry out the calculations in the design of sprinkler irrigation. Zazueta and Treviño (1978) and Scaloppi (1988) presented modified Christiansen coefficients to account for the effects of distance from the first discharge to the entrance of the pipe.
Several researchers (Wu y Gitlin, 1974, 1975; Howell and Hiler, 1974; Keller y Karmeli, 1974 y Perold, 1977) developed analysis methods in which the head loss along the pipe is computed assuming that the discharge from the pipe is continuous. Based on the same assumptions Zazueta (1984, 1985) presented a series of equations useful in the design of sloping tapered multiple outlet pipes. In a similar manner, Anwar (1999) proposed a G factor for computations for a multiple outlet pipe with end flow. Zayani et al. (2001) developed maximum length criteria that satisfy the Christiansen assumptions for pipes in flat and sloping terrains.
Some researchers applied numerical methods to solve the hydraulics of multiple outlet pipes in irrigation. Zazueta (1977) presented an iterative method for microirrigation subunits. Bralts and Segerlind (1985) applied the method of finite differences to the same problem. Mizyed (2002) presented a method based on the Newton-Raphson algorithm.
Some researchers avoided Christiansen’s assumptions. Yitayew and Warrick (1988) used variable discharge function to obtain an analytical solution to the head loss distribution along the pipe. In a similar manner, Valinatzaz (1988) used a power equation to resolve the same problem. Vallesquino (2002) proposed a method that accounts for changes of flow regimes using an equivalent friction factor.
References
Al-Amoud, A. I.. 1995. "Significance of energy losses due to emitter connections in trickle irrigation lines."J. Agric. Engrg. Res., 60(1), 1–5.
Anwar, Arif A.. Factor G for Pipelines with Equally Spaced Multiple Outlets and Outflow Journal of Irrigation and Drainage Engineering, Vol. 125, No. 1, January/February 1999, pp. 34-38.
Benami, A.. New Head-Loss Tables for Sprinkler Laterals. Journal of the Irrigation and Drainage Division, ASCE, Vol. 94, No. 2, June 1968, pp. 185-198.
Bralts V. and Segerlind, L. 1985. Finite element analysis of drip irrigation submain units. Transactions of the ASAE 28(3):809-814.
Christiansen, J. E.. 1942. Irrigation by Sprinkling. California Agricultural Experiment Station Bulletin 670, University of California, Berkeley, CA.
Howel, T. A., and Barinas, F. A. (1980) "Pressure losses across trickle irrigation fittings and emitters."Trans. ASAE, 23(4), 928–933.
Howel, T.A. and E.A. Hiler. 1974. Designing trickle irrigation laterals for uniformity. . American Society of Civil Engineers. Journal of Irrigation and Drainage Division. 100(4), 443-454.
Juana, L. L. Rodrigues-Sinobas and A. Lozada. 2002. Determining minor head losses in drip irrigation laterals 1: Methodology. Journal of Irrigation and Drainage Engineering. Nov-Dec:376-384.
Juana, L. L. Rodrigues-Sinobas and A. Lozada. 2002. Determining minor head losses in drip irrigation laterals II: Experimental study and validation. Journal of Irrigation and Drainage Engineering. Nov-Dec:376-384.
Keller, J. and D. Karmeli. 1974. Trickle irrigation design parameters. Transactions ASE, 17(4):678-683.
Mizyed, Numan. 2002. Numerical analysis to solve the hydraulics of trickle irrigation units. Irrigation and Drainage Systems 16: 53-68.
Perold, R.P. 1977. Design of irrigation pipe laterals with multiple outlets. AMercican Society of Civil Engineers. Journal of Irrigation and Drainage Division 103(2):179-195.
Scaloppi, E.J. 1988. Adjusted F factors for multiple outlet pipes. American Society of Civil Engineers. Journal of Irrigation and Drainage Division 114(1):169-174.
Sotelo, G. 1974. Hidráulica General. Ed. LIMUSA. Mexico City. p. 293.
Stemberg, Y.M.. Analysis of Sprinkler Irrigation Losses Journal of the Irrigation and Drainage Division, ASCE. Vol. 93, No. 4, December 1967, pp. 111-124.
Urbina, J. L. (1976). "Head loss characteristic of trickle irrigation hose with emitters," MS thesis, Utah State University, Logan, Utah.
Yitayew M. and Warrick A.W. 1988. Trickle irrigation lateral hydraulics: II: design and examples. American Society of Civil Engineers. Journal of Irrigation and Drainage Division 114(2):281-288.
Valiantaz J. 1988. Analytical approach for direct drip lateral hydraulic acalculation. American Society of Civil Engineer. Journal of the Irrigation and Drainage Division 124(6):300-305.
Vallesquino P. And Luque–Escamilla. P. 2002. Equivalent friction Factor Method for Hydraulic Calculation in Irrigation Laterals. Journal of Irrigation and Drainage Eng. ASCE 128(5):278–286.
Wu I Pai and Gitlin H.M.. 1974. Drip Irrigation design on uniformity. ASAE transactions, 103(2):179-195.
Wu I Pai and Gitlin H.M.. 1975. Energy gradient line for drip irrigation laterals. ASAE transactions, 17(3): 429-432.
Zayani K. Alouini A. Lebdi F. and Lamaddalena N. (2001). Design of drip line in Irrigation systems: Using the energy Drop Ratio Approach. Trans. ASAE 44(5): 1127 – 1133.
Zazueta F.S. 1977. Uso de Computadoras en el Diseño Hidráulico de Sistemas de Riego por Goteo. II Seminario Latinoamericano sobre Riego por Goteo. SARH, IICA. Región Lagunera, Coah. y Dgo.
Zazueta F.S. 1984 The hydraulics of continuous flow irrigation pipes. Institute of Food and Agricultural Sciences, University of Florida. Journal series paper No. 5087. (Presented at the 1984 ASAE Summer Meeting, Knoxville TN)
Zazueta F.S. y H. Treviño. 1978. Un método simplificado para el diseño de sistemas de riego por goteo con emisores hidráulicamente controlables. II Seminario Nacional sobre Riego por Goteo. SARH. Region Lagunera, Coah. Julio de l978.
Zazueta, F.S. and A.G. Smajstrla. 1995. A simple method to design tapered sloping manifolds. Proc. 5th Int. Irr. Congress. Orlando, FL. ASAE. p. 425-430.