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## Revista Ciencias Técnicas Agropecuarias

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*versión On-line* ISSN 2071-0054

### Rev Cie Téc Agr vol.26 no.1 San José de las Lajas ene.-mar. 2017

**Revista Ciencias Técnicas Agropecuarias, 2****6****(****1****): 57-65, 201**7**, ISSN: 2071-0054**

**ORIGINAL ARTICLE**

**Mathematical Models for Flow Simulation in Cipoletti Weirs Used in Irrigation Systems**

**Modelos matemáticos para la simulación del caudal en vertedores Cipolleti utilizados en sistemas de riego**

**M.Sc. Luis Manuel Sandoval-Mendoza,**^{I} ** Ing. Julio Adrián Miranda-Escobar,**^{I} ** Dr.C. Oscar Brown-Manrique,**^{II} ** Dr.C. Albi Mujica-Cervantes,**^{II} ** Dr.C. Jorge Douglas Bonilla-Rocha,**^{II} ** Dr.C. Yurisbel Gallardo-Ballat,**^{II}

^{I}Universidad de San Carlos de Guatemala (USAC), Facultad de Ingeniería, Guatemala.

^{II}Universidad de Ciego de Ávila (UNICA), Facultad de Ciencias Técnicas, Centro de Estudios Hidrotécnicos, Ciego de Ávila, Cuba.

**ABSTRACT**

This paper presents the results of the research carried out at the Laboratory of Fluid Mechanics and Hydraulics of the University San Carlos de Guatemala, where different weir Cipolleti were built with different values of base (*b*) in the range of 5-60 centimeters. A polynomial model of second order to estimate the parameters *K* and *n* of the exponential equation to estimate the flow and potential for discharge coefficient *C _{d}* function was found. The model to estimate the theoretical flow rate suggested by the hydraulics and hydrometrics manual and the one obtained by SPSS statistical program presented the highest relative errors; however, the rest of the tested models showed a good correlation with the observed flow rates of error less than 2%. The general experimental model relating discharge coefficient obtained reliable values for all evaluated widths and load heights (

*H*) between 4 and 10 centimeters.

**Key words: **calibration, discharge coefficient, hydrodynamic channel, flow measurement.

**RESUMEN**

En este trabajo se presentan los resultados de la investigación desarrollada en el Laboratorio de Mecánica de Fluidos e Hidráulica de la Universidad de San Carlos de Guatemala, donde se construyeron diferentes vertedores Cipolleti con distintos valores de base (*b*) en el rango de 5 a 60 centímetros. Se encontró un modelo polinomial de segundo orden para la estimación de los parámetros *K* y *n* de la ecuación exponencial para la estimación del caudal y una función potencial para el coeficiente de descarga *C _{d}*. El modelo para estimar el caudal teórico sugerido por los manuales de hidráulica e hidrometría y el obtenido mediante el programa estadístico SPSS presentaron los mayores errores relativos; sin embargo el resto de los modelos evaluados mostraron una buena correlación con los caudales observados con porcentajes de error menor al 2%. El modelo experimental general que relaciona el coeficiente de descarga obtuvo valores confiables para todos los anchos evaluados y alturas de carga (

*H*) entre 4 y 10 centímetros.

**Palabras clave:** medición del flujo, calibración, canal hidrodinámico, coeficiente de descarga.

**INTRODUCTION**

Agriculture is an economic activity of high consumption of water resources that requires strategic actions to increase the efficiency in the use of the water based on the automation and modernization of surface irrigation (Olvera *et al.*, 2014) that contributes to the reduction of water losses produced by conduction, filtration and percolation.

The efficient design of surface irrigation can provide the optimal flow^{1} (Bello y Pino, 2000; Durán y García, 2007); but its magnitude must be known so, the farmer can implement flow measurement, not only for water control, but also for the improvement of distribution (Instituto para la Diversificación y Ahorro de la Energía, 2005). This measurement is usually performed with weirs, consisting of a wall intercepting the flow of a liquid free surface, causing an elevation of the water level upstream (Sotelo, 2002); however, the complexity of the hydraulic phenomena in these structures justifies the use of research laboratories (Rodríguez *et al.*, 2015)and mathematical modeling to reproduce flow patterns changes occurring in these devices (Bacolla et al., 2004). Taking into account the matters above mentioned, the objective of this work is to propose mathematical models that allow accurate simulation of the flow in Cipoletti weirs used in irrigation systems.

** **

**METHODS**

The research was conducted at the Laboratory of Fluid Mechanics and Hydraulics of the University San Carlos of Guatemala consisting of a water supply system by pumping and a tank for volumetric capacities. For the development of the experiments eight trapezoidal weirs Cipoletti type with a 3 mm thick iron sheet and coated with anticorrosive paint to lengthen the time of use of the material, were designed and manufactured

Device dimensions were determined according to the proportions of arrival channels available in the laboratory. The slope ratio was set at 1: 4 as being a feature of Cipolleti weirs; however, the value of the base (*b*) in the range of 5 to 60 cm was varied. Weirs of 5 to 15 cm wide base were placed into the basin for testing weirs (small channel) and in the adjacent channel; chutes were installed with 20 to 60 cm of base width as it is shown in Figure 1.

In each weir a bevel was made so that the fluid has less contact with the wall and its output is in a parabolic shape that allows the establishment of an aerated area under the crest. This plays an important role, because it does not allow the flow to slide on the outer face, thereby decreasing the effect of viscosity on the wall of the weir.

In order to fix the weir at the outlet of hydrodynamic channel and to avoid leaks or filtrations that could affect the measurements, it was necessary the placement of 25 mm wide rubber ribbons in the perimeter of the weirs with smaller bases (b<15 cm). In the case of weirs of great width, sikaflex was placed like adhesive and sealer. In small weirs (b <15 cm) assays were performed in the laboratory hydraulic channel and in large weirs (b> 20 cm) the adjacent channel was used. In all measurements of height (H), the flow supplied to achieve stabilization was controlled; only readings which had aired zone under the sheet slope were considered valid.

The distance (d) for taking *H* readings was calculated taking into account the condition *d≥4H*; because of that in weirs of small width the distance was of 51 cm and in the wide wires the distance was of 100 cm; measured from the weir position in longitudinal direction.

For each reading of load H carried out, three volumetric measurements were made to calculate, by means of arithmetic average, the flow that circulates through the weirs and later obtaining the experimental equation of each one of them by means of the following formulation:

where *Q _{obs}* is the observed flow rate (L/s);

*V*the volume captured in the calibrated container (L);

*t*the time to capture the volume in the container (s).

The experimental flow was estimated as in most flowmeters, using the following exponential model:

where *Q _{exp}* is the experimental flow for each type of weir (L/s);

*H*height above the weir crest (cm);

*k*and

*n*coefficients settings.

In the above model, the relationship between *Q* and *H* was linearized from building a graph LOGQ = f (LogH); obtaining a linear model of the type Y=aX-b where Y = LOGQ and X= LogH. Thus the values of *K* and *n *were obtained as follows:

The calibrated model for obtaining actual flow in trapezoidal Cipoletti weirs was determined as follows:

where *Q _{calib}* is the actual flow obtained by calibration of weirs (L/s);

*Q*theoretical flow (L/s);

_{t}*C*

_{d}_{ }discharge coefficient.

Also, the model to obtain the theoretical flow for trapezoidal weirs is:

where *Q _{t}* is the theoretical flow (L/s);

*b*the length of the weir crest (m);

*H*the observed height (m).

By substituting equation (6) into equation (5), it is obtained:

The discharge coefficient was obtained from the construction of a graph of actual flow against theoretical flow that was adjusted to a linear model, where the slope of the line is the value of the discharge coefficient which was used in the generation of a mathematical model from the computer statistical program Statistical Package for Social Sciences (SPSS) with which the different bases (*b*), and the loading height (*H*) were related for simulating the flow (*Q*) responsive to the following model:

where *Q* is the flow rate obtained with the model (L/s); *b* the base of the weir (cm); *H* height of the observed load (cm) and *k*, *n* constants of the model to determine with SPSS program.The general experimental equation for estimating the flow discharged by the Cipolleti weir (*Q _{gen}*) was obtained from deducting a function to calculate the discharge coefficient from the base of the weir that was replaced subsequently in equation (7).

The validation of the proposed model was performed using the average relative error (Erp) which allows comparison between the observed and simulated flow through the different procedures used in research (*Q _{sim}*). The equation used was as follows:

Where:

Erp: average relative error;

Qobs: observed flow (L/s);

Qsim: simulated flow (L/s).

**RESULTS AND DISCUSSION**

**Analysis of K and n parameters for adjusting the exponential model**

Figure 2 shows the results achieved under laboratory conditions for the parameters *K* and *n* regarding the value of the base of the weir. It was found that both parameters relate to the base of the weir (*b*) using a second order polynomial function. In the case of parameter *n* the determination coefficient found was acceptable with value of 0.70; however the parameter *K* reached an excellent fit with a determination coefficient greater than 0,97. These results allowed the construction of the exponential model type Q_{exp}= KH^{n} for the simulation of experimental flows in each weir evaluated.

**Analysis of the discharge coefficient C_{d} for the weir equation calibration**

Figure 3 shows the behavior of the discharge coefficient *C _{d}* for the weir equation calibration. It was shown that the functional relationship of this coefficient with the base of the weir is the potential type with a determination coefficient of 0.979. The exponent of this function has negative slope; it is indicating that to the extent that the value of the base width is increased, it proportionally reduces the discharge coefficient. This coefficient values found in this research oscillate between 1,01 and 1.31; however, other authors such as Arteaga (1993), Roldán

*et al.*(1993) y Pérez (2015) offer discharge coefficients between 0.66 and 0.75 for conditions similar of water depth over the crest, indicating that the Ciplolleti weirs evaluated in this study have a superior ability to discharge flow, which is important in deciding their use in hydraulic works according to the flow that needs to be dewatered.

**Analysis of models for estimating the flow**

In Table 1 *Q _{exp}* and

*Q*models found for estimating flow in Cipolleti weir are exposed. In all cases a determination coefficient was superior to 0,96 indicating excellent fit of the experimental data to the models evaluated. In the model to determine the flow

_{calib}*Q*it was found that the coefficient

_{exp}*K*varied in the range from 0,07 to 0,99 with a growing trend in the extend the value of the base and the exponent

*n*were increased and reached values in the range of 1,51-1,75.

In the model to determine the flow *Q _{calib,}* the exponent remained fixed with a value of 3/2; but the coefficient behaved in a similar way as the model described above did, with values ranging from 0,22 to 1,13.

The model to estimate the flow of the weir obtained using SPSS program (*Q _{SPSS}*) is as follows:

This model has a high determination coefficient and allows estimating the discharge value of a Cipolleti weir for any value of the base (*b*) starting from the measuring of the water height (*H*).

In Table 1 it is found that for estimation of flow *Q _{calib}* and

*Q*a specific function is required for each width of the base; however, this problem was solved by deducting a general experimental equation derived as follows:

_{exp}Substituting equation (11) into equation (7) results in equation (12):

where *Q _{gen}* is the flow obtained with the general experimental equation (L/s);

*b*the base of the weir (cm);

*H*the height of the observed load (cm).

**Analysis of the comparison between the different models evaluated in the investigation**

The result of the comparison among the different models evaluated in research is shown in Figure 4. It was found that the equations derived for estimating the flow in Cipoletti weirs were very accurate as they achieve similar results to those observed flows; only the theoretical flow (*Q _{t}*) and the flow rate obtained using SPSS program (IBM Corporation, 2003) (

*Q*) showed tendencies to underestimation and overestimation of the values of this variable, respectively.

_{SPSS}**Analysis of the errors committed by the models in estimating the flow**

Mistakes made by the models evaluated in estimating the flow is analyzed using the average relative error (Erp). Table 2 shows that the models to determine the flows *Q _{exp}*,

*Q*and

_{calib}*Q*have values lower than 2%, so they can be used reliably to control flows in open conditions. The worst results were found in Q

_{gen}_{t}and Q

_{SPSS}models with an error of 5,49% that exceed the maximum allowable limit of 5%. This behavior indicates that they should not be applied to determine the flow in Cipoletti weirs.

The results achieved in this research are associated with rigorous design and construction of weirs evaluated; as well as with the obtaining of the experimental data. In this respect, Aguilar (2001) referred the need to properly check different parameters such as height and width of the Cipollet weirs, because if there is not precision in the estimation of the expense, it is not reliable. In that same respect Bos *et al.* (1986) y Santos *et al.* (2010), stated that it is necessary to achieve proper design and installation of the weir to measure and regulate the flow of water from irrigation canals; since the correct design and use of these facilities can help to reduce the overexploitation of aquifers and to energy saving by lower pumping requirements bombeo (Instituto para la Diversificación y Ahorro de la Energía, 2005).

This work can contribute to saving water resources by improving the measurement of the flow by Cipolleti weirs, because according to García y Pérez (2004) y Tarjuelo (2005), the efficient use and effective management of those devices is necessary due to the increasing water scarcity and the growing competition for its various usages. Therefore, strategies to achieve sustainability of irrigation systems should consider the installation of measurement systems that allow controlling the amount of water utilized by the user Fernández *et al.* (2009).

** **

**CONLUSIONS**

The adjustment coefficients of exponential model *Q=KH ^{n}* showed an acceptable functional relationship with the width of the Cipolleti weir base, allowing to find a second order polynomial model to estimate reliably the parameters

*K*and

*n*.

The discharge coefficients *C _{d}* were not constant and their values tend to decrease with increased base as evidenced by the potential mathematical model with a negative slope.

The models for estimating the *Q _{exp}*,

*Q*and

_{calib}*Q*flows showed good correlation with the observed flow, which allows errors less than 2%.

_{gen}The suggested model by hydraulic and hydrometric manuals to estimate the theoretical flow in Cipolleti weirs, showed errors higher than 5%.

The general experimental equation relating the loading heights, base and discharge coefficient obtained reliable values for weirs with width smaller than 60 cm and load heights (*H*) between 4 and 10 centimeters.

**NOT****E****S**

*The mention of commercial equipment marks; instruments or specific materials obey identification purposes, not existing any promotional commitment with relationship to them, neither for the authors nor for the editor.

**BIBLIOGRAPHY**

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BELLO, M.A.; PINO, M.T.: “Medición de presión y caudal”, *Boletín INIA*, 28.^{a} ed., p. 20, Punta Arenas, Chile, 2000, ISSN: 0717-4829.

BOS, M.G.; REPLOGLE, J.A.; CLEMMENS, A.J.: *Aforadores de caudal para canales abiertos*, Ed. International Institute for Land Reclamation and Improvement, Wageningen, 293 p., 1986, ISBN: 978-90-70260-92-7.

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FERNÁNDEZ, C.A.; HOLZAPFEL, E.; DEL CALLEJO, I.; BILLIB, M.: *Manejo sostenible del agua para riego en sudamérica*, Ed. Universidad de Buenos Aires. Facultad de Ciencias Veterinarias, Buenos Aires, Argentina, 175 p., 2009, ISBN: 978-987-25074-1-1.

GARCÍA, G.S.; PÉREZ, L.J.R.: “Resultados de la introducción del riego por goteo en la cooperativa de producción agropecuaria cañera «Primer Soviet de América»”, *Revista Ciencias Técnicas Agropecuarias*, 13(1): 47-51, 2004, ISSN: 1010-2760.

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OLVERA, S.M.D.; BAHENA, D.G.; ALPUCHE, G.O.; GARCÍA, M.F.: “La tecnificación del riego ante la escasez del agua para la generación de alimentos. Estudio de caso en Chihuahua, México”, *Ambiente y Desarrollo*, 18(35): 23-36, 2014, ISSN: 0121-760.

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ROLDÁN, J.; PULIDO, I.; CAMACHO, E.; ALCAIDE, M.; LOSADA, A.: *Problemas de hidráulica para riegos*, Ed. Servicios de publicaciones, Universidad de Córdoba, España, 335 p., 1993, ISBN: 84-7801-526-4.

SANTOS, P.L.; DE JUAN, J.A.; PICORNELL, B.M.R.; TARJUELO, J.M.: *El riego y sus tecnologías*, *[en línea]*, Ed. CREA-UCLM, 1.^{a} ed., España, 296 p., 2010, ISBN: 978-84-692-9979-1, *Disponible en: http://www.academia.edu/download/33889207/El_Riego_y_sus_Tecnologias.pdf*, *[Consulta: 26 de noviembre de 2016]*.

SOTELO, Á.G.: *Hidráulica general*, Ed. Limusa, México, 2002, ISBN: 978-968-18-0503-6.

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Received: 13/01/2016

Approved: 14/11/2016

*Luis Manuel Sandoval-Mendoza,* Prof., Universidad de San Carlos de Guatemala (USAC), Facultad de Ingeniería, Guatemala. Email: ingluissandoval@gmail.com