The Darcy-Weisbach equation is the most recommended equation for determining the pressure loss in pressurized pipes because of its wide applicability. However, one of the largest obstacles to implementing this equation is the friction factor (f) calculation. This factor can be precisely determined using the Colebrook equation, which is implicit. Thus, the objective of this study was to compare six explicit equations for calculating the Darcy-Weisbach friction factor with the implicit Colebrook equation based on the relative error. Based on the results, the equations of Vatankhah and Offor & Alabi were the most highly recommended. These six explicit formulas showed a mean relative error of less than ± 1% compared to the Colebrook equation, except for the Swamee and Jain equation, for which the laminar regime generated a mean relative error of 1.83%.

The main disadvantage of trickle irrigation systems is its comparatively high initial cost, which depends on the layout, design, and management of its hydraulic network. Designing the sub-main and lateral lines aiming the emitter uniformity maximization can reduce the microirrigation system costs. This research aimed to compare linear and nonlinear programming models and maximization versus minimization criteria to optimize the crop net benefit, considering the water and energy savings. Two versions of LP and NLP models were developed: the first minimized the equivalent annual cost of the irrigation system considering the pipeline cost and the energy cost; the second maximized the yearly increment in the net benefit (Bn) of the irrigated crop. In both cases, uncertainty about the crop price was considered. The models were applied in a 40 ha citrus orchard in São Paulo State, Brazil. The highest net benefit was found using the NLP model with the maximization criterion. The worst result was obtained with the LP model and the minimization of the total annual cost. The layout and management previously established by the designer are subjective and rarely results in the best solution, although the linear programming model always gets the global optimum. The NLP models get local optimal, but they defined the layout, design, and management of the systems, with more chance to obtain a higher net benefit. The NLP model for maximization showed to be an adequate option for designing microsprinkler irrigation systems, defining the hydraulic network and the operational conditions that maximize Bn and WUE, with the lowest water consumption and lowest energy cost.

Optimal solutions derived from linear programming models depend entirely on input parameters, which may present some imprecision because they come from estimates. Fuzzy linear programming allows the incorporation of these uncertainties in linear models, which can include the flexibility of resources, costs, goals, and constraints. This paper aimed to show new optimal solutions for a model to minimize the equivalent annual cost of micro-irrigation systems on sloping terrains. The Zimmermann-Werner fuzzy linear programming method, whose objective function is diffuse due to the restrictions of the hydraulic network being dispersed, was used. Sixty models were created and all solutions were satisfactory, with an annual cost of the irrigation system lower than the original model. The lowest value was US$ 238.74 ha −1 , which occurred on the 3% slope. A reduction was observed in the annual cost due to the increased use of pipes with a 50mm nominal diameter in the secondary line. Thus, fuzzy linear programming provided better solutions with small modifications to the irrigation system, while maintaining all hydraulic network requirements for proper system operation.

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