Closed-form solutions of the energy balance equation for drip laterals under the Darcy-Weisbach resistance formula

Many studies have investigated easy methods to design drip laterals, as well as the best resistance equation to use, which is fundamental to accurately account for friction losses. This paper addresses both the features of lateral design relationships and the influence of the friction-loss equation on the design variables. First, simple closed-form solutions of the energy balance equation for both sloped and horizontal drip laterals are derived with the simplified Darcy-Weisbach resistance formula by assuming the Darcy friction factor as invariant versus the Reynolds number. Second, an error analysis is performed assuming the friction factor as constant in the design, which is compared to using the Blasius resistance equation that accounts for Reynolds number variation. Third, once these closed-form solutions have been derived, their extension to account for friction-factor variation versus the Reynolds number is demonstrated by introducing an average friction-factor relationship, with errors in computing the design variables that do not exceed 0.27% (i.e., almost the exact solutions obtained when the Blasius equation is used).