Improving the Accuracy of the Trimmer Equation for Estimating Wind Drift and Evaporation Losses in Sprinkler Irrigation Systems Based on Frost and Schwalen's Nomograph

Introduction

One of the most comprehensive approaches employed both internationally and within Iran for estimating wind drift and evaporation losses (WDEL) in sprinkler irrigation systems is the application of the Frost and Schwalen nomograph (1955). This nomograph, derived from the results of more than 700 experiments, incorporates five key variables influencing WDEL. Despite its robustness, the chart’s application is somewhat complex and time-consuming. In 1987, Trimmer conducted extensive analyses of the chart, resulting in the derivation of an empirical equation designed to estimate WDEL more efficiently. The key advantage of this equation lies in its ease of application and compatibility with computer-based simulations. However, the use of Trimmer’s equation requires careful consideration; within certain ranges of the influencing factors, the margin of error increases significantly, making the equation unreliable. Furthermore, under specific conditions, the constants used in the equation vary. Despite these limitations, Trimmer’s equation is currently used in the design of sprinkler irrigation systems without accounting for the aforementioned considerations. The primary objective of this study is to enhance the accuracy and reliability of Trimmer’s equation by incorporating refinements based on the Frost and Schwalen chart. Specifically, by adjusting the constant coefficients in Trimmer’s formulation, this research aims not only to improve its precision but also to enable its application across the entire range of influencing variables presented in the Frost and Schwalen chart.

Method

The parameters required in the Frost and Schwalen nomograph include ambient temperature, relative humidity, wind speed, sprinkler pressure, and nozzle diameter. The ranges of variation for these parameters are as follows: relative humidity: 0 to 100%; ambient temperature: 30 to 110°F (equivalent to -1.1 to 43.3°C); Nozzle diameter: 8.64 to 64.64 inches (equivalent to 3.2 to 25.4 mm); Operating pressure of the sprinkler: 20 to 80 psi (equivalent to 137.9 to 551.6 kPa); Wind speed: 0 to 15 mph (equivalent to 0 to 6.7 m/s). Initially, Trimmer used the Tetens equation (1930) and the parameters of ambient temperature and relative humidity to calculate the saturation vapor pressure deficit. Next, the dimensions of each of the major lines on the chart were measured using a digitizing tablet. By applying similar trigonometric relations and the measured values, Trimmer proposed Eq. (1) to estimate wind drift and evaporation losses:

(1)



Where, , wind drift and evaporation losses (%); , Main nozzle diameter (mm); , Saturation vapor pressure deficit (kPa); , Sprinkler operating pressure (kPa); and , Wind speed at height of 2 m (m/s).

The relative error of the equation is approximately 10% near the central values of the parameters, but the errors increase significantly near the extremes of the chart, reaching over 40% in some cases (Trimmer, 1987). To refine Trimmer’s equation, the Frost and Schwalen nomograph was digitized in this study using Grapher software (Ver. 7.0.1870, Golden, Colorado 80401), and a total of 6,501 data series were extracted from the chart. Of the total dataset, 70% was used for training, and the remaining 30% was allocated for validating the revised equation.

Results

The absolute mean relative error (AMRE) of the Trimmer equation, under the condition of using 70% of the data for training and 30% for validation, was found to be 24.69% and 24.96%, respectively. These values indicate the poor performance of the Trimmer equation in estimating WDEL. Additionally, the P0.25 index was calculated to be 60.37% and 62.22%, respectively. In other words, approximately 40% of the estimates provided by the Trimmer equation have an error greater than 25%. Furthermore, Trimmer (1987) had already reported undesirable error levels in his proposed equation for certain specific parameter ranges (e.g., extreme values of operating conditions), and the findings of the present study confirm those observations. The revised Trimmer equation is presented as Eq. (2):

(2)



The accuracy and reliability of the proposed equation were significantly improved. Specifically, the absolute mean relative error (AMRE) decreased to 4.96% and 4.99% under the use of 70% and 30% of the dataset, respectively. Additionally, the P0.25 index increased to 99.60% and 99.54%, respectively. These results indicate that the developed equation exhibits a highly satisfactory level of accuracy for estimating WDEL.

Conclusions

Given that the accuracy of the revised Trimmer equation in estimating wind drift and evaporation losses in sprinkler irrigation systems is both desirable and of excellent quality compared to the reference data (i.e., the Frost and Schwalen nomograph), it is therefore recommended that, if this method is chosen for estimating losses, either the original chart be used directly or the revised Trimmer equation be employed as a reliable alternative.