Calculation method of fertilizer concentration at outlet based on differential pressure tank considering fertilizer dissolution

Abstract: Differential pressure tank is widely used in the fertigation technology in China because of its low cost, simple and convenient. In this study, a theoretical formula was derived for the fertilizer concentration at outlet based on differential pressure tank considering fertilizer dissolution. The continuity equation for fertilizer transportation and diffusion with source term was used to obtain the differential equation for the fertilizer concentration at outlet. The influence of fertilizer dissolution on the fertilizer concentration at outlet was described by parameters considering the actual process of fertilizer dissolution, and the theoretical formula was obtained for the fertilizer concentration at outlet. In the theoretical formula, one parameter represented the initial dissolution rate of fertilizer in the tank, and the other parameter described the decreasing speed of fertilizer dissolution rate in the tank. When the parameters in the theoretical formula took a specific value, the theoretical formula could be transformed into the classical exponential formula of fertilizer concentration. The variations of source term and fertilizer concentration at outlet with time were further discussed when the parameters took different values. Compared with the experimental data, the theoretical formula considering fertilizer dissolution was suitable for different fertilization experiments and fertilization conditions, and achieved more accurate prediction for fertilizer concentration at outlet than the classical exponential formula and the experimental regression formulas. The classical exponential formula was obviously underestimated for the fertilization experiments with obvious fertilizer dissolution, and the experimental regression formulas were also overestimated or underestimated to some extent for some experimental results. The average values of mean absolute error, root mean square error, geometric mean bias, geometric variance and the fraction of computations within two times the deviation for the theoretical formula were 0.04, 0.06, 0.89, 1.19 and 88.26%, respectively, which verified the accuracy of the theoretical formula for the prediction of fertilizer concentration at outlet. For the classical exponential formula and the experimental regression formulas, the average values of geometric mean bias and geometric variance deviated from the reasonable range, and the deviation for some experiments was very distinct. The average mean absolute error of the theoretical formula for typical cases was 0.020. The mean absolute errors for the classical exponential formula and the experimental regression formulas were 0.102, 0.127 and 0.124, which were 5.1, 6.4 and 6.2 times of the theoretical formula. On this basis, the uniform fertilization method under differential pressure tank was further illustrated considering fertilizer dissolution when the fertilizer concentration in the tank was saturated continuously and the sufficient fertilizer was supplied for long-term dissolution and fertilization. The fertilizer amount dissolved in the tank was always equal to the fertilizer amount flowing out of the tank in this particular case. At this point, the uniform fertilization of constant fertilization flux and optimal fertilizer concentration can be realized by setting an appropriate main pipe flux. The theoretical formula considering fertilizer dissolution calculates the fertilizer concentration at outlet precisely, and provides a feasible scheme for uniform fertilization under differential pressure tank.