Analytical Solution for Optimum Design of Nozzle Discharge Lines Based on Hydraulic Considerations

Abstract

This study focuses on finding the best design of nozzles used in the water discharge pipelines delivering free jets based on hydraulic considerations. An analytical solution is carried out to determine the best design for these nozzles. The solution depends on the derivative method applied to the energy head concerning the nozzle diameter to maximize the power of the jet. The equation derived for computing the best pipe-nozzle diameter ratio D/d in discharge lines applies over practical ranges of relative distance L/D (50-500) and relative roughness e/D (0.01-0.05). An illustrative example is solved and gives D/d = 1.8858 (closely equal to 1.9), the deviation is 0.77%, which is firmly accepted. The equation holds good for D/d from 1.4 to 3.584 and could be applied to compute D/d ratios without the need for a computer program. In the derived equation, D/d is a function of relative distance L/D, and friction factor F with negligible computational errors. The derived equation when compared with the conventional one, shows an increase in the computed power of the jet ranging from 12.5 to 19.23%. The evaluation study reflects the rigidity of the derived equations and their reliability in computing the best pipe-nozzle diameter ratios in discharge lines delivering free jets with reasonable powers, while the conventional formula is approximate. However, many engineering applications of water jet nozzles are used in; impulsive turbines, power-delivering free jets, water filters, flotation tanks, sedimentation tanks, water storage tanks, trickling filters, and other water and wastewater systems.