Optimal Sizing of an Air Vessel in a Long-Distance Water-Supply Pumping System Using the SQP Method

By optimizing the sizing of air vessels, not only is the damage from water hammer prevented, but also the cost of water hammer protection systems is minimized. This paper presents a numerical simulation of an air vessel in a long-distance transfer main system against water hammer resulting from sudden pump stoppage. An optimal design of air vessel parameters is obtained using sequential quadratic programming (SQP) in this case. Three types of connection configuration (between the air vessel and the main pipeline) are tested in the case study, including a vertical cylindrical vessel with a bypass line (configuration 1), a vertical cylindrical vessel with a bypass line equipped with a throttle valve (configuration 2), and a horizontal cylindrical vessel with separated inflow and outflow lines (configuration 3). The variations of optimal volume of the air vessel, $V_T$, are plotted as a function of the allowable maximum pressure head, $H_{\max }^{*}$, with the diameter of the bypass line, $d_b$, or the throttle valve opening angle, $\theta$, respectively. Demonstrating with a case study, this simulation shows that the SQP method was well suited for the optimal design of air vessel parameters in long-distance water-supply transmission pipelines. For configurations 1, 2, and 3, it is beneficial for the optimal design of the air vessel to decrease the $d_b$ value and the $\theta$ value properly. In addition, by plotting the variations of $V_T$ as a function of $H_{\max }^{*}$ with different $d_b$ values or $\theta$ values, it is helpful for the designer to determine the optimal selection of the connecting configuration.