Closure to “Rigid Water Column Model for Simulating the Emptying Process in a Pipeline Using Pressurized Air” by Oscar E. Coronado-Hernández, Vicente S. Fuertes-Miquel, Pedro L. Iglesias-Rey, and Francisco J. Martínez-Solano

Abstract

This closure addresses how the mathematical model developed by the authors is appropriate to simulate emptying processes using pressurized air in pipelines of undulating profiles since it is based on physical equations. The writers thank the discussers for providing information regarding experimental data in the published paper of Laanearu et al. (2012) and for the comments in their discussion.The mathematical model developed by the authors considers three assumptions. These assumptions are (1) a rigid water column model (RWCM) can be used to represent the water behavior, (2) the air-water interface is perpendicular to the pipe direction, and (3) there is a constant friction factor. The discussers agree with Assumptions 1 and 3; however, they disagree with Assumption 2 about the definition of the shape of the air-water interface.The authors assume a piston-flow model to represent the air-water interface considering that the air tank is capable of pressurizing the pipe installation with initial gauge pressures ranging from 10 to 20 m H20 . Laanearu et al. (2012) reported initial values of the air-water front (x i ). The initial length of the water column (L e0 ) was calculated by the authors from the air-water position (with a reference point at x ¼ 0 m) to the location of the discharge valve (x ¼ 271.6 m) [ Fig. 2 and Table 3 of Laanearu et al. (2012)]. In this sense, the initial length of the water column corresponds to the region of pressurized flow of the pipe installation. The part of water corresponding to the free-surface flow was neglected during simulations. The total length of the pipeline (L T ) is 314.1 m (Tijsseling et al. 2016). Initial water columns vary from 91% to 93% regarding the total length of the pipeline installation for all runs. These percentages show that the piston-flow model can be used to represent the behavior of the air-water interface. The remaining percentage of the total length is occupied as either a free-surface flow or pressurized air.The mathematical model presented in the original paper is based on physical equations; therefore, undulating profiles of pipelines can be modeled (Coronado-Hernández et al. 2017;Fuertes-Miquel et al. 2019). The discussers mentioned that the authors did not consider the pipe bridge and the upstream vertical leg to simulate the emptying process. Then, the gravity term (Δz=L e ) was modified in order to consider these pipe branches (Coronado-Hernández et al. 2018a, b). Fig. 1 shows six possible positions of the air-water interface. Positions 5 and 6 were considered in the original paper. Table 1 presents the corresponding values of the gravity term.Results of the mathematical model neglecting the pipe bridge were compared to the scenario considering it. Fig. 2 shows the comparison of the water flow oscillation pattern and gauge pressure pattern for Run 4. The pipe bridge does not affect the behavior of the water movement. A small discrepancy of the gauge pressure pattern was found between 4 and 6 s, which was generated by the pipe bridge. Results co...