**4. Conclusions**

Subatmospheric pressures occur during the emptying process in pipelines with undulating profiles, which is a typical and common operation that engineers have to face. A rigid two-phase flow model was developed for analyzing it, considering *n* possible air pockets, *d* air valves, *p* pipes, and *o* drain valves. The proposed model is validated using an experimental facility of an irregular profile of 7.3 m long and nominal diameter 63 mm (*DN*63), with an air valve located at the high point and several drain valves along the pipeline. The main hydraulic variables could be measured such as flow (both phases), pressure and air-water front position. Comparisons between computed and measured values of the absolute pressure, water velocity and the length of the emptying columns show that the proposed model can predict accurately not only the extreme values but also their patterns. According to the results, the following conclusions can be drawn:

1. The proposed model can be used for planning the emptying operations in pipelines with undulating profiles, considering their limitations.


In real hydraulic pressurized systems, the proposed model can be used for: (i) checking the risk of a system collapse considering factors such as stiffness class, the soil in natural conditions, the type of backfill and the cover depth; and (ii) selecting the air valve sizes depending on the characteristics of each hydraulic system.

**Supplementary Materials:** The following are available online at www.mdpi.com/2073-4441/9/2/98/s1: The following videos show the emptying process for emptying column 1. Videos: Test No. 1, Test No. 2, Test No. 3, Test No. 4, Test No. 5, Test No. 6, Test No. 7, Test No. 8, Test No. 9, and Test No. 10. The behaviour for emptying column 2 is similar to emptying column 1.

**Acknowledgments:** The authors acknowledge: (1) the financial support for the first author covered by Fundación Centro de Estudios Interdisciplinarios Básicos y Aplicados (CEIBA)—Gobernación de Bolívar (Colombia); and (2) the experimental facilities for conducting the tests provided by IST (Instituto Superior Técnico), Civil Engineering Research and Innovation for Sustainability (CERIS), of the University of Lisbon (Portugal).

**Author Contributions:** Helena M. Ramos and Vicente S. Fuertes-Miquel conceived and designed the experimental facility and type of tests; Oscar E. Coronado-Hernández and Mohsen Besharat performed the experimental tests; Oscar E. Coronado-Hernández wrote the first draft paper; and Helena M. Ramos and Vicente S. Fuertes-Miquel revised the paper and completed the submission.

**Conflicts of Interest:** The authors declare no conflict of interest.
