{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:41:17Z","timestamp":1760125277943,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T00:00:00Z","timestamp":1679788800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Water"],"abstract":"<jats:p>The estimation of thermodynamic behavior during filling processes with entrapped air in water pipelines is a complex task as it requires solving a system of algebraic-differential equations. A lot of different numerical methods have been used for this purpose in literature including the rigid water column (RWC) model. The main advantage of the RWC model is its acceptable accuracy with very low computational load. In that context, this research presents the computation of critical points of the physical equations that describe the phenomenon. These points provide information about the final position of the air\u2013water interface. The Newton\u2013Raphson method was then applied to obtain a unique equation that can be used by engineers to directly compute variables such as air pocket pressure and water column length at the end of the hydraulic event. A case study was analyzed to compare the results of the mathematical model with the obtained equation for computing critical points. Both methods provided the same values for the water column length at the end of the hydraulic event. A sensitivity analysis was conducted to identify dependent and non-dependent parameters for evaluating the critical points. The proposed formulation was validated through an experimental set of data.<\/jats:p>","DOI":"10.3390\/w15071304","type":"journal-article","created":{"date-parts":[[2023,3,27]],"date-time":"2023-03-27T01:38:36Z","timestamp":1679881116000},"page":"1304","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Application of Newton\u2013Raphson Method for Computing the Final Air\u2013Water Interface Location in a Pipe Water Filling"],"prefix":"10.3390","volume":"15","author":[{"given":"Dalia M.","family":"Bonilla-Correa","sequence":"first","affiliation":[{"name":"Facultad de Ciencias Exactas y Naturales, Universidad de Cartagena, Cartagena 1310014, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6574-0857","authenticated-orcid":false,"given":"\u00d3scar E.","family":"Coronado-Hern\u00e1ndez","sequence":"additional","affiliation":[{"name":"Facultad de Ingenier\u00eda, Universidad Tecnol\u00f3gica de Bol\u00edvar, Cartagena 131001, Colombia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3524-2555","authenticated-orcid":false,"given":"Vicente S.","family":"Fuertes-Miquel","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda Hidr\u00e1ulica y Medio Ambiente, Universitat Polit\u00e8cnica de Val\u00e8ncia, 46022 Valencia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5222-0679","authenticated-orcid":false,"given":"Mohsen","family":"Besharat","sequence":"additional","affiliation":[{"name":"School of Civil Engineering, University of Leeds, Leeds LS2 9JT, UK"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9028-9711","authenticated-orcid":false,"given":"Helena M.","family":"Ramos","sequence":"additional","affiliation":[{"name":"Department of Civil Engineering, Architecture and Georesources, CERIS, Instituto Superior T\u00e9cnico, University of Lisbon, 1049-001 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Tasca, E., Besharat, M., Ramos, H.M., Luvizotto, E., and Karney, B. (2023). Contribution of Air Management to the Energy Efficiency of Water Pipelines. Sustainability, 15.","DOI":"10.3390\/su15053875"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.compfluid.2015.12.002","article-title":"CFD modeling of transient flow in pressurized pipes","volume":"126","author":"Martins","year":"2016","journal-title":"Comput. Fluids"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"012069","DOI":"10.1088\/1755-1315\/774\/1\/012069","article-title":"Developing a 1D-3D model to investigate the effect of entrapped air on pressure surge during the rapid filling of a pipe","volume":"774","author":"Maddahian","year":"2021","journal-title":"IOP Conf. Ser. Earth Environ. Sci."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1080\/1573062X.2019.1669188","article-title":"Hydraulic Modeling during Filling and Emptying Processes in Pressurized Pipelines: A Literature Review","volume":"16","year":"2019","journal-title":"Urban Water J."},{"key":"ref_5","unstructured":"AWWA (American Water Works Association) (2016). Manual of Water Supply Practices M51\u2014Air Valves: Air Release, Air\/Vacuum and Combination, AWWA."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1080\/00221686.2013.785985","article-title":"Experimental Investigation of Entrapped Air Pocket in a Partially Full Water Pipe","volume":"51","author":"Zhou","year":"2013","journal-title":"J. Hydraul. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"041301","DOI":"10.1115\/1.4043321","article-title":"Rapid Liquid Filling of a Pipe With Venting Entrapped Gas: Analytical and Numerical Solutions","volume":"141","author":"Tijsseling","year":"2019","journal-title":"J. Press. Vessel. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1061\/(ASCE)0733-9429(1996)122:10(534)","article-title":"Filling of pipelines with undulating elevation profiles","volume":"122","author":"Liou","year":"1996","journal-title":"J. Hydraul. Eng."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1080\/00221689909498518","article-title":"Pipeline start-up with entrapped air","volume":"37","author":"Izquierdo","year":"1999","journal-title":"J. Hydraul. Res."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"04016036","DOI":"10.1061\/(ASCE)HY.1943-7900.0001171","article-title":"CFD approach for column separation in water pipelines","volume":"142","author":"Wang","year":"2016","journal-title":"J. Hydraul. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"04017071","DOI":"10.1061\/(ASCE)HY.1943-7900.0001416","article-title":"3d numerical modeling of geyser formation by release of entrapped air from horizontal pipe into vertical shaft","volume":"144","author":"Chan","year":"2018","journal-title":"J. Hydraul. Eng."},{"key":"ref_12","first-page":"1","article-title":"Manhole cover displacement caused by the release of entrapped air pockets","volume":"26","author":"Wang","year":"2018","journal-title":"J. Water Manag. Model."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1080\/00221686.2016.1275046","article-title":"Maximum transient pressures in a rapidly filling pipeline with entrapped air using a CFD model","volume":"55","author":"Martins","year":"2017","journal-title":"J. Hydraul. Res."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"031301","DOI":"10.1115\/1.4031508","article-title":"Improved One-Dimensional Models for Rapid Emptying and Filling of Pipelines","volume":"138","author":"Tijsseling","year":"2016","journal-title":"J. Press. Vessel Technol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"820","DOI":"10.1080\/00221686.2020.1844808","article-title":"Unsteady friction in transient vertical-pipe flow with trapped air","volume":"59","author":"Zhou","year":"2021","journal-title":"J. Hydraul. Res."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1080\/00221686.2018.1475427","article-title":"Rapid air expulsion through an orifice in a vertical water pipe","volume":"57","author":"Zhou","year":"2019","journal-title":"J. Hydraul. Res."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"04020047","DOI":"10.1061\/(ASCE)HY.1943-7900.0001773","article-title":"Expulsion of Entrapped Air in a Rapidly Filling Horizontal Pipe","volume":"146","author":"Zhou","year":"2020","journal-title":"J. Hydraul. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Coronado-Hern\u00e1ndez, O.E., Bonilla-Correa, D.M., Lovo, A., Fuertes-Miquel, V.S., Gatica, G., Linfati, R., and Coronado-Hern\u00e1ndez, J.R. (2022). An Implicit Formulation for Calculating Final Conditions in Drainage Maneuvers in Pressurized Water Installations. Water, 14.","DOI":"10.3390\/w14213364"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1061\/(ASCE)0733-9437(2009)135:1(96)","article-title":"Pivoting Strategies in the Solution of the Saint-Venant Equations","volume":"135","author":"Canelon","year":"2009","journal-title":"J. Irrig. Drain. Eng."},{"key":"ref_20","unstructured":"Martin, C.S. (1976, January 22\u201324). Entrapped Air in Pipelines. Proceedings of the Second International Conference on Pressure Surges, London, UK."},{"key":"ref_21","unstructured":"Chapra, S., and Canale, R. (2015). Numerical Methods for Engineers, Mcgraw-Hill Education, Cop. [7th ed.]."}],"container-title":["Water"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-4441\/15\/7\/1304\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:03:21Z","timestamp":1760123001000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-4441\/15\/7\/1304"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,26]]},"references-count":21,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["w15071304"],"URL":"https:\/\/doi.org\/10.3390\/w15071304","relation":{},"ISSN":["2073-4441"],"issn-type":[{"type":"electronic","value":"2073-4441"}],"subject":[],"published":{"date-parts":[[2023,3,26]]}}}