A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation
Steady-state demand-driven water distribution system (WDS) solution is the bedrock for much research conducted in the field related to WDSs. WDSs are modeled using the Darcy–Weisbach equation with the Swamee–Jain equation. However, the Swamee–Jain equation approximates the Colebrook–White equation,...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-04-01
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| Series: | Water |
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| Online Access: | https://www.mdpi.com/2073-4441/13/9/1163 |
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| author | Mengning Qiu Avi Ostfeld |
| author_facet | Mengning Qiu Avi Ostfeld |
| author_sort | Mengning Qiu |
| collection | DOAJ |
| description | Steady-state demand-driven water distribution system (WDS) solution is the bedrock for much research conducted in the field related to WDSs. WDSs are modeled using the Darcy–Weisbach equation with the Swamee–Jain equation. However, the Swamee–Jain equation approximates the Colebrook–White equation, errors of which are within 1% for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>/</mo><mi>D</mi><mo>∈</mo><mo>[</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mo>,</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>]</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mi>e</mi><mo>∈</mo><mo>[</mo><mn>5000</mn><mo>,</mo><msup><mn>10</mn><mn>8</mn></msup><mo>]</mo></mrow></semantics></math></inline-formula>. A formulation is presented for the solution of WDSs using the Colebrook–White equation. The correctness and efficacy of the head formulation have been demonstrated by applying it to six WDSs with the number of pipes ranges from 454 to 157,044 and the number of nodes ranges from 443 to 150,630. The addition of a physically and fundamentally more accurate WDS solution method can improve the quality of the results achieved in both academic research and industrial application, such as contamination source identification, water hammer analysis, WDS network calibration, sensor placement, and least-cost design and operation of WDSs. |
| first_indexed | 2024-03-10T12:03:35Z |
| format | Article |
| id | doaj.art-335c8ef466f04cb490733c564b6357be |
| institution | Directory Open Access Journal |
| issn | 2073-4441 |
| language | English |
| last_indexed | 2024-03-10T12:03:35Z |
| publishDate | 2021-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Water |
| spelling | doaj.art-335c8ef466f04cb490733c564b6357be2023-11-21T16:43:46ZengMDPI AGWater2073-44412021-04-01139116310.3390/w13091163A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White EquationMengning Qiu0Avi Ostfeld1Faculty of Civil and Environmental Engineering, Technion-Israel Institute of Technology, Haifa 32000, IsraelFaculty of Civil and Environmental Engineering, Technion-Israel Institute of Technology, Haifa 32000, IsraelSteady-state demand-driven water distribution system (WDS) solution is the bedrock for much research conducted in the field related to WDSs. WDSs are modeled using the Darcy–Weisbach equation with the Swamee–Jain equation. However, the Swamee–Jain equation approximates the Colebrook–White equation, errors of which are within 1% for <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>ϵ</mi><mo>/</mo><mi>D</mi><mo>∈</mo><mo>[</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mo>,</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo>]</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>R</mi><mi>e</mi><mo>∈</mo><mo>[</mo><mn>5000</mn><mo>,</mo><msup><mn>10</mn><mn>8</mn></msup><mo>]</mo></mrow></semantics></math></inline-formula>. A formulation is presented for the solution of WDSs using the Colebrook–White equation. The correctness and efficacy of the head formulation have been demonstrated by applying it to six WDSs with the number of pipes ranges from 454 to 157,044 and the number of nodes ranges from 443 to 150,630. The addition of a physically and fundamentally more accurate WDS solution method can improve the quality of the results achieved in both academic research and industrial application, such as contamination source identification, water hammer analysis, WDS network calibration, sensor placement, and least-cost design and operation of WDSs.https://www.mdpi.com/2073-4441/13/9/1163water distribution systemdemand-dependent modelshead formulationcolebrook–white equation |
| spellingShingle | Mengning Qiu Avi Ostfeld A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation Water water distribution system demand-dependent models head formulation colebrook–white equation |
| title | A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation |
| title_full | A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation |
| title_fullStr | A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation |
| title_full_unstemmed | A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation |
| title_short | A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation |
| title_sort | head formulation for the steady state analysis of water distribution systems using an explicit and exact expression of the colebrook white equation |
| topic | water distribution system demand-dependent models head formulation colebrook–white equation |
| url | https://www.mdpi.com/2073-4441/13/9/1163 |
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