Darcy–Weisbach equation

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class="vector-toc-numb">3.1</span> <span>In terms of volumetric flow</span> </div> </a> <ul id="toc-In_terms_of_volumetric_flow-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Shear-stress_form" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Shear-stress_form"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Shear-stress form</span> </div> </a> <ul id="toc-Shear-stress_form-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Darcy_friction_factor" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Darcy_friction_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Darcy friction factor</span> </div> </a> <button aria-controls="toc-Darcy_friction_factor-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Darcy friction factor subsection</span> </button> <ul id="toc-Darcy_friction_factor-sublist" class="vector-toc-list"> <li id="toc-Laminar_regime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Laminar_regime"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Laminar regime</span> </div> </a> <ul id="toc-Laminar_regime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Critical_regime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Critical_regime"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Critical regime</span> </div> </a> <ul id="toc-Critical_regime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Turbulent_regime" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Turbulent_regime"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Turbulent regime</span> </div> </a> <ul id="toc-Turbulent_regime-sublist" class="vector-toc-list"> <li id="toc-Smooth-pipe_regime" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Smooth-pipe_regime"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1</span> <span>Smooth-pipe regime</span> </div> </a> <ul id="toc-Smooth-pipe_regime-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rough-pipe_regime" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Rough-pipe_regime"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.2</span> <span>Rough-pipe regime</span> </div> </a> <ul id="toc-Rough-pipe_regime-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Calculating_the_friction_factor_from_its_parametrization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Calculating_the_friction_factor_from_its_parametrization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Calculating the friction factor from its parametrization</span> </div> </a> <ul id="toc-Calculating_the_friction_factor_from_its_parametrization-sublist" class="vector-toc-list"> <li id="toc-Direct_calculation_when_friction_loss_S_is_known" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Direct_calculation_when_friction_loss_S_is_known"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4.1</span> <span>Direct calculation when friction loss <i><span>S</span></i> is known</span> </div> </a> <ul id="toc-Direct_calculation_when_friction_loss_S_is_known-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Confusion_with_the_Fanning_friction_factor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Confusion_with_the_Fanning_friction_factor"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Confusion with the Fanning friction factor</span> </div> </a> <ul id="toc-Confusion_with_the_Fanning_friction_factor-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Derivation_by_dimensional_analysis" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Derivation_by_dimensional_analysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Derivation by dimensional analysis</span> </div> </a> <ul id="toc-Derivation_by_dimensional_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Practical_application" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Practical_application"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Practical application</span> </div> </a> <ul id="toc-Practical_application-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Advantages" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Advantages"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Advantages</span> </div> </a> <ul id="toc-Advantages-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">11</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">13</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">14</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button 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Available in 23 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-23" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">23 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Darcy-Weisbach-vergelyking" title="Darcy-Weisbach-vergelyking – Afrikaans" lang="af" hreflang="af" data-title="Darcy-Weisbach-vergelyking" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Darsi_d%C3%BCsturu" title="Darsi düsturu – Azerbaijani" lang="az" hreflang="az" data-title="Darsi düsturu" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Equaci%C3%B3_de_Darcy-Weisbach" title="Equació de Darcy-Weisbach – Catalan" lang="ca" hreflang="ca" data-title="Equació de Darcy-Weisbach" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Darcy-Weisbach-Gleichung" title="Darcy-Weisbach-Gleichung – German" lang="de" hreflang="de" data-title="Darcy-Weisbach-Gleichung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n_de_Darcy-Weisbach" title="Ecuación de Darcy-Weisbach – Spanish" lang="es" hreflang="es" data-title="Ecuación de Darcy-Weisbach" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D8%AF%D9%84%D9%87_%D8%AF%D8%A7%D8%B1%D8%B3%DB%8C_%D9%88%DB%8C%D8%B3%D8%A8%D8%A7%D8%AE" title="معادله دارسی ویسباخ – Persian" lang="fa" hreflang="fa" data-title="معادله دارسی ویسباخ" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/%C3%89quation_de_Darcy-Weisbach" title="Équation de Darcy-Weisbach – French" lang="fr" hreflang="fr" data-title="Équation de Darcy-Weisbach" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8B%AC%EC%8B%9C-%EB%B0%94%EC%9D%B4%EC%8A%A4%EB%B0%94%ED%95%98_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="달시-바이스바하 방정식 – Korean" lang="ko" hreflang="ko" data-title="달시-바이스바하 방정식" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Persamaan_Darcy-Weisbach" title="Persamaan Darcy-Weisbach – Indonesian" lang="id" hreflang="id" data-title="Persamaan Darcy-Weisbach" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Equazione_di_Darcy-Weisbach" title="Equazione di Darcy-Weisbach – Italian" lang="it" hreflang="it" data-title="Equazione di Darcy-Weisbach" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A9%D7%95%D7%95%D7%90%D7%AA_%D7%93%D7%A8%D7%A1%D7%99-%D7%95%D7%99%D7%A1%D7%91%D7%9A" title="משוואת דרסי-ויסבך – Hebrew" lang="he" hreflang="he" data-title="משוואת דרסי-ויסבך" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Darcy-Weisbach-vergelijking" title="Darcy-Weisbach-vergelijking – Dutch" lang="nl" hreflang="nl" data-title="Darcy-Weisbach-vergelijking" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%80%E3%83%AB%E3%82%B7%E3%83%BC%E3%83%BB%E3%83%AF%E3%82%A4%E3%82%B9%E3%83%90%E3%83%83%E3%83%8F%E3%81%AE%E5%BC%8F" title="ダルシー・ワイスバッハの式 – Japanese" lang="ja" hreflang="ja" data-title="ダルシー・ワイスバッハの式" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Darcy-Weisbachs_ligning" title="Darcy-Weisbachs ligning – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Darcy-Weisbachs ligning" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/R%C3%B3wnanie_Darcy%E2%80%99ego-Weisbacha" title="Równanie Darcy’ego-Weisbacha – Polish" lang="pl" hreflang="pl" data-title="Równanie Darcy’ego-Weisbacha" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Equa%C3%A7%C3%B5es_expl%C3%ADcitas_para_o_fator_de_atrito_de_Darcy-Weisbach" title="Equações explícitas para o fator de atrito de Darcy-Weisbach – Portuguese" lang="pt" hreflang="pt" data-title="Equações explícitas para o fator de atrito de Darcy-Weisbach" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%94%D0%B0%D1%80%D1%81%D0%B8_%E2%80%94_%D0%92%D0%B5%D0%B9%D1%81%D0%B1%D0%B0%D1%85%D0%B0" title="Формула Дарси — Вейсбаха – Russian" lang="ru" hreflang="ru" data-title="Формула Дарси — Вейсбаха" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Darcyho-Weisbachov_vz%C5%A5ah" title="Darcyho-Weisbachov vzťah – Slovak" lang="sk" hreflang="sk" data-title="Darcyho-Weisbachov vzťah" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Darcy-Weisbachova_ena%C4%8Dba" title="Darcy-Weisbachova enačba – Slovenian" lang="sl" hreflang="sl" data-title="Darcy-Weisbachova enačba" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Darcy-Weisbachs_ekvation" title="Darcy-Weisbachs ekvation – Swedish" lang="sv" hreflang="sv" data-title="Darcy-Weisbachs ekvation" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Darcy-Weisbach_e%C5%9Fitli%C4%9Fi" title="Darcy-Weisbach eşitliği – Turkish" lang="tr" hreflang="tr" data-title="Darcy-Weisbach eşitliği" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D0%BE%D1%80%D0%BC%D1%83%D0%BB%D0%B0_%D0%94%D0%B0%D1%80%D1%81%D1%96_%E2%80%94_%D0%92%D0%B5%D0%B9%D1%81%D0%B1%D0%B0%D1%85%D0%B0" title="Формула Дарсі — Вейсбаха – Ukrainian" lang="uk" hreflang="uk" data-title="Формула Дарсі — Вейсбаха" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%81%94%E8%A5%BF%E2%80%93%E5%A8%81%E6%96%AF%E5%B7%B4%E5%93%88%E6%96%B9%E7%A8%8B%E5%BC%8F" title="達西–威斯巴哈方程式 – Chinese" lang="zh" hreflang="zh" data-title="達西–威斯巴哈方程式" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1142285#sitelinks-wikipedia" 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class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Darcy_friction_factor&redirect=no" class="mw-redirect" title="Darcy friction factor">Darcy friction factor</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Equation in fluid dynamics</div> <p>In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the <b>Darcy–Weisbach equation</b> is an <a href="/wiki/Empirical_research" title="Empirical research">empirical</a> equation that relates the <a href="/wiki/Head_loss" class="mw-redirect" title="Head loss">head loss</a>, or <a href="/wiki/Pressure" title="Pressure">pressure</a> loss, due to <a href="/wiki/Friction" title="Friction">friction</a> along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after <a href="/wiki/Henry_Darcy" title="Henry Darcy">Henry Darcy</a> and <a href="/wiki/Julius_Weisbach" title="Julius Weisbach">Julius Weisbach</a>. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the <a href="/wiki/Moody_diagram" class="mw-redirect" title="Moody diagram">Moody diagram</a> or <a href="/wiki/Colebrook_equation" class="mw-redirect" title="Colebrook equation">Colebrook equation</a>.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The Darcy–Weisbach equation contains a <a href="/wiki/Dimension_analysis" class="mw-redirect" title="Dimension analysis">dimensionless</a> friction factor, known as the <b>Darcy friction factor</b>. This is also variously called the Darcy–Weisbach friction factor, friction factor, resistance coefficient, or flow coefficient.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>a<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Historical_background">Historical background</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=1" title="Edit section: Historical background"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Darcy-Weisbach equation, combined with the <a href="/wiki/Moody_chart" title="Moody chart">Moody chart</a> for calculating head losses in pipes, is traditionally attributed to <a href="/wiki/Henry_Darcy" title="Henry Darcy">Henry Darcy</a>, <a href="/wiki/Julius_Weisbach" title="Julius Weisbach">Julius Weisbach</a>, and <a href="/wiki/Lewis_Ferry_Moody" title="Lewis Ferry Moody">Lewis Ferry Moody</a>. However, the development of these formulas and charts also involved other scientists and engineers over its historical development. Generally, the <a href="/wiki/Bernoulli%27s_equation" class="mw-redirect" title="Bernoulli's equation">Bernoulli's equation</a> would provide the head losses but in terms of quantities not known a priori, such as pressure. Therefore, empirical relationships were sought to correlate the head loss with quantities like pipe diameter and fluid velocity.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Julius Weisbach was certainly not the first to introduce a formula correlating the length and diameter of a pipe to the square of the fluid velocity. <a href="/wiki/Antoine_Ch%C3%A9zy" class="mw-redirect" title="Antoine Chézy">Antoine Chézy</a> (1718-1798), in fact, had published a formula in 1770 that, although referring to open channels (i.e., not under pressure), was formally identical to the one Weisbach would later introduce, provided it was reformulated in terms of the <a href="/wiki/Hydraulic_radius" class="mw-redirect" title="Hydraulic radius">hydraulic radius</a>. However, Chézy's formula was lost until 1800, when <a href="/wiki/Gaspard_de_Prony" title="Gaspard de Prony">Gaspard de Prony</a> (a former student of his) published an account describing his results. It is likely that Weisbach was aware of Chézy's formula through Prony's publications.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Weisbach's formula was proposed in 1845 in the form we still use today: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H=f\cdot {LV^{2} \over {2gD}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>H</mi> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>L</mi> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>g</mi> <mi>D</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta H=f\cdot {LV^{2} \over {2gD}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e144653de082a548daa3846bc82df0094e9fe904" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.446ex; height:6.176ex;" alt="{\displaystyle \Delta H=f\cdot {LV^{2} \over {2gD}}}"></span></dd></dl> <p>where: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta H}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/385a86c9ced1913abd3606f6bfcec2c10c131cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.999ex; height:2.176ex;" alt="{\displaystyle \Delta H}"></span>: head loss.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>: length of the pipe.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span>: <a href="/wiki/Diameter" title="Diameter">diameter</a> of the pipe.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span>: <a href="/wiki/Velocity" title="Velocity">velocity</a> of the fluid.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"></span>: <a href="/wiki/Standard_gravity" title="Standard gravity">acceleration due to gravity</a>.</li></ul> <p>However, the friction factor f was expressed by Weisbach through the following empirical formula: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=\alpha +{\beta \over {\sqrt {V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>α</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>V</mi> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=\alpha +{\beta \over {\sqrt {V}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e867b254ab5a54a8b366722e5e037209c987396" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:13.264ex; height:6.343ex;" alt="{\displaystyle f=\alpha +{\beta \over {\sqrt {V}}}}"></span></dd></dl> <p>with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> depending on the diameter and the type of pipe wall.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Weisbach's work was published in the <a href="/wiki/United_States_of_America" class="mw-redirect" title="United States of America">United States of America</a> in 1848 and soon became well known there. In contrast, it did not initially gain much traction in <a href="/wiki/France" title="France">France</a>, where <a href="/wiki/Prony_equation" title="Prony equation">Prony equation</a>, which had a polynomial form in terms of velocity (often approximated by the square of the velocity), continued to be used. Beyond the historical developments, Weisbach's formula had the objective merit of adhering to <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimensional analysis</a>, resulting in a dimensionless friction factor f. The complexity of f, dependent on the mechanics of the <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layer</a> and the flow regime (laminar, transitional, or turbulent), tended to obscure its dependence on the quantities in Weisbach's formula, leading many researchers to derive irrational and dimensionally inconsistent empirical formulas.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> It was understood not long after Weisbach's work that the friction factor f depended on the flow regime and was independent of the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> (and thus the velocity) only in the case of rough pipes in a turbulent flow regime (Prandtl-von Kármán equation).<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Pressure-loss_equation">Pressure-loss equation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=2" title="Edit section: Pressure-loss equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a cylindrical pipe of uniform diameter <span class="texhtml mvar" style="font-style:italic;">D</span>, flowing full, the pressure loss due to viscous effects <span class="texhtml">Δ<i>p</i></span> is proportional to length <span class="texhtml mvar" style="font-style:italic;">L</span> and can be characterized by the Darcy–Weisbach equation:<sup id="cite_ref-Brown_9-0" class="reference"><a href="#cite_note-Brown-9"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {\rho }{2}}\cdot {\frac {{\langle v\rangle }^{2}}{D_{H}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {\rho }{2}}\cdot {\frac {{\langle v\rangle }^{2}}{D_{H}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/728a9ed8d43a57c5055290a9c4185ec0ac4421f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.537ex; height:6.343ex;" alt="{\displaystyle {\frac {\Delta p}{L}}=f_{\mathrm {D} }\cdot {\frac {\rho }{2}}\cdot {\frac {{\langle v\rangle }^{2}}{D_{H}}},}"></span></dd></dl> <p>where the pressure loss per unit length <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">Δ<i>p</i></span><span class="sr-only">/</span><span class="den"><i>L</i></span></span>⁠</span></span> (SI units: <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">Pa</a>/<a href="/wiki/Metre" title="Metre">m</a>) is a function of: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span>, the density of the fluid (kg/m<sup>3</sup>);</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59e71271d4219161dad0933245747d75b3884c7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.616ex; height:2.509ex;" alt="{\displaystyle D_{H}}"></span>, the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">hydraulic diameter</a> of the pipe (for a pipe of circular section, this equals <span class="texhtml"><i>D</i></span>; otherwise <span class="texhtml"><i>D</i><sub><i>H</i></sub><i> = 4A/P</i></span> for a pipe of cross-sectional area <span class="texhtml mvar" style="font-style:italic;">A</span> and perimeter <span class="texhtml mvar" style="font-style:italic;">P</span>) (m);</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle v\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61604fdde22cb90ecdbdd2813be95b938722f54c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.937ex; height:2.843ex;" alt="{\displaystyle \langle v\rangle }"></span>, the mean <a href="/wiki/Flow_velocity" title="Flow velocity">flow velocity</a>, experimentally measured as the <a href="/wiki/Volumetric_flow_rate" title="Volumetric flow rate">volumetric flow rate</a> <span class="texhtml mvar" style="font-style:italic;">Q</span> per unit cross-sectional <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">wetted area</a> (m/s);</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {D} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {D} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c776760fa101d602d5fe9b662a0c8bc5713a0f09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle f_{\mathrm {D} }}"></span>, the <a href="/wiki/Darcy_friction_factor_formulae" title="Darcy friction factor formulae">Darcy friction factor</a> (also called flow coefficient <span class="texhtml mvar" style="font-style:italic;">λ</span><sup id="cite_ref-Rouse1946_10-0" class="reference"><a href="#cite_note-Rouse1946-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Incopera2002_11-0" class="reference"><a href="#cite_note-Incopera2002-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>).</dd></dl> <p>For <a href="/wiki/Laminar_flow" title="Laminar flow">laminar flow</a> in a circular pipe of diameter <span class="texhtml"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340ae9ef8dca97fe67e399b6b2f21fdf40573683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.868ex; height:2.509ex;" alt="{\displaystyle D_{c}}"></span></span>, the friction factor is inversely proportional to the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> alone (<span class="texhtml"><i>f</i><sub>D</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">64</span><span class="sr-only">/</span><span class="den">Re</span></span>⁠</span></span>) which itself can be expressed in terms of easily measured or published physical quantities (see section below). Making this substitution the Darcy–Weisbach equation is rewritten as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>128</mn> <mi>π</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mi>Q</mi> </mrow> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f507c812765bbf75e32b4a4d321a26da2110a3c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.765ex; height:6.176ex;" alt="{\displaystyle {\frac {\Delta p}{L}}={\frac {128}{\pi }}\cdot {\frac {\mu Q}{D_{c}^{4}}},}"></span></dd></dl> <p>where </p> <dl><dd><span class="texhtml"><i>μ</i></span> is the <a href="/wiki/Dynamic_viscosity" class="mw-redirect" title="Dynamic viscosity">dynamic viscosity</a> of the <a href="/wiki/Fluid" title="Fluid">fluid</a> (Pa·s = N·s/m<sup>2</sup> = kg/(m·s));</dd> <dd><span class="texhtml"><i>Q</i></span> is the <a href="/wiki/Volumetric_flow_rate" title="Volumetric flow rate">volumetric flow rate</a>, used here to measure flow instead of mean velocity according to <span class="texhtml"><i>Q</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">π</span><span class="sr-only">/</span><span class="den">4</span></span>⁠</span><i>D</i><sub>c</sub><sup>2</sup><<i>v</i>></span> (m<sup>3</sup>/s).</dd></dl> <p>Note that this laminar form of Darcy–Weisbach is equivalent to the <a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Hagen–Poiseuille equation</a>, which is analytically derived from the <a href="/wiki/Navier%E2%80%93Stokes_equations" title="Navier–Stokes equations">Navier–Stokes equations</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Head-loss_formula">Head-loss formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=3" title="Edit section: Head-loss formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Hydraulic_head#Head_loss" title="Hydraulic head">head loss</a> <span class="texhtml">Δ<i>h</i></span> (or <span class="texhtml"><i>h</i><sub>f</sub></span>) expresses the pressure loss due to friction in terms of the equivalent height of a column of the working fluid, so the <a href="/wiki/Pressure" title="Pressure">pressure</a> drop is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p=\rho g\,\Delta h,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>=</mo> <mi>ρ</mi> <mi>g</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>h</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p=\rho g\,\Delta h,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8d429297ee7e8159154f2db3c921972f6cbd29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.83ex; height:2.676ex;" alt="{\displaystyle \Delta p=\rho g\,\Delta h,}"></span></dd></dl> <p><b><u>where:</u></b> </p> <dl><dd><b><span class="texhtml">Δ<i>h</i></span></b> = The head loss due to pipe friction over the given length of pipe (SI units: m);<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>b<span class="cite-bracket">]</span></a></sup></dd> <dd><b><span class="texhtml mvar" style="font-style:italic;">g</span></b> = The local acceleration due to <a href="/wiki/Earth%27s_gravity#Variations_on_Earth" class="mw-redirect" title="Earth's gravity">gravity</a> (m/s<sup>2</sup>).</dd></dl> <p>It is useful to present head loss per length of pipe (dimensionless): </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{\rho g}}\cdot {\frac {\Delta p}{L}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>h</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>ρ</mi> <mi>g</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>p</mi> </mrow> <mi>L</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{\rho g}}\cdot {\frac {\Delta p}{L}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182c7916c45f5a6082add0129ad012aa36713231" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.229ex; height:5.843ex;" alt="{\displaystyle S={\frac {\Delta h}{L}}={\frac {1}{\rho g}}\cdot {\frac {\Delta p}{L}},}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">L</span> is the pipe length (<i>m</i>). </p><p>Therefore, the Darcy–Weisbach equation can also be written in terms of head loss:<sup id="cite_ref-Crowe2005_13-0" class="reference"><a href="#cite_note-Crowe2005-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v\rangle }^{2}}{D}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>g</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>D</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v\rangle }^{2}}{D}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60a1b6a9ca49712cbb9dec08b6e028db679db4df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.172ex; height:6.509ex;" alt="{\displaystyle S=f_{\text{D}}\cdot {\frac {1}{2g}}\cdot {\frac {{\langle v\rangle }^{2}}{D}}.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="In_terms_of_volumetric_flow">In terms of volumetric flow</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=4" title="Edit section: In terms of volumetric flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The relationship between mean flow velocity <span class="texhtml"><<i>v</i>></span> and volumetric flow rate <span class="texhtml mvar" style="font-style:italic;">Q</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=A\cdot \langle v\rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>A</mi> <mo>⋅</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=A\cdot \langle v\rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa09d7c74e26d16f45247d47edc569659340948" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.943ex; height:2.843ex;" alt="{\displaystyle Q=A\cdot \langle v\rangle ,}"></span></dd></dl> <p><u><b>where:</b></u> </p> <dl><dd><b><span class="texhtml mvar" style="font-style:italic;">Q</span></b> = The volumetric flow (m<sup>3</sup>/s),</dd> <dd><b><span class="texhtml mvar" style="font-style:italic;">A</span></b> = The cross-sectional wetted area (m<sup>2</sup>).</dd></dl> <p>In a full-flowing, circular pipe of diameter <span class="texhtml mvar" style="font-style:italic;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{c}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340ae9ef8dca97fe67e399b6b2f21fdf40573683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.868ex; height:2.509ex;" alt="{\displaystyle D_{c}}"></span></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v\rangle .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </mrow> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v\rangle .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2f616912a3d47520af31f90c5c09b4cd5d4343b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.667ex; height:4.676ex;" alt="{\displaystyle Q={\frac {\pi }{4}}D_{c}^{2}\langle v\rangle .}"></span></dd></dl> <p>Then the Darcy–Weisbach equation in terms of <span class="texhtml mvar" style="font-style:italic;">Q</span> is </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>8</mn> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>g</mi> </mrow> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e75b0678819e67dc1a0808067cf63ec9203b454b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.385ex; height:6.509ex;" alt="{\displaystyle S=f_{\text{D}}\cdot {\frac {8}{\pi ^{2}g}}\cdot {\frac {Q^{2}}{D_{c}^{5}}}.}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Shear-stress_form">Shear-stress form</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=5" title="Edit section: Shear-stress form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mean <a href="/wiki/Wall_shear_stress" class="mw-redirect" title="Wall shear stress">wall shear stress</a> <span class="texhtml mvar" style="font-style:italic;">τ</span> in a pipe or <a href="/wiki/Open_channel" class="mw-redirect" title="Open channel">open channel</a> is expressed in terms of the Darcy–Weisbach friction factor as<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}\rho {\langle v\rangle }^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>8</mn> </mfrac> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>D</mtext> </mrow> </msub> <mi>ρ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}\rho {\langle v\rangle }^{2}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf4b52969b3b414efb165c0d62bcf0554057b2ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.766ex; height:5.176ex;" alt="{\displaystyle \tau ={\frac {1}{8}}f_{\text{D}}\rho {\langle v\rangle }^{2}.}"></span></dd></dl> <p>The wall <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a> has the <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> of <a href="/wiki/Pascal_(unit)" title="Pascal (unit)">pascals</a> (Pa). </p> <div class="mw-heading mw-heading2"><h2 id="Darcy_friction_factor">Darcy friction factor</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=6" title="Edit section: Darcy friction factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg/220px-Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg.png" decoding="async" width="220" height="137" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/36/Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg/330px-Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/36/Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg/440px-Darcy_Friction_factor_for_Re_between_10_and_10E8_for_values_of_relative_roughness.svg.png 2x" data-file-width="667" data-file-height="415" /></a><figcaption><b>Figure 1.</b> The <a href="/wiki/Darcy_friction_factor" class="mw-redirect" title="Darcy friction factor">Darcy friction factor</a> versus <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> for <span class="texhtml">10 < Re < 10<sup>8</sup></span> for smooth pipe and a range of values of relative roughness <span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">ε</span><span class="sr-only">/</span><span class="den">D</span></span>⁠</span></span>. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).</figcaption></figure> <p>The friction factor <span class="texhtml"><i>f</i><sub>D</sub></span> is not a constant: it depends on such things as the characteristics of the pipe (diameter <span class="texhtml mvar" style="font-style:italic;">D</span> and roughness height <span class="texhtml mvar" style="font-style:italic;">ε</span>), the characteristics of the fluid (its kinematic viscosity <span class="texhtml mvar" style="font-style:italic;">ν</span> [nu]), and the velocity of the fluid flow <span class="texhtml">⟨<i>v</i>⟩</span>. It has been measured to <a href="/wiki/Friction_loss#Measurements" title="Friction loss">high accuracy within certain flow regimes</a> and may be evaluated by the use of various empirical relations, or it may be read from published charts. These charts are often referred to as <a href="/wiki/Moody_diagrams" class="mw-redirect" title="Moody diagrams">Moody diagrams</a>, after <a href="/wiki/Lewis_Ferry_Moody" title="Lewis Ferry Moody">L. F. Moody</a>, and hence the factor itself is sometimes erroneously called the <i>Moody friction factor</i>. It is also sometimes called the <a href="/wiki/Paul_Richard_Heinrich_Blasius" title="Paul Richard Heinrich Blasius">Blasius</a> friction factor, after the approximate formula he proposed. </p><p>Figure 1 shows the value of <span class="texhtml"><i>f</i><sub>D</sub></span> as measured by experimenters for many different fluids, over a wide range of Reynolds numbers, and for pipes of various roughness heights. There are three broad regimes of fluid flow encountered in these data: laminar, critical, and turbulent. </p> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Laminar_regime">Laminar regime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=7" title="Edit section: Laminar regime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For <a href="/wiki/Laminar_flow" title="Laminar flow">laminar (smooth) flows</a>, it is a consequence of <a href="/wiki/Hagen%E2%80%93Poiseuille_equation#Relation_to_the_Darcy–Weisbach_equation" title="Hagen–Poiseuille equation">Poiseuille's law</a> (which stems from an exact classical solution for the fluid flow) that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{\mathrm {D} }={\frac {64}{\mathrm {Re} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>64</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{\mathrm {D} }={\frac {64}{\mathrm {Re} }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd36f45b1885aae29aa55009e1489735f62292a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.951ex; height:5.343ex;" alt="{\displaystyle f_{\mathrm {D} }={\frac {64}{\mathrm {Re} }},}"></span></dd></dl> <p>where <span class="texhtml">Re</span> is the <a href="/wiki/Reynolds_number" title="Reynolds number">Reynolds number</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} ={\frac {\rho }{\mu }}\langle v\rangle D={\frac {\langle v\rangle D}{\nu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mi>μ</mi> </mfrac> </mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> <mi>D</mi> </mrow> <mi>ν</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} ={\frac {\rho }{\mu }}\langle v\rangle D={\frac {\langle v\rangle D}{\nu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d61e7f04daa0531619ab6bde1b005e0561a01c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.383ex; height:6.176ex;" alt="{\displaystyle \mathrm {Re} ={\frac {\rho }{\mu }}\langle v\rangle D={\frac {\langle v\rangle D}{\nu }},}"></span></dd></dl> <p>and where <span class="texhtml mvar" style="font-style:italic;">μ</span> is the viscosity of the fluid and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu ={\frac {\mu }{\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ν</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>ρ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nu ={\frac {\mu }{\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09895668369ce27879e379cd9f7c9cadc63e5974" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:6.568ex; height:5.343ex;" alt="{\displaystyle \nu ={\frac {\mu }{\rho }}}"></span></dd></dl> <p>is known as the <a href="/wiki/Kinematic_viscosity" class="mw-redirect" title="Kinematic viscosity">kinematic viscosity</a>. In this expression for Reynolds number, the characteristic length <span class="texhtml mvar" style="font-style:italic;">D</span> is taken to be the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">hydraulic diameter</a> of the pipe, which, for a cylindrical pipe flowing full, equals the inside diameter. In Figures 1 and 2 of friction factor versus Reynolds number, the regime <span class="texhtml">Re < 2000</span> demonstrates laminar flow; the friction factor is well represented by the above equation.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>c<span class="cite-bracket">]</span></a></sup> </p><p>In effect, the friction loss in the laminar regime is more accurately characterized as being proportional to flow velocity, rather than proportional to the square of that velocity: one could regard the Darcy–Weisbach equation as not truly applicable in the laminar flow regime. </p><p>In laminar flow, friction loss arises from the transfer of momentum from the fluid in the center of the flow to the pipe wall via the viscosity of the fluid; no vortices are present in the flow. Note that the friction loss is insensitive to the pipe roughness height <span class="texhtml mvar" style="font-style:italic;">ε</span>: the flow velocity in the neighborhood of the pipe wall is zero. </p> <div class="mw-heading mw-heading3"><h3 id="Critical_regime">Critical regime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=8" title="Edit section: Critical regime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For Reynolds numbers in the range <span class="texhtml">2000 < Re < 4000</span>, the flow is unsteady (varies grossly with time) and varies from one section of the pipe to another (is not "fully developed"). The flow involves the incipient formation of vortices; it is not well understood. </p> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Turbulent_regime">Turbulent regime</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=9" title="Edit section: Turbulent regime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg/220px-Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg.png" decoding="async" width="220" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/43/Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg/330px-Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/43/Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg/440px-Darcy_Friction_factor_for_Re_between_10E3_and_10E8_for_values_of_relative_roughness.svg.png 2x" data-file-width="667" data-file-height="400" /></a><figcaption><b>Figure 2.</b> The <a href="/wiki/Darcy_friction_factor" class="mw-redirect" title="Darcy friction factor">Darcy friction factor</a> versus Reynolds number for <span class="texhtml">1000 < Re < 10<sup>8</sup></span> for smooth pipe and a range of values of relative roughness <span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">ε</span><span class="sr-only">/</span><span class="den">D</span></span>⁠</span></span>. Data are from Nikuradse (1932, 1933), Colebrook (1939), and McKeon (2004).</figcaption></figure> <p>For Reynolds number greater than 4000, the flow is turbulent; the resistance to flow follows the Darcy–Weisbach equation: it is proportional to the square of the mean flow velocity. Over a domain of many orders of magnitude of <span class="texhtml">Re</span> (<span class="texhtml">4000 < Re < 10<sup>8</sup></span>), the friction factor varies less than one order of magnitude (<span class="texhtml">0.006 < <i>f</i><sub>D</sub> < 0.06</span>). Within the turbulent flow regime, the nature of the flow can be further divided into a regime where the pipe wall is effectively smooth, and one where its roughness height is salient. </p> <div class="mw-heading mw-heading4"><h4 id="Smooth-pipe_regime">Smooth-pipe regime</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=10" title="Edit section: Smooth-pipe regime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="clear:left;" class=""></div> <p>When the pipe surface is smooth (the "smooth pipe" curve in Figure 2), the friction factor's variation with Re can be modeled by the Kármán–Prandtl resistance equation for turbulent flow in smooth pipes<sup id="cite_ref-Rouse1946_10-1" class="reference"><a href="#cite_note-Rouse1946-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> with the parameters suitably adjusted </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=1.930\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.537.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>1.930</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>0.537.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=1.930\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.537.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e8e2bc6c2c068c1a4d3badfea146d9f1e136e17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:36.148ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=1.930\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.537.}"></span></dd></dl> <p>The numbers 1.930 and 0.537 are phenomenological; these specific values provide a fairly good fit to the data.<sup id="cite_ref-McKeon2005_16-0" class="reference"><a href="#cite_note-McKeon2005-16"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> The product <span class="texhtml">Re<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>f</i><sub>D</sub></span></span></span> (called the "friction Reynolds number") can be considered, like the Reynolds number, to be a (dimensionless) parameter of the flow: at fixed values of <span class="texhtml">Re<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>f</i><sub>D</sub></span></span></span>, the friction factor is also fixed. </p><p>In the Kármán–Prandtl resistance equation, <span class="texhtml"><i>f</i><sub>D</sub></span> can be expressed in closed form as an analytic function of <span class="texhtml">Re</span> through the use of the <a href="/wiki/Lambert_W_function" title="Lambert W function">Lambert <span class="texhtml mvar" style="font-style:italic;">W</span> function</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}={\frac {1.930}{\ln(10)}}W\left(10^{\frac {-0.537}{1.930}}{\frac {\ln(10)}{1.930}}\mathrm {Re} \right)=0.838\ W(0.629\ \mathrm {Re} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.930</mn> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi>W</mi> <mrow> <mo>(</mo> <mrow> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>0.537</mn> </mrow> <mn>1.930</mn> </mfrac> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>10</mn> <mo stretchy="false">)</mo> </mrow> <mn>1.930</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.838</mn> <mtext> </mtext> <mi>W</mi> <mo stretchy="false">(</mo> <mn>0.629</mn> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}={\frac {1.930}{\ln(10)}}W\left(10^{\frac {-0.537}{1.930}}{\frac {\ln(10)}{1.930}}\mathrm {Re} \right)=0.838\ W(0.629\ \mathrm {Re} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f074941437a2c8c3d1568598c8ae59604bafe402" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:61.004ex; height:7.009ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}={\frac {1.930}{\ln(10)}}W\left(10^{\frac {-0.537}{1.930}}{\frac {\ln(10)}{1.930}}\mathrm {Re} \right)=0.838\ W(0.629\ \mathrm {Re} )}"></span></dd></dl> <p>In this flow regime, many small vortices are responsible for the transfer of momentum between the bulk of the fluid to the pipe wall. As the friction Reynolds number <span class="texhtml">Re<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>f</i><sub>D</sub></span></span></span> increases, the profile of the fluid velocity approaches the wall asymptotically, thereby transferring more momentum to the pipe wall, as modeled in <a href="/wiki/Blasius_boundary_layer" title="Blasius boundary layer">Blasius boundary layer</a> theory. </p> <div class="mw-heading mw-heading4"><h4 id="Rough-pipe_regime">Rough-pipe regime</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=11" title="Edit section: Rough-pipe regime"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div style="clear:left;" class=""></div> <p>When the pipe surface's roughness height <span class="texhtml mvar" style="font-style:italic;">ε</span> is significant (typically at high Reynolds number), the friction factor departs from the smooth pipe curve, ultimately approaching an asymptotic value ("rough pipe" regime). In this regime, the resistance to flow varies according to the square of the mean flow velocity and is insensitive to Reynolds number. Here, it is useful to employ yet another dimensionless parameter of the flow, the <i>roughness Reynolds number</i><sup id="cite_ref-Nikuradse1933_17-0" class="reference"><a href="#cite_note-Nikuradse1933-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{*}={\frac {1}{\sqrt {8}}}\left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\,\right){\frac {\varepsilon }{D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>8</mn> </msqrt> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>D</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{*}={\frac {1}{\sqrt {8}}}\left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\,\right){\frac {\varepsilon }{D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd426700e159fa7b0701914ddc2c250868370f9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:23.596ex; height:6.176ex;" alt="{\displaystyle R_{*}={\frac {1}{\sqrt {8}}}\left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\,\right){\frac {\varepsilon }{D}}}"></span></dd></dl> <p>where the roughness height <span class="texhtml mvar" style="font-style:italic;">ε</span> is scaled to the pipe diameter <span class="texhtml mvar" style="font-style:italic;">D</span>. </p> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Figure_3_(B_vs_R*).svg" class="mw-file-description"><img alt="Roughness function B vs. friction Reynolds number R∗" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Figure_3_%28B_vs_R%2A%29.svg/220px-Figure_3_%28B_vs_R%2A%29.svg.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Figure_3_%28B_vs_R%2A%29.svg/330px-Figure_3_%28B_vs_R%2A%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Figure_3_%28B_vs_R%2A%29.svg/440px-Figure_3_%28B_vs_R%2A%29.svg.png 2x" data-file-width="577" data-file-height="363" /></a><figcaption><b>Figure 3.</b> Roughness function B vs. friction Reynolds number <span class="texhtml"><i>R</i><sub>∗</sub></span>. The data fall on a single trajectory when plotted in this way. The regime <span class="texhtml"><i>R</i><sub>∗</sub> < 1</span> is effectively that of smooth pipe flow. For large <span class="texhtml"><i>R</i><sub>∗</sub></span>, the roughness function <span class="texhtml mvar" style="font-style:italic;">B</span> approaches a constant value. Phenomenological functions attempting to fit these data, including the Afzal<sup id="cite_ref-Afzal2007_18-0" class="reference"><a href="#cite_note-Afzal2007-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> and Colebrook–White<sup id="cite_ref-Colebrook1939_19-0" class="reference"><a href="#cite_note-Colebrook1939-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> are shown.</figcaption></figure> <p>It is illustrative to plot the roughness function <span class="texhtml mvar" style="font-style:italic;">B</span>:<sup id="cite_ref-Schlichting1955_20-0" class="reference"><a href="#cite_note-Schlichting1955-20"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B(R_{*})={\frac {1}{1.930{\sqrt {f_{\mathrm {D} }}}}}+\log \left({\frac {1.90}{\sqrt {8}}}\cdot {\frac {\varepsilon }{D}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo stretchy="false">(</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1.930</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.90</mn> <msqrt> <mn>8</mn> </msqrt> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>D</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B(R_{*})={\frac {1}{1.930{\sqrt {f_{\mathrm {D} }}}}}+\log \left({\frac {1.90}{\sqrt {8}}}\cdot {\frac {\varepsilon }{D}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5bc2da7f29c8d62eaa6a3f7be4795e2c1fc8260" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.217ex; height:6.843ex;" alt="{\displaystyle B(R_{*})={\frac {1}{1.930{\sqrt {f_{\mathrm {D} }}}}}+\log \left({\frac {1.90}{\sqrt {8}}}\cdot {\frac {\varepsilon }{D}}\right)}"></span></dd></dl> <p>Figure 3 shows <span class="texhtml mvar" style="font-style:italic;">B</span> versus <span class="texhtml"><i>R</i><sub>∗</sub></span> for the rough pipe data of Nikuradse,<sup id="cite_ref-Nikuradse1933_17-1" class="reference"><a href="#cite_note-Nikuradse1933-17"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> Shockling,<sup id="cite_ref-Shockling2006_21-0" class="reference"><a href="#cite_note-Shockling2006-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> and Langelandsvik.<sup id="cite_ref-Langelandsvik2008_22-0" class="reference"><a href="#cite_note-Langelandsvik2008-22"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> </p><p>In this view, the data at different roughness ratio <span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">ε</span><span class="sr-only">/</span><span class="den">D</span></span>⁠</span></span> fall together when plotted against <span class="texhtml"><i>R</i><sub>∗</sub></span>, demonstrating scaling in the variable <span class="texhtml"><i>R</i><sub>∗</sub></span>. The following features are present: </p> <ul><li>When <span class="texhtml"><i>ε</i> = 0</span>, then <span class="texhtml"><i>R</i><sub>∗</sub></span> is identically zero: flow is always in the smooth pipe regime. The data for these points lie to the left extreme of the abscissa and are not within the frame of the graph.</li> <li>When <span class="texhtml"><i>R</i><sub>∗</sub> < 5</span>, the data lie on the line <span class="texhtml"><i>B</i>(<i>R</i><sub>∗</sub>) = <i>R</i><sub>∗</sub></span>; flow is in the smooth pipe regime.</li> <li>When <span class="texhtml"><i>R</i><sub>∗</sub> > 100</span>, the data asymptotically approach a horizontal line; they are independent of <span class="texhtml">Re</span>, <span class="texhtml"><i>f</i><sub>D</sub></span>, and <span class="texhtml mvar" style="font-style:italic;"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">ε</span><span class="sr-only">/</span><span class="den">D</span></span>⁠</span></span>.</li> <li>The intermediate range of <span class="texhtml">5 < <i>R</i><sub>∗</sub> < 100</span> constitutes a transition from one behavior to the other. The data depart from the line <span class="texhtml"><i>B</i>(<i>R</i><sub>∗</sub>) = <i>R</i><sub>∗</sub></span> very slowly, reach a maximum near <span class="texhtml"><i>R</i><sub>∗</sub> = 10</span>, then fall to a constant value.</li></ul> <p>Afzal's fit to these data in the transition from smooth pipe flow to rough pipe flow employs an exponential expression in <span class="texhtml"><i>R</i><sub>∗</sub></span> that ensures proper behavior for <span class="texhtml">1 < <i>R</i><sub>∗</sub> < 50</span> (the transition from the smooth pipe regime to the rough pipe regime):<sup id="cite_ref-Afzal2007_18-1" class="reference"><a href="#cite_note-Afzal2007-18"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Afzal2011_23-0" class="reference"><a href="#cite_note-Afzal2011-23"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Afzal2013_24-0" class="reference"><a href="#cite_note-Afzal2013-24"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.305R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2.51</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>0.305</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>11</mn> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.305R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a955a0bdfea1e54d9763f5ddeb47a992e02f80fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.295ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.305R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}"></span></dd></dl> <p>and </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.930\log \left({\frac {1.90}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.34R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>1.930</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.90</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>0.34</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <mn>11</mn> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.930\log \left({\frac {1.90}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.34R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1682a9dc90bb0f1f0b8e89b7d44d877a100c56c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.266ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.930\log \left({\frac {1.90}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+0.34R_{*}\exp {\frac {-11}{R_{*}}}\right)\right),}"></span></dd></dl> <p>This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for <span class="texhtml"><i>R</i><sub>∗</sub> → ∞</span> along with one further parameter, 11, to govern the transition from smooth to rough flow. It is exhibited in Figure 3. </p><p>The friction factor for another analogous roughness becomes </p><p><br /> </p> <pre> : <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.0\,\log _{10}\left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.305R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>2.0</mn> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2.51</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>0.305</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mrow> <mn>26</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.0\,\log _{10}\left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.305R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0397ffb6a42d8ee62d2c73e1b6cf96f95a7a0db1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.995ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.0\,\log _{10}\left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.305R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}"></span> </pre> <p>and </p> <pre>: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.93\,\log _{10}\left({\frac {1.91}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.34R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>1.93</mn> <mspace width="thinmathspace" /> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1.91</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <mn>0.34</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mspace width="thickmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mrow> <mn>26</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.93\,\log _{10}\left({\frac {1.91}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.34R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd578d2b90431102da1cfbfaf3a6f98b2a6076a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:65.995ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-1.93\,\log _{10}\left({\frac {1.91}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left\{1+0.34R_{*}\;\left(1-\exp {\frac {-R_{*}}{26}}\right)\right\}\right),}"></span> </pre> <p>This function shares the same values for its term in common with the Kármán–Prandtl resistance equation, plus one parameter 0.305 or 0.34 to fit the asymptotic behavior for <span class="texhtml"><i>R</i><sub>∗</sub> → ∞</span> along with one further parameter, 26, to govern the transition from smooth to rough flow. </p><p><br /> The Colebrook–White relation<sup id="cite_ref-Colebrook1939_19-1" class="reference"><a href="#cite_note-Colebrook1939-19"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> fits the friction factor with a function of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+{\frac {R_{*}}{3.3}}\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>2.00</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2.51</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> <mn>3.3</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+{\frac {R_{*}}{3.3}}\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cec64f3a0dabffec79fdc108a5589d3b8ffbeea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:42.664ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left({\frac {2.51}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}\left(1+{\frac {R_{*}}{3.3}}\right)\right).}"></span><sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>d<span class="cite-bracket">]</span></a></sup></dd></dl> <p>This relation has the correct behavior at extreme values of <span class="texhtml"><i>R</i><sub>∗</sub></span>, as shown by the labeled curve in Figure 3: when <span class="texhtml"><i>R</i><sub>∗</sub></span> is small, it is consistent with smooth pipe flow, when large, it is consistent with rough pipe flow. However its performance in the transitional domain overestimates the friction factor by a substantial margin.<sup id="cite_ref-Shockling2006_21-1" class="reference"><a href="#cite_note-Shockling2006-21"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> Colebrook acknowledges the discrepancy with Nikuradze's data but argues that his relation is consistent with the measurements on commercial pipes. Indeed, such pipes are very different from those carefully prepared by Nikuradse: their surfaces are characterized by many different roughness heights and random spatial distribution of roughness points, while those of Nikuradse have surfaces with uniform roughness height, with the points extremely closely packed. </p> <div style="clear:left;" class=""></div> <div class="mw-heading mw-heading3"><h3 id="Calculating_the_friction_factor_from_its_parametrization">Calculating the friction factor from its parametrization</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=12" title="Edit section: Calculating the friction factor from its parametrization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Darcy_friction_factor_formulae" title="Darcy friction factor formulae">Darcy friction factor formulae</a></div> <p>For <a href="/wiki/Turbulent_flow" class="mw-redirect" title="Turbulent flow">turbulent flow</a>, methods for finding the friction factor <span class="texhtml"><i>f</i><sub>D</sub></span> include using a diagram, such as the <a href="/wiki/Moody_chart" title="Moody chart">Moody chart</a>, or solving equations such as the <a href="/wiki/Colebrook%E2%80%93White_equation" class="mw-redirect" title="Colebrook–White equation">Colebrook–White equation</a> (upon which the Moody chart is based), or the <a href="/wiki/Swamee%E2%80%93Jain_equation" class="mw-redirect" title="Swamee–Jain equation">Swamee–Jain equation</a>. While the Colebrook–White relation is, in the general case, an iterative method, the Swamee–Jain equation allows <span class="texhtml"><i>f</i><sub>D</sub></span> to be found directly for full flow in a circular pipe.<sup id="cite_ref-Crowe2005_13-1" class="reference"><a href="#cite_note-Crowe2005-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Direct_calculation_when_friction_loss_S_is_known">Direct calculation when friction loss <i><span class="texhtml mvar" style="font-style:italic;">S</span></i> is known</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=13" title="Edit section: Direct calculation when friction loss S is known"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In typical engineering applications, there will be a set of given or known quantities. The acceleration of gravity <span class="texhtml mvar" style="font-style:italic;">g</span> and the kinematic viscosity of the fluid <span class="texhtml mvar" style="font-style:italic;">ν</span> are known, as are the diameter of the pipe <span class="texhtml mvar" style="font-style:italic;">D</span> and its roughness height <span class="texhtml mvar" style="font-style:italic;">ε</span>. If as well the head loss per unit length <span class="texhtml mvar" style="font-style:italic;">S</span> is a known quantity, then the friction factor <span class="texhtml"><i>f</i><sub>D</sub></span> can be calculated directly from the chosen fitting function. Solving the Darcy–Weisbach equation for <span class="texhtml"><span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>f</i><sub>D</sub></span></span></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {f_{\mathrm {D} }}}={\frac {\sqrt {2gSD}}{\langle v\rangle }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>2</mn> <mi>g</mi> <mi>S</mi> <mi>D</mi> </msqrt> <mrow> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {f_{\mathrm {D} }}}={\frac {\sqrt {2gSD}}{\langle v\rangle }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f32d4649619093472150b32c7160280f9697ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.911ex; height:7.009ex;" alt="{\displaystyle {\sqrt {f_{\mathrm {D} }}}={\frac {\sqrt {2gSD}}{\langle v\rangle }}}"></span></dd></dl> <p>we can now express <span class="texhtml">Re<span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;"><i>f</i><sub>D</sub></span></span></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}={\frac {1}{\nu }}{\sqrt {2g}}{\sqrt {S}}{\sqrt {D^{3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>ν</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>g</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>S</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}={\frac {1}{\nu }}{\sqrt {2g}}{\sqrt {S}}{\sqrt {D^{3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64cea67554a9cf4683cddd119b060dbdc321a3b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.2ex; height:5.176ex;" alt="{\displaystyle \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}={\frac {1}{\nu }}{\sqrt {2g}}{\sqrt {S}}{\sqrt {D^{3}}}}"></span></dd></dl> <p>Expressing the roughness Reynolds number <span class="texhtml"><i>R</i><sub>∗</sub></span>, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R_{*}&={\frac {\varepsilon }{D}}\cdot \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\cdot {\frac {1}{\sqrt {8}}}\\&={\frac {1}{2}}{\frac {\sqrt {g}}{\nu }}\varepsilon {\sqrt {S}}{\sqrt {D}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>D</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>8</mn> </msqrt> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mi>g</mi> </msqrt> <mi>ν</mi> </mfrac> </mrow> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>S</mi> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>D</mi> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{*}&={\frac {\varepsilon }{D}}\cdot \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\cdot {\frac {1}{\sqrt {8}}}\\&={\frac {1}{2}}{\frac {\sqrt {g}}{\nu }}\varepsilon {\sqrt {S}}{\sqrt {D}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fef45e5900ff6603927349efaa5d7b5bd0a5588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.22ex; margin-bottom: -0.284ex; width:24.415ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}R_{*}&={\frac {\varepsilon }{D}}\cdot \mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\cdot {\frac {1}{\sqrt {8}}}\\&={\frac {1}{2}}{\frac {\sqrt {g}}{\nu }}\varepsilon {\sqrt {S}}{\sqrt {D}}\end{aligned}}}"></span></dd></dl> <p>we have the two parameters needed to substitute into the Colebrook–White relation, or any other function, for the friction factor <span class="texhtml"><i>f</i><sub>D</sub></span>, the flow velocity <span class="texhtml">⟨<i>v</i>⟩</span>, and the volumetric flow rate <span class="texhtml mvar" style="font-style:italic;">Q</span>. </p> <div class="mw-heading mw-heading3"><h3 id="Confusion_with_the_Fanning_friction_factor">Confusion with the Fanning friction factor</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=14" title="Edit section: Confusion with the Fanning friction factor"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Darcy–Weisbach friction factor <span class="texhtml"><i>f</i><sub>D</sub></span> is 4 times larger than the <a href="/wiki/Fanning_friction_factor" title="Fanning friction factor">Fanning friction factor</a> <span class="texhtml mvar" style="font-style:italic;">f</span>, so attention must be paid to note which one of these is meant in any "friction factor" chart or equation being used. Of the two, the Darcy–Weisbach factor <span class="texhtml"><i>f</i><sub>D</sub></span> is more commonly used by civil and mechanical engineers, and the Fanning factor <span class="texhtml mvar" style="font-style:italic;">f</span> by chemical engineers, but care should be taken to identify the correct factor regardless of the source of the chart or formula. </p><p>Note that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p=f_{\mathrm {D} }\cdot {\frac {L}{D}}\cdot {\frac {\rho {\langle v\rangle }^{2}}{2}}=f\cdot {\frac {L}{D}}\cdot {2\rho {\langle v\rangle }^{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>=</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>D</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ρ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>D</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>ρ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p=f_{\mathrm {D} }\cdot {\frac {L}{D}}\cdot {\frac {\rho {\langle v\rangle }^{2}}{2}}=f\cdot {\frac {L}{D}}\cdot {2\rho {\langle v\rangle }^{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3207d563c16864c78e4a7f69d46ad0d5ae362531" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:37.83ex; height:6.009ex;" alt="{\displaystyle \Delta p=f_{\mathrm {D} }\cdot {\frac {L}{D}}\cdot {\frac {\rho {\langle v\rangle }^{2}}{2}}=f\cdot {\frac {L}{D}}\cdot {2\rho {\langle v\rangle }^{2}}}"></span></dd></dl> <p>Most charts or tables indicate the type of friction factor, or at least provide the formula for the friction factor with laminar flow. If the formula for laminar flow is <span class="texhtml"><i>f</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">16</span><span class="sr-only">/</span><span class="den">Re</span></span>⁠</span></span>, it is the Fanning factor <span class="texhtml mvar" style="font-style:italic;">f</span>, and if the formula for laminar flow is <span class="texhtml"><i>f</i><sub>D</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">64</span><span class="sr-only">/</span><span class="den">Re</span></span>⁠</span></span>, it is the Darcy–Weisbach factor <span class="texhtml"><i>f</i><sub>D</sub></span>. </p><p>Which friction factor is plotted in a Moody diagram may be determined by inspection if the publisher did not include the formula described above: </p> <ol><li>Observe the value of the friction factor for laminar flow at a Reynolds number of 1000.</li> <li>If the value of the friction factor is 0.064, then the Darcy friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.064 are the numerator in the formula for the laminar Darcy friction factor: <span class="texhtml"><i>f</i><sub>D</sub> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">64</span><span class="sr-only">/</span><span class="den">Re</span></span>⁠</span></span>.</li> <li>If the value of the friction factor is 0.016, then the Fanning friction factor is plotted in the Moody diagram. Note that the nonzero digits in 0.016 are the numerator in the formula for the laminar Fanning friction factor: <span class="texhtml"><i>f</i> = <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">16</span><span class="sr-only">/</span><span class="den">Re</span></span>⁠</span></span>.</li></ol> <p>The procedure above is similar for any available Reynolds number that is an integer power of ten. It is not necessary to remember the value 1000 for this procedure—only that an integer power of ten is of interest for this purpose. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=15" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Historically this equation arose as a variant on the <a href="/wiki/Prony_equation" title="Prony equation">Prony equation</a>; this variant was developed by <a href="/wiki/Henry_Darcy" title="Henry Darcy">Henry Darcy</a> of France, and further refined into the form used today by <a href="/wiki/Julius_Weisbach" title="Julius Weisbach">Julius Weisbach</a> of <a href="/wiki/Saxony" title="Saxony">Saxony</a> in 1845. Initially, data on the variation of <span class="texhtml"><i>f</i><sub>D</sub></span> with velocity was lacking, so the Darcy–Weisbach equation was outperformed at first by the empirical Prony equation in many cases. In later years it was eschewed in many special-case situations in favor of a variety of <a href="/wiki/Empirical_equation" class="mw-redirect" title="Empirical equation">empirical equations</a> valid only for certain flow regimes, notably the <a href="/wiki/Hazen%E2%80%93Williams_equation" title="Hazen–Williams equation">Hazen–Williams equation</a> or the <a href="/wiki/Manning_equation" class="mw-redirect" title="Manning equation">Manning equation</a>, most of which were significantly easier to use in calculations. However, since the advent of the <a href="/wiki/Calculator" title="Calculator">calculator</a>, ease of calculation is no longer a major issue, and so the Darcy–Weisbach equation's generality has made it the preferred one.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Derivation_by_dimensional_analysis">Derivation by dimensional analysis</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=16" title="Edit section: Derivation by dimensional analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Away from the ends of the pipe, the characteristics of the flow are independent of the position along the pipe. The key quantities are then the pressure drop along the pipe per unit length, <span class="texhtml"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1214402035"><span class="sfrac">⁠<span class="tion"><span class="num">Δ<i>p</i></span><span class="sr-only">/</span><span class="den"><i>L</i></span></span>⁠</span></span>, and the volumetric flow rate. The flow rate can be converted to a mean flow velocity <span class="texhtml mvar" style="font-style:italic;">V</span> by dividing by the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">wetted area</a> of the flow (which equals the <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-sectional</a> <a href="/wiki/Area" title="Area">area</a> of the pipe if the pipe is full of fluid). </p><p>Pressure has dimensions of energy per unit volume, therefore the pressure drop between two points must be proportional to the <a href="/wiki/Dynamic_pressure" title="Dynamic pressure">dynamic pressure</a> q. We also know that pressure must be proportional to the length of the pipe between the two points <span class="texhtml mvar" style="font-style:italic;">L</span> as the pressure drop per unit length is a constant. To turn the relationship into a proportionality coefficient of dimensionless quantity, we can divide by the hydraulic diameter of the pipe, <span class="texhtml mvar" style="font-style:italic;">D</span>, which is also constant along the pipe. Therefore, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p\propto {\frac {L}{D}}q={\frac {L}{D}}\cdot {\frac {\rho }{2}}\cdot {\langle v\rangle }^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>D</mi> </mfrac> </mrow> <mi>q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>L</mi> <mi>D</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mn>2</mn> </mfrac> </mrow> <mo>⋅</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⟨</mo> <mi>v</mi> <mo fence="false" stretchy="false">⟩</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta p\propto {\frac {L}{D}}q={\frac {L}{D}}\cdot {\frac {\rho }{2}}\cdot {\langle v\rangle }^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de5b2803e10852362131eb9654e34367ab953b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:25.28ex; height:5.176ex;" alt="{\displaystyle \Delta p\propto {\frac {L}{D}}q={\frac {L}{D}}\cdot {\frac {\rho }{2}}\cdot {\langle v\rangle }^{2}}"></span></dd></dl> <p>The proportionality coefficient is the dimensionless "<a href="/wiki/Darcy_friction_factor_formulae" title="Darcy friction factor formulae">Darcy friction factor</a>" or "flow coefficient". This dimensionless coefficient will be a combination of geometric factors such as <span class="texhtml mvar" style="font-style:italic;">π</span>, the Reynolds number and (outside the laminar regime) the relative roughness of the pipe (the ratio of the <a href="/w/index.php?title=Roughness_height&action=edit&redlink=1" class="new" title="Roughness height (page does not exist)">roughness height</a> to the <a href="/wiki/Hydraulic_diameter" title="Hydraulic diameter">hydraulic diameter</a>). </p><p>Note that the dynamic pressure is not the kinetic energy of the fluid per unit volume,<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2016)">citation needed</span></a></i>]</sup> for the following reasons. Even in the case of <a href="/wiki/Laminar_flow" title="Laminar flow">laminar flow</a>, where all the <a href="/wiki/Streamlines,_streaklines,_and_pathlines" title="Streamlines, streaklines, and pathlines">flow lines</a> are parallel to the length of the pipe, the velocity of the fluid on the inner surface of the pipe is zero due to viscosity, and the velocity in the center of the pipe must therefore be larger than the average velocity obtained by dividing the volumetric flow rate by the wet area. The average kinetic energy then involves the <a href="/wiki/Root-mean-square_speed" class="mw-redirect" title="Root-mean-square speed">root mean-square velocity</a>, which always exceeds the mean velocity. In the case of <a href="/wiki/Turbulent_flow" class="mw-redirect" title="Turbulent flow">turbulent flow</a>, the fluid acquires random velocity components in all directions, including perpendicular to the length of the pipe, and thus turbulence contributes to the kinetic energy per unit volume but not to the average lengthwise velocity of the fluid. </p> <div class="mw-heading mw-heading2"><h2 id="Practical_application">Practical application</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=17" title="Edit section: Practical application"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In a <a href="/wiki/Hydraulic_engineering" title="Hydraulic engineering">hydraulic engineering</a> application, it is typical for the volumetric flow <span class="texhtml mvar" style="font-style:italic;">Q</span> within a pipe (that is, its productivity) and the head loss per unit length <span class="texhtml mvar" style="font-style:italic;">S</span> (the concomitant power consumption) to be the critical important factors. The practical consequence is that, for a fixed volumetric flow rate <span class="texhtml mvar" style="font-style:italic;">Q</span>, head loss <span class="texhtml mvar" style="font-style:italic;">S</span> <i>decreases</i> with the inverse fifth power of the pipe diameter, <span class="texhtml mvar" style="font-style:italic;">D</span>. Doubling the diameter of a pipe of a given schedule (say, ANSI schedule 40) roughly doubles the amount of material required per unit length and thus its installed cost. Meanwhile, the head loss is decreased by a factor of 32 (about a 97% reduction). Thus the energy consumed in moving a given volumetric flow of the fluid is cut down dramatically for a modest increase in capital cost. </p> <div class="mw-heading mw-heading2"><h2 id="Advantages">Advantages</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=18" title="Edit section: Advantages"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Darcy-Weisbach's accuracy and universal applicability makes it the ideal formula for flow in pipes. The advantages of the equation are as follows:<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ul><li>It is based on fundamentals.</li> <li>It is dimensionally consistent.</li> <li>It is useful for any fluid, including oil, gas, brine, and sludges.</li> <li>It can be derived analytically in the laminar flow region.</li> <li>It is useful in the transition region between laminar flow and fully developed turbulent flow.</li> <li>The friction factor variation is well documented.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=19" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli's principle</a></li> <li><a href="/wiki/Darcy_friction_factor_formulae" title="Darcy friction factor formulae">Darcy friction factor formulae</a></li> <li><a href="/wiki/Euler_number_(physics)" title="Euler number (physics)">Euler number</a></li> <li><a href="/wiki/Friction_loss" title="Friction loss">Friction loss</a></li> <li><a href="/wiki/Hazen%E2%80%93Williams_equation" title="Hazen–Williams equation">Hazen–Williams equation</a></li> <li><a href="/wiki/Hagen%E2%80%93Poiseuille_equation" title="Hagen–Poiseuille equation">Hagen–Poiseuille equation</a></li> <li><a href="/wiki/Water_pipe" class="mw-redirect" title="Water pipe">Water pipe</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=20" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">The value of the Darcy friction factor is four times that of the <a href="/wiki/Fanning_friction_factor" title="Fanning friction factor">Fanning friction factor</a>, with which it should not be confused.<sup id="cite_ref-Manning1991_2-0" class="reference"><a href="#cite_note-Manning1991-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">This is related to the <a href="/wiki/Hydraulic_head" title="Hydraulic head">piezometric head</a> along the pipe.</span> </li> <li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text">The data exhibit, however, a systematic departure of up to 50% from the theoretical Hagen–Poiseuille equation in the region of <span class="texhtml">Re > 500</span> up to the onset of critical flow.</span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text">In its originally published form, <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left(2.51{\frac {1}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}+{\frac {1}{3.7}}{\frac {\varepsilon }{D}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mn>2.00</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2.51</mn> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3.7</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ε</mi> <mi>D</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left(2.51{\frac {1}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}+{\frac {1}{3.7}}{\frac {\varepsilon }{D}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a77ac4fad2ba23be2e0c0a9fceb82b4a2f545ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:43.941ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=-2.00\log \left(2.51{\frac {1}{\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}}}+{\frac {1}{3.7}}{\frac {\varepsilon }{D}}\right)}"></span></dd></dl> </span></li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=21" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJones,_Garr_M.2006" class="citation book cs1">Jones, Garr M., ed. (2006). <i>Pumping station design</i> (3rd ed.). Burlington, MA: Butterworth-Heinemann. p. 3.5. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-094106-6" title="Special:BookSources/978-0-08-094106-6"><bdi>978-0-08-094106-6</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/144609617">144609617</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Pumping+station+design&rft.place=Burlington%2C+MA&rft.pages=3.5&rft.edition=3rd&rft.pub=Butterworth-Heinemann&rft.date=2006&rft_id=info%3Aoclcnum%2F144609617&rft.isbn=978-0-08-094106-6&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Manning1991-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-Manning1991_2-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFManningThompson1991" class="citation book cs1">Manning, Francis S.; Thompson, Richard E. (1991). <i>Oilfield Processing of Petroleum. Vol. 1: Natural Gas</i>. PennWell Books. p. 293. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87814-343-2" title="Special:BookSources/0-87814-343-2"><bdi>0-87814-343-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Oilfield+Processing+of+Petroleum.+Vol.+1%3A+Natural+Gas&rft.pages=293&rft.pub=PennWell+Books&rft.date=1991&rft.isbn=0-87814-343-2&rft.aulast=Manning&rft.aufirst=Francis+S.&rft.au=Thompson%2C+Richard+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrown2002">Brown 2002</a>, p. 35-36</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrown2002">Brown 2002</a>, p. 36-37</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrown2002">Brown 2002</a>, p. 35-36</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrown2002">Brown 2002</a>, p. 37</span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><a href="#CITEREFBrown2002">Brown 2002</a>, p. 39</span> </li> <li id="cite_note-Brown-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brown_9-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHowell1970" class="citation book cs1">Howell, Glen (1970-02-01). "3.9.2". <a rel="nofollow" class="external text" href="https://apps.dtic.mil/sti/citations/AD0874542"><i>Aerospace Fluid Component Designers' Handbook</i></a>. Vol. I. Redondo Beach CA: TRW Systems Group. p. 87, equation 3.9.2.1e. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20201020012129/https://apps.dtic.mil/sti/citations/AD0874542">Archived</a> from the original on October 20, 2020 – via Defense Technical Information Center.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=3.9.2&rft.btitle=Aerospace+Fluid+Component+Designers%27+Handbook&rft.place=Redondo+Beach+CA&rft.pages=p.-87%2C+equation+3.9.2.1e&rft.pub=TRW+Systems+Group&rft.date=1970-02-01&rft.aulast=Howell&rft.aufirst=Glen&rft_id=https%3A%2F%2Fapps.dtic.mil%2Fsti%2Fcitations%2FAD0874542&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Rouse1946-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Rouse1946_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Rouse1946_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRouse1946" class="citation book cs1">Rouse, H. (1946). <a rel="nofollow" class="external text" href="https://archive.org/details/in.ernet.dli.2015.140493"><i>Elementary Mechanics of Fluids</i></a>. John Wiley & Sons.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elementary+Mechanics+of+Fluids&rft.pub=John+Wiley+%26+Sons&rft.date=1946&rft.aulast=Rouse&rft.aufirst=H.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fin.ernet.dli.2015.140493&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Incopera2002-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-Incopera2002_11-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFIncoperaDewitt2002" class="citation book cs1">Incopera, Frank P.; Dewitt, David P. (2002). <i>Fundamentals of Heat and Mass Transfer</i> (5th ed.). John Wiley & Sons. p. 470 paragraph 3.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Heat+and+Mass+Transfer&rft.pages=470+paragraph+3&rft.edition=5th&rft.pub=John+Wiley+%26+Sons&rft.date=2002&rft.aulast=Incopera&rft.aufirst=Frank+P.&rft.au=Dewitt%2C+David+P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Crowe2005-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Crowe2005_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Crowe2005_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCroweElgerRobertson2005" class="citation book cs1">Crowe, Clayton T.; Elger, Donald F.; Robertson, John A. (2005). <i>Engineering Fluid Mechanics</i> (8th ed.). John Wiley & Sons. p. 379; Eq. 10:23, 10:24, paragraph 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Engineering+Fluid+Mechanics&rft.pages=p.+379%3B+Eq.+10%3A23%2C+10%3A24%2C+paragraph+4&rft.edition=8th&rft.pub=John+Wiley+%26+Sons&rft.date=2005&rft.aulast=Crowe&rft.aufirst=Clayton+T.&rft.au=Elger%2C+Donald+F.&rft.au=Robertson%2C+John+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=2\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.8\quad {\text{for }}\mathrm {Re} >3000.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mfrac> </mrow> <mo>=</mo> <mn>2</mn> <mi>log</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">D</mi> </mrow> </mrow> </msub> </msqrt> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>0.8</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">R</mi> <mi mathvariant="normal">e</mi> </mrow> <mo>></mo> <mn>3000.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=2\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.8\quad {\text{for }}\mathrm {Re} >3000.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c356b9f4bfa918f08482f68f962fa7433481fec0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.87ex; height:6.509ex;" alt="{\displaystyle {\frac {1}{\sqrt {f_{\mathrm {D} }}}}=2\log \left(\mathrm {Re} {\sqrt {f_{\mathrm {D} }}}\right)-0.8\quad {\text{for }}\mathrm {Re} >3000.}"></span></dd></dl> </span></li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChaudhry2013" class="citation book cs1">Chaudhry, M. H. (2013). <i>Applied Hydraulic Transients</i> (3rd ed.). Springer. p. 45. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4614-8538-4" title="Special:BookSources/978-1-4614-8538-4"><bdi>978-1-4614-8538-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applied+Hydraulic+Transients&rft.pages=45&rft.edition=3rd&rft.pub=Springer&rft.date=2013&rft.isbn=978-1-4614-8538-4&rft.aulast=Chaudhry&rft.aufirst=M.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-McKeon2005-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-McKeon2005_16-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcKeonZagarolaSmits2005" class="citation journal cs1"><a href="/wiki/Beverley_McKeon" title="Beverley McKeon">McKeon, B. J.</a>; Zagarola, M. V; Smits, A. J. (2005). <a rel="nofollow" class="external text" href="http://authors.library.caltech.edu/4467/1/MCKEjfm05.pdf">"A new friction factor relationship for fully developed pipe flow"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Fluid Mechanics</i>. <b>538</b>. Cambridge University Press: 429–443. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005JFM...538..429M">2005JFM...538..429M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112005005501">10.1017/S0022112005005501</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15642454">15642454</a><span class="reference-accessdate">. Retrieved <span class="nowrap">25 June</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=A+new+friction+factor+relationship+for+fully+developed+pipe+flow&rft.volume=538&rft.pages=429-443&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A15642454%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112005005501&rft_id=info%3Abibcode%2F2005JFM...538..429M&rft.aulast=McKeon&rft.aufirst=B.+J.&rft.au=Zagarola%2C+M.+V&rft.au=Smits%2C+A.+J.&rft_id=http%3A%2F%2Fauthors.library.caltech.edu%2F4467%2F1%2FMCKEjfm05.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Nikuradse1933-17"><span class="mw-cite-backlink">^ <a href="#cite_ref-Nikuradse1933_17-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Nikuradse1933_17-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNikuradse1933" class="citation journal cs1">Nikuradse, J. (1933). <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19930093938.pdf">"Strömungsgesetze in rauen Rohren"</a> <span class="cs1-format">(PDF)</span>. <i>V. D. I. Forschungsheft</i>. <b>361</b>. Berlin: 1–22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=V.+D.+I.+Forschungsheft&rft.atitle=Str%C3%B6mungsgesetze+in+rauen+Rohren&rft.volume=361&rft.pages=1-22&rft.date=1933&rft.aulast=Nikuradse&rft.aufirst=J.&rft_id=https%3A%2F%2Fntrs.nasa.gov%2Farchive%2Fnasa%2Fcasi.ntrs.nasa.gov%2F19930093938.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span> In translation, NACA TM 1292. The data are available in <a rel="nofollow" class="external text" href="https://smits.princeton.edu/files/2016/03/Nik-u-vs-y-data.xls">digital form</a><sup class="noprint Inline-Template"><span style="white-space: nowrap;">[<i><a href="/wiki/Wikipedia:Link_rot" title="Wikipedia:Link rot"><span title=" Dead link tagged March 2023">permanent dead link</span></a></i><span style="visibility:hidden; color:transparent; padding-left:2px">‍</span>]</span></sup>.</span> </li> <li id="cite_note-Afzal2007-18"><span class="mw-cite-backlink">^ <a href="#cite_ref-Afzal2007_18-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Afzal2007_18-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAfzal2007" class="citation journal cs1">Afzal, Noor (2007). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/245357256">"Friction Factor Directly From Transitional Roughness in a Turbulent Pipe Flow"</a>. <i>Journal of Fluids Engineering</i>. <b>129</b> (10). ASME: 1255–1267. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.2776961">10.1115/1.2776961</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluids+Engineering&rft.atitle=Friction+Factor+Directly+From+Transitional+Roughness+in+a+Turbulent+Pipe+Flow&rft.volume=129&rft.issue=10&rft.pages=1255-1267&rft.date=2007&rft_id=info%3Adoi%2F10.1115%2F1.2776961&rft.aulast=Afzal&rft.aufirst=Noor&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F245357256&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Colebrook1939-19"><span class="mw-cite-backlink">^ <a href="#cite_ref-Colebrook1939_19-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Colebrook1939_19-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFColebrook1939" class="citation journal cs1">Colebrook, C. F. (February 1939). "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws". <i>Journal of the Institution of Civil Engineers</i>. London. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1680%2Fijoti.1939.14509">10.1680/ijoti.1939.14509</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+the+Institution+of+Civil+Engineers&rft.atitle=Turbulent+flow+in+pipes%2C+with+particular+reference+to+the+transition+region+between+smooth+and+rough+pipe+laws&rft.date=1939-02&rft_id=info%3Adoi%2F10.1680%2Fijoti.1939.14509&rft.aulast=Colebrook&rft.aufirst=C.+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Schlichting1955-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schlichting1955_20-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchlichting1955" class="citation book cs1">Schlichting, H. (1955). <i>Boundary Layer Theory</i>. McGraw-Hill.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Boundary+Layer+Theory&rft.pub=McGraw-Hill&rft.date=1955&rft.aulast=Schlichting&rft.aufirst=H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Shockling2006-21"><span class="mw-cite-backlink">^ <a href="#cite_ref-Shockling2006_21-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Shockling2006_21-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShocklingAllenSmits2006" class="citation journal cs1">Shockling, M. A.; Allen, J. J.; Smits, A. J. (2006). "Roughness effects in turbulent pipe flow". <i>Journal of Fluid Mechanics</i>. <b>564</b>: 267–285. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JFM...564..267S">2006JFM...564..267S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112006001467">10.1017/S0022112006001467</a> (inactive 2024-11-21). <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120958504">120958504</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Roughness+effects+in+turbulent+pipe+flow&rft.volume=564&rft.pages=267-285&rft.date=2006&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120958504%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112006001467&rft_id=info%3Abibcode%2F2006JFM...564..267S&rft.aulast=Shockling&rft.aufirst=M.+A.&rft.au=Allen%2C+J.+J.&rft.au=Smits%2C+A.+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_journal" title="Template:Cite journal">cite journal</a>}}</code>: CS1 maint: DOI inactive as of November 2024 (<a href="/wiki/Category:CS1_maint:_DOI_inactive_as_of_November_2024" title="Category:CS1 maint: DOI inactive as of November 2024">link</a>)</span></span> </li> <li id="cite_note-Langelandsvik2008-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-Langelandsvik2008_22-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLangelandsvikKunkelSmits2008" class="citation journal cs1">Langelandsvik, L. I.; Kunkel, G. J.; Smits, A. J. (2008). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160816195857/http://www.electronicsandbooks.com/eab1/manual/Magazine/J/Journal%20of%20Fluid%20Mechanics/2008%20Volume%20595/S0022112007009305.pdf">"Flow in a commercial steel pipe"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Fluid Mechanics</i>. <b>595</b>. Cambridge University Press: 323–339. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JFM...595..323L">2008JFM...595..323L</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112007009305">10.1017/S0022112007009305</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:59433444">59433444</a>. Archived from <a rel="nofollow" class="external text" href="http://www.electronicsandbooks.com/eab1/manual/Magazine/J/Journal%20of%20Fluid%20Mechanics/2008%20Volume%20595/S0022112007009305.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 16 August 2016<span class="reference-accessdate">. Retrieved <span class="nowrap">25 June</span> 2016</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Flow+in+a+commercial+steel+pipe&rft.volume=595&rft.pages=323-339&rft.date=2008&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A59433444%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112007009305&rft_id=info%3Abibcode%2F2008JFM...595..323L&rft.aulast=Langelandsvik&rft.aufirst=L.+I.&rft.au=Kunkel%2C+G.+J.&rft.au=Smits%2C+A.+J.&rft_id=http%3A%2F%2Fwww.electronicsandbooks.com%2Feab1%2Fmanual%2FMagazine%2FJ%2FJournal%2520of%2520Fluid%2520Mechanics%2F2008%2520Volume%2520595%2FS0022112007009305.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Afzal2011-23"><span class="mw-cite-backlink"><b><a href="#cite_ref-Afzal2011_23-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAfzal2011" class="citation journal cs1">Afzal, Noor (2011). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/245357256">"Erratum: Friction factor directly from transitional roughness in a turbulent pipe flow"</a>. <i>Journal of Fluids Engineering</i>. <b>133</b> (10). ASME: 107001. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1115%2F1.4004961">10.1115/1.4004961</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluids+Engineering&rft.atitle=Erratum%3A+Friction+factor+directly+from+transitional+roughness+in+a+turbulent+pipe+flow&rft.volume=133&rft.issue=10&rft.pages=107001&rft.date=2011&rft_id=info%3Adoi%2F10.1115%2F1.4004961&rft.aulast=Afzal&rft.aufirst=Noor&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F245357256&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-Afzal2013-24"><span class="mw-cite-backlink"><b><a href="#cite_ref-Afzal2013_24-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAfzalSeenaBushra2013" class="citation journal cs1">Afzal, Noor; Seena, Abu; Bushra, A. (2013). <a rel="nofollow" class="external text" href="https://www.academia.edu/25379798">"Turbulent flow in a machine honed rough pipe for large Reynolds numbers: General roughness scaling laws"</a>. <i>Journal of Hydro-environment Research</i>. <b>7</b> (1). Elsevier: 81–90. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jher.2011.08.002">10.1016/j.jher.2011.08.002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Hydro-environment+Research&rft.atitle=Turbulent+flow+in+a+machine+honed+rough+pipe+for+large+Reynolds+numbers%3A+General+roughness+scaling+laws&rft.volume=7&rft.issue=1&rft.pages=81-90&rft.date=2013&rft_id=info%3Adoi%2F10.1016%2Fj.jher.2011.08.002&rft.aulast=Afzal&rft.aufirst=Noor&rft.au=Seena%2C+Abu&rft.au=Bushra%2C+A.&rft_id=https%3A%2F%2Fwww.academia.edu%2F25379798&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2003" class="citation book cs1">Brown, G. O. (2003). <a rel="nofollow" class="external text" href="https://jeplerts.wordpress.com/2008/12/21/henry-darcy-and-his-law-the-history-of-the-darcy-weisbach-equation/">"The History of the Darcy-Weisbach Equation for Pipe Flow Resistance"</a>. In Rogers, J. R.; Fredrich, A. J. (eds.). <a rel="nofollow" class="external text" href="http://ascelibrary.org/doi/abs/10.1061/40650%282003%294"><i>Environmental and Water Resources History</i></a>. American Society of Civil Engineers. pp. 34–43. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2F40650%282003%294">10.1061/40650(2003)4</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7844-0650-2" title="Special:BookSources/978-0-7844-0650-2"><bdi>978-0-7844-0650-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=The+History+of+the+Darcy-Weisbach+Equation+for+Pipe+Flow+Resistance&rft.btitle=Environmental+and+Water+Resources+History&rft.pages=34-43&rft.pub=American+Society+of+Civil+Engineers&rft.date=2003&rft_id=info%3Adoi%2F10.1061%2F40650%282003%294&rft.isbn=978-0-7844-0650-2&rft.aulast=Brown&rft.aufirst=G.+O.&rft_id=https%3A%2F%2Fjeplerts.wordpress.com%2F2008%2F12%2F21%2Fhenry-darcy-and-his-law-the-history-of-the-darcy-weisbach-equation%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></span> </li> </ol></div> <p>18. Afzal, Noor (2013) "Roughness effects of commercial steel pipe in turbulent flow: Universal scaling". Canadian Journal of Civil Engineering 40, 188-193. </p> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=22" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe_Nevers1970" class="citation book cs1">De Nevers (1970). <i>Fluid Mechanics</i>. Addison–Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-01497-1" title="Special:BookSources/0-201-01497-1"><bdi>0-201-01497-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fluid+Mechanics&rft.pub=Addison%E2%80%93Wesley&rft.date=1970&rft.isbn=0-201-01497-1&rft.au=De+Nevers&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShahLondon1978" class="citation book cs1">Shah, R. K.; London, A. L. (1978). "Laminar Flow Forced Convection in Ducts". <i>Supplement 1 to Advances in Heat Transfer</i>. New York: Academic.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Laminar+Flow+Forced+Convection+in+Ducts&rft.btitle=Supplement+1+to+Advances+in+Heat+Transfer&rft.place=New+York&rft.pub=Academic&rft.date=1978&rft.aulast=Shah&rft.aufirst=R.+K.&rft.au=London%2C+A.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRohsenhowHartnettGanić1985" class="citation book cs1">Rohsenhow, W. M.; Hartnett, J. P.; Ganić, E. N. (1985). <i>Handbook of Heat Transfer Fundamentals</i> (2nd ed.). McGraw–Hill Book Company. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-07-053554-X" title="Special:BookSources/0-07-053554-X"><bdi>0-07-053554-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Heat+Transfer+Fundamentals&rft.edition=2nd&rft.pub=McGraw%E2%80%93Hill+Book+Company&rft.date=1985&rft.isbn=0-07-053554-X&rft.aulast=Rohsenhow&rft.aufirst=W.+M.&rft.au=Hartnett%2C+J.+P.&rft.au=Gani%C4%87%2C+E.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrown2002" class="citation journal cs1">Glenn O. Brown (2002). <a rel="nofollow" class="external text" href="https://www.researchgate.net/publication/242138088">"The History of the Darcy-Weisbach Equation for Pipe Flow Resistance"</a>. <i>researchgate.net</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=researchgate.net&rft.atitle=The+History+of+the+Darcy-Weisbach+Equation+for+Pipe+Flow+Resistance&rft.date=2002&rft.au=Glenn+O.+Brown&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F242138088&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADarcy%E2%80%93Weisbach+equation" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Darcy%E2%80%93Weisbach_equation&action=edit&section=23" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm">The History of the Darcy–Weisbach Equation</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110720021255/http://biosystems.okstate.edu/darcy/DarcyWeisbach/Darcy-WeisbachHistory.htm">Archived</a> 2011-07-20 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://www.fxsolver.com/browse/formulas/Darcy+Weisbach+equation+%28head+loss%29">Darcy–Weisbach equation calculator</a></li> <li><a rel="nofollow" class="external text" href="http://www.enggcyclopedia.com/welcome-to-enggcyclopedia/fluid-dynamics/line-sizing-calculator">Pipe pressure drop calculator</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190713054058/http://www.enggcyclopedia.com/welcome-to-enggcyclopedia/fluid-dynamics/line-sizing-calculator">Archived</a> 2019-07-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a> for single phase flows.</li> <li><a rel="nofollow" class="external text" href="http://www.enggcyclopedia.com/welcome-to-enggcyclopedia/fluid-dynamics/pipe-pressure-drop-calculator-phase">Pipe pressure drop calculator for two phase flows.</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190713054058/http://www.enggcyclopedia.com/welcome-to-enggcyclopedia/fluid-dynamics/pipe-pressure-drop-calculator-phase">Archived</a> 2019-07-13 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a></li> <li><a rel="nofollow" class="external text" href="http://pfcalc.sourceforge.net">Open source pipe pressure drop calculator.</a></li> <li><a rel="nofollow" class="external text" href="http://www.sizepipe.com">Web application with pressure drop calculations for pipes and ducts</a></li> <li><a rel="nofollow" class="external text" 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Darcy–Weisbach equation - Wikipedia