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class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Relationship with mass flow rate</span> </div> </a> <ul id="toc-Relationship_with_mass_flow_rate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_quantities" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_quantities"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related quantities</span> </div> </a> <ul id="toc-Related_quantities-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Some_key_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Some_key_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Some key examples</span> </div> </a> <ul id="toc-Some_key_examples-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">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-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">7</span> <span>References</span> </div> </a> <ul id="toc-References-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" 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href="https://az.wikipedia.org/wiki/H%C9%99cmli_ax%C4%B1n%C4%B1_%C3%B6l%C3%A7%C9%99n" title="Həcmli axını ölçən – Azerbaijani" lang="az" hreflang="az" data-title="Həcmli axını ölçən" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9E%D0%B1%D0%B5%D0%BC%D0%B5%D0%BD_%D1%80%D0%B0%D0%B7%D1%85%D0%BE%D0%B4" title="Обемен разход – Bulgarian" lang="bg" hreflang="bg" data-title="Обемен разход" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Zapreminski_protok" title="Zapreminski protok – Bosnian" lang="bs" hreflang="bs" data-title="Zapreminski protok" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Objemov%C3%BD_pr%C5%AFtok" title="Objemový průtok – Czech" lang="cs" hreflang="cs" data-title="Objemový průtok" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Volumenstrom" title="Volumenstrom – German" lang="de" hreflang="de" data-title="Volumenstrom" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B1%CF%81%CE%BF%CF%87%CE%AE" title="Παροχή – Greek" lang="el" hreflang="el" data-title="Παροχή" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Caudal_volum%C3%A9trico" title="Caudal volumétrico – Spanish" lang="es" hreflang="es" data-title="Caudal volumétrico" 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/%D8%AF%D8%A8%DB%8C_%D8%AD%D8%AC%D9%85%DB%8C" title="دبی حجمی – Persian" lang="fa" hreflang="fa" data-title="دبی حجمی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr badge-Q70894304 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/D%C3%A9bit_volumique" title="Débit volumique – French" lang="fr" hreflang="fr" data-title="Débit volumique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Volumni_protok" title="Volumni protok – Croatian" lang="hr" hreflang="hr" data-title="Volumni protok" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Laju_alir_volumetrik" title="Laju alir volumetrik – Indonesian" lang="id" hreflang="id" data-title="Laju alir volumetrik" 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 badge-Q70894304 mw-list-item" title=""><a href="https://it.wikipedia.org/wiki/Portata_volumetrica" title="Portata volumetrica – Italian" lang="it" hreflang="it" data-title="Portata volumetrica" 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%A1%D7%A4%D7%99%D7%A7%D7%94" title="ספיקה – Hebrew" lang="he" hreflang="he" data-title="ספיקה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%A9rfogat%C3%A1ram" title="Térfogatáram – Hungarian" lang="hu" hreflang="hu" data-title="Térfogatáram" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Debiet" title="Debiet – Dutch" lang="nl" hreflang="nl" data-title="Debiet" 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/%E6%B5%81%E9%87%8F" title="流量 – Japanese" lang="ja" hreflang="ja" data-title="流量" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Strumie%C5%84_obj%C4%99to%C5%9Bci" title="Strumień objętości – Polish" lang="pl" hreflang="pl" data-title="Strumień objętości" 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/Fluxo_volum%C3%A9trico" title="Fluxo volumétrico – Portuguese" lang="pt" hreflang="pt" data-title="Fluxo volumétrico" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Debit_volumic_de_curgere" title="Debit volumic de curgere – Romanian" lang="ro" hreflang="ro" data-title="Debit volumic de curgere" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9E%D0%B1%D1%8A%D1%91%D0%BC%D0%BD%D1%8B%D0%B9_%D1%80%D0%B0%D1%81%D1%85%D0%BE%D0%B4" title="Объёмный расход – Russian" lang="ru" hreflang="ru" data-title="Объёмный расход" 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 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output 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src="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/40px-Ambox_important.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/60px-Ambox_important.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/b/b4/Ambox_important.svg/80px-Ambox_important.svg.png 2x" data-file-width="40" data-file-height="40" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs attention from an expert in Physics or Engineering</b>. The specific problem is: <b>Lacks treatment of compressible flow.</b> See the <a href="/wiki/Talk:Volumetric_flow_rate#Compressible_flow?_Surely_there's_no_such_thing." title="Talk:Volumetric flow rate">talk page</a> for details.<span class="hide-when-compact"> <a href="/wiki/Wikipedia:WikiProject_Physics" title="Wikipedia:WikiProject Physics">WikiProject Physics</a> or <a href="/wiki/Wikipedia:WikiProject_Engineering" title="Wikipedia:WikiProject Engineering">WikiProject Engineering</a> may be able to help recruit an expert.</span> <span class="date-container"><i>(<span class="date">May 2023</span>)</i></span></div></td></tr></tbody></table> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Volume of fluid which passes per unit time</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">Not to be confused with <a href="/wiki/Volumetric_flux" title="Volumetric flux">Volumetric flux</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above">Volume flow rate</th></tr><tr><th scope="row" class="infobox-label"><div style="display: inline-block; line-height: 1.2em; padding: .1em 0;">Common symbols</div></th><td class="infobox-data"><span class="texhtml mvar" style="font-style:italic;">Q</span>, <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 {\dot {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49ef9fa9f410331b94e4578bab90e9edda5c919b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.676ex;" alt="{\displaystyle {\dot {V}}}"></span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a></th><td class="infobox-data">m<sup>3</sup>/s</td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Dimensional_analysis#Formulation" title="Dimensional analysis">Dimension</a></th><td class="infobox-data"><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 {\mathsf {L}}^{3}{\mathsf {T}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">L</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathsf {L}}^{3}{\mathsf {T}}^{-1}}</annotation> </semantics> </math></span><img 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//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Branches</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><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 c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><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 \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><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> <mo>=</mo> <mo>−</mo> </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/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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></td><td style="border-top:solid 1px black;"><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 \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><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> <mo>=</mo> </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/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><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 U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><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 H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><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 A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><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(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)">Other</div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>, in particular <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, the <b>volumetric flow rate</b> (also known as <b>volume flow rate</b>, or <b>volume velocity</b>) is the volume of fluid which passes per unit time; usually it is represented by the symbol <span class="texhtml mvar" style="font-style:italic;">Q</span> (sometimes <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 {\dot {V}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {V}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49ef9fa9f410331b94e4578bab90e9edda5c919b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.676ex;" alt="{\displaystyle {\dot {V}}}"></span>). It contrasts with <a href="/wiki/Mass_flow_rate" title="Mass flow rate">mass flow rate</a>, which is the other main type of fluid flow rate. In most contexts a mention of <i>rate of fluid flow</i> is likely to refer to the volumetric rate. In <a href="/wiki/Hydrometry" title="Hydrometry">hydrometry</a>, the volumetric flow rate is known as <i><a href="/wiki/Discharge_(hydrology)" title="Discharge (hydrology)">discharge</a></i>. </p><p>Volumetric flow rate should not be confused with <a href="/wiki/Volumetric_flux" title="Volumetric flux">volumetric flux</a>, as defined by <a href="/wiki/Darcy%27s_law" title="Darcy's law">Darcy's law</a> and represented by the symbol <span class="texhtml mvar" style="font-style:italic;">q</span>, with units of m<sup>3</sup>/(m<sup>2</sup>·s), that is, m·s<sup>−1</sup>. The integration of a <a href="/wiki/Flux" title="Flux">flux</a> over an area gives the volumetric flow rate. </p><p>The <a href="/wiki/SI_unit" class="mw-redirect" title="SI unit">SI unit</a> is <a href="/wiki/Cubic_metre_per_second" title="Cubic metre per second">cubic metres per second</a> (m<sup>3</sup>/s). Another unit used is <a href="/wiki/Standard_cubic_centimetres_per_minute" title="Standard cubic centimetres per minute">standard cubic centimetres per minute</a> (SCCM). In <a href="/wiki/US_customary_units" class="mw-redirect" title="US customary units">US customary units</a> and <a href="/wiki/Imperial_units" title="Imperial units">imperial units</a>, volumetric flow rate is often expressed as <a href="/wiki/Cubic_foot" title="Cubic foot">cubic feet</a> per second (ft<sup>3</sup>/s) or <a href="/wiki/Gallons_per_minute" class="mw-redirect" title="Gallons per minute">gallons per minute</a> (either US or imperial definitions). In <a href="/wiki/Oceanography" title="Oceanography">oceanography</a>, the <a href="/wiki/Sverdrup" title="Sverdrup">sverdrup</a> (symbol: Sv, not to be confused with the <a href="/wiki/Sievert" title="Sievert">sievert</a>) is a non-<a href="/wiki/International_System_of_Units" title="International System of Units">SI</a> <a href="/wiki/Metric_units#Volume_flow_rate" class="mw-redirect" title="Metric units">metric unit</a> of flow, with <span class="nowrap">1 Sv</span> equal to 1 million cubic metres per second (260,000,000 US gal/s);<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> it is equivalent to the SI derived unit cubic <a href="/wiki/Hectometer" class="mw-redirect" title="Hectometer">hectometer</a> per second (symbol: hm<sup>3</sup>/s or hm<sup>3</sup>⋅s<sup>−1</sup>). Named after <a href="/wiki/Harald_Sverdrup_(oceanographer)" title="Harald Sverdrup (oceanographer)">Harald Sverdrup</a>, it is used almost exclusively in <a href="/wiki/Oceanography" title="Oceanography">oceanography</a> to measure the volumetric rate of transport of <a href="/wiki/Ocean_current" title="Ocean current">ocean currents</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Fundamental_definition">Fundamental definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volumetric_flow_rate&action=edit&section=1" title="Edit section: Fundamental definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Volumetric flow rate is defined by the <a href="/wiki/Limit_of_a_function" title="Limit of a function">limit</a><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<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 Q={\dot {V}}=\lim \limits _{\Delta t\to 0}{\frac {\Delta V}{\Delta t}}={\frac {\mathrm {d} V}{\mathrm {d} t}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>V</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <munder> <mo form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">Δ</mi> <mi>V</mi> </mrow> <mrow> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>V</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\dot {V}}=\lim \limits _{\Delta t\to 0}{\frac {\Delta V}{\Delta t}}={\frac {\mathrm {d} V}{\mathrm {d} t}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/796fb06ad03bfc0d7eea1b0e1818781e0b2526d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:26.858ex; height:5.676ex;" alt="{\displaystyle Q={\dot {V}}=\lim \limits _{\Delta t\to 0}{\frac {\Delta V}{\Delta t}}={\frac {\mathrm {d} V}{\mathrm {d} t}},}"></span></dd></dl> <p>that is, the flow of <a href="/wiki/Volume" title="Volume">volume</a> of fluid <span class="texhtml mvar" style="font-style:italic;">V</span> through a surface per unit time <span class="texhtml mvar" style="font-style:italic;">t</span>. </p><p>Since this is only the time derivative of volume, a scalar quantity, the volumetric flow rate is also a scalar quantity. The change in volume is the amount that flows <i>after</i> crossing the boundary for some time duration, not simply the initial amount of volume at the boundary minus the final amount at the boundary, since the change in volume flowing through the area would be zero for steady flow. </p><p>IUPAC<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> prefers the notation <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_{v}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>v</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{v}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86f3050477d3efa6dc855debe33889d811c2a1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.067ex; height:2.009ex;" alt="{\displaystyle q_{v}}"></span><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> 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 q_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7f3fc33576829e4effd50965fce16c1c18c1c5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.712ex; height:2.009ex;" alt="{\displaystyle q_{m}}"></span><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> for volumetric flow and mass flow respectively, to distinguish from the notation <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></span><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> for heat. </p> <div class="mw-heading mw-heading2"><h2 id="Alternative_definition">Alternative definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volumetric_flow_rate&action=edit&section=2" title="Edit section: Alternative definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Volumetric flow rate can also be defined by </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=\mathbf {v} \cdot \mathbf {A} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\mathbf {v} \cdot \mathbf {A} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7525764cb279b7751650b1c5fea78a6599acade4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.693ex; height:2.509ex;" alt="{\displaystyle Q=\mathbf {v} \cdot \mathbf {A} ,}"></span></dd></dl> <p>where </p> <dl><dd><span class="texhtml"><b>v</b></span> = <a href="/wiki/Flow_velocity" title="Flow velocity">flow velocity</a>,</dd> <dd><span class="texhtml"><b>A</b></span> = <a href="/wiki/Cross_section_(geometry)" title="Cross section (geometry)">cross-sectional</a> <a href="/wiki/Vector_area" title="Vector area">vector area</a>/surface.</dd></dl> <p>The above equation is only true for uniform or homogeneous flow velocity and a flat or planar cross section. In general, including spatially variable or non-homogeneous flow velocity and curved surfaces, the equation becomes a <a href="/wiki/Surface_integral" title="Surface integral">surface integral</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 Q=\iint _{A}\mathbf {v} \cdot \mathrm {d} \mathbf {A} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <msub> <mo>∬</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\iint _{A}\mathbf {v} \cdot \mathrm {d} \mathbf {A} .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9ff6773f4effd64b9e1942d8284be4129d2b862" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.356ex; height:5.676ex;" alt="{\displaystyle Q=\iint _{A}\mathbf {v} \cdot \mathrm {d} \mathbf {A} .}"></span></dd></dl> <p>This is the definition used in practice. The <a href="/wiki/Area" title="Area">area</a> required to calculate the volumetric flow rate is real or imaginary, flat or curved, either as a cross-sectional area or a surface. The <a href="/wiki/Vector_area" title="Vector area">vector area</a> is a combination of the magnitude of the area through which the volume passes through, <span class="texhtml mvar" style="font-style:italic;">A</span>, and a <a href="/wiki/Unit_vector" title="Unit vector">unit vector</a> normal to the area, <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 {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}"></span>. The relation is <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 \mathbf {A} =A{\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} =A{\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/547b57365913d6aa226f310afd61f29591007aec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.346ex; height:2.343ex;" alt="{\displaystyle \mathbf {A} =A{\hat {\mathbf {n} }}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volumetric_flow_rate&action=edit&section=3" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The reason for the <a href="/wiki/Dot_product" title="Dot product">dot product</a> is as follows. The only volume flowing <i>through</i> the cross-section is the amount normal to the area, that is, <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to the unit normal. This amount 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=vA\cos \theta ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>v</mi> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=vA\cos \theta ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37babbb764b831a433247df1c4e77fa8cc27d31b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.43ex; height:2.509ex;" alt="{\displaystyle Q=vA\cos \theta ,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">θ</span> is the angle between the unit normal <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 {\hat {\mathbf {n} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">n</mi> </mrow> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {\mathbf {n} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aae87b164ba005e99b51066c46d1eacc7f56564a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:2.343ex;" alt="{\displaystyle {\hat {\mathbf {n} }}}"></span> and the velocity vector <span class="texhtml"><b>v</b></span> of the substance elements. The amount passing through the cross-section is reduced by the factor <span class="texhtml">cos <i>θ</i></span>. As <span class="texhtml mvar" style="font-style:italic;">θ</span> increases less volume passes through. Substance which passes tangential to the area, that is <a href="/wiki/Perpendicular" title="Perpendicular">perpendicular</a> to the unit normal, does not pass through the area. This occurs when <span class="texhtml"><i>θ</i> = <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">π</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span></span> and so this amount of the volumetric flow rate is zero: </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=vA\cos \left({\frac {\pi }{2}}\right)=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mi>v</mi> <mi>A</mi> <mi>cos</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=vA\cos \left({\frac {\pi }{2}}\right)=0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d8c1fd646f59ec05524040afe79df95d0696ffb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.157ex; height:4.843ex;" alt="{\displaystyle Q=vA\cos \left({\frac {\pi }{2}}\right)=0.}"></span></dd></dl> <p>These results are equivalent to the dot product between velocity and the normal direction to the area. </p> <div class="mw-heading mw-heading2"><h2 id="Relationship_with_mass_flow_rate">Relationship with mass flow rate</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volumetric_flow_rate&action=edit&section=4" title="Edit section: Relationship with mass flow rate"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>When the <a href="/wiki/Mass_flow_rate" title="Mass flow rate">mass flow rate</a> is known, and the density can be assumed constant, this is an easy way to get <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}"></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 {\dot {m}}{\rho }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>m</mi> <mo>˙</mo> </mover> </mrow> <mi>ρ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\frac {\dot {m}}{\rho }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/026e0467feea190ced4908e018c7f47fafb9e040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.46ex; height:5.676ex;" alt="{\displaystyle Q={\frac {\dot {m}}{\rho }},}"></span></dd></dl> <p>where </p> <dl><dd><span class="texhtml mvar" style="font-style:italic;">ṁ</span> = <a href="/wiki/Mass_flow_rate" title="Mass flow rate">mass flow rate</a> (in kg/s),</dd> <dd><span class="texhtml mvar" style="font-style:italic;">ρ</span> = <a href="/wiki/Density" title="Density">density</a> (in kg/m<sup>3</sup>).</dd></dl> <div class="mw-heading mw-heading2"><h2 id="Related_quantities">Related quantities</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volumetric_flow_rate&action=edit&section=5" title="Edit section: Related quantities"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In internal combustion engines, the time area integral is considered over the range of valve opening. The time lift integral is given by </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 \int L\,\mathrm {d} \theta ={\frac {RT}{2\pi }}(\cos \theta _{2}-\cos \theta _{1})+{\frac {rT}{2\pi }}(\theta _{2}-\theta _{1}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mi>L</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>θ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>R</mi> <mi>T</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>cos</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>r</mi> <mi>T</mi> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int L\,\mathrm {d} \theta ={\frac {RT}{2\pi }}(\cos \theta _{2}-\cos \theta _{1})+{\frac {rT}{2\pi }}(\theta _{2}-\theta _{1}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9964d60d29b672a9891218fa9c846201222e1ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.151ex; height:5.676ex;" alt="{\displaystyle \int L\,\mathrm {d} \theta ={\frac {RT}{2\pi }}(\cos \theta _{2}-\cos \theta _{1})+{\frac {rT}{2\pi }}(\theta _{2}-\theta _{1}),}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">T</span> is the time per revolution, <span class="texhtml mvar" style="font-style:italic;">R</span> is the distance from the camshaft centreline to the cam tip, <span class="texhtml mvar" style="font-style:italic;">r</span> is the radius of the camshaft (that is, <span class="texhtml"><i>R</i> − <i>r</i></span> is the maximum lift), <span class="texhtml"><i>θ</i><sub>1</sub></span> is the angle where opening begins, and <span class="texhtml"><i>θ</i><sub>2</sub></span> is where the valve closes (seconds, mm, radians). This has to be factored by the width (circumference) of the valve throat. The answer is usually related to the cylinder's swept volume. </p> <div class="mw-heading mw-heading2"><h2 id="Some_key_examples">Some key examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Volumetric_flow_rate&action=edit&section=6" title="Edit section: Some key examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>In <a href="/wiki/Cardiac_physiology" title="Cardiac physiology">cardiac physiology</a>: the <a href="/wiki/Cardiac_output" title="Cardiac output">cardiac output</a></li> <li>In <a href="/wiki/Hydrology" title="Hydrology">hydrology</a>: <a href="/wiki/Discharge_(hydrology)" title="Discharge (hydrology)">discharge</a> <ul><li><a href="/wiki/List_of_rivers_by_discharge" title="List of rivers by discharge">List of rivers by discharge</a></li> <li><a href="/wiki/List_of_waterfalls_by_flow_rate" title="List of waterfalls by flow rate">List of waterfalls by flow rate</a></li> <li><a href="/wiki/Weir#Flow_measurement" title="Weir">Weir § Flow measurement</a></li></ul></li> <li>In <a href="/wiki/Dust_collection_system" class="mw-redirect" title="Dust collection system">dust collection systems</a>: the <a href="/wiki/Air-to-cloth_ratio" title="Air-to-cloth ratio">air-to-cloth ratio</a></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=Volumetric_flow_rate&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bulk_velocity" class="mw-redirect" title="Bulk velocity">Bulk velocity</a></li> <li><a href="/wiki/Flow_measurement" title="Flow measurement">Flow measurement</a></li> <li><a href="/wiki/Flowmeter" class="mw-redirect" title="Flowmeter">Flowmeter</a></li> <li><a href="/wiki/Mass_flow_rate" title="Mass flow rate">Mass flow rate</a></li> <li><a href="/wiki/Orifice_plate" title="Orifice plate">Orifice plate</a></li> <li><a href="/wiki/Poiseuille%27s_law" class="mw-redirect" title="Poiseuille's law">Poiseuille's law</a></li> <li><a href="/wiki/Stokes_flow" title="Stokes flow">Stokes flow</a></li></ul> <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=Volumetric_flow_rate&action=edit&section=8" title="Edit section: References"><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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://oceancurrents.rsmas.miami.edu/glossary.html#S">"Glossary"</a>. <i>Ocean Surface Currents</i>. <a href="/wiki/University_of_Miami" title="University of Miami">University of Miami</a> <a href="/wiki/Rosenstiel_School_of_Marine,_Atmospheric,_and_Earth_Science" title="Rosenstiel School of Marine, Atmospheric, and Earth Science">Rosenstiel School of Marine, Atmospheric, and Earth Science</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-04-15</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Ocean+Surface+Currents&rft.atitle=Glossary&rft_id=https%3A%2F%2Foceancurrents.rsmas.miami.edu%2Fglossary.html%23S&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVolumetric+flow+rate" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110120155822/http://www.ecoworld.com/technology/sverdrups-brine.html">"Sverdrups & Brine"</a>. <i>Ecoworld</i>. Archived from <a rel="nofollow" class="external text" href="http://www.ecoworld.com/technology/sverdrups-brine.html">the original</a> on 20 January 2011<span class="reference-accessdate">. Retrieved <span class="nowrap">12 August</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Ecoworld&rft.atitle=Sverdrups+%26+Brine&rft_id=http%3A%2F%2Fwww.ecoworld.com%2Ftechnology%2Fsverdrups-brine.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVolumetric+flow+rate" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEngineers_Edge,_LLC." class="citation web cs1">Engineers Edge, LLC. <a rel="nofollow" class="external text" href="http://www.engineersedge.com/fluid_flow/volumeetric_flow_rate.htm">"Fluid Volumetric Flow Rate Equation"</a>. Engineers Edge<span class="reference-accessdate">. Retrieved <span class="nowrap">2016-12-01</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Fluid+Volumetric+Flow+Rate+Equation&rft.pub=Engineers+Edge&rft.au=Engineers+Edge%2C+LLC.&rft_id=http%3A%2F%2Fwww.engineersedge.com%2Ffluid_flow%2Fvolumeetric_flow_rate.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVolumetric+flow+rate" 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">International Union of Pure and Applied Chemistry; <a rel="nofollow" class="external free" href="https://iupac.org">https://iupac.org</a></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://goldbook.iupac.org/terms/view/V06642">"Volume flow rate, qv"</a>. <i>The IUPAC Compendium of Chemical Terminology</i>. 2014. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fgoldbook.V06642">10.1351/goldbook.V06642</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Volume+flow+rate%2C+qv&rft.btitle=The+IUPAC+Compendium+of+Chemical+Terminology&rft.date=2014&rft_id=info%3Adoi%2F10.1351%2Fgoldbook.V06642&rft_id=https%3A%2F%2Fgoldbook.iupac.org%2Fterms%2Fview%2FV06642&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVolumetric+flow+rate" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://goldbook.iupac.org/terms/view/M03720">"Mass flow rate, qm"</a>. <i>The IUPAC Compendium of Chemical Terminology</i>. 2014. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fgoldbook.M03720">10.1351/goldbook.M03720</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Mass+flow+rate%2C+qm&rft.btitle=The+IUPAC+Compendium+of+Chemical+Terminology&rft.date=2014&rft_id=info%3Adoi%2F10.1351%2Fgoldbook.M03720&rft_id=https%3A%2F%2Fgoldbook.iupac.org%2Fterms%2Fview%2FM03720&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVolumetric+flow+rate" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation book cs1"><a rel="nofollow" class="external text" href="https://goldbook.iupac.org/terms/view/H02752">"Heat, q, Q"</a>. <i>The IUPAC Compendium of Chemical Terminology</i>. 2014. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1351%2Fgoldbook.H02752">10.1351/goldbook.H02752</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Heat%2C+q%2C+Q&rft.btitle=The+IUPAC+Compendium+of+Chemical+Terminology&rft.date=2014&rft_id=info%3Adoi%2F10.1351%2Fgoldbook.H02752&rft_id=https%3A%2F%2Fgoldbook.iupac.org%2Fterms%2Fview%2FH02752&rfr_id=info%3Asid%2Fen.wikipedia.org%3AVolumetric+flow+rate" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style 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.navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Rivers,_streams_and_springs" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Rivers,_streams_and_springs" title="Template:Rivers, streams and springs"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Rivers,_streams_and_springs" title="Template talk:Rivers, streams and springs"><abbr title="Discuss this 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href="/wiki/Blackwater_river" title="Blackwater river">Blackwater river</a></li> <li><a href="/wiki/Channel_(geography)" title="Channel (geography)">Channel</a></li> <li><a href="/wiki/Channel_pattern" title="Channel pattern">Channel pattern</a></li> <li><a href="/wiki/Channel_types" title="Channel types">Channel types</a></li> <li><a href="/wiki/Confluence" title="Confluence">Confluence</a></li> <li><a href="/wiki/Distributary" title="Distributary">Distributary</a></li> <li><a href="/wiki/Drainage_basin" title="Drainage basin">Drainage basin</a></li> <li><a href="/wiki/Subterranean_river" title="Subterranean river">Subterranean river</a></li> <li><a href="/wiki/River_bifurcation" title="River bifurcation">River bifurcation</a></li> <li><a href="/wiki/River_ecosystem" title="River ecosystem">River ecosystem</a></li> <li><a href="/wiki/River_source" title="River source">River source</a></li> <li><a href="/wiki/Tributary" title="Tributary">Tributary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Stream" title="Stream">Streams</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arroyo_(watercourse)" title="Arroyo (watercourse)">Arroyo</a></li> <li><a href="/wiki/Bourne_(stream)" title="Bourne (stream)">Bourne</a></li> <li><a href="/wiki/Burn_(landform)" title="Burn (landform)">Burn</a></li> <li><a href="/wiki/Chalk_stream" title="Chalk stream">Chalk stream</a></li> <li><a href="/wiki/Coulee" title="Coulee">Coulee</a></li> <li><a href="/wiki/Current_(hydrology)" title="Current (hydrology)">Current</a></li> <li><a href="/wiki/Stream_bed" title="Stream bed">Stream bed</a></li> <li><a href="/wiki/Stream_channel" class="mw-redirect" title="Stream channel">Stream channel</a></li> <li><a href="/wiki/Streamflow" title="Streamflow">Streamflow</a></li> <li><a href="/wiki/Stream_gradient" title="Stream gradient">Stream gradient</a></li> <li><a href="/wiki/Stream_pool" title="Stream pool">Stream pool</a></li> <li><a href="/wiki/Perennial_stream" class="mw-redirect" title="Perennial stream">Perennial stream</a></li> <li><a href="/wiki/Winterbourne_(stream)" title="Winterbourne (stream)">Winterbourne</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Spring_(hydrology)" title="Spring (hydrology)">Springs</a><br />(<a href="/wiki/List_of_springs" title="List of springs">list</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Estavelle" title="Estavelle">Estavelle/Inversac</a></li> <li><a href="/wiki/Geyser" title="Geyser">Geyser</a></li> <li><a href="/wiki/Holy_well" title="Holy well">Holy well</a></li> <li><a href="/wiki/Hot_spring" title="Hot spring">Hot spring</a> <ul><li><a href="/wiki/List_of_hot_springs" title="List of hot springs">list</a></li> <li><a href="/wiki/List_of_hot_springs_in_the_United_States" title="List of hot springs in the United States">list in the US</a></li></ul></li> <li><a href="/wiki/Karst_spring" title="Karst spring">Karst spring</a> <ul><li><a href="/wiki/List_of_karst_springs" title="List of karst springs">list</a></li></ul></li> <li><a href="/wiki/Mineral_spring" title="Mineral spring">Mineral spring</a></li> <li><a href="/wiki/Ponor" title="Ponor">Ponor</a></li> <li><a href="/wiki/Rhythmic_spring" title="Rhythmic spring">Rhythmic spring</a></li> <li><a href="/wiki/Spring_horizon" title="Spring horizon">Spring horizon</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sediment" title="Sediment">Sedimentary processes</a><br />and <a href="/wiki/Erosion" title="Erosion">erosion</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abrasion_(geology)" title="Abrasion (geology)">Abrasion</a></li> <li><a href="/wiki/Anabranch" title="Anabranch">Anabranch</a></li> <li><a href="/wiki/Aggradation" title="Aggradation">Aggradation</a></li> <li><a href="/wiki/Armor_(hydrology)" title="Armor (hydrology)">Armor</a></li> <li><a href="/wiki/Bed_load" title="Bed load">Bed load</a></li> <li><a href="/wiki/Bed_material_load" title="Bed material load">Bed material load</a></li> <li><a href="/wiki/Granular_flow" class="mw-redirect" title="Granular flow">Granular flow</a></li> <li><a href="/wiki/Debris_flow" title="Debris flow">Debris flow</a></li> <li><a href="/wiki/Deposition_(geology)" title="Deposition (geology)">Deposition</a></li> <li><a href="/wiki/Dissolved_load" title="Dissolved load">Dissolved load</a></li> <li><a href="/wiki/Downcutting" title="Downcutting">Downcutting</a></li> <li><a href="/wiki/Erosion" title="Erosion">Erosion</a></li> <li><a href="/wiki/Headward_erosion" title="Headward erosion">Headward erosion</a></li> <li><a href="/wiki/Knickpoint" title="Knickpoint">Knickpoint</a></li> <li><a href="/wiki/Palaeochannel" title="Palaeochannel">Palaeochannel</a></li> <li><a href="/wiki/Progradation" title="Progradation">Progradation</a></li> <li><a href="/wiki/Retrogradation" title="Retrogradation">Retrogradation</a></li> <li><a href="/wiki/Saltation_(geology)" title="Saltation (geology)">Saltation</a></li> <li><a href="/wiki/Secondary_flow#River_bends" title="Secondary flow">Secondary flow</a></li> <li><a href="/wiki/Sediment_transport" title="Sediment transport">Sediment transport</a></li> <li><a href="/wiki/Suspended_load" title="Suspended load">Suspended load</a></li> <li><a href="/wiki/Wash_load" title="Wash load">Wash load</a></li> <li><a href="/wiki/Water_gap" title="Water gap">Water gap</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fluvial#Fluvial_landforms" class="mw-redirect" title="Fluvial">Fluvial landforms</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ait" title="Ait">Ait</a></li> <li><a href="/wiki/Alluvial_fan" title="Alluvial fan">Alluvial fan</a></li> <li><a href="/wiki/Antecedent_drainage_stream" title="Antecedent drainage stream">Antecedent drainage stream</a></li> <li><a href="/wiki/Avulsion_(river)" title="Avulsion (river)">Avulsion</a></li> <li><a href="/wiki/Bank_(geography)" title="Bank (geography)">Bank</a></li> <li><a href="/wiki/Bar_(river_morphology)" title="Bar (river morphology)">Bar</a></li> <li><a href="/wiki/Bayou" title="Bayou">Bayou</a></li> <li><a href="/wiki/Billabong" title="Billabong">Billabong</a></li> <li><a href="/wiki/Canyon" title="Canyon">Canyon</a></li> <li><a href="/wiki/Chine" title="Chine">Chine</a></li> <li><a href="/wiki/Cut_bank" title="Cut bank">Cut bank</a></li> <li><a href="/wiki/Estuary" title="Estuary">Estuary</a></li> <li><a href="/wiki/Floating_island" title="Floating island">Floating island</a></li> <li><a href="/wiki/Fluvial_terrace" title="Fluvial terrace">Fluvial terrace</a></li> <li><a href="/wiki/Gill_(ravine)" title="Gill (ravine)">Gill</a></li> <li><a href="/wiki/Gulch" title="Gulch">Gulch</a></li> <li><a href="/wiki/Gully" title="Gully">Gully</a></li> <li><a href="/wiki/Glen" title="Glen">Glen</a></li> <li><a href="/wiki/Meander_scar" title="Meander scar">Meander scar</a></li> <li><a href="/wiki/Mouth_bar" title="Mouth bar">Mouth bar</a></li> <li><a href="/wiki/Oxbow_lake" title="Oxbow lake">Oxbow lake</a></li> <li><a href="/wiki/Riffle-pool_sequence" title="Riffle-pool sequence">Riffle-pool sequence</a></li> <li><a href="/wiki/Point_bar" title="Point bar">Point bar</a></li> <li><a href="/wiki/Ravine" title="Ravine">Ravine</a></li> <li><a href="/wiki/Rill" title="Rill">Rill</a></li> <li><a href="/wiki/River_island" title="River island">River island</a></li> <li><a href="/wiki/Rock-cut_basin" title="Rock-cut basin">Rock-cut basin</a></li> <li><a href="/wiki/Sedimentary_basin" title="Sedimentary basin">Sedimentary basin</a></li> <li><a href="/wiki/Sedimentary_structures" title="Sedimentary structures">Sedimentary structures</a></li> <li><a href="/wiki/Strath" title="Strath">Strath</a></li> <li><a href="/wiki/Thalweg" title="Thalweg">Thalweg</a></li> <li><a href="/wiki/Valley" title="Valley">River valley</a></li> <li><a href="/wiki/Wadi" title="Wadi">Wadi</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Fluvial#Fluvial_processes" class="mw-redirect" title="Fluvial">Fluvial flow</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Helicoidal_flow" title="Helicoidal flow">Helicoidal flow</a></li> <li><a href="/wiki/International_scale_of_river_difficulty" title="International scale of river difficulty">International scale of river difficulty</a></li> <li><a href="/wiki/Log_jam" title="Log jam">Log jam</a></li> <li><a href="/wiki/Meander" title="Meander">Meander</a></li> <li><a href="/wiki/Plunge_pool" title="Plunge pool">Plunge pool</a></li> <li><a href="/wiki/Rapids" title="Rapids">Rapids</a></li> <li><a href="/wiki/Riffle" title="Riffle">Riffle</a></li> <li><a href="/wiki/Shoal" title="Shoal">Shoal</a></li> <li><a href="/wiki/Stream_capture" title="Stream capture">Stream capture</a></li> <li><a href="/wiki/Waterfall" title="Waterfall">Waterfall</a></li> <li><a href="/wiki/Whitewater" title="Whitewater">Whitewater</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Surface_runoff" title="Surface runoff">Surface runoff</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Agricultural_wastewater" class="mw-redirect" title="Agricultural wastewater">Agricultural wastewater</a></li> <li><a href="/wiki/First_flush" title="First flush">First flush</a></li> <li><a href="/wiki/Urban_runoff" title="Urban runoff">Urban runoff</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Flood" title="Flood">Floods</a> and <a href="/wiki/Stormwater" title="Stormwater">stormwater</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/100-year_flood" title="100-year flood">100-year flood</a></li> <li><a href="/wiki/Crevasse_splay" title="Crevasse splay">Crevasse splay</a></li> <li><a href="/wiki/Flash_flood" title="Flash flood">Flash flood</a></li> <li><a href="/wiki/Flood" title="Flood">Flood</a> <ul><li><a href="/wiki/Urban_flooding" title="Urban flooding">Urban flooding</a></li> <li><a href="/wiki/List_of_non-water_floods" title="List of non-water floods">Non-water flood</a></li></ul></li> <li><a href="/wiki/Flood_barrier" title="Flood barrier">Flood barrier</a></li> <li><a href="/wiki/Flood_control" class="mw-redirect" title="Flood control">Flood control</a></li> <li><a href="/wiki/Flood_forecasting" title="Flood forecasting">Flood forecasting</a></li> <li><a href="/wiki/Flood-meadow" title="Flood-meadow">Flood-meadow</a></li> <li><a href="/wiki/Floodplain" title="Floodplain">Floodplain</a></li> <li><a href="/wiki/Flood_pulse_concept" title="Flood pulse concept">Flood pulse concept</a></li> <li><a href="/wiki/Flooded_grasslands_and_savannas" title="Flooded grasslands and savannas">Flooded grasslands and savannas</a></li> <li><a href="/wiki/Flooding_of_the_Nile" title="Flooding of the Nile">Inundation</a></li> <li><a href="/wiki/Storm_Water_Management_Model" title="Storm Water Management Model">Storm Water Management Model</a></li> <li><a href="/wiki/Return_period" title="Return period">Return period</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Water_pollution#Pollution_from_point_sources" title="Water pollution">Point source pollution</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effluent" title="Effluent">Effluent</a></li> <li><a href="/wiki/Industrial_wastewater" class="mw-redirect" title="Industrial wastewater">Industrial wastewater</a></li> <li><a href="/wiki/Sewage" title="Sewage">Sewage</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">River measurement<br />and modelling</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Baer%27s_law" class="mw-redirect" title="Baer's law">Baer's law</a></li> <li><a href="/wiki/Baseflow" title="Baseflow">Baseflow</a></li> <li><a href="/wiki/Bradshaw_model" title="Bradshaw model">Bradshaw model</a></li> <li><a href="/wiki/Discharge_(hydrology)" title="Discharge (hydrology)">Discharge (hydrology)</a></li> <li><a href="/wiki/Drainage_density" title="Drainage density">Drainage density</a></li> <li><a href="/wiki/Exner_equation" title="Exner equation">Exner equation</a></li> <li><a href="/wiki/Groundwater_model" title="Groundwater model">Groundwater model</a></li> <li><a href="/wiki/Hack%27s_law" title="Hack's law">Hack's law</a></li> <li><a href="/wiki/Hjulstr%C3%B6m_curve" title="Hjulström curve">Hjulström curve</a></li> <li><a href="/wiki/Hydrograph" title="Hydrograph">Hydrograph</a></li> <li><a href="/wiki/Hydrological_model" title="Hydrological model">Hydrological model</a></li> <li><a href="/wiki/Hydrological_transport_model" title="Hydrological transport model">Hydrological transport model</a></li> <li><a href="/wiki/Infiltration_(hydrology)" title="Infiltration (hydrology)">Infiltration (hydrology)</a></li> <li><a href="/wiki/Main_stem" title="Main stem">Main stem</a></li> <li><a href="/wiki/Playfair%27s_law" title="Playfair's law">Playfair's law</a></li> <li><a href="/wiki/Relief_ratio" class="mw-redirect" title="Relief ratio">Relief ratio</a></li> <li><a href="/wiki/River_Continuum_Concept" title="River Continuum Concept">River Continuum Concept</a></li> <li><a href="/wiki/Rouse_number" title="Rouse number">Rouse number</a></li> <li><a href="/wiki/Runoff_curve_number" title="Runoff curve number">Runoff curve number</a></li> <li><a href="/wiki/Runoff_model_(reservoir)" title="Runoff model (reservoir)">Runoff model (reservoir)</a></li> <li><a href="/wiki/Stream_gauge" title="Stream gauge">Stream gauge</a></li> <li><a href="/wiki/Universal_Soil_Loss_Equation" title="Universal Soil Loss Equation">Universal Soil Loss Equation</a></li> <li><a href="/wiki/WAFLEX" title="WAFLEX">WAFLEX</a></li> <li><a href="/wiki/Wetted_perimeter" title="Wetted perimeter">Wetted perimeter</a></li> <li><a class="mw-selflink selflink">Volumetric flow rate</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/River_engineering" title="River engineering">River engineering</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aqueduct_(bridge)" title="Aqueduct (bridge)">Aqueduct</a></li> <li><a href="/wiki/Balancing_lake" title="Balancing lake">Balancing lake</a></li> <li><a href="/wiki/Canal" title="Canal">Canal</a></li> <li><a href="/wiki/Check_dam" title="Check dam">Check dam</a></li> <li><a href="/wiki/Dam" title="Dam">Dam</a></li> <li><a href="/wiki/Drop_structure" title="Drop structure">Drop structure</a></li> <li><a href="/wiki/Daylighting_(streams)" title="Daylighting (streams)">Daylighting</a></li> <li><a href="/wiki/Detention_basin" title="Detention basin">Detention basin</a></li> <li><a href="/wiki/Erosion_control" title="Erosion control">Erosion control</a></li> <li><a href="/wiki/Fish_ladder" title="Fish ladder">Fish ladder</a></li> <li><a href="/wiki/Floodplain_restoration" title="Floodplain restoration">Floodplain restoration</a></li> <li><a href="/wiki/Flume" title="Flume">Flume</a></li> <li><a href="/wiki/Infiltration_basin" title="Infiltration basin">Infiltration basin</a></li> <li><a href="/wiki/Leat" title="Leat">Leat</a></li> <li><a href="/wiki/Levee" title="Levee">Levee</a></li> <li><a href="/wiki/River_morphology" title="River morphology">River morphology</a></li> <li><a href="/wiki/Retention_basin" title="Retention basin">Retention basin</a></li> <li><a href="/wiki/Revetment" title="Revetment">Revetment</a></li> <li><a href="/wiki/Riparian-zone_restoration" title="Riparian-zone restoration">Riparian-zone restoration</a></li> <li><a href="/wiki/Stream_restoration" title="Stream restoration">Stream restoration</a></li> <li><a href="/wiki/Weir" title="Weir">Weir</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">River sports</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Canyoning" title="Canyoning">Canyoning</a></li> <li><a href="/wiki/Fly_fishing" title="Fly fishing">Fly fishing</a></li> <li><a href="/wiki/Rafting" title="Rafting">Rafting</a></li> <li><a href="/wiki/River_surfing" title="River surfing">River surfing</a></li> <li><a href="/wiki/Riverboarding" title="Riverboarding">Riverboarding</a></li> <li><a href="/wiki/Stone_skipping" title="Stone skipping">Stone skipping</a></li> <li><a href="/wiki/Triathlon" title="Triathlon">Triathlon</a></li> <li><a href="/wiki/Whitewater_canoeing" title="Whitewater canoeing">Whitewater canoeing</a></li> <li><a href="/wiki/Whitewater_kayaking" title="Whitewater kayaking">Whitewater kayaking</a></li> <li><a href="/wiki/Whitewater_slalom" class="mw-redirect" title="Whitewater slalom">Whitewater slalom</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aquifer" title="Aquifer">Aquifer</a></li> <li><a href="/wiki/Aquatic_toxicology" title="Aquatic toxicology">Aquatic toxicology</a></li> <li><a href="/wiki/Body_of_water" title="Body of water">Body of water</a></li> <li><a href="/wiki/Hydraulic_civilization" class="mw-redirect" title="Hydraulic civilization">Hydraulic civilization</a></li> <li><a href="/wiki/Limnology" title="Limnology">Limnology</a></li> <li><a href="/wiki/Riparian_zone" title="Riparian zone">Riparian zone</a></li> <li><a href="/wiki/River_valley_civilization" title="River valley civilization">River valley civilization</a></li> <li><a href="/wiki/River_cruise" title="River cruise">River cruise</a></li> <li><a href="/wiki/Sacred_waters" title="Sacred waters">Sacred waters</a></li> <li><a href="/wiki/Surface_water" title="Surface water">Surface water</a></li> <li><a href="/wiki/Wild_river" title="Wild river">Wild river</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/List_of_rivers_by_length" class="mw-redirect" title="List of rivers by length">Rivers by length</a></li> <li><a href="/wiki/List_of_rivers_by_discharge" title="List of rivers by discharge">Rivers by discharge rate</a></li> <li><a href="/wiki/List_of_drainage_basins" class="mw-redirect" title="List of drainage basins">Drainage basins</a></li> <li><a href="/wiki/List_of_whitewater_rivers" title="List of whitewater rivers">Whitewater rivers</a></li> <li><a href="/wiki/List_of_flash_floods" title="List of flash floods">Flash floods</a></li> <li><a href="/wiki/List_of_river_name_etymologies" title="List of river name etymologies">River name etymologies</a></li> <li><a href="/wiki/List_of_countries_without_rivers" title="List of countries without rivers">Countries without rivers</a></li></ul> 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