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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Forchheimer-Gleichung</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Forchheimer-Gleichung</b> (nach <a href="Philipp_Forchheimer" title="Philipp Forchheimer">Philipp Forchheimer</a>) beschreibt den <a href="Druckverlust" title="Druckverlust">Druckverlust</a> in einer <a href="Str%C3%B6mungsmechanik" title="Strömungsmechanik">Strömung</a> (meist in <a href="Por%C3%B6s" class="mw-redirect" title="Porös">porösen</a> Medien). Die Gleichung erweitert das <a href="Darcy-Gesetz" title="Darcy-Gesetz">Darcy-Gesetz</a>, das nur Druckverluste aus der <a href="Dynamische_Viskosit%C3%A4t" class="mw-redirect" title="Dynamische Viskosität">dynamischen Viskosität</a> berücksichtigt, um einen Term für die Druckverluste aus <a href="Turbulente_Str%C3%B6mung" title="Turbulente Strömung">Turbulenz</a>:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {dp}{dx}}={\frac {\eta }{K}}u+{\frac {\rho }{k_{2}}}u^{2}}">
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<mi>η<!-- η --></mi>
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<mi>ρ<!-- ρ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle -{\frac {dp}{dx}}={\frac {\eta }{K}}u+{\frac {\rho }{k_{2}}}u^{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36d565aebaa4be19b8a4f4ded4a5f55ef94fee8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.846ex; height:5.843ex;" alt="{\displaystyle -{\frac {dp}{dx}}={\frac {\eta }{K}}u+{\frac {\rho }{k_{2}}}u^{2}}" loading="lazy"></span></dd></dl>
<p>mit
</p>
<ul><li>dem Druckverlust <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dp}">
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<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle dp}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fbfc7a78f0e04743a4bcdc18bf6dd926e4f215a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.509ex;" alt="{\displaystyle dp}" loading="lazy"></span> über der Strecke <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx}">
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<mi>d</mi>
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<annotation encoding="application/x-tex">{\displaystyle dx}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/845c817e348381a13f3fad5184169ce0e021c685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.546ex; height:2.176ex;" alt="{\displaystyle dx}" loading="lazy"></span></li>
<li>der dynamischen Viskosität <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>η<!-- η --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \eta }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"></span> des strömenden <a href="Fluid" title="Fluid">Fluids</a></li>
<li>der <a href="Permeabilit%C3%A4t_(Geowissenschaften)" title="Permeabilität (Geowissenschaften)">Permeabilität</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
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<mrow class="MJX-TeXAtom-ORD">
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<mi>K</mi>
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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> (Einheit&nbsp;m²) des porösen Mediums</li>
<li>der <a href="Str%C3%B6mungsgeschwindigkeit" title="Strömungsgeschwindigkeit">Strömungsgeschwindigkeit</a> <span class="mwe-math-element mwe-math-element-inline"><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}">
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<mi>u</mi>
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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> des Fluids</li>
<li>der <a href="Dichte" title="Dichte">Dichte</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">
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<mi>ρ<!-- ρ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> des strömenden Fluids</li>
<li>dem nicht-darcyschen Permeabilitätskoeffizienten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
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<mn>2</mn>
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<annotation encoding="application/x-tex">{\displaystyle k_{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}" loading="lazy"></span> (Einheit&nbsp;m) des porösen Mediums.</li></ul>
<p>Die beiden Koeffizienten <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>k</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
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</msub>
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<annotation encoding="application/x-tex">{\displaystyle k_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c51b4ba57ee596d8435fc4ed76703ca3a2fc444a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.265ex; height:2.509ex;" alt="{\displaystyle k_{2}}" loading="lazy"></span> werden meistens experimentell ermittelt und sind nur von der Geometrie des porösen Mediums abhängig, nicht aber vom strömenden Fluid.
</p><p>Für <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{2}\to \infty }">
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<msub>
<mi>k</mi>
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<mo stretchy="false">→<!-- → --></mo>
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<annotation encoding="application/x-tex">{\displaystyle k_{2}\to \infty }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5b070be14c1e0b79c9907df8ae69a55aa724f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.203ex; height:2.509ex;" alt="{\displaystyle k_{2}\to \infty }" loading="lazy"></span> geht die Forchheimer-Gleichung in das Darcy-Gesetz über.
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Keulegan-Carpenter-Zahl" title="Keulegan-Carpenter-Zahl">Keulegan-Carpenter-Zahl</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://books.google.de/books?id=zSJ9ZEQ9GpkC&amp;pg=PA417&amp;lpg=PA417&amp;dq=forchheimer-gleichung&amp;source=bl&amp;ots=b2FTF0GtVw&amp;sig=GimtsXpnetg-tO6IU_c-XeWAU5g&amp;hl=de&amp;sa=X&amp;ei=bXgrULlqkPGyBrqSgKAD#v=onepage&amp;q=forchheimer-gleichung&amp;f=false">H. D. Baehr, K. Stephan: Wärme- und Stoffübertragung, siehe Formel 3.275 auf S. 417</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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