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| In [[convective heat transfer]], the '''Churchill–Bernstein equation''' is used to estimate the surface averaged [[Nusselt number]] for a cylinder in cross flow at various velocities.<ref>Some examples of said system can be seen here: {{Cite web | publisher = Flometrics | title = Cylinder in Cross Flow at Various Velocities | year =1997 | url =http://www.flometrics.com/services/cylinder/cylinder.html | accessdate =10/07/2007 }} {{Dead link|date=October 2010|bot=H3llBot}}</ref> The need for the equation arises from the inability to solve the [[Navier–Stokes equations]] in the [[turbulent flow]] regime, even for a [[Newtonian fluid]]. When the concentration and temperature profiles are independent of one another, the mass-heat transfer analogy can be employed. In the mass-heat transfer analogy, heat transfer [[dimensionless quantities]] are replaced with analogous [[mass transfer]] dimensionless quantities.
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| This equation is named after Stuart W. Churchill and M. Bernstein, who introduced it in 1977. This equation is also called the '''Churchill–Bernstein correlation'''.
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| ==Heat transfer definition==
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| <math>\overline{\mathrm{Nu}}_D \ = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\Pr^{1/3}}{\left[1 + (0.4/\Pr)^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad
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| \Pr\mathrm{Re}_D \ge 0.2 </math>
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| <ref>http://me.queensu.ca/Courses/346/Lec14flowovercylindersspheres.pdf</ref>
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| where:
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| * <math>\overline{\mathrm{Nu}}_D</math> is the surface averaged [[Nusselt number]] with characteristic length of diameter;
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| * <math>\mathrm{Re}_D\,\!</math> is the [[Reynolds number]] with the cylinder diameter as its characteristic length;
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| * <math>\Pr</math> is the [[Prandtl number]].
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| The Churchill–Bernstein equation is valid for a wide range of Reynolds numbers and Prandtl numbers, as long as the product of the two is greater than or equal to 0.2, as defined above. The Churchill–Bernstein equation can be used for any object of cylindrical geometry in which [[boundary layer]]s develop freely, without constraints imposed by other surfaces. Properties of the external free stream fluid are to be evaluated at the [[Film Temperature|film temperature]] in order to account for the variation of the fluid properties at different temperatures. One should not expect much more than 20% accuracy from the above equation due to the wide range of flow conditions that the equation encompasses. The Churchill–Bernstein equation is a [[correlation]] and cannot be derived from principles of [[fluid dynamics]]. The equation yields the surface averaged Nusselt number, which is used to determine the average convective heat transfer coefficient. [[Newton's law of cooling]] can then be invoked to determine the heat loss or gain from the object, fluid and/or surface temperatures, and the area of the object, depending on what information is known.
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| ==Mass transfer definition==
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| <math>\mathrm{Sh}_D = 0.3 + \frac{0.62\mathrm{Re}_D^{1/2}\mathrm{Sc}^{1/3}}{\left[1 + (0.4/\mathrm{Sc})^{2/3} \, \right]^{1/4} \,}\bigg[1 + \bigg(\frac{\mathrm{Re}_D}{282000} \bigg)^{5/8}\bigg]^{4/5} \quad
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| \mathrm{Sc}\,\mathrm{Re}_D \ge 0.2 </math>
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| where: | |
| * <math>\mathrm{Sh}_D</math> is the [[Sherwood number]] related to hydraulic diameter
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| * <math>\mathrm{Sc}</math> is the [[Schmidt number]]
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| Using the mass-heat transfer analogy, the Nusselt number is replaced by the Sherwood number, and the Prandtl number is replaced by the Schmidt number. The same restrictions described in the heat transfer definition are applied to the mass transfer definition. The Sherwood number can be used to find an overall mass transfer coefficient and applied to [[Fick's law of diffusion]] to find concentration profiles and mass transfer fluxes.
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| ==Notes==
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| <references />
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| ==See also ==
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| *[[Prandtl number]]
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| *[[Reynolds number]]
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| ==References==
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| * {{Citation| last1 = Churchill | first1 = S. W. | last2 = Bernstein | first2 = M. | title = A Correlating Equation for Forced Convection From Gases and Liquids to a Circular Cylinder in Crossflow | journal = J. Heat Transfer, Trans. ASME | volume = 99 | pages = 300–306| year = 1977 | doi=10.1115/1.3450685}}
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| * {{cite book | author=[[Frank P. Incropera|Incropera, F.P.]], | coauthors=DeWitt, D.P., Bergman, T.L., and Lavine, A.S. | title=Fundamentals of Heat and Mass Transfer, 6th Ed. | url=http://www.amazon.com/dp/0470055545 | publisher=Wiley | year=2006 | isbn = 978-0-471-45728-2 }}
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| * {{Citation
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| | last =Tammet
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| | last2 =Kulmala
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| | title =Simulating aerosol nucleation bursts in a coniferous forest
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| | date = June 2007
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| | url = http://www.borenv.net/BER/pdfs/ber12/ber12-421.pdf
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| | accessdate =10/07/2007 }}
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| * {{Cite journal | author = Ramachandran Venkatesan; Scott Fogler | title =Comments on Analogies for Correlated Heat and Mass Transfer in Turbulent Flow | journal = [[AIChE Journal]] | volume = 50 | issue = 7 | pages = 1623–1626 | year = 2004 | doi = 10.1002/aic.10146}}
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| * {{Citation
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| | date =
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| | url = http://webserver.dmt.upm.es/~isidoro/bk3/c12/Forced%20and%20natural%20convection.pdf| accessdate =2011-11-30 }}
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| {{DEFAULTSORT:Churchill-Bernstein equation}}
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| [[Category:Heat transfer]]
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| [[Category:Convection]]
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The author's title is Christy Brookins. It's not a typical thing but what I like doing is to climb but I don't have the time lately. Office supervising is where her primary earnings comes from but she's currently applied for another one. Ohio is where her home is.
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