Editing Nusselt number (section)

===Forced convection in turbulent pipe flow===

====Gnielinski correlation====
Gnielinski's correlation for turbulent flow in tubes:<ref name="incropera">{{cite book |author-link=Frank P. Incropera |last1=Incropera |first1=Frank P. |last2=DeWitt |first2=David P. |title=Fundamentals of Heat and Mass Transfer |url=https://archive.org/details/fundamentalsheat00incr_617 |url-access=limited |edition=6th |location=Hoboken |publisher=Wiley |year=2007 |isbn=978-0-471-45728-2 }}</ref>{{rp|pp=490,515}}<ref name="Gnielinski1975">{{cite journal |last=Gnielinski |first=Volker |title=Neue Gleichungen für den Wärme- und den Stoffübergang in turbulent durchströmten Rohren und Kanälen |pages=8–16 |year=1975 |journal=Forsch. Ing.-Wes. |volume=41 |issue=1|doi=10.1007/BF02559682 |s2cid=124105274 }}</ref>

:<math>\mathrm{Nu}_D = \frac{ \left( f/8 \right) \left( \mathrm{Re}_D - 1000 \right) \mathrm{Pr} } {1 + 12.7(f/8)^{1/2} \left( \mathrm{Pr}^{2/3} - 1 \right)}</math>

where f is the [[Darcy friction factor]] that can either be obtained from the [[Moody chart]] or for smooth tubes from correlation developed by Petukhov:{{r|incropera|p=490}}

:<math>f= \left( 0.79 \ln \left(\mathrm{Re}_D \right)-1.64 \right)^{-2}</math>

The Gnielinski Correlation is valid for:{{r|incropera|p=490}}

:<math>0.5 \le \mathrm{Pr} \le 2000</math>
:<math>3000 \le \mathrm{Re}_D \le 5 \times 10^{6}</math>

====Dittus–Boelter equation====
The Dittus–Boelter equation (for turbulent flow) as introduced by W.H. McAdams<ref>{{cite journal |last1=Winterton |first1=R.H.S. |title=Where did the Dittus and Boelter equation come from? |journal=International Journal of Heat and Mass Transfer |date=February 1998 |volume=41 |issue=4–5 |pages=809–810 |doi=10.1016/S0017-9310(97)00177-4 |publisher=Elsevier|bibcode=1998IJHMT..41..809W |url=http://herve.lemonnier.sci.free.fr/TPF/NE/Winterton.pdf}}</ref> is an [[explicit function]] for calculating the Nusselt number. It is easy to solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smooth tubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus–Boelter equation is:

:<math>\mathrm{Nu}_D = 0.023\, \mathrm{Re}_D^{4/5}\, \mathrm{Pr}^{n}</math>

where:
:<math>D</math> is the inside diameter of the circular duct
:<math>\mathrm{Pr}</math> is the [[Prandtl number]]
:<math>n = 0.4</math> for the fluid being heated, and <math>n = 0.3</math> for the fluid being cooled.{{r|incropera|p=493}}

The Dittus–Boelter equation is valid for{{r|incropera|p=514}}
:<math>0.6 \le \mathrm{Pr} \le 160</math>
:<math>\mathrm{Re}_D \gtrsim 10\,000</math>
:<math>\frac{L}{D} \gtrsim 10</math>

The Dittus–Boelter equation is a good approximation where temperature differences between bulk fluid and heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulk fluid average temperature of {{cvt|20|C}}, viscosity {{val|10.07e-4|u=Pa.s}} and a heat transfer surface temperature of {{cvt|40|C}} (viscosity {{val|6.96e-4|u=Pa.s}}, a viscosity correction factor for <math>({\mu} / {\mu_s})</math> can be obtained as 1.45. This increases to 3.57 with a heat transfer surface temperature of {{cvt|100|C}} (viscosity {{val|2.82e-4|u=Pa.s}}), making a significant difference to the Nusselt number and the heat transfer coefficient.

====Sieder–Tate correlation====
The Sieder–Tate correlation for turbulent flow is an [[implicit function]], as it analyzes the system as a nonlinear [[boundary value problem]]. The Sieder–Tate result can be more accurate as it takes into account the change in [[viscosity]] (<math>\mu</math> and <math>\mu_s</math>) due to temperature change between the bulk fluid average temperature and the heat transfer surface temperature, respectively. The Sieder–Tate correlation is normally solved by an iterative process, as the viscosity factor will change as the Nusselt number changes.<ref>{{cite web |url=http://www.profjrwhite.com/math_methods/pdf_files_hw/sgtm3.pdf |title=Temperature Profile in Steam Generator Tube Metal |access-date=23 September 2009 |archive-url=https://web.archive.org/web/20160303224930/http://www.profjrwhite.com/math_methods/pdf_files_hw/sgtm3.pdf |archive-date=3 March 2016 |url-status=dead }}</ref>

:<math>\mathrm{Nu}_D = 0.027\,\mathrm{Re}_D^{4/5}\, \mathrm{Pr}^{1/3}\left(\frac{\mu}{\mu_s}\right)^{0.14}</math>{{r|incropera|p=493}}

where:
:<math>\mu</math> is the fluid viscosity at the bulk fluid temperature
:<math>\mu_s</math> is the fluid viscosity at the heat-transfer boundary surface temperature

The Sieder–Tate correlation is valid for{{r|incropera|p=493}}

:<math>0.7 \le \mathrm{Pr} \le 16\,700</math>
:<math>\mathrm{Re}_D \ge 10\,000</math>
:<math>\frac{L}{D} \gtrsim 10</math>