Editing Heat transfer coefficient (section)

==Convective heat transfer correlations==
Although convective heat transfer can be derived analytically through dimensional analysis, exact analysis of the boundary layer, approximate integral analysis of the boundary layer and analogies between energy and momentum transfer, these analytic approaches may not offer practical solutions to all problems when there are no mathematical models applicable. Therefore, many correlations were developed by various authors to estimate the convective heat transfer coefficient in various cases including natural convection, forced convection for internal flow and forced convection for external flow. These empirical correlations are presented for their particular geometry and flow conditions. As the fluid properties are temperature dependent, they are evaluated at the [[film temperature]] <math>T_f</math>, which is the average of the surface <math>T_s</math> and the surrounding bulk temperature, <math>{{T}_{\infty }}</math>.

:<math>{{T}_{f}}=\frac{{{T}_{s}}+{{T}_{\infty }}}{2}</math>

=== External flow, vertical plane ===
Recommendations by Churchill and Chu provide the following correlation for natural convection adjacent to a vertical plane, both for laminar and turbulent flow.<ref>{{cite journal |last1=Churchill |first1=Stuart W. |last2=Chu |first2=Humbert H.S.|title=Correlating equations for laminar and turbulent free convection from a vertical plate |journal=International Journal of Heat and Mass Transfer |date=November 1975 |volume=18 |issue=11 |pages=1323–1329 |doi=10.1016/0017-9310(75)90243-4|bibcode=1975IJHMT..18.1323C }}</ref><ref>{{cite book |last1=Sukhatme |first1=S. P. |title=A Textbook on Heat Transfer |date=2005 |publisher=Universities Press |isbn=978-8173715440 |pages=257–258 |edition=Fourth}}</ref> ''k'' is the [[thermal conductivity]] of the fluid, ''L'' is the [[characteristic length]] with respect to the direction of gravity, Ra''<sub>L</sub>'' is the [[Rayleigh number]] with respect to this length and Pr is the [[Prandtl number]] (the Rayleigh number can be written as the product of the Grashof number and the Prandtl number).

:<math>h \ = \frac{k}{L}\left({0.825 + \frac{0.387 \mathrm{Ra}_L^{1/6}}{\left(1 + (0.492/\mathrm{Pr})^{9/16} \right)^{8/27} }}\right)^2 \, \quad \mathrm{Ra}_L < 10^{12}</math>

For laminar flows, the following correlation is slightly more accurate. It is observed that a transition from a laminar to a turbulent boundary occurs when Ra''<sub>L</sub>'' exceeds around 10<sup>9</sup>.

:<math>h \ = \frac{k}{L} \left(0.68 + \frac{0.67 \mathrm{Ra}_L^{1/4}}{\left(1 + (0.492/\mathrm{Pr})^{9/16}\right)^{4/9}}\right) \, \quad \mathrm10^{-1} < \mathrm{Ra}_L < 10^9 </math>

=== External flow, vertical cylinders ===
For cylinders with their axes vertical, the expressions for plane surfaces can be used provided the curvature effect is not too significant. This represents the limit where boundary layer thickness is small relative to cylinder diameter <math>D</math>. For fluids with Pr ≤ 0.72, the correlations for vertical plane walls can be used when<ref>{{cite journal | url=http://doi.org/10.1080/01457630801891557 | doi=10.1080/01457630801891557 | title=Free Convection Heat Transfer from Vertical Slender Cylinders: A Review | date=2008 | last1=Popiel | first1=Czeslaw O. | journal=Heat Transfer Engineering | volume=29 | issue=6 | pages=521–536 | bibcode=2008HTrEn..29..521P | url-access=subscription }}</ref>
   
:<math>\frac{D}{L}\ge \frac{35}{\mathrm{Gr}_{L}^{\frac{1}{4}}}</math>

where <math>\mathrm{Gr}_L</math> is the [[Grashof number]].

And in fluids of Pr ≤ 6 when

:<math>\frac{D}{L}\ge \frac{25.1}{\mathrm{Gr}_{L}^{\frac{1}{4}}}</math>

Under these circumstances, the error is limited to up to 5.5%.

=== External flow, horizontal plates ===
W. H. McAdams suggested the following correlations for horizontal plates.<ref>{{cite book |last=McAdams |first=William H. |title=Heat Transmission |date=1954 |publisher=McGraw-Hill |location=New York |page=180 |edition=Third}}</ref> The induced buoyancy will be different depending upon whether the hot surface is facing up or down.

For a hot surface facing up, or a cold surface facing down, for laminar flow:
   
:<math>h \ = \frac{k 0.54 \mathrm{Ra}_L^{1/4}} {L} \, \quad 10^5 < \mathrm{Ra}_L < 2\times 10^7</math>

and for turbulent flow:

:<math>h \ = \frac{k 0.14 \mathrm{Ra}_L^{1/3}} {L} \, \quad 2\times 10^7 < \mathrm{Ra}_L < 3\times 10^{10} .</math>

For a hot surface facing down, or a cold surface facing up, for laminar flow:

:<math>h \ = \frac{k 0.27 \mathrm{Ra}_L^{1/4}} {L} \, \quad 3\times 10^5 < \mathrm{Ra}_L < 3\times 10^{10}.</math>

The characteristic length is the ratio of the plate surface area to perimeter. If the surface is inclined at an angle ''θ'' with the vertical then the equations for a vertical plate by Churchill and Chu may be used for ''θ'' up to 60°; if the boundary layer flow is laminar, the gravitational constant ''g'' is replaced with ''g'' cos&nbsp;''θ'' when calculating the Ra term.

=== External flow, horizontal cylinder ===
For cylinders of sufficient length and negligible end effects, Churchill and Chu has the following correlation for <math>10^{-5}<\mathrm{Ra}_D<10^{12}</math>.

:<math>h \ =  \frac{k} {D}\left({0.6 + \frac{0.387 \mathrm{Ra}_D^{1/6}}{\left(1 + (0.559/\mathrm{Pr})^{9/16} \, \right)^{8/27} \,}}\right)^2</math>

=== External flow, spheres ===
For spheres, T. Yuge has the following correlation for Pr≃1 and <math>1 \le \mathrm{Ra}_D \le 10^5</math>.<ref name="Welty">{{cite book |author1=James R. Welty |author2=Charles E. Wicks |author3=Robert E. Wilson |author4=Gregory L. Rorrer |year=2007 |title=Fundamentals of Momentum, Heat and Mass transfer |publisher=John Wiley and Sons |isbn=978-0470128688|edition=5th }}</ref>

:<math>{\mathrm{Nu}}_D \ = 2 + 0.43 \mathrm{Ra}_D^{1/4}</math>

=== Vertical rectangular enclosure ===
For heat flow between two opposing vertical plates of rectangular enclosures, Catton recommends the following two correlations for smaller aspect ratios.<ref name="Cengel480">{{cite book |last=Çengel |first=Yunus |title=Heat and Mass Transfer |publisher=McGraw-Hill |page=480 |edition=Second}}</ref> The correlations are valid for any value of Prandtl number.

For  <math> 1 <\frac{H}{L} < 2 </math> :

:<math>h \ = \frac{k}{L}0.18 \left(\frac{\mathrm{Pr}}{0.2 + \mathrm{Pr}} \mathrm{Ra}_L \right)^{0.29} \, \quad \mathrm{Ra}_L \mathrm{Pr}/(0.2 + \mathrm{Pr}) > 10^3</math>

where ''H'' is the internal height of the enclosure and ''L'' is the horizontal distance between the two sides of different temperatures.

For  <math> 2 < \frac{H}{L} < 10 </math> :

:<math>h \ = \frac{k}{L}0.22 \left(\frac{\mathrm{Pr}}{0.2 + \mathrm{Pr}} \mathrm{Ra}_L \right)^{0.28} \left(\frac{H}{L} \right)^{-1/4} \, \quad \mathrm{Ra}_L < 10^{10}.</math>

For vertical enclosures with larger aspect ratios, the following two correlations can be used.<ref name="Cengel480" /> For 10 < ''H''/''L'' < 40:

:<math>h \ = \frac{k}{L}0.42 \mathrm{Ra}_L^{1/4} \mathrm{Pr}^{0.012} \left(\frac{H}{L} \right)^{-0.3} \, \quad 1 < \mathrm{Pr} < 2\times10^4, \, \quad 10^4 < \mathrm{Ra}_L < 10^7.</math>

For  <math> 1 < \frac{H}{L} < 40</math> :

:<math>h \ = \frac{k}{L}0.46 \mathrm{Ra}_L^{1/3} \, \quad 1 < \mathrm{Pr} < 20, \, \quad 10^6 < \mathrm{Ra}_L < 10^9.</math>

For all four correlations, fluid properties are evaluated at the average temperature—as opposed to film temperature—<math>(T_1+T_2)/2</math>, where <math>T_1</math> and <math>T_2</math> are the temperatures of the vertical surfaces and <math>T_1 > T_2</math>.

===Forced convection===
See main article [[Nusselt number]] and [[Churchill–Bernstein equation]] for forced convection over a horizontal cylinder.
====Internal flow, laminar flow====
Sieder and Tate give the following correlation to account for entrance effects in laminar flow in tubes where <math>D</math> is the internal diameter, <math>{\mu }_{b}</math> is the fluid viscosity at the bulk mean temperature, <math>{\mu }_{w}</math> is the viscosity at the tube wall surface temperature.<ref name="Welty"/>

:<math>\mathrm{Nu}_{D}={1.86}\cdot{{{\left( \mathrm{Re}\cdot\mathrm{Pr} \right)}^{{}^{1}\!\!\diagup\!\!{}_{3}\;}}}{{\left( \frac{D}{L} \right)}^{{}^{1}\!\!\diagup\!\!{}_{3}\;}}{{\left( \frac{{{\mu }_{b}}}{{{\mu }_{w}}} \right)}^{0.14}}</math>

For fully developed laminar flow, the Nusselt number is constant and equal to 3.66. Mills combines the entrance effects and fully developed flow into one equation

:<math>\mathrm{Nu}_{D}=3.66+\frac{0.065\cdot\mathrm{Re}\cdot\mathrm{Pr}\cdot\frac{D}{L}}{1+0.04\cdot\left( \mathrm{Re}\cdot\mathrm{Pr}\cdot\frac{D}{L}\right)^{2/3}}</math><ref>{{Cite web |url=http://web2.clarkson.edu/projects/subramanian/ch330/notes/Heat%20Transfer%20in%20Flow%20Through%20Conduits.pdf |title=Heat Transfer in Flow Through Conduits |last=Subramanian |first=R. Shankar |website=clarkson.edu}}</ref>

====Internal flow, turbulent flow====
{{See also|Dittus-Boelter equation}}
The Dittus-Bölter correlation (1930) is a common and particularly simple correlation useful for many applications. This correlation is applicable when forced convection is the only mode of heat transfer; i.e., there is no boiling, condensation, significant radiation, etc. The accuracy of this correlation is anticipated to be ±15%.

For a fluid flowing in a straight circular pipe with a [[Reynolds number]] between 10,000 and 120,000 (in the [[turbulent]] pipe flow range), when the fluid's [[Prandtl number]] is between 0.7 and 120, for a location far from the pipe entrance (more than 10 pipe diameters; more than 50 diameters according to many authors<ref>{{cite book |author1=S. S. Kutateladze |author2=V. M. Borishanskii |title=A Concise Encyclopedia of Heat Transfer |publisher=Pergamon Press |year=1966}}</ref>) or other flow disturbances, and when the pipe surface is hydraulically smooth, the heat transfer coefficient between the bulk of the fluid and the pipe surface can be expressed explicitly as:

:<math>{h d \over k}= {0.023} \, \left({j d \over \mu}\right)^{0.8} \, \left({\mu c_p \over k}\right)^n</math>

where:
:<math>d</math> is the [[hydraulic diameter]]
:<math>k</math> is the [[thermal conductivity]] of the bulk fluid
:<math>\mu</math> is the fluid [[viscosity]]
:<math>j</math> is the [[mass flux]]
:<math>c_p</math> is the isobaric [[heat capacity]] of the fluid
:<math>n</math> is 0.4 for heating (wall hotter than the bulk fluid) and 0.33 for cooling (wall cooler than the bulk fluid).<ref>{{cite book |editor=F. Kreith |editor-link=Frank Kreith |title=The CRC Handbook of Thermal Engineering |url=https://archive.org/details/crchandbookofthe00krei |url-access=registration |publisher=CRC Press |year=2000}}</ref>

The fluid properties necessary for the application of this equation are evaluated at the [[bulk temperature]] thus avoiding iteration.

====Forced convection, external flow====
In analyzing the heat transfer associated with the flow past the exterior surface of a solid, the situation is complicated by phenomena such as boundary layer separation. Various authors have correlated charts and graphs for different geometries and flow conditions.
For flow parallel to a plane surface, where <math>x</math> is the distance from the edge and <math>L</math> is the height of the boundary layer, a mean Nusselt number can be calculated using the [[Chilton and Colburn J-factor analogy|Colburn analogy]].<ref name="Welty" />