Nusselt number 1

Nusselt numberIn heat transfer at a boundary (surface) within a fluid, the Nusselt number is the ratio of convective to conductiveheat transfer across (normal to) the boundary. In this context, convection includes both advection and conduction.Named after Wilhelm Nusselt, it is a dimensionless number. The conductive component is measured under the sameconditions as the heat convection but with a (hypothetically) stagnant (or motionless) fluid.A Nusselt number close to one, namely convection and conduction of similar magnitude, is characteristic of "slugflow" or laminar flow. A larger Nusselt number corresponds to more active convection, with turbulent flow typicallyin the 100–1000 range.The convection and conduction heat flows are parallel to each other and to the surface normal of the boundarysurface, and are all perpendicular to the mean fluid flow in the simple case.

where:• L = characteristic length• kf = thermal conductivity of the fluid• h = convective heat transfer coefficientSelection of the characteristic length should be in the direction of growth (or thickness) of the boundary layer. Someexamples of characteristic length are: the outer diameter of a cylinder in (external) cross flow (perpendicular to thecylinder axis), the length of a vertical plate undergoing natural convection, or the diameter of a sphere. For complexshapes, the length may be defined as the volume of the fluid body divided by the surface area. The thermalconductivity of the fluid is typically (but not always) evaluated at the film temperature, which for engineeringpurposes may be calculated as the mean-average of the bulk fluid temperature and wall surface temperature. Forrelations defined as a local Nusselt number, one should take the characteristic length to be the distance from thesurface boundary to the local point of interest. However, to obtain an average Nusselt number, one must integratesaid relation over the entire characteristic length.Typically, for free convection, the average Nusselt number is expressed as a function of the Rayleigh number and thePrandtl number, written as: Nu = f(Ra, Pr). Else, for forced convection, the Nusselt number is generally a function ofthe Reynolds number and the Prandtl number, or Nu = f(Re, Pr). Empirical correlations for a wide variety ofgeometries are available that express the Nusselt number in the aforementioned forms.The mass transfer analog of the Nusselt number is the Sherwood number.

DerivationThe Nusselt Number may be obtained by a non dimensional analysis of the Fourier's law since it is equal to thedimensionless temperature gradient at the surface:

, where q is the heat flux, k is the thermal conductivity and T the fluid temperature.

Indeed if: , and

we arrive at :

then we define :

so the equation become : By integrating over the surface of the body:

Nusselt number 2

, where

Empirical Correlations

Free convection

Free convection at a vertical wall

Cited[1] as coming from Churchill and Chu:

Free convection from horizontal plates

If the characteristic length is defined

where is the surface area of the plate and is its perimeter, then for the top surface of a hot object in a colderenvironment or bottom surface of a cold object in a hotter environment[1]

And for the bottom surface of a hot object in a colder environment or top surface of a cold object in a hotterenvironment[1]

Flat plate in turbulent flow

The local Nusselt number for a turbulant flow is given by [2]

Forced convection in turbulent pipe flow

Gnielinski correlation

Gnielinski is a correlation for turbulent flow in tubes:[2]

where f is the Darcy friction factor that can either be obtained from the Moody chart or for smooth tubes fromcorrelation developed by Petukhov[2] :

The Gnielinski Correlation is valid for[2] :

Nusselt number 3

Dittus-Boelter equation

The Dittus-Boelter equation (for turbulent flow) is an explicit function for calculating the Nusselt number. It is easyto solve but is less accurate when there is a large temperature difference across the fluid. It is tailored to smoothtubes, so use for rough tubes (most commercial applications) is cautioned. The Dittus-Boelter equation is:

where:is the inside diameter of the circular ductis the Prandtl number

for heating of the fluid, and for cooling of the fluid.[1]

The Dittus-Boelter equation is valid for[1]

Example The Dittus-Boelter equation is a good approximation where temperature differences between bulk fluidand heat transfer surface are minimal, avoiding equation complexity and iterative solving. Taking water with a bulkfluid average temperature of 20 °C, viscosity 10.07×10¯⁴ Pa·s and a heat transfer surface temperature of 40 °C(viscosity 6.96×10¯⁴, a viscosity correction factor for can be obtained as 1.45. This increases to 3.57 with aheat transfer surface temperature of 100 °C (viscosity 2.82×10¯⁴ Pa·s), making a significant difference to the Nusseltnumber 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 nonlinearboundary value problem. The Sieder-Tate result can be more accurate as it takes into account the change in viscosity( and ) due to temperature change between the bulk fluid average temperature and the heat transfer surfacetemperature, respectively. The Sieder-Tate correlation is normally solved by an iterative process, as the viscosityfactor will change as the Nusselt number changes.[3]

[1]

where:• is the fluid viscosity at the bulk fluid temperature• is the fluid viscosity at the heat-transfer boundary surface temperatureThe Sieder-Tate correlation is valid for[1]

Nusselt number 4

Forced convection in fully developed laminar pipe flowFor fully developed internal laminar flow, the Nusselt numbers are constant-valued. The values depend on thehydraulic diameter.For internal Flow:

where:• Dh = Hydraulic diameter• kf = thermal conductivity of the fluid• h = convective heat transfer coefficient

Convection with uniform surface heat flux for circular tubes

From Incropera & DeWitt,[4]

Convection with uniform surface temperature for circular tubes

For the case of constant surface temperature,[4]

External Links• Simple derivation of the Nusselt number from Newton's law of cooling [5] (Accessed 23 September 2009)

References[1] Incropera, Frank P.; DeWitt, David P. (2000). Fundamentals of Heat and Mass Transfer (4th ed.). New York: Wiley. p. 493.

ISBN 0471304603.[2] Incropera, Frank P.; DeWitt, David P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). Hoboken: Wiley. pp. 490, 515.

ISBN 9780471457282.[3] "Temperature Profile in Steam Generator Tube Metal" (http:/ / www. profjrwhite. com/ math_methods/ pdf_files_hw/ sgtm3. pdf). . Retrieved

23 September 2009.[4] Incropera, Frank P.; DeWitt, David P. (2002). Fundamentals of Heat and Mass Transfer (5th ed.). Hoboken: Wiley. pp. 486, 487.

ISBN 0471386502.[5] http:/ / www. jhu. edu/ virtlab/ heat/ nusselt/ nusselt. htm

Article Sources and Contributors 5

Article Sources and ContributorsNusselt number  Source: http://en.wikipedia.org/w/index.php?oldid=465266764  Contributors: Ajs1202, Alfy Alf, Bender235, Birdhurst, Chris the speller, Dav2008, Davidkearns, Djd sd,Donk10k, Eusha, Evil saltine, Flyingdreams, Haasl, Hankwang, Hashemabadi, Isucheme, Jdpipe, Karada, Kaszeta, Lcaretto, Mandavi, Michael Hardy, Mikiemike, Mkbnett, Movses, Nardinc,Nick Number, QuantumEngineer, RDT2, Re34646, Rich Farmbrough, Rjwilmsi, Salih, Shadowfax0, Shehal, Skrivande, Slashme, SlaveToTheWage, SpaceFlight89, Svdmolen, Vaughan Pratt,Venny85, Wizard191, Xtobal, 92 ,قلی زادگان anonymous edits

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