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A critical evaluation of analytical solutions and reynolds analogy equations for turbulent heat and mass transfer in smooth tubes

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Abstract

A critical evaluation of recently proposed analytical solutions and Reynolds analogy based correlations for heat transfer with constant properties to turbulent flow of liquids and gases in smooth tubes is presented. A new Reynolds analogy correlation is developed which agrees with the solutions of DeissleR, and Sparrow et al., within ±2 percent. The ability of this equation to correlate constant properties heat transfer data is compared with the equations proposed by FRiend and MetzneR, and Petukhov and Popov. Both the new equation and the Petukhov and Popov equation provide a very good correlation of existing data for.7 <Pr < 50. However, the Petukhov and Popov equation yielded a better correlation of the high Schmidt number mass transfer data of six investigators. Although there is considerable disagreement among these mass transfer data, the Petukhov and Popov equation agrees with the smoothed results of four investigators within ±15 percent. Therefore, this equation is tentatively recommended for use at high Prandtl or Schmidt numbers. The recommended equation is compared to the popular Colburn and Dittus-Boelter empirical equations and is shown to be superior to both equations.

Zusammenfassung

Es wird eine kritische Auswertung analytischer Lösungen und auf der Reynolds-Analogie basierender Beziehungen für den turbulenten Wärmeübergang im glatten Rohr bei konstanten Stoff werten vorgelegt. Eine neu entwikkelte, auf der Reynolds-Analogie basierende Korrelation stimmt mit den Lösungen von Deissler und von Sparrow u. a. innerhalb 2 Prozent überein. Diese Gleichung wird für den Fall konstanter Stoffwerte mit den Gleichungen vonFriend undMetzner und vonPcetukhov undPopov verglichen. Vorhandene Meßwerte werden im Bereich 0,7 <Pr < 50 sowohl von der neuen Gleichung wie von der vonPetukhov und Popov gut wiedergegeben. Bei hohen Schmidt-Zahlen gibt die letztgenannte Gleichung die Meßwerte für den Stoffübergang von sechs Autoren besser wieder. Trotz beträchtlicher Streuungen lassen sich die ausgeglichenen Meßwerte von vier Autoren mit einer Unsicherheit von 15 Prozent durch die Gleichung vonPetukhov undPopov beschreiben. Diese Gleichung wird daher vorläufig für hohe Prandtl- und Schmidt-Zahlen empfohlen. Sie ist auch den weit verbreiteten Beziehungen vonColburn und vonDittus undBoelter überlegen.

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Abbreviations

b :

Defined by Eq. 13

f :

Friction factor, (D/2u 2av )d P/d x

Pr :

Prandtl number evaluated at Tav

Pre :

Effective Prandtl number, ve/αe

Prt :

Turbulent Prandtl number, ɛm/ɛh

q :

Heat flux

r :

Radial coordinate, from centerline

R+ :

Ru*/ν

Re :

Pipe Reynolds number

St :

Stanton number

St c :

Stanton number from Colburn equation

St DB :

Stanton number from Dittus-Boelter equation

St s :

Stanton number given by Sparrow et al. analytical solution (ref. 2)

T :

Fluid temperature

T+ :

T/(qw/ϱcu*)

u :

Fluid velocity

u* :

\(\sqrt {\tau _W /\varrho } \)

u+ :

u/u*

y :

Coordinate distance normal to wall

yb :

Thickness of viscous influenced region

y+ :

yu*/ν

α:

Thermal diffusivity

αe :

α + ɛh

ɛh :

Eddy diffusivity of heat

ɛm :

Eddy diffusivity of momentum

ϑ :

(Tw − Tc)/(Tw -@#@ Tav)

ν :

Kinematic viscosity

τ:

Local time average shear stress

φ :

uc/uav

av:

Mixed-mean value

c:

Quantity evaluated at pipe centerline

w:

Quantity evaluated at wall(y=0)

References

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  1. Research Engineer The Trane Company, La Crosse, Wisconsin

    R. L. Webb (Senior)

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Webb, R.L. A critical evaluation of analytical solutions and reynolds analogy equations for turbulent heat and mass transfer in smooth tubes. Wärme- und Stoffübertragung 4, 197–204 (1971). https://doi.org/10.1007/BF01002474

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  • DOI: https://doi.org/10.1007/BF01002474

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