Advertisement

Wärme - und Stoffübertragung

, Volume 1, Issue 3, pp 177–184 | Cite as

Heat transfer by Hagen-Poiseuille flow in the thermal development region with axial conduction

  • D. K. Hennecke
Article

Abstract

The heat transfer in the region of circular pipes close to the beginning of the heating section is investigated for low-Péclet-number flows with fully developed laminar velocity profile. Axial heat conduction is included and its effect on the temperature distribution is studied not only for the region downstream of the start of heating but also for that upstream. The energy equation is solved numerically by a finite difference method. Results are presented graphically for various Péclet numbers between 1 and 50. The boundary conditions are uniform wall temperature and uniform wall heat flux with step change at a certain cross-section. For the latter case, also some results for the region near the end of the heating section are reported. The solutions are applicable for the corresponding mass transfer situations where axial diffusion is important if the temperature is replaced by the concentration andPe byReSc.

Keywords

Heat Transfer Finite Difference Method Axial Diffusion Wall Heat Flux Circular Pipis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

A, B, C, D

coefficients in Eq. (20)

L

length of heated pipe section

Nu

Nusselt number (2hrw/k)

Pe

Péclet number (2Urwϱcp/k)

Q

heat flow

T

temperature

U

average evlocity

cp

specific heat

h

heat transfer coefficient

k

heat conductivity

q

heat flux

r

radial position

t

time variable

x

axial position

u

velocity

Δr

radial step size

Δx

axial step size

Δt

time step

ϱ

density

Subscripts

b

bulk

cond

conducted

conv

convected

d

development

fd

fully developed

i

value at cross-sectioni

j

value at radial positionj

i, j

value at nodal pointi, j

m

mean value

t

total

w

wall

0

value atx=–∞

1

value atx=+∞

Abbreviations

UWT

uniform wall temperature

UHF

uniform heat flux

Zusammenfassung

Der Wärmeübergang im thermischen Einlaufgebiet wird für den Fall der vollausgebildeten laminaren Rohrströmung mit kleinen Péclet-Zahlen untersucht. Axiale Wärmeleitung wird berücksichtigt, und ihr Einfluß auf die Temperaturverteilung nicht nur im Gebiet stromab vom Querschnitt des Heizbeginns, sondern auch in jenem stromauf, wird ermittelt. Die Energiegleichung wird numerisch mit einem Differenzenverfahren gelöst. Ergebnisse für verschiedene Péclet-Zahlen zwischen 1 und 50 sind graphisch dargestellt. Die Randbedingungen sind gleichförmige Wandtemperatur und gleichförmiger Wärmefluß mit sprunghafter Änderung an einem bestimmten Rohrquerschnitt. Für den letzteren Fall werden auch einige Ergebnisse für das Gebiet in der Nähe des Heizendes präsentiert. Die Lösungen sind für die entsprechenden Stoffübertragungssituationen anwendbar, in denen axiale Diffusion nicht vernachlässigt werden kann, indem man die Temperatur durch die Konzentration undPe durchReSc ersetzt.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Eckert, E. R. G., andR. M. Drake: Heat and Mass Transfer, 2nd ed. New York: McGraw-Hill Book Co., Inc. 1959.Google Scholar
  2. [2]
    Sellars, J. R., M. Tribus, andJ. S. Klein: Trans. ASME, Vol. 78 (1956), pp. 441/448.Google Scholar
  3. [3]
    Siegel, R., E. M. Sparrow, andT. M. Hallman: Appl. Sci. Res., Vol. A 7 (1957), pp. 386/392.Google Scholar
  4. [4]
    Brown, G. M.: AIChE J., Vol. 6 (1960), pp. 179/183.Google Scholar
  5. [5]
    Hsu, C. J.: AIChE J., Vol. 11 (1965), pp. 690/695.Google Scholar
  6. [6]
    Sparrow, E. M., T. M. Hallman, andR. Siegel: Appl. Sci. Res., Vol. A 7 (1957), pp. 37/52.Google Scholar
  7. [7]
    Strunk, M. R., andF. F. Tao: AIChE J., Vol. 10 (1964), pp. 269/273.Google Scholar
  8. [8]
    Wilson, H. A.: Cambridge Philos. Soc. Proc., Vol. 12 (1903–04), pp. 406/423.Google Scholar
  9. [9]
    Harrison, W. B.: ORNL-915 (1954).Google Scholar
  10. [10]
    Bodnarescu, M. V.: VDI-Forschungsheft 450, Ed. B, Bd. 21 (1955), S. 19/27.Google Scholar
  11. [11]
    Millsaps, K., andK. Pohlhausen: Proc. of the Conf. on Diff. Equations (J. B. Diaz andL. E. Payne, eds.), Univ. of Maryland Bookstore, College Park, Md. (1956), pp. 271/294.Google Scholar
  12. [12]
    Pahor, S., andJ. Strnad: Z. angew. Math. Phys., Vol. 7 (1956), pp. 536/538.Google Scholar
  13. [13]
    Schneider, P. J.: Heat Transfer and Fluid Mech. Inst. Stanford, Calif.: Stanford Univ. Press 1956; and Trans. ASME, Vol. 79 (1957), pp. 766/773.Google Scholar
  14. [14]
    Singh, S. N.: Appl. Sci. Res., Vol. A 7 (1957), pp. 237/250.Google Scholar
  15. [15]
    Singh, S. N.: Appl. Sci. Res., Vol. A 7 (1957), pp. 325/340.Google Scholar
  16. [16]
    Labuntsov, B. S.: Sov. Phys. Doklady, Vol. 3 (1958), pp. 33/35.Google Scholar
  17. [17]
    Agrawal, H.: Appl. Sci. Res., Vol. A 9 (1960), pp. 177/196.Google Scholar
  18. [18]
    Pahor, S., andJ. Strand: Appl. Sci. Res., Vol. A 10 (1961), pp. 81/84.Google Scholar
  19. [19]
    Gill, W. N., andS. M. Lee: AIChE J., Vol. 8 (1962), pp. 303/309.Google Scholar
  20. [20]
    Stein, R. P.: In: Advances in Heat Transfer, Vol. III,T. F. Irvine andJ. P. Hartnett, eds., New York: Academic Press 1966.Google Scholar
  21. [21]
    Hsu, C. J.: Appl. Sci. Res., Vol. 17 (1967), pp. 359/376.Google Scholar
  22. [22]
    McMordie, R. K., andA. F. Emery: Trans. ASME, J. Heat Transfer Vol. 89C (1967), pp. 11/16.Google Scholar
  23. [23]
    Burchill, W. E., R. P. Stein, andB. G. Jones: ASME Paper 67-WA/HT-26.Google Scholar
  24. [24]
    Johnson, H. A., J. P. Hartnett, andJ. W. Clabaugh: Trans. ASME, Vol. 76 (1954), pp. 513/517.Google Scholar
  25. [25]
    Petukhov, B. S., andA. J. Yushin: Sov. Phys. Doklady, Vol. 6 (1961), pp. 159/161.Google Scholar
  26. [26]
    Tratz, H.: Bundesministerium f. wiss. Forschung, Bericht K 67-05, 1967.Google Scholar
  27. [27]
    Holtz, R. E.: AIChE J., Vol. 11 (1965), pp. 1151/1153.Google Scholar
  28. [28]
    Emery, A. F., andD. A. Bailey: Trans. ASME, J. Heat Transfer, Vol. 89C (1967), pp. 272/273.Google Scholar
  29. [29]
    Kays, W. M.: Trans. ASME, Vol. 77 (1955), pp. 1265/1274.Google Scholar
  30. [30]
    Grigull, U., andH. Tratz: Int. J. Heat Mass Transfer, Vol. 8 (1965), pp. 669/678.Google Scholar
  31. [31]
    Forsythe, G. E., andW. R. Wasow: Finite-Difference Methods for Partial Differential Equations. Section 22. John Wiley & Sons, Inc. 1960.Google Scholar
  32. [32]
    Hennecke, D. K.: Heat Transfer Lab., Dept. of Mech. Eng., Univ. of Minnesota, HTL TR No. 78 (1968).Google Scholar

Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • D. K. Hennecke
    • 1
  1. 1.MinneapolisUSA

Personalised recommendations