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Free-forced convection from a heated longitudinal horizontal cylinder embedded in a porous medium

Freie erzwungene Konvektion von einem beheizten schlanken horizontalen Zylinder, eingebettet in ein poröses Medium

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Abstract

The flow and heat transfer from a heated semi-infinite horizontal circular cylinder which is moving with a constant speed into a porous medium is considered. It is assumed that the Grashof and Reynolds numbers are large so that the governing equations are the three dimensional boundary-layer equations. A numerical procedure for solving these equations is described and the asymptotic solutions which are valid both near and distant from the leading edge of the cylinder are presented. The range of validity of these asymptotic solutions is discussed and the results are compared in detail with the full numerical solution. The problem is of practical importance, for example in the drilling of pipes into a geothermal reservoir.

Zusammenfassung

Es wird die Strömung und der Wärmeübergang an einem beheizten, halbunendlichen horizontalen Kreiszylinder betrachtet, der mit konstanter Geschwindigkeit sich in ein poröses Medium bewegt. Dabei wird angenommen, daß die Grashof- und Reynolds-Zahlen groß sind, so daß die Bestimmungsgleichungen von den dreidimensionalen Grenzschichtgleichungen gebildet werden. Es wird ein numerisches Verfahren zur Lösung dieser Gleichungen beschrieben und eine asymptotische Lösung präsentiert, die sowohl in der Nähe als auch in großem Abstand von dem vorderen Ende des Zylinders gültig ist. Der Gültigkeitsbereich dieser asymptotischen Lösungen wird diskutiert und die Ergebnisse werden im Detail mit vollständigen numerischen Lösungen verglichen. Das Problem ist z.B. beim Eindringen von Rohrleitungen in geothermische Reservoire von praktischer Wichtigkeit.

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Abbreviations

a :

radius of cylinder

Gr :

Grashof number (=gχ(Tw-T∞a/ν2)

g :

acceleration due to gravity

χ :

permeability in the porous medium

Nu :

local Nusselt number

\(\overline {Nu} \) :

total heat flux from cylinder

q w :

heat flux from cylinder

r :

radial co ordinate

Ra :

Rayleigh number (=g β χ(Tw - Tt8) a/α ν)

Re :

Reynolds number (=U t8 a/ν)

T :

temperature

u, v, w :

speeds inx, ϕ, r directions

x :

axial co ordinate

α:

equivalent thermal diffusivity

β :

thermal expansion coefficient

ɛ :

ratioGr/Re

η :

similarity variable

θ :

dimensionless temperature (=(T- T)/(T w- T∞)

ν :

kinematic viscosity

ϕ :

azimuthal co ordinate

w :

cylinder surface

∞:

free stream

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Authors and Affiliations

  1. Department of Applied Mathematical Studies, The University of Leeds, LS29JT, Leeds, England

    D. B. Ingham

  2. Faculty of Mathematics, University of Cluj, R-3400, Cluj, C.P. 253, Romania

    I. Pop

Authors
  1. D. B. Ingham
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  2. I. Pop
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Ingham, D.B., Pop, I. Free-forced convection from a heated longitudinal horizontal cylinder embedded in a porous medium. Warme- und Stoffubertragung 20, 283–289 (1986). https://doi.org/10.1007/BF01002419

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  • DOI: https://doi.org/10.1007/BF01002419

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