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Fully developed laminar mixed convection in uniformly heated pipes

Ausgebildete Laminarmischkonvektion in gleichförmig beheizten Rohren

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Abstract

Approximate solutions are carried out for fully developed laminar mixed convection in uniformly heated horizontal pipes. The problem considered is steady state and validity of the Boussinesq approximation is assumed. The analysis reveals that the relevant parameters are the Prandtl number and the product of the Rayleigh number based on temperature gradient along the pipe wall and of the Reynolds number based on pressure gradient along the pipe axis. Solutions for the velocity and temperature fields are obtained by a spectral Galerkin method, which proves capable of giving rapid convergence and, by implication, good accuracy, as demonstrated by the comparison with other numerical predictions. The results are presented in terms of fluid flow patterns, isotherms, and graphs of mass flow rate, local and average Nusselt number for various values of the parameters.

Zusammenfassung

Es werden Näherungslösungen für ausgebildete Laminarmischkonvektion in gleichförmig beheizten Rohren angeboten. Die Boussinesq-Approximation geht bei der Ermittlung des Stationärzustandes als Voraussetzung ein. Die Untersuchung zeigt, daß als relevante Parameter die Prandtl-Zahl und das Produkt aus Rayleigh-Zahl (basierend auf dem Temperaturgradienten entlang der Rohrwand) und Reynolds-Zahl (basie-rend auf dem Druckgradienten entlang der Rohrachse) auftreten. Lösungen für die Geschwindigkeits- und Temperaturfelder werden mit Hilfe einer spektralen Galerkinmethode erhalten, die rasche Konvergenz zeigt und auf gute Genauigkeit schließen läßt, wie durch Vergleich mit anderen numerischen Vorhersagen gezeigt werden kann. Die Ergebnisse werden in Form von Strömungsbildern, von Isothermen und von Diagrammdarstellungen des Massenstroms, der lokalen und der gemittelten Nußelt-Zahl als Funktion verschiedener Parameter angeboten.

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Abbreviations

a :

pipe radius

A, B :

dummy functions

c :

specific heat

c ij :

normalizing factors

d ij :

normalizing factors

f :

dimensionless stream function

f ij :

expansion coefficients forf N,M

f N,M :

approximate solution forf

F ij :

trial functions forf N,M

g :

acceleration due to gravity

G ij :

trial functions forW N,M and θN,M

h :

local heat transfer coefficient

\(\bar h\) :

average heat transfer coefficient

J :

Jacobian operator

I i :

Bessel function of orderi

J i :

modified Bessel function of orderi

k :

thermal conductivity

m :

mass flow rate

N :

number of radial modes

M :

number of circumferential modes

Nu :

local Nusselt number

\(\overline {Nu}\) :

average Nusselt number

p :

pressure

p′:

pressure distribution in the cross section defined by Eq. (2)

P :

dimensionless pressure

Pr :

Prandtl number

q :

heat flux

Q :

heat flow per unit length

r :

radial coordinate

R :

dimensionless radial coordinate

Ra :

Rayleigh number

Re :

Reynolds number

T :

temperature

T′:

temperature distribution in the cross section defined by Eq. (1)

u, v, w :

radial, tangential and axial velocity

U, V, W :

dimensionless radial, tangential and axial velocity

w i,j :

expansion coefficient forW N,M

x :

dummy variable

z :

axial coordinate

β :

coefficient of thermal expansion

γ :

axial pressure gradient

μ :

convergence parameter

∇:

gradient operator

Δ:

Laplace operator

Δ2 :

biharmonic operator

θ :

dimensionless temperature

θ i,j :

expansion coefficient forθ N,M

λ ij :

eigenvalues

μ :

viscosity

ν ij :

eigenvalues

ρ :

density

τ :

axial temperature gradient

ψ :

tangential coordinate

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Authors and Affiliations

  1. Dipartimento di Energetica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133, Milano, Italia

    A. Muzzio & P. Parolini

Authors
  1. A. Muzzio
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  2. P. Parolini
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Muzzio, A., Parolini, P. Fully developed laminar mixed convection in uniformly heated pipes. Warme - Und Stoffubertragung 29, 487–494 (1994). https://doi.org/10.1007/BF01539501

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  • DOI: https://doi.org/10.1007/BF01539501

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