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Heat transfer in laminar entrance region of a flat channel for the temperature boundary condition of the third kind

Wärmeübertragung bei der laminaren Einlaufströmung im ebenen Spalt mit der Randbedingung dritter Art

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Abstract

To explore the influence of the developing flow in a flat channel on the laminar forced convection heat transfer, the non-linear momentum and the linear energy equation are solved successively by employing the Galerkin-Kantorowich method of variational calculus. Assuming constant fluid properties, negligible axial diffusion and temperature boundary condition of the third kind, semi analytic solution for velocity and temperature is derived. The local Nusselt numbers are tabulated for various values of Biot and Prandtl number.

Zusammenfassung

Um den Einfluß der sich ausbildenden Strömung auf die laminare erzwungene Konvektion in einem ebenen Spalt zu untersuchen, werden die nicht-lineare Impuls- und lineare Energiegleichung mit dem Galerkin-Kantorowitsch-Verfahren der Variationsrechnung hintereinander gelöst. Unter der Annahme der konstanten Stoffwerte, der vernachlässigbaren axialen Diffusion und der Temperatur-Randbedingung dritter Art wird eine semi analytische Lösung für die Geschwindigkeit und Temperatur abgeleitet. Die lokalen Nusselt-Zahlen sind für verschiedene Biot- und Prandtl-Zahlen in den Tabellen wiedergegeben.

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Abbreviations

A:

channel cross section

Bi:

Biot number, Eq. (5)

F:

a vector, Eq. (7)

Nu:

Nusselt number, Eq. (27)

Pe:

Peclet number, Eq. (5)

Pr:

Prandtl number, Eq. (5)

Re:

Reynolds number, Eq. (5)

T:

temperature

c:

half channel height, Fig. 1

cp :

specific heat at constant pressure

dh :

hydraulic diameter

f:

a function, Eq. (7)

h:

heat transfer coefficient, Eq. (27)

k:

over-all heat transfer coefficient, Eq. (4)

p:

pressure

s:

characteristic value, Eq. (7)

v:

velocity

x, y, z:

cartesian coordinate

β:

a parameter, Eq. (6)

λ:

thermal conductivity

ν:

kinematic viscosity

ρ:

density

[ ]:

matrix

{ }:

column vector

a:

ambient

d:

fully developed

e:

entrance value at x=0

ent:

entrance region, Eq. (19)

i, j:

running index

inf:

asymptotic value at x → ∞

m:

mean value

w:

wall

x, y, z:

cartesian coordinate direction

−:

dimensionless quantity, Eq. (5)

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Authors and Affiliations

  1. Institut für Reaktorsicherheit, Glockengasse 2, 5000, Köln 1

    V. Javeri

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Javeri, V. Heat transfer in laminar entrance region of a flat channel for the temperature boundary condition of the third kind. Warme- und Stoffubertragung 10, 137–143 (1977). https://doi.org/10.1007/BF01682707

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  • DOI: https://doi.org/10.1007/BF01682707

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