Abstract
Thermal entrance region heat transfer for laminar forced convection of power-law fluids inside a circular tube and parallel plate channel for uniform wall temperature is solved exactly, and as many eigenvalues and eigenfunctions as needed for the solution are determined automatically and with high accuracy by using the recently advanced Sign-Count method. Results are presented for the local and average Nusselt number over a wide range of the Graetz number in both graphical and tabular forms. The present benchmark results are utilized to critically examine the accuracy of the approximate Leveque solution.
Zusammenfassung
Es werden exakte Lösungen für den Wärmetransport in der thermischen Einlaufzone in runden Rohren und zwischen parallelen Platten für Fluide nach dem Exponentialansatz bei laminarer Zwangskonvektion und mit gleichmäßiger Wandtemperatur angegeben. Unter Benutzung der jüngst verbesserten Sign-Count-Methode werden so viele Eigenwerte und Eigenfunktionen, wie für die Lösung benötigt, automatisch und mit großer Genauigkeit bestimmt. Ergebnisse werden in graphischer wie tabellarischer Form über einen weiten Bereich der Graetz-Zahl für die örtliche und mittlere Nusselt-Zahl vorgestellt. Die vorliegenden richtungsweisenden Ergebnisse werden dazu benützt, um die Genauigkeit der Levequeschen Näherungslösung kritisch zu prüfen.
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Abbreviations
- b :
-
radius of circular duct or half the spacing between parallel plates
- C :
-
\(\frac{{1 + (2 + p)n}}{{1 + n}}\)
- D h :
-
hydraulic diameter=4b for parallel plate, 2b for circular tube
- f(r), F(r) :
-
temperature distribution at the inlet, dimensional and dimensionless, respectively
- g (r), G (R) :
-
energy generation, dimensional and dimensionless, respectively
- h (z) :
-
heat transfer coefficient
- H i(Z):
-
defined by Eq. (11b)
- k :
-
thermal conductivity
- l0,l N :
-
reference lengths to nondimensionalizer andz coordinates respectively (l 0=b andl N= Dh)
- n :
-
power-law index
- Nu av :
-
average Nusselt number=h av Dh/k
- Nu (Z):
-
local Nusselt number=h (z)D h/k
- p :
-
0 for parallel-plate duct, 1 for circular duct
- r :
-
radial or normal coordinate
- R :
-
\(\frac{r}{{l_0 }}\) = dimensionless radial coordinate
- T(r, z) :
-
fluid temperature
- T av (z):
-
average fluid temperature
- T * :
-
reference temperature
- ΔT :
-
reference temperature difference
- U(R):
-
\(\frac{{w(r)}}{{w_{av} }}\) = normalized velocity profile
- U *(R):
-
\(1 - \left( {\frac{{l_0 }}{b}R} \right)^{\frac{{n + 1}}{n}} \)
- w (r) :
-
fully developed velocity profile
- w av :
-
average velocity
- z:
-
axial coordinate
- Z :
-
\(\frac{{\alpha z}}{{w_{av} l_N^2 }}\) = dimensionless axial coordinate
- α :
-
thermal diffusivity
- θ (R,Z) :
-
\(\frac{{T(r,z) - T^* }}{{\Delta {\rm T}}}\) = dimensionless temperature
- θ w :
-
\(\frac{{T_w (z) - T^* }}{{\Delta {\rm T}}}\) = dimensionless wall temperature
- μ i :
-
eigenvalues of the eigenvalue problem (10)
- ψ (μi,R):
-
eigenfunctions of the eigenvalue problem (10)
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Cotta, R.M., Özi§ik, M.N. Laminar forced convection of power-law non-Newtonian fluids inside ducts. Wärme- und Stoffübertragung 20, 211–218 (1986). https://doi.org/10.1007/BF01303453
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DOI: https://doi.org/10.1007/BF01303453