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Viscous dissipation effects on fully developed combined free and forced non-Newtonian convection in a vertical tube

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Abstract

An investigation which includes the simultaneous effects of viscous dissipation and combined free and forced laminar non-Newtonian convection is presented. The problem under consideration is that of fully developed upflow in a vertical, circular tube which is heated with a constant wall heat flux. All properties are assumed to be constants in the analysis except for a temperature dependent density in the body force term which generates the free convection effects. The coupled continuity, momentum, and energy equations are solved using a finite difference technique. Numerical solutions are presented as a function of the parameters of the problem-flow behavior index n, Grashof number over Reynolds number ratio Gr/Re, and the Eckert number-Prandtl number product E Pr. The results show that heating due to viscous dissipation distorts the velocity profile, increases the friction factor, and decreases the Nusselt number.

Zusammenfassung

Für kombinierte erzwungene und freie laminare Konvektion einer nicht-Newton'schen Flüssigkeit wird eine Untersuchung unter Einschluß der Dissipation durch Zähigkeit vorgelegt. Es handelt sich um ausgebildete Aufwärtsströmung im senkrechten Heizrohr bei konstanter WandWärmestromdichte. Die Stoffwerte werden als konstant angenommen mit Ausnahme der Dichte im Auftriebsterm. Die gekoppelten Gleichungen der Kontinuität, der Impuls- und der Energieübertragung werden mit einer Methode der finiten Differenzen gelöst. Als Parameter treten auf der Exponent für das nicht-Newton'sche Verhalten, der Quotient Grashof-Zahl durch Reynolds-Zahl und das Produkt Eckert-Zahl und Prandtl-Zahl. Es zeigt sich, daß die dissipative Heizung das Geschwindigkeitsprofil zerstört, die Reibung erhöht und den Wärmeübergang herabsetzt.

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Abbreviations

A:

tube cross-sectional area

cp :

constant pressure specific heat

D:

tube diameter

E:

Eckert number, kv 2m /cpqw″ R

f:

local friction factor,τ w/1/2 ρv 2m

g:

acceleration due to gravity

Gr:

power law Grashof number,ρ 2gβiqw″ R2n+2v 2−2nm /k m2

h:

local convective heat transfer coefficient, k(∂T/∂r)w(Tw−Tm)

k:

thermal conductivity

m:

power law consistency index

n:

power law flow behavior index

Nu:

local Nusselt number, h D/k

p:

pressure

P:

dimensionless pressure, (p +ρ igz)/ρ v 2m Pr

Pr:

power law Prandtl number, (cpm/k)(vm/R)n−1

qw″:

wall heat flux, k(∂T/∂r)w

r:

radial coordinate

¯r:

dimensionless radial coordinate, r/R

R:

tube radius

Re:

power law Reynolds number,ρ v 2−nm Rn/m

T:

temperature

Tm :

mean temperature,\({{\mathop \smallint \limits_{\rm A} Tv_z dA} \mathord{\left/ {\vphantom {{\mathop \smallint \limits_{\rm A} Tv_z dA} {\mathop \smallint \limits_{\rm A} }}} \right. \kern-\nulldelimiterspace} {\mathop \smallint \limits_{\rm A} }}v_z dA\)

U:

dimensionless axial velocity, vz/vm

vm :

mean axial velocity, (1/A)\(\mathop \smallint \limits_{\rm A} v_z dA\)

vz :

axial velocity

w:

mass flow rate

z:

axial coordinate

\(\bar z\) :

dimensionless axial coordinate,\(\frac{{z/R}}{{Re Pr}}\)

β :

coefficient of volumetric expansion

θ :

dimensionless temperature, k(T−Ti)/Rqw″

ρ :

density

τ w :

wall shear stress, m(−dvz/dr) nw

i:

inlet

m:

mean

w:

wall

References

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Authors and Affiliations

  1. Heat Transfer Research, Inc., 1000 South Fremont Avenue, 91802, Alhambra, California, USA

    W. J. Marner (Assistant Professor)

  2. Research Specialties, Inc., 625 South Ninth Street, 57701, Rapid City, South Dakota, USA

    H. Hovland (Graduate Assistant)

Authors
  1. W. J. Marner
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  2. H. Hovland
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Marner, W.J., Hovland, H. Viscous dissipation effects on fully developed combined free and forced non-Newtonian convection in a vertical tube. Warme- und Stoffubertragung 6, 199–204 (1973). https://doi.org/10.1007/BF02575265

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  • DOI: https://doi.org/10.1007/BF02575265

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