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Heat transfer by Hagen-Poiseuille flow in the thermal development region with axial conduction

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Abstract

The heat transfer in the region of circular pipes close to the beginning of the heating section is investigated for low-Péclet-number flows with fully developed laminar velocity profile. Axial heat conduction is included and its effect on the temperature distribution is studied not only for the region downstream of the start of heating but also for that upstream. The energy equation is solved numerically by a finite difference method. Results are presented graphically for various Péclet numbers between 1 and 50. The boundary conditions are uniform wall temperature and uniform wall heat flux with step change at a certain cross-section. For the latter case, also some results for the region near the end of the heating section are reported. The solutions are applicable for the corresponding mass transfer situations where axial diffusion is important if the temperature is replaced by the concentration andPe byReSc.

Zusammenfassung

Der Wärmeübergang im thermischen Einlaufgebiet wird für den Fall der vollausgebildeten laminaren Rohrströmung mit kleinen Péclet-Zahlen untersucht. Axiale Wärmeleitung wird berücksichtigt, und ihr Einfluß auf die Temperaturverteilung nicht nur im Gebiet stromab vom Querschnitt des Heizbeginns, sondern auch in jenem stromauf, wird ermittelt. Die Energiegleichung wird numerisch mit einem Differenzenverfahren gelöst. Ergebnisse für verschiedene Péclet-Zahlen zwischen 1 und 50 sind graphisch dargestellt. Die Randbedingungen sind gleichförmige Wandtemperatur und gleichförmiger Wärmefluß mit sprunghafter Änderung an einem bestimmten Rohrquerschnitt. Für den letzteren Fall werden auch einige Ergebnisse für das Gebiet in der Nähe des Heizendes präsentiert. Die Lösungen sind für die entsprechenden Stoffübertragungssituationen anwendbar, in denen axiale Diffusion nicht vernachlässigt werden kann, indem man die Temperatur durch die Konzentration undPe durchReSc ersetzt.

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Abbreviations

A, B, C, D :

coefficients in Eq. (20)

L :

length of heated pipe section

Nu :

Nusselt number (2hr w/k)

Pe :

Péclet number (2Ur w ϱc p/k)

Q :

heat flow

T :

temperature

U :

average evlocity

c p :

specific heat

h :

heat transfer coefficient

k :

heat conductivity

q :

heat flux

r :

radial position

t :

time variable

x :

axial position

u :

velocity

Δr :

radial step size

Δx :

axial step size

Δt :

time step

ϱ :

density

b:

bulk

cond:

conducted

conv:

convected

d:

development

fd:

fully developed

i :

value at cross-sectioni

j :

value at radial positionj

i, j :

value at nodal pointi, j

m:

mean value

t:

total

w:

wall

0:

value atx=–∞

1:

value atx=+∞

UWT:

uniform wall temperature

UHF:

uniform heat flux

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  1. Minneapolis, Minnesota, USA

    D. K. Hennecke

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  1. D. K. Hennecke
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Superscripts dimensionless quantity (Eqs. (10, 11, 16))

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Hennecke, D.K. Heat transfer by Hagen-Poiseuille flow in the thermal development region with axial conduction. Wärme- und Stoffübertragung 1, 177–184 (1968). https://doi.org/10.1007/BF00751149

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