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Unsteady thermal entrance heat transfer in laminar pipe flows with step change in ambient temperature

Instationärer Wärmeübergang im thermischen Einlauf laminarer Rohrströmungen bei plötzlicher Änderung der Umgebungstemperatur

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Abstract

A fully implicit upwind finite difference numerical scheme has been proposed to investigate the characteristics of thermal entrance heat transfer in laminar pipe flows subject to a step change in ambient temperature. In order to demonstrate the results more clearly, a modified Nusselt number is introduced. The unsteady axial variations of modified Nusselt number, bulk fluid temperature, and wall temperature and the transient temperature profiles at certain axial locations are presented graphically for various outside heat transfer coefficients. The effects of the outside heat transfer coefficient on the heat transport processes in the flow are examined in detail. The results can be comprehensively explained by the interaction between the upstream convective heat transfer and the diffusion heat transfer in the radial direction. Steady state is reached when the axial convection balances the radial diffusion.

Zusammenfassung

Eine vollständig implizite Differenzen-Methode mit gegen den Strom gerichteten Schritten (upwind) wird angewandt, um die Zustandsänderung im thermischen Einlauf einer laminaren Rohrströmung als Folge eines Sprunges der Umgebungstemperatur zu berechnen.

Um die Ergebnisse klarer darzustellen, wird eine modifizierte Nusselt-Zahl eingeführt. Die instationäre axiale Änderung dieser Zahl, der mittleren Flüssigkeits- und der Wandtemperatur und der Temperaturprofile in der Strömung werden für verschiedene äußere Wärmeübergangszahlen graphisch dargestellt. Der Einfluß des äußeren Wärmeüberganges auf den in der Rohrströmung wird untersucht. Die Ergebnisse lassen sich durch das Zusammenwirken des im thermischen Einlauf beginnenden konvektiven Wärmetransportes und der radialen Wärmeleitung erklären. Der stationäre Zustand ist erreicht, wenn der axiale konvektive Transport gleich der radialen Wärmeleitung wird.

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Abbreviations

a ij :

coefficients defined in Eq. (6)

h :

heat transfer coefficient inside the pipe

h e :

modified local heat transfer coefficient; Eq. (9)

i, j, m :

indices for finite difference discretization

J :

total number of grid points in radial direction

k :

thermal conductivity of the fluid in the pipe

Nu :

local Nusselt number;h (2 R)/k

Nu e :

modified local Nusselt number; Eq. (10)

Nu o :

outside Nusselt number;UR/k

Pe :

Peclet number;u m (2 R)/α

q″ w :

wall heat flux

r :

radial coordinate

R :

pipe radius

t :

time

T :

temperature

u :

axial velocity

U :

outside heat transfer coefficient; Eq. (2)

x :

axial coordinate

α :

thermal diffusivity of the fluid in the pipe

Δη :

dimensionless radial interval

Δξ :

dimensionless axial interval

Δτ :

dimensionless time step

η :

dimensionless radial coordinate

θ :

dimensionless temperature difference

ξ :

dimensionless axial coordinate

τ :

dimensionless time

a :

ambient

b :

bulk

e :

entrance

m :

mean

o :

outside

w :

wall

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Authors and Affiliations

  1. School of Mechanical Engineering, Purdue University, West Lafayette, Indiana

    T. F. Lin (Graduate Student, Visiting Ass. Prof.),  K. H. Hawks &  W. Leidenfrost

  2. Department of Mechanical Engineering, Rochester Institute of Technology, Rochester, New York

    T. F. Lin (Graduate Student, Visiting Ass. Prof.)

Authors
  1. T. F. Lin
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  2. K. H. Hawks
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  3. W. Leidenfrost
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Lin, T.F., Hawks, K.H. & Leidenfrost, W. Unsteady thermal entrance heat transfer in laminar pipe flows with step change in ambient temperature. Warme- und Stoffubertragung 17, 125–132 (1983). https://doi.org/10.1007/BF02570522

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  • DOI: https://doi.org/10.1007/BF02570522

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