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Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe

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Abstract

A numerical solution, for incompressible, steady-state, laminar flow heat transfer in the combined entrance region of a circular tube is presented for the case of constant wall heat flux and constant wall temperature. The development of velocity profile is obtained from Sparrow's entrance region solution. This velocity distribution is used in solving the energy equation numerically to obtain temperature profiles. Variation of the heat transfer coefficient for these two different boundary conditions for the early stages of boundary layer formation on the pipe wall is obtained. Local Nusselt numbers are calculated and the results are compared with those given byUlrichson andSchmitz. The effect of the thermal boundary conditions is studied by comparing the uniform wall heat flux results with uniform wall temperature.

Zusammenfassung

Für inkompressible, stationäre, laminare Rohrströmung wird der Wärmeübergang im gemeinsamen Einlauf für konstante Wärmestromdichte und konstante Wandtemperatur numerisch berechnet. Die Entwicklung der Geschwindigkeitsprofile wird einer Lösung vonSparrow entnommen, mit deren Hilfe die Temperaturprofile berechnet werden. Hieraus erhält man den Verlauf des Wärmeübergangskoeffizienten am Beginn der Grenzschichtbildung. Die berechneten örtlichen Nusselt-Zahlen werden mit jenen nachUlrichson undSchmitz verglichen. Die Wirkung der thermischen Grenzschicht wird durch Vergleich der Ergebnisse für konstante Wärmestromdichte mit jenen für konstante Wandtemperatur untersucht.

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Abbreviations

A :

Surface area, m2

C p :

Specific heat at constant pressure, kcal/kg grd

J 0 :

Zero order Bessel functions of the first kind

T :

Temperature, °C

d :

Diameter of pipe, m

k :

Thermal conductivity, kcal/m h grd

m :

Number of increment inx direction

n :

Number of increment iny direction

q/A :

Heat flow rate per unit area, kcal/h m2

r :

Radial distance measured from centre of pipe, m

r 0 :

Radius of pipe, m

u :

Local axial velocity component, m/s

ū :

Average axial velocity, m/s

v :

Local radial velocity component, m/s

x :

Axial coordinate measured from inlet of pipe

x*:

Axial stretched coordinate measured from inlet of pipe

y :

Radial distance measured from pipe wall

ΔX, ΔY :

Dimensionless increments in axial and radial directions

Nu (=hd/k):

Nusselt number

Pr (=μC p/k):

Prandtl number

Re (=ūd/v):

Reynolds number

U (=u/ū):

Dimensionless velocity

X (=4x/d/Re):

Dimensionless axial coordinate

Y (=y/d):

Dimensionless radial coordinate

θ:

Fluid density, kg/m3

β:

Eigenvalues

μ:

Dynamic viscosity, kg/m·s

ν:

Kinematic viscosity, m2/s

b:

Refers to bulk, or mixed mean temperature

i:

inlet

w:

surface

T:

constant wall temperature boundary condition

q :

constant heat flux boundary condition

References

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  4. Heaton, H. S., W. C. Reynolds andW. M. Kays: Rep. No. AHT-5, Thermosciences Division, Dept. of Mech. Engng. Stanford University. Stanford (1962).

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  6. Manohar, R.: Int. J. Heat Mass Transfer Vol. 12 (1969).

  7. Sparrow, E. M., S. H. Lin andT. S. Lundgren: The Physics of Fluids, Vol. 7 (1964) No. 3,

  8. Jeffreys, S. H., andB. S. Jeffreys: Methods of Mathematical Physics, Cambridge Press 1962.

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Authors and Affiliations

  1. Ankara

    S. Kakaç & M. R. Özgü

Authors
  1. S. Kakaç
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  2. M. R. Özgü
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Kakaç, S., Özgü, M.R. Analysis of laminar flow forced convection heat transfer in the entrance region of a circular pipe. Wärme- und Stoffübertragung 2, 240–245 (1969). https://doi.org/10.1007/BF00751357

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  • DOI: https://doi.org/10.1007/BF00751357

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