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Wärme- und Stoffübertragung

, Volume 2, Issue 4, pp 240–245 | Cite as

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

  • S. Kakaç
  • M. R. Özgü
Article

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.

Keywords

Heat Transfer Heat Transfer Coefficient Nusselt Number Wall Temperature Force Convection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Nomenclature

A

Surface area, m2

Cp

Specific heat at constant pressure, kcal/kg grd

J0

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

r0

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 (=μCp/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

Subscripts

b

Refers to bulk, or mixed mean temperature

i

inlet

w

surface

T

constant wall temperature boundary condition

q

constant heat flux boundary condition

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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References

  1. [1]
    Ulrichson, D. L., andR. A. Schmitz: Int. J. Heat Mass Transfer Vol. 8 (1965) pp. 253/258.CrossRefGoogle Scholar
  2. [2]
    Roy, D. N.: Trans. ASME, J. of Heat Transfer Vol. 98 (1965) p. 425.CrossRefGoogle Scholar
  3. [3]
    Kays, W. M.: Trans. Am. Soc. Mech. Engrs. 77 (1955) pp. 1265/1274.Google Scholar
  4. [4]
    Heaton, H. S., W. C. Reynolds andW. M. Kays: Rep. No. AHT-5, Thermosciences Division, Dept. of Mech. Engng. Stanford University. Stanford (1962).Google Scholar
  5. [5]
    Basworth, R. T., andH. C. Ward: Paper no. 169, presented at the Annual Meeting of the Amer. Inst. Chem. Engrs. Chicago (1962).Google Scholar
  6. [6]
    Manohar, R.: Int. J. Heat Mass Transfer Vol. 12 (1969).Google Scholar
  7. [7]
    Sparrow, E. M., S. H. Lin andT. S. Lundgren: The Physics of Fluids, Vol. 7 (1964) No. 3,Google Scholar
  8. [8]
    Jeffreys, S. H., andB. S. Jeffreys: Methods of Mathematical Physics, Cambridge Press 1962.Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • S. Kakaç
    • 1
  • M. R. Özgü
    • 1
  1. 1.Ankara

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