Skip to main content
SpringerLink
Log in

Turbulent heat transfer in the thermal entrance region of a pipe with uniform heat flux

  • Published:
Applied Scientific Research, Section A Aims and scope Submit manuscript

Summary

An analysis has been performed using a method similar to Graetz’s formulation for the laminar thermal entry region. The fluid is assumed to have a fully developed turbulent velocity profile throughout the length of the pipe. Local and fully developed Nusselt numbers are presented for fluids with Prandtl numbers ranging from 0.7 to 100 for Reynolds numbers between 50000 and 500000. A thermal entrance length is defined as the heated length required to bring the local Nusselt number to within 5 percent of the fully developed value. This length is found to decrease with increasing Prandtl number, dropping from about 10 diameters for a Prandtl number of 0.7 to less than one diameter for a Prandtl number of 100. Comparison is made with the results of Deissler, who used an integral method-boundary layer approach, and also with available experimental data. The effect of the thermal boundary conditions was studied by comparing the present uniform heat flux results with those of previous investigators who considered uniform wall temperature.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

a 2 :

thermal diffusivity,k/ρc p

A n :

quantity used in calculating the Nusselt number from equation (18), C n φ n (r +0 )/G(r +0 ), dimensionless

C n :

coefficient in the series expansion ofϑ 2, dimensionless

c p :

specific heat at constant pressure

D :

pipe diameter

G(r+):

radial variation of the fully developed temperature profile, dimension-less

h :

local heat transfer coefficient,q/(T w −T b )

k :

thermal conductivity

Nu :

local Nusselt number,hD/k, dimensionless;Nu ∞, fully developed Nusselt number

Pr :

Prandtl number,v/a 2 =c p μ/k, dimensionless

q :

local heat transfer rate at wall to fluid

r :

radial coordinate;r 0, pipe radius

r + :

dimensionless radial coordinate, (r√τ 0/ρ)/v;r +0 dimensionless pipe radius, (r 0√ρ 0/ρ)/v

Re :

Reynolds number,Dū/v, dimensionless

T :

temperature;T i , entering fluid temperature;T w , wall temperature;T b , bulk temperature

u :

velocity component inx direction;ū, mean velocity

u + :

dimensionless velocity inx direction,u/(√τ 0/ρ)

x :

coordinate measuring axial distance from tube entrance

x + :

dimensionless axial coordinate,x/r 0

y :

coordinate measuring normal distance from tube wall,r 0 −r

y + :

dimensionless coordinate normal to wall,y√τ 0/ρ)/v

β 2 n :

eigenvalues of eq. (12b), dimensionless

γ :

dimensionless total thermal diffusivity, (a 2 +te h )/ν

ε h ,ε m :

eddy diffusivities for heat and momentum

ϑ :

dimensionless temperature, (T −T i )/(qr 0/k)

ν :

kinematic viscosity

ρ :

fluid density

τ 0 :

shear stress at wall

ϕ :

eigenfunctions of eq. (12b), dimensionless

χ :

exponentially decaying portion of entrance region temperature profile, dimensionless

References

  1. Jakob, M., Heat Transfer, John Wiley and Sons, New York 1949, Vol. I, p. 451–461.

    Google Scholar 

  2. Latzko, H., Heat transfer in a turbulent liquid or gas stream, NACA TM 1068, 1944. See also ref. 1), p. 476–480.

  3. Berry, V. J., Appl. Sci. Res.A4 (1953) 61.

    Google Scholar 

  4. Sleicher, C. A. and M. Tribus, Heat transfer in a pipe with turbulent flow and arbitrary wall temperature distribution. 1956 Heat Transfer and Fluid Mechanics Institute Preprints, Stanford University Press, Stanford, California, pp. 59–78.

    Google Scholar 

  5. Beckers, H. L., Appl. Sci. Res.A6 (1956) 147.

    MathSciNet  Google Scholar 

  6. Deissler, R. G., Trans. Amer. Soc. Mech. Engrs77 (1955) 1221. See also NACA TN 3016, 1953.

    Google Scholar 

  7. Deissler, R. G., Analysis of turbulent heat transfer, mass transfer and friction in smooth tubes at high Prandtl and Schmidt numbers, NACA TN 3145, 1954.

  8. Wolf, H. and J. Lehman, Project Squid Research, Jet Propulsion Center, Purdue Univ. private communication.

  9. Hartnett, J. P., Trans. Amer. Soc. Mech. Engrs77 (1955) 1211.

    Google Scholar 

  10. McAdams, W. H., Heat Transmission, 3rd ed., McGraw-Hill Book Co. Inc., New York 1954, p. 219.

    Google Scholar 

  11. Boelter, L. M. K., D. Young and H. W. Iverson, An investigation of aircraft heaters - XXVII, NACA TN 1451, 1948.

Download references

Author information

Authors and Affiliations

  1. NACA Lewis Laboratory, Cleveland, Ohio, U.S.A.

    E. M. Sparrow, T. M. Hallman & R. Siegel

Authors
  1. E. M. Sparrow
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. T. M. Hallman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Siegel
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sparrow, E.M., Hallman, T.M. & Siegel, R. Turbulent heat transfer in the thermal entrance region of a pipe with uniform heat flux. Appl. sci. Res. 7, 37–52 (1957). https://doi.org/10.1007/BF03184700

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF03184700

Keywords

Search

Navigation