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An exact mathematical solution for entrance-region laminar heat transfer with axial conduction

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Summary

A relatively simple mathematical scheme is proposed by which the entrance-region temperature solution for laminar flow heat transfer with axial conduction can be rigorously obtained.

For Poiseuille pipe flow (parabolic velocity profile) with uniform wall heat flux, the accurate first twelve eigenvalues, eigenfunctions, and the coefficients of the series expansion which are required in the temperature solution have been determined for Peclet numbers of 5, 10, 20, 30, 50, and 100. In addition, asymptotic expressions for the eigenvalues and the eigenfunction,R n (1), are derived. It is possible to use the asymptotic equation to predict, with satisfactory accuracy, even the first few eigenvalues for all the Peclet numbers considered.

By employing the computed eigenvalues and the relevant constants, the effect of axial conduction on the entrance-region temperature profile and local Nusselt numbers has been examined and reported.

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Abbreviations

C :

specific heat at constant pressure

C n :

coefficients of series expansion in Eq. (11)

Nu :

Nusselt number\(\left( { = \frac{{q_w }}{{T_w - T_b }}\frac{{2r_0 }}{k}} \right)\)

N Pr :

Prandtl number (=Cμ/k)

N Re :

Reynolds number (=2r 0 ūρ/μ)

Pe :

Peclet number (=N Re N Pr )

R n :

eigenfunctions for (12) and (13)

R n (1):

eigenfunction atξ=1

T :

temperature

T 0 :

fluid temperature forx<0

T f :

fully developed temperature

T w :

temperature at tube wall

k :

thermal conductivity of fluid

q w :

heat flux at tube wall

r :

radial coordinate distance

r 0 :

radius of a circular tube

t*:

=T−T f

u m :

maximum fluid velocity

ū :

average fluid velocity

x :

axial coordinate distance

β n :

eigenvalues of (12) and (13)

Δ :

=x/(r 0 N Re N Pr )

η :

a parameter

Θ :

=(T−T 0)/(q w r 0/k)

θ :

=t*/(q w r 0/k)

μ :

viscosity of fluid

ξ :

=r/r 0

ρ :

density of fluid

References

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Authors and Affiliations

  1. Brookhaven National Laboratory Upton, New York, USA

    Chia-Jung Hsu

Authors
  1. Chia-Jung Hsu
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Additional information

This work was performed under the auspices of the U.S. Atomic Energy Commission.

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Hsu, CJ. An exact mathematical solution for entrance-region laminar heat transfer with axial conduction. Appl. Sci. Res. 17, 359–376 (1967). https://doi.org/10.1007/BF02116438

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  • DOI: https://doi.org/10.1007/BF02116438

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